LMFDB - Embedded newform 1560.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("1560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.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:	$92.0429796090$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 76x^{2} + 50x + 1156 $ x^4 - x^3 - 76x^2 + 50x + 1156
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.17243$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -5.00000 q^{5} -22.2653 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -5.00000 q^{5} -22.2653 q^{7} +9.00000 q^{9} -55.6961 q^{11} +13.0000 q^{13} -15.0000 q^{15} -95.2572 q^{17} +43.8922 q^{19} -66.7958 q^{21} -89.2572 q^{23} +25.0000 q^{25} +27.0000 q^{27} -46.5080 q^{29} +60.6158 q^{31} -167.088 q^{33} +111.326 q^{35} +122.749 q^{37} +39.0000 q^{39} -23.8967 q^{41} +287.968 q^{43} -45.0000 q^{45} +249.506 q^{47} +152.742 q^{49} -285.772 q^{51} -267.778 q^{53} +278.480 q^{55} +131.677 q^{57} -321.327 q^{59} -553.199 q^{61} -200.387 q^{63} -65.0000 q^{65} +582.008 q^{67} -267.772 q^{69} +806.776 q^{71} +105.491 q^{73} +75.0000 q^{75} +1240.09 q^{77} +1328.46 q^{79} +81.0000 q^{81} -63.3323 q^{83} +476.286 q^{85} -139.524 q^{87} -8.82072 q^{89} -289.449 q^{91} +181.847 q^{93} -219.461 q^{95} +116.349 q^{97} -501.264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9	$ 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9} - 7 q^{11} + 52 q^{13} - 60 q^{15} - 73 q^{17} + 68 q^{19} + 45 q^{21} - 49 q^{23} + 100 q^{25} + 108 q^{27} - 66 q^{29} + 230 q^{31} - 21 q^{33} - 75 q^{35} + 303 q^{37} + 156 q^{39} + 155 q^{41} + 336 q^{43} - 180 q^{45} + 178 q^{47} + 377 q^{49} - 219 q^{51} + 349 q^{53} + 35 q^{55} + 204 q^{57} - 360 q^{59} + 223 q^{61} + 135 q^{63} - 260 q^{65} + 588 q^{67} - 147 q^{69} - 83 q^{71} + 754 q^{73} + 300 q^{75} + 1401 q^{77} + 869 q^{79} + 324 q^{81} - 1044 q^{83} + 365 q^{85} - 198 q^{87} + 1017 q^{89} + 195 q^{91} + 690 q^{93} - 340 q^{95} + 1367 q^{97} - 63 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9 - 7 * q^11 + 52 * q^13 - 60 * q^15 - 73 * q^17 + 68 * q^19 + 45 * q^21 - 49 * q^23 + 100 * q^25 + 108 * q^27 - 66 * q^29 + 230 * q^31 - 21 * q^33 - 75 * q^35 + 303 * q^37 + 156 * q^39 + 155 * q^41 + 336 * q^43 - 180 * q^45 + 178 * q^47 + 377 * q^49 - 219 * q^51 + 349 * q^53 + 35 * q^55 + 204 * q^57 - 360 * q^59 + 223 * q^61 + 135 * q^63 - 260 * q^65 + 588 * q^67 - 147 * q^69 - 83 * q^71 + 754 * q^73 + 300 * q^75 + 1401 * q^77 + 869 * q^79 + 324 * q^81 - 1044 * q^83 + 365 * q^85 - 198 * q^87 + 1017 * q^89 + 195 * q^91 + 690 * q^93 - 340 * q^95 + 1367 * q^97 - 63 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−22.2653	−1.20221	−0.601106	−	0.799169i	$-0.705273\pi$
$7$	−22.2653	−1.20221	−0.601106	+	0.799169i	$0.705273\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−55.6961	−1.52664	−0.763318	−	0.646023i	$-0.776431\pi$
$11$	−55.6961	−1.52664	−0.763318	+	0.646023i	$0.776431\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−95.2572	−1.35902	−0.679508	−	0.733668i	$-0.737807\pi$
$17$	−95.2572	−1.35902	−0.679508	+	0.733668i	$0.737807\pi$
$18$	0	0
$19$	43.8922	0.529977	0.264988	−	0.964252i	$-0.414632\pi$
$19$	43.8922	0.529977	0.264988	+	0.964252i	$0.414632\pi$
$20$	0	0
$21$	−66.7958	−0.694097
$22$	0	0
$23$	−89.2572	−0.809192	−0.404596	−	0.914496i	$-0.632588\pi$
$23$	−89.2572	−0.809192	−0.404596	+	0.914496i	$0.632588\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−46.5080	−0.297804	−0.148902	−	0.988852i	$-0.547574\pi$
$29$	−46.5080	−0.297804	−0.148902	+	0.988852i	$0.547574\pi$
$30$	0	0
$31$	60.6158	0.351191	0.175595	−	0.984462i	$-0.443815\pi$
$31$	60.6158	0.351191	0.175595	+	0.984462i	$0.443815\pi$
$32$	0	0
$33$	−167.088	−0.881403
$34$	0	0
$35$	111.326	0.537645
$36$	0	0
$37$	122.749	0.545401	0.272701	−	0.962099i	$-0.412083\pi$
$37$	122.749	0.545401	0.272701	+	0.962099i	$0.412083\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−23.8967	−0.0910252	−0.0455126	−	0.998964i	$-0.514492\pi$
$41$	−23.8967	−0.0910252	−0.0455126	+	0.998964i	$0.514492\pi$
$42$	0	0
$43$	287.968	1.02127	0.510636	−	0.859797i	$-0.329410\pi$
$43$	287.968	1.02127	0.510636	+	0.859797i	$0.329410\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	249.506	0.774346	0.387173	−	0.922007i	$-0.373452\pi$
$47$	249.506	0.774346	0.387173	+	0.922007i	$0.373452\pi$
$48$	0	0
$49$	152.742	0.445313
$50$	0	0
$51$	−285.772	−0.784628
$52$	0	0
$53$	−267.778	−0.694003	−0.347002	−	0.937865i	$-0.612800\pi$
$53$	−267.778	−0.694003	−0.347002	+	0.937865i	$0.612800\pi$
$54$	0	0
$55$	278.480	0.682732
$56$	0	0
$57$	131.677	0.305982
$58$	0	0
$59$	−321.327	−0.709037	−0.354519	−	0.935049i	$-0.615355\pi$
$59$	−321.327	−0.709037	−0.354519	+	0.935049i	$0.615355\pi$
$60$	0	0
