LMFDB - Embedded newform 1840.4.a.z.1.6

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6502x^{5} - 4928x^{4} - 87343x^{3} + 42737x^{2} + 286800x + 53104 $ x^9 - 3x^8 - 148x^7 + 278x^6 + 6502x^5 - 4928x^4 - 87343x^3 + 42737x^2 + 286800x + 53104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.99459$ 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)

$f(q)$	$=$	$q+2.99459 q^{3} -5.00000 q^{5} +22.8972 q^{7} -18.0324 q^{9} +O(q^{10})$	$q+2.99459 q^{3} -5.00000 q^{5} +22.8972 q^{7} -18.0324 q^{9} -26.8752 q^{11} -29.4243 q^{13} -14.9730 q^{15} -40.2791 q^{17} +155.515 q^{19} +68.5678 q^{21} +23.0000 q^{23} +25.0000 q^{25} -134.854 q^{27} -101.685 q^{29} +167.152 q^{31} -80.4802 q^{33} -114.486 q^{35} +87.1326 q^{37} -88.1137 q^{39} -57.3785 q^{41} +434.915 q^{43} +90.1621 q^{45} -305.393 q^{47} +181.281 q^{49} -120.620 q^{51} -722.286 q^{53} +134.376 q^{55} +465.704 q^{57} -592.779 q^{59} +319.629 q^{61} -412.892 q^{63} +147.121 q^{65} -37.7775 q^{67} +68.8756 q^{69} -958.725 q^{71} -514.025 q^{73} +74.8648 q^{75} -615.366 q^{77} -373.423 q^{79} +83.0432 q^{81} -1065.93 q^{83} +201.396 q^{85} -304.506 q^{87} +806.921 q^{89} -673.734 q^{91} +500.554 q^{93} -777.576 q^{95} -1298.95 q^{97} +484.624 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9	$ 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9} - 22 q^{11} + 23 q^{13} - 15 q^{15} - 135 q^{17} - 102 q^{19} + 54 q^{21} + 207 q^{23} + 225 q^{25} + 363 q^{27} + 280 q^{29} - 168 q^{31} + 28 q^{33} + 125 q^{35} + 153 q^{37} + 5 q^{39} - 502 q^{41} - 110 q^{43} - 310 q^{45} + 153 q^{47} + 764 q^{49} - 924 q^{51} - 273 q^{53} + 110 q^{55} + 748 q^{57} - 827 q^{59} + 1976 q^{61} - 2237 q^{63} - 115 q^{65} - 1613 q^{67} + 69 q^{69} - 1370 q^{71} + 425 q^{73} + 75 q^{75} + 2006 q^{77} - 2624 q^{79} + 1729 q^{81} - 2505 q^{83} + 675 q^{85} - 1591 q^{87} + 1120 q^{89} - 2392 q^{91} + 4401 q^{93} + 510 q^{95} + 2026 q^{97} - 3206 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9 - 22 * q^11 + 23 * q^13 - 15 * q^15 - 135 * q^17 - 102 * q^19 + 54 * q^21 + 207 * q^23 + 225 * q^25 + 363 * q^27 + 280 * q^29 - 168 * q^31 + 28 * q^33 + 125 * q^35 + 153 * q^37 + 5 * q^39 - 502 * q^41 - 110 * q^43 - 310 * q^45 + 153 * q^47 + 764 * q^49 - 924 * q^51 - 273 * q^53 + 110 * q^55 + 748 * q^57 - 827 * q^59 + 1976 * q^61 - 2237 * q^63 - 115 * q^65 - 1613 * q^67 + 69 * q^69 - 1370 * q^71 + 425 * q^73 + 75 * q^75 + 2006 * q^77 - 2624 * q^79 + 1729 * q^81 - 2505 * q^83 + 675 * q^85 - 1591 * q^87 + 1120 * q^89 - 2392 * q^91 + 4401 * q^93 + 510 * q^95 + 2026 * q^97 - 3206 * 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
$3$	2.99459	0.576310	0.288155	−	0.957584i	$-0.406958\pi$
$3$	2.99459	0.576310	0.288155	+	0.957584i	$0.406958\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	22.8972	1.23633	0.618166	−	0.786047i	$-0.287876\pi$
$7$	22.8972	1.23633	0.618166	+	0.786047i	$0.287876\pi$
$8$	0	0
$9$	−18.0324	−0.667867
$10$	0	0
$11$	−26.8752	−0.736651	−0.368326	−	0.929697i	$-0.620069\pi$
$11$	−26.8752	−0.736651	−0.368326	+	0.929697i	$0.620069\pi$
$12$	0	0
$13$	−29.4243	−0.627756	−0.313878	−	0.949463i	$-0.601628\pi$
$13$	−29.4243	−0.627756	−0.313878	+	0.949463i	$0.601628\pi$
$14$	0	0
$15$	−14.9730	−0.257733
$16$	0	0
$17$	−40.2791	−0.574655	−0.287327	−	0.957832i	$-0.592767\pi$
$17$	−40.2791	−0.574655	−0.287327	+	0.957832i	$0.592767\pi$
$18$	0	0
$19$	155.515	1.87777	0.938885	−	0.344232i	$-0.111861\pi$
$19$	155.515	1.87777	0.938885	+	0.344232i	$0.111861\pi$
$20$	0	0
$21$	68.5678	0.712510
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−134.854	−0.961208
$28$	0	0
$29$	−101.685	−0.651120	−0.325560	−	0.945521i	$-0.605553\pi$
$29$	−101.685	−0.651120	−0.325560	+	0.945521i	$0.605553\pi$
$30$	0	0
$31$	167.152	0.968434	0.484217	−	0.874948i	$-0.339104\pi$
$31$	167.152	0.968434	0.484217	+	0.874948i	$0.339104\pi$
$32$	0	0
$33$	−80.4802	−0.424539
$34$	0	0
$35$	−114.486	−0.552905
$36$	0	0
$37$	87.1326	0.387149	0.193575	−	0.981086i	$-0.437992\pi$
$37$	87.1326	0.387149	0.193575	+	0.981086i	$0.437992\pi$
$38$	0	0
$39$	−88.1137	−0.361782
$40$	0	0
$41$	−57.3785	−0.218561	−0.109281	−	0.994011i	$-0.534855\pi$
$41$	−57.3785	−0.218561	−0.109281	+	0.994011i	$0.534855\pi$
$42$	0	0
$43$	434.915	1.54242	0.771209	−	0.636583i	$-0.219653\pi$
$43$	434.915	1.54242	0.771209	+	0.636583i	$0.219653\pi$
$44$	0	0
$45$	90.1621	0.298679
$46$	0	0
$47$	−305.393	−0.947792	−0.473896	−	0.880581i	$-0.657153\pi$
$47$	−305.393	−0.947792	−0.473896	+	0.880581i	$0.657153\pi$
$48$	0	0
$49$	181.281	0.528517
$50$	0	0
$51$	−120.620	−0.331179
$52$	0	0
$53$	−722.286	−1.87195	−0.935977	−	0.352061i	$-0.885481\pi$
$53$	−722.286	−1.87195	−0.935977	+	0.352061i	$0.885481\pi$
$54$	0	0
$55$	134.376	0.329441
$56$	0	0
$57$	465.704	1.08218
$58$	0	0
