LMFDB - Embedded newform 1380.4.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 513x^{3} + 983x^{2} + 42916x - 124026 $ x^5 - x^4 - 513x^3 + 983x^2 + 42916*x - 124026
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.00340$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000 q^{5} +13.6558 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +5.00000 q^{5} +13.6558 q^{7} +9.00000 q^{9} +47.9766 q^{11} +34.0324 q^{13} -15.0000 q^{15} +95.6324 q^{17} -35.9322 q^{19} -40.9673 q^{21} -23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +117.634 q^{29} +307.332 q^{31} -143.930 q^{33} +68.2789 q^{35} +180.162 q^{37} -102.097 q^{39} -82.1275 q^{41} +264.366 q^{43} +45.0000 q^{45} -480.393 q^{47} -156.520 q^{49} -286.897 q^{51} -199.476 q^{53} +239.883 q^{55} +107.797 q^{57} -818.992 q^{59} -10.8543 q^{61} +122.902 q^{63} +170.162 q^{65} -535.097 q^{67} +69.0000 q^{69} +715.815 q^{71} -767.424 q^{73} -75.0000 q^{75} +655.158 q^{77} -591.821 q^{79} +81.0000 q^{81} +701.775 q^{83} +478.162 q^{85} -352.903 q^{87} +1112.75 q^{89} +464.739 q^{91} -921.995 q^{93} -179.661 q^{95} +736.502 q^{97} +431.790 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 15 q^{3} + 25 q^{5} - 25 q^{7} + 45 q^{9}+O(q^{10}) $ 5 * q - 15 * q^3 + 25 * q^5 - 25 * q^7 + 45 * q^9	$ 5 q - 15 q^{3} + 25 q^{5} - 25 q^{7} + 45 q^{9} - 4 q^{11} + 88 q^{13} - 75 q^{15} + 141 q^{17} - 16 q^{19} + 75 q^{21} - 115 q^{23} + 125 q^{25} - 135 q^{27} + 95 q^{29} - 9 q^{31} + 12 q^{33} - 125 q^{35} + 179 q^{37} - 264 q^{39} + 73 q^{41} - 100 q^{43} + 225 q^{45} - 242 q^{47} + 16 q^{49} - 423 q^{51} + 653 q^{53} - 20 q^{55} + 48 q^{57} - 683 q^{59} + 482 q^{61} - 225 q^{63} + 440 q^{65} + 159 q^{67} + 345 q^{69} - 789 q^{71} + 1112 q^{73} - 375 q^{75} - 68 q^{77} + 340 q^{79} + 405 q^{81} + 33 q^{83} + 705 q^{85} - 285 q^{87} + 1344 q^{89} - 504 q^{91} + 27 q^{93} - 80 q^{95} + 1902 q^{97} - 36 q^{99}+O(q^{100}) $ 5 * q - 15 * q^3 + 25 * q^5 - 25 * q^7 + 45 * q^9 - 4 * q^11 + 88 * q^13 - 75 * q^15 + 141 * q^17 - 16 * q^19 + 75 * q^21 - 115 * q^23 + 125 * q^25 - 135 * q^27 + 95 * q^29 - 9 * q^31 + 12 * q^33 - 125 * q^35 + 179 * q^37 - 264 * q^39 + 73 * q^41 - 100 * q^43 + 225 * q^45 - 242 * q^47 + 16 * q^49 - 423 * q^51 + 653 * q^53 - 20 * q^55 + 48 * q^57 - 683 * q^59 + 482 * q^61 - 225 * q^63 + 440 * q^65 + 159 * q^67 + 345 * q^69 - 789 * q^71 + 1112 * q^73 - 375 * q^75 - 68 * q^77 + 340 * q^79 + 405 * q^81 + 33 * q^83 + 705 * q^85 - 285 * q^87 + 1344 * q^89 - 504 * q^91 + 27 * q^93 - 80 * q^95 + 1902 * q^97 - 36 * 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
$7$	13.6558	0.737343	0.368671	−	0.929560i	$-0.379813\pi$
$7$	13.6558	0.737343	0.368671	+	0.929560i	$0.379813\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	47.9766	1.31504	0.657522	−	0.753435i	$-0.271604\pi$
$11$	47.9766	1.31504	0.657522	+	0.753435i	$0.271604\pi$
$12$	0	0
$13$	34.0324	0.726069	0.363035	−	0.931776i	$-0.381741\pi$
$13$	34.0324	0.726069	0.363035	+	0.931776i	$0.381741\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	95.6324	1.36437	0.682184	−	0.731180i	$-0.261030\pi$
$17$	95.6324	1.36437	0.682184	+	0.731180i	$0.261030\pi$
$18$	0	0
$19$	−35.9322	−0.433864	−0.216932	−	0.976187i	$-0.569605\pi$
$19$	−35.9322	−0.433864	−0.216932	+	0.976187i	$0.569605\pi$
$20$	0	0
$21$	−40.9673	−0.425705
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	117.634	0.753246	0.376623	−	0.926367i	$-0.377085\pi$
$29$	117.634	0.753246	0.376623	+	0.926367i	$0.377085\pi$
$30$	0	0
$31$	307.332	1.78059	0.890297	−	0.455380i	$-0.150497\pi$
$31$	307.332	1.78059	0.890297	+	0.455380i	$0.150497\pi$
$32$	0	0
$33$	−143.930	−0.759242
$34$	0	0
$35$	68.2789	0.329750
$36$	0	0
$37$	180.162	0.800498	0.400249	−	0.916406i	$-0.368924\pi$
$37$	180.162	0.800498	0.400249	+	0.916406i	$0.368924\pi$
$38$	0	0
$39$	−102.097	−0.419196
$40$	0	0
$41$	−82.1275	−0.312833	−0.156417	−	0.987691i	$-0.549994\pi$
$41$	−82.1275	−0.312833	−0.156417	+	0.987691i	$0.549994\pi$
$42$	0	0
$43$	264.366	0.937570	0.468785	−	0.883312i	$-0.344692\pi$
$43$	264.366	0.937570	0.468785	+	0.883312i	$0.344692\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−480.393	−1.49090	−0.745452	−	0.666559i	$-0.767766\pi$
$47$	−480.393	−1.49090	−0.745452	+	0.666559i	$0.767766\pi$
$48$	0	0
$49$	−156.520	−0.456326
$50$	0	0
$51$	−286.897	−0.787718
$52$	0	0
$53$	−199.476	−0.516983	−0.258492	−	0.966013i	$-0.583225\pi$
$53$	−199.476	−0.516983	−0.258492	+	0.966013i	$0.583225\pi$
$54$	0	0
$55$	239.883	0.588106
$56$	0	0
$57$	107.797	0.250491
$58$	0	0
$59$	−818.992	−1.80718	−0.903590	−	0.428398i	$-0.859078\pi$
$59$	−818.992	−1.80718	−0.903590	+	0.428398i	$0.859078\pi$
$60$	0	0
$61$	−10.8543	−0.0227829	−0.0113914	−	0.999935i	$-0.503626\pi$
