LMFDB - Embedded newform 76.4.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,4,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, 4, names="a")

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

chi := DirichletCharacter("76.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
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:	$4.48414516044$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ 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)

$f(q)$	$=$	$q+0.372281 q^{3} -16.8614 q^{5} -3.51087 q^{7} -26.8614 q^{9} +O(q^{10})$	$q+0.372281 q^{3} -16.8614 q^{5} -3.51087 q^{7} -26.8614 q^{9} -21.1386 q^{11} +14.0951 q^{13} -6.27719 q^{15} -17.2337 q^{17} -19.0000 q^{19} -1.30703 q^{21} -171.965 q^{23} +159.307 q^{25} -20.0516 q^{27} +264.198 q^{29} +185.783 q^{31} -7.86950 q^{33} +59.1983 q^{35} +212.978 q^{37} +5.24734 q^{39} -157.168 q^{41} -258.557 q^{43} +452.921 q^{45} -293.562 q^{47} -330.674 q^{49} -6.41578 q^{51} +215.791 q^{53} +356.426 q^{55} -7.07335 q^{57} -537.030 q^{59} -280.149 q^{61} +94.3070 q^{63} -237.663 q^{65} +147.291 q^{67} -64.0192 q^{69} -913.446 q^{71} -678.826 q^{73} +59.3070 q^{75} +74.2150 q^{77} -608.277 q^{79} +717.793 q^{81} +282.440 q^{83} +290.584 q^{85} +98.3561 q^{87} -214.217 q^{89} -49.4861 q^{91} +69.1634 q^{93} +320.367 q^{95} +1670.50 q^{97} +567.812 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{3} - 5 q^{5} - 30 q^{7} - 25 q^{9}+O(q^{10}) $ 2 * q - 5 * q^3 - 5 * q^5 - 30 * q^7 - 25 * q^9	$ 2 q - 5 q^{3} - 5 q^{5} - 30 q^{7} - 25 q^{9} - 71 q^{11} - 35 q^{13} - 70 q^{15} - 38 q^{19} + 141 q^{21} - 5 q^{23} + 175 q^{25} + 115 q^{27} + 155 q^{29} - 88 q^{31} + 260 q^{33} - 255 q^{35} + 380 q^{37} + 269 q^{39} - 142 q^{41} + 155 q^{43} + 475 q^{45} - 455 q^{47} + 28 q^{49} - 99 q^{51} - 275 q^{53} - 235 q^{55} + 95 q^{57} - 873 q^{59} + 445 q^{61} + 45 q^{63} - 820 q^{65} + 645 q^{67} - 961 q^{69} - 1712 q^{71} - 990 q^{73} - 25 q^{75} + 1395 q^{77} - 1274 q^{79} - 58 q^{81} - 90 q^{83} + 495 q^{85} + 685 q^{87} - 888 q^{89} + 1251 q^{91} + 1540 q^{93} + 95 q^{95} + 710 q^{97} + 475 q^{99}+O(q^{100}) $ 2 * q - 5 * q^3 - 5 * q^5 - 30 * q^7 - 25 * q^9 - 71 * q^11 - 35 * q^13 - 70 * q^15 - 38 * q^19 + 141 * q^21 - 5 * q^23 + 175 * q^25 + 115 * q^27 + 155 * q^29 - 88 * q^31 + 260 * q^33 - 255 * q^35 + 380 * q^37 + 269 * q^39 - 142 * q^41 + 155 * q^43 + 475 * q^45 - 455 * q^47 + 28 * q^49 - 99 * q^51 - 275 * q^53 - 235 * q^55 + 95 * q^57 - 873 * q^59 + 445 * q^61 + 45 * q^63 - 820 * q^65 + 645 * q^67 - 961 * q^69 - 1712 * q^71 - 990 * q^73 - 25 * q^75 + 1395 * q^77 - 1274 * q^79 - 58 * q^81 - 90 * q^83 + 495 * q^85 + 685 * q^87 - 888 * q^89 + 1251 * q^91 + 1540 * q^93 + 95 * q^95 + 710 * q^97 + 475 * 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
$3$	0.372281	0.0716456	0.0358228	−	0.999358i	$-0.488595\pi$
$3$	0.372281	0.0716456	0.0358228	+	0.999358i	$0.488595\pi$
$4$	0	0
$5$	−16.8614	−1.50813	−0.754065	−	0.656800i	$-0.771910\pi$
$5$	−16.8614	−1.50813	−0.754065	+	0.656800i	$0.771910\pi$
$6$	0	0
$7$	−3.51087	−0.189569	−0.0947847	−	0.995498i	$-0.530216\pi$
$7$	−3.51087	−0.189569	−0.0947847	+	0.995498i	$0.530216\pi$
$8$	0	0
$9$	−26.8614	−0.994867
$10$	0	0
$11$	−21.1386	−0.579411	−0.289706	−	0.957116i	$-0.593557\pi$
$11$	−21.1386	−0.579411	−0.289706	+	0.957116i	$0.593557\pi$
$12$	0	0
$13$	14.0951	0.300714	0.150357	−	0.988632i	$-0.451958\pi$
$13$	14.0951	0.300714	0.150357	+	0.988632i	$0.451958\pi$
$14$	0	0
$15$	−6.27719	−0.108051
$16$	0	0
$17$	−17.2337	−0.245870	−0.122935	−	0.992415i	$-0.539231\pi$
$17$	−17.2337	−0.245870	−0.122935	+	0.992415i	$0.539231\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−1.30703	−0.0135818
$22$	0	0
$23$	−171.965	−1.55900	−0.779502	−	0.626400i	$-0.784528\pi$
$23$	−171.965	−1.55900	−0.779502	+	0.626400i	$0.784528\pi$
$24$	0	0
$25$	159.307	1.27446
$26$	0	0
$27$	−20.0516	−0.142923
$28$	0	0
$29$	264.198	1.69174	0.845869	−	0.533391i	$-0.179083\pi$
$29$	264.198	1.69174	0.845869	+	0.533391i	$0.179083\pi$
$30$	0	0
$31$	185.783	1.07637	0.538186	−	0.842826i	$-0.319110\pi$
$31$	185.783	1.07637	0.538186	+	0.842826i	$0.319110\pi$
$32$	0	0
$33$	−7.86950	−0.0415123
$34$	0	0
$35$	59.1983	0.285895
$36$	0	0
$37$	212.978	0.946308	0.473154	−	0.880980i	$-0.343115\pi$
$37$	212.978	0.946308	0.473154	+	0.880980i	$0.343115\pi$
$38$	0	0
$39$	5.24734	0.0215448
$40$	0	0
$41$	−157.168	−0.598673	−0.299336	−	0.954148i	$-0.596765\pi$
$41$	−157.168	−0.598673	−0.299336	+	0.954148i	$0.596765\pi$
$42$	0	0
$43$	−258.557	−0.916967	−0.458483	−	0.888703i	$-0.651607\pi$
$43$	−258.557	−0.916967	−0.458483	+	0.888703i	$0.651607\pi$
$44$	0	0
$45$	452.921	1.50039
$46$	0	0
$47$	−293.562	−0.911074	−0.455537	−	0.890217i	$-0.650553\pi$
$47$	−293.562	−0.911074	−0.455537	+	0.890217i	$0.650553\pi$
$48$	0	0
$49$	−330.674	−0.964063
$50$	0	0
$51$	−6.41578	−0.0176155
$52$	0	0
