LMFDB - Embedded newform 76.6.a.a.1.1

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			1.1
Root			$7.75086$ 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)

$f(q)$	$=$	$q-26.4151 q^{3} +50.4185 q^{5} +117.462 q^{7} +454.756 q^{9} +O(q^{10})$	$q-26.4151 q^{3} +50.4185 q^{5} +117.462 q^{7} +454.756 q^{9} -736.100 q^{11} +393.662 q^{13} -1331.81 q^{15} -1999.73 q^{17} +361.000 q^{19} -3102.77 q^{21} -2742.34 q^{23} -582.972 q^{25} -5593.56 q^{27} +6935.15 q^{29} -5987.95 q^{31} +19444.1 q^{33} +5922.26 q^{35} -1945.37 q^{37} -10398.6 q^{39} -18304.8 q^{41} -8002.92 q^{43} +22928.1 q^{45} -12245.8 q^{47} -3009.68 q^{49} +52822.9 q^{51} +4155.62 q^{53} -37113.1 q^{55} -9535.84 q^{57} +14405.3 q^{59} -10708.7 q^{61} +53416.6 q^{63} +19847.8 q^{65} +14728.1 q^{67} +72439.1 q^{69} -9729.09 q^{71} +46960.7 q^{73} +15399.3 q^{75} -86463.7 q^{77} -6745.18 q^{79} +37248.6 q^{81} -49891.3 q^{83} -100823. q^{85} -183192. q^{87} -37071.9 q^{89} +46240.3 q^{91} +158172. q^{93} +18201.1 q^{95} +181017. q^{97} -334746. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−26.4151	−1.69453	−0.847264	−	0.531172i	$-0.821752\pi$
$3$	−26.4151	−1.69453	−0.847264	+	0.531172i	$0.821752\pi$
$4$	0	0
$5$	50.4185	0.901914	0.450957	−	0.892546i	$-0.351083\pi$
$5$	50.4185	0.901914	0.450957	+	0.892546i	$0.351083\pi$
$6$	0	0
$7$	117.462	0.906050	0.453025	−	0.891498i	$-0.350345\pi$
$7$	117.462	0.906050	0.453025	+	0.891498i	$0.350345\pi$
$8$	0	0
$9$	454.756	1.87143
$10$	0	0
$11$	−736.100	−1.83423	−0.917117	−	0.398618i	$-0.869490\pi$
$11$	−736.100	−1.83423	−0.917117	+	0.398618i	$0.869490\pi$
$12$	0	0
$13$	393.662	0.646048	0.323024	−	0.946391i	$-0.395301\pi$
$13$	393.662	0.646048	0.323024	+	0.946391i	$0.395301\pi$
$14$	0	0
$15$	−1331.81	−1.52832
$16$	0	0
$17$	−1999.73	−1.67822	−0.839109	−	0.543964i	$-0.816923\pi$
$17$	−1999.73	−1.67822	−0.839109	+	0.543964i	$0.816923\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	−3102.77	−1.53533
$22$	0	0
$23$	−2742.34	−1.08094	−0.540470	−	0.841363i	$-0.681754\pi$
$23$	−2742.34	−1.08094	−0.540470	+	0.841363i	$0.681754\pi$
$24$	0	0
$25$	−582.972	−0.186551
$26$	0	0
$27$	−5593.56	−1.47665
$28$	0	0
$29$	6935.15	1.53130	0.765651	−	0.643257i	$-0.222417\pi$
$29$	6935.15	1.53130	0.765651	+	0.643257i	$0.222417\pi$
$30$	0	0
$31$	−5987.95	−1.11911	−0.559557	−	0.828792i	$-0.689029\pi$
$31$	−5987.95	−1.11911	−0.559557	+	0.828792i	$0.689029\pi$
$32$	0	0
$33$	19444.1	3.10816
$34$	0	0
$35$	5922.26	0.817179
$36$	0	0
$37$	−1945.37	−0.233613	−0.116807	−	0.993155i	$-0.537266\pi$
$37$	−1945.37	−0.233613	−0.116807	+	0.993155i	$0.537266\pi$
$38$	0	0
$39$	−10398.6	−1.09475
$40$	0	0
$41$	−18304.8	−1.70061	−0.850304	−	0.526291i	$-0.823582\pi$
$41$	−18304.8	−1.70061	−0.850304	+	0.526291i	$0.823582\pi$
$42$	0	0
$43$	−8002.92	−0.660051	−0.330025	−	0.943972i	$-0.607057\pi$
$43$	−8002.92	−0.660051	−0.330025	+	0.943972i	$0.607057\pi$
$44$	0	0
$45$	22928.1	1.68786
$46$	0	0
$47$	−12245.8	−0.808614	−0.404307	−	0.914623i	$-0.632487\pi$
$47$	−12245.8	−0.808614	−0.404307	+	0.914623i	$0.632487\pi$
$48$	0	0
$49$	−3009.68	−0.179073
$50$	0	0
$51$	52822.9	2.84379
$52$	0	0
$53$	4155.62	0.203210	0.101605	−	0.994825i	$-0.467602\pi$
$53$	4155.62	0.203210	0.101605	+	0.994825i	$0.467602\pi$
$54$	0	0
$55$	−37113.1	−1.65432
$56$	0	0
$57$	−9535.84	−0.388751
$58$	0	0
$59$	14405.3	0.538758	0.269379	−	0.963034i	$-0.413182\pi$
$59$	14405.3	0.538758	0.269379	+	0.963034i	$0.413182\pi$
$60$	0	0
$61$	−10708.7	−0.368479	−0.184239	−	0.982881i	$-0.558982\pi$
$61$	−10708.7	−0.368479	−0.184239	+	0.982881i	$0.558982\pi$
$62$	0	0
$63$	53416.6	1.69561
$64$	0	0
$65$	19847.8	0.582680
$66$	0	0
$67$	14728.1	0.400828	0.200414	−	0.979711i	$-0.435771\pi$
$67$	14728.1	0.400828	0.200414	+	0.979711i	$0.435771\pi$
$68$	0	0
$69$	72439.1	1.83168
$70$	0	0
$71$	−9729.09	−0.229048	−0.114524	−	0.993420i	$-0.536534\pi$
$71$	−9729.09	−0.229048	−0.114524	+	0.993420i	$0.536534\pi$
$72$	0	0
$73$	46960.7	1.03140	0.515700	−	0.856769i	$-0.327532\pi$
$73$	46960.7	1.03140	0.515700	+	0.856769i	$0.327532\pi$
$74$	0	0
$75$	15399.3	0.316116
$76$	0	0
$77$	−86463.7	−1.66191
$78$	0	0
$79$	−6745.18	−0.121598	−0.0607989	−	0.998150i	$-0.519365\pi$
$79$	−6745.18	−0.121598	−0.0607989	+	0.998150i	$0.519365\pi$
$80$	0	0
$81$	37248.6	0.630808
$82$	0	0
$83$	−49891.3	−0.794931	−0.397465	−	0.917617i	$-0.630110\pi$
$83$	−49891.3	−0.794931	−0.397465	+	0.917617i	$0.630110\pi$
$84$	0	0
$85$	−100823.	−1.51361
$86$	0	0
$87$	−183192.	−2.59483
$88$	0	0
$89$	−37071.9	−0.496100	−0.248050	−	0.968747i	$-0.579790\pi$