$61$	−10.8543	−0.0227829	−0.0113914	+	0.999935i	$0.503626\pi$
$62$	0	0
$63$	122.902	0.245781
$64$	0	0
$65$	170.162	0.324708
$66$	0	0
$67$	−535.097	−0.975710	−0.487855	−	0.872925i	$-0.662220\pi$
$67$	−535.097	−0.975710	−0.487855	+	0.872925i	$0.662220\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	715.815	1.19650	0.598251	−	0.801309i	$-0.295863\pi$
$71$	715.815	1.19650	0.598251	+	0.801309i	$0.295863\pi$
$72$	0	0
$73$	−767.424	−1.23041	−0.615207	−	0.788366i	$-0.710928\pi$
$73$	−767.424	−1.23041	−0.615207	+	0.788366i	$0.710928\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	655.158	0.969639
$78$	0	0
$79$	−591.821	−0.842850	−0.421425	−	0.906863i	$-0.638470\pi$
$79$	−591.821	−0.842850	−0.421425	+	0.906863i	$0.638470\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	701.775	0.928070	0.464035	−	0.885817i	$-0.346401\pi$
$83$	701.775	0.928070	0.464035	+	0.885817i	$0.346401\pi$
$84$	0	0
$85$	478.162	0.610164
$86$	0	0
$87$	−352.903	−0.434887
$88$	0	0
$89$	1112.75	1.32530	0.662648	−	0.748931i	$-0.269433\pi$
$89$	1112.75	1.32530	0.662648	+	0.748931i	$0.269433\pi$
$90$	0	0
$91$	464.739	0.535362
$92$	0	0
$93$	−921.995	−1.02803
$94$	0	0
$95$	−179.661	−0.194030
$96$	0	0
$97$	736.502	0.770932	0.385466	−	0.922722i	$-0.374041\pi$
$97$	736.502	0.770932	0.385466	+	0.922722i	$0.374041\pi$
$98$	0	0
$99$	431.790	0.438348
$100$	0	0
$101$	235.942	0.232446	0.116223	−	0.993223i	$-0.462921\pi$
$101$	235.942	0.232446	0.116223	+	0.993223i	$0.462921\pi$
$102$	0	0
$103$	807.617	0.772591	0.386296	−	0.922375i	$-0.373754\pi$
$103$	807.617	0.772591	0.386296	+	0.922375i	$0.373754\pi$
$104$	0	0
$105$	−204.837	−0.190381
$106$	0	0
$107$	−3.32891	−0.00300765	−0.00150382	−	0.999999i	$-0.500479\pi$
$107$	−3.32891	−0.00300765	−0.00150382	+	0.999999i	$0.500479\pi$
$108$	0	0
$109$	−629.635	−0.553285	−0.276642	−	0.960973i	$-0.589222\pi$
$109$	−629.635	−0.553285	−0.276642	+	0.960973i	$0.589222\pi$
$110$	0	0
$111$	−540.485	−0.462168
$112$	0	0
$113$	−1000.71	−0.833087	−0.416543	−	0.909116i	$-0.636759\pi$
$113$	−1000.71	−0.833087	−0.416543	+	0.909116i	$0.636759\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	306.292	0.242023
$118$	0	0
$119$	1305.93	1.00601
$120$	0	0
$121$	970.756	0.729343
$122$	0	0
$123$	246.383	0.180614
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	640.496	0.447518	0.223759	−	0.974644i	$-0.428167\pi$
$127$	640.496	0.447518	0.223759	+	0.974644i	$0.428167\pi$
$128$	0	0
$129$	−793.099	−0.541306
$130$	0	0
$131$	−150.594	−0.100439	−0.0502193	−	0.998738i	$-0.515992\pi$
$131$	−150.594	−0.100439	−0.0502193	+	0.998738i	$0.515992\pi$
$132$	0	0
$133$	−490.682	−0.319906
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−540.804	−0.337255	−0.168628	−	0.985680i	$-0.553934\pi$
$137$	−540.804	−0.337255	−0.168628	+	0.985680i	$0.553934\pi$
$138$	0	0
$139$	−400.968	−0.244674	−0.122337	−	0.992489i	$-0.539039\pi$
$139$	−400.968	−0.244674	−0.122337	+	0.992489i	$0.539039\pi$
$140$	0	0
$141$	1441.18	0.860774
$142$	0	0
$143$	1632.76	0.954813
$144$	0	0
$145$	588.171	0.336862
$146$	0	0
$147$	469.559	0.263460
$148$	0	0
$149$	669.004	0.367832	0.183916	−	0.982942i	$-0.441123\pi$
$149$	669.004	0.367832	0.183916	+	0.982942i	$0.441123\pi$
$150$	0	0
$151$	−1244.67	−0.670793	−0.335396	−	0.942077i	$-0.608870\pi$
$151$	−1244.67	−0.670793	−0.335396	+	0.942077i	$0.608870\pi$
$152$	0	0
$153$	860.692	0.454789
$154$	0	0
$155$	1536.66	0.796306
$156$	0	0
$157$	−1101.72	−0.560044	−0.280022	−	0.959994i	$-0.590342\pi$
$157$	−1101.72	−0.560044	−0.280022	+	0.959994i	$0.590342\pi$
$158$	0	0
$159$	598.428	0.298480
$160$	0	0
$161$	−314.083	−0.153747
$162$	0	0
$163$	849.526	0.408221	0.204111	−	0.978948i	$-0.434570\pi$
$163$	849.526	0.408221	0.204111	+	0.978948i	$0.434570\pi$
$164$	0	0
$165$	−719.649	−0.339543
$166$	0	0
$167$	1439.59	0.667059	0.333530	−	0.942740i	$-0.391760\pi$
$167$	1439.59	0.667059	0.333530	+	0.942740i	$0.391760\pi$
$168$	0	0
$169$	−1038.79	−0.472824
$170$	0	0
$171$	−323.390	−0.144621
$172$	0	0
$173$	1953.00	0.858286	0.429143	−	0.903237i	$-0.358816\pi$
$173$	1953.00	0.858286	0.429143	+	0.903237i	$0.358816\pi$
$174$	0	0
$175$	341.394	0.147469
$176$	0	0
$177$	2456.97	1.04338
$178$	0	0
$179$	−310.551	−0.129674	−0.0648370	−	0.997896i	$-0.520653\pi$
$179$	−310.551	−0.129674	−0.0648370	+	0.997896i	$0.520653\pi$
$180$	0	0
$181$	1880.00	0.772039	0.386019	−	0.922491i	$-0.373850\pi$
$181$	1880.00	0.772039	0.386019	+	0.922491i	$0.373850\pi$
$182$	0	0
$183$	32.5630	0.0131537
$184$	0	0
$185$	900.809	0.357994
$186$	0	0
$187$	4588.12	1.79421
$188$	0	0
$189$	−368.706	−0.141902
$190$	0	0
$191$	184.884	0.0700404	0.0350202	−	0.999387i	$-0.488850\pi$
$191$	184.884	0.0700404	0.0350202	+	0.999387i	$0.488850\pi$
$192$	0	0
$193$	4164.18	1.55308	0.776539	−	0.630069i	$-0.216973\pi$
