LMFDB - Embedded newform 1840.4.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + \cdots + 57408 $ x^10 - 5x^9 - 192x^8 + 762x^7 + 12246x^6 - 33828x^5 - 298243x^4 + 383603x^3 + 2423016x^2 + 864576*x + 57408
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.17442$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.17442 q^{3} +5.00000 q^{5} +21.9366 q^{7} +11.1235 q^{9} +O(q^{10})$	$q-6.17442 q^{3} +5.00000 q^{5} +21.9366 q^{7} +11.1235 q^{9} -33.1304 q^{11} -2.67228 q^{13} -30.8721 q^{15} +90.0398 q^{17} -77.7108 q^{19} -135.446 q^{21} +23.0000 q^{23} +25.0000 q^{25} +98.0281 q^{27} -128.012 q^{29} -98.6899 q^{31} +204.561 q^{33} +109.683 q^{35} +198.611 q^{37} +16.4998 q^{39} +371.407 q^{41} +203.912 q^{43} +55.6176 q^{45} -182.075 q^{47} +138.214 q^{49} -555.944 q^{51} +476.502 q^{53} -165.652 q^{55} +479.820 q^{57} -337.495 q^{59} -352.182 q^{61} +244.012 q^{63} -13.3614 q^{65} -771.762 q^{67} -142.012 q^{69} -421.766 q^{71} +867.989 q^{73} -154.361 q^{75} -726.767 q^{77} +870.474 q^{79} -905.602 q^{81} -860.946 q^{83} +450.199 q^{85} +790.402 q^{87} +81.6610 q^{89} -58.6208 q^{91} +609.353 q^{93} -388.554 q^{95} -293.715 q^{97} -368.526 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) $ 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9	$ 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100}) $ 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9 + 56 * q^11 + 49 * q^13 - 25 * q^15 + 240 * q^17 - 88 * q^19 + 346 * q^21 + 230 * q^23 + 250 * q^25 - 449 * q^27 + 319 * q^29 + 109 * q^31 + 504 * q^33 - 70 * q^35 + 580 * q^37 - 107 * q^39 + 259 * q^41 - 330 * q^43 + 695 * q^45 - 227 * q^47 + 630 * q^49 + 192 * q^51 - 186 * q^53 + 280 * q^55 + 1708 * q^57 - 262 * q^59 + 1000 * q^61 - 722 * q^63 + 245 * q^65 - 354 * q^67 - 115 * q^69 - 599 * q^71 + 355 * q^73 - 125 * q^75 + 1776 * q^77 + 1068 * q^79 + 3490 * q^81 - 754 * q^83 + 1200 * q^85 - 2675 * q^87 + 1740 * q^89 - 690 * q^91 + 1669 * q^93 - 440 * q^95 + 2592 * q^97 - 916 * 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
$3$	−6.17442	−1.18827	−0.594134	−	0.804366i	$-0.702505\pi$
$3$	−6.17442	−1.18827	−0.594134	+	0.804366i	$0.702505\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	21.9366	1.18446	0.592232	−	0.805767i	$-0.298247\pi$
$7$	21.9366	1.18446	0.592232	+	0.805767i	$0.298247\pi$
$8$	0	0
$9$	11.1235	0.411982
$10$	0	0
$11$	−33.1304	−0.908107	−0.454053	−	0.890974i	$-0.650023\pi$
$11$	−33.1304	−0.908107	−0.454053	+	0.890974i	$0.650023\pi$
$12$	0	0
$13$	−2.67228	−0.0570122	−0.0285061	−	0.999594i	$-0.509075\pi$
$13$	−2.67228	−0.0570122	−0.0285061	+	0.999594i	$0.509075\pi$
$14$	0	0
$15$	−30.8721	−0.531410
$16$	0	0
$17$	90.0398	1.28458	0.642290	−	0.766462i	$-0.277985\pi$
$17$	90.0398	1.28458	0.642290	+	0.766462i	$0.277985\pi$
$18$	0	0
$19$	−77.7108	−0.938321	−0.469160	−	0.883113i	$-0.655443\pi$
$19$	−77.7108	−0.938321	−0.469160	+	0.883113i	$0.655443\pi$
$20$	0	0
$21$	−135.446	−1.40746
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	98.0281	0.698723
$28$	0	0
$29$	−128.012	−0.819699	−0.409849	−	0.912153i	$-0.634419\pi$
$29$	−128.012	−0.819699	−0.409849	+	0.912153i	$0.634419\pi$
$30$	0	0
$31$	−98.6899	−0.571782	−0.285891	−	0.958262i	$-0.592289\pi$
$31$	−98.6899	−0.571782	−0.285891	+	0.958262i	$0.592289\pi$
$32$	0	0
$33$	204.561	1.07907
$34$	0	0
$35$	109.683	0.529709
$36$	0	0
$37$	198.611	0.882473	0.441237	−	0.897391i	$-0.354540\pi$
$37$	198.611	0.882473	0.441237	+	0.897391i	$0.354540\pi$
$38$	0	0
$39$	16.4998	0.0677458
$40$	0	0
$41$	371.407	1.41473	0.707366	−	0.706847i	$-0.249883\pi$
$41$	371.407	1.41473	0.707366	+	0.706847i	$0.249883\pi$
$42$	0	0
$43$	203.912	0.723170	0.361585	−	0.932339i	$-0.382236\pi$
$43$	203.912	0.723170	0.361585	+	0.932339i	$0.382236\pi$
$44$	0	0
$45$	55.6176	0.184244
$46$	0	0
$47$	−182.075	−0.565072	−0.282536	−	0.959257i	$-0.591176\pi$
$47$	−182.075	−0.565072	−0.282536	+	0.959257i	$0.591176\pi$
$48$	0	0
$49$	138.214	0.402956
$50$	0	0
$51$	−555.944	−1.52643
$52$	0	0
$53$	476.502	1.23495	0.617477	−	0.786589i	$-0.288155\pi$
$53$	476.502	1.23495	0.617477	+	0.786589i	$0.288155\pi$
$54$	0	0
$55$	−165.652	−0.406118
$56$	0	0
$57$	479.820	1.11498
$58$	0	0
$59$	−337.495	−0.744713	−0.372357	−	0.928090i	$-0.621450\pi$
$59$	−337.495	−0.744713	−0.372357	+	0.928090i	$0.621450\pi$
$60$	0	0
$61$	−352.182	−0.739219	−0.369609	−	0.929187i	$-0.620508\pi$
$61$	−352.182	−0.739219	−0.369609	+	0.929187i	$0.620508\pi$
$62$	0	0
$63$	244.012	0.487978
$64$	0	0
$65$	−13.3614	−0.0254966
$66$	0	0
$67$	−771.762	−1.40725	−0.703625	−	0.710572i	$-0.748436\pi$
$67$	−771.762	−1.40725	−0.703625	+	0.710572i	$0.748436\pi$
