LMFDB - Embedded newform 1840.4.a.l.1.2

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:	$1$
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} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.50148$ 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)

$f(q)$	$=$	$q-5.50148 q^{3} -5.00000 q^{5} -0.0500526 q^{7} +3.26627 q^{9} +O(q^{10})$	$q-5.50148 q^{3} -5.00000 q^{5} -0.0500526 q^{7} +3.26627 q^{9} -35.1757 q^{11} +86.1100 q^{13} +27.5074 q^{15} +8.82793 q^{17} -106.317 q^{19} +0.275364 q^{21} -23.0000 q^{23} +25.0000 q^{25} +130.571 q^{27} -280.754 q^{29} +117.077 q^{31} +193.518 q^{33} +0.250263 q^{35} +93.3088 q^{37} -473.732 q^{39} -58.0579 q^{41} +508.101 q^{43} -16.3313 q^{45} +407.914 q^{47} -342.997 q^{49} -48.5666 q^{51} +316.251 q^{53} +175.878 q^{55} +584.899 q^{57} -129.500 q^{59} +299.008 q^{61} -0.163485 q^{63} -430.550 q^{65} -596.326 q^{67} +126.534 q^{69} +692.108 q^{71} +842.098 q^{73} -137.537 q^{75} +1.76063 q^{77} +1161.06 q^{79} -806.521 q^{81} -1121.69 q^{83} -44.1396 q^{85} +1544.56 q^{87} +409.092 q^{89} -4.31003 q^{91} -644.098 q^{93} +531.583 q^{95} -1123.62 q^{97} -114.893 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9	$ 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9} - 93 q^{11} + 32 q^{13} + 20 q^{15} + 108 q^{17} - 185 q^{19} + 302 q^{21} - 92 q^{23} + 100 q^{25} - 259 q^{27} + 294 q^{29} + 211 q^{31} - 237 q^{33} + 130 q^{35} + 5 q^{37} + 421 q^{39} - 369 q^{41} + 100 q^{43} - 320 q^{45} + 363 q^{47} + 600 q^{49} + 315 q^{51} + 21 q^{53} + 465 q^{55} - 160 q^{57} + 33 q^{59} - 307 q^{61} - 167 q^{63} - 160 q^{65} - 725 q^{67} + 92 q^{69} + 1257 q^{71} + 509 q^{73} - 100 q^{75} - 1962 q^{77} - 1202 q^{79} - 992 q^{81} + 1377 q^{83} - 540 q^{85} + 2829 q^{87} - 984 q^{89} + 995 q^{91} - 1843 q^{93} + 925 q^{95} + 137 q^{97} - 2013 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9 - 93 * q^11 + 32 * q^13 + 20 * q^15 + 108 * q^17 - 185 * q^19 + 302 * q^21 - 92 * q^23 + 100 * q^25 - 259 * q^27 + 294 * q^29 + 211 * q^31 - 237 * q^33 + 130 * q^35 + 5 * q^37 + 421 * q^39 - 369 * q^41 + 100 * q^43 - 320 * q^45 + 363 * q^47 + 600 * q^49 + 315 * q^51 + 21 * q^53 + 465 * q^55 - 160 * q^57 + 33 * q^59 - 307 * q^61 - 167 * q^63 - 160 * q^65 - 725 * q^67 + 92 * q^69 + 1257 * q^71 + 509 * q^73 - 100 * q^75 - 1962 * q^77 - 1202 * q^79 - 992 * q^81 + 1377 * q^83 - 540 * q^85 + 2829 * q^87 - 984 * q^89 + 995 * q^91 - 1843 * q^93 + 925 * q^95 + 137 * q^97 - 2013 * 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
$3$	−5.50148	−1.05876	−0.529380	−	0.848385i	$-0.677575\pi$
$3$	−5.50148	−1.05876	−0.529380	+	0.848385i	$0.677575\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−0.0500526	−0.00270259	−0.00135129	−	0.999999i	$-0.500430\pi$
$7$	−0.0500526	−0.00270259	−0.00135129	+	0.999999i	$0.500430\pi$
$8$	0	0
$9$	3.26627	0.120973
$10$	0	0
$11$	−35.1757	−0.964169	−0.482085	−	0.876125i	$-0.660120\pi$
$11$	−35.1757	−0.964169	−0.482085	+	0.876125i	$0.660120\pi$
$12$	0	0
$13$	86.1100	1.83712	0.918562	−	0.395277i	$-0.129351\pi$
$13$	86.1100	1.83712	0.918562	+	0.395277i	$0.129351\pi$
$14$	0	0
$15$	27.5074	0.473492
$16$	0	0
$17$	8.82793	0.125946	0.0629731	−	0.998015i	$-0.479942\pi$
$17$	8.82793	0.125946	0.0629731	+	0.998015i	$0.479942\pi$
$18$	0	0
$19$	−106.317	−1.28372	−0.641861	−	0.766821i	$-0.721837\pi$
$19$	−106.317	−1.28372	−0.641861	+	0.766821i	$0.721837\pi$
$20$	0	0
$21$	0.275364	0.00286139
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	130.571	0.930679
$28$	0	0
$29$	−280.754	−1.79775	−0.898873	−	0.438209i	$-0.855613\pi$
$29$	−280.754	−1.79775	−0.898873	+	0.438209i	$0.855613\pi$
$30$	0	0
$31$	117.077	0.678313	0.339156	−	0.940730i	$-0.389858\pi$
$31$	117.077	0.678313	0.339156	+	0.940730i	$0.389858\pi$
$32$	0	0
$33$	193.518	1.02082
$34$	0	0
$35$	0.250263	0.00120863
$36$	0	0
$37$	93.3088	0.414591	0.207296	−	0.978278i	$-0.433534\pi$
$37$	93.3088	0.414591	0.207296	+	0.978278i	$0.433534\pi$
$38$	0	0
$39$	−473.732	−1.94507
$40$	0	0
$41$	−58.0579	−0.221149	−0.110575	−	0.993868i	$-0.535269\pi$
$41$	−58.0579	−0.221149	−0.110575	+	0.993868i	$0.535269\pi$
$42$	0	0
$43$	508.101	1.80197	0.900985	−	0.433851i	$-0.142845\pi$
$43$	508.101	1.80197	0.900985	+	0.433851i	$0.142845\pi$
$44$	0	0
$45$	−16.3313	−0.0541007
$46$	0	0
$47$	407.914	1.26597	0.632983	−	0.774166i	$-0.281830\pi$
$47$	407.914	1.26597	0.632983	+	0.774166i	$0.281830\pi$
$48$	0	0
$49$	−342.997	−0.999993
$50$	0	0
$51$	−48.5666	−0.133347
$52$	0	0
$53$	316.251	0.819629	0.409815	−	0.912169i	$-0.365593\pi$
$53$	316.251	0.819629	0.409815	+	0.912169i	$0.365593\pi$
$54$	0	0
$55$	175.878	0.431190
$56$	0	0
$57$	584.899	1.35915
$58$	0	0
$59$	−129.500	−0.285754	−0.142877	−	0.989740i	$-0.545635\pi$
$59$	−129.500	−0.285754	−0.142877	+	0.989740i	$0.545635\pi$
$60$	0	0