$89$	−37071.9	−0.496100	−0.248050	+	0.968747i	$0.579790\pi$
$90$	0	0
$91$	46240.3	0.585352
$92$	0	0
$93$	158172.	1.89637
$94$	0	0
$95$	18201.1	0.206913
$96$	0	0
$97$	181017.	1.95339	0.976695	−	0.214631i	$-0.0688549\pi$
$97$	181017.	1.95339	0.976695	+	0.214631i	$0.0688549\pi$
$98$	0	0
$99$	−334746.	−3.43263
$100$	0	0
$101$	−140069.	−1.36627	−0.683137	−	0.730290i	$-0.739385\pi$
$101$	−140069.	−1.36627	−0.683137	+	0.730290i	$0.739385\pi$
$102$	0	0
$103$	−17513.2	−0.162657	−0.0813284	−	0.996687i	$-0.525916\pi$
$103$	−17513.2	−0.162657	−0.0813284	+	0.996687i	$0.525916\pi$
$104$	0	0
$105$	−156437.	−1.38473
$106$	0	0
$107$	107700.	0.909406	0.454703	−	0.890643i	$-0.349745\pi$
$107$	107700.	0.909406	0.454703	+	0.890643i	$0.349745\pi$
$108$	0	0
$109$	−156244.	−1.25961	−0.629806	−	0.776752i	$-0.716866\pi$
$109$	−156244.	−1.25961	−0.629806	+	0.776752i	$0.716866\pi$
$110$	0	0
$111$	51387.0	0.395864
$112$	0	0
$113$	−28681.8	−0.211305	−0.105653	−	0.994403i	$-0.533693\pi$
$113$	−28681.8	−0.211305	−0.105653	+	0.994403i	$0.533693\pi$
$114$	0	0
$115$	−138265.	−0.974915
$116$	0	0
$117$	179020.	1.20903
$118$	0	0
$119$	−234892.	−1.52055
$120$	0	0
$121$	380791.	2.36442
$122$	0	0
$123$	483522.	2.88173
$124$	0	0
$125$	−186950.	−1.07017
$126$	0	0
$127$	51483.4	0.283242	0.141621	−	0.989921i	$-0.454769\pi$
$127$	51483.4	0.283242	0.141621	+	0.989921i	$0.454769\pi$
$128$	0	0
$129$	211398.	1.11847
$130$	0	0
$131$	318922.	1.62370	0.811850	−	0.583866i	$-0.198461\pi$
$131$	318922.	1.62370	0.811850	+	0.583866i	$0.198461\pi$
$132$	0	0
$133$	42403.8	0.207862
$134$	0	0
$135$	−282019.	−1.33182
$136$	0	0
$137$	−104328.	−0.474897	−0.237449	−	0.971400i	$-0.576311\pi$
$137$	−104328.	−0.474897	−0.237449	+	0.971400i	$0.576311\pi$
$138$	0	0
$139$	−132371.	−0.581109	−0.290554	−	0.956859i	$-0.593840\pi$
$139$	−132371.	−0.581109	−0.290554	+	0.956859i	$0.593840\pi$
$140$	0	0
$141$	323473.	1.37022
$142$	0	0
$143$	−289774.	−1.18500
$144$	0	0
$145$	349660.	1.38110
$146$	0	0
$147$	79500.9	0.303444
$148$	0	0
$149$	187442.	0.691672	0.345836	−	0.938295i	$-0.387595\pi$
$149$	187442.	0.691672	0.345836	+	0.938295i	$0.387595\pi$
$150$	0	0
$151$	−494518.	−1.76498	−0.882491	−	0.470330i	$-0.844135\pi$
$151$	−494518.	−1.76498	−0.882491	+	0.470330i	$0.844135\pi$
$152$	0	0
$153$	−909388.	−3.14066
$154$	0	0
$155$	−301904.	−1.00934
$156$	0	0
$157$	543621.	1.76014	0.880070	−	0.474844i	$-0.157495\pi$
$157$	543621.	1.76014	0.880070	+	0.474844i	$0.157495\pi$
$158$	0	0
$159$	−109771.	−0.344346
$160$	0	0
$161$	−322121.	−0.979386
$162$	0	0
$163$	394609.	1.16332	0.581658	−	0.813433i	$-0.302404\pi$
$163$	394609.	1.16332	0.581658	+	0.813433i	$0.302404\pi$
$164$	0	0
$165$	980344.	2.80329
$166$	0	0
$167$	717473.	1.99074	0.995369	−	0.0961230i	$-0.0306442\pi$
$167$	717473.	1.99074	0.995369	+	0.0961230i	$0.0306442\pi$
$168$	0	0
$169$	−216324.	−0.582622
$170$	0	0
$171$	164167.	0.429334
$172$	0	0
$173$	385124.	0.978329	0.489165	−	0.872191i	$-0.337302\pi$
$173$	385124.	0.978329	0.489165	+	0.872191i	$0.337302\pi$
$174$	0	0
$175$	−68477.1	−0.169025
$176$	0	0
$177$	−380518.	−0.912940
$178$	0	0
$179$	−477897.	−1.11481	−0.557406	−	0.830240i	$-0.688203\pi$
$179$	−477897.	−1.11481	−0.557406	+	0.830240i	$0.688203\pi$
$180$	0	0
$181$	116299.	0.263864	0.131932	−	0.991259i	$-0.457882\pi$
$181$	116299.	0.263864	0.131932	+	0.991259i	$0.457882\pi$
$182$	0	0
$183$	282872.	0.624398
$184$	0	0
$185$	−98082.5	−0.210699
$186$	0	0
$187$	1.47200e6	3.07824
$188$	0	0
$189$	−657031.	−1.33792
$190$	0	0
$191$	119655.	0.237327	0.118664	−	0.992935i	$-0.462139\pi$
$191$	119655.	0.237327	0.118664	+	0.992935i	$0.462139\pi$
$192$	0	0
$193$	−514668.	−0.994566	−0.497283	−	0.867588i	$-0.665669\pi$
$193$	−514668.	−0.994566	−0.497283	+	0.867588i	$0.665669\pi$
$194$	0	0
$195$	−524282.	−0.987367
$196$	0	0
$197$	−680374.	−1.24906	−0.624528	−	0.781002i	$-0.714709\pi$
$197$	−680374.	−1.24906	−0.624528	+	0.781002i	$0.714709\pi$
$198$	0	0
$199$	466682.	0.835389	0.417694	−	0.908588i	$-0.362838\pi$
$199$	466682.	0.835389	0.417694	+	0.908588i	$0.362838\pi$
$200$	0	0
$201$	−389043.	−0.679215
$202$	0	0
$203$	814616.	1.38744
$204$	0	0
$205$	−922899.	−1.53380
$206$	0	0
$207$	−1.24710e6	−2.02290
$208$	0	0
$209$	−265732.	−0.420802
$210$	0	0
$211$	397850.	0.615195	0.307598	−	0.951517i	$-0.400475\pi$
$211$	397850.	0.615195	0.307598	+	0.951517i	$0.400475\pi$
$212$	0	0
$213$	256995.	0.388128
$214$	0	0
$215$	−403496.	−0.595309
$216$	0	0
$217$	−703357.	−1.01397
$218$	0	0
$219$	−1.24047e6	−1.74774
$220$	0	0