$61$	−553.199	−1.16115	−0.580573	−	0.814208i	$-0.697171\pi$
$61$	−553.199	−1.16115	−0.580573	+	0.814208i	$0.697171\pi$
$62$	0	0
$63$	−200.387	−0.400737
$64$	0	0
$65$	−65.0000	−0.124035
$66$	0	0
$67$	582.008	1.06125	0.530623	−	0.847608i	$-0.321958\pi$
$67$	582.008	1.06125	0.530623	+	0.847608i	$0.321958\pi$
$68$	0	0
$69$	−267.772	−0.467187
$70$	0	0
$71$	806.776	1.34854	0.674272	−	0.738483i	$-0.264457\pi$
$71$	806.776	1.34854	0.674272	+	0.738483i	$0.264457\pi$
$72$	0	0
$73$	105.491	0.169135	0.0845674	−	0.996418i	$-0.473049\pi$
$73$	105.491	0.169135	0.0845674	+	0.996418i	$0.473049\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	1240.09	1.83534
$78$	0	0
$79$	1328.46	1.89194	0.945969	−	0.324258i	$-0.105114\pi$
$79$	1328.46	1.89194	0.945969	+	0.324258i	$0.105114\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−63.3323	−0.0837545	−0.0418772	−	0.999123i	$-0.513334\pi$
$83$	−63.3323	−0.0837545	−0.0418772	+	0.999123i	$0.513334\pi$
$84$	0	0
$85$	476.286	0.607770
$86$	0	0
$87$	−139.524	−0.171937
$88$	0	0
$89$	−8.82072	−0.0105056	−0.00525278	−	0.999986i	$-0.501672\pi$
$89$	−8.82072	−0.0105056	−0.00525278	+	0.999986i	$0.501672\pi$
$90$	0	0
$91$	−289.449	−0.333434
$92$	0	0
$93$	181.847	0.202760
$94$	0	0
$95$	−219.461	−0.237013
$96$	0	0
$97$	116.349	0.121788	0.0608942	−	0.998144i	$-0.480605\pi$
$97$	116.349	0.121788	0.0608942	+	0.998144i	$0.480605\pi$
$98$	0	0
$99$	−501.264	−0.508879
$100$	0	0
$101$	−168.341	−0.165847	−0.0829233	−	0.996556i	$-0.526426\pi$
$101$	−168.341	−0.165847	−0.0829233	+	0.996556i	$0.526426\pi$
$102$	0	0
$103$	−1964.51	−1.87931	−0.939656	−	0.342120i	$-0.888855\pi$
$103$	−1964.51	−1.87931	−0.939656	+	0.342120i	$0.888855\pi$
$104$	0	0
$105$	333.979	0.310410
$106$	0	0
$107$	1409.04	1.27306	0.636530	−	0.771252i	$-0.280369\pi$
$107$	1409.04	1.27306	0.636530	+	0.771252i	$0.280369\pi$
$108$	0	0
$109$	227.504	0.199917	0.0999585	−	0.994992i	$-0.468129\pi$
$109$	227.504	0.199917	0.0999585	+	0.994992i	$0.468129\pi$
$110$	0	0
$111$	368.248	0.314888
$112$	0	0
$113$	1118.16	0.930861	0.465430	−	0.885085i	$-0.345900\pi$
$113$	1118.16	0.930861	0.465430	+	0.885085i	$0.345900\pi$
$114$	0	0
$115$	446.286	0.361882
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	2120.93	1.63382
$120$	0	0
$121$	1771.05	1.33062
$122$	0	0
$123$	−71.6900	−0.0525534
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1716.79	1.19953	0.599764	−	0.800177i	$-0.295261\pi$
$127$	1716.79	1.19953	0.599764	+	0.800177i	$0.295261\pi$
$128$	0	0
$129$	863.903	0.589631
$130$	0	0
$131$	−955.139	−0.637029	−0.318515	−	0.947918i	$-0.603184\pi$
$131$	−955.139	−0.637029	−0.318515	+	0.947918i	$0.603184\pi$
$132$	0	0
$133$	−977.272	−0.637144
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	371.018	0.231374	0.115687	−	0.993286i	$-0.463093\pi$
$137$	371.018	0.231374	0.115687	+	0.993286i	$0.463093\pi$
$138$	0	0
$139$	−1877.82	−1.14586	−0.572929	−	0.819605i	$-0.694193\pi$
$139$	−1877.82	−1.14586	−0.572929	+	0.819605i	$0.694193\pi$
$140$	0	0
$141$	748.519	0.447069
$142$	0	0
$143$	−724.049	−0.423413
$144$	0	0
$145$	232.540	0.133182
$146$	0	0
$147$	458.227	0.257102
$148$	0	0
$149$	2161.41	1.18839	0.594193	−	0.804322i	$-0.297472\pi$
$149$	2161.41	1.18839	0.594193	+	0.804322i	$0.297472\pi$
$150$	0	0
$151$	−116.710	−0.0628987	−0.0314494	−	0.999505i	$-0.510012\pi$
$151$	−116.710	−0.0628987	−0.0314494	+	0.999505i	$0.510012\pi$
$152$	0	0
$153$	−857.315	−0.453005
$154$	0	0
$155$	−303.079	−0.157057
$156$	0	0
$157$	1058.27	0.537959	0.268979	−	0.963146i	$-0.413314\pi$
$157$	1058.27	0.537959	0.268979	+	0.963146i	$0.413314\pi$
$158$	0	0
$159$	−803.334	−0.400683
$160$	0	0
$161$	1987.34	0.972820
$162$	0	0
$163$	−576.331	−0.276943	−0.138472	−	0.990366i	$-0.544219\pi$
$163$	−576.331	−0.276943	−0.138472	+	0.990366i	$0.544219\pi$
$164$	0	0
$165$	835.441	0.394176
$166$	0	0
$167$	−1302.72	−0.603640	−0.301820	−	0.953365i	$-0.597594\pi$
$167$	−1302.72	−0.603640	−0.301820	+	0.953365i	$0.597594\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	395.030	0.176659
$172$	0	0
$173$	2393.83	1.05202	0.526010	−	0.850478i	$-0.323687\pi$
$173$	2393.83	1.05202	0.526010	+	0.850478i	$0.323687\pi$
$174$	0	0
$175$	−556.632	−0.240442
$176$	0	0
$177$	−963.981	−0.409363
$178$	0	0
$179$	−55.3932	−0.0231301	−0.0115650	−	0.999933i	$-0.503681\pi$
$179$	−55.3932	−0.0231301	−0.0115650	+	0.999933i	$0.503681\pi$
$180$	0	0
$181$	−501.931	−0.206123	−0.103061	−	0.994675i	$-0.532864\pi$
$181$	−501.931	−0.206123	−0.103061	+	0.994675i	$0.532864\pi$
$182$	0	0
$183$	−1659.60	−0.670388
$184$	0	0
$185$	−613.746	−0.243911
$186$	0	0
$187$	5305.45	2.07472
$188$	0	0
$189$	−601.162	−0.231366
$190$	0	0
$191$	−1318.48	−0.499484	−0.249742	−	0.968312i	$-0.580346\pi$
$191$	−1318.48	−0.499484	−0.249742	+	0.968312i	$0.580346\pi$
$192$	0	0