$68$	0	0
$69$	−142.012	−0.247771
$70$	0	0
$71$	−421.766	−0.704992	−0.352496	−	0.935813i	$-0.614667\pi$
$71$	−421.766	−0.704992	−0.352496	+	0.935813i	$0.614667\pi$
$72$	0	0
$73$	867.989	1.39165	0.695825	−	0.718212i	$-0.255039\pi$
$73$	867.989	1.39165	0.695825	+	0.718212i	$0.255039\pi$
$74$	0	0
$75$	−154.361	−0.237654
$76$	0	0
$77$	−726.767	−1.07562
$78$	0	0
$79$	870.474	1.23970	0.619848	−	0.784722i	$-0.287194\pi$
$79$	870.474	1.23970	0.619848	+	0.784722i	$0.287194\pi$
$80$	0	0
$81$	−905.602	−1.24225
$82$	0	0
$83$	−860.946	−1.13857	−0.569284	−	0.822141i	$-0.692779\pi$
$83$	−860.946	−1.13857	−0.569284	+	0.822141i	$0.692779\pi$
$84$	0	0
$85$	450.199	0.574482
$86$	0	0
$87$	790.402	0.974022
$88$	0	0
$89$	81.6610	0.0972590	0.0486295	−	0.998817i	$-0.484515\pi$
$89$	81.6610	0.0972590	0.0486295	+	0.998817i	$0.484515\pi$
$90$	0	0
$91$	−58.6208	−0.0675289
$92$	0	0
$93$	609.353	0.679430
$94$	0	0
$95$	−388.554	−0.419630
$96$	0	0
$97$	−293.715	−0.307446	−0.153723	−	0.988114i	$-0.549126\pi$
$97$	−293.715	−0.307446	−0.153723	+	0.988114i	$0.549126\pi$
$98$	0	0
$99$	−368.526	−0.374124
$100$	0	0
$101$	143.719	0.141590	0.0707951	−	0.997491i	$-0.477446\pi$
$101$	143.719	0.141590	0.0707951	+	0.997491i	$0.477446\pi$
$102$	0	0
$103$	150.039	0.143532	0.0717660	−	0.997421i	$-0.477137\pi$
$103$	150.039	0.143532	0.0717660	+	0.997421i	$0.477137\pi$
$104$	0	0
$105$	−677.229	−0.629436
$106$	0	0
$107$	−564.748	−0.510245	−0.255122	−	0.966909i	$-0.582116\pi$
$107$	−564.748	−0.510245	−0.255122	+	0.966909i	$0.582116\pi$
$108$	0	0
$109$	−777.109	−0.682876	−0.341438	−	0.939904i	$-0.610914\pi$
$109$	−777.109	−0.682876	−0.341438	+	0.939904i	$0.610914\pi$
$110$	0	0
$111$	−1226.31	−1.04862
$112$	0	0
$113$	1929.39	1.60621	0.803105	−	0.595838i	$-0.203180\pi$
$113$	1929.39	1.60621	0.803105	+	0.595838i	$0.203180\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−29.7252	−0.0234880
$118$	0	0
$119$	1975.17	1.52154
$120$	0	0
$121$	−233.380	−0.175342
$122$	0	0
$123$	−2293.22	−1.68108
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	2324.03	1.62381	0.811905	−	0.583789i	$-0.198431\pi$
$127$	2324.03	1.62381	0.811905	+	0.583789i	$0.198431\pi$
$128$	0	0
$129$	−1259.04	−0.859320
$130$	0	0
$131$	−2305.30	−1.53752	−0.768760	−	0.639537i	$-0.779126\pi$
$131$	−2305.30	−1.53752	−0.768760	+	0.639537i	$0.779126\pi$
$132$	0	0
$133$	−1704.71	−1.11141
$134$	0	0
$135$	490.141	0.312478
$136$	0	0
$137$	−2427.90	−1.51408	−0.757042	−	0.653366i	$-0.773356\pi$
$137$	−2427.90	−1.51408	−0.757042	+	0.653366i	$0.773356\pi$
$138$	0	0
$139$	1729.58	1.05541	0.527703	−	0.849429i	$-0.323054\pi$
$139$	1729.58	1.05541	0.527703	+	0.849429i	$0.323054\pi$
$140$	0	0
$141$	1124.21	0.671457
$142$	0	0
$143$	88.5337	0.0517732
$144$	0	0
$145$	−640.061	−0.366580
$146$	0	0
$147$	−853.391	−0.478820
$148$	0	0
$149$	3152.73	1.73344	0.866718	−	0.498798i	$-0.166225\pi$
$149$	3152.73	1.73344	0.866718	+	0.498798i	$0.166225\pi$
$150$	0	0
$151$	−932.680	−0.502652	−0.251326	−	0.967903i	$-0.580867\pi$
$151$	−932.680	−0.502652	−0.251326	+	0.967903i	$0.580867\pi$
$152$	0	0
$153$	1001.56	0.529224
$154$	0	0
$155$	−493.449	−0.255708
$156$	0	0
$157$	−1207.97	−0.614057	−0.307028	−	0.951700i	$-0.599335\pi$
$157$	−1207.97	−0.614057	−0.307028	+	0.951700i	$0.599335\pi$
$158$	0	0
$159$	−2942.12	−1.46746
$160$	0	0
$161$	504.542	0.246978
$162$	0	0
$163$	2192.09	1.05336	0.526681	−	0.850063i	$-0.323436\pi$
$163$	2192.09	1.05336	0.526681	+	0.850063i	$0.323436\pi$
$164$	0	0
$165$	1022.80	0.482577
$166$	0	0
$167$	−1371.88	−0.635683	−0.317842	−	0.948144i	$-0.602958\pi$
$167$	−1371.88	−0.635683	−0.317842	+	0.948144i	$0.602958\pi$
$168$	0	0
$169$	−2189.86	−0.996750
$170$	0	0
$171$	−864.418	−0.386571
$172$	0	0
$173$	291.322	0.128028	0.0640139	−	0.997949i	$-0.479610\pi$
$173$	291.322	0.128028	0.0640139	+	0.997949i	$0.479610\pi$
$174$	0	0
$175$	548.415	0.236893
$176$	0	0
$177$	2083.84	0.884919
$178$	0	0
$179$	3895.09	1.62644	0.813220	−	0.581956i	$-0.197713\pi$
$179$	3895.09	1.62644	0.813220	+	0.581956i	$0.197713\pi$
$180$	0	0
$181$	2361.33	0.969702	0.484851	−	0.874597i	$-0.338874\pi$
$181$	2361.33	0.969702	0.484851	+	0.874597i	$0.338874\pi$
$182$	0	0
$183$	2174.52	0.878390
$184$	0	0
$185$	993.057	0.394654
$186$	0	0
$187$	−2983.05	−1.16654
$188$	0	0
$189$	2150.40	0.827613
$190$	0	0
$191$	227.757	0.0862823	0.0431412	−	0.999069i	$-0.486263\pi$
$191$	227.757	0.0862823	0.0431412	+	0.999069i	$0.486263\pi$
$192$	0	0
$193$	470.841	0.175606	0.0878028	−	0.996138i	$-0.472015\pi$
$193$	470.841	0.175606	0.0878028	+	0.996138i	$0.472015\pi$
$194$	0	0
$195$	82.4991	0.0302968
$196$	0	0
