LMFDB - Embedded newform 1840.4.a.x.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$108.563514411$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 123x^{6} + 335x^{5} + 4492x^{4} - 7035x^{3} - 45582x^{2} + 36684x + 124632 $ x^8 - 4x^7 - 123x^6 + 335x^5 + 4492x^4 - 7035x^3 - 45582x^2 + 36684*x + 124632
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.8
Root			$-7.52581$ 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)

$f(q)$	$=$	$q+9.52581 q^{3} +5.00000 q^{5} -21.2263 q^{7} +63.7410 q^{9} +O(q^{10})$	$q+9.52581 q^{3} +5.00000 q^{5} -21.2263 q^{7} +63.7410 q^{9} +56.1882 q^{11} -14.1012 q^{13} +47.6290 q^{15} -3.13924 q^{17} +34.2862 q^{19} -202.198 q^{21} +23.0000 q^{23} +25.0000 q^{25} +349.988 q^{27} +62.4426 q^{29} -163.897 q^{31} +535.238 q^{33} -106.132 q^{35} +79.7510 q^{37} -134.325 q^{39} +327.933 q^{41} +7.29898 q^{43} +318.705 q^{45} +238.049 q^{47} +107.557 q^{49} -29.9038 q^{51} +299.115 q^{53} +280.941 q^{55} +326.604 q^{57} -3.32727 q^{59} -492.381 q^{61} -1352.99 q^{63} -70.5059 q^{65} +991.503 q^{67} +219.094 q^{69} -808.403 q^{71} +567.816 q^{73} +238.145 q^{75} -1192.67 q^{77} +674.957 q^{79} +1612.91 q^{81} -800.050 q^{83} -15.6962 q^{85} +594.817 q^{87} -215.037 q^{89} +299.316 q^{91} -1561.25 q^{93} +171.431 q^{95} -1515.14 q^{97} +3581.49 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9}+O(q^{10}) $ 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9	$ 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9} + 35 q^{11} - 54 q^{13} + 60 q^{15} - 11 q^{17} + 107 q^{19} - 148 q^{21} + 184 q^{23} + 200 q^{25} + 405 q^{27} + 37 q^{29} + 290 q^{31} + 289 q^{33} + 155 q^{35} - 172 q^{37} + 311 q^{39} + 308 q^{41} + 538 q^{43} + 310 q^{45} + 1035 q^{47} + 585 q^{49} + 387 q^{51} + 46 q^{53} + 175 q^{55} + 286 q^{57} + 1256 q^{59} + 399 q^{61} + 1234 q^{63} - 270 q^{65} + 1598 q^{67} + 276 q^{69} + 750 q^{71} + 177 q^{73} + 300 q^{75} - 1228 q^{77} + 1292 q^{79} + 380 q^{81} + 2094 q^{83} - 55 q^{85} + 2245 q^{87} + 484 q^{89} + 679 q^{91} - 2111 q^{93} + 535 q^{95} - 2225 q^{97} + 2117 q^{99}+O(q^{100}) $ 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9 + 35 * q^11 - 54 * q^13 + 60 * q^15 - 11 * q^17 + 107 * q^19 - 148 * q^21 + 184 * q^23 + 200 * q^25 + 405 * q^27 + 37 * q^29 + 290 * q^31 + 289 * q^33 + 155 * q^35 - 172 * q^37 + 311 * q^39 + 308 * q^41 + 538 * q^43 + 310 * q^45 + 1035 * q^47 + 585 * q^49 + 387 * q^51 + 46 * q^53 + 175 * q^55 + 286 * q^57 + 1256 * q^59 + 399 * q^61 + 1234 * q^63 - 270 * q^65 + 1598 * q^67 + 276 * q^69 + 750 * q^71 + 177 * q^73 + 300 * q^75 - 1228 * q^77 + 1292 * q^79 + 380 * q^81 + 2094 * q^83 - 55 * q^85 + 2245 * q^87 + 484 * q^89 + 679 * q^91 - 2111 * q^93 + 535 * q^95 - 2225 * q^97 + 2117 * 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
$3$	9.52581	1.83324	0.916621	−	0.399757i	$-0.130905\pi$
$3$	9.52581	1.83324	0.916621	+	0.399757i	$0.130905\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−21.2263	−1.14611	−0.573057	−	0.819515i	$-0.694243\pi$
$7$	−21.2263	−1.14611	−0.573057	+	0.819515i	$0.694243\pi$
$8$	0	0
$9$	63.7410	2.36078
$10$	0	0
$11$	56.1882	1.54012	0.770062	−	0.637969i	$-0.220225\pi$
$11$	56.1882	1.54012	0.770062	+	0.637969i	$0.220225\pi$
$12$	0	0
$13$	−14.1012	−0.300843	−0.150422	−	0.988622i	$-0.548063\pi$
$13$	−14.1012	−0.300843	−0.150422	+	0.988622i	$0.548063\pi$
$14$	0	0
$15$	47.6290	0.819851
$16$	0	0
$17$	−3.13924	−0.0447869	−0.0223935	−	0.999749i	$-0.507129\pi$
$17$	−3.13924	−0.0447869	−0.0223935	+	0.999749i	$0.507129\pi$
$18$	0	0
$19$	34.2862	0.413989	0.206995	−	0.978342i	$-0.433632\pi$
