LMFDB - Embedded newform 1840.4.a.s.1.5

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 $ x^6 - 2x^5 - 81x^4 + 161x^3 + 1520x^2 - 3915*x + 588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-6.38774$ 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)

$f(q)$	$=$	$q+6.38774 q^{3} -5.00000 q^{5} +5.78952 q^{7} +13.8032 q^{9} +O(q^{10})$	$q+6.38774 q^{3} -5.00000 q^{5} +5.78952 q^{7} +13.8032 q^{9} +3.04007 q^{11} -41.8142 q^{13} -31.9387 q^{15} +89.8459 q^{17} -102.477 q^{19} +36.9819 q^{21} +23.0000 q^{23} +25.0000 q^{25} -84.2976 q^{27} -77.9931 q^{29} +17.8575 q^{31} +19.4192 q^{33} -28.9476 q^{35} +236.953 q^{37} -267.098 q^{39} -318.152 q^{41} +31.9573 q^{43} -69.0161 q^{45} -239.636 q^{47} -309.482 q^{49} +573.913 q^{51} +498.695 q^{53} -15.2004 q^{55} -654.596 q^{57} -276.837 q^{59} -86.9389 q^{61} +79.9140 q^{63} +209.071 q^{65} -1019.33 q^{67} +146.918 q^{69} -1020.60 q^{71} -104.709 q^{73} +159.694 q^{75} +17.6006 q^{77} +1052.52 q^{79} -911.158 q^{81} +1006.59 q^{83} -449.230 q^{85} -498.200 q^{87} -1098.26 q^{89} -242.084 q^{91} +114.069 q^{93} +512.384 q^{95} -489.424 q^{97} +41.9628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9	$ 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9} + 3 q^{11} - 28 q^{13} + 10 q^{15} + 24 q^{17} + 3 q^{19} + 60 q^{21} + 138 q^{23} + 150 q^{25} + 97 q^{27} + 76 q^{29} + 381 q^{31} - 3 q^{33} + 140 q^{35} + 131 q^{37} - 41 q^{39} - 95 q^{41} + 202 q^{43} - 20 q^{45} + 119 q^{47} - 578 q^{49} + 997 q^{51} + 137 q^{53} - 15 q^{55} - 894 q^{57} + 39 q^{59} - 573 q^{61} + 355 q^{63} + 140 q^{65} + 563 q^{67} - 46 q^{69} + 83 q^{71} - 1799 q^{73} - 50 q^{75} - 1410 q^{77} + 1636 q^{79} - 2006 q^{81} + 1191 q^{83} - 120 q^{85} + 1821 q^{87} - 1370 q^{89} + 1251 q^{91} - 2215 q^{93} - 15 q^{95} - 3021 q^{97} + 525 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9 + 3 * q^11 - 28 * q^13 + 10 * q^15 + 24 * q^17 + 3 * q^19 + 60 * q^21 + 138 * q^23 + 150 * q^25 + 97 * q^27 + 76 * q^29 + 381 * q^31 - 3 * q^33 + 140 * q^35 + 131 * q^37 - 41 * q^39 - 95 * q^41 + 202 * q^43 - 20 * q^45 + 119 * q^47 - 578 * q^49 + 997 * q^51 + 137 * q^53 - 15 * q^55 - 894 * q^57 + 39 * q^59 - 573 * q^61 + 355 * q^63 + 140 * q^65 + 563 * q^67 - 46 * q^69 + 83 * q^71 - 1799 * q^73 - 50 * q^75 - 1410 * q^77 + 1636 * q^79 - 2006 * q^81 + 1191 * q^83 - 120 * q^85 + 1821 * q^87 - 1370 * q^89 + 1251 * q^91 - 2215 * q^93 - 15 * q^95 - 3021 * q^97 + 525 * 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
$3$	6.38774	1.22932	0.614661	−	0.788792i	$-0.289293\pi$
$3$	6.38774	1.22932	0.614661	+	0.788792i	$0.289293\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	5.78952	0.312604	0.156302	−	0.987709i	$-0.450043\pi$
$7$	5.78952	0.312604	0.156302	+	0.987709i	$0.450043\pi$
$8$	0	0
$9$	13.8032	0.511231
$10$	0	0
$11$	3.04007	0.0833288	0.0416644	−	0.999132i	$-0.486734\pi$
$11$	3.04007	0.0833288	0.0416644	+	0.999132i	$0.486734\pi$
$12$	0	0
$13$	−41.8142	−0.892090	−0.446045	−	0.895011i	$-0.647168\pi$
$13$	−41.8142	−0.892090	−0.446045	+	0.895011i	$0.647168\pi$
$14$	0	0
$15$	−31.9387	−0.549769
$16$	0	0
$17$	89.8459	1.28181	0.640907	−	0.767618i	$-0.278558\pi$
$17$	89.8459	1.28181	0.640907	+	0.767618i	$0.278558\pi$
$18$	0	0
$19$	−102.477	−1.23736	−0.618679	−	0.785644i	$-0.712332\pi$
$19$	−102.477	−1.23736	−0.618679	+	0.785644i	$0.712332\pi$
$20$	0	0
$21$	36.9819	0.384291
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−84.2976	−0.600855
$28$	0	0
$29$	−77.9931	−0.499412	−0.249706	−	0.968322i	$-0.580334\pi$
$29$	−77.9931	−0.499412	−0.249706	+	0.968322i	$0.580334\pi$
$30$	0	0
$31$	17.8575	0.103461	0.0517307	−	0.998661i	$-0.483526\pi$
$31$	17.8575	0.103461	0.0517307	+	0.998661i	$0.483526\pi$
$32$	0	0
$33$	19.4192	0.102438
$34$	0	0
$35$	−28.9476	−0.139801
$36$	0	0
$37$	236.953	1.05283	0.526417	−	0.850226i	$-0.323535\pi$
$37$	236.953	1.05283	0.526417	+	0.850226i	$0.323535\pi$
$38$	0	0
$39$	−267.098	−1.09667
$40$	0	0
$41$	−318.152	−1.21188	−0.605939	−	0.795511i	$-0.707202\pi$
$41$	−318.152	−1.21188	−0.605939	+	0.795511i	$0.707202\pi$
$42$	0	0
$43$	31.9573	0.113336	0.0566679	−	0.998393i	$-0.481952\pi$
$43$	31.9573	0.113336	0.0566679	+	0.998393i	$0.481952\pi$
$44$	0	0
$45$	−69.0161	−0.228629
$46$	0	0
$47$	−239.636	−0.743713	−0.371856	−	0.928290i	$-0.621279\pi$
$47$	−239.636	−0.743713	−0.371856	+	0.928290i	$0.621279\pi$
$48$	0	0
$49$	−309.482	−0.902278
$50$	0	0
$51$	573.913	1.57576
$52$	0	0
$53$	498.695	1.29247	0.646236	−	0.763137i	$-0.276342\pi$
$53$	498.695	1.29247	0.646236	+	0.763137i	$0.276342\pi$
$54$	0	0
$55$	−15.2004	−0.0372658
$56$	0	0
$57$	−654.596	−1.52111
$58$	0	0
$59$	−276.837	−0.610867	−0.305434	−	0.952213i	$-0.598801\pi$
$59$	−276.837	−0.610867	−0.305434	+	0.952213i	$0.598801\pi$