$61$	299.008	0.627607	0.313804	−	0.949488i	$-0.398397\pi$
$61$	299.008	0.627607	0.313804	+	0.949488i	$0.398397\pi$
$62$	0	0
$63$	−0.163485	−0.000326940 0
$64$	0	0
$65$	−430.550	−0.821587
$66$	0	0
$67$	−596.326	−1.08736	−0.543678	−	0.839294i	$-0.682969\pi$
$67$	−596.326	−1.08736	−0.543678	+	0.839294i	$0.682969\pi$
$68$	0	0
$69$	126.534	0.220767
$70$	0	0
$71$	692.108	1.15687	0.578437	−	0.815727i	$-0.303663\pi$
$71$	692.108	1.15687	0.578437	+	0.815727i	$0.303663\pi$
$72$	0	0
$73$	842.098	1.35014	0.675069	−	0.737755i	$-0.264114\pi$
$73$	842.098	1.35014	0.675069	+	0.737755i	$0.264114\pi$
$74$	0	0
$75$	−137.537	−0.211752
$76$	0	0
$77$	1.76063	0.00260575
$78$	0	0
$79$	1161.06	1.65354	0.826769	−	0.562542i	$-0.190177\pi$
$79$	1161.06	1.65354	0.826769	+	0.562542i	$0.190177\pi$
$80$	0	0
$81$	−806.521	−1.10634
$82$	0	0
$83$	−1121.69	−1.48340	−0.741699	−	0.670733i	$-0.765980\pi$
$83$	−1121.69	−1.48340	−0.741699	+	0.670733i	$0.765980\pi$
$84$	0	0
$85$	−44.1396	−0.0563249
$86$	0	0
$87$	1544.56	1.90338
$88$	0	0
$89$	409.092	0.487232	0.243616	−	0.969872i	$-0.421666\pi$
$89$	409.092	0.487232	0.243616	+	0.969872i	$0.421666\pi$
$90$	0	0
$91$	−4.31003	−0.00496499
$92$	0	0
$93$	−644.098	−0.718170
$94$	0	0
$95$	531.583	0.574098
$96$	0	0
$97$	−1123.62	−1.17615	−0.588075	−	0.808807i	$-0.700114\pi$
$97$	−1123.62	−1.17615	−0.588075	+	0.808807i	$0.700114\pi$
$98$	0	0
$99$	−114.893	−0.116638
$100$	0	0
$101$	−981.824	−0.967278	−0.483639	−	0.875268i	$-0.660685\pi$
$101$	−981.824	−0.967278	−0.483639	+	0.875268i	$0.660685\pi$
$102$	0	0
$103$	28.9575	0.0277016	0.0138508	−	0.999904i	$-0.495591\pi$
$103$	28.9575	0.0277016	0.0138508	+	0.999904i	$0.495591\pi$
$104$	0	0
$105$	−1.37682	−0.00127965
$106$	0	0
$107$	187.874	0.169742	0.0848712	−	0.996392i	$-0.472952\pi$
$107$	187.874	0.169742	0.0848712	+	0.996392i	$0.472952\pi$
$108$	0	0
$109$	2142.02	1.88228	0.941139	−	0.338018i	$-0.109757\pi$
$109$	2142.02	1.88228	0.941139	+	0.338018i	$0.109757\pi$
$110$	0	0
$111$	−513.337	−0.438953
$112$	0	0
$113$	−1668.67	−1.38916	−0.694582	−	0.719414i	$-0.744411\pi$
$113$	−1668.67	−1.38916	−0.694582	+	0.719414i	$0.744411\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	281.258	0.222242
$118$	0	0
$119$	−0.441861	−0.000340381 0
$120$	0	0
$121$	−93.6729	−0.0703779
$122$	0	0
$123$	319.404	0.234144
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	354.326	0.247570	0.123785	−	0.992309i	$-0.460497\pi$
$127$	354.326	0.247570	0.123785	+	0.992309i	$0.460497\pi$
$128$	0	0
$129$	−2795.31	−1.90785
$130$	0	0
$131$	922.236	0.615085	0.307543	−	0.951534i	$-0.400493\pi$
$131$	922.236	0.615085	0.307543	+	0.951534i	$0.400493\pi$
$132$	0	0
$133$	5.32143	0.00346937
$134$	0	0
$135$	−652.853	−0.416212
$136$	0	0
$137$	−1664.96	−1.03830	−0.519149	−	0.854684i	$-0.673751\pi$
$137$	−1664.96	−1.03830	−0.519149	+	0.854684i	$0.673751\pi$
$138$	0	0
$139$	−2407.80	−1.46926	−0.734628	−	0.678470i	$-0.762643\pi$
$139$	−2407.80	−1.46926	−0.734628	+	0.678470i	$0.762643\pi$
$140$	0	0
$141$	−2244.13	−1.34035
$142$	0	0
$143$	−3028.97	−1.77130
$144$	0	0
$145$	1403.77	0.803977
$146$	0	0
$147$	1886.99	1.05875
$148$	0	0
$149$	1838.74	1.01098	0.505488	−	0.862834i	$-0.331312\pi$
$149$	1838.74	1.01098	0.505488	+	0.862834i	$0.331312\pi$
$150$	0	0
$151$	1276.86	0.688139	0.344070	−	0.938944i	$-0.388194\pi$
$151$	1276.86	0.688139	0.344070	+	0.938944i	$0.388194\pi$
$152$	0	0
$153$	28.8344	0.0152361
$154$	0	0
$155$	−585.386	−0.303351
$156$	0	0
$157$	−1288.13	−0.654804	−0.327402	−	0.944885i	$-0.606173\pi$
$157$	−1288.13	−0.654804	−0.327402	+	0.944885i	$0.606173\pi$
$158$	0	0
$159$	−1739.85	−0.867791
$160$	0	0
$161$	1.15121	0.000563529 0
$162$	0	0
$163$	995.942	0.478578	0.239289	−	0.970948i	$-0.423086\pi$
$163$	995.942	0.478578	0.239289	+	0.970948i	$0.423086\pi$
$164$	0	0
$165$	−967.591	−0.456526
$166$	0	0
$167$	1568.39	0.726742	0.363371	−	0.931645i	$-0.381626\pi$
$167$	1568.39	0.726742	0.363371	+	0.931645i	$0.381626\pi$
$168$	0	0
$169$	5217.93	2.37502
$170$	0	0
$171$	−347.259	−0.155295
$172$	0	0
$173$	2478.94	1.08942	0.544711	−	0.838624i	$-0.316639\pi$
$173$	2478.94	1.08942	0.544711	+	0.838624i	$0.316639\pi$
$174$	0	0
$175$	−1.25132	−0.000540518 0
$176$	0	0
$177$	712.444	0.302545
$178$	0	0
$179$	−3740.99	−1.56210	−0.781048	−	0.624471i	$-0.785314\pi$
$179$	−3740.99	−1.56210	−0.781048	+	0.624471i	$0.785314\pi$
$180$	0	0
$181$	−3841.87	−1.57770	−0.788851	−	0.614584i	$-0.789324\pi$
$181$	−3841.87	−1.57770	−0.788851	+	0.614584i	$0.789324\pi$
$182$	0	0
$183$	−1644.99	−0.664486
$184$	0	0
$185$	−466.544	−0.185411
$186$	0	0
$187$	−310.528	−0.121433
$188$	0	0
$189$	−6.53540	−0.00251524
$190$	0	0
$191$	−2097.13	−0.794465	−0.397233	−	0.917718i	$-0.630029\pi$
$191$	−2097.13	−0.794465	−0.397233	+	0.917718i	$0.630029\pi$
$192$	0	0
$193$	1264.65	0.471666	0.235833	−	0.971794i	$-0.424218\pi$