$221$	−787216.	−1.08421
$222$	0	0
$223$	899258.	1.21094	0.605469	−	0.795869i	$-0.292985\pi$
$223$	899258.	1.21094	0.605469	+	0.795869i	$0.292985\pi$
$224$	0	0
$225$	−265110.	−0.349117
$226$	0	0
$227$	−110485.	−0.142311	−0.0711554	−	0.997465i	$-0.522669\pi$
$227$	−110485.	−0.142311	−0.0711554	+	0.997465i	$0.522669\pi$
$228$	0	0
$229$	−1.33652e6	−1.68417	−0.842085	−	0.539344i	$-0.818672\pi$
$229$	−1.33652e6	−1.68417	−0.842085	+	0.539344i	$0.818672\pi$
$230$	0	0
$231$	2.28395e6	2.81615
$232$	0	0
$233$	−120754.	−0.145717	−0.0728586	−	0.997342i	$-0.523212\pi$
$233$	−120754.	−0.145717	−0.0728586	+	0.997342i	$0.523212\pi$
$234$	0	0
$235$	−617413.	−0.729300
$236$	0	0
$237$	178174.	0.206051
$238$	0	0
$239$	−288249.	−0.326417	−0.163209	−	0.986592i	$-0.552184\pi$
$239$	−288249.	−0.326417	−0.163209	+	0.986592i	$0.552184\pi$
$240$	0	0
$241$	−709988.	−0.787424	−0.393712	−	0.919234i	$-0.628809\pi$
$241$	−709988.	−0.787424	−0.393712	+	0.919234i	$0.628809\pi$
$242$	0	0
$243$	375312.	0.407733
$244$	0	0
$245$	−151744.	−0.161508
$246$	0	0
$247$	142112.	0.148214
$248$	0	0
$249$	1.31788e6	1.34703
$250$	0	0
$251$	−1.56345e6	−1.56639	−0.783193	−	0.621778i	$-0.786410\pi$
$251$	−1.56345e6	−1.56639	−0.783193	+	0.621778i	$0.786410\pi$
$252$	0	0
$253$	2.01863e6	1.98270
$254$	0	0
$255$	2.66325e6	2.56485
$256$	0	0
$257$	−824516.	−0.778693	−0.389346	−	0.921091i	$-0.627299\pi$
$257$	−824516.	−0.778693	−0.389346	+	0.921091i	$0.627299\pi$
$258$	0	0
$259$	−228507.	−0.211665
$260$	0	0
$261$	3.15380e6	2.86572
$262$	0	0
$263$	−395085.	−0.352209	−0.176105	−	0.984371i	$-0.556350\pi$
$263$	−395085.	−0.352209	−0.176105	+	0.984371i	$0.556350\pi$
$264$	0	0
$265$	209520.	0.183278
$266$	0	0
$267$	979256.	0.840655
$268$	0	0
$269$	670319.	0.564808	0.282404	−	0.959296i	$-0.408868\pi$
$269$	670319.	0.564808	0.282404	+	0.959296i	$0.408868\pi$
$270$	0	0
$271$	250941.	0.207562	0.103781	−	0.994600i	$-0.466906\pi$
$271$	250941.	0.207562	0.103781	+	0.994600i	$0.466906\pi$
$272$	0	0
$273$	−1.22144e6	−0.991895
$274$	0	0
$275$	429126.	0.342179
$276$	0	0
$277$	1.09157e6	0.854777	0.427389	−	0.904068i	$-0.359434\pi$
$277$	1.09157e6	0.854777	0.427389	+	0.904068i	$0.359434\pi$
$278$	0	0
$279$	−2.72306e6	−2.09434
$280$	0	0
$281$	−745172.	−0.562977	−0.281488	−	0.959565i	$-0.590828\pi$
$281$	−745172.	−0.562977	−0.281488	+	0.959565i	$0.590828\pi$
$282$	0	0
$283$	2.42623e6	1.80080	0.900402	−	0.435058i	$-0.143272\pi$
$283$	2.42623e6	1.80080	0.900402	+	0.435058i	$0.143272\pi$
$284$	0	0
$285$	−480783.	−0.350620
$286$	0	0
$287$	−2.15011e6	−1.54084
$288$	0	0
$289$	2.57905e6	1.81641
$290$	0	0
$291$	−4.78157e6	−3.31007
$292$	0	0
$293$	−1.06475e6	−0.724568	−0.362284	−	0.932068i	$-0.618003\pi$
$293$	−1.06475e6	−0.724568	−0.362284	+	0.932068i	$0.618003\pi$
$294$	0	0
$295$	726296.	0.485913
$296$	0	0
$297$	4.11742e6	2.70853
$298$	0	0
$299$	−1.07955e6	−0.698339
$300$	0	0
$301$	−940039.	−0.598039
$302$	0	0
$303$	3.69993e6	2.31519
$304$	0	0
$305$	−539918.	−0.332336
$306$	0	0
$307$	939760.	0.569077	0.284539	−	0.958665i	$-0.408160\pi$
$307$	939760.	0.569077	0.284539	+	0.958665i	$0.408160\pi$
$308$	0	0
$309$	462612.	0.275626
$310$	0	0
$311$	1.58564e6	0.929616	0.464808	−	0.885412i	$-0.346123\pi$
$311$	1.58564e6	0.929616	0.464808	+	0.885412i	$0.346123\pi$
$312$	0	0
$313$	−178425.	−0.102943	−0.0514713	−	0.998674i	$-0.516391\pi$
$313$	−178425.	−0.102943	−0.0514713	+	0.998674i	$0.516391\pi$
$314$	0	0
$315$	2.69319e6	1.52929
$316$	0	0
$317$	−3.01072e6	−1.68276	−0.841380	−	0.540443i	$-0.818257\pi$
$317$	−3.01072e6	−1.68276	−0.841380	+	0.540443i	$0.818257\pi$
$318$	0	0
$319$	−5.10496e6	−2.80877
$320$	0	0
$321$	−2.84491e6	−1.54101
$322$	0	0
$323$	−721901.	−0.385010
$324$	0	0
$325$	−229494.	−0.120521
$326$	0	0
$327$	4.12720e6	2.13445
$328$	0	0
$329$	−1.43841e6	−0.732645
$330$	0	0
$331$	−820811.	−0.411788	−0.205894	−	0.978574i	$-0.566010\pi$
$331$	−820811.	−0.411788	−0.205894	+	0.978574i	$0.566010\pi$
$332$	0	0
$333$	−884668.	−0.437190
$334$	0	0
$335$	742567.	0.361513
$336$	0	0
$337$	−2.81923e6	−1.35225	−0.676123	−	0.736789i	$-0.736341\pi$
$337$	−2.81923e6	−1.35225	−0.676123	+	0.736789i	$0.736341\pi$
$338$	0	0
$339$	757631.	0.358062
$340$	0	0
$341$	4.40773e6	2.05272
$342$	0	0
$343$	−2.32771e6	−1.06830
$344$	0	0
$345$	3.65227e6	1.65202
$346$	0	0
$347$	406596.	0.181276	0.0906378	−	0.995884i	$-0.471109\pi$
$347$	406596.	0.181276	0.0906378	+	0.995884i	$0.471109\pi$
$348$	0	0
$349$	−1.61502e6	−0.709766	−0.354883	−	0.934911i	$-0.615479\pi$
$349$	−1.61502e6	−0.709766	−0.354883	+	0.934911i	$0.615479\pi$