$60$	0	0
$61$	−86.9389	−0.182482	−0.0912409	−	0.995829i	$-0.529083\pi$
$61$	−86.9389	−0.182482	−0.0912409	+	0.995829i	$0.529083\pi$
$62$	0	0
$63$	79.9140	0.159813
$64$	0	0
$65$	209.071	0.398955
$66$	0	0
$67$	−1019.33	−1.85866	−0.929332	−	0.369246i	$-0.879616\pi$
$67$	−1019.33	−1.85866	−0.929332	+	0.369246i	$0.879616\pi$
$68$	0	0
$69$	146.918	0.256331
$70$	0	0
$71$	−1020.60	−1.70596	−0.852978	−	0.521947i	$-0.825206\pi$
$71$	−1020.60	−1.70596	−0.852978	+	0.521947i	$0.825206\pi$
$72$	0	0
$73$	−104.709	−0.167880	−0.0839398	−	0.996471i	$-0.526750\pi$
$73$	−104.709	−0.167880	−0.0839398	+	0.996471i	$0.526750\pi$
$74$	0	0
$75$	159.694	0.245864
$76$	0	0
$77$	17.6006	0.0260490
$78$	0	0
$79$	1052.52	1.49896	0.749482	−	0.662025i	$-0.230303\pi$
$79$	1052.52	1.49896	0.749482	+	0.662025i	$0.230303\pi$
$80$	0	0
$81$	−911.158	−1.24987
$82$	0	0
$83$	1006.59	1.33118	0.665589	−	0.746318i	$-0.268180\pi$
$83$	1006.59	1.33118	0.665589	+	0.746318i	$0.268180\pi$
$84$	0	0
$85$	−449.230	−0.573245
$86$	0	0
$87$	−498.200	−0.613938
$88$	0	0
$89$	−1098.26	−1.30804	−0.654022	−	0.756476i	$-0.726919\pi$
$89$	−1098.26	−1.30804	−0.654022	+	0.756476i	$0.726919\pi$
$90$	0	0
$91$	−242.084	−0.278871
$92$	0	0
$93$	114.069	0.127187
$94$	0	0
$95$	512.384	0.553364
$96$	0	0
$97$	−489.424	−0.512304	−0.256152	−	0.966637i	$-0.582455\pi$
$97$	−489.424	−0.512304	−0.256152	+	0.966637i	$0.582455\pi$
$98$	0	0
$99$	41.9628	0.0426002
$100$	0	0
$101$	−434.531	−0.428094	−0.214047	−	0.976823i	$-0.568665\pi$
$101$	−434.531	−0.428094	−0.214047	+	0.976823i	$0.568665\pi$
$102$	0	0
$103$	−409.330	−0.391577	−0.195789	−	0.980646i	$-0.562727\pi$
$103$	−409.330	−0.391577	−0.195789	+	0.980646i	$0.562727\pi$
$104$	0	0
$105$	−184.910	−0.171860
$106$	0	0
$107$	1944.48	1.75683	0.878413	−	0.477902i	$-0.158603\pi$
$107$	1944.48	1.75683	0.878413	+	0.477902i	$0.158603\pi$
$108$	0	0
$109$	322.594	0.283476	0.141738	−	0.989904i	$-0.454731\pi$
$109$	322.594	0.283476	0.141738	+	0.989904i	$0.454731\pi$
$110$	0	0
$111$	1513.60	1.29427
$112$	0	0
$113$	−2353.00	−1.95886	−0.979432	−	0.201777i	$-0.935328\pi$
$113$	−2353.00	−1.95886	−0.979432	+	0.201777i	$0.935328\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−577.171	−0.456064
$118$	0	0
$119$	520.164	0.400701
$120$	0	0
$121$	−1321.76	−0.993056
$122$	0	0
$123$	−2032.27	−1.48979
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	133.612	0.0933557	0.0466779	−	0.998910i	$-0.485137\pi$
$127$	133.612	0.0933557	0.0466779	+	0.998910i	$0.485137\pi$
$128$	0	0
$129$	204.135	0.139326
$130$	0	0
$131$	775.474	0.517202	0.258601	−	0.965984i	$-0.416739\pi$
$131$	775.474	0.517202	0.258601	+	0.965984i	$0.416739\pi$
$132$	0	0
$133$	−593.292	−0.386804
$134$	0	0
$135$	421.488	0.268710
$136$	0	0
$137$	987.824	0.616026	0.308013	−	0.951382i	$-0.400336\pi$
$137$	987.824	0.616026	0.308013	+	0.951382i	$0.400336\pi$
$138$	0	0
$139$	−2051.26	−1.25170	−0.625848	−	0.779945i	$-0.715247\pi$
$139$	−2051.26	−1.25170	−0.625848	+	0.779945i	$0.715247\pi$
$140$	0	0
$141$	−1530.73	−0.914262
$142$	0	0
$143$	−127.118	−0.0743368
$144$	0	0
$145$	389.966	0.223344
$146$	0	0
$147$	−1976.89	−1.10919
$148$	0	0
$149$	−2554.07	−1.40428	−0.702141	−	0.712038i	$-0.747772\pi$
$149$	−2554.07	−1.40428	−0.702141	+	0.712038i	$0.747772\pi$
$150$	0	0
$151$	1312.77	0.707495	0.353748	−	0.935341i	$-0.384907\pi$
$151$	1312.77	0.707495	0.353748	+	0.935341i	$0.384907\pi$
$152$	0	0
$153$	1240.16	0.655303
$154$	0	0
$155$	−89.2875	−0.0462693
$156$	0	0
$157$	−3430.46	−1.74382	−0.871912	−	0.489662i	$-0.837120\pi$
$157$	−3430.46	−1.74382	−0.871912	+	0.489662i	$0.837120\pi$
$158$	0	0
$159$	3185.54	1.58886
$160$	0	0
$161$	133.159	0.0651825
$162$	0	0
$163$	−190.995	−0.0917785	−0.0458892	−	0.998947i	$-0.514612\pi$
$163$	−190.995	−0.0917785	−0.0458892	+	0.998947i	$0.514612\pi$
$164$	0	0
$165$	−97.0960	−0.0458116
$166$	0	0
$167$	−621.559	−0.288010	−0.144005	−	0.989577i	$-0.545998\pi$
$167$	−621.559	−0.288010	−0.144005	+	0.989577i	$0.545998\pi$
$168$	0	0
$169$	−448.574	−0.204176
$170$	0	0
$171$	−1414.51	−0.632576
$172$	0	0
$173$	3722.24	1.63582	0.817910	−	0.575346i	$-0.195133\pi$
$173$	3722.24	1.63582	0.817910	+	0.575346i	$0.195133\pi$
$174$	0	0
$175$	144.738	0.0625209
$176$	0	0
$177$	−1768.37	−0.750952
$178$	0	0
$179$	−170.146	−0.0710464	−0.0355232	−	0.999369i	$-0.511310\pi$
$179$	−170.146	−0.0710464	−0.0355232	+	0.999369i	$0.511310\pi$
$180$	0	0
$181$	−80.7800	−0.0331731	−0.0165865	−	0.999862i	$-0.505280\pi$
$181$	−80.7800	−0.0331731	−0.0165865	+	0.999862i	$0.505280\pi$
$182$	0	0
$183$	−555.343	−0.224329
$184$	0	0
$185$	−1184.77	−0.470842
$186$	0	0
$187$	273.138	0.106812
$188$	0	0
$189$	−488.042	−0.187830
$190$	0	0
$191$	1105.26	0.418712	0.209356	−	0.977839i	$-0.432863\pi$
$191$	1105.26	0.418712	0.209356	+	0.977839i	$0.432863\pi$