$193$	1264.65	0.471666	0.235833	+	0.971794i	$0.424218\pi$
$194$	0	0
$195$	2368.66	0.869863
$196$	0	0
$197$	−850.129	−0.307458	−0.153729	−	0.988113i	$-0.549128\pi$
$197$	−850.129	−0.307458	−0.153729	+	0.988113i	$0.549128\pi$
$198$	0	0
$199$	203.089	0.0723447	0.0361724	−	0.999346i	$-0.488483\pi$
$199$	203.089	0.0723447	0.0361724	+	0.999346i	$0.488483\pi$
$200$	0	0
$201$	3280.68	1.15125
$202$	0	0
$203$	14.0525	0.00485857
$204$	0	0
$205$	290.289	0.0989009
$206$	0	0
$207$	−75.1241	−0.0252246
$208$	0	0
$209$	3739.76	1.23773
$210$	0	0
$211$	−5612.25	−1.83111	−0.915553	−	0.402197i	$-0.868247\pi$
$211$	−5612.25	−1.83111	−0.915553	+	0.402197i	$0.868247\pi$
$212$	0	0
$213$	−3807.62	−1.22485
$214$	0	0
$215$	−2540.51	−0.805865
$216$	0	0
$217$	−5.86003	−0.00183320
$218$	0	0
$219$	−4632.78	−1.42947
$220$	0	0
$221$	760.172	0.231379
$222$	0	0
$223$	557.909	0.167535	0.0837676	−	0.996485i	$-0.473305\pi$
$223$	557.909	0.167535	0.0837676	+	0.996485i	$0.473305\pi$
$224$	0	0
$225$	81.6567	0.0241946
$226$	0	0
$227$	684.858	0.200245	0.100123	−	0.994975i	$-0.468077\pi$
$227$	684.858	0.200245	0.100123	+	0.994975i	$0.468077\pi$
$228$	0	0
$229$	369.484	0.106621	0.0533104	−	0.998578i	$-0.483023\pi$
$229$	369.484	0.106621	0.0533104	+	0.998578i	$0.483023\pi$
$230$	0	0
$231$	−9.68609	−0.00275887
$232$	0	0
$233$	−5119.84	−1.43954	−0.719768	−	0.694215i	$-0.755752\pi$
$233$	−5119.84	−1.43954	−0.719768	+	0.694215i	$0.755752\pi$
$234$	0	0
$235$	−2039.57	−0.566157
$236$	0	0
$237$	−6387.55	−1.75070
$238$	0	0
$239$	−1665.03	−0.450635	−0.225317	−	0.974285i	$-0.572342\pi$
$239$	−1665.03	−0.450635	−0.225317	+	0.974285i	$0.572342\pi$
$240$	0	0
$241$	5401.56	1.44376	0.721878	−	0.692020i	$-0.243279\pi$
$241$	5401.56	1.44376	0.721878	+	0.692020i	$0.243279\pi$
$242$	0	0
$243$	911.649	0.240668
$244$	0	0
$245$	1714.99	0.447210
$246$	0	0
$247$	−9154.93	−2.35836
$248$	0	0
$249$	6170.98	1.57056
$250$	0	0
$251$	−4246.35	−1.06784	−0.533919	−	0.845535i	$-0.679281\pi$
$251$	−4246.35	−1.06784	−0.533919	+	0.845535i	$0.679281\pi$
$252$	0	0
$253$	809.040	0.201043
$254$	0	0
$255$	242.833	0.0596345
$256$	0	0
$257$	−5585.56	−1.35571	−0.677855	−	0.735196i	$-0.737090\pi$
$257$	−5585.56	−1.35571	−0.677855	+	0.735196i	$0.737090\pi$
$258$	0	0
$259$	−4.67035	−0.00112047
$260$	0	0
$261$	−917.016	−0.217478
$262$	0	0
$263$	2131.01	0.499634	0.249817	−	0.968293i	$-0.419630\pi$
$263$	2131.01	0.499634	0.249817	+	0.968293i	$0.419630\pi$
$264$	0	0
$265$	−1581.25	−0.366549
$266$	0	0
$267$	−2250.61	−0.515862
$268$	0	0
$269$	−8003.32	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$269$	−8003.32	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$270$	0	0
$271$	−1349.33	−0.302458	−0.151229	−	0.988499i	$-0.548323\pi$
$271$	−1349.33	−0.302458	−0.151229	+	0.988499i	$0.548323\pi$
$272$	0	0
$273$	23.7115	0.00525673
$274$	0	0
$275$	−879.392	−0.192834
$276$	0	0
$277$	−3762.76	−0.816181	−0.408091	−	0.912941i	$-0.633805\pi$
$277$	−3762.76	−0.816181	−0.408091	+	0.912941i	$0.633805\pi$
$278$	0	0
$279$	382.405	0.0820574
$280$	0	0
$281$	2885.36	0.612548	0.306274	−	0.951943i	$-0.400918\pi$
$281$	2885.36	0.612548	0.306274	+	0.951943i	$0.400918\pi$
$282$	0	0
$283$	−4809.11	−1.01015	−0.505074	−	0.863076i	$-0.668535\pi$
$283$	−4809.11	−1.01015	−0.505074	+	0.863076i	$0.668535\pi$
$284$	0	0
$285$	−2924.49	−0.607832
$286$	0	0
$287$	2.90595	0.000597675 0
$288$	0	0
$289$	−4835.07	−0.984138
$290$	0	0
$291$	6181.58	1.24526
$292$	0	0
$293$	−5362.69	−1.06925	−0.534627	−	0.845088i	$-0.679548\pi$
$293$	−5362.69	−1.06925	−0.534627	+	0.845088i	$0.679548\pi$
$294$	0	0
$295$	647.502	0.127793
$296$	0	0
$297$	−4592.91	−0.897332
$298$	0	0
$299$	−1980.53	−0.383067
$300$	0	0
$301$	−25.4318	−0.00486998
$302$	0	0
$303$	5401.48	1.02412
$304$	0	0
$305$	−1495.04	−0.280675
$306$	0	0
$307$	−6374.74	−1.18510	−0.592549	−	0.805534i	$-0.701879\pi$
$307$	−6374.74	−1.18510	−0.592549	+	0.805534i	$0.701879\pi$
$308$	0	0
$309$	−159.309	−0.0293293
$310$	0	0
$311$	−37.2799	−0.00679727	−0.00339863	−	0.999994i	$-0.501082\pi$
$311$	−37.2799	−0.00679727	−0.00339863	+	0.999994i	$0.501082\pi$
$312$	0	0
$313$	−3787.66	−0.683998	−0.341999	−	0.939700i	$-0.611104\pi$
$313$	−3787.66	−0.683998	−0.341999	+	0.939700i	$0.611104\pi$
$314$	0	0
$315$	0.817426	0.000146212 0
$316$	0	0
$317$	−5633.36	−0.998110	−0.499055	−	0.866570i	$-0.666319\pi$
$317$	−5633.36	−0.998110	−0.499055	+	0.866570i	$0.666319\pi$
$318$	0	0
$319$	9875.70	1.73333
$320$	0	0
$321$	−1033.58	−0.179716
$322$	0	0
$323$	−938.556	−0.161680
$324$	0	0
$325$	2152.75	0.367425
$326$	0	0
$327$	−11784.3	−1.99288
$328$	0	0
$329$	−20.4172	−0.00342138
$330$	0	0
$331$	−4566.91	−0.758369	−0.379185	−	0.925321i	$-0.623796\pi$
$331$	−4566.91	−0.758369	−0.379185	+	0.925321i	$0.623796\pi$
$332$	0	0
$333$	304.771	0.0501543
$334$	0	0
$335$	2981.63	0.486280
$336$	0	0