$19$	34.2862	0.413989	0.206995	+	0.978342i	$0.433632\pi$
$20$	0	0
$21$	−202.198	−2.10111
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	349.988	2.49464
$28$	0	0
$29$	62.4426	0.399838	0.199919	−	0.979812i	$-0.435932\pi$
$29$	62.4426	0.399838	0.199919	+	0.979812i	$0.435932\pi$
$30$	0	0
$31$	−163.897	−0.949573	−0.474787	−	0.880101i	$-0.657475\pi$
$31$	−163.897	−0.949573	−0.474787	+	0.880101i	$0.657475\pi$
$32$	0	0
$33$	535.238	2.82342
$34$	0	0
$35$	−106.132	−0.512558
$36$	0	0
$37$	79.7510	0.354351	0.177176	−	0.984179i	$-0.443304\pi$
$37$	79.7510	0.354351	0.177176	+	0.984179i	$0.443304\pi$
$38$	0	0
$39$	−134.325	−0.551519
$40$	0	0
$41$	327.933	1.24914	0.624568	−	0.780970i	$-0.285275\pi$
$41$	327.933	1.24914	0.624568	+	0.780970i	$0.285275\pi$
$42$	0	0
$43$	7.29898	0.0258857	0.0129428	−	0.999916i	$-0.495880\pi$
$43$	7.29898	0.0258857	0.0129428	+	0.999916i	$0.495880\pi$
$44$	0	0
$45$	318.705	1.05577
$46$	0	0
$47$	238.049	0.738789	0.369394	−	0.929273i	$-0.379565\pi$
$47$	238.049	0.738789	0.369394	+	0.929273i	$0.379565\pi$
$48$	0	0
$49$	107.557	0.313579
$50$	0	0
$51$	−29.9038	−0.0821053
$52$	0	0
$53$	299.115	0.775219	0.387609	−	0.921824i	$-0.373301\pi$
$53$	299.115	0.775219	0.387609	+	0.921824i	$0.373301\pi$
$54$	0	0
$55$	280.941	0.688765
$56$	0	0
$57$	326.604	0.758942
$58$	0	0
$59$	−3.32727	−0.00734193	−0.00367096	−	0.999993i	$-0.501169\pi$
$59$	−3.32727	−0.00734193	−0.00367096	+	0.999993i	$0.501169\pi$
$60$	0	0
$61$	−492.381	−1.03349	−0.516746	−	0.856139i	$-0.672857\pi$
$61$	−492.381	−1.03349	−0.516746	+	0.856139i	$0.672857\pi$
$62$	0	0
$63$	−1352.99	−2.70572
$64$	0	0
$65$	−70.5059	−0.134541
$66$	0	0
$67$	991.503	1.80793	0.903965	−	0.427606i	$-0.140643\pi$
$67$	991.503	1.80793	0.903965	+	0.427606i	$0.140643\pi$
$68$	0	0
$69$	219.094	0.382258
$70$	0	0
$71$	−808.403	−1.35126	−0.675632	−	0.737239i	$-0.736129\pi$
$71$	−808.403	−1.35126	−0.675632	+	0.737239i	$0.736129\pi$
$72$	0	0
$73$	567.816	0.910382	0.455191	−	0.890394i	$-0.349571\pi$
$73$	567.816	0.910382	0.455191	+	0.890394i	$0.349571\pi$
$74$	0	0
$75$	238.145	0.366649
$76$	0	0
$77$	−1192.67	−1.76516
$78$	0	0
$79$	674.957	0.961248	0.480624	−	0.876927i	$-0.340410\pi$
$79$	674.957	0.961248	0.480624	+	0.876927i	$0.340410\pi$
$80$	0	0
$81$	1612.91	2.21250
$82$	0	0
$83$	−800.050	−1.05804	−0.529018	−	0.848611i	$-0.677440\pi$
$83$	−800.050	−1.05804	−0.529018	+	0.848611i	$0.677440\pi$
$84$	0	0
$85$	−15.6962	−0.0200293
$86$	0	0
$87$	594.817	0.733000
$88$	0	0
$89$	−215.037	−0.256111	−0.128055	−	0.991767i	$-0.540873\pi$
$89$	−215.037	−0.256111	−0.128055	+	0.991767i	$0.540873\pi$
$90$	0	0
$91$	299.316	0.344801
$92$	0	0
$93$	−1561.25	−1.74080
$94$	0	0
$95$	171.431	0.185142
$96$	0	0
$97$	−1515.14	−1.58597	−0.792984	−	0.609243i	$-0.791473\pi$
$97$	−1515.14	−1.58597	−0.792984	+	0.609243i	$0.791473\pi$
$98$	0	0
$99$	3581.49	3.63589
$100$	0	0
$101$	−302.904	−0.298416	−0.149208	−	0.988806i	$-0.547672\pi$
$101$	−302.904	−0.298416	−0.149208	+	0.988806i	$0.547672\pi$
$102$	0	0
$103$	492.361	0.471008	0.235504	−	0.971873i	$-0.424326\pi$
$103$	492.361	0.471008	0.235504	+	0.971873i	$0.424326\pi$
$104$	0	0
$105$	−1010.99	−0.939643
$106$	0	0
$107$	289.238	0.261324	0.130662	−	0.991427i	$-0.458290\pi$
$107$	289.238	0.261324	0.130662	+	0.991427i	$0.458290\pi$
$108$	0	0
$109$	1325.06	1.16439	0.582193	−	0.813051i	$-0.302195\pi$
$109$	1325.06	1.16439	0.582193	+	0.813051i	$0.302195\pi$
$110$	0	0
$111$	759.693	0.649611
$112$	0	0
$113$	−1836.68	−1.52903	−0.764515	−	0.644605i	$-0.777022\pi$
$113$	−1836.68	−1.52903	−0.764515	+	0.644605i	$0.777022\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−898.823	−0.710224