$193$	−289.509	−0.107976	−0.0539878	−	0.998542i	$-0.517193\pi$
$193$	−289.509	−0.107976	−0.0539878	+	0.998542i	$0.517193\pi$
$194$	0	0
$195$	−195.000	−0.0716115
$196$	0	0
$197$	415.846	0.150395	0.0751974	−	0.997169i	$-0.476041\pi$
$197$	415.846	0.150395	0.0751974	+	0.997169i	$0.476041\pi$
$198$	0	0
$199$	−847.607	−0.301936	−0.150968	−	0.988539i	$-0.548239\pi$
$199$	−847.607	−0.301936	−0.150968	+	0.988539i	$0.548239\pi$
$200$	0	0
$201$	1746.02	0.612711
$202$	0	0
$203$	1035.51	0.358023
$204$	0	0
$205$	119.483	0.0407077
$206$	0	0
$207$	−803.315	−0.269731
$208$	0	0
$209$	−2444.62	−0.809081
$210$	0	0
$211$	3852.98	1.25711	0.628554	−	0.777766i	$-0.283647\pi$
$211$	3852.98	1.25711	0.628554	+	0.777766i	$0.283647\pi$
$212$	0	0
$213$	2420.33	0.778583
$214$	0	0
$215$	−1439.84	−0.456727
$216$	0	0
$217$	−1349.63	−0.422206
$218$	0	0
$219$	316.474	0.0976500
$220$	0	0
$221$	−1238.34	−0.376923
$222$	0	0
$223$	5764.36	1.73099	0.865494	−	0.500919i	$-0.167005\pi$
$223$	5764.36	1.73099	0.865494	+	0.500919i	$0.167005\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−4216.33	−1.23281	−0.616404	−	0.787430i	$-0.711411\pi$
$227$	−4216.33	−1.23281	−0.616404	+	0.787430i	$0.711411\pi$
$228$	0	0
$229$	−1792.45	−0.517242	−0.258621	−	0.965979i	$-0.583268\pi$
$229$	−1792.45	−0.517242	−0.258621	+	0.965979i	$0.583268\pi$
$230$	0	0
$231$	3720.26	1.05963
$232$	0	0
$233$	−3799.48	−1.06829	−0.534146	−	0.845392i	$-0.679367\pi$
$233$	−3799.48	−1.06829	−0.534146	+	0.845392i	$0.679367\pi$
$234$	0	0
$235$	−1247.53	−0.346298
$236$	0	0
$237$	3985.37	1.09231
$238$	0	0
$239$	2987.31	0.808506	0.404253	−	0.914647i	$-0.367532\pi$
$239$	2987.31	0.808506	0.404253	+	0.914647i	$0.367532\pi$
$240$	0	0
$241$	−6261.39	−1.67357	−0.836787	−	0.547528i	$-0.815569\pi$
$241$	−6261.39	−1.67357	−0.836787	+	0.547528i	$0.815569\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−763.712	−0.199150
$246$	0	0
$247$	570.598	0.146989
$248$	0	0
$249$	−189.997	−0.0483557
$250$	0	0
$251$	4813.26	1.21040	0.605199	−	0.796074i	$-0.293093\pi$
$251$	4813.26	1.21040	0.605199	+	0.796074i	$0.293093\pi$
$252$	0	0
$253$	4971.28	1.23534
$254$	0	0
$255$	1428.86	0.350896
$256$	0	0
$257$	−730.802	−0.177378	−0.0886891	−	0.996059i	$-0.528268\pi$
$257$	−730.802	−0.177378	−0.0886891	+	0.996059i	$0.528268\pi$
$258$	0	0
$259$	−2733.05	−0.655688
$260$	0	0
$261$	−418.572	−0.0992680
$262$	0	0
$263$	2684.95	0.629510	0.314755	−	0.949173i	$-0.398078\pi$
$263$	2684.95	0.629510	0.314755	+	0.949173i	$0.398078\pi$
$264$	0	0
$265$	1338.89	0.310368
$266$	0	0
$267$	−26.4622	−0.00606538
$268$	0	0
$269$	−5611.89	−1.27198	−0.635991	−	0.771696i	$-0.719408\pi$
$269$	−5611.89	−1.27198	−0.635991	+	0.771696i	$0.719408\pi$
$270$	0	0
$271$	6149.07	1.37834	0.689168	−	0.724601i	$-0.257976\pi$
$271$	6149.07	1.37834	0.689168	+	0.724601i	$0.257976\pi$
$272$	0	0
$273$	−868.346	−0.192508
$274$	0	0
$275$	−1392.40	−0.305327
$276$	0	0
$277$	−4361.77	−0.946113	−0.473056	−	0.881032i	$-0.656849\pi$
$277$	−4361.77	−0.946113	−0.473056	+	0.881032i	$0.656849\pi$
$278$	0	0
$279$	545.542	0.117064
$280$	0	0
$281$	−2043.81	−0.433892	−0.216946	−	0.976184i	$-0.569610\pi$
$281$	−2043.81	−0.433892	−0.216946	+	0.976184i	$0.569610\pi$
$282$	0	0
$283$	−394.485	−0.0828612	−0.0414306	−	0.999141i	$-0.513192\pi$
$283$	−394.485	−0.0828612	−0.0414306	+	0.999141i	$0.513192\pi$
$284$	0	0
$285$	−658.383	−0.136839
$286$	0	0
$287$	532.066	0.109432
$288$	0	0
$289$	4160.94	0.846924
$290$	0	0
$291$	349.048	0.0703146
$292$	0	0
$293$	−743.808	−0.148306	−0.0741531	−	0.997247i	$-0.523625\pi$
$293$	−743.808	−0.148306	−0.0741531	+	0.997247i	$0.523625\pi$
$294$	0	0
$295$	1606.63	0.317091
$296$	0	0
$297$	−1503.79	−0.293801
$298$	0	0
$299$	−1160.34	−0.224430
$300$	0	0
$301$	−6411.68	−1.22778
$302$	0	0
$303$	−505.022	−0.0957516
$304$	0	0
$305$	2766.00	0.519280
$306$	0	0
$307$	8999.26	1.67301	0.836507	−	0.547957i	$-0.184594\pi$
$307$	8999.26	1.67301	0.836507	+	0.547957i	$0.184594\pi$
$308$	0	0
$309$	−5893.54	−1.08502
$310$	0	0
$311$	1391.75	0.253759	0.126880	−	0.991918i	$-0.459504\pi$
$311$	1391.75	0.253759	0.126880	+	0.991918i	$0.459504\pi$
$312$	0	0
$313$	3179.85	0.574236	0.287118	−	0.957895i	$-0.407303\pi$
$313$	3179.85	0.574236	0.287118	+	0.957895i	$0.407303\pi$
$314$	0	0
$315$	1001.94	0.179215
$316$	0	0
$317$	2938.42	0.520625	0.260313	−	0.965524i	$-0.416174\pi$
$317$	2938.42	0.520625	0.260313	+	0.965524i	$0.416174\pi$
$318$	0	0
$319$	2590.31	0.454638
$320$	0	0
$321$	4227.13	0.735002
$322$	0	0
$323$	−4181.05	−0.720247
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	682.512	0.115422
$328$	0	0
$329$	−5555.33	−0.930927
$330$	0	0
$331$	10198.9	1.69360	0.846798	−	0.531914i	$-0.178527\pi$
$331$	10198.9	1.69360	0.846798	+	0.531914i	$0.178527\pi$
$332$	0	0
$333$	1104.74	0.181800