$53$	215.791	0.559266	0.279633	−	0.960107i	$-0.409787\pi$
$53$	215.791	0.559266	0.279633	+	0.960107i	$0.409787\pi$
$54$	0	0
$55$	356.426	0.873828
$56$	0	0
$57$	−7.07335	−0.0164366
$58$	0	0
$59$	−537.030	−1.18501	−0.592503	−	0.805568i	$-0.701860\pi$
$59$	−537.030	−1.18501	−0.592503	+	0.805568i	$0.701860\pi$
$60$	0	0
$61$	−280.149	−0.588024	−0.294012	−	0.955802i	$-0.594990\pi$
$61$	−280.149	−0.588024	−0.294012	+	0.955802i	$0.594990\pi$
$62$	0	0
$63$	94.3070	0.188596
$64$	0	0
$65$	−237.663	−0.453515
$66$	0	0
$67$	147.291	0.268574	0.134287	−	0.990942i	$-0.457126\pi$
$67$	147.291	0.268574	0.134287	+	0.990942i	$0.457126\pi$
$68$	0	0
$69$	−64.0192	−0.111696
$70$	0	0
$71$	−913.446	−1.52685	−0.763423	−	0.645899i	$-0.776483\pi$
$71$	−913.446	−1.52685	−0.763423	+	0.645899i	$0.776483\pi$
$72$	0	0
$73$	−678.826	−1.08836	−0.544182	−	0.838967i	$-0.683160\pi$
$73$	−678.826	−1.08836	−0.544182	+	0.838967i	$0.683160\pi$
$74$	0	0
$75$	59.3070	0.0913092
$76$	0	0
$77$	74.2150	0.109839
$78$	0	0
$79$	−608.277	−0.866285	−0.433143	−	0.901325i	$-0.642595\pi$
$79$	−608.277	−0.866285	−0.433143	+	0.901325i	$0.642595\pi$
$80$	0	0
$81$	717.793	0.984627
$82$	0	0
$83$	282.440	0.373516	0.186758	−	0.982406i	$-0.440202\pi$
$83$	282.440	0.373516	0.186758	+	0.982406i	$0.440202\pi$
$84$	0	0
$85$	290.584	0.370803
$86$	0	0
$87$	98.3561	0.121206
$88$	0	0
$89$	−214.217	−0.255135	−0.127567	−	0.991830i	$-0.540717\pi$
$89$	−214.217	−0.255135	−0.127567	+	0.991830i	$0.540717\pi$
$90$	0	0
$91$	−49.4861	−0.0570061
$92$	0	0
$93$	69.1634	0.0771173
$94$	0	0
$95$	320.367	0.345989
$96$	0	0
$97$	1670.50	1.74860	0.874299	−	0.485387i	$-0.161321\pi$
$97$	1670.50	1.74860	0.874299	+	0.485387i	$0.161321\pi$
$98$	0	0
$99$	567.812	0.576437
$100$	0	0
$101$	166.316	0.163852	0.0819258	−	0.996638i	$-0.473893\pi$
$101$	166.316	0.163852	0.0819258	+	0.996638i	$0.473893\pi$
$102$	0	0
$103$	1219.78	1.16688	0.583440	−	0.812156i	$-0.301706\pi$
$103$	1219.78	1.16688	0.583440	+	0.812156i	$0.301706\pi$
$104$	0	0
$105$	22.0384	0.0204831
$106$	0	0
$107$	−1736.73	−1.56912	−0.784561	−	0.620052i	$-0.787112\pi$
$107$	−1736.73	−1.56912	−0.784561	+	0.620052i	$0.787112\pi$
$108$	0	0
$109$	−18.2367	−0.0160253	−0.00801266	−	0.999968i	$-0.502551\pi$
$109$	−18.2367	−0.0160253	−0.00801266	+	0.999968i	$0.502551\pi$
$110$	0	0
$111$	79.2878	0.0677988
$112$	0	0
$113$	−1586.59	−1.32083	−0.660414	−	0.750902i	$-0.729619\pi$
$113$	−1586.59	−1.32083	−0.660414	+	0.750902i	$0.729619\pi$
$114$	0	0
$115$	2899.57	2.35118
$116$	0	0
$117$	−378.614	−0.299170
$118$	0	0
$119$	60.5053	0.0466094
$120$	0	0
$121$	−884.160	−0.664282
$122$	0	0
$123$	−58.5109	−0.0428923
$124$	0	0
$125$	−578.465	−0.413916
$126$	0	0
$127$	2327.85	1.62648	0.813240	−	0.581928i	$-0.197701\pi$
$127$	2327.85	1.62648	0.813240	+	0.581928i	$0.197701\pi$
$128$	0	0
$129$	−96.2559	−0.0656966
$130$	0	0
$131$	631.753	0.421347	0.210674	−	0.977556i	$-0.432434\pi$
$131$	631.753	0.421347	0.210674	+	0.977556i	$0.432434\pi$
$132$	0	0
$133$	66.7066	0.0434902
$134$	0	0
$135$	338.098	0.215547
$136$	0	0
$137$	1098.25	0.684892	0.342446	−	0.939537i	$-0.388745\pi$
$137$	1098.25	0.684892	0.342446	+	0.939537i	$0.388745\pi$
$138$	0	0
$139$	1523.38	0.929577	0.464788	−	0.885422i	$-0.346130\pi$
$139$	1523.38	0.929577	0.464788	+	0.885422i	$0.346130\pi$
$140$	0	0
$141$	−109.288	−0.0652744
$142$	0	0
$143$	−297.950	−0.174237
$144$	0	0
$145$	−4454.75	−2.55136
$146$	0	0
$147$	−123.104	−0.0690709
$148$	0	0
$149$	−3160.49	−1.73770	−0.868849	−	0.495077i	$-0.835140\pi$
$149$	−3160.49	−1.73770	−0.868849	+	0.495077i	$0.835140\pi$
$150$	0	0
$151$	−1930.72	−1.04053	−0.520263	−	0.854006i	$-0.674166\pi$
$151$	−1930.72	−1.04053	−0.520263	+	0.854006i	$0.674166\pi$
$152$	0	0
$153$	462.921	0.244608
$154$	0	0
$155$	−3132.55	−1.62331
$156$	0	0
$157$	1818.29	0.924303	0.462152	−	0.886801i	$-0.347078\pi$
$157$	1818.29	0.924303	0.462152	+	0.886801i	$0.347078\pi$
$158$	0	0
$159$	80.3348	0.0400690
$160$	0	0
$161$	603.746	0.295540
$162$	0	0
$163$	−1259.17	−0.605068	−0.302534	−	0.953139i	$-0.597833\pi$
$163$	−1259.17	−0.605068	−0.302534	+	0.953139i	$0.597833\pi$
$164$	0	0
$165$	132.691	0.0626059
$166$	0	0
$167$	−1315.17	−0.609405	−0.304703	−	0.952447i	$-0.598557\pi$
$167$	−1315.17	−0.609405	−0.304703	+	0.952447i	$0.598557\pi$
$168$	0	0
$169$	−1998.33	−0.909571
$170$	0	0
$171$	510.367	0.228238
$172$	0	0
$173$	2407.00	1.05781	0.528904	−	0.848682i	$-0.322603\pi$
$173$	2407.00	1.05781	0.528904	+	0.848682i	$0.322603\pi$
$174$	0	0
$175$	−559.307	−0.241598
$176$	0	0
$177$	−199.926	−0.0849004
$178$	0	0
$179$	1082.32	0.451935	0.225968	−	0.974135i	$-0.427446\pi$
$179$	1082.32	0.451935	0.225968	+	0.974135i	$0.427446\pi$
$180$	0	0
$181$	4317.41	1.77299	0.886494	−	0.462741i	$-0.153134\pi$
$181$	4317.41	1.77299	0.886494	+	0.462741i	$0.153134\pi$
$182$	0	0
$183$	−104.294	−0.0421293
$184$	0	0
$185$	−3591.11	−1.42716
$186$	0	0
$187$	364.296	0.142460