$193$	4164.18	1.55308	0.776539	+	0.630069i	$0.216973\pi$
$194$	0	0
$195$	−510.486	−0.187470
$196$	0	0
$197$	109.057	0.0394416	0.0197208	−	0.999806i	$-0.493722\pi$
$197$	109.057	0.0394416	0.0197208	+	0.999806i	$0.493722\pi$
$198$	0	0
$199$	−4359.12	−1.55281	−0.776407	−	0.630232i	$-0.782960\pi$
$199$	−4359.12	−1.55281	−0.776407	+	0.630232i	$0.782960\pi$
$200$	0	0
$201$	1605.29	0.563326
$202$	0	0
$203$	1606.39	0.555401
$204$	0	0
$205$	−410.638	−0.139903
$206$	0	0
$207$	−207.000	−0.0695048
$208$	0	0
$209$	−1723.90	−0.570550
$210$	0	0
$211$	500.714	0.163368	0.0816839	−	0.996658i	$-0.473970\pi$
$211$	500.714	0.163368	0.0816839	+	0.996658i	$0.473970\pi$
$212$	0	0
$213$	−2147.45	−0.690801
$214$	0	0
$215$	1321.83	0.419294
$216$	0	0
$217$	4196.86	1.31291
$218$	0	0
$219$	2302.27	0.710380
$220$	0	0
$221$	3254.60	0.990626
$222$	0	0
$223$	1383.14	0.415346	0.207673	−	0.978198i	$-0.433411\pi$
$223$	1383.14	0.415346	0.207673	+	0.978198i	$0.433411\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3455.44	−1.01033	−0.505166	−	0.863022i	$-0.668569\pi$
$227$	−3455.44	−1.01033	−0.505166	+	0.863022i	$0.668569\pi$
$228$	0	0
$229$	4211.51	1.21530	0.607651	−	0.794204i	$-0.292112\pi$
$229$	4211.51	1.21530	0.607651	+	0.794204i	$0.292112\pi$
$230$	0	0
$231$	−1965.47	−0.559821
$232$	0	0
$233$	3847.61	1.08182	0.540912	−	0.841079i	$-0.318079\pi$
$233$	3847.61	1.08182	0.540912	+	0.841079i	$0.318079\pi$
$234$	0	0
$235$	−2401.96	−0.666753
$236$	0	0
$237$	1775.46	0.486619
$238$	0	0
$239$	−6600.52	−1.78641	−0.893206	−	0.449648i	$-0.851549\pi$
$239$	−6600.52	−1.78641	−0.893206	+	0.449648i	$0.851549\pi$
$240$	0	0
$241$	3048.77	0.814890	0.407445	−	0.913230i	$-0.366420\pi$
$241$	3048.77	0.814890	0.407445	+	0.913230i	$0.366420\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−782.598	−0.204075
$246$	0	0
$247$	−1222.86	−0.315015
$248$	0	0
$249$	−2105.33	−0.535822
$250$	0	0
$251$	3110.31	0.782154	0.391077	−	0.920358i	$-0.372103\pi$
$251$	3110.31	0.782154	0.391077	+	0.920358i	$0.372103\pi$
$252$	0	0
$253$	−1103.46	−0.274206
$254$	0	0
$255$	−1434.49	−0.352278
$256$	0	0
$257$	3084.56	0.748676	0.374338	−	0.927292i	$-0.377870\pi$
$257$	3084.56	0.748676	0.374338	+	0.927292i	$0.377870\pi$
$258$	0	0
$259$	2460.25	0.590241
$260$	0	0
$261$	1058.71	0.251082
$262$	0	0
$263$	7344.25	1.72192	0.860962	−	0.508669i	$-0.169862\pi$
$263$	7344.25	1.72192	0.860962	+	0.508669i	$0.169862\pi$
$264$	0	0
$265$	−997.379	−0.231202
$266$	0	0
$267$	−3338.25	−0.765160
$268$	0	0
$269$	7486.98	1.69699	0.848493	−	0.529206i	$-0.177510\pi$
$269$	7486.98	1.69699	0.848493	+	0.529206i	$0.177510\pi$
$270$	0	0
$271$	−5456.35	−1.22306	−0.611531	−	0.791221i	$-0.709446\pi$
$271$	−5456.35	−1.22306	−0.611531	+	0.791221i	$0.709446\pi$
$272$	0	0
$273$	−1394.22	−0.309091
$274$	0	0
$275$	1199.42	0.263009
$276$	0	0
$277$	1758.95	0.381534	0.190767	−	0.981635i	$-0.438902\pi$
$277$	1758.95	0.381534	0.190767	+	0.981635i	$0.438902\pi$
$278$	0	0
$279$	2765.99	0.593531
$280$	0	0
$281$	4766.31	1.01187	0.505933	−	0.862573i	$-0.331148\pi$
$281$	4766.31	1.01187	0.505933	+	0.862573i	$0.331148\pi$
$282$	0	0
$283$	2415.38	0.507348	0.253674	−	0.967290i	$-0.418361\pi$
$283$	2415.38	0.507348	0.253674	+	0.967290i	$0.418361\pi$
$284$	0	0
$285$	538.983	0.112023
$286$	0	0
$287$	−1121.52	−0.230665
$288$	0	0
$289$	4232.56	0.861501
$290$	0	0
$291$	−2209.51	−0.445098
$292$	0	0
$293$	3734.69	0.744652	0.372326	−	0.928102i	$-0.378560\pi$
$293$	3734.69	0.744652	0.372326	+	0.928102i	$0.378560\pi$
$294$	0	0
$295$	−4094.96	−0.808196
$296$	0	0
$297$	−1295.37	−0.253081
$298$	0	0
$299$	−782.746	−0.151396
$300$	0	0
$301$	3610.13	0.691310
$302$	0	0
$303$	−707.825	−0.134203
$304$	0	0
$305$	−54.2717	−0.0101888
$306$	0	0
$307$	450.046	0.0836661	0.0418331	−	0.999125i	$-0.486680\pi$
$307$	450.046	0.0836661	0.0418331	+	0.999125i	$0.486680\pi$
$308$	0	0
$309$	−2422.85	−0.446056
$310$	0	0
$311$	3766.19	0.686692	0.343346	−	0.939209i	$-0.388440\pi$
$311$	3766.19	0.686692	0.343346	+	0.939209i	$0.388440\pi$
$312$	0	0
$313$	−2619.23	−0.472995	−0.236498	−	0.971632i	$-0.576000\pi$
$313$	−2619.23	−0.472995	−0.236498	+	0.971632i	$0.576000\pi$
$314$	0	0
$315$	614.510	0.109917
$316$	0	0
$317$	−4619.18	−0.818419	−0.409210	−	0.912440i	$-0.634196\pi$
$317$	−4619.18	−0.818419	−0.409210	+	0.912440i	$0.634196\pi$
$318$	0	0
$319$	5643.69	0.990552
$320$	0	0
$321$	9.98673	0.00173646
$322$	0	0
$323$	−3436.28	−0.591950
$324$	0	0
$325$	850.811	0.145214
$326$	0	0
$327$	1888.90	0.319439
$328$	0	0
$329$	−6560.14	−1.09931
$330$	0	0
$331$	−6921.02	−1.14929	−0.574643	−	0.818404i	$-0.694859\pi$
$331$	−6921.02	−1.14929	−0.574643	+	0.818404i	$0.694859\pi$
$332$	0	0
$333$	1621.46	0.266833
$334$	0	0
$335$	−2675.49	−0.436351
$336$	0	0