$334$	0	0
$335$	−2910.04	−0.474604
$336$	0	0
$337$	1461.50	0.236240	0.118120	−	0.992999i	$-0.462313\pi$
$337$	1461.50	0.236240	0.118120	+	0.992999i	$0.462313\pi$
$338$	0	0
$339$	3354.47	0.537433
$340$	0	0
$341$	−3376.06	−0.536140
$342$	0	0
$343$	4236.14	0.666851
$344$	0	0
$345$	1338.86	0.208932
$346$	0	0
$347$	−3910.96	−0.605048	−0.302524	−	0.953142i	$-0.597829\pi$
$347$	−3910.96	−0.605048	−0.302524	+	0.953142i	$0.597829\pi$
$348$	0	0
$349$	8338.28	1.27890	0.639452	−	0.768831i	$-0.279161\pi$
$349$	8338.28	1.27890	0.639452	+	0.768831i	$0.279161\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−1175.69	−0.177268	−0.0886340	−	0.996064i	$-0.528250\pi$
$353$	−1175.69	−0.177268	−0.0886340	+	0.996064i	$0.528250\pi$
$354$	0	0
$355$	−4033.88	−0.603088
$356$	0	0
$357$	6362.78	0.943289
$358$	0	0
$359$	−2015.83	−0.296354	−0.148177	−	0.988961i	$-0.547341\pi$
$359$	−2015.83	−0.296354	−0.148177	+	0.988961i	$0.547341\pi$
$360$	0	0
$361$	−4932.48	−0.719125
$362$	0	0
$363$	5313.15	0.768232
$364$	0	0
$365$	−527.457	−0.0756394
$366$	0	0
$367$	6957.37	0.989568	0.494784	−	0.869016i	$-0.335247\pi$
$367$	6957.37	0.989568	0.494784	+	0.869016i	$0.335247\pi$
$368$	0	0
$369$	−215.070	−0.0303417
$370$	0	0
$371$	5962.15	0.834339
$372$	0	0
$373$	−11633.5	−1.61491	−0.807455	−	0.589929i	$-0.799156\pi$
$373$	−11633.5	−1.61491	−0.807455	+	0.589929i	$0.799156\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−604.604	−0.0825959
$378$	0	0
$379$	−11302.9	−1.53191	−0.765954	−	0.642895i	$-0.777733\pi$
$379$	−11302.9	−1.53191	−0.765954	+	0.642895i	$0.777733\pi$
$380$	0	0
$381$	5150.36	0.692548
$382$	0	0
$383$	9649.38	1.28736	0.643682	−	0.765293i	$-0.277406\pi$
$383$	9649.38	1.28736	0.643682	+	0.765293i	$0.277406\pi$
$384$	0	0
$385$	−6200.44	−0.820789
$386$	0	0
$387$	2591.71	0.340424
$388$	0	0
$389$	14375.7	1.87372	0.936858	−	0.349709i	$-0.113720\pi$
$389$	14375.7	1.87372	0.936858	+	0.349709i	$0.113720\pi$
$390$	0	0
$391$	8502.40	1.09970
$392$	0	0
$393$	−2865.42	−0.367789
$394$	0	0
$395$	−6642.28	−0.846100
$396$	0	0
$397$	12724.0	1.60856	0.804280	−	0.594250i	$-0.202551\pi$
$397$	12724.0	1.60856	0.804280	+	0.594250i	$0.202551\pi$
$398$	0	0
$399$	−2931.81	−0.367855
$400$	0	0
$401$	−8113.82	−1.01044	−0.505218	−	0.862992i	$-0.668588\pi$
$401$	−8113.82	−1.01044	−0.505218	+	0.862992i	$0.668588\pi$
$402$	0	0
$403$	788.005	0.0974028
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−6836.65	−0.832629
$408$	0	0
$409$	9338.80	1.12903	0.564516	−	0.825422i	$-0.309063\pi$
$409$	9338.80	1.12903	0.564516	+	0.825422i	$0.309063\pi$
$410$	0	0
$411$	1113.05	0.133584
$412$	0	0
$413$	7154.43	0.852413
$414$	0	0
$415$	316.661	0.0374561
$416$	0	0
$417$	−5633.45	−0.661562
$418$	0	0
$419$	−14127.8	−1.64723	−0.823614	−	0.567151i	$-0.808045\pi$
$419$	−14127.8	−1.64723	−0.823614	+	0.567151i	$0.808045\pi$
$420$	0	0
$421$	−330.548	−0.0382659	−0.0191329	−	0.999817i	$-0.506091\pi$
$421$	−330.548	−0.0382659	−0.0191329	+	0.999817i	$0.506091\pi$
$422$	0	0
$423$	2245.56	0.258115
$424$	0	0
$425$	−2381.43	−0.271803
$426$	0	0
$427$	12317.1	1.39594
$428$	0	0
$429$	−2172.15	−0.244457
$430$	0	0
$431$	17565.4	1.96310	0.981548	−	0.191218i	$-0.0612436\pi$
$431$	17565.4	1.96310	0.981548	+	0.191218i	$0.0612436\pi$
$432$	0	0
$433$	−4826.36	−0.535658	−0.267829	−	0.963466i	$-0.586306\pi$
$433$	−4826.36	−0.535658	−0.267829	+	0.963466i	$0.586306\pi$
$434$	0	0
$435$	697.619	0.0768926
$436$	0	0
$437$	−3917.70	−0.428853
$438$	0	0
$439$	−4020.53	−0.437105	−0.218553	−	0.975825i	$-0.570134\pi$
$439$	−4020.53	−0.437105	−0.218553	+	0.975825i	$0.570134\pi$
$440$	0	0
$441$	1374.68	0.148438
$442$	0	0
$443$	1613.00	0.172993	0.0864965	−	0.996252i	$-0.472433\pi$
$443$	1613.00	0.172993	0.0864965	+	0.996252i	$0.472433\pi$
$444$	0	0
$445$	44.1036	0.00469823
$446$	0	0
$447$	6484.23	0.686115
$448$	0	0
$449$	−6481.63	−0.681263	−0.340631	−	0.940197i	$-0.610641\pi$
$449$	−6481.63	−0.681263	−0.340631	+	0.940197i	$0.610641\pi$
$450$	0	0
$451$	1330.95	0.138962
$452$	0	0
$453$	−350.129	−0.0363146
$454$	0	0
$455$	1447.24	0.149116
$456$	0	0
$457$	−3456.20	−0.353773	−0.176886	−	0.984231i	$-0.556602\pi$
$457$	−3456.20	−0.353773	−0.176886	+	0.984231i	$0.556602\pi$
$458$	0	0
$459$	−2571.95	−0.261543
$460$	0	0
$461$	7897.63	0.797894	0.398947	−	0.916974i	$-0.369376\pi$
$461$	7897.63	0.797894	0.398947	+	0.916974i	$0.369376\pi$
$462$	0	0
$463$	14980.2	1.50364	0.751821	−	0.659367i	$-0.229176\pi$
$463$	14980.2	1.50364	0.751821	+	0.659367i	$0.229176\pi$
$464$	0	0
$465$	−909.237	−0.0906771
$466$	0	0
$467$	−19174.4	−1.89997	−0.949986	−	0.312292i	$-0.898903\pi$
$467$	−19174.4	−1.89997	−0.949986	+	0.312292i	$0.898903\pi$
$468$	0	0
$469$	−12958.6	−1.27584
$470$	0	0
$471$	3174.82	0.310591
$472$	0	0
$473$	−16038.7	−1.55911
$474$	0	0