$59$	−592.779	−1.30802	−0.654010	−	0.756486i	$-0.726915\pi$
$59$	−592.779	−1.30802	−0.654010	+	0.756486i	$0.726915\pi$
$60$	0	0
$61$	319.629	0.670890	0.335445	−	0.942060i	$-0.391113\pi$
$61$	319.629	0.670890	0.335445	+	0.942060i	$0.391113\pi$
$62$	0	0
$63$	−412.892	−0.825706
$64$	0	0
$65$	147.121	0.280741
$66$	0	0
$67$	−37.7775	−0.0688843	−0.0344422	−	0.999407i	$-0.510965\pi$
$67$	−37.7775	−0.0688843	−0.0344422	+	0.999407i	$0.510965\pi$
$68$	0	0
$69$	68.8756	0.120169
$70$	0	0
$71$	−958.725	−1.60253	−0.801266	−	0.598309i	$-0.795840\pi$
$71$	−958.725	−1.60253	−0.801266	+	0.598309i	$0.795840\pi$
$72$	0	0
$73$	−514.025	−0.824138	−0.412069	−	0.911153i	$-0.635194\pi$
$73$	−514.025	−0.824138	−0.412069	+	0.911153i	$0.635194\pi$
$74$	0	0
$75$	74.8648	0.115262
$76$	0	0
$77$	−615.366	−0.910746
$78$	0	0
$79$	−373.423	−0.531814	−0.265907	−	0.963999i	$-0.585671\pi$
$79$	−373.423	−0.531814	−0.265907	+	0.963999i	$0.585671\pi$
$80$	0	0
$81$	83.0432	0.113914
$82$	0	0
$83$	−1065.93	−1.40966	−0.704828	−	0.709378i	$-0.748976\pi$
$83$	−1065.93	−1.40966	−0.704828	+	0.709378i	$0.748976\pi$
$84$	0	0
$85$	201.396	0.256993
$86$	0	0
$87$	−304.506	−0.375247
$88$	0	0
$89$	806.921	0.961050	0.480525	−	0.876981i	$-0.340446\pi$
$89$	806.921	0.961050	0.480525	+	0.876981i	$0.340446\pi$
$90$	0	0
$91$	−673.734	−0.776115
$92$	0	0
$93$	500.554	0.558118
$94$	0	0
$95$	−777.576	−0.839764
$96$	0	0
$97$	−1298.95	−1.35967	−0.679835	−	0.733365i	$-0.737949\pi$
$97$	−1298.95	−1.35967	−0.679835	+	0.733365i	$0.737949\pi$
$98$	0	0
$99$	484.624	0.491985
$100$	0	0
$101$	−232.434	−0.228991	−0.114495	−	0.993424i	$-0.536525\pi$
$101$	−232.434	−0.228991	−0.114495	+	0.993424i	$0.536525\pi$
$102$	0	0
$103$	1474.07	1.41014	0.705070	−	0.709138i	$-0.250915\pi$
$103$	1474.07	1.41014	0.705070	+	0.709138i	$0.250915\pi$
$104$	0	0
$105$	−342.839	−0.318644
$106$	0	0
$107$	289.033	0.261139	0.130570	−	0.991439i	$-0.458319\pi$
$107$	289.033	0.261139	0.130570	+	0.991439i	$0.458319\pi$
$108$	0	0
$109$	459.550	0.403825	0.201912	−	0.979404i	$-0.435284\pi$
$109$	459.550	0.403825	0.201912	+	0.979404i	$0.435284\pi$
$110$	0	0
$111$	260.927	0.223118
$112$	0	0
$113$	−555.656	−0.462582	−0.231291	−	0.972885i	$-0.574295\pi$
$113$	−555.656	−0.462582	−0.231291	+	0.972885i	$0.574295\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	530.591	0.419258
$118$	0	0
$119$	−922.279	−0.710464
$120$	0	0
$121$	−608.726	−0.457345
$122$	0	0
$123$	−171.825	−0.125959
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−624.068	−0.436040	−0.218020	−	0.975944i	$-0.569960\pi$
$127$	−624.068	−0.436040	−0.218020	+	0.975944i	$0.569960\pi$
$128$	0	0
$129$	1302.39	0.888910
$130$	0	0
$131$	128.057	0.0854074	0.0427037	−	0.999088i	$-0.486403\pi$
$131$	128.057	0.0854074	0.0427037	+	0.999088i	$0.486403\pi$
$132$	0	0
$133$	3560.86	2.32155
$134$	0	0
$135$	674.269	0.429865
$136$	0	0
$137$	2189.93	1.36568	0.682840	−	0.730568i	$-0.260745\pi$
$137$	2189.93	1.36568	0.682840	+	0.730568i	$0.260745\pi$
$138$	0	0
$139$	235.986	0.144000	0.0720002	−	0.997405i	$-0.477062\pi$
$139$	235.986	0.144000	0.0720002	+	0.997405i	$0.477062\pi$
$140$	0	0
$141$	−914.529	−0.546222
$142$	0	0
$143$	790.782	0.462437
$144$	0	0
$145$	508.426	0.291190
$146$	0	0
$147$	542.864	0.304590
$148$	0	0
$149$	−108.600	−0.0597104	−0.0298552	−	0.999554i	$-0.509505\pi$
$149$	−108.600	−0.0597104	−0.0298552	+	0.999554i	$0.509505\pi$
$150$	0	0
$151$	−3587.95	−1.93366	−0.966831	−	0.255416i	$-0.917788\pi$
$151$	−3587.95	−1.93366	−0.966831	+	0.255416i	$0.917788\pi$
$152$	0	0
$153$	726.330	0.383793
$154$	0	0
$155$	−835.762	−0.433097
$156$	0	0
$157$	−923.521	−0.469459	−0.234729	−	0.972061i	$-0.575420\pi$
$157$	−923.521	−0.469459	−0.234729	+	0.972061i	$0.575420\pi$
$158$	0	0
$159$	−2162.95	−1.07883
$160$	0	0
$161$	526.635	0.257793
$162$	0	0
$163$	659.640	0.316975	0.158488	−	0.987361i	$-0.449338\pi$
$163$	659.640	0.316975	0.158488	+	0.987361i	$0.449338\pi$
$164$	0	0
$165$	402.401	0.189860
$166$	0	0
$167$	−3847.47	−1.78279	−0.891397	−	0.453224i	$-0.850274\pi$
$167$	−3847.47	−1.78279	−0.891397	+	0.453224i	$0.850274\pi$
$168$	0	0
$169$	−1331.21	−0.605922
$170$	0	0
$171$	−2804.31	−1.25410
$172$	0	0
$173$	1172.41	0.515241	0.257620	−	0.966246i	$-0.417062\pi$
$173$	1172.41	0.515241	0.257620	+	0.966246i	$0.417062\pi$
$174$	0	0
$175$	572.430	0.247266
$176$	0	0
$177$	−1775.13	−0.753825
$178$	0	0
$179$	−988.750	−0.412864	−0.206432	−	0.978461i	$-0.566185\pi$
$179$	−988.750	−0.412864	−0.206432	+	0.978461i	$0.566185\pi$
$180$	0	0
$181$	−319.537	−0.131221	−0.0656105	−	0.997845i	$-0.520899\pi$
$181$	−319.537	−0.131221	−0.0656105	+	0.997845i	$0.520899\pi$
$182$	0	0
$183$	957.158	0.386640
$184$	0	0
$185$	−435.663	−0.173138
$186$	0	0
$187$	1082.51	0.423320
$188$	0	0
$189$	−3087.77	−1.18837
$190$	0	0