$118$	0	0
$119$	66.6346	0.0513309
$120$	0	0
$121$	1826.11	1.37198
$122$	0	0
$123$	3123.83	2.28997
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	2314.11	1.61688	0.808442	−	0.588575i	$-0.200311\pi$
$127$	2314.11	1.61688	0.808442	+	0.588575i	$0.200311\pi$
$128$	0	0
$129$	69.5287	0.0474547
$130$	0	0
$131$	2197.24	1.46545	0.732725	−	0.680524i	$-0.238248\pi$
$131$	2197.24	1.46545	0.732725	+	0.680524i	$0.238248\pi$
$132$	0	0
$133$	−727.770	−0.474479
$134$	0	0
$135$	1749.94	1.11564
$136$	0	0
$137$	2485.82	1.55020	0.775102	−	0.631836i	$-0.217698\pi$
$137$	2485.82	1.55020	0.775102	+	0.631836i	$0.217698\pi$
$138$	0	0
$139$	461.790	0.281788	0.140894	−	0.990025i	$-0.455002\pi$
$139$	461.790	0.281788	0.140894	+	0.990025i	$0.455002\pi$
$140$	0	0
$141$	2267.61	1.35438
$142$	0	0
$143$	−792.319	−0.463336
$144$	0	0
$145$	312.213	0.178813
$146$	0	0
$147$	1024.57	0.574866
$148$	0	0
$149$	2636.72	1.44972	0.724862	−	0.688894i	$-0.241904\pi$
$149$	2636.72	1.44972	0.724862	+	0.688894i	$0.241904\pi$
$150$	0	0
$151$	312.292	0.168305	0.0841523	−	0.996453i	$-0.473182\pi$
$151$	312.292	0.168305	0.0841523	+	0.996453i	$0.473182\pi$
$152$	0	0
$153$	−200.098	−0.105732
$154$	0	0
$155$	−819.485	−0.424662
$156$	0	0
$157$	−3007.07	−1.52860	−0.764301	−	0.644860i	$-0.776916\pi$
$157$	−3007.07	−1.52860	−0.764301	+	0.644860i	$0.776916\pi$
$158$	0	0
$159$	2849.31	1.42116
$160$	0	0
$161$	−488.206	−0.238981
$162$	0	0
$163$	1746.64	0.839310	0.419655	−	0.907684i	$-0.362151\pi$
$163$	1746.64	0.839310	0.419655	+	0.907684i	$0.362151\pi$
$164$	0	0
$165$	2676.19	1.26267
$166$	0	0
$167$	−562.051	−0.260436	−0.130218	−	0.991485i	$-0.541568\pi$
$167$	−562.051	−0.260436	−0.130218	+	0.991485i	$0.541568\pi$
$168$	0	0
$169$	−1998.16	−0.909493
$170$	0	0
$171$	2185.44	0.977337
$172$	0	0
$173$	−1338.38	−0.588182	−0.294091	−	0.955777i	$-0.595017\pi$
$173$	−1338.38	−0.588182	−0.294091	+	0.955777i	$0.595017\pi$
$174$	0	0
$175$	−530.658	−0.229223
$176$	0	0
$177$	−31.6949	−0.0134595
$178$	0	0
$179$	1381.37	0.576805	0.288403	−	0.957509i	$-0.406876\pi$
$179$	1381.37	0.576805	0.288403	+	0.957509i	$0.406876\pi$
$180$	0	0
$181$	3755.24	1.54213	0.771064	−	0.636758i	$-0.219725\pi$
$181$	3755.24	1.54213	0.771064	+	0.636758i	$0.219725\pi$
$182$	0	0
$183$	−4690.33	−1.89464
$184$	0	0
$185$	398.755	0.158471
$186$	0	0
$187$	−176.388	−0.0689774
$188$	0	0
$189$	−7428.96	−2.85914
$190$	0	0
$191$	−1063.78	−0.402997	−0.201498	−	0.979489i	$-0.564581\pi$
$191$	−1063.78	−0.402997	−0.201498	+	0.979489i	$0.564581\pi$
$192$	0	0
$193$	−5199.63	−1.93926	−0.969631	−	0.244574i	$-0.921352\pi$
$193$	−5199.63	−1.93926	−0.969631	+	0.244574i	$0.921352\pi$
$194$	0	0
$195$	−671.626	−0.246647
$196$	0	0
$197$	−2559.32	−0.925603	−0.462801	−	0.886462i	$-0.653156\pi$
$197$	−2559.32	−0.925603	−0.462801	+	0.886462i	$0.653156\pi$
$198$	0	0
$199$	−1408.85	−0.501863	−0.250931	−	0.968005i	$-0.580737\pi$
$199$	−1408.85	−0.501863	−0.250931	+	0.968005i	$0.580737\pi$
$200$	0	0
$201$	9444.86	3.31437
$202$	0	0
$203$	−1325.43	−0.458260
$204$	0	0
$205$	1639.67	0.558631
$206$	0	0
$207$	1466.04	0.492256
$208$	0	0
$209$	1926.48	0.637595
$210$	0	0
$211$	3478.54	1.13494	0.567470	−	0.823394i	$-0.307922\pi$
$211$	3478.54	1.13494	0.567470	+	0.823394i	$0.307922\pi$
$212$	0	0
$213$	−7700.69	−2.47720
$214$	0	0
$215$	36.4949	0.0115764
$216$	0	0
$217$	3478.93	1.08832
$218$	0	0
$219$	5408.91	1.66895
$220$	0	0
$221$	44.2670	0.0134738
$222$	0	0
$223$	−229.611	−0.0689500	−0.0344750	−	0.999406i	$-0.510976\pi$
$223$	−229.611	−0.0689500	−0.0344750	+	0.999406i	$0.510976\pi$
$224$	0	0
$225$	1593.53	0.472156
$226$	0	0