$337$	1748.86	0.282690	0.141345	−	0.989960i	$-0.454857\pi$
$337$	1748.86	0.282690	0.141345	+	0.989960i	$0.454857\pi$
$338$	0	0
$339$	3002.13	0.480983
$340$	0	0
$341$	14744.7	2.34156
$342$	0	0
$343$	−6821.33	−1.07381
$344$	0	0
$345$	345.000	0.0538382
$346$	0	0
$347$	−4134.05	−0.639560	−0.319780	−	0.947492i	$-0.603609\pi$
$347$	−4134.05	−0.639560	−0.319780	+	0.947492i	$0.603609\pi$
$348$	0	0
$349$	−6046.09	−0.927334	−0.463667	−	0.886009i	$-0.653467\pi$
$349$	−6046.09	−0.927334	−0.463667	+	0.886009i	$0.653467\pi$
$350$	0	0
$351$	−918.876	−0.139732
$352$	0	0
$353$	−608.413	−0.0917353	−0.0458677	−	0.998948i	$-0.514605\pi$
$353$	−608.413	−0.0917353	−0.0458677	+	0.998948i	$0.514605\pi$
$354$	0	0
$355$	3579.08	0.535092
$356$	0	0
$357$	−3917.80	−0.580819
$358$	0	0
$359$	−2988.05	−0.439285	−0.219642	−	0.975580i	$-0.570489\pi$
$359$	−2988.05	−0.439285	−0.219642	+	0.975580i	$0.570489\pi$
$360$	0	0
$361$	−5567.88	−0.811762
$362$	0	0
$363$	−2912.27	−0.421086
$364$	0	0
$365$	−3837.12	−0.550258
$366$	0	0
$367$	−6579.99	−0.935892	−0.467946	−	0.883757i	$-0.655006\pi$
$367$	−6579.99	−0.935892	−0.467946	+	0.883757i	$0.655006\pi$
$368$	0	0
$369$	−739.148	−0.104278
$370$	0	0
$371$	−2724.00	−0.381194
$372$	0	0
$373$	870.787	0.120878	0.0604392	−	0.998172i	$-0.480750\pi$
$373$	870.787	0.120878	0.0604392	+	0.998172i	$0.480750\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	4003.38	0.546909
$378$	0	0
$379$	−9648.72	−1.30771	−0.653854	−	0.756621i	$-0.726849\pi$
$379$	−9648.72	−1.30771	−0.653854	+	0.756621i	$0.726849\pi$
$380$	0	0
$381$	−1921.49	−0.258375
$382$	0	0
$383$	−2550.50	−0.340272	−0.170136	−	0.985421i	$-0.554421\pi$
$383$	−2550.50	−0.340272	−0.170136	+	0.985421i	$0.554421\pi$
$384$	0	0
$385$	3275.79	0.433636
$386$	0	0
$387$	2379.30	0.312523
$388$	0	0
$389$	6509.79	0.848481	0.424241	−	0.905549i	$-0.360541\pi$
$389$	6509.79	0.848481	0.424241	+	0.905549i	$0.360541\pi$
$390$	0	0
$391$	−2199.55	−0.284490
$392$	0	0
$393$	451.782	0.0579883
$394$	0	0
$395$	−2959.11	−0.376934
$396$	0	0
$397$	5211.78	0.658871	0.329436	−	0.944178i	$-0.393142\pi$
$397$	5211.78	0.658871	0.329436	+	0.944178i	$0.393142\pi$
$398$	0	0
$399$	1472.05	0.184698
$400$	0	0
$401$	9685.77	1.20620	0.603098	−	0.797667i	$-0.293933\pi$
$401$	9685.77	1.20620	0.603098	+	0.797667i	$0.293933\pi$
$402$	0	0
$403$	10459.2	1.29283
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	8643.55	1.05269
$408$	0	0
$409$	−12010.0	−1.45197	−0.725986	−	0.687709i	$-0.758616\pi$
$409$	−12010.0	−1.45197	−0.725986	+	0.687709i	$0.758616\pi$
$410$	0	0
$411$	1622.41	0.194714
$412$	0	0
$413$	−11184.0	−1.33251
$414$	0	0
$415$	3508.88	0.415046
$416$	0	0
$417$	1202.90	0.141262
$418$	0	0
$419$	2414.12	0.281474	0.140737	−	0.990047i	$-0.455053\pi$
$419$	2414.12	0.281474	0.140737	+	0.990047i	$0.455053\pi$
$420$	0	0
$421$	−11314.5	−1.30982	−0.654909	−	0.755708i	$-0.727293\pi$
$421$	−11314.5	−1.30982	−0.654909	+	0.755708i	$0.727293\pi$
$422$	0	0
$423$	−4323.54	−0.496968
$424$	0	0
$425$	2390.81	0.272874
$426$	0	0
$427$	−148.225	−0.0167988
$428$	0	0
$429$	−4898.28	−0.551262
$430$	0	0
$431$	9871.51	1.10323	0.551617	−	0.834098i	$-0.314011\pi$
$431$	9871.51	1.10323	0.551617	+	0.834098i	$0.314011\pi$
$432$	0	0
$433$	−4473.76	−0.496524	−0.248262	−	0.968693i	$-0.579859\pi$
$433$	−4473.76	−0.496524	−0.248262	+	0.968693i	$0.579859\pi$
$434$	0	0
$435$	−1764.51	−0.194487
$436$	0	0
$437$	826.440	0.0904668
$438$	0	0
$439$	−16934.1	−1.84105	−0.920523	−	0.390689i	$-0.872237\pi$
$439$	−16934.1	−1.84105	−0.920523	+	0.390689i	$0.872237\pi$
$440$	0	0
$441$	−1408.68	−0.152109
$442$	0	0
$443$	−12828.2	−1.37582	−0.687908	−	0.725798i	$-0.741471\pi$
$443$	−12828.2	−1.37582	−0.687908	+	0.725798i	$0.741471\pi$
$444$	0	0
$445$	5563.75	0.592690
$446$	0	0
$447$	−2007.01	−0.212368
$448$	0	0
$449$	−2847.47	−0.299288	−0.149644	−	0.988740i	$-0.547813\pi$
$449$	−2847.47	−0.299288	−0.149644	+	0.988740i	$0.547813\pi$
$450$	0	0
$451$	−3940.20	−0.411390
$452$	0	0
$453$	3734.01	0.387282
$454$	0	0
$455$	2323.70	0.239421
$456$	0	0
$457$	8194.73	0.838804	0.419402	−	0.907801i	$-0.362240\pi$
$457$	8194.73	0.838804	0.419402	+	0.907801i	$0.362240\pi$
$458$	0	0
$459$	−2582.07	−0.262573
$460$	0	0
$461$	−9193.70	−0.928836	−0.464418	−	0.885616i	$-0.653736\pi$
$461$	−9193.70	−0.928836	−0.464418	+	0.885616i	$0.653736\pi$
$462$	0	0
$463$	−5568.62	−0.558954	−0.279477	−	0.960152i	$-0.590161\pi$
$463$	−5568.62	−0.558954	−0.279477	+	0.960152i	$0.590161\pi$
$464$	0	0
$465$	−4609.98	−0.459747
$466$	0	0
$467$	15984.9	1.58393	0.791964	−	0.610567i	$-0.209059\pi$
$467$	15984.9	1.58393	0.791964	+	0.610567i	$0.209059\pi$
$468$	0	0
$469$	−7307.17	−0.719432
$470$	0	0
$471$	3305.16	0.323341
$472$	0	0
$473$	12683.4	1.23295
$474$	0	0
$475$	−898.304	−0.0867727
$476$	0	0
$477$	−1795.28	−0.172328