$188$	0	0
$189$	70.3986	0.0270939
$190$	0	0
$191$	3208.27	1.21540	0.607702	−	0.794165i	$-0.292091\pi$
$191$	3208.27	1.21540	0.607702	+	0.794165i	$0.292091\pi$
$192$	0	0
$193$	−4206.70	−1.56894	−0.784469	−	0.620168i	$-0.787064\pi$
$193$	−4206.70	−1.56894	−0.784469	+	0.620168i	$0.787064\pi$
$194$	0	0
$195$	−88.4775	−0.0324924
$196$	0	0
$197$	4789.05	1.73201	0.866004	−	0.500037i	$-0.166680\pi$
$197$	4789.05	1.73201	0.866004	+	0.500037i	$0.166680\pi$
$198$	0	0
$199$	−3504.05	−1.24822	−0.624110	−	0.781337i	$-0.714538\pi$
$199$	−3504.05	−1.24822	−0.624110	+	0.781337i	$0.714538\pi$
$200$	0	0
$201$	54.8336	0.0192421
$202$	0	0
$203$	−927.567	−0.320702
$204$	0	0
$205$	2650.08	0.902877
$206$	0	0
$207$	4619.21	1.55100
$208$	0	0
$209$	401.633	0.132926
$210$	0	0
$211$	−3066.90	−1.00063	−0.500317	−	0.865842i	$-0.666783\pi$
$211$	−3066.90	−1.00063	−0.500317	+	0.865842i	$0.666783\pi$
$212$	0	0
$213$	−340.059	−0.109392
$214$	0	0
$215$	4359.63	1.38290
$216$	0	0
$217$	−652.259	−0.204047
$218$	0	0
$219$	−252.714	−0.0779765
$220$	0	0
$221$	−242.910	−0.0739363
$222$	0	0
$223$	−2117.06	−0.635733	−0.317867	−	0.948135i	$-0.602966\pi$
$223$	−2117.06	−0.635733	−0.317867	+	0.948135i	$0.602966\pi$
$224$	0	0
$225$	−4279.21	−1.26791
$226$	0	0
$227$	6476.34	1.89361	0.946806	−	0.321804i	$-0.104289\pi$
$227$	6476.34	1.89361	0.946806	+	0.321804i	$0.104289\pi$
$228$	0	0
$229$	3686.92	1.06392	0.531962	−	0.846768i	$-0.321455\pi$
$229$	3686.92	1.06392	0.531962	+	0.846768i	$0.321455\pi$
$230$	0	0
$231$	27.6288	0.00786946
$232$	0	0
$233$	1119.42	0.314744	0.157372	−	0.987539i	$-0.449698\pi$
$233$	1119.42	0.314744	0.157372	+	0.987539i	$0.449698\pi$
$234$	0	0
$235$	4949.88	1.37402
$236$	0	0
$237$	−226.450	−0.0620655
$238$	0	0
$239$	−6621.48	−1.79208	−0.896041	−	0.443971i	$-0.853569\pi$
$239$	−6621.48	−1.79208	−0.896041	+	0.443971i	$0.853569\pi$
$240$	0	0
$241$	2784.49	0.744253	0.372127	−	0.928182i	$-0.378629\pi$
$241$	2784.49	0.744253	0.372127	+	0.928182i	$0.378629\pi$
$242$	0	0
$243$	808.614	0.213468
$244$	0	0
$245$	5575.62	1.45393
$246$	0	0
$247$	−267.807	−0.0689884
$248$	0	0
$249$	105.147	0.0267608
$250$	0	0
$251$	−4670.19	−1.17442	−0.587210	−	0.809434i	$-0.699774\pi$
$251$	−4670.19	−1.17442	−0.587210	+	0.809434i	$0.699774\pi$
$252$	0	0
$253$	3635.09	0.903305
$254$	0	0
$255$	108.179	0.0265664
$256$	0	0
$257$	5167.86	1.25433	0.627164	−	0.778887i	$-0.284216\pi$
$257$	5167.86	1.25433	0.627164	+	0.778887i	$0.284216\pi$
$258$	0	0
$259$	−747.740	−0.179391
$260$	0	0
$261$	−7096.74	−1.68305
$262$	0	0
$263$	−3254.16	−0.762966	−0.381483	−	0.924376i	$-0.624587\pi$
$263$	−3254.16	−0.762966	−0.381483	+	0.924376i	$0.624587\pi$
$264$	0	0
$265$	−3638.53	−0.843446
$266$	0	0
$267$	−79.7492	−0.0182793
$268$	0	0
$269$	−1547.77	−0.350814	−0.175407	−	0.984496i	$-0.556124\pi$
$269$	−1547.77	−0.350814	−0.175407	+	0.984496i	$0.556124\pi$
$270$	0	0
$271$	−4969.53	−1.11394	−0.556969	−	0.830533i	$-0.688036\pi$
$271$	−4969.53	−1.11394	−0.556969	+	0.830533i	$0.688036\pi$
$272$	0	0
$273$	−18.4228	−0.00408423
$274$	0	0
$275$	−3367.53	−0.738435
$276$	0	0
$277$	279.862	0.0607049	0.0303525	−	0.999539i	$-0.490337\pi$
$277$	279.862	0.0607049	0.0303525	+	0.999539i	$0.490337\pi$
$278$	0	0
$279$	−4990.38	−1.07085
$280$	0	0
$281$	2092.54	0.444236	0.222118	−	0.975020i	$-0.428703\pi$
$281$	2092.54	0.444236	0.222118	+	0.975020i	$0.428703\pi$
$282$	0	0
$283$	−3939.01	−0.827384	−0.413692	−	0.910417i	$-0.635761\pi$
$283$	−3939.01	−0.827384	−0.413692	+	0.910417i	$0.635761\pi$
$284$	0	0
$285$	119.267	0.0247886
$286$	0	0
$287$	551.799	0.113490
$288$	0	0
$289$	−4616.00	−0.939548
$290$	0	0
$291$	621.898	0.125279
$292$	0	0
$293$	965.927	0.192594	0.0962970	−	0.995353i	$-0.469300\pi$
$293$	965.927	0.192594	0.0962970	+	0.995353i	$0.469300\pi$
$294$	0	0
$295$	9055.08	1.78714
$296$	0	0
$297$	423.863	0.0828114
$298$	0	0
$299$	−2423.86	−0.468814
$300$	0	0
$301$	907.761	0.173829
$302$	0	0
$303$	61.9162	0.0117392
$304$	0	0
$305$	4723.71	0.886816
$306$	0	0
$307$	−9718.56	−1.80673	−0.903367	−	0.428867i	$-0.858913\pi$
$307$	−9718.56	−1.80673	−0.903367	+	0.428867i	$0.858913\pi$
$308$	0	0
$309$	454.102	0.0836019
$310$	0	0
$311$	3449.04	0.628865	0.314433	−	0.949280i	$-0.398186\pi$
$311$	3449.04	0.628865	0.314433	+	0.949280i	$0.398186\pi$
$312$	0	0
$313$	23.3728	0.00422079	0.00211039	−	0.999998i	$-0.499328\pi$
$313$	23.3728	0.00422079	0.00211039	+	0.999998i	$0.499328\pi$
$314$	0	0
$315$	−1590.15	−0.284428
$316$	0	0
$317$	567.966	0.100631	0.0503157	−	0.998733i	$-0.483977\pi$
$317$	567.966	0.100631	0.0503157	+	0.998733i	$0.483977\pi$
$318$	0	0
$319$	−5584.78	−0.980212
$320$	0	0
$321$	−646.552	−0.112421
$322$	0	0
$323$	327.440	0.0564064
$324$	0	0
$325$	2245.45	0.383246
$326$	0	0
$327$	−6.78918	−0.00114814
$328$	0	0
$329$	1030.66	0.172712
$330$	0	0
$331$	−553.324	−0.0918835	−0.0459418	−	0.998944i	$-0.514629\pi$