$197$	2499.94	0.904130	0.452065	−	0.891985i	$-0.350688\pi$
$197$	2499.94	0.904130	0.452065	+	0.891985i	$0.350688\pi$
$198$	0	0
$199$	1194.18	0.425392	0.212696	−	0.977118i	$-0.431776\pi$
$199$	1194.18	0.425392	0.212696	+	0.977118i	$0.431776\pi$
$200$	0	0
$201$	4765.18	1.67219
$202$	0	0
$203$	−2808.15	−0.970904
$204$	0	0
$205$	1857.04	0.632688
$206$	0	0
$207$	255.841	0.0859042
$208$	0	0
$209$	2574.59	0.852095
$210$	0	0
$211$	−1594.12	−0.520112	−0.260056	−	0.965594i	$-0.583741\pi$
$211$	−1594.12	−0.520112	−0.260056	+	0.965594i	$0.583741\pi$
$212$	0	0
$213$	2604.16	0.837720
$214$	0	0
$215$	1019.56	0.323411
$216$	0	0
$217$	−2164.92	−0.677255
$218$	0	0
$219$	−5359.33	−1.65365
$220$	0	0
$221$	−240.612	−0.0732367
$222$	0	0
$223$	1853.84	0.556693	0.278346	−	0.960481i	$-0.410214\pi$
$223$	1853.84	0.556693	0.278346	+	0.960481i	$0.410214\pi$
$224$	0	0
$225$	278.088	0.0823964
$226$	0	0
$227$	3534.54	1.03346	0.516730	−	0.856148i	$-0.327149\pi$
$227$	3534.54	1.03346	0.516730	+	0.856148i	$0.327149\pi$
$228$	0	0
$229$	3341.14	0.964144	0.482072	−	0.876132i	$-0.339884\pi$
$229$	3341.14	0.964144	0.482072	+	0.876132i	$0.339884\pi$
$230$	0	0
$231$	4487.37	1.27813
$232$	0	0
$233$	−739.757	−0.207996	−0.103998	−	0.994578i	$-0.533164\pi$
$233$	−739.757	−0.207996	−0.103998	+	0.994578i	$0.533164\pi$
$234$	0	0
$235$	−910.375	−0.252708
$236$	0	0
$237$	−5374.68	−1.47309
$238$	0	0
$239$	5290.77	1.43193	0.715966	−	0.698135i	$-0.245987\pi$
$239$	5290.77	1.43193	0.715966	+	0.698135i	$0.245987\pi$
$240$	0	0
$241$	6797.92	1.81698	0.908491	−	0.417905i	$-0.137236\pi$
$241$	6797.92	1.81698	0.908491	+	0.417905i	$0.137236\pi$
$242$	0	0
$243$	2944.81	0.777407
$244$	0	0
$245$	691.069	0.180207
$246$	0	0
$247$	207.665	0.0534957
$248$	0	0
$249$	5315.84	1.35292
$250$	0	0
$251$	4440.75	1.11672	0.558361	−	0.829598i	$-0.311430\pi$
$251$	4440.75	1.11672	0.558361	+	0.829598i	$0.311430\pi$
$252$	0	0
$253$	−761.998	−0.189353
$254$	0	0
$255$	−2779.72	−0.682638
$256$	0	0
$257$	2785.11	0.675993	0.337996	−	0.941147i	$-0.390251\pi$
$257$	2785.11	0.675993	0.337996	+	0.941147i	$0.390251\pi$
$258$	0	0
$259$	4356.86	1.04526
$260$	0	0
$261$	−1423.95	−0.337701
$262$	0	0
$263$	5116.28	1.19956	0.599779	−	0.800166i	$-0.295255\pi$
$263$	5116.28	1.19956	0.599779	+	0.800166i	$0.295255\pi$
$264$	0	0
$265$	2382.51	0.552288
$266$	0	0
$267$	−504.210	−0.115570
$268$	0	0
$269$	3363.46	0.762355	0.381178	−	0.924502i	$-0.375519\pi$
$269$	3363.46	0.762355	0.381178	+	0.924502i	$0.375519\pi$
$270$	0	0
$271$	2496.77	0.559660	0.279830	−	0.960050i	$-0.409722\pi$
$271$	2496.77	0.559660	0.279830	+	0.960050i	$0.409722\pi$
$272$	0	0
$273$	361.950	0.0802425
$274$	0	0
$275$	−828.259	−0.181621
$276$	0	0
$277$	−2713.70	−0.588629	−0.294315	−	0.955709i	$-0.595091\pi$
$277$	−2713.70	−0.588629	−0.294315	+	0.955709i	$0.595091\pi$
$278$	0	0
$279$	−1097.78	−0.235564
$280$	0	0
$281$	−7953.73	−1.68854	−0.844271	−	0.535917i	$-0.819966\pi$
$281$	−7953.73	−1.68854	−0.844271	+	0.535917i	$0.819966\pi$
$282$	0	0
$283$	5286.06	1.11033	0.555165	−	0.831740i	$-0.312655\pi$
$283$	5286.06	1.11033	0.555165	+	0.831740i	$0.312655\pi$
$284$	0	0
$285$	2399.10	0.498633
$286$	0	0
$287$	8147.40	1.67570
$288$	0	0
$289$	3194.16	0.650145
$290$	0	0
$291$	1813.52	0.365328
$292$	0	0
$293$	5991.72	1.19468	0.597338	−	0.801990i	$-0.296225\pi$
$293$	5991.72	1.19468	0.597338	+	0.801990i	$0.296225\pi$
$294$	0	0
$295$	−1687.47	−0.333046
$296$	0	0
$297$	−3247.71	−0.634515
$298$	0	0
$299$	−61.4625	−0.0118879
$300$	0	0
$301$	4473.14	0.856569
$302$	0	0
$303$	−887.384	−0.168247
$304$	0	0
$305$	−1760.91	−0.330589
$306$	0	0
$307$	1493.77	0.277701	0.138850	−	0.990313i	$-0.455659\pi$
$307$	1493.77	0.277701	0.138850	+	0.990313i	$0.455659\pi$
$308$	0	0
$309$	−926.405	−0.170555
$310$	0	0
$311$	1469.27	0.267892	0.133946	−	0.990989i	$-0.457235\pi$
$311$	1469.27	0.267892	0.133946	+	0.990989i	$0.457235\pi$
$312$	0	0
$313$	6250.50	1.12875	0.564375	−	0.825518i	$-0.309117\pi$
$313$	6250.50	1.12875	0.564375	+	0.825518i	$0.309117\pi$
$314$	0	0
$315$	1220.06	0.218230
$316$	0	0
$317$	1611.24	0.285478	0.142739	−	0.989760i	$-0.454409\pi$
$317$	1611.24	0.285478	0.142739	+	0.989760i	$0.454409\pi$
$318$	0	0
$319$	4241.09	0.744374
$320$	0	0
$321$	3486.99	0.606308
$322$	0	0
$323$	−6997.07	−1.20535
$324$	0	0
$325$	−66.8071	−0.0114024
$326$	0	0
$327$	4798.20	0.811440
$328$	0	0
$329$	−3994.10	−0.669307
$330$	0	0
$331$	−6080.55	−1.00972	−0.504860	−	0.863201i	$-0.668456\pi$
$331$	−6080.55	−1.00972	−0.504860	+	0.863201i	$0.668456\pi$
$332$	0	0
$333$	2209.26	0.363563
$334$	0	0
$335$	−3858.81	−0.629341
$336$	0	0
$337$	−6967.42	−1.12623	−0.563115	−	0.826379i	$-0.690397\pi$