$192$	0	0
$193$	−5280.24	−1.96933	−0.984663	−	0.174464i	$-0.944181\pi$
$193$	−5280.24	−1.96933	−0.984663	+	0.174464i	$0.944181\pi$
$194$	0	0
$195$	1335.49	0.490444
$196$	0	0
$197$	1592.44	0.575922	0.287961	−	0.957642i	$-0.407023\pi$
$197$	1592.44	0.575922	0.287961	+	0.957642i	$0.407023\pi$
$198$	0	0
$199$	−3585.69	−1.27730	−0.638651	−	0.769497i	$-0.720507\pi$
$199$	−3585.69	−1.27730	−0.638651	+	0.769497i	$0.720507\pi$
$200$	0	0
$201$	−6511.19	−2.28489
$202$	0	0
$203$	−451.542	−0.156119
$204$	0	0
$205$	1590.76	0.541968
$206$	0	0
$207$	317.474	0.106599
$208$	0	0
$209$	−311.537	−0.103108
$210$	0	0
$211$	−140.259	−0.0457623	−0.0228812	−	0.999738i	$-0.507284\pi$
$211$	−140.259	−0.0457623	−0.0228812	+	0.999738i	$0.507284\pi$
$212$	0	0
$213$	−6519.33	−2.09717
$214$	0	0
$215$	−159.786	−0.0506853
$216$	0	0
$217$	103.386	0.0323425
$218$	0	0
$219$	−668.851	−0.206378
$220$	0	0
$221$	−3756.83	−1.14349
$222$	0	0
$223$	457.275	0.137316	0.0686579	−	0.997640i	$-0.478128\pi$
$223$	457.275	0.137316	0.0686579	+	0.997640i	$0.478128\pi$
$224$	0	0
$225$	345.081	0.102246
$226$	0	0
$227$	207.632	0.0607093	0.0303546	−	0.999539i	$-0.490336\pi$
$227$	207.632	0.0607093	0.0303546	+	0.999539i	$0.490336\pi$
$228$	0	0
$229$	−2372.26	−0.684555	−0.342278	−	0.939599i	$-0.611198\pi$
$229$	−2372.26	−0.684555	−0.342278	+	0.939599i	$0.611198\pi$
$230$	0	0
$231$	112.428	0.0320225
$232$	0	0
$233$	6027.61	1.69477	0.847385	−	0.530978i	$-0.178175\pi$
$233$	6027.61	1.69477	0.847385	+	0.530978i	$0.178175\pi$
$234$	0	0
$235$	1198.18	0.332599
$236$	0	0
$237$	6723.25	1.84271
$238$	0	0
$239$	−4764.03	−1.28937	−0.644686	−	0.764448i	$-0.723012\pi$
$239$	−4764.03	−1.28937	−0.644686	+	0.764448i	$0.723012\pi$
$240$	0	0
$241$	−195.211	−0.0521771	−0.0260885	−	0.999660i	$-0.508305\pi$
$241$	−195.211	−0.0521771	−0.0260885	+	0.999660i	$0.508305\pi$
$242$	0	0
$243$	−3544.21	−0.935642
$244$	0	0
$245$	1547.41	0.403511
$246$	0	0
$247$	4284.99	1.10384
$248$	0	0
$249$	6429.85	1.63645
$250$	0	0
$251$	5890.81	1.48137	0.740686	−	0.671851i	$-0.234501\pi$
$251$	5890.81	1.48137	0.740686	+	0.671851i	$0.234501\pi$
$252$	0	0
$253$	69.9217	0.0173753
$254$	0	0
$255$	−2869.56	−0.704702
$256$	0	0
$257$	4390.17	1.06557	0.532784	−	0.846251i	$-0.321146\pi$
$257$	4390.17	1.06557	0.532784	+	0.846251i	$0.321146\pi$
$258$	0	0
$259$	1371.84	0.329121
$260$	0	0
$261$	−1076.56	−0.255315
$262$	0	0
$263$	−1403.84	−0.329142	−0.164571	−	0.986365i	$-0.552624\pi$
$263$	−1403.84	−0.329142	−0.164571	+	0.986365i	$0.552624\pi$
$264$	0	0
$265$	−2493.48	−0.578011
$266$	0	0
$267$	−7015.43	−1.60801
$268$	0	0
$269$	2725.26	0.617704	0.308852	−	0.951110i	$-0.400055\pi$
$269$	2725.26	0.617704	0.308852	+	0.951110i	$0.400055\pi$
$270$	0	0
$271$	−230.366	−0.0516375	−0.0258188	−	0.999667i	$-0.508219\pi$
$271$	−230.366	−0.0516375	−0.0258188	+	0.999667i	$0.508219\pi$
$272$	0	0
$273$	−1546.37	−0.342822
$274$	0	0
$275$	76.0019	0.0166658
$276$	0	0
$277$	5516.86	1.19666	0.598332	−	0.801248i	$-0.295830\pi$
$277$	5516.86	1.19666	0.598332	+	0.801248i	$0.295830\pi$
$278$	0	0
$279$	246.491	0.0528926
$280$	0	0
$281$	−5384.04	−1.14301	−0.571504	−	0.820600i	$-0.693640\pi$
$281$	−5384.04	−1.14301	−0.571504	+	0.820600i	$0.693640\pi$
$282$	0	0
$283$	−2449.05	−0.514420	−0.257210	−	0.966355i	$-0.582803\pi$
$283$	−2449.05	−0.514420	−0.257210	+	0.966355i	$0.582803\pi$
$284$	0	0
$285$	3272.98	0.680262
$286$	0	0
$287$	−1841.94	−0.378838
$288$	0	0
$289$	3159.29	0.643047
$290$	0	0
$291$	−3126.31	−0.629786
$292$	0	0
$293$	5257.98	1.04838	0.524189	−	0.851602i	$-0.324369\pi$
$293$	5257.98	1.04838	0.524189	+	0.851602i	$0.324369\pi$
$294$	0	0
$295$	1384.19	0.273188
$296$	0	0
$297$	−256.271	−0.0500685
$298$	0	0
$299$	−961.726	−0.186014
$300$	0	0
$301$	185.017	0.0354293
$302$	0	0
$303$	−2775.67	−0.526265
$304$	0	0
$305$	434.695	0.0816083
$306$	0	0
$307$	5730.30	1.06530	0.532648	−	0.846337i	$-0.321197\pi$
$307$	5730.30	1.06530	0.532648	+	0.846337i	$0.321197\pi$
$308$	0	0
$309$	−2614.69	−0.481374
$310$	0	0
$311$	6887.88	1.25587	0.627936	−	0.778265i	$-0.283900\pi$
$311$	6887.88	1.25587	0.627936	+	0.778265i	$0.283900\pi$
$312$	0	0
$313$	−4574.19	−0.826033	−0.413017	−	0.910723i	$-0.635525\pi$
$313$	−4574.19	−0.826033	−0.413017	+	0.910723i	$0.635525\pi$
$314$	0	0
$315$	−399.570	−0.0714705
$316$	0	0
$317$	−6481.61	−1.14840	−0.574201	−	0.818714i	$-0.694687\pi$
$317$	−6481.61	−1.14840	−0.574201	+	0.818714i	$0.694687\pi$
$318$	0	0
$319$	−237.105	−0.0416154
$320$	0	0
$321$	12420.9	2.15970
$322$	0	0
$323$	−9207.13	−1.58606
$324$	0	0
$325$	−1045.35	−0.178418
$326$	0	0
$327$	2060.65	0.348483
$328$	0	0
$329$	−1387.38	−0.232488
$330$	0	0
$331$	5626.40	0.934305	0.467153	−	0.884177i	$-0.345280\pi$
$331$	5626.40	0.934305	0.467153	+	0.884177i	$0.345280\pi$
$332$	0	0