$337$	787.487	0.127291	0.0636456	−	0.997973i	$-0.479727\pi$
$337$	787.487	0.127291	0.0636456	+	0.997973i	$0.479727\pi$
$338$	0	0
$339$	9180.17	1.47079
$340$	0	0
$341$	−4118.27	−0.654008
$342$	0	0
$343$	34.3360	0.00540516
$344$	0	0
$345$	−632.670	−0.0987299
$346$	0	0
$347$	3738.52	0.578370	0.289185	−	0.957273i	$-0.406616\pi$
$347$	3738.52	0.578370	0.289185	+	0.957273i	$0.406616\pi$
$348$	0	0
$349$	425.210	0.0652177	0.0326088	−	0.999468i	$-0.489618\pi$
$349$	425.210	0.0652177	0.0326088	+	0.999468i	$0.489618\pi$
$350$	0	0
$351$	11243.4	1.70977
$352$	0	0
$353$	355.990	0.0536755	0.0268377	−	0.999640i	$-0.491456\pi$
$353$	355.990	0.0536755	0.0268377	+	0.999640i	$0.491456\pi$
$354$	0	0
$355$	−3460.54	−0.517370
$356$	0	0
$357$	2.43089	0.000360382 0
$358$	0	0
$359$	3713.17	0.545888	0.272944	−	0.962030i	$-0.412003\pi$
$359$	3713.17	0.545888	0.272944	+	0.962030i	$0.412003\pi$
$360$	0	0
$361$	4444.24	0.647942
$362$	0	0
$363$	515.340	0.0745133
$364$	0	0
$365$	−4210.49	−0.603800
$366$	0	0
$367$	−3908.40	−0.555904	−0.277952	−	0.960595i	$-0.589656\pi$
$367$	−3908.40	−0.555904	−0.277952	+	0.960595i	$0.589656\pi$
$368$	0	0
$369$	−189.632	−0.0267530
$370$	0	0
$371$	−15.8292	−0.00221512
$372$	0	0
$373$	13956.5	1.93737	0.968685	−	0.248293i	$-0.0798697\pi$
$373$	13956.5	1.93737	0.968685	+	0.248293i	$0.0798697\pi$
$374$	0	0
$375$	687.685	0.0946984
$376$	0	0
$377$	−24175.7	−3.30268
$378$	0	0
$379$	343.162	0.0465093	0.0232547	−	0.999730i	$-0.492597\pi$
$379$	343.162	0.0465093	0.0232547	+	0.999730i	$0.492597\pi$
$380$	0	0
$381$	−1949.32	−0.262117
$382$	0	0
$383$	13870.9	1.85057	0.925285	−	0.379273i	$-0.123826\pi$
$383$	13870.9	1.85057	0.925285	+	0.379273i	$0.123826\pi$
$384$	0	0
$385$	−8.80317	−0.00116533
$386$	0	0
$387$	1659.59	0.217989
$388$	0	0
$389$	80.0920	0.0104391	0.00521957	−	0.999986i	$-0.498339\pi$
$389$	80.0920	0.0104391	0.00521957	+	0.999986i	$0.498339\pi$
$390$	0	0
$391$	−203.042	−0.0262616
$392$	0	0
$393$	−5073.66	−0.651228
$394$	0	0
$395$	−5805.30	−0.739484
$396$	0	0
$397$	−5077.38	−0.641880	−0.320940	−	0.947099i	$-0.603999\pi$
$397$	−5077.38	−0.641880	−0.320940	+	0.947099i	$0.603999\pi$
$398$	0	0
$399$	−29.2757	−0.00367323
$400$	0	0
$401$	1588.41	0.197809	0.0989045	−	0.995097i	$-0.468466\pi$
$401$	1588.41	0.197809	0.0989045	+	0.995097i	$0.468466\pi$
$402$	0	0
$403$	10081.5	1.24614
$404$	0	0
$405$	4032.60	0.494770
$406$	0	0
$407$	−3282.20	−0.399736
$408$	0	0
$409$	4704.53	0.568763	0.284381	−	0.958711i	$-0.408212\pi$
$409$	4704.53	0.568763	0.284381	+	0.958711i	$0.408212\pi$
$410$	0	0
$411$	9159.73	1.09931
$412$	0	0
$413$	6.48184	0.000772277 0
$414$	0	0
$415$	5608.47	0.663395
$416$	0	0
$417$	13246.4	1.55559
$418$	0	0
$419$	−8774.62	−1.02307	−0.511537	−	0.859261i	$-0.670924\pi$
$419$	−8774.62	−1.02307	−0.511537	+	0.859261i	$0.670924\pi$
$420$	0	0
$421$	−12089.9	−1.39959	−0.699794	−	0.714345i	$-0.746725\pi$
$421$	−12089.9	−1.39959	−0.699794	+	0.714345i	$0.746725\pi$
$422$	0	0
$423$	1332.36	0.153147
$424$	0	0
$425$	220.698	0.0251893
$426$	0	0
$427$	−14.9661	−0.00169616
$428$	0	0
$429$	16663.8	1.87538
$430$	0	0
$431$	901.917	0.100798	0.0503989	−	0.998729i	$-0.483951\pi$
$431$	901.917	0.100798	0.0503989	+	0.998729i	$0.483951\pi$
$432$	0	0
$433$	11009.9	1.22194	0.610972	−	0.791652i	$-0.290779\pi$
$433$	11009.9	1.22194	0.610972	+	0.791652i	$0.290779\pi$
$434$	0	0
$435$	−7722.80	−0.851218
$436$	0	0
$437$	2445.28	0.267675
$438$	0	0
$439$	−10645.7	−1.15738	−0.578690	−	0.815548i	$-0.696436\pi$
$439$	−10645.7	−1.15738	−0.578690	+	0.815548i	$0.696436\pi$
$440$	0	0
$441$	−1120.32	−0.120972
$442$	0	0
$443$	−3031.48	−0.325124	−0.162562	−	0.986698i	$-0.551976\pi$
$443$	−3031.48	−0.325124	−0.162562	+	0.986698i	$0.551976\pi$
$444$	0	0
$445$	−2045.46	−0.217897
$446$	0	0
$447$	−10115.8	−1.07038
$448$	0	0
$449$	−2924.54	−0.307388	−0.153694	−	0.988118i	$-0.549117\pi$
$449$	−2924.54	−0.307388	−0.153694	+	0.988118i	$0.549117\pi$
$450$	0	0
$451$	2042.22	0.213225
$452$	0	0
$453$	−7024.59	−0.728574
$454$	0	0
$455$	21.5502	0.00222041
$456$	0	0
$457$	16680.1	1.70736	0.853681	−	0.520797i	$-0.174365\pi$
$457$	16680.1	1.70736	0.853681	+	0.520797i	$0.174365\pi$
$458$	0	0
$459$	1152.67	0.117216
$460$	0	0
$461$	10948.6	1.10613	0.553065	−	0.833138i	$-0.313458\pi$
$461$	10948.6	1.10613	0.553065	+	0.833138i	$0.313458\pi$
$462$	0	0
$463$	−9101.09	−0.913528	−0.456764	−	0.889588i	$-0.650992\pi$
$463$	−9101.09	−0.913528	−0.456764	+	0.889588i	$0.650992\pi$
$464$	0	0
$465$	3220.49	0.321176
$466$	0	0
$467$	−12728.2	−1.26122	−0.630609	−	0.776101i	$-0.717195\pi$
$467$	−12728.2	−1.26122	−0.630609	+	0.776101i	$0.717195\pi$
$468$	0	0
$469$	29.8477	0.00293868
$470$	0	0
$471$	7086.63	0.693280
$472$	0	0
$473$	−17872.8	−1.73740
$474$	0	0
$475$	−2657.92	−0.256744
$476$	0	0
$477$	1032.96	0.0991529
$478$	0	0