$337$	−6967.42	−1.12623	−0.563115	+	0.826379i	$0.690397\pi$
$338$	0	0
$339$	−11912.9	−1.90861
$340$	0	0
$341$	3269.63	0.519239
$342$	0	0
$343$	−4492.31	−0.707177
$344$	0	0
$345$	−710.059	−0.110807
$346$	0	0
$347$	−2543.17	−0.393442	−0.196721	−	0.980460i	$-0.563029\pi$
$347$	−2543.17	−0.393442	−0.196721	+	0.980460i	$0.563029\pi$
$348$	0	0
$349$	−11483.6	−1.76132	−0.880660	−	0.473749i	$-0.842901\pi$
$349$	−11483.6	−1.76132	−0.880660	+	0.473749i	$0.842901\pi$
$350$	0	0
$351$	−261.959	−0.0398357
$352$	0	0
$353$	6453.14	0.972992	0.486496	−	0.873683i	$-0.338275\pi$
$353$	6453.14	0.972992	0.486496	+	0.873683i	$0.338275\pi$
$354$	0	0
$355$	−2108.83	−0.315282
$356$	0	0
$357$	−12195.5	−1.80800
$358$	0	0
$359$	6208.99	0.912808	0.456404	−	0.889773i	$-0.349137\pi$
$359$	6208.99	0.912808	0.456404	+	0.889773i	$0.349137\pi$
$360$	0	0
$361$	−820.024	−0.119555
$362$	0	0
$363$	1440.99	0.208353
$364$	0	0
$365$	4339.94	0.622365
$366$	0	0
$367$	3598.65	0.511848	0.255924	−	0.966697i	$-0.417620\pi$
$367$	3598.65	0.511848	0.255924	+	0.966697i	$0.417620\pi$
$368$	0	0
$369$	4131.35	0.582845
$370$	0	0
$371$	10452.8	1.46276
$372$	0	0
$373$	1618.39	0.224656	0.112328	−	0.993671i	$-0.464169\pi$
$373$	1618.39	0.224656	0.112328	+	0.993671i	$0.464169\pi$
$374$	0	0
$375$	−771.803	−0.106282
$376$	0	0
$377$	342.085	0.0467328
$378$	0	0
$379$	2510.13	0.340202	0.170101	−	0.985427i	$-0.445591\pi$
$379$	2510.13	0.340202	0.170101	+	0.985427i	$0.445591\pi$
$380$	0	0
$381$	−14349.5	−1.92952
$382$	0	0
$383$	953.240	0.127176	0.0635879	−	0.997976i	$-0.479746\pi$
$383$	953.240	0.127176	0.0635879	+	0.997976i	$0.479746\pi$
$384$	0	0
$385$	−3633.83	−0.481032
$386$	0	0
$387$	2268.22	0.297933
$388$	0	0
$389$	1006.49	0.131185	0.0655924	−	0.997847i	$-0.479106\pi$
$389$	1006.49	0.131185	0.0655924	+	0.997847i	$0.479106\pi$
$390$	0	0
$391$	2070.92	0.267853
$392$	0	0
$393$	14233.9	1.82699
$394$	0	0
$395$	4352.37	0.554409
$396$	0	0
$397$	3036.84	0.383916	0.191958	−	0.981403i	$-0.438516\pi$
$397$	3036.84	0.383916	0.191958	+	0.981403i	$0.438516\pi$
$398$	0	0
$399$	10525.6	1.32065
$400$	0	0
$401$	−4616.11	−0.574856	−0.287428	−	0.957802i	$-0.592800\pi$
$401$	−4616.11	−0.574856	−0.287428	+	0.957802i	$0.592800\pi$
$402$	0	0
$403$	263.727	0.0325985
$404$	0	0
$405$	−4528.01	−0.555552
$406$	0	0
$407$	−6580.07	−0.801380
$408$	0	0
$409$	3422.80	0.413806	0.206903	−	0.978361i	$-0.433661\pi$
$409$	3422.80	0.413806	0.206903	+	0.978361i	$0.433661\pi$
$410$	0	0
$411$	14990.9	1.79914
$412$	0	0
$413$	−7403.48	−0.882086
$414$	0	0
$415$	−4304.73	−0.509183
$416$	0	0
$417$	−10679.2	−1.25411
$418$	0	0
$419$	11498.3	1.34064	0.670318	−	0.742074i	$-0.266158\pi$
$419$	11498.3	1.34064	0.670318	+	0.742074i	$0.266158\pi$
$420$	0	0
$421$	−1937.02	−0.224239	−0.112119	−	0.993695i	$-0.535764\pi$
$421$	−1937.02	−0.224239	−0.112119	+	0.993695i	$0.535764\pi$
$422$	0	0
$423$	−2025.31	−0.232799
$424$	0	0
$425$	2250.99	0.256916
$426$	0	0
$427$	−7725.68	−0.875578
$428$	0	0
$429$	−546.645	−0.0615204
$430$	0	0
$431$	17070.8	1.90782	0.953912	−	0.300087i	$-0.0970157\pi$
$431$	17070.8	1.90782	0.953912	+	0.300087i	$0.0970157\pi$
$432$	0	0
$433$	−5170.63	−0.573868	−0.286934	−	0.957950i	$-0.592636\pi$
$433$	−5170.63	−0.573868	−0.286934	+	0.957950i	$0.592636\pi$
$434$	0	0
$435$	3952.01	0.435596
$436$	0	0
$437$	−1787.35	−0.195653
$438$	0	0
$439$	3137.77	0.341133	0.170567	−	0.985346i	$-0.445440\pi$
$439$	3137.77	0.341133	0.170567	+	0.985346i	$0.445440\pi$
$440$	0	0
$441$	1537.42	0.166011
$442$	0	0
$443$	6092.44	0.653410	0.326705	−	0.945126i	$-0.394062\pi$
$443$	6092.44	0.653410	0.326705	+	0.945126i	$0.394062\pi$
$444$	0	0
$445$	408.305	0.0434955
$446$	0	0
$447$	−19466.3	−2.05979
$448$	0	0
$449$	15872.4	1.66830	0.834150	−	0.551537i	$-0.185959\pi$
$449$	15872.4	1.66830	0.834150	+	0.551537i	$0.185959\pi$
$450$	0	0
$451$	−12304.8	−1.28473
$452$	0	0
$453$	5758.76	0.597285
$454$	0	0
$455$	−293.104	−0.0301998
$456$	0	0
$457$	−5422.54	−0.555045	−0.277523	−	0.960719i	$-0.589513\pi$
$457$	−5422.54	−0.555045	−0.277523	+	0.960719i	$0.589513\pi$
$458$	0	0
$459$	8826.43	0.897566
$460$	0	0
$461$	5129.70	0.518251	0.259126	−	0.965844i	$-0.416566\pi$
$461$	5129.70	0.518251	0.259126	+	0.965844i	$0.416566\pi$
$462$	0	0
$463$	2516.02	0.252547	0.126273	−	0.991995i	$-0.459698\pi$
$463$	2516.02	0.252547	0.126273	+	0.991995i	$0.459698\pi$
$464$	0	0
$465$	3046.77	0.303850
$466$	0	0
$467$	4758.80	0.471544	0.235772	−	0.971808i	$-0.424238\pi$
$467$	4758.80	0.471544	0.235772	+	0.971808i	$0.424238\pi$
$468$	0	0
$469$	−16929.8	−1.66684
$470$	0	0
$471$	7458.55	0.729664
$472$	0	0
$473$	−6755.68	−0.656716
$474$	0	0
$475$	−1942.77	−0.187664