$478$	0	0
$479$	1450.17	0.138330	0.0691648	−	0.997605i	$-0.477967\pi$
$479$	1450.17	0.138330	0.0691648	+	0.997605i	$0.477967\pi$
$480$	0	0
$481$	6131.34	0.581217
$482$	0	0
$483$	942.249	0.0887656
$484$	0	0
$485$	3682.51	0.344771
$486$	0	0
$487$	11068.0	1.02985	0.514925	−	0.857235i	$-0.327820\pi$
$487$	11068.0	1.02985	0.514925	+	0.857235i	$0.327820\pi$
$488$	0	0
$489$	−2548.58	−0.235687
$490$	0	0
$491$	−16208.2	−1.48975	−0.744875	−	0.667205i	$-0.767491\pi$
$491$	−16208.2	−1.48975	−0.744875	+	0.667205i	$0.767491\pi$
$492$	0	0
$493$	11249.6	1.02771
$494$	0	0
$495$	2158.95	0.196035
$496$	0	0
$497$	9775.02	0.882232
$498$	0	0
$499$	−1240.11	−0.111253	−0.0556263	−	0.998452i	$-0.517716\pi$
$499$	−1240.11	−0.111253	−0.0556263	+	0.998452i	$0.517716\pi$
$500$	0	0
$501$	−4318.77	−0.385127
$502$	0	0
$503$	−4916.41	−0.435809	−0.217904	−	0.975970i	$-0.569922\pi$
$503$	−4916.41	−0.435809	−0.217904	+	0.975970i	$0.569922\pi$
$504$	0	0
$505$	1179.71	0.103953
$506$	0	0
$507$	3116.38	0.272985
$508$	0	0
$509$	−5147.30	−0.448232	−0.224116	−	0.974562i	$-0.571949\pi$
$509$	−5147.30	−0.448232	−0.224116	+	0.974562i	$0.571949\pi$
$510$	0	0
$511$	−10479.8	−0.907237
$512$	0	0
$513$	970.169	0.0834971
$514$	0	0
$515$	4038.09	0.345513
$516$	0	0
$517$	−23047.6	−1.96061
$518$	0	0
$519$	−5858.99	−0.495532
$520$	0	0
$521$	−274.228	−0.0230598	−0.0115299	−	0.999934i	$-0.503670\pi$
$521$	−274.228	−0.0230598	−0.0115299	+	0.999934i	$0.503670\pi$
$522$	0	0
$523$	5721.87	0.478394	0.239197	−	0.970971i	$-0.423116\pi$
$523$	5721.87	0.478394	0.239197	+	0.970971i	$0.423116\pi$
$524$	0	0
$525$	−1024.18	−0.0851410
$526$	0	0
$527$	29390.9	2.42939
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−7370.92	−0.602393
$532$	0	0
$533$	−2795.00	−0.227139
$534$	0	0
$535$	−16.6446	−0.00134506
$536$	0	0
$537$	931.652	0.0748674
$538$	0	0
$539$	−7509.28	−0.600089
$540$	0	0
$541$	−6049.38	−0.480745	−0.240373	−	0.970681i	$-0.577270\pi$
$541$	−6049.38	−0.480745	−0.240373	+	0.970681i	$0.577270\pi$
$542$	0	0
$543$	−5639.99	−0.445737
$544$	0	0
$545$	−3148.17	−0.247437
$546$	0	0
$547$	−4315.89	−0.337356	−0.168678	−	0.985671i	$-0.553950\pi$
$547$	−4315.89	−0.337356	−0.168678	+	0.985671i	$0.553950\pi$
$548$	0	0
$549$	−97.6891	−0.00759430
$550$	0	0
$551$	−4226.86	−0.326806
$552$	0	0
$553$	−8081.78	−0.621469
$554$	0	0
$555$	−2702.43	−0.206688
$556$	0	0
$557$	23211.5	1.76572	0.882859	−	0.469639i	$-0.155616\pi$
$557$	23211.5	1.76572	0.882859	+	0.469639i	$0.155616\pi$
$558$	0	0
$559$	8997.03	0.680740
$560$	0	0
$561$	−13764.4	−1.03589
$562$	0	0
$563$	−15566.7	−1.16529	−0.582643	−	0.812728i	$-0.697982\pi$
$563$	−15566.7	−1.16529	−0.582643	+	0.812728i	$0.697982\pi$
$564$	0	0
$565$	−5003.55	−0.372568
$566$	0	0
$567$	1106.12	0.0819270
$568$	0	0
$569$	1653.76	0.121844	0.0609219	−	0.998143i	$-0.480596\pi$
$569$	1653.76	0.121844	0.0609219	+	0.998143i	$0.480596\pi$
$570$	0	0
$571$	−18759.8	−1.37491	−0.687455	−	0.726227i	$-0.741272\pi$
$571$	−18759.8	−1.37491	−0.687455	+	0.726227i	$0.741272\pi$
$572$	0	0
$573$	−554.651	−0.0404379
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	9350.16	0.674614	0.337307	−	0.941395i	$-0.390484\pi$
$577$	9350.16	0.674614	0.337307	+	0.941395i	$0.390484\pi$
$578$	0	0
$579$	−12492.5	−0.896670
$580$	0	0
$581$	9583.29	0.684306
$582$	0	0
$583$	−9570.18	−0.679856
$584$	0	0
$585$	1531.46	0.108236
$586$	0	0
$587$	−6527.68	−0.458988	−0.229494	−	0.973310i	$-0.573707\pi$
$587$	−6527.68	−0.458988	−0.229494	+	0.973310i	$0.573707\pi$
$588$	0	0
$589$	−11043.1	−0.772535
$590$	0	0
$591$	−327.171	−0.0227716
$592$	0	0
$593$	4272.14	0.295845	0.147922	−	0.988999i	$-0.452741\pi$
$593$	4272.14	0.295845	0.147922	+	0.988999i	$0.452741\pi$
$594$	0	0
$595$	6529.67	0.449900
$596$	0	0
$597$	13077.4	0.896518
$598$	0	0
$599$	−7983.83	−0.544592	−0.272296	−	0.962214i	$-0.587783\pi$
$599$	−7983.83	−0.544592	−0.272296	+	0.962214i	$0.587783\pi$
$600$	0	0
$601$	−25749.2	−1.74764	−0.873819	−	0.486251i	$-0.838364\pi$
$601$	−25749.2	−1.74764	−0.873819	+	0.486251i	$0.838364\pi$
$602$	0	0
$603$	−4815.88	−0.325237
$604$	0	0
$605$	4853.78	0.326172
$606$	0	0
$607$	−22431.1	−1.49992	−0.749961	−	0.661482i	$-0.769928\pi$
$607$	−22431.1	−1.49992	−0.749961	+	0.661482i	$0.769928\pi$
$608$	0	0
$609$	−4819.16	−0.320661
$610$	0	0
$611$	−16348.9	−1.08250
$612$	0	0
$613$	−10331.5	−0.680725	−0.340362	−	0.940294i	$-0.610550\pi$
$613$	−10331.5	−0.680725	−0.340362	+	0.940294i	$0.610550\pi$
$614$	0	0
$615$	1231.91	0.0807732
$616$	0	0
$617$	3390.63	0.221235	0.110617	−	0.993863i	$-0.464717\pi$
$617$	3390.63	0.221235	0.110617	+	0.993863i	$0.464717\pi$
$618$	0	0
$619$	−4942.95	−0.320960	−0.160480	−	0.987039i	$-0.551304\pi$
$619$	−4942.95	−0.320960	−0.160480	+	0.987039i	$0.551304\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	15195.5	0.977197