$475$	1097.30	0.105995
$476$	0	0
$477$	−2410.00	−0.231334
$478$	0	0
$479$	17754.1	1.69354	0.846770	−	0.531959i	$-0.178544\pi$
$479$	17754.1	1.69354	0.846770	+	0.531959i	$0.178544\pi$
$480$	0	0
$481$	1595.74	0.151267
$482$	0	0
$483$	5962.01	0.561658
$484$	0	0
$485$	−581.746	−0.0544655
$486$	0	0
$487$	2357.56	0.219366	0.109683	−	0.993967i	$-0.465016\pi$
$487$	2357.56	0.219366	0.109683	+	0.993967i	$0.465016\pi$
$488$	0	0
$489$	−1728.99	−0.159893
$490$	0	0
$491$	1624.47	0.149310	0.0746551	−	0.997209i	$-0.476214\pi$
$491$	1624.47	0.149310	0.0746551	+	0.997209i	$0.476214\pi$
$492$	0	0
$493$	4430.22	0.404720
$494$	0	0
$495$	2506.32	0.227577
$496$	0	0
$497$	−17963.1	−1.62124
$498$	0	0
$499$	−755.920	−0.0678149	−0.0339074	−	0.999425i	$-0.510795\pi$
$499$	−755.920	−0.0678149	−0.0339074	+	0.999425i	$0.510795\pi$
$500$	0	0
$501$	−3908.17	−0.348512
$502$	0	0
$503$	−6209.03	−0.550392	−0.275196	−	0.961388i	$-0.588743\pi$
$503$	−6209.03	−0.550392	−0.275196	+	0.961388i	$0.588743\pi$
$504$	0	0
$505$	841.703	0.0741689
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	7726.96	0.672872	0.336436	−	0.941706i	$-0.390778\pi$
$509$	7726.96	0.672872	0.336436	+	0.941706i	$0.390778\pi$
$510$	0	0
$511$	−2348.80	−0.203336
$512$	0	0
$513$	1185.09	0.101994
$514$	0	0
$515$	9822.56	0.840454
$516$	0	0
$517$	−13896.5	−1.18214
$518$	0	0
$519$	7181.49	0.607384
$520$	0	0
$521$	−463.642	−0.0389875	−0.0194938	−	0.999810i	$-0.506205\pi$
$521$	−463.642	−0.0389875	−0.0194938	+	0.999810i	$0.506205\pi$
$522$	0	0
$523$	579.506	0.0484513	0.0242256	−	0.999707i	$-0.492288\pi$
$523$	579.506	0.0484513	0.0242256	+	0.999707i	$0.492288\pi$
$524$	0	0
$525$	−1669.90	−0.138819
$526$	0	0
$527$	−5774.09	−0.477274
$528$	0	0
$529$	−4200.15	−0.345208
$530$	0	0
$531$	−2891.94	−0.236346
$532$	0	0
$533$	−310.657	−0.0252459
$534$	0	0
$535$	−7045.22	−0.569330
$536$	0	0
$537$	−166.180	−0.0133541
$538$	0	0
$539$	−8507.14	−0.679831
$540$	0	0
$541$	−9392.76	−0.746444	−0.373222	−	0.927742i	$-0.621747\pi$
$541$	−9392.76	−0.746444	−0.373222	+	0.927742i	$0.621747\pi$
$542$	0	0
$543$	−1505.79	−0.119005
$544$	0	0
$545$	−1137.52	−0.0894056
$546$	0	0
$547$	15393.0	1.20321	0.601607	−	0.798792i	$-0.294527\pi$
$547$	15393.0	1.20321	0.601607	+	0.798792i	$0.294527\pi$
$548$	0	0
$549$	−4978.79	−0.387049
$550$	0	0
$551$	−2041.34	−0.157829
$552$	0	0
$553$	−29578.5	−2.27451
$554$	0	0
$555$	−1841.24	−0.140822
$556$	0	0
$557$	5333.16	0.405697	0.202848	−	0.979210i	$-0.434980\pi$
$557$	5333.16	0.405697	0.202848	+	0.979210i	$0.434980\pi$
$558$	0	0
$559$	3743.58	0.283250
$560$	0	0
$561$	15916.4	1.19784
$562$	0	0
$563$	−4384.85	−0.328241	−0.164120	−	0.986440i	$-0.552479\pi$
$563$	−4384.85	−0.328241	−0.164120	+	0.986440i	$0.552479\pi$
$564$	0	0
$565$	−5590.78	−0.416294
$566$	0	0
$567$	−1803.49	−0.133579
$568$	0	0
$569$	−18796.6	−1.38488	−0.692438	−	0.721477i	$-0.743463\pi$
$569$	−18796.6	−1.38488	−0.692438	+	0.721477i	$0.743463\pi$
$570$	0	0
$571$	−24939.2	−1.82780	−0.913898	−	0.405945i	$-0.866943\pi$
$571$	−24939.2	−1.82780	−0.913898	+	0.405945i	$0.866943\pi$
$572$	0	0
$573$	−3955.43	−0.288377
$574$	0	0
$575$	−2231.43	−0.161838
$576$	0	0
$577$	5471.91	0.394798	0.197399	−	0.980323i	$-0.436751\pi$
$577$	5471.91	0.394798	0.197399	+	0.980323i	$0.436751\pi$
$578$	0	0
$579$	−868.526	−0.0623397
$580$	0	0
$581$	1410.11	0.100691
$582$	0	0
$583$	14914.2	1.05949
$584$	0	0
$585$	−585.000	−0.0413449
$586$	0	0
$587$	−3510.36	−0.246828	−0.123414	−	0.992355i	$-0.539384\pi$
$587$	−3510.36	−0.246828	−0.123414	+	0.992355i	$0.539384\pi$
$588$	0	0
$589$	2660.56	0.186123
$590$	0	0
$591$	1247.54	0.0868305
$592$	0	0
$593$	9525.51	0.659639	0.329820	−	0.944044i	$-0.393012\pi$
$593$	9525.51	0.659639	0.329820	+	0.944044i	$0.393012\pi$
$594$	0	0
$595$	−10604.6	−0.730669
$596$	0	0
$597$	−2542.82	−0.174323
$598$	0	0
$599$	−10043.5	−0.685083	−0.342541	−	0.939503i	$-0.611288\pi$
$599$	−10043.5	−0.685083	−0.342541	+	0.939503i	$0.611288\pi$
$600$	0	0
$601$	6834.21	0.463849	0.231925	−	0.972734i	$-0.425498\pi$
$601$	6834.21	0.463849	0.231925	+	0.972734i	$0.425498\pi$
$602$	0	0
$603$	5238.07	0.353749
$604$	0	0
$605$	−8855.25	−0.595070
$606$	0	0
$607$	−26135.4	−1.74761	−0.873807	−	0.486274i	$-0.838356\pi$
$607$	−26135.4	−1.74761	−0.873807	+	0.486274i	$0.838356\pi$
$608$	0	0
$609$	3106.54	0.206705
$610$	0	0
$611$	3243.58	0.214765
$612$	0	0
$613$	−6501.08	−0.428346	−0.214173	−	0.976796i	$-0.568706\pi$
$613$	−6501.08	−0.428346	−0.214173	+	0.976796i	$0.568706\pi$
$614$	0	0
$615$	358.450	0.0235026
$616$	0	0
$617$	−15752.8	−1.02785	−0.513927	−	0.857834i	$-0.671810\pi$
$617$	−15752.8	−1.02785	−0.513927	+	0.857834i	$0.671810\pi$
$618$	0	0
$619$	8568.35	0.556367	0.278183	−	0.960528i	$-0.410268\pi$
$619$	8568.35	0.556367	0.278183	+	0.960528i	$0.410268\pi$