$476$	0	0
$477$	5300.38	0.508779
$478$	0	0
$479$	−2104.82	−0.200775	−0.100388	−	0.994948i	$-0.532008\pi$
$479$	−2104.82	−0.200775	−0.100388	+	0.994948i	$0.532008\pi$
$480$	0	0
$481$	−530.746	−0.0503117
$482$	0	0
$483$	−3115.25	−0.293476
$484$	0	0
$485$	−1468.58	−0.137494
$486$	0	0
$487$	1436.18	0.133634	0.0668169	−	0.997765i	$-0.478716\pi$
$487$	1436.18	0.133634	0.0668169	+	0.997765i	$0.478716\pi$
$488$	0	0
$489$	−13534.9	−1.25168
$490$	0	0
$491$	7724.79	0.710010	0.355005	−	0.934864i	$-0.384479\pi$
$491$	7724.79	0.710010	0.355005	+	0.934864i	$0.384479\pi$
$492$	0	0
$493$	−11526.2	−1.05297
$494$	0	0
$495$	−1842.63	−0.167313
$496$	0	0
$497$	−9252.11	−0.835038
$498$	0	0
$499$	−5121.48	−0.459456	−0.229728	−	0.973255i	$-0.573784\pi$
$499$	−5121.48	−0.459456	−0.229728	+	0.973255i	$0.573784\pi$
$500$	0	0
$501$	8470.56	0.755362
$502$	0	0
$503$	20669.0	1.83218	0.916090	−	0.400972i	$-0.131328\pi$
$503$	20669.0	1.83218	0.916090	+	0.400972i	$0.131328\pi$
$504$	0	0
$505$	718.597	0.0633210
$506$	0	0
$507$	13521.1	1.18441
$508$	0	0
$509$	−13411.0	−1.16785	−0.583923	−	0.811809i	$-0.698483\pi$
$509$	−13411.0	−1.16785	−0.583923	+	0.811809i	$0.698483\pi$
$510$	0	0
$511$	19040.7	1.64836
$512$	0	0
$513$	−7617.85	−0.655626
$514$	0	0
$515$	750.196	0.0641895
$516$	0	0
$517$	6032.21	0.513145
$518$	0	0
$519$	−1798.75	−0.152132
$520$	0	0
$521$	−4898.23	−0.411891	−0.205945	−	0.978563i	$-0.566027\pi$
$521$	−4898.23	−0.411891	−0.205945	+	0.978563i	$0.566027\pi$
$522$	0	0
$523$	−21416.0	−1.79055	−0.895274	−	0.445517i	$-0.853020\pi$
$523$	−21416.0	−1.79055	−0.895274	+	0.445517i	$0.853020\pi$
$524$	0	0
$525$	−3386.15	−0.281492
$526$	0	0
$527$	−8886.02	−0.734499
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−3754.13	−0.306809
$532$	0	0
$533$	−992.505	−0.0806570
$534$	0	0
$535$	−2823.74	−0.228188
$536$	0	0
$537$	−24049.9	−1.93265
$538$	0	0
$539$	−4579.07	−0.365927
$540$	0	0
$541$	−11496.3	−0.913610	−0.456805	−	0.889567i	$-0.651006\pi$
$541$	−11496.3	−0.913610	−0.456805	+	0.889567i	$0.651006\pi$
$542$	0	0
$543$	−14579.8	−1.15227
$544$	0	0
$545$	−3885.54	−0.305391
$546$	0	0
$547$	2909.16	0.227398	0.113699	−	0.993515i	$-0.463730\pi$
$547$	2909.16	0.227398	0.113699	+	0.993515i	$0.463730\pi$
$548$	0	0
$549$	−3917.51	−0.304545
$550$	0	0
$551$	9947.94	0.769140
$552$	0	0
$553$	19095.2	1.46838
$554$	0	0
$555$	−6131.56	−0.468955
$556$	0	0
$557$	−18845.3	−1.43358	−0.716788	−	0.697291i	$-0.754388\pi$
$557$	−18845.3	−1.43358	−0.716788	+	0.697291i	$0.754388\pi$
$558$	0	0
$559$	−544.911	−0.0412295
$560$	0	0
$561$	18418.6	1.38616
$562$	0	0
$563$	−6793.51	−0.508548	−0.254274	−	0.967132i	$-0.581836\pi$
$563$	−6793.51	−0.508548	−0.254274	+	0.967132i	$0.581836\pi$
$564$	0	0
$565$	9646.95	0.718319
$566$	0	0
$567$	−19865.8	−1.47140
$568$	0	0
$569$	−16769.2	−1.23550	−0.617751	−	0.786373i	$-0.711956\pi$
$569$	−16769.2	−1.23550	−0.617751	+	0.786373i	$0.711956\pi$
$570$	0	0
$571$	12223.0	0.895826	0.447913	−	0.894077i	$-0.352168\pi$
$571$	12223.0	0.895826	0.447913	+	0.894077i	$0.352168\pi$
$572$	0	0
$573$	−1406.27	−0.102527
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	12373.6	0.892753	0.446377	−	0.894845i	$-0.352714\pi$
$577$	12373.6	0.892753	0.446377	+	0.894845i	$0.352714\pi$
$578$	0	0
$579$	−2907.17	−0.208667
$580$	0	0
$581$	−18886.2	−1.34859
$582$	0	0
$583$	−15786.7	−1.12147
$584$	0	0
$585$	−148.626	−0.0105042
$586$	0	0
$587$	−25037.7	−1.76050	−0.880252	−	0.474506i	$-0.842627\pi$
$587$	−25037.7	−1.76050	−0.880252	+	0.474506i	$0.842627\pi$
$588$	0	0
$589$	7669.28	0.536514
$590$	0	0
$591$	−15435.7	−1.07435
$592$	0	0
$593$	16211.8	1.12267	0.561333	−	0.827590i	$-0.310289\pi$
$593$	16211.8	1.12267	0.561333	+	0.827590i	$0.310289\pi$
$594$	0	0
$595$	9875.83	0.680453
$596$	0	0
$597$	−7373.36	−0.505480
$598$	0	0
$599$	−9680.02	−0.660292	−0.330146	−	0.943930i	$-0.607098\pi$
$599$	−9680.02	−0.660292	−0.330146	+	0.943930i	$0.607098\pi$
$600$	0	0
$601$	−55.1340	−0.00374203	−0.00187102	−	0.999998i	$-0.500596\pi$
$601$	−55.1340	−0.00374203	−0.00187102	+	0.999998i	$0.500596\pi$
$602$	0	0
$603$	−8584.70	−0.579761
$604$	0	0
$605$	−1166.90	−0.0784152
$606$	0	0
$607$	−2153.87	−0.144025	−0.0720123	−	0.997404i	$-0.522942\pi$
$607$	−2153.87	−0.144025	−0.0720123	+	0.997404i	$0.522942\pi$
$608$	0	0
$609$	17338.7	1.15369
$610$	0	0
$611$	486.556	0.0322160
$612$	0	0
$613$	−2011.64	−0.132544	−0.0662718	−	0.997802i	$-0.521110\pi$
$613$	−2011.64	−0.132544	−0.0662718	+	0.997802i	$0.521110\pi$
$614$	0	0
$615$	−11466.1	−0.751803
$616$	0	0
$617$	2133.99	0.139240	0.0696202	−	0.997574i	$-0.477821\pi$
$617$	2133.99	0.139240	0.0696202	+	0.997574i	$0.477821\pi$