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	5171.71	0.329407
$628$	0	0
$629$	17229.3	1.09217
$630$	0	0
$631$	22735.8	1.43439	0.717195	−	0.696873i	$-0.245426\pi$
$631$	22735.8	1.43439	0.717195	+	0.696873i	$0.245426\pi$
$632$	0	0
$633$	−1502.14	−0.0943204
$634$	0	0
$635$	3202.48	0.200136
$636$	0	0
$637$	−5326.75	−0.331324
$638$	0	0
$639$	6442.34	0.398834
$640$	0	0
$641$	−19811.7	−1.22077	−0.610387	−	0.792103i	$-0.708986\pi$
$641$	−19811.7	−1.22077	−0.610387	+	0.792103i	$0.708986\pi$
$642$	0	0
$643$	6938.94	0.425576	0.212788	−	0.977098i	$-0.431746\pi$
$643$	6938.94	0.425576	0.212788	+	0.977098i	$0.431746\pi$
$644$	0	0
$645$	−3965.50	−0.242079
$646$	0	0
$647$	18937.0	1.15068	0.575341	−	0.817913i	$-0.304869\pi$
$647$	18937.0	1.15068	0.575341	+	0.817913i	$0.304869\pi$
$648$	0	0
$649$	−39292.4	−2.37652
$650$	0	0
$651$	−12590.6	−0.758008
$652$	0	0
$653$	4523.81	0.271103	0.135552	−	0.990770i	$-0.456719\pi$
$653$	4523.81	0.271103	0.135552	+	0.990770i	$0.456719\pi$
$654$	0	0
$655$	−752.971	−0.0449176
$656$	0	0
$657$	−6906.82	−0.410138
$658$	0	0
$659$	16108.7	0.952210	0.476105	−	0.879388i	$-0.342048\pi$
$659$	16108.7	0.952210	0.476105	+	0.879388i	$0.342048\pi$
$660$	0	0
$661$	20839.3	1.22626	0.613128	−	0.789984i	$-0.289911\pi$
$661$	20839.3	1.22626	0.613128	+	0.789984i	$0.289911\pi$
$662$	0	0
$663$	−9763.81	−0.571938
$664$	0	0
$665$	−2453.41	−0.143066
$666$	0	0
$667$	−2705.59	−0.157063
$668$	0	0
$669$	−4149.43	−0.239800
$670$	0	0
$671$	−520.755	−0.0299605
$672$	0	0
$673$	−3036.65	−0.173929	−0.0869645	−	0.996211i	$-0.527717\pi$
$673$	−3036.65	−0.173929	−0.0869645	+	0.996211i	$0.527717\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	19968.4	1.13360	0.566801	−	0.823855i	$-0.308181\pi$
$677$	19968.4	1.13360	0.566801	+	0.823855i	$0.308181\pi$
$678$	0	0
$679$	10057.5	0.568441
$680$	0	0
$681$	10366.3	0.583316
$682$	0	0
$683$	−16729.1	−0.937218	−0.468609	−	0.883406i	$-0.655245\pi$
$683$	−16729.1	−0.937218	−0.468609	+	0.883406i	$0.655245\pi$
$684$	0	0
$685$	−2704.02	−0.150825
$686$	0	0
$687$	−12634.5	−0.701655
$688$	0	0
$689$	−6788.65	−0.375366
$690$	0	0
$691$	−255.322	−0.0140563	−0.00702815	−	0.999975i	$-0.502237\pi$
$691$	−255.322	−0.0140563	−0.00702815	+	0.999975i	$0.502237\pi$
$692$	0	0
$693$	5896.42	0.323213
$694$	0	0
$695$	−2004.84	−0.109421
$696$	0	0
$697$	−7854.05	−0.426820
$698$	0	0
$699$	−11542.8	−0.624592
$700$	0	0
$701$	19926.3	1.07362	0.536809	−	0.843704i	$-0.319630\pi$
$701$	19926.3	1.07362	0.536809	+	0.843704i	$0.319630\pi$
$702$	0	0
$703$	−6473.61	−0.347307
$704$	0	0
$705$	7205.89	0.384950
$706$	0	0
$707$	3221.97	0.171393
$708$	0	0
$709$	28717.9	1.52119	0.760596	−	0.649226i	$-0.224907\pi$
$709$	28717.9	1.52119	0.760596	+	0.649226i	$0.224907\pi$
$710$	0	0
$711$	−5326.39	−0.280950
$712$	0	0
$713$	−7068.63	−0.371280
$714$	0	0
$715$	8163.80	0.427006
$716$	0	0
$717$	19801.6	1.03139
$718$	0	0
$719$	−26321.9	−1.36529	−0.682644	−	0.730751i	$-0.739170\pi$
$719$	−26321.9	−1.36529	−0.682644	+	0.730751i	$0.739170\pi$
$720$	0	0
$721$	11028.6	0.569665
$722$	0	0
$723$	−9146.31	−0.470477
$724$	0	0
$725$	2940.86	0.150649
$726$	0	0
$727$	−14495.2	−0.739473	−0.369737	−	0.929137i	$-0.620552\pi$
$727$	−14495.2	−0.739473	−0.369737	+	0.929137i	$0.620552\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	25282.0	1.27919
$732$	0	0
$733$	−13653.3	−0.687989	−0.343995	−	0.938972i	$-0.611780\pi$
$733$	−13653.3	−0.687989	−0.343995	+	0.938972i	$0.611780\pi$
$734$	0	0
$735$	2347.80	0.117823
$736$	0	0
$737$	−25672.2	−1.28310
$738$	0	0
$739$	5063.98	0.252072	0.126036	−	0.992026i	$-0.459774\pi$
$739$	5063.98	0.252072	0.126036	+	0.992026i	$0.459774\pi$
$740$	0	0
$741$	3668.58	0.181874
$742$	0	0
$743$	36968.3	1.82535	0.912675	−	0.408685i	$-0.134013\pi$
$743$	36968.3	1.82535	0.912675	+	0.408685i	$0.134013\pi$
$744$	0	0
$745$	3345.02	0.164499
$746$	0	0
$747$	6315.98	0.309357
$748$	0	0
$749$	−45.4589	−0.00221767
$750$	0	0
$751$	19856.2	0.964798	0.482399	−	0.875952i	$-0.339765\pi$
$751$	19856.2	0.964798	0.482399	+	0.875952i	$0.339765\pi$
$752$	0	0
$753$	−9330.92	−0.451577
$754$	0	0
$755$	−6223.34	−0.299988
$756$	0	0
$757$	27974.4	1.34312	0.671562	−	0.740948i	$-0.265624\pi$
$757$	27974.4	1.34312	0.671562	+	0.740948i	$0.265624\pi$
$758$	0	0
$759$	3310.39	0.158313
$760$	0	0
$761$	17103.7	0.814728	0.407364	−	0.913266i	$-0.366448\pi$
$761$	17103.7	0.814728	0.407364	+	0.913266i	$0.366448\pi$
$762$	0	0
$763$	−8598.15	−0.407961
$764$	0	0
$765$	4303.46	0.203388
$766$	0	0
$767$	−27872.3	−1.31214
$768$	0	0
$769$	−11977.7	−0.561672	−0.280836	−	0.959756i	$-0.590612\pi$
$769$	−11977.7	−0.561672	−0.280836	+	0.959756i	$0.590612\pi$
$770$	0	0
$771$	−9253.68	−0.432248
$772$	0	0