$333$	3270.72	0.538241
$334$	0	0
$335$	5096.63	0.831220
$336$	0	0
$337$	1655.51	0.267601	0.133800	−	0.991008i	$-0.457282\pi$
$337$	1655.51	0.267601	0.133800	+	0.991008i	$0.457282\pi$
$338$	0	0
$339$	−15030.3	−2.40807
$340$	0	0
$341$	54.2881	0.00862131
$342$	0	0
$343$	−3777.55	−0.594661
$344$	0	0
$345$	−734.590	−0.114635
$346$	0	0
$347$	−3969.00	−0.614026	−0.307013	−	0.951705i	$-0.599330\pi$
$347$	−3969.00	−0.614026	−0.307013	+	0.951705i	$0.599330\pi$
$348$	0	0
$349$	−2786.18	−0.427337	−0.213669	−	0.976906i	$-0.568541\pi$
$349$	−2786.18	−0.427337	−0.213669	+	0.976906i	$0.568541\pi$
$350$	0	0
$351$	3524.83	0.536016
$352$	0	0
$353$	−635.463	−0.0958138	−0.0479069	−	0.998852i	$-0.515255\pi$
$353$	−635.463	−0.0958138	−0.0479069	+	0.998852i	$0.515255\pi$
$354$	0	0
$355$	5103.00	0.762927
$356$	0	0
$357$	3322.68	0.492590
$358$	0	0
$359$	−10070.3	−1.48047	−0.740237	−	0.672346i	$-0.765287\pi$
$359$	−10070.3	−1.48047	−0.740237	+	0.672346i	$0.765287\pi$
$360$	0	0
$361$	3642.51	0.531056
$362$	0	0
$363$	−8443.05	−1.22079
$364$	0	0
$365$	523.543	0.0750780
$366$	0	0
$367$	10473.3	1.48964	0.744822	−	0.667263i	$-0.232534\pi$
$367$	10473.3	1.48964	0.744822	+	0.667263i	$0.232534\pi$
$368$	0	0
$369$	−4391.52	−0.619549
$370$	0	0
$371$	2887.20	0.404033
$372$	0	0
$373$	5752.25	0.798499	0.399250	−	0.916842i	$-0.369271\pi$
$373$	5752.25	0.798499	0.399250	+	0.916842i	$0.369271\pi$
$374$	0	0
$375$	−798.468	−0.109954
$376$	0	0
$377$	3261.22	0.445521
$378$	0	0
$379$	105.211	0.0142595	0.00712974	−	0.999975i	$-0.497731\pi$
$379$	105.211	0.0142595	0.00712974	+	0.999975i	$0.497731\pi$
$380$	0	0
$381$	853.481	0.114764
$382$	0	0
$383$	7728.47	1.03109	0.515544	−	0.856863i	$-0.327590\pi$
$383$	7728.47	1.03109	0.515544	+	0.856863i	$0.327590\pi$
$384$	0	0
$385$	−88.0028	−0.0116494
$386$	0	0
$387$	441.113	0.0579407
$388$	0	0
$389$	−10791.3	−1.40653	−0.703266	−	0.710927i	$-0.748276\pi$
$389$	−10791.3	−1.40653	−0.703266	+	0.710927i	$0.748276\pi$
$390$	0	0
$391$	2066.46	0.267277
$392$	0	0
$393$	4953.52	0.635807
$394$	0	0
$395$	−5262.62	−0.670357
$396$	0	0
$397$	12131.5	1.53366	0.766831	−	0.641849i	$-0.221832\pi$
$397$	12131.5	1.53366	0.766831	+	0.641849i	$0.221832\pi$
$398$	0	0
$399$	−3789.79	−0.475506
$400$	0	0
$401$	−11439.2	−1.42455	−0.712275	−	0.701901i	$-0.752335\pi$
$401$	−11439.2	−1.42455	−0.712275	+	0.701901i	$0.752335\pi$
$402$	0	0
$403$	−746.697	−0.0922968
$404$	0	0
$405$	4555.79	0.558961
$406$	0	0
$407$	720.356	0.0877315
$408$	0	0
$409$	−6923.37	−0.837013	−0.418507	−	0.908214i	$-0.637446\pi$
$409$	−6923.37	−0.837013	−0.418507	+	0.908214i	$0.637446\pi$
$410$	0	0
$411$	6309.96	0.757293
$412$	0	0
$413$	−1602.75	−0.190960
$414$	0	0
$415$	−5032.96	−0.595321
$416$	0	0
$417$	−13102.9	−1.53874
$418$	0	0
$419$	−11554.9	−1.34724	−0.673622	−	0.739076i	$-0.735262\pi$
$419$	−11554.9	−1.34724	−0.673622	+	0.739076i	$0.735262\pi$
$420$	0	0
$421$	−3922.75	−0.454117	−0.227058	−	0.973881i	$-0.572911\pi$
$421$	−3922.75	−0.454117	−0.227058	+	0.973881i	$0.572911\pi$
$422$	0	0
$423$	−3307.75	−0.380209
$424$	0	0
$425$	2246.15	0.256363
$426$	0	0
$427$	−503.334	−0.0570446
$428$	0	0
$429$	−811.998	−0.0913838
$430$	0	0
$431$	−7190.68	−0.803626	−0.401813	−	0.915722i	$-0.631620\pi$
$431$	−7190.68	−0.803626	−0.401813	+	0.915722i	$0.631620\pi$
$432$	0	0
$433$	2292.30	0.254413	0.127206	−	0.991876i	$-0.459399\pi$
$433$	2292.30	0.254413	0.127206	+	0.991876i	$0.459399\pi$
$434$	0	0
$435$	2491.00	0.274562
$436$	0	0
$437$	−2356.97	−0.258007
$438$	0	0
$439$	11453.9	1.24525	0.622627	−	0.782519i	$-0.286065\pi$
$439$	11453.9	1.24525	0.622627	+	0.782519i	$0.286065\pi$
$440$	0	0
$441$	−4271.84	−0.461272
$442$	0	0
$443$	−15928.7	−1.70835	−0.854173	−	0.519990i	$-0.825936\pi$
$443$	−15928.7	−1.70835	−0.854173	+	0.519990i	$0.825936\pi$
$444$	0	0
$445$	5491.32	0.584975
$446$	0	0
$447$	−16314.8	−1.72631
$448$	0	0
$449$	−13852.4	−1.45598	−0.727988	−	0.685590i	$-0.759544\pi$
$449$	−13852.4	−1.45598	−0.727988	+	0.685590i	$0.759544\pi$
$450$	0	0
$451$	−967.205	−0.100984
$452$	0	0
$453$	8385.64	0.869739
$454$	0	0
$455$	1210.42	0.124715
$456$	0	0
$457$	1993.34	0.204036	0.102018	−	0.994783i	$-0.467470\pi$
$457$	1993.34	0.204036	0.102018	+	0.994783i	$0.467470\pi$
$458$	0	0
$459$	−7573.79	−0.770184
$460$	0	0
$461$	−2772.98	−0.280153	−0.140076	−	0.990141i	$-0.544735\pi$
$461$	−2772.98	−0.280153	−0.140076	+	0.990141i	$0.544735\pi$
$462$	0	0
$463$	13282.8	1.33327	0.666635	−	0.745385i	$-0.267734\pi$
$463$	13282.8	1.33327	0.666635	+	0.745385i	$0.267734\pi$
$464$	0	0
$465$	−570.345	−0.0568799
$466$	0	0
$467$	−7196.43	−0.713086	−0.356543	−	0.934279i	$-0.616045\pi$
$467$	−7196.43	−0.713086	−0.356543	+	0.934279i	$0.616045\pi$
$468$	0	0
$469$	−5901.40	−0.581027
$470$	0	0
$471$	−21912.9	−2.14372
$472$	0	0
$473$	97.1524	0.00944413