$227$	4471.24	1.30734	0.653670	−	0.756779i	$-0.273228\pi$
$227$	4471.24	1.30734	0.653670	+	0.756779i	$0.273228\pi$
$228$	0	0
$229$	−4034.97	−1.16436	−0.582180	−	0.813060i	$-0.697800\pi$
$229$	−4034.97	−1.16436	−0.582180	+	0.813060i	$0.697800\pi$
$230$	0	0
$231$	−11361.1	−3.23597
$232$	0	0
$233$	−3856.27	−1.08426	−0.542130	−	0.840294i	$-0.682382\pi$
$233$	−3856.27	−1.08426	−0.542130	+	0.840294i	$0.682382\pi$
$234$	0	0
$235$	1190.25	0.330396
$236$	0	0
$237$	6429.51	1.76220
$238$	0	0
$239$	−3960.16	−1.07181	−0.535903	−	0.844280i	$-0.680029\pi$
$239$	−3960.16	−1.07181	−0.535903	+	0.844280i	$0.680029\pi$
$240$	0	0
$241$	−1090.81	−0.291558	−0.145779	−	0.989317i	$-0.546569\pi$
$241$	−1090.81	−0.291558	−0.145779	+	0.989317i	$0.546569\pi$
$242$	0	0
$243$	5914.60	1.56141
$244$	0	0
$245$	537.787	0.140237
$246$	0	0
$247$	−483.476	−0.124546
$248$	0	0
$249$	−7621.13	−1.93964
$250$	0	0
$251$	−2150.83	−0.540874	−0.270437	−	0.962738i	$-0.587168\pi$
$251$	−2150.83	−0.540874	−0.270437	+	0.962738i	$0.587168\pi$
$252$	0	0
$253$	1292.33	0.321138
$254$	0	0
$255$	−149.519	−0.0367186
$256$	0	0
$257$	−6347.55	−1.54066	−0.770330	−	0.637646i	$-0.779908\pi$
$257$	−6347.55	−1.54066	−0.770330	+	0.637646i	$0.779908\pi$
$258$	0	0
$259$	−1692.82	−0.406127
$260$	0	0
$261$	3980.16	0.943929
$262$	0	0
$263$	−3918.60	−0.918750	−0.459375	−	0.888242i	$-0.651927\pi$
$263$	−3918.60	−0.918750	−0.459375	+	0.888242i	$0.651927\pi$
$264$	0	0
$265$	1495.57	0.346688
$266$	0	0
$267$	−2048.40	−0.469513
$268$	0	0
$269$	4688.97	1.06279	0.531397	−	0.847123i	$-0.321667\pi$
$269$	4688.97	1.06279	0.531397	+	0.847123i	$0.321667\pi$
$270$	0	0
$271$	3748.06	0.840143	0.420071	−	0.907491i	$-0.362005\pi$
$271$	3748.06	0.840143	0.420071	+	0.907491i	$0.362005\pi$
$272$	0	0
$273$	2851.23	0.632104
$274$	0	0
$275$	1404.70	0.308025
$276$	0	0
$277$	6622.16	1.43641	0.718207	−	0.695829i	$-0.244963\pi$
$277$	6622.16	1.43641	0.718207	+	0.695829i	$0.244963\pi$
$278$	0	0
$279$	−10447.0	−2.24173
$280$	0	0
$281$	−3082.33	−0.654364	−0.327182	−	0.944961i	$-0.606099\pi$
$281$	−3082.33	−0.654364	−0.327182	+	0.944961i	$0.606099\pi$
$282$	0	0
$283$	2389.03	0.501814	0.250907	−	0.968011i	$-0.419271\pi$
$283$	2389.03	0.501814	0.250907	+	0.968011i	$0.419271\pi$
$284$	0	0
$285$	1633.02	0.339409
$286$	0	0
$287$	−6960.83	−1.43165
$288$	0	0
$289$	−4903.15	−0.997994
$290$	0	0
$291$	−14432.9	−2.90746
$292$	0	0
$293$	1805.01	0.359897	0.179948	−	0.983676i	$-0.442407\pi$
$293$	1805.01	0.359897	0.179948	+	0.983676i	$0.442407\pi$
$294$	0	0
$295$	−16.6364	−0.00328341
$296$	0	0
$297$	19665.2	3.84205
$298$	0	0
$299$	−324.327	−0.0627302
$300$	0	0
$301$	−154.931	−0.0296680
$302$	0	0
$303$	−2885.40	−0.547070
$304$	0	0
$305$	−2461.91	−0.462191
$306$	0	0
$307$	1226.82	0.228072	0.114036	−	0.993477i	$-0.463622\pi$
$307$	1226.82	0.228072	0.114036	+	0.993477i	$0.463622\pi$
$308$	0	0
$309$	4690.14	0.863472
$310$	0	0
$311$	−3731.83	−0.680427	−0.340214	−	0.940348i	$-0.610499\pi$
$311$	−3731.83	−0.680427	−0.340214	+	0.940348i	$0.610499\pi$
$312$	0	0
$313$	−3685.41	−0.665533	−0.332766	−	0.943009i	$-0.607982\pi$
$313$	−3685.41	−0.665533	−0.332766	+	0.943009i	$0.607982\pi$
$314$	0	0
$315$	−6764.94	−1.21004
$316$	0	0
$317$	−9633.52	−1.70685	−0.853427	−	0.521213i	$-0.825480\pi$
$317$	−9633.52	−1.70685	−0.853427	+	0.521213i	$0.825480\pi$
$318$	0	0
$319$	3508.54	0.615800
$320$	0	0
$321$	2755.22	0.479070
$322$	0	0
$323$	−107.633	−0.0185413
$324$	0	0
$325$	−352.529	−0.0601687
$326$	0	0
$327$	12622.3	2.13460
$328$	0	0
$329$	−5052.92	−0.846736
$330$	0	0
$331$	−8287.71	−1.37623	−0.688117	−	0.725599i	$-0.741563\pi$