$191$	−812.796	−0.307916	−0.153958	−	0.988077i	$-0.549202\pi$
$191$	−812.796	−0.307916	−0.153958	+	0.988077i	$0.549202\pi$
$192$	0	0
$193$	−1670.32	−0.622967	−0.311483	−	0.950252i	$-0.600826\pi$
$193$	−1670.32	−0.622967	−0.311483	+	0.950252i	$0.600826\pi$
$194$	0	0
$195$	440.569	0.161794
$196$	0	0
$197$	1097.29	0.396845	0.198422	−	0.980117i	$-0.436418\pi$
$197$	1097.29	0.396845	0.198422	+	0.980117i	$0.436418\pi$
$198$	0	0
$199$	−3494.02	−1.24465	−0.622323	−	0.782761i	$-0.713811\pi$
$199$	−3494.02	−1.24465	−0.622323	+	0.782761i	$0.713811\pi$
$200$	0	0
$201$	−113.128	−0.0396987
$202$	0	0
$203$	−2328.31	−0.805001
$204$	0	0
$205$	286.892	0.0977436
$206$	0	0
$207$	−414.746	−0.139260
$208$	0	0
$209$	−4179.49	−1.38326
$210$	0	0
$211$	−1427.79	−0.465844	−0.232922	−	0.972495i	$-0.574829\pi$
$211$	−1427.79	−0.465844	−0.232922	+	0.972495i	$0.574829\pi$
$212$	0	0
$213$	−2870.99	−0.923554
$214$	0	0
$215$	−2174.58	−0.689790
$216$	0	0
$217$	3827.32	1.19731
$218$	0	0
$219$	−1539.30	−0.474959
$220$	0	0
$221$	1185.19	0.360743
$222$	0	0
$223$	−1566.40	−0.470377	−0.235189	−	0.971950i	$-0.575571\pi$
$223$	−1566.40	−0.470377	−0.235189	+	0.971950i	$0.575571\pi$
$224$	0	0
$225$	−450.810	−0.133573
$226$	0	0
$227$	4183.56	1.22323	0.611614	−	0.791156i	$-0.290521\pi$
$227$	4183.56	1.22323	0.611614	+	0.791156i	$0.290521\pi$
$228$	0	0
$229$	2308.76	0.666232	0.333116	−	0.942886i	$-0.391900\pi$
$229$	2308.76	0.666232	0.333116	+	0.942886i	$0.391900\pi$
$230$	0	0
$231$	−1842.77	−0.524872
$232$	0	0
$233$	1926.16	0.541575	0.270788	−	0.962639i	$-0.412716\pi$
$233$	1926.16	0.541575	0.270788	+	0.962639i	$0.412716\pi$
$234$	0	0
$235$	1526.97	0.423865
$236$	0	0
$237$	−1118.25	−0.306490
$238$	0	0
$239$	−1547.27	−0.418765	−0.209382	−	0.977834i	$-0.567145\pi$
$239$	−1547.27	−0.418765	−0.209382	+	0.977834i	$0.567145\pi$
$240$	0	0
$241$	−1106.94	−0.295870	−0.147935	−	0.988997i	$-0.547263\pi$
$241$	−1106.94	−0.295870	−0.147935	+	0.988997i	$0.547263\pi$
$242$	0	0
$243$	3889.73	1.02686
$244$	0	0
$245$	−906.407	−0.236360
$246$	0	0
$247$	−4575.92	−1.17878
$248$	0	0
$249$	−3192.04	−0.812399
$250$	0	0
$251$	−6333.81	−1.59278	−0.796388	−	0.604786i	$-0.793259\pi$
$251$	−6333.81	−1.59278	−0.796388	+	0.604786i	$0.793259\pi$
$252$	0	0
$253$	−618.129	−0.153602
$254$	0	0
$255$	603.098	0.148108
$256$	0	0
$257$	−7923.13	−1.92308	−0.961539	−	0.274669i	$-0.911432\pi$
$257$	−7923.13	−1.92308	−0.961539	+	0.274669i	$0.911432\pi$
$258$	0	0
$259$	1995.09	0.478645
$260$	0	0
$261$	1833.63	0.434862
$262$	0	0
$263$	7519.51	1.76301	0.881507	−	0.472170i	$-0.156529\pi$
$263$	7519.51	1.76301	0.881507	+	0.472170i	$0.156529\pi$
$264$	0	0
$265$	3611.43	0.837163
$266$	0	0
$267$	2416.40	0.553862
$268$	0	0
$269$	−6333.19	−1.43547	−0.717735	−	0.696316i	$-0.754821\pi$
$269$	−6333.19	−1.43547	−0.717735	+	0.696316i	$0.754821\pi$
$270$	0	0
$271$	7075.42	1.58598	0.792991	−	0.609233i	$-0.208523\pi$
$271$	7075.42	1.58598	0.792991	+	0.609233i	$0.208523\pi$
$272$	0	0
$273$	−2017.56	−0.447283
$274$	0	0
$275$	−671.879	−0.147330
$276$	0	0
$277$	−3011.89	−0.653311	−0.326655	−	0.945144i	$-0.605922\pi$
$277$	−3011.89	−0.653311	−0.326655	+	0.945144i	$0.605922\pi$
$278$	0	0
$279$	−3014.16	−0.646786
$280$	0	0
$281$	−3421.71	−0.726412	−0.363206	−	0.931709i	$-0.618318\pi$
$281$	−3421.71	−0.726412	−0.363206	+	0.931709i	$0.618318\pi$
$282$	0	0
$283$	1110.98	0.233361	0.116681	−	0.993169i	$-0.462775\pi$
$283$	1110.98	0.233361	0.116681	+	0.993169i	$0.462775\pi$
$284$	0	0
$285$	−2328.52	−0.483964
$286$	0	0
$287$	−1313.81	−0.270214
$288$	0	0
$289$	−3290.59	−0.669772
$290$	0	0
$291$	−3889.81	−0.783591
$292$	0	0
$293$	7375.12	1.47051	0.735254	−	0.677791i	$-0.237063\pi$
$293$	7375.12	1.47051	0.735254	+	0.677791i	$0.237063\pi$
$294$	0	0
$295$	2963.89	0.584964
$296$	0	0
$297$	3624.22	0.708075
$298$	0	0
$299$	−676.759	−0.130896
$300$	0	0
$301$	9958.34	1.90694
$302$	0	0
$303$	−696.046	−0.131970
$304$	0	0
$305$	−1598.14	−0.300031
$306$	0	0
$307$	640.417	0.119057	0.0595286	−	0.998227i	$-0.481040\pi$
$307$	640.417	0.119057	0.0595286	+	0.998227i	$0.481040\pi$
$308$	0	0
$309$	4414.24	0.812677
$310$	0	0
$311$	9787.95	1.78464	0.892321	−	0.451401i	$-0.149075\pi$
$311$	9787.95	1.78464	0.892321	+	0.451401i	$0.149075\pi$
$312$	0	0
$313$	−374.358	−0.0676037	−0.0338018	−	0.999429i	$-0.510762\pi$
$313$	−374.358	−0.0676037	−0.0338018	+	0.999429i	$0.510762\pi$
$314$	0	0
$315$	2064.46	0.369267
$316$	0	0
$317$	−2644.28	−0.468510	−0.234255	−	0.972175i	$-0.575265\pi$
$317$	−2644.28	−0.468510	−0.234255	+	0.972175i	$0.575265\pi$
$318$	0	0
$319$	2732.81	0.479648
$320$	0	0
$321$	865.536	0.150497
$322$	0	0
$323$	−6264.02	−1.07907
$324$	0	0
$325$	−735.607	−0.125551
$326$	0	0
$327$	1376.17	0.232728
$328$	0	0
$329$	−6992.65	−1.17179
$330$	0	0
$331$	−9089.66	−1.50941	−0.754703	−	0.656067i	$-0.772219\pi$