$350$	0	0
$351$	−2.20197e6	−0.953990
$352$	0	0
$353$	1.08174e6	0.462045	0.231023	−	0.972948i	$-0.425793\pi$
$353$	1.08174e6	0.462045	0.231023	+	0.972948i	$0.425793\pi$
$354$	0	0
$355$	−490526.	−0.206582
$356$	0	0
$357$	6.20469e6	2.57661
$358$	0	0
$359$	−245971.	−0.100727	−0.0503636	−	0.998731i	$-0.516038\pi$
$359$	−245971.	−0.100727	−0.0503636	+	0.998731i	$0.516038\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−1.00586e7	−4.00657
$364$	0	0
$365$	2.36769e6	0.930235
$366$	0	0
$367$	2.88270e6	1.11721	0.558603	−	0.829435i	$-0.311337\pi$
$367$	2.88270e6	1.11721	0.558603	+	0.829435i	$0.311337\pi$
$368$	0	0
$369$	−8.32421e6	−3.18256
$370$	0	0
$371$	488127.	0.184119
$372$	0	0
$373$	−131540.	−0.0489538	−0.0244769	−	0.999700i	$-0.507792\pi$
$373$	−131540.	−0.0489538	−0.0244769	+	0.999700i	$0.507792\pi$
$374$	0	0
$375$	4.93831e6	1.81343
$376$	0	0
$377$	2.73010e6	0.989294
$378$	0	0
$379$	−916724.	−0.327824	−0.163912	−	0.986475i	$-0.552411\pi$
$379$	−916724.	−0.327824	−0.163912	+	0.986475i	$0.552411\pi$
$380$	0	0
$381$	−1.35994e6	−0.479962
$382$	0	0
$383$	−765638.	−0.266702	−0.133351	−	0.991069i	$-0.542574\pi$
$383$	−765638.	−0.266702	−0.133351	+	0.991069i	$0.542574\pi$
$384$	0	0
$385$	−4.35937e6	−1.49890
$386$	0	0
$387$	−3.63938e6	−1.23524
$388$	0	0
$389$	1.76287e6	0.590670	0.295335	−	0.955394i	$-0.404569\pi$
$389$	1.76287e6	0.590670	0.295335	+	0.955394i	$0.404569\pi$
$390$	0	0
$391$	5.48393e6	1.81405
$392$	0	0
$393$	−8.42434e6	−2.75141
$394$	0	0
$395$	−340082.	−0.109671
$396$	0	0
$397$	−3.88958e6	−1.23859	−0.619293	−	0.785160i	$-0.712581\pi$
$397$	−3.88958e6	−1.23859	−0.619293	+	0.785160i	$0.712581\pi$
$398$	0	0
$399$	−1.12010e6	−0.352228
$400$	0	0
$401$	−4.10595e6	−1.27513	−0.637563	−	0.770398i	$-0.720057\pi$
$401$	−4.10595e6	−1.27513	−0.637563	+	0.770398i	$0.720057\pi$
$402$	0	0
$403$	−2.35723e6	−0.723001
$404$	0	0
$405$	1.87802e6	0.568934
$406$	0	0
$407$	1.43198e6	0.428501
$408$	0	0
$409$	2.65576e6	0.785020	0.392510	−	0.919748i	$-0.371607\pi$
$409$	2.65576e6	0.785020	0.392510	+	0.919748i	$0.371607\pi$
$410$	0	0
$411$	2.75583e6	0.804727
$412$	0	0
$413$	1.69208e6	0.488142
$414$	0	0
$415$	−2.51544e6	−0.716959
$416$	0	0
$417$	3.49660e6	0.984705
$418$	0	0
$419$	4.97316e6	1.38388	0.691938	−	0.721957i	$-0.256757\pi$
$419$	4.97316e6	1.38388	0.691938	+	0.721957i	$0.256757\pi$
$420$	0	0
$421$	6.07891e6	1.67156	0.835778	−	0.549068i	$-0.185017\pi$
$421$	6.07891e6	1.67156	0.835778	+	0.549068i	$0.185017\pi$
$422$	0	0
$423$	−5.56884e6	−1.51326
$424$	0	0
$425$	1.16579e6	0.313073
$426$	0	0
$427$	−1.25787e6	−0.333860
$428$	0	0
$429$	7.65441e6	2.00802
$430$	0	0
$431$	2.92330e6	0.758020	0.379010	−	0.925393i	$-0.376265\pi$
$431$	2.92330e6	0.758020	0.379010	+	0.925393i	$0.376265\pi$
$432$	0	0
$433$	−2.41856e6	−0.619923	−0.309961	−	0.950749i	$-0.600316\pi$
$433$	−2.41856e6	−0.619923	−0.309961	+	0.950749i	$0.600316\pi$
$434$	0	0
$435$	−9.23629e6	−2.34032
$436$	0	0
$437$	−989984.	−0.247985
$438$	0	0
$439$	5.67777e6	1.40610	0.703051	−	0.711140i	$-0.251821\pi$
$439$	5.67777e6	1.40610	0.703051	+	0.711140i	$0.251821\pi$
$440$	0	0
$441$	−1.36867e6	−0.335122
$442$	0	0
$443$	−3.99470e6	−0.967108	−0.483554	−	0.875315i	$-0.660654\pi$
$443$	−3.99470e6	−0.967108	−0.483554	+	0.875315i	$0.660654\pi$
$444$	0	0
$445$	−1.86911e6	−0.447440
$446$	0	0
$447$	−4.95128e6	−1.17206
$448$	0	0
$449$	6.26357e6	1.46624	0.733122	−	0.680097i	$-0.238063\pi$
$449$	6.26357e6	1.46624	0.733122	+	0.680097i	$0.238063\pi$
$450$	0	0
$451$	1.34741e7	3.11931
$452$	0	0
$453$	1.30627e7	2.99081
$454$	0	0
$455$	2.33137e6	0.527937
$456$	0	0
$457$	2.71215e6	0.607468	0.303734	−	0.952757i	$-0.401767\pi$
$457$	2.71215e6	0.607468	0.303734	+	0.952757i	$0.401767\pi$
$458$	0	0
$459$	1.11856e7	2.47815
$460$	0	0
$461$	−1.15615e6	−0.253373	−0.126687	−	0.991943i	$-0.540434\pi$
$461$	−1.15615e6	−0.253373	−0.126687	+	0.991943i	$0.540434\pi$
$462$	0	0
$463$	−3.61856e6	−0.784483	−0.392242	−	0.919862i	$-0.628300\pi$
$463$	−3.61856e6	−0.784483	−0.392242	+	0.919862i	$0.628300\pi$
$464$	0	0
$465$	7.97481e6	1.71036
$466$	0	0
$467$	1.88251e6	0.399433	0.199717	−	0.979854i	$-0.435998\pi$
$467$	1.88251e6	0.399433	0.199717	+	0.979854i	$0.435998\pi$
$468$	0	0
$469$	1.72999e6	0.363170
$470$	0	0
$471$	−1.43598e7	−2.98261
$472$	0	0
$473$	5.89095e6	1.21069
$474$	0	0
$475$	−210453.	−0.0427978
$476$	0	0
$477$	1.88979e6	0.380293
$478$	0	0
$479$	−6.80502e6	−1.35516	−0.677580	−	0.735449i	$-0.736971\pi$
$479$	−6.80502e6	−1.35516	−0.677580	+	0.735449i	$0.736971\pi$
$480$	0	0