$331$	−8287.71	−1.37623	−0.688117	+	0.725599i	$0.741563\pi$
$332$	0	0
$333$	5083.41	0.836544
$334$	0	0
$335$	4957.51	0.808531
$336$	0	0
$337$	5658.66	0.914679	0.457340	−	0.889292i	$-0.348802\pi$
$337$	5658.66	0.914679	0.457340	+	0.889292i	$0.348802\pi$
$338$	0	0
$339$	−17495.9	−2.80308
$340$	0	0
$341$	−9209.07	−1.46246
$342$	0	0
$343$	4997.58	0.786717
$344$	0	0
$345$	1095.47	0.170951
$346$	0	0
$347$	9804.99	1.51689	0.758443	−	0.651739i	$-0.225960\pi$
$347$	9804.99	1.51689	0.758443	+	0.651739i	$0.225960\pi$
$348$	0	0
$349$	10251.2	1.57231	0.786153	−	0.618032i	$-0.212070\pi$
$349$	10251.2	1.57231	0.786153	+	0.618032i	$0.212070\pi$
$350$	0	0
$351$	−4935.24	−0.750495
$352$	0	0
$353$	−11033.7	−1.66365	−0.831823	−	0.555041i	$-0.812703\pi$
$353$	−11033.7	−1.66365	−0.831823	+	0.555041i	$0.812703\pi$
$354$	0	0
$355$	−4042.01	−0.604304
$356$	0	0
$357$	634.748	0.0941021
$358$	0	0
$359$	4212.47	0.619292	0.309646	−	0.950852i	$-0.399789\pi$
$359$	4212.47	0.619292	0.309646	+	0.950852i	$0.399789\pi$
$360$	0	0
$361$	−5683.46	−0.828613
$362$	0	0
$363$	17395.2	2.51518
$364$	0	0
$365$	2839.08	0.407135
$366$	0	0
$367$	9576.10	1.36204	0.681019	−	0.732266i	$-0.261537\pi$
$367$	9576.10	1.36204	0.681019	+	0.732266i	$0.261537\pi$
$368$	0	0
$369$	20902.8	2.94893
$370$	0	0
$371$	−6349.11	−0.888489
$372$	0	0
$373$	−9431.83	−1.30928	−0.654640	−	0.755941i	$-0.727180\pi$
$373$	−9431.83	−1.30928	−0.654640	+	0.755941i	$0.727180\pi$
$374$	0	0
$375$	1190.73	0.163970
$376$	0	0
$377$	−880.515	−0.120289
$378$	0	0
$379$	10098.7	1.36869	0.684347	−	0.729156i	$-0.260087\pi$
$379$	10098.7	1.36869	0.684347	+	0.729156i	$0.260087\pi$
$380$	0	0
$381$	22043.8	2.96414
$382$	0	0
$383$	−7606.73	−1.01485	−0.507423	−	0.861697i	$-0.669402\pi$
$383$	−7606.73	−1.01485	−0.507423	+	0.861697i	$0.669402\pi$
$384$	0	0
$385$	−5963.35	−0.789403
$386$	0	0
$387$	465.245	0.0611104
$388$	0	0
$389$	−1957.62	−0.255155	−0.127578	−	0.991829i	$-0.540720\pi$
$389$	−1957.62	−0.255155	−0.127578	+	0.991829i	$0.540720\pi$
$390$	0	0
$391$	−72.2025	−0.00933872
$392$	0	0
$393$	20930.5	2.68653
$394$	0	0
$395$	3374.78	0.429883
$396$	0	0
$397$	−5195.21	−0.656776	−0.328388	−	0.944543i	$-0.606505\pi$
$397$	−5195.21	−0.656776	−0.328388	+	0.944543i	$0.606505\pi$
$398$	0	0
$399$	−6932.60	−0.869835
$400$	0	0
$401$	−4789.37	−0.596433	−0.298217	−	0.954498i	$-0.596392\pi$
$401$	−4789.37	−0.596433	−0.298217	+	0.954498i	$0.596392\pi$
$402$	0	0
$403$	2311.14	0.285673
$404$	0	0
$405$	8064.55	0.989459
$406$	0	0
$407$	4481.06	0.545745
$408$	0	0
$409$	−11023.6	−1.33272	−0.666358	−	0.745632i	$-0.732148\pi$
$409$	−11023.6	−1.33272	−0.666358	+	0.745632i	$0.732148\pi$
$410$	0	0
$411$	23679.5	2.84190
$412$	0	0
$413$	70.6258	0.00841469
$414$	0	0
$415$	−4000.25	−0.473168
$416$	0	0
$417$	4398.92	0.516585
$418$	0	0
$419$	−10435.9	−1.21677	−0.608387	−	0.793641i	$-0.708183\pi$
$419$	−10435.9	−1.21677	−0.608387	+	0.793641i	$0.708183\pi$
$420$	0	0
$421$	16583.8	1.91982	0.959911	−	0.280305i	$-0.0904356\pi$
$421$	16583.8	1.91982	0.959911	+	0.280305i	$0.0904356\pi$
$422$	0	0
$423$	15173.5	1.74412
$424$	0	0
$425$	−78.4810	−0.00895738
$426$	0	0
$427$	10451.5	1.18450
$428$	0	0
$429$	−7547.48	−0.849408
$430$	0	0
$431$	13994.9	1.56406	0.782030	−	0.623241i	$-0.214185\pi$
$431$	13994.9	1.56406	0.782030	+	0.623241i	$0.214185\pi$
$432$	0	0
$433$	13047.2	1.44806	0.724029	−	0.689769i	$-0.242288\pi$
$433$	13047.2	1.44806	0.724029	+	0.689769i	$0.242288\pi$
$434$	0	0
$435$	2974.08	0.327808
$436$	0	0
$437$	788.582	0.0863227
$438$	0	0
$439$	−6555.57	−0.712711	−0.356356	−	0.934350i	$-0.615981\pi$
$439$	−6555.57	−0.712711	−0.356356	+	0.934350i	$0.615981\pi$