$620$	0	0
$621$	−2409.95	−0.155729
$622$	0	0
$623$	196.396	0.0126299
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−7333.87	−0.467123
$628$	0	0
$629$	−11692.8	−0.741209
$630$	0	0
$631$	−258.571	−0.0163131	−0.00815653	−	0.999967i	$-0.502596\pi$
$631$	−258.571	−0.0163131	−0.00815653	+	0.999967i	$0.502596\pi$
$632$	0	0
$633$	11558.9	0.725792
$634$	0	0
$635$	−8583.93	−0.536445
$636$	0	0
$637$	1985.65	0.123508
$638$	0	0
$639$	7260.98	0.449515
$640$	0	0
$641$	−9478.60	−0.584060	−0.292030	−	0.956409i	$-0.594331\pi$
$641$	−9478.60	−0.584060	−0.292030	+	0.956409i	$0.594331\pi$
$642$	0	0
$643$	−12654.8	−0.776139	−0.388070	−	0.921630i	$-0.626858\pi$
$643$	−12654.8	−0.776139	−0.388070	+	0.921630i	$0.626858\pi$
$644$	0	0
$645$	−4319.52	−0.263691
$646$	0	0
$647$	−27880.9	−1.69415	−0.847073	−	0.531477i	$-0.821637\pi$
$647$	−27880.9	−1.69415	−0.847073	+	0.531477i	$0.821637\pi$
$648$	0	0
$649$	17896.6	1.08244
$650$	0	0
$651$	−4048.88	−0.243761
$652$	0	0
$653$	−246.844	−0.0147929	−0.00739643	−	0.999973i	$-0.502354\pi$
$653$	−246.844	−0.0147929	−0.00739643	+	0.999973i	$0.502354\pi$
$654$	0	0
$655$	4775.69	0.284888
$656$	0	0
$657$	949.423	0.0563783
$658$	0	0
$659$	23976.0	1.41726	0.708629	−	0.705581i	$-0.249314\pi$
$659$	23976.0	1.41726	0.708629	+	0.705581i	$0.249314\pi$
$660$	0	0
$661$	−21529.7	−1.26688	−0.633441	−	0.773791i	$-0.718358\pi$
$661$	−21529.7	−1.26688	−0.633441	+	0.773791i	$0.718358\pi$
$662$	0	0
$663$	−3715.03	−0.217617
$664$	0	0
$665$	4886.36	0.284940
$666$	0	0
$667$	4151.17	0.240981
$668$	0	0
$669$	17293.1	0.999387
$670$	0	0
$671$	30811.0	1.77265
$672$	0	0
$673$	25850.2	1.48061	0.740305	−	0.672271i	$-0.234681\pi$
$673$	25850.2	1.48061	0.740305	+	0.672271i	$0.234681\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	16615.1	0.943237	0.471619	−	0.881803i	$-0.343670\pi$
$677$	16615.1	0.943237	0.471619	+	0.881803i	$0.343670\pi$
$678$	0	0
$679$	−2590.55	−0.146415
$680$	0	0
$681$	−12649.0	−0.711763
$682$	0	0
$683$	413.017	0.0231386	0.0115693	−	0.999933i	$-0.496317\pi$
$683$	413.017	0.0231386	0.0115693	+	0.999933i	$0.496317\pi$
$684$	0	0
$685$	−1855.09	−0.103473
$686$	0	0
$687$	−5377.35	−0.298630
$688$	0	0
$689$	−3481.12	−0.192482
$690$	0	0
$691$	9396.25	0.517294	0.258647	−	0.965972i	$-0.416723\pi$
$691$	9396.25	0.517294	0.258647	+	0.965972i	$0.416723\pi$
$692$	0	0
$693$	11160.8	0.611780
$694$	0	0
$695$	9389.08	0.512444
$696$	0	0
$697$	2276.33	0.123705
$698$	0	0
$699$	−11398.4	−0.616779
$700$	0	0
$701$	15445.5	0.832193	0.416097	−	0.909320i	$-0.363398\pi$
$701$	15445.5	0.832193	0.416097	+	0.909320i	$0.363398\pi$
$702$	0	0
$703$	5387.73	0.289050
$704$	0	0
$705$	−3742.60	−0.199935
$706$	0	0
$707$	3748.15	0.199383
$708$	0	0
$709$	−16616.9	−0.880198	−0.440099	−	0.897949i	$-0.645057\pi$
$709$	−16616.9	−0.880198	−0.440099	+	0.897949i	$0.645057\pi$
$710$	0	0
$711$	11956.1	0.630646
$712$	0	0
$713$	−5410.40	−0.284181
$714$	0	0
$715$	3620.24	0.189356
$716$	0	0
$717$	8961.92	0.466791
$718$	0	0
$719$	−5037.76	−0.261303	−0.130651	−	0.991428i	$-0.541707\pi$
$719$	−5037.76	−0.261303	−0.130651	+	0.991428i	$0.541707\pi$
$720$	0	0
$721$	43740.4	2.25933
$722$	0	0
$723$	−18784.2	−0.966239
$724$	0	0
$725$	−1162.70	−0.0595608
$726$	0	0
$727$	23564.5	1.20214	0.601072	−	0.799195i	$-0.294740\pi$
$727$	23564.5	1.20214	0.601072	+	0.799195i	$0.294740\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−27431.0	−1.38792
$732$	0	0
$733$	−2663.10	−0.134193	−0.0670967	−	0.997746i	$-0.521374\pi$
$733$	−2663.10	−0.134193	−0.0670967	+	0.997746i	$0.521374\pi$
$734$	0	0
$735$	−2291.13	−0.114979
$736$	0	0
$737$	−32415.5	−1.62014
$738$	0	0
$739$	−30184.9	−1.50253	−0.751265	−	0.660001i	$-0.770556\pi$
$739$	−30184.9	−1.50253	−0.751265	+	0.660001i	$0.770556\pi$
$740$	0	0
$741$	1711.80	0.0848642
$742$	0	0
$743$	−21409.6	−1.05712	−0.528560	−	0.848896i	$-0.677268\pi$
$743$	−21409.6	−1.05712	−0.528560	+	0.848896i	$0.677268\pi$
$744$	0	0
$745$	−10807.0	−0.531462
$746$	0	0
$747$	−569.991	−0.0279182
$748$	0	0
$749$	−31372.8	−1.53049
$750$	0	0
$751$	7487.65	0.363819	0.181910	−	0.983315i	$-0.441772\pi$
$751$	7487.65	0.363819	0.181910	+	0.983315i	$0.441772\pi$
$752$	0	0
$753$	14439.8	0.698824
$754$	0	0
$755$	583.549	0.0281292
$756$	0	0
$757$	−16991.2	−0.815792	−0.407896	−	0.913028i	$-0.633737\pi$
$757$	−16991.2	−0.815792	−0.407896	+	0.913028i	$0.633737\pi$
$758$	0	0
$759$	14913.8	0.713225
$760$	0	0
$761$	−24012.5	−1.14383	−0.571914	−	0.820314i	$-0.693799\pi$
$761$	−24012.5	−1.14383	−0.571914	+	0.820314i	$0.693799\pi$
$762$	0	0
$763$	−5065.44	−0.240342
$764$	0	0
$765$	4286.58	0.202590
$766$	0	0
$767$	−4177.25	−0.196652
$768$	0	0
$769$	−565.525	−0.0265193	−0.0132597	−	0.999912i	$-0.504221\pi$
$769$	−565.525	−0.0265193	−0.0132597	+	0.999912i	$0.504221\pi$