$481$	−765816.	−0.150925
$482$	0	0
$483$	8.50884e6	1.65960
$484$	0	0
$485$	9.12659e6	1.76179
$486$	0	0
$487$	717935.	0.137171	0.0685856	−	0.997645i	$-0.478151\pi$
$487$	717935.	0.137171	0.0685856	+	0.997645i	$0.478151\pi$
$488$	0	0
$489$	−1.04236e7	−1.97127
$490$	0	0
$491$	−5.33377e6	−0.998459	−0.499230	−	0.866470i	$-0.666384\pi$
$491$	−5.33377e6	−0.998459	−0.499230	+	0.866470i	$0.666384\pi$
$492$	0	0
$493$	−1.38684e7	−2.56986
$494$	0	0
$495$	−1.68774e7	−3.09594
$496$	0	0
$497$	−1.14280e6	−0.207529
$498$	0	0
$499$	−2.34464e6	−0.421526	−0.210763	−	0.977537i	$-0.567595\pi$
$499$	−2.34464e6	−0.421526	−0.210763	+	0.977537i	$0.567595\pi$
$500$	0	0
$501$	−1.89521e7	−3.37336
$502$	0	0
$503$	−4.28715e6	−0.755524	−0.377762	−	0.925903i	$-0.623306\pi$
$503$	−4.28715e6	−0.755524	−0.377762	+	0.925903i	$0.623306\pi$
$504$	0	0
$505$	−7.06206e6	−1.23226
$506$	0	0
$507$	5.71420e6	0.987270
$508$	0	0
$509$	1.04612e7	1.78973	0.894864	−	0.446339i	$-0.147272\pi$
$509$	1.04612e7	1.78973	0.894864	+	0.446339i	$0.147272\pi$
$510$	0	0
$511$	5.51610e6	0.934501
$512$	0	0
$513$	−2.01928e6	−0.338768
$514$	0	0
$515$	−882989.	−0.146702
$516$	0	0
$517$	9.01410e6	1.48319
$518$	0	0
$519$	−1.01731e7	−1.65781
$520$	0	0
$521$	3.49778e6	0.564545	0.282273	−	0.959334i	$-0.408912\pi$
$521$	3.49778e6	0.564545	0.282273	+	0.959334i	$0.408912\pi$
$522$	0	0
$523$	−3.56526e6	−0.569950	−0.284975	−	0.958535i	$-0.591985\pi$
$523$	−3.56526e6	−0.569950	−0.284975	+	0.958535i	$0.591985\pi$
$524$	0	0
$525$	1.80883e6	0.286417
$526$	0	0
$527$	1.19743e7	1.87812
$528$	0	0
$529$	1.08408e6	0.168431
$530$	0	0
$531$	6.55092e6	1.00824
$532$	0	0
$533$	−7.20588e6	−1.09867
$534$	0	0
$535$	5.43009e6	0.820206
$536$	0	0
$537$	1.26237e7	1.88908
$538$	0	0
$539$	2.21542e6	0.328462
$540$	0	0
$541$	−5.03584e6	−0.739740	−0.369870	−	0.929084i	$-0.620598\pi$
$541$	−5.03584e6	−0.739740	−0.369870	+	0.929084i	$0.620598\pi$
$542$	0	0
$543$	−3.07205e6	−0.447125
$544$	0	0
$545$	−7.87759e6	−1.13606
$546$	0	0
$547$	2.93665e6	0.419647	0.209823	−	0.977739i	$-0.432711\pi$
$547$	2.93665e6	0.419647	0.209823	+	0.977739i	$0.432711\pi$
$548$	0	0
$549$	−4.86986e6	−0.689581
$550$	0	0
$551$	2.50359e6	0.351305
$552$	0	0
$553$	−792302.	−0.110174
$554$	0	0
$555$	2.59086e6	0.357035
$556$	0	0
$557$	−650124.	−0.0887887	−0.0443944	−	0.999014i	$-0.514136\pi$
$557$	−650124.	−0.0887887	−0.0443944	+	0.999014i	$0.514136\pi$
$558$	0	0
$559$	−3.15044e6	−0.426424
$560$	0	0
$561$	−3.88829e7	−5.21617
$562$	0	0
$563$	1.08380e7	1.44104	0.720521	−	0.693433i	$-0.243903\pi$
$563$	1.08380e7	1.44104	0.720521	+	0.693433i	$0.243903\pi$
$564$	0	0
$565$	−1.44609e6	−0.190579
$566$	0	0
$567$	4.37529e6	0.571543
$568$	0	0
$569$	4.07511e6	0.527665	0.263832	−	0.964569i	$-0.415013\pi$
$569$	4.07511e6	0.527665	0.263832	+	0.964569i	$0.415013\pi$
$570$	0	0
$571$	−6.50701e6	−0.835202	−0.417601	−	0.908631i	$-0.637129\pi$
$571$	−6.50701e6	−0.835202	−0.417601	+	0.908631i	$0.637129\pi$
$572$	0	0
$573$	−3.16070e6	−0.402157
$574$	0	0
$575$	1.59871e6	0.201651
$576$	0	0
$577$	7.71744e6	0.965015	0.482507	−	0.875892i	$-0.339726\pi$
$577$	7.71744e6	0.965015	0.482507	+	0.875892i	$0.339726\pi$
$578$	0	0
$579$	1.35950e7	1.68532
$580$	0	0
$581$	−5.86033e6	−0.720247
$582$	0	0
$583$	−3.05895e6	−0.372735
$584$	0	0
$585$	9.02593e6	1.09044
$586$	0	0
$587$	265516.	0.0318050	0.0159025	−	0.999874i	$-0.494938\pi$
$587$	265516.	0.0318050	0.0159025	+	0.999874i	$0.494938\pi$
$588$	0	0
$589$	−2.16165e6	−0.256742
$590$	0	0
$591$	1.79721e7	2.11656
$592$	0	0
$593$	2.11004e6	0.246407	0.123204	−	0.992381i	$-0.460683\pi$
$593$	2.11004e6	0.246407	0.123204	+	0.992381i	$0.460683\pi$
$594$	0	0
$595$	−1.18429e7	−1.37140
$596$	0	0
$597$	−1.23274e7	−1.41559
$598$	0	0
$599$	−9.93148e6	−1.13096	−0.565480	−	0.824762i	$-0.691309\pi$
$599$	−9.93148e6	−1.13096	−0.565480	+	0.824762i	$0.691309\pi$
$600$	0	0
$601$	−1.40245e6	−0.158381	−0.0791903	−	0.996860i	$-0.525233\pi$
$601$	−1.40245e6	−0.158381	−0.0791903	+	0.996860i	$0.525233\pi$
$602$	0	0
$603$	6.69768e6	0.750120
$604$	0	0
$605$	1.91989e7	2.13250
$606$	0	0
$607$	−1.57081e7	−1.73042	−0.865212	−	0.501406i	$-0.832816\pi$
$607$	−1.57081e7	−1.73042	−0.865212	+	0.501406i	$0.832816\pi$
$608$	0	0
$609$	−2.15181e7	−2.35105
$610$	0	0
$611$	−4.82069e6	−0.522403
$612$	0	0
$613$	−4.71450e6	−0.506739	−0.253370	−	0.967370i	$-0.581539\pi$
$613$	−4.71450e6	−0.506739	−0.253370	+	0.967370i	$0.581539\pi$
$614$	0	0
$615$	2.43784e7	2.59907
$616$	0	0
$617$	8.13407e6	0.860191	0.430096	−	0.902783i	$-0.358480\pi$
$617$	8.13407e6	0.860191	0.430096	+	0.902783i	$0.358480\pi$