$440$	0	0
$441$	6855.82	0.740290
$442$	0	0
$443$	3556.19	0.381399	0.190700	−	0.981648i	$-0.438924\pi$
$443$	3556.19	0.381399	0.190700	+	0.981648i	$0.438924\pi$
$444$	0	0
$445$	−1075.18	−0.114536
$446$	0	0
$447$	25116.9	2.65770
$448$	0	0
$449$	−12514.8	−1.31539	−0.657696	−	0.753284i	$-0.728469\pi$
$449$	−12514.8	−1.31539	−0.657696	+	0.753284i	$0.728469\pi$
$450$	0	0
$451$	18426.0	1.92383
$452$	0	0
$453$	2974.84	0.308543
$454$	0	0
$455$	1496.58	0.154200
$456$	0	0
$457$	−10295.1	−1.05380	−0.526899	−	0.849928i	$-0.676645\pi$
$457$	−10295.1	−1.05380	−0.526899	+	0.849928i	$0.676645\pi$
$458$	0	0
$459$	−1098.70	−0.111727
$460$	0	0
$461$	1465.30	0.148039	0.0740195	−	0.997257i	$-0.476417\pi$
$461$	1465.30	0.148039	0.0740195	+	0.997257i	$0.476417\pi$
$462$	0	0
$463$	−10614.7	−1.06546	−0.532728	−	0.846286i	$-0.678833\pi$
$463$	−10614.7	−1.06546	−0.532728	+	0.846286i	$0.678833\pi$
$464$	0	0
$465$	−7806.26	−0.778509
$466$	0	0
$467$	8440.24	0.836333	0.418167	−	0.908370i	$-0.362673\pi$
$467$	8440.24	0.836333	0.418167	+	0.908370i	$0.362673\pi$
$468$	0	0
$469$	−21046.0	−2.07210
$470$	0	0
$471$	−28644.8	−2.80230
$472$	0	0
$473$	410.116	0.0398672
$474$	0	0
$475$	857.155	0.0827978
$476$	0	0
$477$	19065.9	1.83012
$478$	0	0
$479$	−4886.49	−0.466116	−0.233058	−	0.972463i	$-0.574873\pi$
$479$	−4886.49	−0.466116	−0.233058	+	0.972463i	$0.574873\pi$
$480$	0	0
$481$	−1124.58	−0.106604
$482$	0	0
$483$	−4650.55	−0.438111
$484$	0	0
$485$	−7575.68	−0.709266
$486$	0	0
$487$	−16342.4	−1.52063	−0.760315	−	0.649554i	$-0.774955\pi$
$487$	−16342.4	−1.52063	−0.760315	+	0.649554i	$0.774955\pi$
$488$	0	0
$489$	16638.2	1.53866
$490$	0	0
$491$	7757.89	0.713052	0.356526	−	0.934285i	$-0.383961\pi$
$491$	7757.89	0.713052	0.356526	+	0.934285i	$0.383961\pi$
$492$	0	0
$493$	−196.022	−0.0179075
$494$	0	0
$495$	17907.5	1.62602
$496$	0	0
$497$	17159.4	1.54870
$498$	0	0
$499$	−6699.37	−0.601012	−0.300506	−	0.953780i	$-0.597156\pi$
$499$	−6699.37	−0.601012	−0.300506	+	0.953780i	$0.597156\pi$
$500$	0	0
$501$	−5353.99	−0.477442
$502$	0	0
$503$	−18427.3	−1.63347	−0.816734	−	0.577014i	$-0.804218\pi$
$503$	−18427.3	−1.63347	−0.816734	+	0.577014i	$0.804218\pi$
$504$	0	0
$505$	−1514.52	−0.133456
$506$	0	0
$507$	−19034.1	−1.66732
$508$	0	0
$509$	973.677	0.0847888	0.0423944	−	0.999101i	$-0.486501\pi$
$509$	973.677	0.0847888	0.0423944	+	0.999101i	$0.486501\pi$
$510$	0	0
$511$	−12052.7	−1.04340
$512$	0	0
$513$	11999.8	1.03275
$514$	0	0
$515$	2461.81	0.210641
$516$	0	0
$517$	13375.6	1.13783
$518$	0	0
$519$	−12749.2	−1.07828
$520$	0	0
$521$	−2293.80	−0.192885	−0.0964426	−	0.995339i	$-0.530746\pi$
$521$	−2293.80	−0.192885	−0.0964426	+	0.995339i	$0.530746\pi$
$522$	0	0
$523$	−21652.7	−1.81033	−0.905167	−	0.425056i	$-0.860254\pi$
$523$	−21652.7	−1.81033	−0.905167	+	0.425056i	$0.860254\pi$
$524$	0	0
$525$	−5054.95	−0.420221
$526$	0	0
$527$	514.512	0.0425285
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−212.084	−0.0173327
$532$	0	0
$533$	−4624.25	−0.375794
$534$	0	0
$535$	1446.19	0.116868
$536$	0	0
$537$	13158.6	1.05742
$538$	0	0
$539$	6043.46	0.482950
$540$	0	0
$541$	15167.2	1.20534	0.602671	−	0.797990i	$-0.294103\pi$
$541$	15167.2	1.20534	0.602671	+	0.797990i	$0.294103\pi$
$542$	0	0
$543$	35771.7	2.82709
$544$	0	0
$545$	6625.32	0.520729
$546$	0	0
$547$	−7317.69	−0.571996	−0.285998	−	0.958230i	$-0.592325\pi$
$547$	−7317.69	−0.571996	−0.285998	+	0.958230i	$0.592325\pi$
$548$	0	0
$549$	−31384.9	−2.43984
$550$	0	0
$551$	2140.92	0.165529
$552$	0	0
$553$	−14326.9	−1.10170
$554$	0	0
$555$	3798.47	0.290515
$556$	0	0
$557$	−6504.05	−0.494767	−0.247384	−	0.968918i	$-0.579571\pi$