$479$	858.175	0.0818602	0.0409301	−	0.999162i	$-0.486968\pi$
$479$	858.175	0.0818602	0.0409301	+	0.999162i	$0.486968\pi$
$480$	0	0
$481$	8034.82	0.761656
$482$	0	0
$483$	−6.33336	−0.000596642 0
$484$	0	0
$485$	5618.11	0.525990
$486$	0	0
$487$	−1777.12	−0.165357	−0.0826784	−	0.996576i	$-0.526347\pi$
$487$	−1777.12	−0.165357	−0.0826784	+	0.996576i	$0.526347\pi$
$488$	0	0
$489$	−5479.15	−0.506699
$490$	0	0
$491$	−13458.2	−1.23699	−0.618495	−	0.785789i	$-0.712257\pi$
$491$	−13458.2	−1.23699	−0.618495	+	0.785789i	$0.712257\pi$
$492$	0	0
$493$	−2478.47	−0.226419
$494$	0	0
$495$	574.465	0.0521622
$496$	0	0
$497$	−34.6418	−0.00312656
$498$	0	0
$499$	−8470.02	−0.759860	−0.379930	−	0.925015i	$-0.624052\pi$
$499$	−8470.02	−0.759860	−0.379930	+	0.925015i	$0.624052\pi$
$500$	0	0
$501$	−8628.48	−0.769445
$502$	0	0
$503$	3558.11	0.315404	0.157702	−	0.987487i	$-0.449591\pi$
$503$	3558.11	0.315404	0.157702	+	0.987487i	$0.449591\pi$
$504$	0	0
$505$	4909.12	0.432580
$506$	0	0
$507$	−28706.3	−2.51458
$508$	0	0
$509$	−11039.9	−0.961370	−0.480685	−	0.876893i	$-0.659612\pi$
$509$	−11039.9	−0.961370	−0.480685	+	0.876893i	$0.659612\pi$
$510$	0	0
$511$	−42.1492	−0.00364887
$512$	0	0
$513$	−13881.8	−1.19473
$514$	0	0
$515$	−144.787	−0.0123885
$516$	0	0
$517$	−14348.6	−1.22060
$518$	0	0
$519$	−13637.8	−1.15344
$520$	0	0
$521$	5846.79	0.491656	0.245828	−	0.969314i	$-0.420940\pi$
$521$	5846.79	0.491656	0.245828	+	0.969314i	$0.420940\pi$
$522$	0	0
$523$	−17121.8	−1.43152	−0.715759	−	0.698348i	$-0.753919\pi$
$523$	−17121.8	−1.43152	−0.715759	+	0.698348i	$0.753919\pi$
$524$	0	0
$525$	6.88409	0.000572279 0
$526$	0	0
$527$	1033.55	0.0854310
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−422.983	−0.0345685
$532$	0	0
$533$	−4999.36	−0.406278
$534$	0	0
$535$	−939.369	−0.0759111
$536$	0	0
$537$	20581.0	1.65388
$538$	0	0
$539$	12065.2	0.964162
$540$	0	0
$541$	12373.9	0.983352	0.491676	−	0.870778i	$-0.336384\pi$
$541$	12373.9	0.983352	0.491676	+	0.870778i	$0.336384\pi$
$542$	0	0
$543$	21136.0	1.67041
$544$	0	0
$545$	−10710.1	−0.841781
$546$	0	0
$547$	16142.0	1.26176	0.630881	−	0.775880i	$-0.282694\pi$
$547$	16142.0	1.26176	0.630881	+	0.775880i	$0.282694\pi$
$548$	0	0
$549$	976.640	0.0759234
$550$	0	0
$551$	29848.8	2.30781
$552$	0	0
$553$	−58.1141	−0.00446883
$554$	0	0
$555$	2566.68	0.196306
$556$	0	0
$557$	8855.85	0.673671	0.336835	−	0.941564i	$-0.390643\pi$
$557$	8855.85	0.673671	0.336835	+	0.941564i	$0.390643\pi$
$558$	0	0
$559$	43752.6	3.31044
$560$	0	0
$561$	1708.36	0.128569
$562$	0	0
$563$	25526.5	1.91086	0.955428	−	0.295223i	$-0.0953939\pi$
$563$	25526.5	1.91086	0.955428	+	0.295223i	$0.0953939\pi$
$564$	0	0
$565$	8343.36	0.621253
$566$	0	0
$567$	40.3685	0.00298998
$568$	0	0
$569$	5758.98	0.424304	0.212152	−	0.977237i	$-0.431953\pi$
$569$	5758.98	0.424304	0.212152	+	0.977237i	$0.431953\pi$
$570$	0	0
$571$	14978.9	1.09781	0.548903	−	0.835886i	$-0.315045\pi$
$571$	14978.9	1.09781	0.548903	+	0.835886i	$0.315045\pi$
$572$	0	0
$573$	11537.3	0.841148
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	19218.9	1.38664	0.693322	−	0.720627i	$-0.256146\pi$
$577$	19218.9	1.38664	0.693322	+	0.720627i	$0.256146\pi$
$578$	0	0
$579$	−6957.45	−0.499381
$580$	0	0
$581$	56.1438	0.00400901
$582$	0	0
$583$	−11124.3	−0.790261
$584$	0	0
$585$	−1406.29	−0.0993897
$586$	0	0
$587$	4648.60	0.326863	0.163431	−	0.986555i	$-0.447744\pi$
$587$	4648.60	0.326863	0.163431	+	0.986555i	$0.447744\pi$
$588$	0	0
$589$	−12447.3	−0.870765
$590$	0	0
$591$	4676.97	0.325524
$592$	0	0
$593$	−22294.2	−1.54387	−0.771933	−	0.635703i	$-0.780710\pi$
$593$	−22294.2	−1.54387	−0.771933	+	0.635703i	$0.780710\pi$
$594$	0	0
$595$	2.20930	0.000152223 0
$596$	0	0
$597$	−1117.29	−0.0765957
$598$	0	0
$599$	−14695.1	−1.00238	−0.501191	−	0.865336i	$-0.667105\pi$
$599$	−14695.1	−1.00238	−0.501191	+	0.865336i	$0.667105\pi$
$600$	0	0
$601$	−4381.79	−0.297399	−0.148699	−	0.988882i	$-0.547509\pi$
$601$	−4381.79	−0.297399	−0.148699	+	0.988882i	$0.547509\pi$
$602$	0	0
$603$	−1947.76	−0.131541
$604$	0	0
$605$	468.365	0.0314739
$606$	0	0
$607$	−14325.9	−0.957943	−0.478971	−	0.877831i	$-0.658990\pi$
$607$	−14325.9	−0.957943	−0.478971	+	0.877831i	$0.658990\pi$
$608$	0	0
$609$	−77.3093	−0.00514406
$610$	0	0
$611$	35125.5	2.32574
$612$	0	0
$613$	12551.9	0.827028	0.413514	−	0.910498i	$-0.364301\pi$
$613$	12551.9	0.827028	0.413514	+	0.910498i	$0.364301\pi$
$614$	0	0
$615$	−1597.02	−0.104712
$616$	0	0
$617$	5556.60	0.362561	0.181281	−	0.983431i	$-0.441976\pi$
$617$	5556.60	0.362561	0.181281	+	0.983431i	$0.441976\pi$
$618$	0	0
$619$	−16504.3	−1.07167	−0.535834	−	0.844323i	$-0.680003\pi$
$619$	−16504.3	−1.07167	−0.535834	+	0.844323i	$0.680003\pi$
$620$	0	0
$621$	−3003.12	−0.194060
$622$	0	0
$623$	−20.4761	−0.00131679
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−20574.2	−1.31045
$628$	0	0