$618$	0	0
$619$	−9.01047e6	−0.945193	−0.472597	−	0.881279i	$-0.656683\pi$
$619$	−9.01047e6	−0.945193	−0.472597	+	0.881279i	$0.656683\pi$
$620$	0	0
$621$	1.53394e7	1.59617
$622$	0	0
$623$	−4.35453e6	−0.449492
$624$	0	0
$625$	−7.60398e6	−0.778647
$626$	0	0
$627$	7.01933e6	0.713061
$628$	0	0
$629$	3.89020e6	0.392054
$630$	0	0
$631$	356846.	0.0356785	0.0178393	−	0.999841i	$-0.494321\pi$
$631$	356846.	0.0356785	0.0178393	+	0.999841i	$0.494321\pi$
$632$	0	0
$633$	−1.05092e7	−1.04247
$634$	0	0
$635$	2.59572e6	0.255460
$636$	0	0
$637$	−1.18480e6	−0.115690
$638$	0	0
$639$	−4.42437e6	−0.428646
$640$	0	0
$641$	−1.48947e7	−1.43181	−0.715906	−	0.698196i	$-0.753986\pi$
$641$	−1.48947e7	−1.43181	−0.715906	+	0.698196i	$0.753986\pi$
$642$	0	0
$643$	7.47747e6	0.713227	0.356613	−	0.934252i	$-0.383931\pi$
$643$	7.47747e6	0.713227	0.356613	+	0.934252i	$0.383931\pi$
$644$	0	0
$645$	1.06584e7	1.00877
$646$	0	0
$647$	−1.65813e7	−1.55725	−0.778623	−	0.627492i	$-0.784081\pi$
$647$	−1.65813e7	−1.55725	−0.778623	+	0.627492i	$0.784081\pi$
$648$	0	0
$649$	−1.06038e7	−0.988208
$650$	0	0
$651$	1.85792e7	1.71821
$652$	0	0
$653$	805018.	0.0738793	0.0369396	−	0.999317i	$-0.488239\pi$
$653$	805018.	0.0738793	0.0369396	+	0.999317i	$0.488239\pi$
$654$	0	0
$655$	1.60796e7	1.46444
$656$	0	0
$657$	2.13557e7	1.93019
$658$	0	0
$659$	−3.38938e6	−0.304024	−0.152012	−	0.988379i	$-0.548575\pi$
$659$	−3.38938e6	−0.304024	−0.152012	+	0.988379i	$0.548575\pi$
$660$	0	0
$661$	1.26082e7	1.12240	0.561202	−	0.827679i	$-0.310339\pi$
$661$	1.26082e7	1.12240	0.561202	+	0.827679i	$0.310339\pi$
$662$	0	0
$663$	2.07944e7	1.83722
$664$	0	0
$665$	2.13794e6	0.187474
$666$	0	0
$667$	−1.90185e7	−1.65524
$668$	0	0
$669$	−2.37540e7	−2.05197
$670$	0	0
$671$	7.88268e6	0.675877
$672$	0	0
$673$	9.74804e6	0.829621	0.414810	−	0.909908i	$-0.363848\pi$
$673$	9.74804e6	0.829621	0.414810	+	0.909908i	$0.363848\pi$
$674$	0	0
$675$	3.26089e6	0.275472
$676$	0	0
$677$	1.46184e7	1.22583	0.612913	−	0.790151i	$-0.289998\pi$
$677$	1.46184e7	1.22583	0.612913	+	0.790151i	$0.289998\pi$
$678$	0	0
$679$	2.12626e7	1.76987
$680$	0	0
$681$	2.91847e6	0.241150
$682$	0	0
$683$	−777842.	−0.0638028	−0.0319014	−	0.999491i	$-0.510156\pi$
$683$	−777842.	−0.0638028	−0.0319014	+	0.999491i	$0.510156\pi$
$684$	0	0
$685$	−5.26007e6	−0.428317
$686$	0	0
$687$	3.53042e7	2.85387
$688$	0	0
$689$	1.63591e6	0.131284
$690$	0	0
$691$	−1.28544e7	−1.02413	−0.512066	−	0.858946i	$-0.671120\pi$
$691$	−1.28544e7	−1.02413	−0.512066	+	0.858946i	$0.671120\pi$
$692$	0	0
$693$	−3.93199e7	−3.11014
$694$	0	0
$695$	−6.67397e6	−0.524110
$696$	0	0
$697$	3.66045e7	2.85399
$698$	0	0
$699$	3.18972e6	0.246922
$700$	0	0
$701$	−4.22611e6	−0.324823	−0.162411	−	0.986723i	$-0.551927\pi$
$701$	−4.22611e6	−0.324823	−0.162411	+	0.986723i	$0.551927\pi$
$702$	0	0
$703$	−702278.	−0.0535945
$704$	0	0
$705$	1.63090e7	1.23582
$706$	0	0
$707$	−1.64528e7	−1.23791
$708$	0	0
$709$	2.83805e6	0.212033	0.106017	−	0.994364i	$-0.466190\pi$
$709$	2.83805e6	0.212033	0.106017	+	0.994364i	$0.466190\pi$
$710$	0	0
$711$	−3.06741e6	−0.227561
$712$	0	0
$713$	1.64210e7	1.20969
$714$	0	0
$715$	−1.46100e7	−1.06877
$716$	0	0
$717$	7.61412e6	0.553123
$718$	0	0
$719$	−2.06802e7	−1.49187	−0.745936	−	0.666017i	$-0.767998\pi$
$719$	−2.06802e7	−1.49187	−0.745936	+	0.666017i	$0.767998\pi$
$720$	0	0
$721$	−2.05713e6	−0.147375
$722$	0	0
$723$	1.87544e7	1.33431
$724$	0	0
$725$	−4.04300e6	−0.285666
$726$	0	0
$727$	−6.59677e6	−0.462909	−0.231454	−	0.972846i	$-0.574348\pi$
$727$	−6.59677e6	−0.462909	−0.231454	+	0.972846i	$0.574348\pi$
$728$	0	0
$729$	−1.89653e7	−1.32172
$730$	0	0
$731$	1.60037e7	1.10771
$732$	0	0
$733$	−2.81462e7	−1.93491	−0.967454	−	0.253047i	$-0.918567\pi$
$733$	−2.81462e7	−1.93491	−0.967454	+	0.253047i	$0.918567\pi$
$734$	0	0
$735$	4.00832e6	0.273681
$736$	0	0
$737$	−1.08413e7	−0.735213
$738$	0	0
$739$	−1.88834e7	−1.27195	−0.635973	−	0.771711i	$-0.719401\pi$
$739$	−1.88834e7	−1.27195	−0.635973	+	0.771711i	$0.719401\pi$
$740$	0	0
$741$	−3.75390e6	−0.251152
$742$	0	0
$743$	−1.34081e7	−0.891037	−0.445519	−	0.895273i	$-0.646981\pi$
$743$	−1.34081e7	−0.891037	−0.445519	+	0.895273i	$0.646981\pi$
$744$	0	0
$745$	9.45052e6	0.623828
$746$	0	0
$747$	−2.26884e7	−1.48765
$748$	0	0
$749$	1.26507e7	0.823967
$750$	0	0
$751$	6.58370e6	0.425961	0.212981	−	0.977056i	$-0.431683\pi$
$751$	6.58370e6	0.425961	0.212981	+	0.977056i	$0.431683\pi$
$752$	0	0
$753$	4.12986e7	2.65429
$754$	0	0
$755$	−2.49329e7	−1.59186
$756$	0	0