$557$	−6504.05	−0.494767	−0.247384	+	0.968918i	$0.579571\pi$
$558$	0	0
$559$	−102.924	−0.00778753
$560$	0	0
$561$	−1680.24	−0.126452
$562$	0	0
$563$	−19869.4	−1.48738	−0.743690	−	0.668525i	$-0.766926\pi$
$563$	−19869.4	−1.48738	−0.743690	+	0.668525i	$0.766926\pi$
$564$	0	0
$565$	−9183.41	−0.683803
$566$	0	0
$567$	−34236.2	−2.53577
$568$	0	0
$569$	18326.9	1.35027	0.675134	−	0.737695i	$-0.264086\pi$
$569$	18326.9	1.35027	0.675134	+	0.737695i	$0.264086\pi$
$570$	0	0
$571$	11066.0	0.811031	0.405515	−	0.914088i	$-0.367092\pi$
$571$	11066.0	0.811031	0.405515	+	0.914088i	$0.367092\pi$
$572$	0	0
$573$	−10133.4	−0.738791
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	5269.66	0.380206	0.190103	−	0.981764i	$-0.439118\pi$
$577$	5269.66	0.380206	0.190103	+	0.981764i	$0.439118\pi$
$578$	0	0
$579$	−49530.7	−3.55514
$580$	0	0
$581$	16982.1	1.21263
$582$	0	0
$583$	16806.7	1.19393
$584$	0	0
$585$	−4494.12	−0.317622
$586$	0	0
$587$	−7331.62	−0.515517	−0.257758	−	0.966209i	$-0.582984\pi$
$587$	−7331.62	−0.515517	−0.257758	+	0.966209i	$0.582984\pi$
$588$	0	0
$589$	−5619.41	−0.393113
$590$	0	0
$591$	−24379.6	−1.69685
$592$	0	0
$593$	−20671.3	−1.43148	−0.715741	−	0.698366i	$-0.753911\pi$
$593$	−20671.3	−1.43148	−0.715741	+	0.698366i	$0.753911\pi$
$594$	0	0
$595$	333.173	0.0229559
$596$	0	0
$597$	−13420.4	−0.920036
$598$	0	0
$599$	6253.79	0.426583	0.213291	−	0.976989i	$-0.431582\pi$
$599$	6253.79	0.426583	0.213291	+	0.976989i	$0.431582\pi$
$600$	0	0
$601$	−27756.4	−1.88387	−0.941937	−	0.335788i	$-0.890997\pi$
$601$	−27756.4	−1.88387	−0.941937	+	0.335788i	$0.890997\pi$
$602$	0	0
$603$	63199.4	4.26812
$604$	0	0
$605$	9130.55	0.613570
$606$	0	0
$607$	−4257.20	−0.284670	−0.142335	−	0.989819i	$-0.545461\pi$
$607$	−4257.20	−0.284670	−0.142335	+	0.989819i	$0.545461\pi$
$608$	0	0
$609$	−12625.8	−0.840102
$610$	0	0
$611$	−3356.78	−0.222260
$612$	0	0
$613$	−23697.3	−1.56138	−0.780688	−	0.624921i	$-0.785131\pi$
$613$	−23697.3	−1.56138	−0.780688	+	0.624921i	$0.785131\pi$
$614$	0	0
$615$	15619.2	1.02411
$616$	0	0
$617$	13379.3	0.872985	0.436492	−	0.899708i	$-0.356221\pi$
$617$	13379.3	0.872985	0.436492	+	0.899708i	$0.356221\pi$
$618$	0	0
$619$	13590.5	0.882471	0.441236	−	0.897391i	$-0.354540\pi$
$619$	13590.5	0.882471	0.441236	+	0.897391i	$0.354540\pi$
$620$	0	0
$621$	8049.72	0.520168
$622$	0	0
$623$	4564.44	0.293532
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	18351.3	1.16887
$628$	0	0
$629$	−250.358	−0.0158703
$630$	0	0
$631$	2059.75	0.129948	0.0649742	−	0.997887i	$-0.479303\pi$
$631$	2059.75	0.129948	0.0649742	+	0.997887i	$0.479303\pi$
$632$	0	0
$633$	33135.9	2.08062
$634$	0	0
$635$	11570.6	0.723093
$636$	0	0
$637$	−1516.69	−0.0943380
$638$	0	0
$639$	−51528.4	−3.19004
$640$	0	0
$641$	−9079.81	−0.559487	−0.279743	−	0.960075i	$-0.590249\pi$
$641$	−9079.81	−0.559487	−0.279743	+	0.960075i	$0.590249\pi$
$642$	0	0
$643$	16311.1	1.00038	0.500192	−	0.865915i	$-0.333263\pi$
$643$	16311.1	1.00038	0.500192	+	0.865915i	$0.333263\pi$
$644$	0	0
$645$	347.643	0.0212224
$646$	0	0
$647$	−25134.1	−1.52724	−0.763619	−	0.645667i	$-0.776580\pi$
$647$	−25134.1	−1.52724	−0.763619	+	0.645667i	$0.776580\pi$
$648$	0	0
$649$	−186.953	−0.0113075
$650$	0	0
$651$	33139.7	1.99515
$652$	0	0
$653$	6959.32	0.417059	0.208529	−	0.978016i	$-0.433132\pi$
$653$	6959.32	0.417059	0.208529	+	0.978016i	$0.433132\pi$
$654$	0	0
$655$	10986.2	0.655370
$656$	0	0
$657$	36193.2	2.14921
$658$	0	0
$659$	9055.19	0.535266	0.267633	−	0.963521i	$-0.413759\pi$
$659$	9055.19	0.535266	0.267633	+	0.963521i	$0.413759\pi$
$660$	0	0
$661$	29686.7	1.74687	0.873435	−	0.486941i	$-0.161887\pi$