$331$	−553.324	−0.0918835	−0.0459418	+	0.998944i	$0.514629\pi$
$332$	0	0
$333$	−5720.90	−0.941451
$334$	0	0
$335$	−2483.53	−0.405044
$336$	0	0
$337$	7328.60	1.18461	0.592306	−	0.805713i	$-0.298218\pi$
$337$	7328.60	1.18461	0.592306	+	0.805713i	$0.298218\pi$
$338$	0	0
$339$	−590.657	−0.0946315
$340$	0	0
$341$	−3927.18	−0.623662
$342$	0	0
$343$	2365.18	0.372326
$344$	0	0
$345$	1079.45	0.168452
$346$	0	0
$347$	4828.56	0.747005	0.373502	−	0.927629i	$-0.378157\pi$
$347$	4828.56	0.747005	0.373502	+	0.927629i	$0.378157\pi$
$348$	0	0
$349$	−10256.6	−1.57313	−0.786564	−	0.617508i	$-0.788142\pi$
$349$	−10256.6	−1.57313	−0.786564	+	0.617508i	$0.788142\pi$
$350$	0	0
$351$	−282.629	−0.0429790
$352$	0	0
$353$	2003.83	0.302133	0.151067	−	0.988524i	$-0.451729\pi$
$353$	2003.83	0.302133	0.151067	+	0.988524i	$0.451729\pi$
$354$	0	0
$355$	15402.0	2.30268
$356$	0	0
$357$	22.5250	0.00333935
$358$	0	0
$359$	4841.62	0.711786	0.355893	−	0.934527i	$-0.384177\pi$
$359$	4841.62	0.711786	0.355893	+	0.934527i	$0.384177\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−329.156	−0.0475929
$364$	0	0
$365$	11446.0	1.64139
$366$	0	0
$367$	7004.47	0.996268	0.498134	−	0.867100i	$-0.334019\pi$
$367$	7004.47	0.996268	0.498134	+	0.867100i	$0.334019\pi$
$368$	0	0
$369$	4221.77	0.595600
$370$	0	0
$371$	−757.614	−0.106020
$372$	0	0
$373$	−11718.1	−1.62665	−0.813324	−	0.581812i	$-0.802344\pi$
$373$	−11718.1	−1.62665	−0.813324	+	0.581812i	$0.802344\pi$
$374$	0	0
$375$	−215.352	−0.0296552
$376$	0	0
$377$	3723.90	0.508728
$378$	0	0
$379$	−4807.97	−0.651633	−0.325817	−	0.945433i	$-0.605639\pi$
$379$	−4807.97	−0.651633	−0.325817	+	0.945433i	$0.605639\pi$
$380$	0	0
$381$	866.614	0.116530
$382$	0	0
$383$	5998.09	0.800230	0.400115	−	0.916465i	$-0.368970\pi$
$383$	5998.09	0.800230	0.400115	+	0.916465i	$0.368970\pi$
$384$	0	0
$385$	−1251.37	−0.165651
$386$	0	0
$387$	6945.20	0.912260
$388$	0	0
$389$	7480.25	0.974971	0.487486	−	0.873131i	$-0.337914\pi$
$389$	7480.25	0.974971	0.487486	+	0.873131i	$0.337914\pi$
$390$	0	0
$391$	2963.58	0.383312
$392$	0	0
$393$	235.190	0.0301877
$394$	0	0
$395$	10256.4	1.30647
$396$	0	0
$397$	−10366.4	−1.31052	−0.655258	−	0.755406i	$-0.727440\pi$
$397$	−10366.4	−1.31052	−0.655258	+	0.755406i	$0.727440\pi$
$398$	0	0
$399$	24.8336	0.00311588
$400$	0	0
$401$	801.467	0.0998088	0.0499044	−	0.998754i	$-0.484108\pi$
$401$	801.467	0.0998088	0.0499044	+	0.998754i	$0.484108\pi$
$402$	0	0
$403$	2618.62	0.323680
$404$	0	0
$405$	−12103.0	−1.48495
$406$	0	0
$407$	−4502.06	−0.548302
$408$	0	0
$409$	−9396.88	−1.13605	−0.568027	−	0.823010i	$-0.692293\pi$
$409$	−9396.88	−1.13605	−0.568027	+	0.823010i	$0.692293\pi$
$410$	0	0
$411$	408.860	0.0490695
$412$	0	0
$413$	1885.44	0.224641
$414$	0	0
$415$	−4762.34	−0.563311
$416$	0	0
$417$	567.125	0.0666001
$418$	0	0
$419$	−5721.15	−0.667056	−0.333528	−	0.942740i	$-0.608239\pi$
$419$	−5721.15	−0.667056	−0.333528	+	0.942740i	$0.608239\pi$
$420$	0	0
$421$	678.987	0.0786029	0.0393014	−	0.999227i	$-0.487487\pi$
$421$	678.987	0.0786029	0.0393014	+	0.999227i	$0.487487\pi$
$422$	0	0
$423$	7885.50	0.906398
$424$	0	0
$425$	−2745.45	−0.313350
$426$	0	0
$427$	983.569	0.111471
$428$	0	0
$429$	−110.921	−0.0124833
$430$	0	0
$431$	−16238.9	−1.81485	−0.907423	−	0.420219i	$-0.861953\pi$
$431$	−16238.9	−1.81485	−0.907423	+	0.420219i	$0.861953\pi$
$432$	0	0
$433$	−12712.1	−1.41087	−0.705433	−	0.708777i	$-0.749247\pi$
$433$	−12712.1	−1.41087	−0.705433	+	0.708777i	$0.749247\pi$
$434$	0	0
$435$	−1658.42	−0.182794
$436$	0	0
$437$	3267.33	0.357660
$438$	0	0
$439$	15433.5	1.67791	0.838955	−	0.544200i	$-0.183167\pi$
$439$	15433.5	1.67791	0.838955	+	0.544200i	$0.183167\pi$
$440$	0	0
$441$	8882.36	0.959115
$442$	0	0
$443$	−14185.4	−1.52137	−0.760685	−	0.649122i	$-0.775137\pi$
$443$	−14185.4	−1.52137	−0.760685	+	0.649122i	$0.775137\pi$
$444$	0	0
$445$	3612.01	0.384777
$446$	0	0
$447$	−1176.59	−0.124498
$448$	0	0
$449$	−3674.17	−0.386180	−0.193090	−	0.981181i	$-0.561851\pi$
$449$	−3674.17	−0.386180	−0.193090	+	0.981181i	$0.561851\pi$
$450$	0	0
$451$	3322.32	0.346878
$452$	0	0
$453$	−718.770	−0.0745491
$454$	0	0
$455$	834.405	0.0859726
$456$	0	0
$457$	−2247.26	−0.230027	−0.115013	−	0.993364i	$-0.536691\pi$
$457$	−2247.26	−0.230027	−0.115013	+	0.993364i	$0.536691\pi$
$458$	0	0
$459$	345.563	0.0351405
$460$	0	0
$461$	6336.44	0.640168	0.320084	−	0.947389i	$-0.396289\pi$
$461$	6336.44	0.640168	0.320084	+	0.947389i	$0.396289\pi$
$462$	0	0
$463$	−14249.5	−1.43030	−0.715152	−	0.698969i	$-0.753643\pi$
$463$	−14249.5	−1.43030	−0.715152	+	0.698969i	$0.753643\pi$
$464$	0	0
$465$	−1166.19	−0.116303
$466$	0	0
$467$	7043.41	0.697923	0.348961	−	0.937137i	$-0.386534\pi$
$467$	7043.41	0.697923	0.348961	+	0.937137i	$0.386534\pi$
$468$	0	0
$469$	−517.120	−0.0509134
$470$	0	0
$471$	676.917	0.0662222
$472$	0	0
$473$	5465.53	0.531301
$474$	0	0
$475$	−3026.83	−0.292380
$476$	0	0
$477$	−5796.44	−0.556396
$478$	0	0