$629$	823.723	0.0522162
$630$	0	0
$631$	−20737.5	−1.30832	−0.654158	−	0.756358i	$-0.726977\pi$
$631$	−20737.5	−1.30832	−0.654158	+	0.756358i	$0.726977\pi$
$632$	0	0
$633$	30875.7	1.93870
$634$	0	0
$635$	−1771.63	−0.110716
$636$	0	0
$637$	−29535.5	−1.83711
$638$	0	0
$639$	2260.61	0.139950
$640$	0	0
$641$	−5138.54	−0.316630	−0.158315	−	0.987389i	$-0.550606\pi$
$641$	−5138.54	−0.316630	−0.158315	+	0.987389i	$0.550606\pi$
$642$	0	0
$643$	20288.1	1.24430	0.622149	−	0.782899i	$-0.286260\pi$
$643$	20288.1	1.24430	0.622149	+	0.782899i	$0.286260\pi$
$644$	0	0
$645$	13976.5	0.853218
$646$	0	0
$647$	22823.9	1.38686	0.693432	−	0.720522i	$-0.256098\pi$
$647$	22823.9	1.38686	0.693432	+	0.720522i	$0.256098\pi$
$648$	0	0
$649$	4555.26	0.275516
$650$	0	0
$651$	32.2388	0.00194092
$652$	0	0
$653$	27010.0	1.61866	0.809329	−	0.587356i	$-0.199831\pi$
$653$	27010.0	1.61866	0.809329	+	0.587356i	$0.199831\pi$
$654$	0	0
$655$	−4611.18	−0.275074
$656$	0	0
$657$	2750.52	0.163330
$658$	0	0
$659$	4630.23	0.273700	0.136850	−	0.990592i	$-0.456302\pi$
$659$	4630.23	0.273700	0.136850	+	0.990592i	$0.456302\pi$
$660$	0	0
$661$	−12644.2	−0.744028	−0.372014	−	0.928227i	$-0.621333\pi$
$661$	−12644.2	−0.744028	−0.372014	+	0.928227i	$0.621333\pi$
$662$	0	0
$663$	−4182.07	−0.244975
$664$	0	0
$665$	−26.6072	−0.00155155
$666$	0	0
$667$	6457.33	0.374856
$668$	0	0
$669$	−3069.32	−0.177380
$670$	0	0
$671$	−10517.8	−0.605120
$672$	0	0
$673$	−10099.8	−0.578482	−0.289241	−	0.957256i	$-0.593403\pi$
$673$	−10099.8	−0.578482	−0.289241	+	0.957256i	$0.593403\pi$
$674$	0	0
$675$	3264.27	0.186136
$676$	0	0
$677$	−10253.3	−0.582075	−0.291037	−	0.956712i	$-0.594000\pi$
$677$	−10253.3	−0.582075	−0.291037	+	0.956712i	$0.594000\pi$
$678$	0	0
$679$	56.2402	0.00317865
$680$	0	0
$681$	−3767.73	−0.212012
$682$	0	0
$683$	5525.56	0.309560	0.154780	−	0.987949i	$-0.450533\pi$
$683$	5525.56	0.309560	0.154780	+	0.987949i	$0.450533\pi$
$684$	0	0
$685$	8324.79	0.464341
$686$	0	0
$687$	−2032.71	−0.112886
$688$	0	0
$689$	27232.3	1.50576
$690$	0	0
$691$	−15754.4	−0.867332	−0.433666	−	0.901074i	$-0.642780\pi$
$691$	−15754.4	−0.867332	−0.433666	+	0.901074i	$0.642780\pi$
$692$	0	0
$693$	5.75070	0.000315225 0
$694$	0	0
$695$	12039.0	0.657072
$696$	0	0
$697$	−512.530	−0.0278529
$698$	0	0
$699$	28166.7	1.52412
$700$	0	0
$701$	10304.0	0.555176	0.277588	−	0.960700i	$-0.410465\pi$
$701$	10304.0	0.555176	0.277588	+	0.960700i	$0.410465\pi$
$702$	0	0
$703$	−9920.29	−0.532220
$704$	0	0
$705$	11220.7	0.599424
$706$	0	0
$707$	49.1429	0.00261415
$708$	0	0
$709$	−4916.67	−0.260436	−0.130218	−	0.991485i	$-0.541568\pi$
$709$	−4916.67	−0.260436	−0.130218	+	0.991485i	$0.541568\pi$
$710$	0	0
$711$	3792.33	0.200033
$712$	0	0
$713$	−2692.78	−0.141438
$714$	0	0
$715$	15144.9	0.792149
$716$	0	0
$717$	9160.11	0.477114
$718$	0	0
$719$	3323.08	0.172364	0.0861821	−	0.996279i	$-0.472533\pi$
$719$	3323.08	0.172364	0.0861821	+	0.996279i	$0.472533\pi$
$720$	0	0
$721$	−1.44940	−7.48660e−5 0
$722$	0	0
$723$	−29716.6	−1.52859
$724$	0	0
$725$	−7018.84	−0.359549
$726$	0	0
$727$	29941.1	1.52745	0.763724	−	0.645543i	$-0.223369\pi$
$727$	29941.1	1.52745	0.763724	+	0.645543i	$0.223369\pi$
$728$	0	0
$729$	16760.6	0.851529
$730$	0	0
$731$	4485.48	0.226951
$732$	0	0
$733$	−16395.5	−0.826170	−0.413085	−	0.910693i	$-0.635549\pi$
$733$	−16395.5	−0.826170	−0.413085	+	0.910693i	$0.635549\pi$
$734$	0	0
$735$	−9434.97	−0.473488
$736$	0	0
$737$	20976.2	1.04840
$738$	0	0
$739$	24881.9	1.23856	0.619279	−	0.785171i	$-0.287425\pi$
$739$	24881.9	1.23856	0.619279	+	0.785171i	$0.287425\pi$
$740$	0	0
$741$	50365.6	2.49693
$742$	0	0
$743$	8845.30	0.436746	0.218373	−	0.975865i	$-0.429925\pi$
$743$	8845.30	0.436746	0.218373	+	0.975865i	$0.429925\pi$
$744$	0	0
$745$	−9193.70	−0.452122
$746$	0	0
$747$	−3663.75	−0.179451
$748$	0	0
$749$	−9.40358	−0.000458744 0
$750$	0	0
$751$	−33958.9	−1.65004	−0.825020	−	0.565104i	$-0.808836\pi$
$751$	−33958.9	−1.65004	−0.825020	+	0.565104i	$0.808836\pi$
$752$	0	0
$753$	23361.2	1.13058
$754$	0	0
$755$	−6384.28	−0.307745
$756$	0	0
$757$	−22718.8	−1.09079	−0.545395	−	0.838179i	$-0.683621\pi$
$757$	−22718.8	−1.09079	−0.545395	+	0.838179i	$0.683621\pi$
$758$	0	0
$759$	−4450.92	−0.212856
$760$	0	0
$761$	21813.2	1.03906	0.519532	−	0.854451i	$-0.326106\pi$
$761$	21813.2	1.03906	0.519532	+	0.854451i	$0.326106\pi$
$762$	0	0
$763$	−107.214	−0.00508703
$764$	0	0
$765$	−144.172	−0.00681378
$766$	0	0
$767$	−11151.3	−0.524966
$768$	0	0
$769$	−15098.4	−0.708011	−0.354006	−	0.935243i	$-0.615181\pi$
$769$	−15098.4	−0.708011	−0.354006	+	0.935243i	$0.615181\pi$
$770$	0	0
$771$	30728.8	1.43537
$772$	0	0
$773$	4343.28	0.202092	0.101046	−	0.994882i	$-0.467781\pi$
$773$	4343.28	0.202092	0.101046	+	0.994882i	$0.467781\pi$
$774$	0	0
$775$	2926.93	0.135663
$776$	0	0
$777$	25.6938	0.00118631
$778$	0	0