$474$	0	0
$475$	−2561.92	−0.247472
$476$	0	0
$477$	6883.60	0.660752
$478$	0	0
$479$	6023.00	0.574525	0.287263	−	0.957852i	$-0.407255\pi$
$479$	6023.00	0.574525	0.287263	+	0.957852i	$0.407255\pi$
$480$	0	0
$481$	−9908.01	−0.939223
$482$	0	0
$483$	850.584	0.0801303
$484$	0	0
$485$	2447.12	0.229109
$486$	0	0
$487$	16386.8	1.52476	0.762380	−	0.647130i	$-0.224031\pi$
$487$	16386.8	1.52476	0.762380	+	0.647130i	$0.224031\pi$
$488$	0	0
$489$	−1220.03	−0.112825
$490$	0	0
$491$	1991.00	0.182999	0.0914993	−	0.995805i	$-0.470834\pi$
$491$	1991.00	0.182999	0.0914993	+	0.995805i	$0.470834\pi$
$492$	0	0
$493$	−7007.37	−0.640154
$494$	0	0
$495$	−209.814	−0.0190514
$496$	0	0
$497$	−5908.78	−0.533290
$498$	0	0
$499$	−3012.35	−0.270244	−0.135122	−	0.990829i	$-0.543143\pi$
$499$	−3012.35	−0.270244	−0.135122	+	0.990829i	$0.543143\pi$
$500$	0	0
$501$	−3970.35	−0.354057
$502$	0	0
$503$	−13147.4	−1.16543	−0.582716	−	0.812676i	$-0.698010\pi$
$503$	−13147.4	−1.16543	−0.582716	+	0.812676i	$0.698010\pi$
$504$	0	0
$505$	2172.66	0.191449
$506$	0	0
$507$	−2865.37	−0.250997
$508$	0	0
$509$	−1885.12	−0.164158	−0.0820790	−	0.996626i	$-0.526156\pi$
$509$	−1885.12	−0.164158	−0.0820790	+	0.996626i	$0.526156\pi$
$510$	0	0
$511$	−606.212	−0.0524799
$512$	0	0
$513$	8638.55	0.743472
$514$	0	0
$515$	2046.65	0.175119
$516$	0	0
$517$	−728.511	−0.0619727
$518$	0	0
$519$	23776.7	2.01095
$520$	0	0
$521$	8179.75	0.687833	0.343917	−	0.939000i	$-0.388246\pi$
$521$	8179.75	0.687833	0.343917	+	0.939000i	$0.388246\pi$
$522$	0	0
$523$	2525.47	0.211149	0.105575	−	0.994411i	$-0.466332\pi$
$523$	2525.47	0.211149	0.105575	+	0.994411i	$0.466332\pi$
$524$	0	0
$525$	924.548	0.0768583
$526$	0	0
$527$	1604.42	0.132618
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−3821.25	−0.312294
$532$	0	0
$533$	13303.3	1.08110
$534$	0	0
$535$	−9722.42	−0.785677
$536$	0	0
$537$	−1086.85	−0.0873389
$538$	0	0
$539$	−940.847	−0.0751858
$540$	0	0
$541$	10062.6	0.799680	0.399840	−	0.916585i	$-0.369066\pi$
$541$	10062.6	0.799680	0.399840	+	0.916585i	$0.369066\pi$
$542$	0	0
$543$	−516.002	−0.0407804
$544$	0	0
$545$	−1612.97	−0.126774
$546$	0	0
$547$	4559.71	0.356415	0.178208	−	0.983993i	$-0.442970\pi$
$547$	4559.71	0.356415	0.178208	+	0.983993i	$0.442970\pi$
$548$	0	0
$549$	−1200.04	−0.0932903
$550$	0	0
$551$	7992.49	0.617952
$552$	0	0
$553$	6093.60	0.468583
$554$	0	0
$555$	−7567.98	−0.578816
$556$	0	0
$557$	−17074.5	−1.29887	−0.649433	−	0.760419i	$-0.724994\pi$
$557$	−17074.5	−1.29887	−0.649433	+	0.760419i	$0.724994\pi$
$558$	0	0
$559$	−1336.27	−0.101106
$560$	0	0
$561$	1744.74	0.131306
$562$	0	0
$563$	2166.16	0.162154	0.0810772	−	0.996708i	$-0.474164\pi$
$563$	2166.16	0.162154	0.0810772	+	0.996708i	$0.474164\pi$
$564$	0	0
$565$	11765.0	0.876030
$566$	0	0
$567$	−5275.16	−0.390716
$568$	0	0
$569$	−8376.91	−0.617185	−0.308593	−	0.951194i	$-0.599858\pi$
$569$	−8376.91	−0.617185	−0.308593	+	0.951194i	$0.599858\pi$
$570$	0	0
$571$	−5555.57	−0.407169	−0.203585	−	0.979057i	$-0.565259\pi$
$571$	−5555.57	−0.407169	−0.203585	+	0.979057i	$0.565259\pi$
$572$	0	0
$573$	7060.13	0.514732
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−16122.8	−1.16326	−0.581630	−	0.813454i	$-0.697585\pi$
$577$	−16122.8	−1.16326	−0.581630	+	0.813454i	$0.697585\pi$
$578$	0	0
$579$	−33728.8	−2.42094
$580$	0	0
$581$	5827.68	0.416132
$582$	0	0
$583$	1516.07	0.107700
$584$	0	0
$585$	2885.85	0.203958
$586$	0	0
$587$	26083.7	1.83405	0.917027	−	0.398826i	$-0.130582\pi$
$587$	26083.7	1.83405	0.917027	+	0.398826i	$0.130582\pi$
$588$	0	0
$589$	−1829.98	−0.128019
$590$	0	0
$591$	10172.1	0.707993
$592$	0	0
$593$	−12843.5	−0.889410	−0.444705	−	0.895677i	$-0.646691\pi$
$593$	−12843.5	−0.889410	−0.444705	+	0.895677i	$0.646691\pi$
$594$	0	0
$595$	−2600.82	−0.179199
$596$	0	0
$597$	−22904.5	−1.57021
$598$	0	0
$599$	−3857.91	−0.263155	−0.131578	−	0.991306i	$-0.542004\pi$
$599$	−3857.91	−0.263155	−0.131578	+	0.991306i	$0.542004\pi$
$600$	0	0
$601$	−1568.28	−0.106442	−0.0532210	−	0.998583i	$-0.516949\pi$
$601$	−1568.28	−0.106442	−0.0532210	+	0.998583i	$0.516949\pi$
$602$	0	0
$603$	−14070.0	−0.950206
$604$	0	0
$605$	6608.79	0.444108
$606$	0	0
$607$	13332.7	0.891528	0.445764	−	0.895150i	$-0.352932\pi$
$607$	13332.7	0.891528	0.445764	+	0.895150i	$0.352932\pi$
$608$	0	0
$609$	−2884.34	−0.191920
$610$	0	0
$611$	10020.2	0.663459
$612$	0	0
$613$	17285.0	1.13889	0.569443	−	0.822031i	$-0.307159\pi$
$613$	17285.0	1.13889	0.569443	+	0.822031i	$0.307159\pi$
$614$	0	0
$615$	10161.4	0.666253
$616$	0	0
$617$	17342.5	1.13157	0.565787	−	0.824551i	$-0.308572\pi$
$617$	17342.5	1.13157	0.565787	+	0.824551i	$0.308572\pi$
$618$	0	0
$619$	−10636.7	−0.690670	−0.345335	−	0.938479i	$-0.612235\pi$
$619$	−10636.7	−0.690670	−0.345335	+	0.938479i	$0.612235\pi$