$479$	−13038.5	−1.24372	−0.621860	−	0.783128i	$-0.713623\pi$
$479$	−13038.5	−1.24372	−0.621860	+	0.783128i	$0.713623\pi$
$480$	0	0
$481$	3001.95	0.284568
$482$	0	0
$483$	224.763	0.0211741
$484$	0	0
$485$	−28167.1	−2.63711
$486$	0	0
$487$	18977.6	1.76583	0.882913	−	0.469536i	$-0.155579\pi$
$487$	18977.6	1.76583	0.882913	+	0.469536i	$0.155579\pi$
$488$	0	0
$489$	−468.767	−0.0433505
$490$	0	0
$491$	3177.78	0.292080	0.146040	−	0.989279i	$-0.453347\pi$
$491$	3177.78	0.292080	0.146040	+	0.989279i	$0.453347\pi$
$492$	0	0
$493$	−4553.11	−0.415947
$494$	0	0
$495$	−9574.11	−0.869342
$496$	0	0
$497$	3206.99	0.289443
$498$	0	0
$499$	−9180.49	−0.823597	−0.411799	−	0.911275i	$-0.635099\pi$
$499$	−9180.49	−0.823597	−0.411799	+	0.911275i	$0.635099\pi$
$500$	0	0
$501$	−489.612	−0.0436612
$502$	0	0
$503$	10409.3	0.922723	0.461362	−	0.887212i	$-0.347361\pi$
$503$	10409.3	0.922723	0.461362	+	0.887212i	$0.347361\pi$
$504$	0	0
$505$	−2804.32	−0.247110
$506$	0	0
$507$	−743.940	−0.0651668
$508$	0	0
$509$	−6173.36	−0.537582	−0.268791	−	0.963198i	$-0.586624\pi$
$509$	−6173.36	−0.537582	−0.268791	+	0.963198i	$0.586624\pi$
$510$	0	0
$511$	2383.27	0.206321
$512$	0	0
$513$	380.980	0.0327889
$514$	0	0
$515$	−20567.2	−1.75981
$516$	0	0
$517$	6205.50	0.527887
$518$	0	0
$519$	896.081	0.0757873
$520$	0	0
$521$	660.989	0.0555825	0.0277912	−	0.999614i	$-0.491153\pi$
$521$	660.989	0.0555825	0.0277912	+	0.999614i	$0.491153\pi$
$522$	0	0
$523$	19526.4	1.63257	0.816283	−	0.577652i	$-0.196031\pi$
$523$	19526.4	1.63257	0.816283	+	0.577652i	$0.196031\pi$
$524$	0	0
$525$	−208.220	−0.0173094
$526$	0	0
$527$	−3201.72	−0.264647
$528$	0	0
$529$	17404.8	1.43049
$530$	0	0
$531$	14425.4	1.17892
$532$	0	0
$533$	−2215.30	−0.180029
$534$	0	0
$535$	29283.7	2.36644
$536$	0	0
$537$	402.927	0.0323791
$538$	0	0
$539$	6989.98	0.558589
$540$	0	0
$541$	−9259.01	−0.735815	−0.367907	−	0.929862i	$-0.619926\pi$
$541$	−9259.01	−0.735815	−0.367907	+	0.929862i	$0.619926\pi$
$542$	0	0
$543$	1607.29	0.127027
$544$	0	0
$545$	307.496	0.0241683
$546$	0	0
$547$	11947.9	0.933924	0.466962	−	0.884277i	$-0.345348\pi$
$547$	11947.9	0.933924	0.466962	+	0.884277i	$0.345348\pi$
$548$	0	0
$549$	7525.20	0.585005
$550$	0	0
$551$	−5019.77	−0.388111
$552$	0	0
$553$	2135.58	0.164221
$554$	0	0
$555$	−1336.90	−0.102249
$556$	0	0
$557$	−3317.62	−0.252373	−0.126187	−	0.992007i	$-0.540274\pi$
$557$	−3317.62	−0.252373	−0.126187	+	0.992007i	$0.540274\pi$
$558$	0	0
$559$	−3644.38	−0.275744
$560$	0	0
$561$	135.621	0.0102066
$562$	0	0
$563$	21580.3	1.61546	0.807729	−	0.589555i	$-0.200697\pi$
$563$	21580.3	1.61546	0.807729	+	0.589555i	$0.200697\pi$
$564$	0	0
$565$	26752.1	1.99198
$566$	0	0
$567$	−2520.08	−0.186655
$568$	0	0
$569$	11040.1	0.813401	0.406700	−	0.913562i	$-0.366679\pi$
$569$	11040.1	0.813401	0.406700	+	0.913562i	$0.366679\pi$
$570$	0	0
$571$	−19559.0	−1.43348	−0.716742	−	0.697338i	$-0.754368\pi$
$571$	−19559.0	−1.43348	−0.716742	+	0.697338i	$0.754368\pi$
$572$	0	0
$573$	1194.38	0.0870784
$574$	0	0
$575$	−27395.2	−1.98688
$576$	0	0
$577$	−4148.41	−0.299308	−0.149654	−	0.988738i	$-0.547816\pi$
$577$	−4148.41	−0.299308	−0.149654	+	0.988738i	$0.547816\pi$
$578$	0	0
$579$	−1566.08	−0.112407
$580$	0	0
$581$	−991.612	−0.0708072
$582$	0	0
$583$	−4561.51	−0.324045
$584$	0	0
$585$	6383.97	0.451187
$586$	0	0
$587$	−23530.2	−1.65451	−0.827253	−	0.561829i	$-0.810098\pi$
$587$	−23530.2	−1.65451	−0.827253	+	0.561829i	$0.810098\pi$
$588$	0	0
$589$	−3529.87	−0.246937
$590$	0	0
$591$	1782.87	0.124091
$592$	0	0
$593$	−3378.46	−0.233957	−0.116979	−	0.993134i	$-0.537321\pi$
$593$	−3378.46	−0.233957	−0.116979	+	0.993134i	$0.537321\pi$
$594$	0	0
$595$	−1020.20	−0.0702930
$596$	0	0
$597$	−1304.49	−0.0894294
$598$	0	0
$599$	15748.7	1.07424	0.537122	−	0.843504i	$-0.319511\pi$
$599$	15748.7	1.07424	0.537122	+	0.843504i	$0.319511\pi$
$600$	0	0
$601$	17980.4	1.22036	0.610180	−	0.792263i	$-0.291097\pi$
$601$	17980.4	1.22036	0.610180	+	0.792263i	$0.291097\pi$
$602$	0	0
$603$	−3956.44	−0.267195
$604$	0	0
$605$	14908.2	1.00182
$606$	0	0
$607$	−4314.26	−0.288485	−0.144243	−	0.989542i	$-0.546075\pi$
$607$	−4314.26	−0.288485	−0.144243	+	0.989542i	$0.546075\pi$
$608$	0	0
$609$	−345.316	−0.0229769
$610$	0	0
$611$	−4137.79	−0.273972
$612$	0	0
$613$	−3891.03	−0.256374	−0.128187	−	0.991750i	$-0.540916\pi$
$613$	−3891.03	−0.256374	−0.128187	+	0.991750i	$0.540916\pi$
$614$	0	0
$615$	986.576	0.0646871
$616$	0	0
$617$	6001.52	0.391592	0.195796	−	0.980645i	$-0.437271\pi$
$617$	6001.52	0.391592	0.195796	+	0.980645i	$0.437271\pi$
$618$	0	0
$619$	16195.8	1.05164	0.525820	−	0.850596i	$-0.323759\pi$
$619$	16195.8	1.05164	0.525820	+	0.850596i	$0.323759\pi$
$620$	0	0
$621$	3448.16	0.222818
$622$	0	0
$623$	752.091	0.0483658
$624$	0	0
$625$	−10159.6	−0.650217
$626$	0	0
$627$	149.521	0.00952357
$628$	0	0
$629$	−3670.40	−0.232668
$630$	0	0