$331$	−9089.66	−1.50941	−0.754703	+	0.656067i	$0.772219\pi$
$332$	0	0
$333$	−1571.21	−0.258564
$334$	0	0
$335$	188.887	0.0308060
$336$	0	0
$337$	−6303.41	−1.01890	−0.509449	−	0.860501i	$-0.670151\pi$
$337$	−6303.41	−1.01890	−0.509449	+	0.860501i	$0.670151\pi$
$338$	0	0
$339$	−1663.96	−0.266590
$340$	0	0
$341$	−4492.25	−0.713399
$342$	0	0
$343$	−3702.90	−0.582909
$344$	0	0
$345$	−344.378	−0.0537411
$346$	0	0
$347$	−960.696	−0.148625	−0.0743125	−	0.997235i	$-0.523676\pi$
$347$	−960.696	−0.148625	−0.0743125	+	0.997235i	$0.523676\pi$
$348$	0	0
$349$	−4844.91	−0.743101	−0.371551	−	0.928413i	$-0.621174\pi$
$349$	−4844.91	−0.743101	−0.371551	+	0.928413i	$0.621174\pi$
$350$	0	0
$351$	3967.97	0.603404
$352$	0	0
$353$	−11353.4	−1.71184	−0.855919	−	0.517110i	$-0.827008\pi$
$353$	−11353.4	−1.71184	−0.855919	+	0.517110i	$0.827008\pi$
$354$	0	0
$355$	4793.63	0.716674
$356$	0	0
$357$	−2761.85	−0.409447
$358$	0	0
$359$	11361.3	1.67027	0.835133	−	0.550048i	$-0.185391\pi$
$359$	11361.3	1.67027	0.835133	+	0.550048i	$0.185391\pi$
$360$	0	0
$361$	17326.0	2.52602
$362$	0	0
$363$	−1822.89	−0.263572
$364$	0	0
$365$	2570.13	0.368566
$366$	0	0
$367$	6945.37	0.987862	0.493931	−	0.869501i	$-0.335560\pi$
$367$	6945.37	0.987862	0.493931	+	0.869501i	$0.335560\pi$
$368$	0	0
$369$	1034.67	0.145970
$370$	0	0
$371$	−16538.3	−2.31436
$372$	0	0
$373$	−5882.04	−0.816516	−0.408258	−	0.912866i	$-0.633864\pi$
$373$	−5882.04	−0.816516	−0.408258	+	0.912866i	$0.633864\pi$
$374$	0	0
$375$	−374.324	−0.0515467
$376$	0	0
$377$	2992.02	0.408745
$378$	0	0
$379$	−9939.28	−1.34709	−0.673544	−	0.739147i	$-0.735229\pi$
$379$	−9939.28	−1.34709	−0.673544	+	0.739147i	$0.735229\pi$
$380$	0	0
$381$	−1868.83	−0.251294
$382$	0	0
$383$	7569.55	1.00988	0.504942	−	0.863153i	$-0.331514\pi$
$383$	7569.55	1.00988	0.504942	+	0.863153i	$0.331514\pi$
$384$	0	0
$385$	3076.83	0.407298
$386$	0	0
$387$	−7842.57	−1.03013
$388$	0	0
$389$	6399.32	0.834084	0.417042	−	0.908887i	$-0.363067\pi$
$389$	6399.32	0.834084	0.417042	+	0.908887i	$0.363067\pi$
$390$	0	0
$391$	−926.420	−0.119824
$392$	0	0
$393$	383.478	0.0492211
$394$	0	0
$395$	1867.11	0.237835
$396$	0	0
$397$	5771.37	0.729614	0.364807	−	0.931083i	$-0.381135\pi$
$397$	5771.37	0.729614	0.364807	+	0.931083i	$0.381135\pi$
$398$	0	0
$399$	10663.3	1.33793
$400$	0	0
$401$	2024.30	0.252092	0.126046	−	0.992024i	$-0.459771\pi$
$401$	2024.30	0.252092	0.126046	+	0.992024i	$0.459771\pi$
$402$	0	0
$403$	−4918.34	−0.607941
$404$	0	0
$405$	−415.216	−0.0509438
$406$	0	0
$407$	−2341.70	−0.285194
$408$	0	0
$409$	−7439.28	−0.899386	−0.449693	−	0.893183i	$-0.648467\pi$
$409$	−7439.28	−0.899386	−0.449693	+	0.893183i	$0.648467\pi$
$410$	0	0
$411$	6557.94	0.787055
$412$	0	0
$413$	−13573.0	−1.61715
$414$	0	0
$415$	5329.67	0.630418
$416$	0	0
$417$	706.681	0.0829888
$418$	0	0
$419$	2075.10	0.241945	0.120973	−	0.992656i	$-0.461399\pi$
$419$	2075.10	0.241945	0.120973	+	0.992656i	$0.461399\pi$
$420$	0	0
$421$	1614.70	0.186925	0.0934627	−	0.995623i	$-0.470206\pi$
$421$	1614.70	0.186925	0.0934627	+	0.995623i	$0.470206\pi$
$422$	0	0
$423$	5506.98	0.632999
$424$	0	0
$425$	−1006.98	−0.114931
$426$	0	0
$427$	7318.60	0.829443
$428$	0	0
$429$	2368.07	0.266507
$430$	0	0
$431$	8595.26	0.960601	0.480301	−	0.877104i	$-0.340528\pi$
$431$	8595.26	0.960601	0.480301	+	0.877104i	$0.340528\pi$
$432$	0	0
$433$	−17005.2	−1.88734	−0.943671	−	0.330885i	$-0.892653\pi$
$433$	−17005.2	−1.88734	−0.943671	+	0.330885i	$0.892653\pi$
$434$	0	0
$435$	1522.53	0.167815
$436$	0	0
$437$	3576.85	0.391542
$438$	0	0
$439$	6630.03	0.720807	0.360403	−	0.932797i	$-0.382639\pi$
$439$	6630.03	0.720807	0.360403	+	0.932797i	$0.382639\pi$
$440$	0	0
$441$	−3268.94	−0.352979
$442$	0	0
$443$	1682.59	0.180457	0.0902284	−	0.995921i	$-0.471240\pi$
$443$	1682.59	0.180457	0.0902284	+	0.995921i	$0.471240\pi$
$444$	0	0
$445$	−4034.60	−0.429795
$446$	0	0
$447$	−325.212	−0.0344117
$448$	0	0
$449$	7223.21	0.759208	0.379604	−	0.925149i	$-0.376060\pi$
$449$	7223.21	0.759208	0.379604	+	0.925149i	$0.376060\pi$
$450$	0	0
$451$	1542.06	0.161003
$452$	0	0
$453$	−10744.4	−1.11439
$454$	0	0
$455$	3368.67	0.347089
$456$	0	0
$457$	9564.92	0.979055	0.489527	−	0.871988i	$-0.337169\pi$
$457$	9564.92	0.979055	0.489527	+	0.871988i	$0.337169\pi$
$458$	0	0
$459$	5431.79	0.552363
$460$	0	0
$461$	6253.73	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$461$	6253.73	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$462$	0	0
$463$	−6856.21	−0.688197	−0.344098	−	0.938934i	$-0.611815\pi$
$463$	−6856.21	−0.688197	−0.344098	+	0.938934i	$0.611815\pi$
$464$	0	0
$465$	−2502.77	−0.249598
$466$	0	0
$467$	4342.26	0.430270	0.215135	−	0.976584i	$-0.430981\pi$
$467$	4342.26	0.430270	0.215135	+	0.976584i	$0.430981\pi$
$468$	0	0
$469$	−864.998	−0.0851639
$470$	0	0