$618$	0	0
$619$	14880.5	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$619$	14880.5	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$620$	0	0
$621$	2254.65	0.145694
$622$	0	0
$623$	1791.36	0.115200
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−15896.6	−1.01252
$628$	0	0
$629$	17882.9	1.13361
$630$	0	0
$631$	12083.2	0.762324	0.381162	−	0.924508i	$-0.375524\pi$
$631$	12083.2	0.762324	0.381162	+	0.924508i	$0.375524\pi$
$632$	0	0
$633$	9842.75	0.618032
$634$	0	0
$635$	11620.1	0.726190
$636$	0	0
$637$	−369.347	−0.0229734
$638$	0	0
$639$	−4691.52	−0.290444
$640$	0	0
$641$	−18029.7	−1.11097	−0.555483	−	0.831528i	$-0.687467\pi$
$641$	−18029.7	−1.11097	−0.555483	+	0.831528i	$0.687467\pi$
$642$	0	0
$643$	−6862.56	−0.420891	−0.210445	−	0.977606i	$-0.567491\pi$
$643$	−6862.56	−0.420891	−0.210445	+	0.977606i	$0.567491\pi$
$644$	0	0
$645$	−6295.20	−0.384300
$646$	0	0
$647$	16193.7	0.983990	0.491995	−	0.870598i	$-0.336268\pi$
$647$	16193.7	0.983990	0.491995	+	0.870598i	$0.336268\pi$
$648$	0	0
$649$	11181.3	0.676279
$650$	0	0
$651$	13367.1	0.804761
$652$	0	0
$653$	28473.8	1.70638	0.853188	−	0.521603i	$-0.174666\pi$
$653$	28473.8	1.70638	0.853188	+	0.521603i	$0.174666\pi$
$654$	0	0
$655$	−11526.5	−0.687600
$656$	0	0
$657$	9655.09	0.573335
$658$	0	0
$659$	−23211.3	−1.37206	−0.686028	−	0.727575i	$-0.740647\pi$
$659$	−23211.3	−1.37206	−0.686028	+	0.727575i	$0.740647\pi$
$660$	0	0
$661$	3168.68	0.186456	0.0932280	−	0.995645i	$-0.470281\pi$
$661$	3168.68	0.186456	0.0932280	+	0.995645i	$0.470281\pi$
$662$	0	0
$663$	1485.64	0.0870248
$664$	0	0
$665$	−8523.55	−0.497036
$666$	0	0
$667$	−2944.28	−0.170919
$668$	0	0
$669$	−11446.4	−0.661501
$670$	0	0
$671$	11667.9	0.671290
$672$	0	0
$673$	33652.0	1.92747	0.963736	−	0.266856i	$-0.0859850\pi$
$673$	33652.0	1.92747	0.963736	+	0.266856i	$0.0859850\pi$
$674$	0	0
$675$	2450.70	0.139745
$676$	0	0
$677$	2495.05	0.141644	0.0708218	−	0.997489i	$-0.477438\pi$
$677$	2495.05	0.141644	0.0708218	+	0.997489i	$0.477438\pi$
$678$	0	0
$679$	−6443.11	−0.364159
$680$	0	0
$681$	−21823.7	−1.22803
$682$	0	0
$683$	−26126.2	−1.46368	−0.731839	−	0.681477i	$-0.761338\pi$
$683$	−26126.2	−1.46368	−0.731839	+	0.681477i	$0.761338\pi$
$684$	0	0
$685$	−12139.5	−0.677119
$686$	0	0
$687$	−20629.6	−1.14566
$688$	0	0
$689$	−1273.35	−0.0704074
$690$	0	0
$691$	−33535.3	−1.84623	−0.923114	−	0.384525i	$-0.874365\pi$
$691$	−33535.3	−1.84623	−0.923114	+	0.384525i	$0.874365\pi$
$692$	0	0
$693$	−8084.20	−0.443136
$694$	0	0
$695$	8647.92	0.471992
$696$	0	0
$697$	33441.4	1.81734
$698$	0	0
$699$	4567.57	0.247155
$700$	0	0
$701$	−1725.01	−0.0929424	−0.0464712	−	0.998920i	$-0.514798\pi$
$701$	−1725.01	−0.0929424	−0.0464712	+	0.998920i	$0.514798\pi$
$702$	0	0
$703$	−15434.3	−0.828043
$704$	0	0
$705$	5621.04	0.300285
$706$	0	0
$707$	3152.71	0.167709
$708$	0	0
$709$	33188.8	1.75801	0.879006	−	0.476811i	$-0.158207\pi$
$709$	33188.8	1.75801	0.879006	+	0.476811i	$0.158207\pi$
$710$	0	0
$711$	9682.74	0.510733
$712$	0	0
$713$	−2269.87	−0.119225
$714$	0	0
$715$	442.669	0.0231537
$716$	0	0
$717$	−32667.5	−1.70152
$718$	0	0
$719$	10183.7	0.528216	0.264108	−	0.964493i	$-0.414923\pi$
$719$	10183.7	0.528216	0.264108	+	0.964493i	$0.414923\pi$
$720$	0	0
$721$	3291.35	0.170009
$722$	0	0
$723$	−41973.2	−2.15906
$724$	0	0
$725$	−3200.30	−0.163940
$726$	0	0
$727$	−7926.17	−0.404354	−0.202177	−	0.979349i	$-0.564802\pi$
$727$	−7926.17	−0.404354	−0.202177	+	0.979349i	$0.564802\pi$
$728$	0	0
$729$	6268.73	0.318485
$730$	0	0
$731$	18360.2	0.928970
$732$	0	0
$733$	−29287.2	−1.47578	−0.737891	−	0.674920i	$-0.764178\pi$
$733$	−29287.2	−1.47578	−0.737891	+	0.674920i	$0.764178\pi$
$734$	0	0
$735$	−4266.96	−0.214135
$736$	0	0
$737$	25568.7	1.27793
$738$	0	0
$739$	9561.78	0.475962	0.237981	−	0.971270i	$-0.423514\pi$
$739$	9561.78	0.475962	0.237981	+	0.971270i	$0.423514\pi$
$740$	0	0
$741$	−1282.21	−0.0635673
$742$	0	0
$743$	−31343.2	−1.54760	−0.773802	−	0.633428i	$-0.781647\pi$
$743$	−31343.2	−1.54760	−0.773802	+	0.633428i	$0.781647\pi$
$744$	0	0
$745$	15763.7	0.775216
$746$	0	0
$747$	−9576.75	−0.469069
$748$	0	0
$749$	−12388.6	−0.604367
$750$	0	0
$751$	13193.4	0.641059	0.320529	−	0.947239i	$-0.396139\pi$
$751$	13193.4	0.641059	0.320529	+	0.947239i	$0.396139\pi$
$752$	0	0
$753$	−27419.1	−1.32697
$754$	0	0
$755$	−4663.40	−0.224793
$756$	0	0
$757$	−8920.60	−0.428302	−0.214151	−	0.976801i	$-0.568698\pi$
$757$	−8920.60	−0.428302	−0.214151	+	0.976801i	$0.568698\pi$
$758$	0	0
$759$	4704.90	0.225003
$760$	0	0
$761$	22345.1	1.06440	0.532200	−	0.846619i	$-0.321366\pi$
$761$	22345.1	1.06440	0.532200	+	0.846619i	$0.321366\pi$