$773$	24394.3	1.13506	0.567531	−	0.823352i	$-0.307899\pi$
$773$	24394.3	1.13506	0.567531	+	0.823352i	$0.307899\pi$
$774$	0	0
$775$	7683.30	0.356119
$776$	0	0
$777$	−7380.75	−0.340776
$778$	0	0
$779$	2951.02	0.135727
$780$	0	0
$781$	34342.4	1.57345
$782$	0	0
$783$	−3176.13	−0.144962
$784$	0	0
$785$	−5508.60	−0.250459
$786$	0	0
$787$	−19601.9	−0.887841	−0.443920	−	0.896066i	$-0.646413\pi$
$787$	−19601.9	−0.887841	−0.443920	+	0.896066i	$0.646413\pi$
$788$	0	0
$789$	−22032.8	−0.994153
$790$	0	0
$791$	−13665.5	−0.614271
$792$	0	0
$793$	−369.400	−0.0165420
$794$	0	0
$795$	2992.14	0.133485
$796$	0	0
$797$	38023.2	1.68990	0.844951	−	0.534844i	$-0.179630\pi$
$797$	38023.2	1.68990	0.844951	+	0.534844i	$0.179630\pi$
$798$	0	0
$799$	−45941.1	−2.03414
$800$	0	0
$801$	10014.8	0.441765
$802$	0	0
$803$	−36818.4	−1.61805
$804$	0	0
$805$	−1570.41	−0.0687576
$806$	0	0
$807$	−22460.9	−0.979756
$808$	0	0
$809$	−22391.1	−0.973091	−0.486545	−	0.873655i	$-0.661743\pi$
$809$	−22391.1	−0.973091	−0.486545	+	0.873655i	$0.661743\pi$
$810$	0	0
$811$	−2001.50	−0.0866609	−0.0433305	−	0.999061i	$-0.513797\pi$
$811$	−2001.50	−0.0866609	−0.0433305	+	0.999061i	$0.513797\pi$
$812$	0	0
$813$	16369.1	0.706135
$814$	0	0
$815$	4247.63	0.182562
$816$	0	0
$817$	−9499.26	−0.406777
$818$	0	0
$819$	4182.65	0.178454
$820$	0	0
$821$	−2786.93	−0.118471	−0.0592354	−	0.998244i	$-0.518866\pi$
$821$	−2786.93	−0.118471	−0.0592354	+	0.998244i	$0.518866\pi$
$822$	0	0
$823$	−31848.2	−1.34891	−0.674457	−	0.738314i	$-0.735622\pi$
$823$	−31848.2	−1.34891	−0.674457	+	0.738314i	$0.735622\pi$
$824$	0	0
$825$	−3598.25	−0.151848
$826$	0	0
$827$	31847.5	1.33911	0.669556	−	0.742761i	$-0.266484\pi$
$827$	31847.5	1.33911	0.669556	+	0.742761i	$0.266484\pi$
$828$	0	0
$829$	−30639.2	−1.28365	−0.641824	−	0.766852i	$-0.721822\pi$
$829$	−30639.2	−1.28365	−0.641824	+	0.766852i	$0.721822\pi$
$830$	0	0
$831$	−5276.84	−0.220279
$832$	0	0
$833$	−14968.4	−0.622596
$834$	0	0
$835$	7197.95	0.298318
$836$	0	0
$837$	−8297.96	−0.342676
$838$	0	0
$839$	−22892.4	−0.941996	−0.470998	−	0.882134i	$-0.656106\pi$
$839$	−22892.4	−0.941996	−0.470998	+	0.882134i	$0.656106\pi$
$840$	0	0
$841$	−10551.2	−0.432620
$842$	0	0
$843$	−14298.9	−0.584201
$844$	0	0
$845$	−5193.97	−0.211453
$846$	0	0
$847$	13256.4	0.537776
$848$	0	0
$849$	−7246.15	−0.292918
$850$	0	0
$851$	−4143.72	−0.166915
$852$	0	0
$853$	7317.06	0.293706	0.146853	−	0.989158i	$-0.453086\pi$
$853$	7317.06	0.293706	0.146853	+	0.989158i	$0.453086\pi$
$854$	0	0
$855$	−1616.95	−0.0646766
$856$	0	0
$857$	−25354.6	−1.01061	−0.505307	−	0.862940i	$-0.668621\pi$
$857$	−25354.6	−1.01061	−0.505307	+	0.862940i	$0.668621\pi$
$858$	0	0
$859$	11644.2	0.462508	0.231254	−	0.972893i	$-0.425717\pi$
$859$	11644.2	0.462508	0.231254	+	0.972893i	$0.425717\pi$
$860$	0	0
$861$	3364.55	0.133175
$862$	0	0
$863$	−22902.6	−0.903374	−0.451687	−	0.892176i	$-0.649178\pi$
$863$	−22902.6	−0.903374	−0.451687	+	0.892176i	$0.649178\pi$
$864$	0	0
$865$	9764.98	0.383837
$866$	0	0
$867$	−12697.7	−0.497388
$868$	0	0
$869$	−28393.6	−1.10838
$870$	0	0
$871$	−18210.7	−0.708433
$872$	0	0
$873$	6628.52	0.256977
$874$	0	0
$875$	1706.97	0.0659499
$876$	0	0
$877$	36004.5	1.38630	0.693151	−	0.720793i	$-0.256222\pi$
$877$	36004.5	1.38630	0.693151	+	0.720793i	$0.256222\pi$
$878$	0	0
$879$	−11204.1	−0.429925
$880$	0	0
$881$	−26708.2	−1.02136	−0.510682	−	0.859769i	$-0.670607\pi$
$881$	−26708.2	−1.02136	−0.510682	+	0.859769i	$0.670607\pi$
$882$	0	0
$883$	47331.3	1.80388	0.901939	−	0.431864i	$-0.142144\pi$
$883$	47331.3	1.80388	0.901939	+	0.431864i	$0.142144\pi$
$884$	0	0
$885$	12284.9	0.466612
$886$	0	0
$887$	17265.1	0.653559	0.326780	−	0.945101i	$-0.394037\pi$
$887$	17265.1	0.653559	0.326780	+	0.945101i	$0.394037\pi$
$888$	0	0
$889$	8746.47	0.329975
$890$	0	0
$891$	3886.11	0.146116
$892$	0	0
$893$	17261.6	0.646849
$894$	0	0
$895$	−1552.75	−0.0579920
$896$	0	0
$897$	2348.24	0.0874084
$898$	0	0
$899$	36152.8	1.34123
$900$	0	0
$901$	−19076.4	−0.705356
$902$	0	0
$903$	−10830.4	−0.399128
$904$	0	0
$905$	9399.98	0.345266
$906$	0	0
$907$	−30591.5	−1.11993	−0.559965	−	0.828517i	$-0.689185\pi$
$907$	−30591.5	−1.11993	−0.559965	+	0.828517i	$0.689185\pi$
$908$	0	0
$909$	2123.47	0.0774821
$910$	0	0
$911$	−38809.5	−1.41143	−0.705716	−	0.708495i	$-0.749375\pi$
$911$	−38809.5	−1.41143	−0.705716	+	0.708495i	$0.749375\pi$
$912$	0	0
$913$	33668.8	1.22045
$914$	0	0
$915$	162.815	0.00588252
$916$	0	0
$917$	−2056.48	−0.0740578
$918$	0	0
$919$	−10135.7	−0.363816	−0.181908	−	0.983316i	$-0.558227\pi$
$919$	−10135.7	−0.363816	−0.181908	+	0.983316i	$0.558227\pi$
$920$	0	0
$921$	−1350.14	−0.0483047
$922$	0	0
$923$	24360.9	0.868743
$924$	0	0
$925$	4504.05	0.160100
$926$	0	0
$927$	7268.56	0.257530
$928$	0	0
$929$	41764.3	1.47497	0.737483	−	0.675366i	$-0.236014\pi$