$471$	−2765.57	−0.270553
$472$	0	0
$473$	−11688.4	−1.13622
$474$	0	0
$475$	3887.88	0.375554
$476$	0	0
$477$	13024.6	1.25022
$478$	0	0
$479$	−17828.7	−1.70066	−0.850329	−	0.526252i	$-0.823597\pi$
$479$	−17828.7	−1.70066	−0.850329	+	0.526252i	$0.823597\pi$
$480$	0	0
$481$	−2563.81	−0.243035
$482$	0	0
$483$	1577.06	0.148569
$484$	0	0
$485$	6494.73	0.608063
$486$	0	0
$487$	−16432.7	−1.52903	−0.764515	−	0.644606i	$-0.777021\pi$
$487$	−16432.7	−1.52903	−0.764515	+	0.644606i	$0.777021\pi$
$488$	0	0
$489$	1975.35	0.182676
$490$	0	0
$491$	2870.09	0.263799	0.131899	−	0.991263i	$-0.457892\pi$
$491$	2870.09	0.263799	0.131899	+	0.991263i	$0.457892\pi$
$492$	0	0
$493$	4095.80	0.374169
$494$	0	0
$495$	−2423.12	−0.220023
$496$	0	0
$497$	−21952.1	−1.98126
$498$	0	0
$499$	−8652.75	−0.776253	−0.388126	−	0.921606i	$-0.626878\pi$
$499$	−8652.75	−0.776253	−0.388126	+	0.921606i	$0.626878\pi$
$500$	0	0
$501$	−11521.6	−1.02744
$502$	0	0
$503$	−1620.84	−0.143677	−0.0718386	−	0.997416i	$-0.522887\pi$
$503$	−1620.84	−0.143677	−0.0718386	+	0.997416i	$0.522887\pi$
$504$	0	0
$505$	1162.17	0.102408
$506$	0	0
$507$	−3986.44	−0.349199
$508$	0	0
$509$	−5090.09	−0.443250	−0.221625	−	0.975132i	$-0.571136\pi$
$509$	−5090.09	−0.443250	−0.221625	+	0.975132i	$0.571136\pi$
$510$	0	0
$511$	−11769.7	−1.01891
$512$	0	0
$513$	−20971.8	−1.80493
$514$	0	0
$515$	−7370.35	−0.630634
$516$	0	0
$517$	8207.50	0.698192
$518$	0	0
$519$	3510.89	0.296938
$520$	0	0
$521$	−1344.78	−0.113083	−0.0565414	−	0.998400i	$-0.518007\pi$
$521$	−1344.78	−0.113083	−0.0565414	+	0.998400i	$0.518007\pi$
$522$	0	0
$523$	12620.5	1.05517	0.527585	−	0.849502i	$-0.323097\pi$
$523$	12620.5	1.05517	0.527585	+	0.849502i	$0.323097\pi$
$524$	0	0
$525$	1714.19	0.142502
$526$	0	0
$527$	−6732.76	−0.556515
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	10689.2	0.873584
$532$	0	0
$533$	1688.32	0.137203
$534$	0	0
$535$	−1445.17	−0.116785
$536$	0	0
$537$	−2960.90	−0.237937
$538$	0	0
$539$	−4871.97	−0.389333
$540$	0	0
$541$	13950.6	1.10865	0.554326	−	0.832299i	$-0.312976\pi$
$541$	13950.6	1.10865	0.554326	+	0.832299i	$0.312976\pi$
$542$	0	0
$543$	−956.883	−0.0756239
$544$	0	0
$545$	−2297.75	−0.180596
$546$	0	0
$547$	−20474.7	−1.60043	−0.800215	−	0.599713i	$-0.795281\pi$
$547$	−20474.7	−1.60043	−0.800215	+	0.599713i	$0.795281\pi$
$548$	0	0
$549$	−5763.68	−0.448065
$550$	0	0
$551$	−15813.6	−1.22265
$552$	0	0
$553$	−8550.33	−0.657499
$554$	0	0
$555$	−1304.63	−0.0997813
$556$	0	0
$557$	19141.3	1.45609	0.728047	−	0.685528i	$-0.240428\pi$
$557$	19141.3	1.45609	0.728047	+	0.685528i	$0.240428\pi$
$558$	0	0
$559$	−12797.1	−0.968262
$560$	0	0
$561$	3241.67	0.243963
$562$	0	0
$563$	−5130.93	−0.384090	−0.192045	−	0.981386i	$-0.561512\pi$
$563$	−5130.93	−0.384090	−0.192045	+	0.981386i	$0.561512\pi$
$564$	0	0
$565$	2778.28	0.206873
$566$	0	0
$567$	1901.46	0.140835
$568$	0	0
$569$	−15969.3	−1.17657	−0.588283	−	0.808655i	$-0.700196\pi$
$569$	−15969.3	−1.17657	−0.588283	+	0.808655i	$0.700196\pi$
$570$	0	0
$571$	19204.2	1.40748	0.703741	−	0.710456i	$-0.251511\pi$
$571$	19204.2	1.40748	0.703741	+	0.710456i	$0.251511\pi$
$572$	0	0
$573$	−2433.99	−0.177455
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−7424.24	−0.535659	−0.267829	−	0.963466i	$-0.586306\pi$
$577$	−7424.24	−0.535659	−0.267829	+	0.963466i	$0.586306\pi$
$578$	0	0
$579$	−5001.94	−0.359022
$580$	0	0
$581$	−24406.9	−1.74280
$582$	0	0
$583$	19411.5	1.37898
$584$	0	0
$585$	−2652.95	−0.187498
$586$	0	0
$587$	17748.5	1.24797	0.623984	−	0.781437i	$-0.285513\pi$
$587$	17748.5	1.24797	0.623984	+	0.781437i	$0.285513\pi$
$588$	0	0
$589$	25994.7	1.81850
$590$	0	0
$591$	3285.93	0.228706
$592$	0	0
$593$	−8872.62	−0.614426	−0.307213	−	0.951641i	$-0.599396\pi$
$593$	−8872.62	−0.614426	−0.307213	+	0.951641i	$0.599396\pi$
$594$	0	0
$595$	4611.40	0.317729
$596$	0	0
$597$	−10463.2	−0.717301
$598$	0	0
$599$	8675.69	0.591785	0.295892	−	0.955221i	$-0.404383\pi$
$599$	8675.69	0.591785	0.295892	+	0.955221i	$0.404383\pi$
$600$	0	0
$601$	13282.9	0.901530	0.450765	−	0.892643i	$-0.351151\pi$
$601$	13282.9	0.901530	0.450765	+	0.892643i	$0.351151\pi$
$602$	0	0
$603$	681.219	0.0460056
$604$	0	0
$605$	3043.63	0.204531
$606$	0	0
$607$	−6282.28	−0.420082	−0.210041	−	0.977693i	$-0.567360\pi$
$607$	−6282.28	−0.420082	−0.210041	+	0.977693i	$0.567360\pi$
$608$	0	0
$609$	−6972.33	−0.463930
$610$	0	0
$611$	8985.98	0.594982
$612$	0	0
$613$	−9644.23	−0.635443	−0.317722	−	0.948184i	$-0.602918\pi$
$613$	−9644.23	−0.635443	−0.317722	+	0.948184i	$0.602918\pi$
$614$	0	0
$615$	859.126	0.0563306
$616$	0	0
$617$	−5792.37	−0.377945	−0.188972	−	0.981982i	$-0.560516\pi$
$617$	−5792.37	−0.377945	−0.188972	+	0.981982i	$0.560516\pi$
$618$	0	0
$619$	−19858.1	−1.28944	−0.644719	−	0.764419i	$-0.723026\pi$
$619$	−19858.1	−1.28944	−0.644719	+	0.764419i	$0.723026\pi$