$779$	6172.52	0.283894
$780$	0	0
$781$	−24345.4	−1.11542
$782$	0	0
$783$	−36658.2	−1.67312
$784$	0	0
$785$	6440.66	0.292837
$786$	0	0
$787$	−4399.04	−0.199249	−0.0996243	−	0.995025i	$-0.531764\pi$
$787$	−4399.04	−0.199249	−0.0996243	+	0.995025i	$0.531764\pi$
$788$	0	0
$789$	−11723.7	−0.528992
$790$	0	0
$791$	83.5215	0.00375434
$792$	0	0
$793$	25747.6	1.15299
$794$	0	0
$795$	8699.23	0.388088
$796$	0	0
$797$	16911.6	0.751619	0.375809	−	0.926697i	$-0.377365\pi$
$797$	16911.6	0.751619	0.375809	+	0.926697i	$0.377365\pi$
$798$	0	0
$799$	3601.03	0.159444
$800$	0	0
$801$	1336.20	0.0589418
$802$	0	0
$803$	−29621.3	−1.30176
$804$	0	0
$805$	−5.75605	−0.000252018 0
$806$	0	0
$807$	44030.1	1.92061
$808$	0	0
$809$	−21310.5	−0.926128	−0.463064	−	0.886325i	$-0.653250\pi$
$809$	−21310.5	−0.926128	−0.463064	+	0.886325i	$0.653250\pi$
$810$	0	0
$811$	5999.06	0.259748	0.129874	−	0.991531i	$-0.458543\pi$
$811$	5999.06	0.259748	0.129874	+	0.991531i	$0.458543\pi$
$812$	0	0
$813$	7423.32	0.320230
$814$	0	0
$815$	−4979.71	−0.214026
$816$	0	0
$817$	−54019.6	−2.31323
$818$	0	0
$819$	−14.0777	−0.000600629 0
$820$	0	0
$821$	−34257.7	−1.45628	−0.728138	−	0.685431i	$-0.759614\pi$
$821$	−34257.7	−1.45628	−0.728138	+	0.685431i	$0.759614\pi$
$822$	0	0
$823$	46224.1	1.95780	0.978902	−	0.204331i	$-0.0655020\pi$
$823$	46224.1	1.95780	0.978902	+	0.204331i	$0.0655020\pi$
$824$	0	0
$825$	4837.95	0.204165
$826$	0	0
$827$	−37548.2	−1.57881	−0.789407	−	0.613870i	$-0.789612\pi$
$827$	−37548.2	−1.57881	−0.789407	+	0.613870i	$0.789612\pi$
$828$	0	0
$829$	−30920.8	−1.29544	−0.647721	−	0.761877i	$-0.724278\pi$
$829$	−30920.8	−1.29544	−0.647721	+	0.761877i	$0.724278\pi$
$830$	0	0
$831$	20700.7	0.864140
$832$	0	0
$833$	−3027.96	−0.125945
$834$	0	0
$835$	−7841.97	−0.325009
$836$	0	0
$837$	15286.9	0.631291
$838$	0	0
$839$	−4014.99	−0.165212	−0.0826060	−	0.996582i	$-0.526324\pi$
$839$	−4014.99	−0.165212	−0.0826060	+	0.996582i	$0.526324\pi$
$840$	0	0
$841$	54433.6	2.23189
$842$	0	0
$843$	−15873.7	−0.648542
$844$	0	0
$845$	−26089.6	−1.06214
$846$	0	0
$847$	4.68858	0.000190202 0
$848$	0	0
$849$	26457.2	1.06950
$850$	0	0
$851$	−2146.10	−0.0864483
$852$	0	0
$853$	−4140.42	−0.166196	−0.0830981	−	0.996541i	$-0.526481\pi$
$853$	−4140.42	−0.166196	−0.0830981	+	0.996541i	$0.526481\pi$
$854$	0	0
$855$	1736.29	0.0694502
$856$	0	0
$857$	−21402.7	−0.853093	−0.426547	−	0.904466i	$-0.640270\pi$
$857$	−21402.7	−0.853093	−0.426547	+	0.904466i	$0.640270\pi$
$858$	0	0
$859$	−32876.0	−1.30584	−0.652919	−	0.757427i	$-0.726456\pi$
$859$	−32876.0	−1.30584	−0.652919	+	0.757427i	$0.726456\pi$
$860$	0	0
$861$	−15.9870	−0.000632794 0
$862$	0	0
$863$	−29257.0	−1.15402	−0.577010	−	0.816737i	$-0.695781\pi$
$863$	−29257.0	−1.15402	−0.577010	+	0.816737i	$0.695781\pi$
$864$	0	0
$865$	−12394.7	−0.487204
$866$	0	0
$867$	26600.0	1.04197
$868$	0	0
$869$	−40841.0	−1.59429
$870$	0	0
$871$	−51349.6	−1.99761
$872$	0	0
$873$	−3670.05	−0.142282
$874$	0	0
$875$	6.25658	0.000241727 0
$876$	0	0
$877$	−7545.57	−0.290531	−0.145266	−	0.989393i	$-0.546404\pi$
$877$	−7545.57	−0.290531	−0.145266	+	0.989393i	$0.546404\pi$
$878$	0	0
$879$	29502.7	1.13208
$880$	0	0
$881$	19544.5	0.747414	0.373707	−	0.927547i	$-0.378087\pi$
$881$	19544.5	0.747414	0.373707	+	0.927547i	$0.378087\pi$
$882$	0	0
$883$	−28251.9	−1.07673	−0.538365	−	0.842712i	$-0.680958\pi$
$883$	−28251.9	−1.07673	−0.538365	+	0.842712i	$0.680958\pi$
$884$	0	0
$885$	−3562.22	−0.135302
$886$	0	0
$887$	−8880.90	−0.336180	−0.168090	−	0.985772i	$-0.553760\pi$
$887$	−8880.90	−0.336180	−0.168090	+	0.985772i	$0.553760\pi$
$888$	0	0
$889$	−17.7349	−0.000669079 0
$890$	0	0
$891$	28369.9	1.06670
$892$	0	0
$893$	−43368.1	−1.62515
$894$	0	0
$895$	18705.0	0.698590
$896$	0	0
$897$	10895.8	0.405576
$898$	0	0
$899$	−32869.9	−1.21943
$900$	0	0
$901$	2791.84	0.103229
$902$	0	0
$903$	139.913	0.00515614
$904$	0	0
$905$	19209.4	0.705570
$906$	0	0
$907$	−15437.3	−0.565147	−0.282574	−	0.959246i	$-0.591188\pi$
$907$	−15437.3	−0.565147	−0.282574	+	0.959246i	$0.591188\pi$
$908$	0	0
$909$	−3206.90	−0.117014
$910$	0	0
$911$	−33155.3	−1.20580	−0.602900	−	0.797817i	$-0.705988\pi$
$911$	−33155.3	−1.20580	−0.602900	+	0.797817i	$0.705988\pi$
$912$	0	0
$913$	39456.3	1.43025
$914$	0	0
$915$	8224.93	0.297167
$916$	0	0
$917$	−46.1604	−0.00166232
$918$	0	0
$919$	12375.1	0.444196	0.222098	−	0.975024i	$-0.428709\pi$
$919$	12375.1	0.444196	0.222098	+	0.975024i	$0.428709\pi$
$920$	0	0
$921$	35070.5	1.25474
$922$	0	0
$923$	59597.4	2.12532
$924$	0	0
$925$	2332.72	0.0829183
$926$	0	0
$927$	94.5828	0.00335114
$928$	0	0
$929$	49693.8	1.75501	0.877504	−	0.479570i	$-0.159207\pi$
$929$	49693.8	1.75501	0.877504	+	0.479570i	$0.159207\pi$
$930$	0	0
$931$	36466.4	1.28371
$932$	0	0
$933$	205.095	0.00719668
$934$	0	0
$935$	1552.64	0.0543067
$936$	0	0