$631$	−29691.7	−1.87323	−0.936615	−	0.350361i	$-0.886059\pi$
$631$	−29691.7	−1.87323	−0.936615	+	0.350361i	$0.886059\pi$
$632$	0	0
$633$	−1141.75	−0.0716910
$634$	0	0
$635$	−39250.8	−2.45294
$636$	0	0
$637$	−4660.88	−0.289907
$638$	0	0
$639$	24536.4	1.51901
$640$	0	0
$641$	19562.7	1.20543	0.602716	−	0.797956i	$-0.294085\pi$
$641$	19562.7	1.20543	0.602716	+	0.797956i	$0.294085\pi$
$642$	0	0
$643$	20169.7	1.23704	0.618518	−	0.785770i	$-0.287733\pi$
$643$	20169.7	1.23704	0.618518	+	0.785770i	$0.287733\pi$
$644$	0	0
$645$	1623.01	0.0990790
$646$	0	0
$647$	−19276.8	−1.17133	−0.585663	−	0.810555i	$-0.699166\pi$
$647$	−19276.8	−1.17133	−0.585663	+	0.810555i	$0.699166\pi$
$648$	0	0
$649$	11352.1	0.686606
$650$	0	0
$651$	−242.824	−0.0146191
$652$	0	0
$653$	−17256.6	−1.03416	−0.517078	−	0.855938i	$-0.672980\pi$
$653$	−17256.6	−1.03416	−0.517078	+	0.855938i	$0.672980\pi$
$654$	0	0
$655$	−10652.2	−0.635446
$656$	0	0
$657$	18234.2	1.08278
$658$	0	0
$659$	−15812.2	−0.934681	−0.467340	−	0.884078i	$-0.654788\pi$
$659$	−15812.2	−0.934681	−0.467340	+	0.884078i	$0.654788\pi$
$660$	0	0
$661$	−21264.8	−1.25130	−0.625648	−	0.780106i	$-0.715165\pi$
$661$	−21264.8	−1.25130	−0.625648	+	0.780106i	$0.715165\pi$
$662$	0	0
$663$	−90.4310	−0.00529721
$664$	0	0
$665$	−1124.77	−0.0655889
$666$	0	0
$667$	−45432.8	−2.63743
$668$	0	0
$669$	−788.140	−0.0455475
$670$	0	0
$671$	5921.96	0.340708
$672$	0	0
$673$	10290.0	0.589374	0.294687	−	0.955594i	$-0.404785\pi$
$673$	10290.0	0.589374	0.294687	+	0.955594i	$0.404785\pi$
$674$	0	0
$675$	−3194.36	−0.182150
$676$	0	0
$677$	−3204.13	−0.181897	−0.0909487	−	0.995856i	$-0.528990\pi$
$677$	−3204.13	−0.181897	−0.0909487	+	0.995856i	$0.528990\pi$
$678$	0	0
$679$	−5864.93	−0.331481
$680$	0	0
$681$	2411.02	0.135669
$682$	0	0
$683$	22703.2	1.27191	0.635954	−	0.771727i	$-0.280607\pi$
$683$	22703.2	1.27191	0.635954	+	0.771727i	$0.280607\pi$
$684$	0	0
$685$	−18518.1	−1.03291
$686$	0	0
$687$	1372.57	0.0762254
$688$	0	0
$689$	3041.59	0.168179
$690$	0	0
$691$	14141.0	0.778509	0.389255	−	0.921130i	$-0.372733\pi$
$691$	14141.0	0.778509	0.389255	+	0.921130i	$0.372733\pi$
$692$	0	0
$693$	−1993.52	−0.109275
$694$	0	0
$695$	−25686.3	−1.40192
$696$	0	0
$697$	2708.59	0.147195
$698$	0	0
$699$	416.738	0.0225500
$700$	0	0
$701$	−8699.08	−0.468701	−0.234351	−	0.972152i	$-0.575296\pi$
$701$	−8699.08	−0.468701	−0.234351	+	0.972152i	$0.575296\pi$
$702$	0	0
$703$	−4046.59	−0.217098
$704$	0	0
$705$	1842.75	0.0984423
$706$	0	0
$707$	−583.913	−0.0310613
$708$	0	0
$709$	−19611.9	−1.03884	−0.519422	−	0.854518i	$-0.673853\pi$
$709$	−19611.9	−1.03884	−0.519422	+	0.854518i	$0.673853\pi$
$710$	0	0
$711$	16339.2	0.861838
$712$	0	0
$713$	−31948.0	−1.67807
$714$	0	0
$715$	5023.86	0.262772
$716$	0	0
$717$	−2465.05	−0.128395
$718$	0	0
$719$	3735.43	0.193752	0.0968762	−	0.995296i	$-0.469115\pi$
$719$	3735.43	0.193752	0.0968762	+	0.995296i	$0.469115\pi$
$720$	0	0
$721$	−4282.50	−0.221205
$722$	0	0
$723$	1036.62	0.0533225
$724$	0	0
$725$	42088.6	2.15605
$726$	0	0
$727$	−4091.05	−0.208705	−0.104353	−	0.994540i	$-0.533277\pi$
$727$	−4091.05	−0.208705	−0.104353	+	0.994540i	$0.533277\pi$
$728$	0	0
$729$	−19079.4	−0.969333
$730$	0	0
$731$	4455.89	0.225454
$732$	0	0
$733$	21206.4	1.06859	0.534294	−	0.845299i	$-0.320577\pi$
$733$	21206.4	1.06859	0.534294	+	0.845299i	$0.320577\pi$
$734$	0	0
$735$	2075.70	0.104168
$736$	0	0
$737$	−3113.52	−0.155615
$738$	0	0
$739$	8939.54	0.444988	0.222494	−	0.974934i	$-0.428580\pi$
$739$	8939.54	0.444988	0.222494	+	0.974934i	$0.428580\pi$
$740$	0	0
$741$	−99.6995	−0.00494271
$742$	0	0
$743$	26137.1	1.29055	0.645273	−	0.763952i	$-0.276743\pi$
$743$	26137.1	1.29055	0.645273	+	0.763952i	$0.276743\pi$
$744$	0	0
$745$	53290.2	2.62068
$746$	0	0
$747$	−7586.74	−0.371599
$748$	0	0
$749$	6097.44	0.297458
$750$	0	0
$751$	−11466.6	−0.557155	−0.278578	−	0.960414i	$-0.589863\pi$
$751$	−11466.6	−0.557155	−0.278578	+	0.960414i	$0.589863\pi$
$752$	0	0
$753$	−1738.62	−0.0841421
$754$	0	0
$755$	32554.6	1.56925
$756$	0	0
$757$	−8936.66	−0.429073	−0.214537	−	0.976716i	$-0.568824\pi$
$757$	−8936.66	−0.429073	−0.214537	+	0.976716i	$0.568824\pi$
$758$	0	0
$759$	1353.28	0.0647178
$760$	0	0
$761$	−9971.09	−0.474969	−0.237485	−	0.971391i	$-0.576323\pi$
$761$	−9971.09	−0.474969	−0.237485	+	0.971391i	$0.576323\pi$
$762$	0	0
$763$	64.0268	0.00303791
$764$	0	0
$765$	−7805.50	−0.368900
$766$	0	0
$767$	−7569.49	−0.356347
$768$	0	0
$769$	25560.0	1.19859	0.599296	−	0.800527i	$-0.295447\pi$
$769$	25560.0	1.19859	0.599296	+	0.800527i	$0.295447\pi$
$770$	0	0
$771$	1923.90	0.0898670
$772$	0	0
$773$	−15918.8	−0.740698	−0.370349	−	0.928893i	$-0.620762\pi$
$773$	−15918.8	−0.740698	−0.370349	+	0.928893i	$0.620762\pi$
$774$	0	0
$775$	29596.5	1.37179
$776$	0	0
$777$	−278.370	−0.0128526
$778$	0	0
$779$	2986.20	0.137345
$780$	0	0
$781$	19309.0	0.884672
$782$	0	0
$783$	−5297.60	−0.241789
$784$	0	0
$785$	−30659.0	−1.39397