$620$	0	0
$621$	−3101.64	−0.200426
$622$	0	0
$623$	18476.2	1.18818
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−12515.9	−0.797187
$628$	0	0
$629$	−3509.63	−0.222477
$630$	0	0
$631$	14742.6	0.930103	0.465051	−	0.885284i	$-0.346036\pi$
$631$	14742.6	0.930103	0.465051	+	0.885284i	$0.346036\pi$
$632$	0	0
$633$	−4275.65	−0.268470
$634$	0	0
$635$	3120.34	0.195003
$636$	0	0
$637$	−5334.08	−0.331780
$638$	0	0
$639$	17288.1	1.07028
$640$	0	0
$641$	15028.1	0.926010	0.463005	−	0.886356i	$-0.346771\pi$
$641$	15028.1	0.926010	0.463005	+	0.886356i	$0.346771\pi$
$642$	0	0
$643$	−18830.0	−1.15487	−0.577435	−	0.816437i	$-0.695946\pi$
$643$	−18830.0	−1.15487	−0.577435	+	0.816437i	$0.695946\pi$
$644$	0	0
$645$	−6511.97	−0.397533
$646$	0	0
$647$	−9640.11	−0.585768	−0.292884	−	0.956148i	$-0.594615\pi$
$647$	−9640.11	−0.585768	−0.292884	+	0.956148i	$0.594615\pi$
$648$	0	0
$649$	15931.0	0.963555
$650$	0	0
$651$	11461.3	0.690019
$652$	0	0
$653$	20960.6	1.25613	0.628063	−	0.778162i	$-0.283848\pi$
$653$	20960.6	1.25613	0.628063	+	0.778162i	$0.283848\pi$
$654$	0	0
$655$	−640.283	−0.0381953
$656$	0	0
$657$	9269.12	0.550415
$658$	0	0
$659$	2346.87	0.138727	0.0693634	−	0.997591i	$-0.477903\pi$
$659$	2346.87	0.138727	0.0693634	+	0.997591i	$0.477903\pi$
$660$	0	0
$661$	14586.0	0.858290	0.429145	−	0.903236i	$-0.358815\pi$
$661$	14586.0	0.858290	0.429145	+	0.903236i	$0.358815\pi$
$662$	0	0
$663$	3549.15	0.207900
$664$	0	0
$665$	−17804.3	−1.03823
$666$	0	0
$667$	−2338.76	−0.135768
$668$	0	0
$669$	−4690.74	−0.271083
$670$	0	0
$671$	−8590.07	−0.494212
$672$	0	0
$673$	26363.8	1.51003	0.755016	−	0.655707i	$-0.227629\pi$
$673$	26363.8	1.51003	0.755016	+	0.655707i	$0.227629\pi$
$674$	0	0
$675$	−3371.34	−0.192242
$676$	0	0
$677$	−7216.39	−0.409673	−0.204836	−	0.978796i	$-0.565666\pi$
$677$	−7216.39	−0.409673	−0.204836	+	0.978796i	$0.565666\pi$
$678$	0	0
$679$	−29742.2	−1.68100
$680$	0	0
$681$	12528.1	0.704958
$682$	0	0
$683$	−5503.00	−0.308296	−0.154148	−	0.988048i	$-0.549263\pi$
$683$	−5503.00	−0.308296	−0.154148	+	0.988048i	$0.549263\pi$
$684$	0	0
$685$	−10949.6	−0.610751
$686$	0	0
$687$	6913.80	0.383956
$688$	0	0
$689$	21252.7	1.17513
$690$	0	0
$691$	19321.6	1.06372	0.531859	−	0.846833i	$-0.321494\pi$
$691$	19321.6	1.06372	0.531859	+	0.846833i	$0.321494\pi$
$692$	0	0
$693$	11096.5	0.608257
$694$	0	0
$695$	−1179.93	−0.0643989
$696$	0	0
$697$	2311.16	0.125597
$698$	0	0
$699$	5768.07	0.312115
$700$	0	0
$701$	27447.4	1.47885	0.739426	−	0.673238i	$-0.235097\pi$
$701$	27447.4	1.47885	0.739426	+	0.673238i	$0.235097\pi$
$702$	0	0
$703$	13550.4	0.726977
$704$	0	0
$705$	4572.65	0.244278
$706$	0	0
$707$	−5322.09	−0.283109
$708$	0	0
$709$	21302.1	1.12837	0.564187	−	0.825647i	$-0.309190\pi$
$709$	21302.1	1.12837	0.564187	+	0.825647i	$0.309190\pi$
$710$	0	0
$711$	6733.71	0.355181
$712$	0	0
$713$	3844.51	0.201933
$714$	0	0
$715$	−3953.91	−0.206808
$716$	0	0
$717$	−4633.45	−0.241338
$718$	0	0
$719$	−16554.9	−0.858684	−0.429342	−	0.903142i	$-0.641254\pi$
$719$	−16554.9	−0.858684	−0.429342	+	0.903142i	$0.641254\pi$
$720$	0	0
$721$	33752.1	1.74340
$722$	0	0
$723$	−3314.85	−0.170513
$724$	0	0
$725$	−2542.13	−0.130224
$726$	0	0
$727$	19887.5	1.01456	0.507281	−	0.861781i	$-0.330651\pi$
$727$	19887.5	1.01456	0.507281	+	0.861781i	$0.330651\pi$
$728$	0	0
$729$	9405.99	0.477874
$730$	0	0
$731$	−17518.0	−0.886357
$732$	0	0
$733$	−18472.2	−0.930812	−0.465406	−	0.885097i	$-0.654092\pi$
$733$	−18472.2	−0.930812	−0.465406	+	0.885097i	$0.654092\pi$
$734$	0	0
$735$	−2714.32	−0.136217
$736$	0	0
$737$	1015.28	0.0507437
$738$	0	0
$739$	8178.41	0.407101	0.203550	−	0.979064i	$-0.434752\pi$
$739$	8178.41	0.407101	0.203550	+	0.979064i	$0.434752\pi$
$740$	0	0
$741$	−13703.0	−0.679343
$742$	0	0
$743$	24289.3	1.19931	0.599656	−	0.800258i	$-0.295304\pi$
$743$	24289.3	1.19931	0.599656	+	0.800258i	$0.295304\pi$
$744$	0	0
$745$	542.999	0.0267033
$746$	0	0
$747$	19221.4	0.941464
$748$	0	0
$749$	6618.05	0.322855
$750$	0	0
$751$	−1506.66	−0.0732075	−0.0366037	−	0.999330i	$-0.511654\pi$
$751$	−1506.66	−0.0732075	−0.0366037	+	0.999330i	$0.511654\pi$
$752$	0	0
$753$	−18967.2	−0.917932
$754$	0	0
$755$	17939.7	0.864760
$756$	0	0
$757$	28882.5	1.38673	0.693364	−	0.720588i	$-0.256128\pi$
$757$	28882.5	1.38673	0.693364	+	0.720588i	$0.256128\pi$
$758$	0	0
$759$	−1851.04	−0.0885226
$760$	0	0
$761$	35803.3	1.70548	0.852739	−	0.522337i	$-0.174940\pi$
$761$	35803.3	1.70548	0.852739	+	0.522337i	$0.174940\pi$
$762$	0	0
$763$	10522.4	0.499262
$764$	0	0
$765$	−3631.65	−0.171637
$766$	0	0
$767$	17442.1	0.821117
$768$	0	0
$769$	−19984.6	−0.937142	−0.468571	−	0.883426i	$-0.655231\pi$
$769$	−19984.6	−0.937142	−0.468571	+	0.883426i	$0.655231\pi$
$770$	0	0
$771$	−23726.5	−1.10829
$772$	0	0