$762$	0	0
$763$	−17047.1	−0.808842
$764$	0	0
$765$	5007.80	0.236676
$766$	0	0
$767$	901.882	0.0424577
$768$	0	0
$769$	−2435.99	−0.114232	−0.0571159	−	0.998368i	$-0.518190\pi$
$769$	−2435.99	−0.114232	−0.0571159	+	0.998368i	$0.518190\pi$
$770$	0	0
$771$	−17196.4	−0.803261
$772$	0	0
$773$	−573.293	−0.0266752	−0.0133376	−	0.999911i	$-0.504246\pi$
$773$	−573.293	−0.0266752	−0.0133376	+	0.999911i	$0.504246\pi$
$774$	0	0
$775$	−2467.25	−0.114356
$776$	0	0
$777$	−26901.1	−1.24205
$778$	0	0
$779$	−28862.4	−1.32747
$780$	0	0
$781$	13973.3	0.640208
$782$	0	0
$783$	−12548.8	−0.572743
$784$	0	0
$785$	−6039.87	−0.274614
$786$	0	0
$787$	19415.7	0.879411	0.439706	−	0.898142i	$-0.355083\pi$
$787$	19415.7	0.879411	0.439706	+	0.898142i	$0.355083\pi$
$788$	0	0
$789$	−31590.1	−1.42540
$790$	0	0
$791$	42324.2	1.90250
$792$	0	0
$793$	941.131	0.0421445
$794$	0	0
$795$	−14710.6	−0.656267
$796$	0	0
$797$	−9404.39	−0.417968	−0.208984	−	0.977919i	$-0.567016\pi$
$797$	−9404.39	−0.417968	−0.208984	+	0.977919i	$0.567016\pi$
$798$	0	0
$799$	−16394.0	−0.725879
$800$	0	0
$801$	908.358	0.0400690
$802$	0	0
$803$	−28756.8	−1.26377
$804$	0	0
$805$	2522.71	0.110452
$806$	0	0
$807$	−20767.4	−0.905883
$808$	0	0
$809$	29231.5	1.27036	0.635182	−	0.772362i	$-0.280925\pi$
$809$	29231.5	1.27036	0.635182	+	0.772362i	$0.280925\pi$
$810$	0	0
$811$	20710.9	0.896744	0.448372	−	0.893847i	$-0.352004\pi$
$811$	20710.9	0.896744	0.448372	+	0.893847i	$0.352004\pi$
$812$	0	0
$813$	−15416.1	−0.665027
$814$	0	0
$815$	10960.5	0.471077
$816$	0	0
$817$	−15846.2	−0.678565
$818$	0	0
$819$	−652.069	−0.0278207
$820$	0	0
$821$	−8033.67	−0.341507	−0.170753	−	0.985314i	$-0.554620\pi$
$821$	−8033.67	−0.341507	−0.170753	+	0.985314i	$0.554620\pi$
$822$	0	0
$823$	−4956.98	−0.209951	−0.104975	−	0.994475i	$-0.533476\pi$
$823$	−4956.98	−0.209951	−0.104975	+	0.994475i	$0.533476\pi$
$824$	0	0
$825$	5114.02	0.215815
$826$	0	0
$827$	31454.3	1.32258	0.661290	−	0.750130i	$-0.270009\pi$
$827$	31454.3	1.32258	0.661290	+	0.750130i	$0.270009\pi$
$828$	0	0
$829$	23752.7	0.995132	0.497566	−	0.867426i	$-0.334227\pi$
$829$	23752.7	0.995132	0.497566	+	0.867426i	$0.334227\pi$
$830$	0	0
$831$	16755.5	0.699450
$832$	0	0
$833$	12444.7	0.517629
$834$	0	0
$835$	−6859.39	−0.284286
$836$	0	0
$837$	−9674.39	−0.399517
$838$	0	0
$839$	24668.3	1.01507	0.507535	−	0.861631i	$-0.330557\pi$
$839$	24668.3	1.01507	0.507535	+	0.861631i	$0.330557\pi$
$840$	0	0
$841$	−8001.88	−0.328094
$842$	0	0
$843$	49109.7	2.00644
$844$	0	0
$845$	−10949.3	−0.445760
$846$	0	0
$847$	−5119.56	−0.207686
$848$	0	0
$849$	−32638.4	−1.31937
$850$	0	0
$851$	4568.06	0.184008
$852$	0	0
$853$	23883.2	0.958670	0.479335	−	0.877632i	$-0.340878\pi$
$853$	23883.2	0.958670	0.479335	+	0.877632i	$0.340878\pi$
$854$	0	0
$855$	−4322.09	−0.172880
$856$	0	0
$857$	46020.0	1.83432	0.917160	−	0.398519i	$-0.130476\pi$
$857$	46020.0	1.83432	0.917160	+	0.398519i	$0.130476\pi$
$858$	0	0
$859$	3261.83	0.129560	0.0647800	−	0.997900i	$-0.479365\pi$
$859$	3261.83	0.129560	0.0647800	+	0.997900i	$0.479365\pi$
$860$	0	0
$861$	−50305.5	−1.99118
$862$	0	0
$863$	25378.4	1.00103	0.500517	−	0.865727i	$-0.333143\pi$
$863$	25378.4	1.00103	0.500517	+	0.865727i	$0.333143\pi$
$864$	0	0
$865$	1456.61	0.0572558
$866$	0	0
$867$	−19722.1	−0.772547
$868$	0	0
$869$	−28839.1	−1.12578
$870$	0	0
$871$	2062.37	0.0802303
$872$	0	0
$873$	−3267.15	−0.126662
$874$	0	0
$875$	2742.07	0.105942
$876$	0	0
$877$	7704.25	0.296641	0.148320	−	0.988939i	$-0.452613\pi$
$877$	7704.25	0.296641	0.148320	+	0.988939i	$0.452613\pi$
$878$	0	0
$879$	−36995.4	−1.41959
$880$	0	0
$881$	29599.4	1.13193	0.565965	−	0.824429i	$-0.308504\pi$
$881$	29599.4	1.13193	0.565965	+	0.824429i	$0.308504\pi$
$882$	0	0
$883$	39617.5	1.50989	0.754945	−	0.655788i	$-0.227663\pi$
$883$	39617.5	1.50989	0.754945	+	0.655788i	$0.227663\pi$
$884$	0	0
$885$	10419.2	0.395748
$886$	0	0
$887$	−37469.4	−1.41837	−0.709187	−	0.705020i	$-0.750938\pi$
$887$	−37469.4	−1.41837	−0.709187	+	0.705020i	$0.750938\pi$
$888$	0	0
$889$	50981.2	1.92335
$890$	0	0
$891$	30002.9	1.12810
$892$	0	0
$893$	14149.2	0.530218
$894$	0	0
$895$	19475.5	0.727366
$896$	0	0
$897$	379.496	0.0141260
$898$	0	0
$899$	12633.5	0.468689
$900$	0	0
$901$	42904.1	1.58640
$902$	0	0
$903$	−27619.0	−1.01783
$904$	0	0
$905$	11806.6	0.433664
$906$	0	0
$907$	−25747.8	−0.942604	−0.471302	−	0.881972i	$-0.656216\pi$
$907$	−25747.8	−0.942604	−0.471302	+	0.881972i	$0.656216\pi$
$908$	0	0
$909$	1598.66	0.0583326
$910$	0	0
$911$	29919.0	1.08810	0.544050	−	0.839053i	$-0.316890\pi$
$911$	29919.0	1.08810	0.544050	+	0.839053i	$0.316890\pi$
$912$	0	0
$913$	28523.4	1.03394