$929$	41764.3	1.47497	0.737483	+	0.675366i	$0.236014\pi$
$930$	0	0
$931$	5624.09	0.197983
$932$	0	0
$933$	−11298.6	−0.396462
$934$	0	0
$935$	22940.6	0.802393
$936$	0	0
$937$	24662.0	0.859843	0.429921	−	0.902866i	$-0.358541\pi$
$937$	24662.0	0.859843	0.429921	+	0.902866i	$0.358541\pi$
$938$	0	0
$939$	7857.68	0.273084
$940$	0	0
$941$	−39223.1	−1.35881	−0.679404	−	0.733765i	$-0.737761\pi$
$941$	−39223.1	−1.35881	−0.679404	+	0.733765i	$0.737761\pi$
$942$	0	0
$943$	1888.93	0.0652302
$944$	0	0
$945$	−1843.53	−0.0634604
$946$	0	0
$947$	46893.7	1.60912	0.804562	−	0.593869i	$-0.202400\pi$
$947$	46893.7	1.60912	0.804562	+	0.593869i	$0.202400\pi$
$948$	0	0
$949$	−26117.3	−0.893366
$950$	0	0
$951$	13857.5	0.472515
$952$	0	0
$953$	−53639.1	−1.82323	−0.911617	−	0.411041i	$-0.865165\pi$
$953$	−53639.1	−1.82323	−0.911617	+	0.411041i	$0.865165\pi$
$954$	0	0
$955$	924.419	0.0313230
$956$	0	0
$957$	−16931.1	−0.571896
$958$	0	0
$959$	−7385.10	−0.248673
$960$	0	0
$961$	64661.8	2.17052
$962$	0	0
$963$	−29.9602	−0.00100255
$964$	0	0
$965$	20820.9	0.694558
$966$	0	0
$967$	28648.0	0.952698	0.476349	−	0.879256i	$-0.341960\pi$
$967$	28648.0	0.952698	0.476349	+	0.879256i	$0.341960\pi$
$968$	0	0
$969$	10308.8	0.341762
$970$	0	0
$971$	−16770.6	−0.554268	−0.277134	−	0.960831i	$-0.589385\pi$
$971$	−16770.6	−0.554268	−0.277134	+	0.960831i	$0.589385\pi$
$972$	0	0
$973$	−5475.53	−0.180408
$974$	0	0
$975$	−2552.43	−0.0838392
$976$	0	0
$977$	−32422.1	−1.06169	−0.530847	−	0.847468i	$-0.678126\pi$
$977$	−32422.1	−1.06169	−0.530847	+	0.847468i	$0.678126\pi$
$978$	0	0
$979$	53386.0	1.74282
$980$	0	0
$981$	−5666.71	−0.184428
$982$	0	0
$983$	−33466.4	−1.08587	−0.542937	−	0.839774i	$-0.682688\pi$
$983$	−33466.4	−1.08587	−0.542937	+	0.839774i	$0.682688\pi$
$984$	0	0
$985$	545.286	0.0176388
$986$	0	0
$987$	19680.4	0.634685
$988$	0	0
$989$	−6080.43	−0.195497
$990$	0	0
$991$	−33699.5	−1.08022	−0.540112	−	0.841593i	$-0.681618\pi$
$991$	−33699.5	−1.08022	−0.540112	+	0.841593i	$0.681618\pi$
$992$	0	0
$993$	20763.1	0.663541
$994$	0	0
$995$	−21795.6	−0.694440
$996$	0	0
$997$	−8355.43	−0.265415	−0.132708	−	0.991155i	$-0.542367\pi$
$997$	−8355.43	−0.265415	−0.132708	+	0.991155i	$0.542367\pi$
$998$	0	0
$999$	−4864.37	−0.154056

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.f.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.f.1.4	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000 q^{5} +13.6558 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +5.00000 q^{5} +13.6558 q^{7} +9.00000 q^{9} +47.9766 q^{11} +34.0324 q^{13} -15.0000 q^{15} +95.6324 q^{17} -35.9322 q^{19} -40.9673 q^{21} -23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +117.634 q^{29} +307.332 q^{31} -143.930 q^{33} +68.2789 q^{35} +180.162 q^{37} -102.097 q^{39} -82.1275 q^{41} +264.366 q^{43} +45.0000 q^{45} -480.393 q^{47} -156.520 q^{49} -286.897 q^{51} -199.476 q^{53} +239.883 q^{55} +107.797 q^{57} -818.992 q^{59} -10.8543 q^{61} +122.902 q^{63} +170.162 q^{65} -535.097 q^{67} +69.0000 q^{69} +715.815 q^{71} -767.424 q^{73} -75.0000 q^{75} +655.158 q^{77} -591.821 q^{79} +81.0000 q^{81} +701.775 q^{83} +478.162 q^{85} -352.903 q^{87} +1112.75 q^{89} +464.739 q^{91} -921.995 q^{93} -179.661 q^{95} +736.502 q^{97} +431.790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} + 25 q^{5} - 25 q^{7} + 45 q^{9}+O(q^{10}) \) 5 * q - 15 * q^3 + 25 * q^5 - 25 * q^7 + 45 * q^9	\( 5 q - 15 q^{3} + 25 q^{5} - 25 q^{7} + 45 q^{9} - 4 q^{11} + 88 q^{13} - 75 q^{15} + 141 q^{17} - 16 q^{19} + 75 q^{21} - 115 q^{23} + 125 q^{25} - 135 q^{27} + 95 q^{29} - 9 q^{31} + 12 q^{33} - 125 q^{35} + 179 q^{37} - 264 q^{39} + 73 q^{41} - 100 q^{43} + 225 q^{45} - 242 q^{47} + 16 q^{49} - 423 q^{51} + 653 q^{53} - 20 q^{55} + 48 q^{57} - 683 q^{59} + 482 q^{61} - 225 q^{63} + 440 q^{65} + 159 q^{67} + 345 q^{69} - 789 q^{71} + 1112 q^{73} - 375 q^{75} - 68 q^{77} + 340 q^{79} + 405 q^{81} + 33 q^{83} + 705 q^{85} - 285 q^{87} + 1344 q^{89} - 504 q^{91} + 27 q^{93} - 80 q^{95} + 1902 q^{97} - 36 q^{99}+O(q^{100}) \) 5 * q - 15 * q^3 + 25 * q^5 - 25 * q^7 + 45 * q^9 - 4 * q^11 + 88 * q^13 - 75 * q^15 + 141 * q^17 - 16 * q^19 + 75 * q^21 - 115 * q^23 + 125 * q^25 - 135 * q^27 + 95 * q^29 - 9 * q^31 + 12 * q^33 - 125 * q^35 + 179 * q^37 - 264 * q^39 + 73 * q^41 - 100 * q^43 + 225 * q^45 - 242 * q^47 + 16 * q^49 - 423 * q^51 + 653 * q^53 - 20 * q^55 + 48 * q^57 - 683 * q^59 + 482 * q^61 - 225 * q^63 + 440 * q^65 + 159 * q^67 + 345 * q^69 - 789 * q^71 + 1112 * q^73 - 375 * q^75 - 68 * q^77 + 340 * q^79 + 405 * q^81 + 33 * q^83 + 705 * q^85 - 285 * q^87 + 1344 * q^89 - 504 * q^91 + 27 * q^93 - 80 * q^95 + 1902 * q^97 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more