$773$	−324.416	−0.0150950	−0.00754751	−	0.999972i	$-0.502402\pi$
$773$	−324.416	−0.0150950	−0.00754751	+	0.999972i	$0.502402\pi$
$774$	0	0
$775$	4178.81	0.193687
$776$	0	0
$777$	5974.49	0.275848
$778$	0	0
$779$	−8923.22	−0.410408
$780$	0	0
$781$	25765.9	1.18051
$782$	0	0
$783$	13712.6	0.625862
$784$	0	0
$785$	4617.60	0.209948
$786$	0	0
$787$	17434.6	0.789678	0.394839	−	0.918750i	$-0.370800\pi$
$787$	17434.6	0.789678	0.394839	+	0.918750i	$0.370800\pi$
$788$	0	0
$789$	22517.9	1.01604
$790$	0	0
$791$	−12723.0	−0.571905
$792$	0	0
$793$	−9404.85	−0.421155
$794$	0	0
$795$	10814.8	0.482465
$796$	0	0
$797$	13395.1	0.595330	0.297665	−	0.954670i	$-0.403792\pi$
$797$	13395.1	0.595330	0.297665	+	0.954670i	$0.403792\pi$
$798$	0	0
$799$	12301.0	0.544653
$800$	0	0
$801$	−14550.7	−0.641854
$802$	0	0
$803$	13814.5	0.607103
$804$	0	0
$805$	−2633.18	−0.115289
$806$	0	0
$807$	−18965.3	−0.827275
$808$	0	0
$809$	−34459.3	−1.49756	−0.748779	−	0.662820i	$-0.769360\pi$
$809$	−34459.3	−1.49756	−0.748779	+	0.662820i	$0.769360\pi$
$810$	0	0
$811$	24076.3	1.04246	0.521229	−	0.853417i	$-0.325474\pi$
$811$	24076.3	1.04246	0.521229	+	0.853417i	$0.325474\pi$
$812$	0	0
$813$	21188.0	0.914017
$814$	0	0
$815$	−3298.20	−0.141756
$816$	0	0
$817$	67635.9	2.89630
$818$	0	0
$819$	12149.0	0.518342
$820$	0	0
$821$	34057.5	1.44776	0.723882	−	0.689924i	$-0.242356\pi$
$821$	34057.5	1.44776	0.723882	+	0.689924i	$0.242356\pi$
$822$	0	0
$823$	−43552.6	−1.84465	−0.922327	−	0.386411i	$-0.873714\pi$
$823$	−43552.6	−1.84465	−0.922327	+	0.386411i	$0.873714\pi$
$824$	0	0
$825$	−2012.00	−0.0849079
$826$	0	0
$827$	20044.3	0.842817	0.421409	−	0.906871i	$-0.361536\pi$
$827$	20044.3	0.842817	0.421409	+	0.906871i	$0.361536\pi$
$828$	0	0
$829$	−5752.50	−0.241004	−0.120502	−	0.992713i	$-0.538450\pi$
$829$	−5752.50	−0.241004	−0.120502	+	0.992713i	$0.538450\pi$
$830$	0	0
$831$	−9019.39	−0.376509
$832$	0	0
$833$	−7301.86	−0.303715
$834$	0	0
$835$	19237.4	0.797290
$836$	0	0
$837$	−22541.1	−0.930867
$838$	0	0
$839$	−43495.5	−1.78979	−0.894893	−	0.446280i	$-0.852748\pi$
$839$	−43495.5	−1.78979	−0.894893	+	0.446280i	$0.852748\pi$
$840$	0	0
$841$	−14049.1	−0.576043
$842$	0	0
$843$	−10246.6	−0.418638
$844$	0	0
$845$	6656.06	0.270977
$846$	0	0
$847$	−13938.1	−0.565430
$848$	0	0
$849$	3326.95	0.134488
$850$	0	0
$851$	2004.05	0.0807262
$852$	0	0
$853$	−8312.13	−0.333648	−0.166824	−	0.985987i	$-0.553351\pi$
$853$	−8312.13	−0.333648	−0.166824	+	0.985987i	$0.553351\pi$
$854$	0	0
$855$	14021.6	0.560851
$856$	0	0
$857$	11565.4	0.460987	0.230493	−	0.973074i	$-0.425966\pi$
$857$	11565.4	0.460987	0.230493	+	0.973074i	$0.425966\pi$
$858$	0	0
$859$	4774.43	0.189641	0.0948205	−	0.995494i	$-0.469772\pi$
$859$	4774.43	0.189641	0.0948205	+	0.995494i	$0.469772\pi$
$860$	0	0
$861$	−3934.31	−0.155727
$862$	0	0
$863$	29965.2	1.18196	0.590978	−	0.806688i	$-0.298742\pi$
$863$	29965.2	1.18196	0.590978	+	0.806688i	$0.298742\pi$
$864$	0	0
$865$	−5862.05	−0.230423
$866$	0	0
$867$	−9853.98	−0.385996
$868$	0	0
$869$	10035.8	0.391762
$870$	0	0
$871$	1111.57	0.0432426
$872$	0	0
$873$	23423.1	0.908079
$874$	0	0
$875$	−2862.15	−0.110581
$876$	0	0
$877$	39968.1	1.53891	0.769457	−	0.638699i	$-0.220527\pi$
$877$	39968.1	1.53891	0.769457	+	0.638699i	$0.220527\pi$
$878$	0	0
$879$	22085.5	0.847468
$880$	0	0
$881$	21666.8	0.828573	0.414287	−	0.910146i	$-0.364031\pi$
$881$	21666.8	0.828573	0.414287	+	0.910146i	$0.364031\pi$
$882$	0	0
$883$	−21867.2	−0.833396	−0.416698	−	0.909045i	$-0.636813\pi$
$883$	−21867.2	−0.833396	−0.416698	+	0.909045i	$0.636813\pi$
$884$	0	0
$885$	8875.65	0.337121
$886$	0	0
$887$	−15460.5	−0.585245	−0.292623	−	0.956228i	$-0.594528\pi$
$887$	−15460.5	−0.585245	−0.292623	+	0.956228i	$0.594528\pi$
$888$	0	0
$889$	−14289.4	−0.539091
$890$	0	0
$891$	−2231.80	−0.0839148
$892$	0	0
$893$	−47493.3	−1.77973
$894$	0	0
$895$	4943.75	0.184638
$896$	0	0
$897$	−2026.62	−0.0754367
$898$	0	0
$899$	−16996.9	−0.630567
$900$	0	0
$901$	29093.1	1.07573
$902$	0	0
$903$	29821.2	1.09899
$904$	0	0
$905$	1597.68	0.0586838
$906$	0	0
$907$	−54212.9	−1.98469	−0.992343	−	0.123512i	$-0.960584\pi$
$907$	−54212.9	−1.98469	−0.992343	+	0.123512i	$0.960584\pi$
$908$	0	0
$909$	4191.35	0.152935
$910$	0	0
$911$	7648.48	0.278162	0.139081	−	0.990281i	$-0.455585\pi$
$911$	7648.48	0.278162	0.139081	+	0.990281i	$0.455585\pi$
$912$	0	0
$913$	28647.2	1.03843
$914$	0	0
$915$	−4785.79	−0.172911
$916$	0	0
$917$	2932.14	0.105592
$918$	0	0
$919$	−4694.32	−0.168500	−0.0842500	−	0.996445i	$-0.526849\pi$
$919$	−4694.32	−0.168500	−0.0842500	+	0.996445i	$0.526849\pi$
$920$	0	0
$921$	1917.79	0.0686138
$922$	0	0
$923$	28209.8	1.00600
$924$	0	0
$925$	2178.32	0.0774298
$926$	0	0
$927$	−26581.0	−0.941787
$928$	0	0
$929$	−20288.1	−0.716504	−0.358252	−	0.933625i	$-0.616627\pi$