$914$	0	0
$915$	10872.6	0.392828
$916$	0	0
$917$	−50570.5	−1.82114
$918$	0	0
$919$	47257.1	1.69626	0.848132	−	0.529785i	$-0.177727\pi$
$919$	47257.1	1.69626	0.848132	+	0.529785i	$0.177727\pi$
$920$	0	0
$921$	−9223.19	−0.329983
$922$	0	0
$923$	1127.08	0.0401931
$924$	0	0
$925$	4965.29	0.176495
$926$	0	0
$927$	1668.96	0.0591326
$928$	0	0
$929$	−30206.4	−1.06678	−0.533390	−	0.845869i	$-0.679082\pi$
$929$	−30206.4	−1.06678	−0.533390	+	0.845869i	$0.679082\pi$
$930$	0	0
$931$	−10740.7	−0.378102
$932$	0	0
$933$	−9071.87	−0.318328
$934$	0	0
$935$	−14915.2	−0.521691
$936$	0	0
$937$	−14738.9	−0.513872	−0.256936	−	0.966428i	$-0.582713\pi$
$937$	−14738.9	−0.513872	−0.256936	+	0.966428i	$0.582713\pi$
$938$	0	0
$939$	−38593.2	−1.34126
$940$	0	0
$941$	−32326.0	−1.11987	−0.559935	−	0.828537i	$-0.689174\pi$
$941$	−32326.0	−1.11987	−0.559935	+	0.828537i	$0.689174\pi$
$942$	0	0
$943$	8542.36	0.294992
$944$	0	0
$945$	10752.0	0.370120
$946$	0	0
$947$	23893.5	0.819888	0.409944	−	0.912111i	$-0.365548\pi$
$947$	23893.5	0.819888	0.409944	+	0.912111i	$0.365548\pi$
$948$	0	0
$949$	−2319.51	−0.0793410
$950$	0	0
$951$	−9948.50	−0.339224
$952$	0	0
$953$	−34843.8	−1.18437	−0.592184	−	0.805803i	$-0.701734\pi$
$953$	−34843.8	−1.18437	−0.592184	+	0.805803i	$0.701734\pi$
$954$	0	0
$955$	1138.79	0.0385866
$956$	0	0
$957$	−26186.3	−0.884517
$958$	0	0
$959$	−53259.8	−1.79338
$960$	0	0
$961$	−20051.3	−0.673066
$962$	0	0
$963$	−6281.98	−0.210212
$964$	0	0
$965$	2354.20	0.0785332
$966$	0	0
$967$	1172.91	0.0390056	0.0195028	−	0.999810i	$-0.493792\pi$
$967$	1172.91	0.0390056	0.0195028	+	0.999810i	$0.493792\pi$
$968$	0	0
$969$	43202.9	1.43228
$970$	0	0
$971$	−27955.0	−0.923912	−0.461956	−	0.886903i	$-0.652852\pi$
$971$	−27955.0	−0.923912	−0.461956	+	0.886903i	$0.652852\pi$
$972$	0	0
$973$	37941.2	1.25009
$974$	0	0
$975$	412.495	0.0135492
$976$	0	0
$977$	34189.8	1.11958	0.559790	−	0.828635i	$-0.310882\pi$
$977$	34189.8	1.11958	0.559790	+	0.828635i	$0.310882\pi$
$978$	0	0
$979$	−2705.46	−0.0883216
$980$	0	0
$981$	−8644.18	−0.281333
$982$	0	0
$983$	−24828.5	−0.805601	−0.402800	−	0.915288i	$-0.631963\pi$
$983$	−24828.5	−0.805601	−0.402800	+	0.915288i	$0.631963\pi$
$984$	0	0
$985$	12499.7	0.404339
$986$	0	0
$987$	24661.3	0.795317
$988$	0	0
$989$	4689.98	0.150791
$990$	0	0
$991$	−32896.9	−1.05449	−0.527247	−	0.849712i	$-0.676776\pi$
$991$	−32896.9	−1.05449	−0.527247	+	0.849712i	$0.676776\pi$
$992$	0	0
$993$	37543.9	1.19982
$994$	0	0
$995$	5970.89	0.190241
$996$	0	0
$997$	21939.9	0.696935	0.348467	−	0.937321i	$-0.386702\pi$
$997$	21939.9	0.696935	0.348467	+	0.937321i	$0.386702\pi$
$998$	0	0
$999$	19469.5	0.616605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.ba.1.3		10
4.3	odd	2		920.4.a.h.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.h.1.8	✓	10	4.3	odd	2
1840.4.a.ba.1.3		10	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-6.17442 q^{3} +5.00000 q^{5} +21.9366 q^{7} +11.1235 q^{9} +O(q^{10})\)	\(q-6.17442 q^{3} +5.00000 q^{5} +21.9366 q^{7} +11.1235 q^{9} -33.1304 q^{11} -2.67228 q^{13} -30.8721 q^{15} +90.0398 q^{17} -77.7108 q^{19} -135.446 q^{21} +23.0000 q^{23} +25.0000 q^{25} +98.0281 q^{27} -128.012 q^{29} -98.6899 q^{31} +204.561 q^{33} +109.683 q^{35} +198.611 q^{37} +16.4998 q^{39} +371.407 q^{41} +203.912 q^{43} +55.6176 q^{45} -182.075 q^{47} +138.214 q^{49} -555.944 q^{51} +476.502 q^{53} -165.652 q^{55} +479.820 q^{57} -337.495 q^{59} -352.182 q^{61} +244.012 q^{63} -13.3614 q^{65} -771.762 q^{67} -142.012 q^{69} -421.766 q^{71} +867.989 q^{73} -154.361 q^{75} -726.767 q^{77} +870.474 q^{79} -905.602 q^{81} -860.946 q^{83} +450.199 q^{85} +790.402 q^{87} +81.6610 q^{89} -58.6208 q^{91} +609.353 q^{93} -388.554 q^{95} -293.715 q^{97} -368.526 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) \) 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9	\( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100}) \) 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9 + 56 * q^11 + 49 * q^13 - 25 * q^15 + 240 * q^17 - 88 * q^19 + 346 * q^21 + 230 * q^23 + 250 * q^25 - 449 * q^27 + 319 * q^29 + 109 * q^31 + 504 * q^33 - 70 * q^35 + 580 * q^37 - 107 * q^39 + 259 * q^41 - 330 * q^43 + 695 * q^45 - 227 * q^47 + 630 * q^49 + 192 * q^51 - 186 * q^53 + 280 * q^55 + 1708 * q^57 - 262 * q^59 + 1000 * q^61 - 722 * q^63 + 245 * q^65 - 354 * q^67 - 115 * q^69 - 599 * q^71 + 355 * q^73 - 125 * q^75 + 1776 * q^77 + 1068 * q^79 + 3490 * q^81 - 754 * q^83 + 1200 * q^85 - 2675 * q^87 + 1740 * q^89 - 690 * q^91 + 1669 * q^93 - 440 * q^95 + 2592 * q^97 - 916 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more