$770$	0	0
$771$	−2192.41	−0.102409
$772$	0	0
$773$	−17843.7	−0.830265	−0.415132	−	0.909761i	$-0.636265\pi$
$773$	−17843.7	−0.830265	−0.415132	+	0.909761i	$0.636265\pi$
$774$	0	0
$775$	1515.39	0.0702382
$776$	0	0
$777$	−8199.14	−0.378562
$778$	0	0
$779$	−1048.88	−0.0482413
$780$	0	0
$781$	−44934.2	−2.05874
$782$	0	0
$783$	−1255.71	−0.0573124
$784$	0	0
$785$	−5291.37	−0.240582
$786$	0	0
$787$	19759.6	0.894988	0.447494	−	0.894287i	$-0.352317\pi$
$787$	19759.6	0.894988	0.447494	+	0.894287i	$0.352317\pi$
$788$	0	0
$789$	8054.85	0.363448
$790$	0	0
$791$	−24896.0	−1.11909
$792$	0	0
$793$	−7191.59	−0.322044
$794$	0	0
$795$	4016.67	0.179191
$796$	0	0
$797$	7289.55	0.323976	0.161988	−	0.986793i	$-0.448209\pi$
$797$	7289.55	0.323976	0.161988	+	0.986793i	$0.448209\pi$
$798$	0	0
$799$	−23767.3	−1.05235
$800$	0	0
$801$	−79.3865	−0.00350185
$802$	0	0
$803$	−5875.46	−0.258207
$804$	0	0
$805$	−9936.68	−0.435058
$806$	0	0
$807$	−16835.7	−0.734379
$808$	0	0
$809$	−18096.2	−0.786438	−0.393219	−	0.919445i	$-0.628639\pi$
$809$	−18096.2	−0.786438	−0.393219	+	0.919445i	$0.628639\pi$
$810$	0	0
$811$	−14445.7	−0.625472	−0.312736	−	0.949840i	$-0.601246\pi$
$811$	−14445.7	−0.625472	−0.312736	+	0.949840i	$0.601246\pi$
$812$	0	0
$813$	18447.2	0.795783
$814$	0	0
$815$	2881.65	0.123853
$816$	0	0
$817$	12639.5	0.541250
$818$	0	0
$819$	−2605.04	−0.111145
$820$	0	0
$821$	−19845.5	−0.843620	−0.421810	−	0.906684i	$-0.638605\pi$
$821$	−19845.5	−0.843620	−0.421810	+	0.906684i	$0.638605\pi$
$822$	0	0
$823$	10131.4	0.429112	0.214556	−	0.976712i	$-0.431170\pi$
$823$	10131.4	0.429112	0.214556	+	0.976712i	$0.431170\pi$
$824$	0	0
$825$	−4177.20	−0.176281
$826$	0	0
$827$	46514.3	1.95582	0.977908	−	0.209037i	$-0.0670328\pi$
$827$	46514.3	1.95582	0.977908	+	0.209037i	$0.0670328\pi$
$828$	0	0
$829$	27639.1	1.15795	0.578977	−	0.815344i	$-0.303452\pi$
$829$	27639.1	1.15795	0.578977	+	0.815344i	$0.303452\pi$
$830$	0	0
$831$	−13085.3	−0.546238
$832$	0	0
$833$	−14549.8	−0.605187
$834$	0	0
$835$	6513.62	0.269956
$836$	0	0
$837$	1636.63	0.0675867
$838$	0	0
$839$	−45766.3	−1.88323	−0.941614	−	0.336695i	$-0.890691\pi$
$839$	−45766.3	−1.88323	−0.941614	+	0.336695i	$0.890691\pi$
$840$	0	0
$841$	−22226.0	−0.911313
$842$	0	0
$843$	−6131.44	−0.250508
$844$	0	0
$845$	−845.000	−0.0344010
$846$	0	0
$847$	−39432.9	−1.59968
$848$	0	0
$849$	−1183.46	−0.0478399
$850$	0	0
$851$	−10956.3	−0.441335
$852$	0	0
$853$	39928.2	1.60271	0.801357	−	0.598186i	$-0.204112\pi$
$853$	39928.2	1.60271	0.801357	+	0.598186i	$0.204112\pi$
$854$	0	0
$855$	−1975.15	−0.0790043
$856$	0	0
$857$	5722.79	0.228106	0.114053	−	0.993475i	$-0.463617\pi$
$857$	5722.79	0.228106	0.114053	+	0.993475i	$0.463617\pi$
$858$	0	0
$859$	−16309.9	−0.647830	−0.323915	−	0.946086i	$-0.604999\pi$
$859$	−16309.9	−0.647830	−0.323915	+	0.946086i	$0.604999\pi$
$860$	0	0
$861$	1596.20	0.0631804
$862$	0	0
$863$	−9868.97	−0.389274	−0.194637	−	0.980875i	$-0.562353\pi$
$863$	−9868.97	−0.389274	−0.194637	+	0.980875i	$0.562353\pi$
$864$	0	0
$865$	−11969.1	−0.470478
$866$	0	0
$867$	12482.8	0.488972
$868$	0	0
$869$	−73989.8	−2.88830
$870$	0	0
$871$	7566.10	0.294337
$872$	0	0
$873$	1047.14	0.0405962
$874$	0	0
$875$	2783.16	0.107529
$876$	0	0
$877$	−12934.5	−0.498023	−0.249011	−	0.968501i	$-0.580106\pi$
$877$	−12934.5	−0.498023	−0.249011	+	0.968501i	$0.580106\pi$
$878$	0	0
$879$	−2231.42	−0.0856246
$880$	0	0
$881$	−9505.35	−0.363500	−0.181750	−	0.983345i	$-0.558176\pi$
$881$	−9505.35	−0.363500	−0.181750	+	0.983345i	$0.558176\pi$
$882$	0	0
$883$	32758.8	1.24849	0.624247	−	0.781227i	$-0.285406\pi$
$883$	32758.8	1.24849	0.624247	+	0.781227i	$0.285406\pi$
$884$	0	0
$885$	4819.90	0.183073
$886$	0	0
$887$	−48177.8	−1.82374	−0.911868	−	0.410484i	$-0.865360\pi$
$887$	−48177.8	−1.82374	−0.911868	+	0.410484i	$0.865360\pi$
$888$	0	0
$889$	−38224.7	−1.44209
$890$	0	0
$891$	−4511.38	−0.169626
$892$	0	0
$893$	10951.4	0.410385
$894$	0	0
$895$	276.966	0.0103441
$896$	0	0
$897$	−3481.03	−0.129574
$898$	0	0
$899$	−2819.12	−0.104586
$900$	0	0
$901$	25507.8	0.943161
$902$	0	0
$903$	−19235.0	−0.708862
$904$	0	0
$905$	2509.66	0.0921810
$906$	0	0
$907$	26931.8	0.985949	0.492975	−	0.870044i	$-0.335910\pi$
$907$	26931.8	0.985949	0.492975	+	0.870044i	$0.335910\pi$
$908$	0	0
$909$	−1515.07	−0.0552822
$910$	0	0
$911$	−35167.0	−1.27896	−0.639481	−	0.768807i	$-0.720851\pi$
$911$	−35167.0	−1.27896	−0.639481	+	0.768807i	$0.720851\pi$
$912$	0	0
$913$	3527.36	0.127863
$914$	0	0
$915$	8297.99	0.299807
$916$	0	0
$917$	21266.4	0.765844
$918$	0	0
$919$	−6940.10	−0.249111	−0.124555	−	0.992213i	$-0.539750\pi$
$919$	−6940.10	−0.249111	−0.124555	+	0.992213i	$0.539750\pi$
$920$	0	0
$921$	26997.8	0.965915
$922$	0	0
$923$	10488.1	0.374019
$924$	0	0
$925$	3068.73	0.109080
$926$	0	0