$620$	0	0
$621$	−1938.84	−0.125287
$622$	0	0
$623$	−6358.42	−0.408900
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−1990.02	−0.126752
$628$	0	0
$629$	21289.3	1.34954
$630$	0	0
$631$	19106.3	1.20540	0.602702	−	0.797966i	$-0.294091\pi$
$631$	19106.3	1.20540	0.602702	+	0.797966i	$0.294091\pi$
$632$	0	0
$633$	−895.940	−0.0562566
$634$	0	0
$635$	−668.062	−0.0417500
$636$	0	0
$637$	12940.7	0.804914
$638$	0	0
$639$	−14087.6	−0.872137
$640$	0	0
$641$	12081.8	0.744466	0.372233	−	0.928139i	$-0.378592\pi$
$641$	12081.8	0.744466	0.372233	+	0.928139i	$0.378592\pi$
$642$	0	0
$643$	1075.78	0.0659794	0.0329897	−	0.999456i	$-0.489497\pi$
$643$	1075.78	0.0659794	0.0329897	+	0.999456i	$0.489497\pi$
$644$	0	0
$645$	−1020.67	−0.0623085
$646$	0	0
$647$	−1762.84	−0.107117	−0.0535584	−	0.998565i	$-0.517056\pi$
$647$	−1762.84	−0.107117	−0.0535584	+	0.998565i	$0.517056\pi$
$648$	0	0
$649$	−841.607	−0.0509028
$650$	0	0
$651$	660.405	0.0397593
$652$	0	0
$653$	−22927.7	−1.37401	−0.687007	−	0.726651i	$-0.741076\pi$
$653$	−22927.7	−1.37401	−0.687007	+	0.726651i	$0.741076\pi$
$654$	0	0
$655$	−3877.37	−0.231300
$656$	0	0
$657$	−1445.32	−0.0858252
$658$	0	0
$659$	21744.5	1.28535	0.642674	−	0.766140i	$-0.277825\pi$
$659$	21744.5	1.28535	0.642674	+	0.766140i	$0.277825\pi$
$660$	0	0
$661$	12575.4	0.739982	0.369991	−	0.929035i	$-0.379361\pi$
$661$	12575.4	0.739982	0.369991	+	0.929035i	$0.379361\pi$
$662$	0	0
$663$	−23997.7	−1.40572
$664$	0	0
$665$	2966.46	0.172984
$666$	0	0
$667$	−1793.84	−0.104135
$668$	0	0
$669$	2920.96	0.168805
$670$	0	0
$671$	−264.301	−0.0152060
$672$	0	0
$673$	−8350.98	−0.478316	−0.239158	−	0.970981i	$-0.576871\pi$
$673$	−8350.98	−0.478316	−0.239158	+	0.970981i	$0.576871\pi$
$674$	0	0
$675$	−2107.44	−0.120171
$676$	0	0
$677$	12815.0	0.727507	0.363754	−	0.931495i	$-0.381495\pi$
$677$	12815.0	0.727507	0.363754	+	0.931495i	$0.381495\pi$
$678$	0	0
$679$	−2833.53	−0.160149
$680$	0	0
$681$	1326.30	0.0746312
$682$	0	0
$683$	−20368.0	−1.14108	−0.570542	−	0.821268i	$-0.693267\pi$
$683$	−20368.0	−1.14108	−0.570542	+	0.821268i	$0.693267\pi$
$684$	0	0
$685$	−4939.12	−0.275495
$686$	0	0
$687$	−15153.4	−0.841538
$688$	0	0
$689$	−20852.5	−1.15300
$690$	0	0
$691$	1766.94	0.0972759	0.0486379	−	0.998816i	$-0.484512\pi$
$691$	1766.94	0.0972759	0.0486379	+	0.998816i	$0.484512\pi$
$692$	0	0
$693$	242.945	0.0133170
$694$	0	0
$695$	10256.3	0.559775
$696$	0	0
$697$	−28584.6	−1.55340
$698$	0	0
$699$	38502.8	2.08342
$700$	0	0
$701$	7068.17	0.380829	0.190414	−	0.981704i	$-0.439017\pi$
$701$	7068.17	0.380829	0.190414	+	0.981704i	$0.439017\pi$
$702$	0	0
$703$	−24282.2	−1.30273
$704$	0	0
$705$	7653.66	0.408870
$706$	0	0
$707$	−2515.73	−0.133824
$708$	0	0
$709$	−5015.87	−0.265691	−0.132845	−	0.991137i	$-0.542411\pi$
$709$	−5015.87	−0.265691	−0.132845	+	0.991137i	$0.542411\pi$
$710$	0	0
$711$	14528.2	0.766316
$712$	0	0
$713$	410.723	0.0215732
$714$	0	0
$715$	635.591	0.0332444
$716$	0	0
$717$	−30431.4	−1.58505
$718$	0	0
$719$	−15104.0	−0.783426	−0.391713	−	0.920088i	$-0.628117\pi$
$719$	−15104.0	−0.783426	−0.391713	+	0.920088i	$0.628117\pi$
$720$	0	0
$721$	−2369.82	−0.122409
$722$	0	0
$723$	−1246.96	−0.0641424
$724$	0	0
$725$	−1949.83	−0.0998825
$726$	0	0
$727$	22043.5	1.12455	0.562275	−	0.826950i	$-0.309926\pi$
$727$	22043.5	1.12455	0.562275	+	0.826950i	$0.309926\pi$
$728$	0	0
$729$	1961.79	0.0996694
$730$	0	0
$731$	2871.23	0.145275
$732$	0	0
$733$	−17431.2	−0.878355	−0.439178	−	0.898400i	$-0.644730\pi$
$733$	−17431.2	−0.878355	−0.439178	+	0.898400i	$0.644730\pi$
$734$	0	0
$735$	9884.44	0.496045
$736$	0	0
$737$	−3098.83	−0.154880
$738$	0	0
$739$	39019.7	1.94230	0.971152	−	0.238463i	$-0.0766436\pi$
$739$	39019.7	1.94230	0.971152	+	0.238463i	$0.0766436\pi$
$740$	0	0
$741$	27371.4	1.35697
$742$	0	0
$743$	−2408.03	−0.118899	−0.0594495	−	0.998231i	$-0.518935\pi$
$743$	−2408.03	−0.118899	−0.0594495	+	0.998231i	$0.518935\pi$
$744$	0	0
$745$	12770.4	0.628014
$746$	0	0
$747$	13894.2	0.680539
$748$	0	0
$749$	11257.6	0.549192
$750$	0	0
$751$	13056.5	0.634406	0.317203	−	0.948358i	$-0.397256\pi$
$751$	13056.5	0.634406	0.317203	+	0.948358i	$0.397256\pi$
$752$	0	0
$753$	37628.9	1.82108
$754$	0	0
$755$	−6563.85	−0.316401
$756$	0	0
$757$	−16457.2	−0.790156	−0.395078	−	0.918648i	$-0.629282\pi$
$757$	−16457.2	−0.790156	−0.395078	+	0.918648i	$0.629282\pi$
$758$	0	0
$759$	446.642	0.0213598
$760$	0	0
$761$	20147.2	0.959705	0.479853	−	0.877349i	$-0.340690\pi$
$761$	20147.2	0.959705	0.479853	+	0.877349i	$0.340690\pi$
$762$	0	0
$763$	1867.66	0.0886160
$764$	0	0
$765$	−6200.82	−0.293060
$766$	0	0
$767$	11575.7	0.544949
$768$	0	0
$769$	17051.3	0.799591	0.399795	−	0.916604i	$-0.369081\pi$
$769$	17051.3	0.799591	0.399795	+	0.916604i	$0.369081\pi$
$770$	0	0
$771$	28043.2	1.30993
$772$	0	0