$786$	0	0
$787$	−11278.0	−0.510822	−0.255411	−	0.966833i	$-0.582211\pi$
$787$	−11278.0	−0.510822	−0.255411	+	0.966833i	$0.582211\pi$
$788$	0	0
$789$	−1211.46	−0.0546631
$790$	0	0
$791$	5570.31	0.250389
$792$	0	0
$793$	−3948.73	−0.176827
$794$	0	0
$795$	−1354.56	−0.0604292
$796$	0	0
$797$	−23845.2	−1.05978	−0.529888	−	0.848068i	$-0.677766\pi$
$797$	−23845.2	−1.05978	−0.529888	+	0.848068i	$0.677766\pi$
$798$	0	0
$799$	5059.16	0.224005
$800$	0	0
$801$	5754.18	0.253825
$802$	0	0
$803$	14349.4	0.630611
$804$	0	0
$805$	−10180.0	−0.445712
$806$	0	0
$807$	−576.204	−0.0251343
$808$	0	0
$809$	12729.7	0.553216	0.276608	−	0.960983i	$-0.410790\pi$
$809$	12729.7	0.553216	0.276608	+	0.960983i	$0.410790\pi$
$810$	0	0
$811$	−5859.98	−0.253726	−0.126863	−	0.991920i	$-0.540491\pi$
$811$	−5859.98	−0.253726	−0.126863	+	0.991920i	$0.540491\pi$
$812$	0	0
$813$	−1850.06	−0.0798088
$814$	0	0
$815$	21231.5	0.912522
$816$	0	0
$817$	4912.58	0.210367
$818$	0	0
$819$	1329.27	0.0567135
$820$	0	0
$821$	12778.2	0.543195	0.271598	−	0.962411i	$-0.412448\pi$
$821$	12778.2	0.543195	0.271598	+	0.962411i	$0.412448\pi$
$822$	0	0
$823$	−15511.0	−0.656963	−0.328482	−	0.944510i	$-0.606537\pi$
$823$	−15511.0	−0.656963	−0.328482	+	0.944510i	$0.606537\pi$
$824$	0	0
$825$	−1253.67	−0.0529056
$826$	0	0
$827$	−12501.0	−0.525639	−0.262819	−	0.964845i	$-0.584652\pi$
$827$	−12501.0	−0.525639	−0.262819	+	0.964845i	$0.584652\pi$
$828$	0	0
$829$	−2013.80	−0.0843694	−0.0421847	−	0.999110i	$-0.513432\pi$
$829$	−2013.80	−0.0843694	−0.0421847	+	0.999110i	$0.513432\pi$
$830$	0	0
$831$	104.187	0.00434924
$832$	0	0
$833$	5698.73	0.237034
$834$	0	0
$835$	22175.6	0.919063
$836$	0	0
$837$	−3725.24	−0.153839
$838$	0	0
$839$	16805.2	0.691513	0.345757	−	0.938324i	$-0.387622\pi$
$839$	16805.2	0.691513	0.345757	+	0.938324i	$0.387622\pi$
$840$	0	0
$841$	45411.7	1.86198
$842$	0	0
$843$	779.013	0.0318275
$844$	0	0
$845$	33694.6	1.37175
$846$	0	0
$847$	3104.17	0.125928
$848$	0	0
$849$	−1466.42	−0.0592784
$850$	0	0
$851$	−36624.7	−1.47530
$852$	0	0
$853$	−16648.7	−0.668279	−0.334139	−	0.942524i	$-0.608446\pi$
$853$	−16648.7	−0.668279	−0.334139	+	0.942524i	$0.608446\pi$
$854$	0	0
$855$	−8605.50	−0.344213
$856$	0	0
$857$	12094.5	0.482078	0.241039	−	0.970515i	$-0.422512\pi$
$857$	12094.5	0.482078	0.241039	+	0.970515i	$0.422512\pi$
$858$	0	0
$859$	10296.1	0.408962	0.204481	−	0.978871i	$-0.434449\pi$
$859$	10296.1	0.408962	0.204481	+	0.978871i	$0.434449\pi$
$860$	0	0
$861$	205.424	0.00813106
$862$	0	0
$863$	27367.0	1.07947	0.539736	−	0.841834i	$-0.318524\pi$
$863$	27367.0	1.07947	0.539736	+	0.841834i	$0.318524\pi$
$864$	0	0
$865$	−40585.4	−1.59531
$866$	0	0
$867$	−1718.45	−0.0673145
$868$	0	0
$869$	12858.1	0.501936
$870$	0	0
$871$	2076.08	0.0807638
$872$	0	0
$873$	−44872.1	−1.73962
$874$	0	0
$875$	2030.92	0.0784658
$876$	0	0
$877$	−29880.2	−1.15049	−0.575247	−	0.817979i	$-0.695094\pi$
$877$	−29880.2	−1.15049	−0.575247	+	0.817979i	$0.695094\pi$
$878$	0	0
$879$	359.597	0.0137985
$880$	0	0
$881$	−28639.8	−1.09523	−0.547617	−	0.836729i	$-0.684465\pi$
$881$	−28639.8	−1.09523	−0.547617	+	0.836729i	$0.684465\pi$
$882$	0	0
$883$	−16394.7	−0.624832	−0.312416	−	0.949945i	$-0.601138\pi$
$883$	−16394.7	−0.624832	−0.312416	+	0.949945i	$0.601138\pi$
$884$	0	0
$885$	3371.04	0.128041
$886$	0	0
$887$	−16041.2	−0.607226	−0.303613	−	0.952795i	$-0.598193\pi$
$887$	−16041.2	−0.607226	−0.303613	+	0.952795i	$0.598193\pi$
$888$	0	0
$889$	−8172.78	−0.308331
$890$	0	0
$891$	−15173.1	−0.570504
$892$	0	0
$893$	5577.69	0.209015
$894$	0	0
$895$	−18249.4	−0.681577
$896$	0	0
$897$	−902.357	−0.0335884
$898$	0	0
$899$	49083.4	1.82094
$900$	0	0
$901$	−3718.87	−0.137507
$902$	0	0
$903$	337.942	0.0124541
$904$	0	0
$905$	−72797.6	−2.67390
$906$	0	0
$907$	−10953.4	−0.400994	−0.200497	−	0.979694i	$-0.564256\pi$
$907$	−10953.4	−0.400994	−0.200497	+	0.979694i	$0.564256\pi$
$908$	0	0
$909$	−4467.47	−0.163011
$910$	0	0
$911$	12739.1	0.463299	0.231649	−	0.972799i	$-0.425588\pi$
$911$	12739.1	0.463299	0.231649	+	0.972799i	$0.425588\pi$
$912$	0	0
$913$	−5970.39	−0.216419
$914$	0	0
$915$	1758.55	0.0635364
$916$	0	0
$917$	−2218.00	−0.0798745
$918$	0	0
$919$	−10544.1	−0.378475	−0.189238	−	0.981931i	$-0.560602\pi$
$919$	−10544.1	−0.378475	−0.189238	+	0.981931i	$0.560602\pi$
$920$	0	0
$921$	−3618.04	−0.129445
$922$	0	0
$923$	−12875.1	−0.459143
$924$	0	0
$925$	33928.9	1.20603
$926$	0	0
$927$	−32765.1	−1.16089
$928$	0	0
$929$	−28682.8	−1.01297	−0.506487	−	0.862247i	$-0.669056\pi$
$929$	−28682.8	−1.01297	−0.506487	+	0.862247i	$0.669056\pi$
$930$	0	0
$931$	6282.80	0.221171
$932$	0	0
$933$	1284.01	0.0450554
$934$	0	0
$935$	−6142.54	−0.214848
$936$	0	0
$937$	−39763.3	−1.38635	−0.693175	−	0.720769i	$-0.743789\pi$
$937$	−39763.3	−1.38635	−0.693175	+	0.720769i	$0.743789\pi$
$938$	0	0
$939$	8.70124	0.000302401 0
$940$	0	0
$941$	27875.7	0.965696	0.482848	−	0.875704i	$-0.339602\pi$