$927$	−17680.6	−0.626438
$928$	0	0
$929$	−55614.4	−1.96410	−0.982050	−	0.188621i	$-0.939598\pi$
$929$	−55614.4	−1.96410	−0.982050	+	0.188621i	$0.939598\pi$
$930$	0	0
$931$	6704.20	0.236006
$932$	0	0
$933$	4175.26	0.146508
$934$	0	0
$935$	−26527.3	−0.927844
$936$	0	0
$937$	37504.0	1.30758	0.653789	−	0.756676i	$-0.273178\pi$
$937$	37504.0	1.30758	0.653789	+	0.756676i	$0.273178\pi$
$938$	0	0
$939$	9539.55	0.331535
$940$	0	0
$941$	−8768.15	−0.303755	−0.151877	−	0.988399i	$-0.548532\pi$
$941$	−8768.15	−0.303755	−0.151877	+	0.988399i	$0.548532\pi$
$942$	0	0
$943$	2132.95	0.0736569
$944$	0	0
$945$	3005.81	0.103470
$946$	0	0
$947$	24086.2	0.826500	0.413250	−	0.910618i	$-0.364393\pi$
$947$	24086.2	0.826500	0.413250	+	0.910618i	$0.364393\pi$
$948$	0	0
$949$	1371.39	0.0469096
$950$	0	0
$951$	8815.26	0.300583
$952$	0	0
$953$	35827.4	1.21780	0.608901	−	0.793246i	$-0.291611\pi$
$953$	35827.4	1.21780	0.608901	+	0.793246i	$0.291611\pi$
$954$	0	0
$955$	6592.38	0.223376
$956$	0	0
$957$	7770.93	0.262485
$958$	0	0
$959$	−8260.81	−0.278160
$960$	0	0
$961$	−26116.7	−0.876665
$962$	0	0
$963$	12681.4	0.424353
$964$	0	0
$965$	1447.54	0.0482882
$966$	0	0
$967$	54336.0	1.80696	0.903479	−	0.428632i	$-0.141004\pi$
$967$	54336.0	1.80696	0.903479	+	0.428632i	$0.141004\pi$
$968$	0	0
$969$	−12543.1	−0.415835
$970$	0	0
$971$	33582.9	1.10991	0.554957	−	0.831879i	$-0.312735\pi$
$971$	33582.9	1.10991	0.554957	+	0.831879i	$0.312735\pi$
$972$	0	0
$973$	41810.1	1.37756
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	4317.60	0.141384	0.0706921	−	0.997498i	$-0.477479\pi$
$977$	4317.60	0.141384	0.0706921	+	0.997498i	$0.477479\pi$
$978$	0	0
$979$	491.279	0.0160382
$980$	0	0
$981$	2047.54	0.0666390
$982$	0	0
$983$	45012.2	1.46050	0.730248	−	0.683182i	$-0.239404\pi$
$983$	45012.2	1.46050	0.730248	+	0.683182i	$0.239404\pi$
$984$	0	0
$985$	−2079.23	−0.0672586
$986$	0	0
$987$	−16666.0	−0.537471
$988$	0	0
$989$	−25703.2	−0.826405
$990$	0	0
$991$	37193.0	1.19221	0.596103	−	0.802908i	$-0.296715\pi$
$991$	37193.0	1.19221	0.596103	+	0.802908i	$0.296715\pi$
$992$	0	0
$993$	30596.6	0.977798
$994$	0	0
$995$	4238.03	0.135030
$996$	0	0
$997$	25436.9	0.808019	0.404010	−	0.914755i	$-0.367616\pi$
$997$	25436.9	0.808019	0.404010	+	0.914755i	$0.367616\pi$
$998$	0	0
$999$	3314.23	0.104963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.m.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.m.1.1	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -5.00000 q^{5} -22.2653 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -5.00000 q^{5} -22.2653 q^{7} +9.00000 q^{9} -55.6961 q^{11} +13.0000 q^{13} -15.0000 q^{15} -95.2572 q^{17} +43.8922 q^{19} -66.7958 q^{21} -89.2572 q^{23} +25.0000 q^{25} +27.0000 q^{27} -46.5080 q^{29} +60.6158 q^{31} -167.088 q^{33} +111.326 q^{35} +122.749 q^{37} +39.0000 q^{39} -23.8967 q^{41} +287.968 q^{43} -45.0000 q^{45} +249.506 q^{47} +152.742 q^{49} -285.772 q^{51} -267.778 q^{53} +278.480 q^{55} +131.677 q^{57} -321.327 q^{59} -553.199 q^{61} -200.387 q^{63} -65.0000 q^{65} +582.008 q^{67} -267.772 q^{69} +806.776 q^{71} +105.491 q^{73} +75.0000 q^{75} +1240.09 q^{77} +1328.46 q^{79} +81.0000 q^{81} -63.3323 q^{83} +476.286 q^{85} -139.524 q^{87} -8.82072 q^{89} -289.449 q^{91} +181.847 q^{93} -219.461 q^{95} +116.349 q^{97} -501.264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9	\( 4 q + 12 q^{3} - 20 q^{5} + 15 q^{7} + 36 q^{9} - 7 q^{11} + 52 q^{13} - 60 q^{15} - 73 q^{17} + 68 q^{19} + 45 q^{21} - 49 q^{23} + 100 q^{25} + 108 q^{27} - 66 q^{29} + 230 q^{31} - 21 q^{33} - 75 q^{35} + 303 q^{37} + 156 q^{39} + 155 q^{41} + 336 q^{43} - 180 q^{45} + 178 q^{47} + 377 q^{49} - 219 q^{51} + 349 q^{53} + 35 q^{55} + 204 q^{57} - 360 q^{59} + 223 q^{61} + 135 q^{63} - 260 q^{65} + 588 q^{67} - 147 q^{69} - 83 q^{71} + 754 q^{73} + 300 q^{75} + 1401 q^{77} + 869 q^{79} + 324 q^{81} - 1044 q^{83} + 365 q^{85} - 198 q^{87} + 1017 q^{89} + 195 q^{91} + 690 q^{93} - 340 q^{95} + 1367 q^{97} - 63 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 - 20 * q^5 + 15 * q^7 + 36 * q^9 - 7 * q^11 + 52 * q^13 - 60 * q^15 - 73 * q^17 + 68 * q^19 + 45 * q^21 - 49 * q^23 + 100 * q^25 + 108 * q^27 - 66 * q^29 + 230 * q^31 - 21 * q^33 - 75 * q^35 + 303 * q^37 + 156 * q^39 + 155 * q^41 + 336 * q^43 - 180 * q^45 + 178 * q^47 + 377 * q^49 - 219 * q^51 + 349 * q^53 + 35 * q^55 + 204 * q^57 - 360 * q^59 + 223 * q^61 + 135 * q^63 - 260 * q^65 + 588 * q^67 - 147 * q^69 - 83 * q^71 + 754 * q^73 + 300 * q^75 + 1401 * q^77 + 869 * q^79 + 324 * q^81 - 1044 * q^83 + 365 * q^85 - 198 * q^87 + 1017 * q^89 + 195 * q^91 + 690 * q^93 - 340 * q^95 + 1367 * q^97 - 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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