$773$	36283.3	1.68825	0.844125	−	0.536146i	$-0.180120\pi$
$773$	36283.3	1.68825	0.844125	+	0.536146i	$0.180120\pi$
$774$	0	0
$775$	446.438	0.0206923
$776$	0	0
$777$	8762.99	0.404595
$778$	0	0
$779$	32603.2	1.49953
$780$	0	0
$781$	−3102.70	−0.142155
$782$	0	0
$783$	6574.63	0.300074
$784$	0	0
$785$	17152.3	0.779862
$786$	0	0
$787$	17235.7	0.780670	0.390335	−	0.920673i	$-0.372359\pi$
$787$	17235.7	0.780670	0.390335	+	0.920673i	$0.372359\pi$
$788$	0	0
$789$	−8967.35	−0.404621
$790$	0	0
$791$	−13622.7	−0.612349
$792$	0	0
$793$	3635.28	0.162790
$794$	0	0
$795$	−15927.7	−0.710562
$796$	0	0
$797$	26857.3	1.19365	0.596823	−	0.802373i	$-0.296430\pi$
$797$	26857.3	1.19365	0.596823	+	0.802373i	$0.296430\pi$
$798$	0	0
$799$	−21530.3	−0.953302
$800$	0	0
$801$	−15159.6	−0.668712
$802$	0	0
$803$	−318.322	−0.0139892
$804$	0	0
$805$	−665.794	−0.0291505
$806$	0	0
$807$	17408.3	0.759356
$808$	0	0
$809$	−3922.75	−0.170478	−0.0852389	−	0.996361i	$-0.527165\pi$
$809$	−3922.75	−0.170478	−0.0852389	+	0.996361i	$0.527165\pi$
$810$	0	0
$811$	32637.4	1.41314	0.706568	−	0.707645i	$-0.250242\pi$
$811$	32637.4	1.41314	0.706568	+	0.707645i	$0.250242\pi$
$812$	0	0
$813$	−1471.52	−0.0634791
$814$	0	0
$815$	954.976	0.0410446
$816$	0	0
$817$	−3274.88	−0.140237
$818$	0	0
$819$	−3341.54	−0.142568
$820$	0	0
$821$	6375.47	0.271017	0.135509	−	0.990776i	$-0.456733\pi$
$821$	6375.47	0.271017	0.135509	+	0.990776i	$0.456733\pi$
$822$	0	0
$823$	43501.6	1.84249	0.921246	−	0.388981i	$-0.127173\pi$
$823$	43501.6	1.84249	0.921246	+	0.388981i	$0.127173\pi$
$824$	0	0
$825$	485.480	0.0204876
$826$	0	0
$827$	40508.7	1.70330	0.851648	−	0.524114i	$-0.175603\pi$
$827$	40508.7	1.70330	0.851648	+	0.524114i	$0.175603\pi$
$828$	0	0
$829$	26594.0	1.11417	0.557085	−	0.830456i	$-0.311920\pi$
$829$	26594.0	1.11417	0.557085	+	0.830456i	$0.311920\pi$
$830$	0	0
$831$	35240.3	1.47108
$832$	0	0
$833$	−27805.7	−1.15655
$834$	0	0
$835$	3107.79	0.128802
$836$	0	0
$837$	−1505.34	−0.0621652
$838$	0	0
$839$	−8885.84	−0.365642	−0.182821	−	0.983146i	$-0.558523\pi$
$839$	−8885.84	−0.365642	−0.182821	+	0.983146i	$0.558523\pi$
$840$	0	0
$841$	−18306.1	−0.750587
$842$	0	0
$843$	−34391.9	−1.40512
$844$	0	0
$845$	2242.87	0.0913101
$846$	0	0
$847$	−7652.34	−0.310434
$848$	0	0
$849$	−15643.9	−0.632388
$850$	0	0
$851$	5449.93	0.219531
$852$	0	0
$853$	−812.221	−0.0326025	−0.0163012	−	0.999867i	$-0.505189\pi$
$853$	−812.221	−0.0326025	−0.0163012	+	0.999867i	$0.505189\pi$
$854$	0	0
$855$	7072.56	0.282896
$856$	0	0
$857$	8502.24	0.338893	0.169446	−	0.985539i	$-0.445802\pi$
$857$	8502.24	0.338893	0.169446	+	0.985539i	$0.445802\pi$
$858$	0	0
$859$	37323.1	1.48248	0.741239	−	0.671241i	$-0.234239\pi$
$859$	37323.1	1.48248	0.741239	+	0.671241i	$0.234239\pi$
$860$	0	0
$861$	−11765.9	−0.465714
$862$	0	0
$863$	12734.0	0.502282	0.251141	−	0.967951i	$-0.419194\pi$
$863$	12734.0	0.502282	0.251141	+	0.967951i	$0.419194\pi$
$864$	0	0
$865$	−18611.2	−0.731561
$866$	0	0
$867$	20180.7	0.790512
$868$	0	0
$869$	3199.75	0.124907
$870$	0	0
$871$	42622.3	1.65810
$872$	0	0
$873$	−6755.63	−0.261906
$874$	0	0
$875$	−723.689	−0.0279602
$876$	0	0
$877$	−41366.0	−1.59274	−0.796368	−	0.604812i	$-0.793248\pi$
$877$	−41366.0	−1.59274	−0.796368	+	0.604812i	$0.793248\pi$
$878$	0	0
$879$	33586.6	1.28879
$880$	0	0
$881$	−7609.60	−0.291003	−0.145502	−	0.989358i	$-0.546480\pi$
$881$	−7609.60	−0.291003	−0.145502	+	0.989358i	$0.546480\pi$
$882$	0	0
$883$	−15972.5	−0.608739	−0.304369	−	0.952554i	$-0.598446\pi$
$883$	−15972.5	−0.608739	−0.304369	+	0.952554i	$0.598446\pi$
$884$	0	0
$885$	8841.83	0.335836
$886$	0	0
$887$	−39795.1	−1.50641	−0.753207	−	0.657784i	$-0.771494\pi$
$887$	−39795.1	−1.50641	−0.753207	+	0.657784i	$0.771494\pi$
$888$	0	0
$889$	773.551	0.0291834
$890$	0	0
$891$	−2769.99	−0.104151
$892$	0	0
$893$	24557.2	0.920239
$894$	0	0
$895$	850.730	0.0317729
$896$	0	0
$897$	−6143.26	−0.228670
$898$	0	0
$899$	−1392.76	−0.0516699
$900$	0	0
$901$	44805.7	1.65671
$902$	0	0
$903$	1181.84	0.0435539
$904$	0	0
$905$	403.900	0.0148355
$906$	0	0
$907$	37348.3	1.36729	0.683644	−	0.729815i	$-0.260394\pi$
$907$	37348.3	1.36729	0.683644	+	0.729815i	$0.260394\pi$
$908$	0	0
$909$	−5997.93	−0.218855
$910$	0	0
$911$	−13218.4	−0.480729	−0.240365	−	0.970683i	$-0.577267\pi$
$911$	−13218.4	−0.480729	−0.240365	+	0.970683i	$0.577267\pi$
$912$	0	0
$913$	3060.11	0.110926
$914$	0	0
$915$	2776.72	0.100323
$916$	0	0
$917$	4489.62	0.161680
$918$	0	0
$919$	40709.0	1.46122	0.730612	−	0.682792i	$-0.239235\pi$
$919$	40709.0	1.46122	0.730612	+	0.682792i	$0.239235\pi$
$920$	0	0
$921$	36603.7	1.30959
$922$	0	0
$923$	42675.5	1.52187
$924$	0	0
$925$	5923.83	0.210567
$926$	0	0
$927$	−5650.07	−0.200186
$928$	0	0
$929$	−34275.6	−1.21049	−0.605246	−	0.796039i	$-0.706925\pi$