$937$	−42717.9	−1.48936	−0.744682	−	0.667420i	$-0.767399\pi$
$937$	−42717.9	−1.48936	−0.744682	+	0.667420i	$0.767399\pi$
$938$	0	0
$939$	20837.8	0.724190
$940$	0	0
$941$	20273.7	0.702344	0.351172	−	0.936311i	$-0.385783\pi$
$941$	20273.7	0.702344	0.351172	+	0.936311i	$0.385783\pi$
$942$	0	0
$943$	1335.33	0.0461128
$944$	0	0
$945$	32.6770	0.00112485
$946$	0	0
$947$	−20905.2	−0.717347	−0.358674	−	0.933463i	$-0.616771\pi$
$947$	−20905.2	−0.717347	−0.358674	+	0.933463i	$0.616771\pi$
$948$	0	0
$949$	72513.0	2.48037
$950$	0	0
$951$	30991.8	1.05676
$952$	0	0
$953$	39537.9	1.34392	0.671960	−	0.740587i	$-0.265452\pi$
$953$	39537.9	1.34392	0.671960	+	0.740587i	$0.265452\pi$
$954$	0	0
$955$	10485.6	0.355296
$956$	0	0
$957$	−54330.9	−1.83518
$958$	0	0
$959$	83.3355	0.00280609
$960$	0	0
$961$	−16083.9	−0.539892
$962$	0	0
$963$	613.646	0.0205342
$964$	0	0
$965$	−6323.26	−0.210936
$966$	0	0
$967$	−26356.2	−0.876482	−0.438241	−	0.898858i	$-0.644398\pi$
$967$	−26356.2	−0.876482	−0.438241	+	0.898858i	$0.644398\pi$
$968$	0	0
$969$	5163.44	0.171180
$970$	0	0
$971$	−11885.8	−0.392827	−0.196413	−	0.980521i	$-0.562929\pi$
$971$	−11885.8	−0.392827	−0.196413	+	0.980521i	$0.562929\pi$
$972$	0	0
$973$	120.517	0.00397080
$974$	0	0
$975$	−11843.3	−0.389015
$976$	0	0
$977$	−11798.1	−0.386341	−0.193171	−	0.981165i	$-0.561877\pi$
$977$	−11798.1	−0.386341	−0.193171	+	0.981165i	$0.561877\pi$
$978$	0	0
$979$	−14390.1	−0.469774
$980$	0	0
$981$	6996.41	0.227705
$982$	0	0
$983$	5991.38	0.194400	0.0972001	−	0.995265i	$-0.469011\pi$
$983$	5991.38	0.194400	0.0972001	+	0.995265i	$0.469011\pi$
$984$	0	0
$985$	4250.65	0.137499
$986$	0	0
$987$	112.325	0.00362242
$988$	0	0
$989$	−11686.3	−0.375737
$990$	0	0
$991$	18597.3	0.596129	0.298065	−	0.954546i	$-0.403659\pi$
$991$	18597.3	0.596129	0.298065	+	0.954546i	$0.403659\pi$
$992$	0	0
$993$	25124.8	0.802931
$994$	0	0
$995$	−1015.44	−0.0323535
$996$	0	0
$997$	21172.0	0.672541	0.336271	−	0.941765i	$-0.390834\pi$
$997$	21172.0	0.672541	0.336271	+	0.941765i	$0.390834\pi$
$998$	0	0
$999$	12183.4	0.385851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.l.1.2		4
4.3	odd	2		230.4.a.i.1.3	✓	4
12.11	even	2		2070.4.a.bi.1.2		4
20.3	even	4		1150.4.b.m.599.3		8
20.7	even	4		1150.4.b.m.599.6		8
20.19	odd	2		1150.4.a.o.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.3	✓	4	4.3	odd	2
1150.4.a.o.1.2		4	20.19	odd	2
1150.4.b.m.599.3		8	20.3	even	4
1150.4.b.m.599.6		8	20.7	even	4
1840.4.a.l.1.2		4	1.1	even	1	trivial
2070.4.a.bi.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-5.50148 q^{3} -5.00000 q^{5} -0.0500526 q^{7} +3.26627 q^{9} +O(q^{10})\)	\(q-5.50148 q^{3} -5.00000 q^{5} -0.0500526 q^{7} +3.26627 q^{9} -35.1757 q^{11} +86.1100 q^{13} +27.5074 q^{15} +8.82793 q^{17} -106.317 q^{19} +0.275364 q^{21} -23.0000 q^{23} +25.0000 q^{25} +130.571 q^{27} -280.754 q^{29} +117.077 q^{31} +193.518 q^{33} +0.250263 q^{35} +93.3088 q^{37} -473.732 q^{39} -58.0579 q^{41} +508.101 q^{43} -16.3313 q^{45} +407.914 q^{47} -342.997 q^{49} -48.5666 q^{51} +316.251 q^{53} +175.878 q^{55} +584.899 q^{57} -129.500 q^{59} +299.008 q^{61} -0.163485 q^{63} -430.550 q^{65} -596.326 q^{67} +126.534 q^{69} +692.108 q^{71} +842.098 q^{73} -137.537 q^{75} +1.76063 q^{77} +1161.06 q^{79} -806.521 q^{81} -1121.69 q^{83} -44.1396 q^{85} +1544.56 q^{87} +409.092 q^{89} -4.31003 q^{91} -644.098 q^{93} +531.583 q^{95} -1123.62 q^{97} -114.893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9	\( 4 q - 4 q^{3} - 20 q^{5} - 26 q^{7} + 64 q^{9} - 93 q^{11} + 32 q^{13} + 20 q^{15} + 108 q^{17} - 185 q^{19} + 302 q^{21} - 92 q^{23} + 100 q^{25} - 259 q^{27} + 294 q^{29} + 211 q^{31} - 237 q^{33} + 130 q^{35} + 5 q^{37} + 421 q^{39} - 369 q^{41} + 100 q^{43} - 320 q^{45} + 363 q^{47} + 600 q^{49} + 315 q^{51} + 21 q^{53} + 465 q^{55} - 160 q^{57} + 33 q^{59} - 307 q^{61} - 167 q^{63} - 160 q^{65} - 725 q^{67} + 92 q^{69} + 1257 q^{71} + 509 q^{73} - 100 q^{75} - 1962 q^{77} - 1202 q^{79} - 992 q^{81} + 1377 q^{83} - 540 q^{85} + 2829 q^{87} - 984 q^{89} + 995 q^{91} - 1843 q^{93} + 925 q^{95} + 137 q^{97} - 2013 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 20 * q^5 - 26 * q^7 + 64 * q^9 - 93 * q^11 + 32 * q^13 + 20 * q^15 + 108 * q^17 - 185 * q^19 + 302 * q^21 - 92 * q^23 + 100 * q^25 - 259 * q^27 + 294 * q^29 + 211 * q^31 - 237 * q^33 + 130 * q^35 + 5 * q^37 + 421 * q^39 - 369 * q^41 + 100 * q^43 - 320 * q^45 + 363 * q^47 + 600 * q^49 + 315 * q^51 + 21 * q^53 + 465 * q^55 - 160 * q^57 + 33 * q^59 - 307 * q^61 - 167 * q^63 - 160 * q^65 - 725 * q^67 + 92 * q^69 + 1257 * q^71 + 509 * q^73 - 100 * q^75 - 1962 * q^77 - 1202 * q^79 - 992 * q^81 + 1377 * q^83 - 540 * q^85 + 2829 * q^87 - 984 * q^89 + 995 * q^91 - 1843 * q^93 + 925 * q^95 + 137 * q^97 - 2013 * q^99

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