$941$	27875.7	0.965696	0.482848	+	0.875704i	$0.339602\pi$
$942$	0	0
$943$	27027.4	0.933333
$944$	0	0
$945$	−1187.02	−0.0408611
$946$	0	0
$947$	54572.7	1.87262	0.936312	−	0.351169i	$-0.114216\pi$
$947$	54572.7	1.87262	0.936312	+	0.351169i	$0.114216\pi$
$948$	0	0
$949$	−9568.12	−0.327286
$950$	0	0
$951$	211.443	0.00720979
$952$	0	0
$953$	−8065.62	−0.274157	−0.137078	−	0.990560i	$-0.543771\pi$
$953$	−8065.62	−0.274157	−0.137078	+	0.990560i	$0.543771\pi$
$954$	0	0
$955$	−54096.0	−1.83299
$956$	0	0
$957$	−2079.11	−0.0702279
$958$	0	0
$959$	−3855.83	−0.129835
$960$	0	0
$961$	4724.14	0.158576
$962$	0	0
$963$	46651.0	1.56107
$964$	0	0
$965$	70930.9	2.36616
$966$	0	0
$967$	14289.8	0.475210	0.237605	−	0.971362i	$-0.423638\pi$
$967$	14289.8	0.475210	0.237605	+	0.971362i	$0.423638\pi$
$968$	0	0
$969$	121.900	0.00404127
$970$	0	0
$971$	−14397.3	−0.475832	−0.237916	−	0.971286i	$-0.576464\pi$
$971$	−14397.3	−0.475832	−0.237916	+	0.971286i	$0.576464\pi$
$972$	0	0
$973$	−5348.39	−0.176219
$974$	0	0
$975$	835.938	0.0274579
$976$	0	0
$977$	−41896.2	−1.37193	−0.685966	−	0.727634i	$-0.740620\pi$
$977$	−41896.2	−1.37193	−0.685966	+	0.727634i	$0.740620\pi$
$978$	0	0
$979$	4528.26	0.147828
$980$	0	0
$981$	489.863	0.0159431
$982$	0	0
$983$	19373.7	0.628612	0.314306	−	0.949322i	$-0.398228\pi$
$983$	19373.7	0.628612	0.314306	+	0.949322i	$0.398228\pi$
$984$	0	0
$985$	−80750.1	−2.61209
$986$	0	0
$987$	383.696	0.0123740
$988$	0	0
$989$	44462.6	1.42955
$990$	0	0
$991$	−13250.6	−0.424741	−0.212371	−	0.977189i	$-0.568118\pi$
$991$	−13250.6	−0.424741	−0.212371	+	0.977189i	$0.568118\pi$
$992$	0	0
$993$	−205.992	−0.00658305
$994$	0	0
$995$	59083.3	1.88248
$996$	0	0
$997$	−4129.59	−0.131179	−0.0655895	−	0.997847i	$-0.520893\pi$
$997$	−4129.59	−0.131179	−0.0655895	+	0.997847i	$0.520893\pi$
$998$	0	0
$999$	−4270.55	−0.135250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.4.a.a.1.2	✓	2
3.2	odd	2		684.4.a.g.1.2		2
4.3	odd	2		304.4.a.f.1.1		2
5.2	odd	4		1900.4.c.b.1749.2		4
5.3	odd	4		1900.4.c.b.1749.3		4
5.4	even	2		1900.4.a.b.1.1		2
8.3	odd	2		1216.4.a.h.1.2		2
8.5	even	2		1216.4.a.o.1.1		2
19.18	odd	2		1444.4.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.a.a.1.2	✓	2	1.1	even	1	trivial
304.4.a.f.1.1		2	4.3	odd	2
684.4.a.g.1.2		2	3.2	odd	2
1216.4.a.h.1.2		2	8.3	odd	2
1216.4.a.o.1.1		2	8.5	even	2
1444.4.a.d.1.1		2	19.18	odd	2
1900.4.a.b.1.1		2	5.4	even	2
1900.4.c.b.1749.2		4	5.2	odd	4
1900.4.c.b.1749.3		4	5.3	odd	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.a (trivial)

\(f(q)\)	\(=\)	\(q+0.372281 q^{3} -16.8614 q^{5} -3.51087 q^{7} -26.8614 q^{9} +O(q^{10})\)	\(q+0.372281 q^{3} -16.8614 q^{5} -3.51087 q^{7} -26.8614 q^{9} -21.1386 q^{11} +14.0951 q^{13} -6.27719 q^{15} -17.2337 q^{17} -19.0000 q^{19} -1.30703 q^{21} -171.965 q^{23} +159.307 q^{25} -20.0516 q^{27} +264.198 q^{29} +185.783 q^{31} -7.86950 q^{33} +59.1983 q^{35} +212.978 q^{37} +5.24734 q^{39} -157.168 q^{41} -258.557 q^{43} +452.921 q^{45} -293.562 q^{47} -330.674 q^{49} -6.41578 q^{51} +215.791 q^{53} +356.426 q^{55} -7.07335 q^{57} -537.030 q^{59} -280.149 q^{61} +94.3070 q^{63} -237.663 q^{65} +147.291 q^{67} -64.0192 q^{69} -913.446 q^{71} -678.826 q^{73} +59.3070 q^{75} +74.2150 q^{77} -608.277 q^{79} +717.793 q^{81} +282.440 q^{83} +290.584 q^{85} +98.3561 q^{87} -214.217 q^{89} -49.4861 q^{91} +69.1634 q^{93} +320.367 q^{95} +1670.50 q^{97} +567.812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} - 5 q^{5} - 30 q^{7} - 25 q^{9}+O(q^{10}) \) 2 * q - 5 * q^3 - 5 * q^5 - 30 * q^7 - 25 * q^9	\( 2 q - 5 q^{3} - 5 q^{5} - 30 q^{7} - 25 q^{9} - 71 q^{11} - 35 q^{13} - 70 q^{15} - 38 q^{19} + 141 q^{21} - 5 q^{23} + 175 q^{25} + 115 q^{27} + 155 q^{29} - 88 q^{31} + 260 q^{33} - 255 q^{35} + 380 q^{37} + 269 q^{39} - 142 q^{41} + 155 q^{43} + 475 q^{45} - 455 q^{47} + 28 q^{49} - 99 q^{51} - 275 q^{53} - 235 q^{55} + 95 q^{57} - 873 q^{59} + 445 q^{61} + 45 q^{63} - 820 q^{65} + 645 q^{67} - 961 q^{69} - 1712 q^{71} - 990 q^{73} - 25 q^{75} + 1395 q^{77} - 1274 q^{79} - 58 q^{81} - 90 q^{83} + 495 q^{85} + 685 q^{87} - 888 q^{89} + 1251 q^{91} + 1540 q^{93} + 95 q^{95} + 710 q^{97} + 475 q^{99}+O(q^{100}) \) 2 * q - 5 * q^3 - 5 * q^5 - 30 * q^7 - 25 * q^9 - 71 * q^11 - 35 * q^13 - 70 * q^15 - 38 * q^19 + 141 * q^21 - 5 * q^23 + 175 * q^25 + 115 * q^27 + 155 * q^29 - 88 * q^31 + 260 * q^33 - 255 * q^35 + 380 * q^37 + 269 * q^39 - 142 * q^41 + 155 * q^43 + 475 * q^45 - 455 * q^47 + 28 * q^49 - 99 * q^51 - 275 * q^53 - 235 * q^55 + 95 * q^57 - 873 * q^59 + 445 * q^61 + 45 * q^63 - 820 * q^65 + 645 * q^67 - 961 * q^69 - 1712 * q^71 - 990 * q^73 - 25 * q^75 + 1395 * q^77 - 1274 * q^79 - 58 * q^81 - 90 * q^83 + 495 * q^85 + 685 * q^87 - 888 * q^89 + 1251 * q^91 + 1540 * q^93 + 95 * q^95 + 710 * q^97 + 475 * q^99

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