$929$	−34275.6	−1.21049	−0.605246	+	0.796039i	$0.706925\pi$
$930$	0	0
$931$	31714.7	1.11644
$932$	0	0
$933$	43998.0	1.54387
$934$	0	0
$935$	−1365.69	−0.0477678
$936$	0	0
$937$	−9681.42	−0.337544	−0.168772	−	0.985655i	$-0.553980\pi$
$937$	−9681.42	−0.337544	−0.168772	+	0.985655i	$0.553980\pi$
$938$	0	0
$939$	−29218.7	−1.01546
$940$	0	0
$941$	−29631.7	−1.02653	−0.513266	−	0.858230i	$-0.671564\pi$
$941$	−29631.7	−1.02653	−0.513266	+	0.858230i	$0.671564\pi$
$942$	0	0
$943$	−7317.49	−0.252694
$944$	0	0
$945$	2440.21	0.0840000
$946$	0	0
$947$	13770.7	0.472531	0.236265	−	0.971689i	$-0.424077\pi$
$947$	13770.7	0.472531	0.236265	+	0.971689i	$0.424077\pi$
$948$	0	0
$949$	4378.30	0.149764
$950$	0	0
$951$	−41402.9	−1.41176
$952$	0	0
$953$	−42244.4	−1.43592	−0.717959	−	0.696085i	$-0.754924\pi$
$953$	−42244.4	−1.43592	−0.717959	+	0.696085i	$0.754924\pi$
$954$	0	0
$955$	−5526.31	−0.187254
$956$	0	0
$957$	−1514.56	−0.0511587
$958$	0	0
$959$	5719.02	0.192572
$960$	0	0
$961$	−29472.1	−0.989296
$962$	0	0
$963$	26840.2	0.898144
$964$	0	0
$965$	26401.2	0.880710
$966$	0	0
$967$	8247.06	0.274258	0.137129	−	0.990553i	$-0.456212\pi$
$967$	8247.06	0.274258	0.137129	+	0.990553i	$0.456212\pi$
$968$	0	0
$969$	−58812.8	−1.94978
$970$	0	0
$971$	58514.6	1.93390	0.966952	−	0.254957i	$-0.0820613\pi$
$971$	58514.6	1.93390	0.966952	+	0.254957i	$0.0820613\pi$
$972$	0	0
$973$	−11875.8	−0.391286
$974$	0	0
$975$	−6677.45	−0.219333
$976$	0	0
$977$	−16995.2	−0.556526	−0.278263	−	0.960505i	$-0.589759\pi$
$977$	−16995.2	−0.556526	−0.278263	+	0.960505i	$0.589759\pi$
$978$	0	0
$979$	−3338.81	−0.108998
$980$	0	0
$981$	4452.84	0.144922
$982$	0	0
$983$	34851.6	1.13082	0.565409	−	0.824811i	$-0.308718\pi$
$983$	34851.6	1.13082	0.565409	+	0.824811i	$0.308718\pi$
$984$	0	0
$985$	−7962.20	−0.257560
$986$	0	0
$987$	−8862.20	−0.285802
$988$	0	0
$989$	735.017	0.0236321
$990$	0	0
$991$	6949.64	0.222768	0.111384	−	0.993777i	$-0.464472\pi$
$991$	6949.64	0.222768	0.111384	+	0.993777i	$0.464472\pi$
$992$	0	0
$993$	35940.0	1.14856
$994$	0	0
$995$	17928.5	0.571227
$996$	0	0
$997$	20280.1	0.644210	0.322105	−	0.946704i	$-0.395610\pi$
$997$	20280.1	0.644210	0.322105	+	0.946704i	$0.395610\pi$
$998$	0	0
$999$	−19974.6	−0.632600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.s.1.5		6
4.3	odd	2		920.4.a.b.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.b.1.2	✓	6	4.3	odd	2
1840.4.a.s.1.5		6	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)

\(f(q)\)	\(=\)	\(q+6.38774 q^{3} -5.00000 q^{5} +5.78952 q^{7} +13.8032 q^{9} +O(q^{10})\)	\(q+6.38774 q^{3} -5.00000 q^{5} +5.78952 q^{7} +13.8032 q^{9} +3.04007 q^{11} -41.8142 q^{13} -31.9387 q^{15} +89.8459 q^{17} -102.477 q^{19} +36.9819 q^{21} +23.0000 q^{23} +25.0000 q^{25} -84.2976 q^{27} -77.9931 q^{29} +17.8575 q^{31} +19.4192 q^{33} -28.9476 q^{35} +236.953 q^{37} -267.098 q^{39} -318.152 q^{41} +31.9573 q^{43} -69.0161 q^{45} -239.636 q^{47} -309.482 q^{49} +573.913 q^{51} +498.695 q^{53} -15.2004 q^{55} -654.596 q^{57} -276.837 q^{59} -86.9389 q^{61} +79.9140 q^{63} +209.071 q^{65} -1019.33 q^{67} +146.918 q^{69} -1020.60 q^{71} -104.709 q^{73} +159.694 q^{75} +17.6006 q^{77} +1052.52 q^{79} -911.158 q^{81} +1006.59 q^{83} -449.230 q^{85} -498.200 q^{87} -1098.26 q^{89} -242.084 q^{91} +114.069 q^{93} +512.384 q^{95} -489.424 q^{97} +41.9628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9	\( 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9} + 3 q^{11} - 28 q^{13} + 10 q^{15} + 24 q^{17} + 3 q^{19} + 60 q^{21} + 138 q^{23} + 150 q^{25} + 97 q^{27} + 76 q^{29} + 381 q^{31} - 3 q^{33} + 140 q^{35} + 131 q^{37} - 41 q^{39} - 95 q^{41} + 202 q^{43} - 20 q^{45} + 119 q^{47} - 578 q^{49} + 997 q^{51} + 137 q^{53} - 15 q^{55} - 894 q^{57} + 39 q^{59} - 573 q^{61} + 355 q^{63} + 140 q^{65} + 563 q^{67} - 46 q^{69} + 83 q^{71} - 1799 q^{73} - 50 q^{75} - 1410 q^{77} + 1636 q^{79} - 2006 q^{81} + 1191 q^{83} - 120 q^{85} + 1821 q^{87} - 1370 q^{89} + 1251 q^{91} - 2215 q^{93} - 15 q^{95} - 3021 q^{97} + 525 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9 + 3 * q^11 - 28 * q^13 + 10 * q^15 + 24 * q^17 + 3 * q^19 + 60 * q^21 + 138 * q^23 + 150 * q^25 + 97 * q^27 + 76 * q^29 + 381 * q^31 - 3 * q^33 + 140 * q^35 + 131 * q^37 - 41 * q^39 - 95 * q^41 + 202 * q^43 - 20 * q^45 + 119 * q^47 - 578 * q^49 + 997 * q^51 + 137 * q^53 - 15 * q^55 - 894 * q^57 + 39 * q^59 - 573 * q^61 + 355 * q^63 + 140 * q^65 + 563 * q^67 - 46 * q^69 + 83 * q^71 - 1799 * q^73 - 50 * q^75 - 1410 * q^77 + 1636 * q^79 - 2006 * q^81 + 1191 * q^83 - 120 * q^85 + 1821 * q^87 - 1370 * q^89 + 1251 * q^91 - 2215 * q^93 - 15 * q^95 - 3021 * q^97 + 525 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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