$929$	−20288.1	−0.716504	−0.358252	+	0.933625i	$0.616627\pi$
$930$	0	0
$931$	28192.0	0.992434
$932$	0	0
$933$	29310.9	1.02851
$934$	0	0
$935$	−5412.54	−0.189315
$936$	0	0
$937$	−27047.1	−0.943000	−0.471500	−	0.881866i	$-0.656287\pi$
$937$	−27047.1	−0.943000	−0.471500	+	0.881866i	$0.656287\pi$
$938$	0	0
$939$	−1121.05	−0.0389606
$940$	0	0
$941$	21616.3	0.748855	0.374427	−	0.927256i	$-0.377839\pi$
$941$	21616.3	0.748855	0.374427	+	0.927256i	$0.377839\pi$
$942$	0	0
$943$	−1319.70	−0.0455732
$944$	0	0
$945$	15438.9	0.531456
$946$	0	0
$947$	−15208.5	−0.521869	−0.260935	−	0.965356i	$-0.584031\pi$
$947$	−15208.5	−0.521869	−0.260935	+	0.965356i	$0.584031\pi$
$948$	0	0
$949$	15124.8	0.517358
$950$	0	0
$951$	−7918.54	−0.270007
$952$	0	0
$953$	46043.1	1.56504	0.782520	−	0.622625i	$-0.213934\pi$
$953$	46043.1	1.56504	0.782520	+	0.622625i	$0.213934\pi$
$954$	0	0
$955$	4063.98	0.137704
$956$	0	0
$957$	8183.65	0.276426
$958$	0	0
$959$	50143.2	1.68843
$960$	0	0
$961$	−1851.05	−0.0621347
$962$	0	0
$963$	−5211.96	−0.174406
$964$	0	0
$965$	8351.62	0.278599
$966$	0	0
$967$	4878.54	0.162237	0.0811185	−	0.996704i	$-0.474151\pi$
$967$	4878.54	0.162237	0.0811185	+	0.996704i	$0.474151\pi$
$968$	0	0
$969$	−18758.2	−0.621878
$970$	0	0
$971$	−28973.3	−0.957567	−0.478783	−	0.877933i	$-0.658922\pi$
$971$	−28973.3	−0.957567	−0.478783	+	0.877933i	$0.658922\pi$
$972$	0	0
$973$	5403.41	0.178032
$974$	0	0
$975$	−2202.84	−0.0723564
$976$	0	0
$977$	−59286.5	−1.94140	−0.970698	−	0.240303i	$-0.922753\pi$
$977$	−59286.5	−1.94140	−0.970698	+	0.240303i	$0.922753\pi$
$978$	0	0
$979$	−21686.1	−0.707959
$980$	0	0
$981$	−8286.80	−0.269701
$982$	0	0
$983$	43868.4	1.42338	0.711691	−	0.702493i	$-0.247930\pi$
$983$	43868.4	1.42338	0.711691	+	0.702493i	$0.247930\pi$
$984$	0	0
$985$	−5486.43	−0.177474
$986$	0	0
$987$	−20940.1	−0.675311
$988$	0	0
$989$	10003.0	0.321616
$990$	0	0
$991$	−53277.1	−1.70777	−0.853886	−	0.520460i	$-0.825760\pi$
$991$	−53277.1	−1.70777	−0.853886	+	0.520460i	$0.825760\pi$
$992$	0	0
$993$	−27219.8	−0.869885
$994$	0	0
$995$	17470.1	0.556622
$996$	0	0
$997$	25936.1	0.823875	0.411937	−	0.911212i	$-0.364852\pi$
$997$	25936.1	0.823875	0.411937	+	0.911212i	$0.364852\pi$
$998$	0	0
$999$	−11750.2	−0.372131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.z.1.6		9
4.3	odd	2		920.4.a.e.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.e.1.4	✓	9	4.3	odd	2
1840.4.a.z.1.6		9	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)

\(f(q)\)	\(=\)	\(q+2.99459 q^{3} -5.00000 q^{5} +22.8972 q^{7} -18.0324 q^{9} +O(q^{10})\)	\(q+2.99459 q^{3} -5.00000 q^{5} +22.8972 q^{7} -18.0324 q^{9} -26.8752 q^{11} -29.4243 q^{13} -14.9730 q^{15} -40.2791 q^{17} +155.515 q^{19} +68.5678 q^{21} +23.0000 q^{23} +25.0000 q^{25} -134.854 q^{27} -101.685 q^{29} +167.152 q^{31} -80.4802 q^{33} -114.486 q^{35} +87.1326 q^{37} -88.1137 q^{39} -57.3785 q^{41} +434.915 q^{43} +90.1621 q^{45} -305.393 q^{47} +181.281 q^{49} -120.620 q^{51} -722.286 q^{53} +134.376 q^{55} +465.704 q^{57} -592.779 q^{59} +319.629 q^{61} -412.892 q^{63} +147.121 q^{65} -37.7775 q^{67} +68.8756 q^{69} -958.725 q^{71} -514.025 q^{73} +74.8648 q^{75} -615.366 q^{77} -373.423 q^{79} +83.0432 q^{81} -1065.93 q^{83} +201.396 q^{85} -304.506 q^{87} +806.921 q^{89} -673.734 q^{91} +500.554 q^{93} -777.576 q^{95} -1298.95 q^{97} +484.624 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9	\( 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9} - 22 q^{11} + 23 q^{13} - 15 q^{15} - 135 q^{17} - 102 q^{19} + 54 q^{21} + 207 q^{23} + 225 q^{25} + 363 q^{27} + 280 q^{29} - 168 q^{31} + 28 q^{33} + 125 q^{35} + 153 q^{37} + 5 q^{39} - 502 q^{41} - 110 q^{43} - 310 q^{45} + 153 q^{47} + 764 q^{49} - 924 q^{51} - 273 q^{53} + 110 q^{55} + 748 q^{57} - 827 q^{59} + 1976 q^{61} - 2237 q^{63} - 115 q^{65} - 1613 q^{67} + 69 q^{69} - 1370 q^{71} + 425 q^{73} + 75 q^{75} + 2006 q^{77} - 2624 q^{79} + 1729 q^{81} - 2505 q^{83} + 675 q^{85} - 1591 q^{87} + 1120 q^{89} - 2392 q^{91} + 4401 q^{93} + 510 q^{95} + 2026 q^{97} - 3206 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9 - 22 * q^11 + 23 * q^13 - 15 * q^15 - 135 * q^17 - 102 * q^19 + 54 * q^21 + 207 * q^23 + 225 * q^25 + 363 * q^27 + 280 * q^29 - 168 * q^31 + 28 * q^33 + 125 * q^35 + 153 * q^37 + 5 * q^39 - 502 * q^41 - 110 * q^43 - 310 * q^45 + 153 * q^47 + 764 * q^49 - 924 * q^51 - 273 * q^53 + 110 * q^55 + 748 * q^57 - 827 * q^59 + 1976 * q^61 - 2237 * q^63 - 115 * q^65 - 1613 * q^67 + 69 * q^69 - 1370 * q^71 + 425 * q^73 + 75 * q^75 + 2006 * q^77 - 2624 * q^79 + 1729 * q^81 - 2505 * q^83 + 675 * q^85 - 1591 * q^87 + 1120 * q^89 - 2392 * q^91 + 4401 * q^93 + 510 * q^95 + 2026 * q^97 - 3206 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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