$757$	1.88527e7	1.19573	0.597866	−	0.801596i	$-0.296015\pi$
$757$	1.88527e7	1.19573	0.597866	+	0.801596i	$0.296015\pi$
$758$	0	0
$759$	−5.33224e7	−3.35974
$760$	0	0
$761$	−2.85989e6	−0.179014	−0.0895071	−	0.995986i	$-0.528529\pi$
$761$	−2.85989e6	−0.179014	−0.0895071	+	0.995986i	$0.528529\pi$
$762$	0	0
$763$	−1.83527e7	−1.14127
$764$	0	0
$765$	−4.58500e7	−2.83260
$766$	0	0
$767$	5.67083e6	0.348063
$768$	0	0
$769$	7.27604e6	0.443689	0.221845	−	0.975082i	$-0.428792\pi$
$769$	7.27604e6	0.443689	0.221845	+	0.975082i	$0.428792\pi$
$770$	0	0
$771$	2.17796e7	1.31952
$772$	0	0
$773$	−3.66144e6	−0.220396	−0.110198	−	0.993910i	$-0.535148\pi$
$773$	−3.66144e6	−0.220396	−0.110198	+	0.993910i	$0.535148\pi$
$774$	0	0
$775$	3.49081e6	0.208772
$776$	0	0
$777$	6.03602e6	0.358673
$778$	0	0
$779$	−6.60802e6	−0.390146
$780$	0	0
$781$	7.16158e6	0.420128
$782$	0	0
$783$	−3.87922e7	−2.26120
$784$	0	0
$785$	2.74086e7	1.58749
$786$	0	0
$787$	7.11841e6	0.409681	0.204841	−	0.978795i	$-0.434332\pi$
$787$	7.11841e6	0.409681	0.204841	+	0.978795i	$0.434332\pi$
$788$	0	0
$789$	1.04362e7	0.596828
$790$	0	0
$791$	−3.36902e6	−0.191453
$792$	0	0
$793$	−4.21561e6	−0.238055
$794$	0	0
$795$	−5.53449e6	−0.310570
$796$	0	0
$797$	−8.68702e6	−0.484423	−0.242212	−	0.970223i	$-0.577873\pi$
$797$	−8.68702e6	−0.484423	−0.242212	+	0.970223i	$0.577873\pi$
$798$	0	0
$799$	2.44882e7	1.35703
$800$	0	0
$801$	−1.68587e7	−0.928414
$802$	0	0
$803$	−3.45677e7	−1.89183
$804$	0	0
$805$	−1.62408e7	−0.883322
$806$	0	0
$807$	−1.77065e7	−0.957083
$808$	0	0
$809$	−2.53472e7	−1.36163	−0.680815	−	0.732456i	$-0.738374\pi$
$809$	−2.53472e7	−1.36163	−0.680815	+	0.732456i	$0.738374\pi$
$810$	0	0
$811$	2.31637e6	0.123668	0.0618338	−	0.998086i	$-0.480305\pi$
$811$	2.31637e6	0.123668	0.0618338	+	0.998086i	$0.480305\pi$
$812$	0	0
$813$	−6.62861e6	−0.351719
$814$	0	0
$815$	1.98956e7	1.04921
$816$	0	0
$817$	−2.88905e6	−0.151426
$818$	0	0
$819$	2.10281e7	1.09544
$820$	0	0
$821$	3.51834e7	1.82171	0.910857	−	0.412722i	$-0.135422\pi$
$821$	3.51834e7	1.82171	0.910857	+	0.412722i	$0.135422\pi$
$822$	0	0
$823$	3.79443e7	1.95275	0.976376	−	0.216078i	$-0.0693265\pi$
$823$	3.79443e7	1.95275	0.976376	+	0.216078i	$0.0693265\pi$
$824$	0	0
$825$	−1.13354e7	−0.579831
$826$	0	0
$827$	1.57194e7	0.799228	0.399614	−	0.916683i	$-0.369144\pi$
$827$	1.57194e7	0.799228	0.399614	+	0.916683i	$0.369144\pi$
$828$	0	0
$829$	−1.53889e7	−0.777719	−0.388859	−	0.921297i	$-0.627131\pi$
$829$	−1.53889e7	−0.777719	−0.388859	+	0.921297i	$0.627131\pi$
$830$	0	0
$831$	−2.88340e7	−1.44844
$832$	0	0
$833$	6.01854e6	0.300523
$834$	0	0
$835$	3.61739e7	1.79548
$836$	0	0
$837$	3.34940e7	1.65254
$838$	0	0
$839$	1.76431e7	0.865306	0.432653	−	0.901561i	$-0.357578\pi$
$839$	1.76431e7	0.865306	0.432653	+	0.901561i	$0.357578\pi$
$840$	0	0
$841$	2.75851e7	1.34488
$842$	0	0
$843$	1.96838e7	0.953980
$844$	0	0
$845$	−1.09067e7	−0.525475
$846$	0	0
$847$	4.47285e7	2.14228
$848$	0	0
$849$	−6.40892e7	−3.05151
$850$	0	0
$851$	5.33486e6	0.252522
$852$	0	0
$853$	2.98567e6	0.140498	0.0702488	−	0.997530i	$-0.477621\pi$
$853$	2.98567e6	0.140498	0.0702488	+	0.997530i	$0.477621\pi$
$854$	0	0
$855$	8.27706e6	0.387223
$856$	0	0
$857$	−1.93515e7	−0.900041	−0.450020	−	0.893018i	$-0.648583\pi$
$857$	−1.93515e7	−0.900041	−0.450020	+	0.893018i	$0.648583\pi$
$858$	0	0
$859$	−2.00171e7	−0.925591	−0.462796	−	0.886465i	$-0.653154\pi$
$859$	−2.00171e7	−0.925591	−0.462796	+	0.886465i	$0.653154\pi$
$860$	0	0
$861$	5.67954e7	2.61099
$862$	0	0
$863$	−2.19015e7	−1.00103	−0.500516	−	0.865727i	$-0.666856\pi$
$863$	−2.19015e7	−1.00103	−0.500516	+	0.865727i	$0.666856\pi$
$864$	0	0
$865$	1.94174e7	0.882369
$866$	0	0
$867$	−6.81258e7	−3.07797
$868$	0	0
$869$	4.96512e6	0.223039
$870$	0	0
$871$	5.79787e6	0.258954
$872$	0	0
$873$	8.23185e7	3.65562
$874$	0	0
$875$	−2.19596e7	−0.969625
$876$	0	0
$877$	−3.99674e7	−1.75472	−0.877358	−	0.479836i	$-0.840696\pi$
$877$	−3.99674e7	−1.75472	−0.877358	+	0.479836i	$0.840696\pi$
$878$	0	0
$879$	2.81255e7	1.22780
$880$	0	0
$881$	9.05757e6	0.393162	0.196581	−	0.980488i	$-0.437016\pi$
$881$	9.05757e6	0.393162	0.196581	+	0.980488i	$0.437016\pi$
$882$	0	0
$883$	2.76140e7	1.19187	0.595933	−	0.803034i	$-0.296783\pi$
$883$	2.76140e7	1.19187	0.595933	+	0.803034i	$0.296783\pi$
$884$	0	0
$885$	−1.91852e7	−0.823393
$886$	0	0
$887$	−2.44354e7	−1.04282	−0.521411	−	0.853306i	$-0.674594\pi$
$887$	−2.44354e7	−1.04282	−0.521411	+	0.853306i	$0.674594\pi$
$888$	0	0
$889$	6.04734e6	0.256632
$890$	0	0
$891$	−2.74186e7	−1.15705
$892$	0	0
$893$	−4.42072e6	−0.185509
$894$	0	0
$895$	−2.40948e7	−1.00546
$896$	0	0
$897$	2.85165e7	1.18335
$898$	0	0
$899$	−4.15273e7	−1.71370
$900$	0	0
$901$	−8.31009e6	−0.341031
$902$	0	0
$903$	2.48312e7	1.01339
$904$	0	0
$905$	5.86364e6	0.237983
$906$	0	0
$907$	2.62509e7	1.05956	0.529780	−	0.848135i	$-0.322274\pi$
$907$	2.62509e7	1.05956	0.529780	+	0.848135i	$0.322274\pi$
$908$	0	0
$909$	−6.36972e7	−2.55688
$910$	0	0
$911$	−2.95953e7	−1.18148	−0.590741	−	0.806861i	$-0.701165\pi$
$911$	−2.95953e7	−1.18148	−0.590741	+	0.806861i	$0.701165\pi$
$912$	0	0
$913$	3.67249e7	1.45809
$914$	0	0
$915$	1.42620e7	0.563153
$916$	0	0
$917$	3.74612e7	1.47115
$918$	0	0
$919$	−3.70829e7	−1.44839	−0.724194	−	0.689596i	$-0.757788\pi$
$919$	−3.70829e7	−1.44839	−0.724194	+	0.689596i	$0.757788\pi$
$920$	0	0
$921$	−2.48238e7	−0.964317
$922$	0	0
$923$	−3.82997e6	−0.147976
$924$	0	0
$925$	1.13410e6	0.0435808
$926$	0	0
$927$	−7.96423e6	−0.304400
$928$	0	0
$929$	4.54045e7	1.72607	0.863037	−	0.505140i	$-0.168559\pi$
$929$	4.54045e7	1.72607	0.863037	+	0.505140i	$0.168559\pi$
$930$	0	0
$931$	−1.08649e6	−0.0410822
$932$	0	0
$933$	−4.18848e7	−1.57526
$934$	0	0
$935$	7.42159e7	2.77631
$936$	0	0
$937$	−2.70789e7	−1.00759	−0.503794	−	0.863824i	$-0.668063\pi$
$937$	−2.70789e7	−1.00759	−0.503794	+	0.863824i	$0.668063\pi$
$938$	0	0
$939$	4.71312e6	0.174439
$940$	0	0
$941$	−2.33175e7	−0.858438	−0.429219	−	0.903200i	$-0.641211\pi$
$941$	−2.33175e7	−0.858438	−0.429219	+	0.903200i	$0.641211\pi$
$942$	0	0
$943$	5.01978e7	1.83826
$944$	0	0
$945$	−3.31265e7	−1.20669
$946$	0	0
$947$	−3.64273e7	−1.31993	−0.659966	−	0.751295i	$-0.729429\pi$
$947$	−3.64273e7	−1.31993	−0.659966	+	0.751295i	$0.729429\pi$
$948$	0	0
$949$	1.84866e7	0.666334
$950$	0	0
$951$	7.95284e7	2.85149
$952$	0	0
$953$	667679.	0.0238142	0.0119071	−	0.999929i	$-0.496210\pi$
$953$	667679.	0.0238142	0.0119071	+	0.999929i	$0.496210\pi$
$954$	0	0
$955$	6.03283e6	0.214049
$956$	0	0
$957$	1.34848e8	4.75953
$958$	0	0
$959$	−1.22546e7	−0.430281
$960$	0	0
$961$	7.22640e6	0.252414
$962$	0	0
$963$	4.89774e7	1.70188
$964$	0	0
$965$	−2.59488e7	−0.897013
$966$	0	0
$967$	−3.04350e7	−1.04666	−0.523332	−	0.852129i	$-0.675311\pi$
$967$	−3.04350e7	−1.04666	−0.523332	+	0.852129i	$0.675311\pi$
$968$	0	0
$969$	1.90691e7	0.652409
$970$	0	0
$971$	−3.05747e6	−0.104067	−0.0520336	−	0.998645i	$-0.516570\pi$
$971$	−3.05747e6	−0.104067	−0.0520336	+	0.998645i	$0.516570\pi$
$972$	0	0
$973$	−1.55486e7	−0.526513
$974$	0	0
$975$	6.06210e6	0.204226
$976$	0	0
$977$	−2.13637e7	−0.716045	−0.358023	−	0.933713i	$-0.616549\pi$
$977$	−2.13637e7	−0.716045	−0.358023	+	0.933713i	$0.616549\pi$
$978$	0	0
$979$	2.72886e7	0.909964
$980$	0	0
$981$	−7.10529e7	−2.35727
$982$	0	0
$983$	−2.28275e7	−0.753484	−0.376742	−	0.926318i	$-0.622956\pi$
$983$	−2.28275e7	−0.753484	−0.376742	+	0.926318i	$0.622956\pi$
$984$	0	0
$985$	−3.43034e7	−1.12654
$986$	0	0
$987$	3.79958e7	1.24149
$988$	0	0
$989$	2.19467e7	0.713475
$990$	0	0
$991$	−4.82171e7	−1.55961	−0.779806	−	0.626021i	$-0.784682\pi$
$991$	−4.82171e7	−1.55961	−0.779806	+	0.626021i	$0.784682\pi$
$992$	0	0
$993$	2.16818e7	0.697786
$994$	0	0
$995$	2.35294e7	0.753449
$996$	0	0
$997$	−1.86762e7	−0.595047	−0.297523	−	0.954715i	$-0.596161\pi$
$997$	−1.86762e7	−0.595047	−0.297523	+	0.954715i	$0.596161\pi$
$998$	0	0
$999$	1.08815e7	0.344966

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.a.a.1.1	✓	3	1.1	even	1	trivial
304.6.a.j.1.3		3	4.3	odd	2
684.6.a.b.1.1		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)

\(f(q)\)	\(=\)	\(q-26.4151 q^{3} +50.4185 q^{5} +117.462 q^{7} +454.756 q^{9} +O(q^{10})\)	\(q-26.4151 q^{3} +50.4185 q^{5} +117.462 q^{7} +454.756 q^{9} -736.100 q^{11} +393.662 q^{13} -1331.81 q^{15} -1999.73 q^{17} +361.000 q^{19} -3102.77 q^{21} -2742.34 q^{23} -582.972 q^{25} -5593.56 q^{27} +6935.15 q^{29} -5987.95 q^{31} +19444.1 q^{33} +5922.26 q^{35} -1945.37 q^{37} -10398.6 q^{39} -18304.8 q^{41} -8002.92 q^{43} +22928.1 q^{45} -12245.8 q^{47} -3009.68 q^{49} +52822.9 q^{51} +4155.62 q^{53} -37113.1 q^{55} -9535.84 q^{57} +14405.3 q^{59} -10708.7 q^{61} +53416.6 q^{63} +19847.8 q^{65} +14728.1 q^{67} +72439.1 q^{69} -9729.09 q^{71} +46960.7 q^{73} +15399.3 q^{75} -86463.7 q^{77} -6745.18 q^{79} +37248.6 q^{81} -49891.3 q^{83} -100823. q^{85} -183192. q^{87} -37071.9 q^{89} +46240.3 q^{91} +158172. q^{93} +18201.1 q^{95} +181017. q^{97} -334746. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{3} - 9 q^{5} - 13 q^{7} + 189 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more