$661$	29686.7	1.74687	0.873435	+	0.486941i	$0.161887\pi$
$662$	0	0
$663$	421.679	0.0247008
$664$	0	0
$665$	−3638.85	−0.212193
$666$	0	0
$667$	1436.18	0.0833720
$668$	0	0
$669$	−2187.23	−0.126402
$670$	0	0
$671$	−27666.0	−1.59171
$672$	0	0
$673$	20085.3	1.15042	0.575210	−	0.818006i	$-0.304920\pi$
$673$	20085.3	1.15042	0.575210	+	0.818006i	$0.304920\pi$
$674$	0	0
$675$	8749.70	0.498927
$676$	0	0
$677$	12918.6	0.733387	0.366694	−	0.930342i	$-0.380490\pi$
$677$	12918.6	0.733387	0.366694	+	0.930342i	$0.380490\pi$
$678$	0	0
$679$	32160.8	1.81770
$680$	0	0
$681$	42592.1	2.39667
$682$	0	0
$683$	−19672.9	−1.10214	−0.551072	−	0.834458i	$-0.685781\pi$
$683$	−19672.9	−1.10214	−0.551072	+	0.834458i	$0.685781\pi$
$684$	0	0
$685$	12429.1	0.693272
$686$	0	0
$687$	−38436.4	−2.13455
$688$	0	0
$689$	−4217.87	−0.233219
$690$	0	0
$691$	−3646.77	−0.200766	−0.100383	−	0.994949i	$-0.532007\pi$
$691$	−3646.77	−0.200766	−0.100383	+	0.994949i	$0.532007\pi$
$692$	0	0
$693$	−76021.9	−4.16715
$694$	0	0
$695$	2308.95	0.126019
$696$	0	0
$697$	−1029.46	−0.0559450
$698$	0	0
$699$	−36734.1	−1.98771
$700$	0	0
$701$	7874.56	0.424277	0.212138	−	0.977240i	$-0.431957\pi$
$701$	7874.56	0.424277	0.212138	+	0.977240i	$0.431957\pi$
$702$	0	0
$703$	2734.36	0.146697
$704$	0	0
$705$	11338.1	0.605697
$706$	0	0
$707$	6429.54	0.342019
$708$	0	0
$709$	−36787.2	−1.94862	−0.974309	−	0.225214i	$-0.927692\pi$
$709$	−36787.2	−1.94862	−0.974309	+	0.225214i	$0.927692\pi$
$710$	0	0
$711$	43022.4	2.26929
$712$	0	0
$713$	−3769.63	−0.198000
$714$	0	0
$715$	−3961.60	−0.207210
$716$	0	0
$717$	−37723.7	−1.96488
$718$	0	0
$719$	−7004.84	−0.363333	−0.181667	−	0.983360i	$-0.558149\pi$
$719$	−7004.84	−0.363333	−0.181667	+	0.983360i	$0.558149\pi$
$720$	0	0
$721$	−10451.0	−0.539829
$722$	0	0
$723$	−10390.9	−0.534496
$724$	0	0
$725$	1561.07	0.0799676
$726$	0	0
$727$	−36412.4	−1.85758	−0.928792	−	0.370603i	$-0.879151\pi$
$727$	−36412.4	−1.85758	−0.928792	+	0.370603i	$0.879151\pi$
$728$	0	0
$729$	12792.8	0.649939
$730$	0	0
$731$	−22.9133	−0.00115934
$732$	0	0
$733$	−12791.4	−0.644558	−0.322279	−	0.946645i	$-0.604449\pi$
$733$	−12791.4	−0.644558	−0.322279	+	0.946645i	$0.604449\pi$
$734$	0	0
$735$	5122.86	0.257088
$736$	0	0
$737$	55710.7	2.78444
$738$	0	0
$739$	31635.5	1.57474	0.787368	−	0.616483i	$-0.211443\pi$
$739$	31635.5	1.57474	0.787368	+	0.616483i	$0.211443\pi$
$740$	0	0
$741$	−4605.50	−0.228323
$742$	0	0
$743$	5757.37	0.284277	0.142138	−	0.989847i	$-0.454602\pi$
$743$	5757.37	0.284277	0.142138	+	0.989847i	$0.454602\pi$
$744$	0	0
$745$	13183.6	0.648336
$746$	0	0
$747$	−50996.0	−2.49779
$748$	0	0
$749$	−6139.46	−0.299507
$750$	0	0
$751$	−5493.59	−0.266929	−0.133465	−	0.991054i	$-0.542610\pi$
$751$	−5493.59	−0.266929	−0.133465	+	0.991054i	$0.542610\pi$
$752$	0	0
$753$	−20488.4	−0.991552
$754$	0	0
$755$	1561.46	0.0752681
$756$	0	0
$757$	−21969.0	−1.05479	−0.527394	−	0.849621i	$-0.676831\pi$
$757$	−21969.0	−1.05479	−0.527394	+	0.849621i	$0.676831\pi$
$758$	0	0
$759$	12310.5	0.588724
$760$	0	0
$761$	−8764.05	−0.417472	−0.208736	−	0.977972i	$-0.566935\pi$
$761$	−8764.05	−0.417472	−0.208736	+	0.977972i	$0.566935\pi$
$762$	0	0
$763$	−28126.2	−1.33452
$764$	0	0
$765$	−1000.49	−0.0472848
$766$	0	0
$767$	46.9184	0.00220877
$768$	0	0
$769$	17138.2	0.803664	0.401832	−	0.915713i	$-0.368374\pi$
$769$	17138.2	0.803664	0.401832	+	0.915713i	$0.368374\pi$
$770$	0	0
$771$	−60465.6	−2.82440
$772$	0	0
$773$	10744.3	0.499929	0.249965	−	0.968255i	$-0.419581\pi$
$773$	10744.3	0.499929	0.249965	+	0.968255i	$0.419581\pi$
$774$	0	0
$775$	−4097.43	−0.189915
$776$	0	0
$777$	−16125.5	−0.744529
$778$	0	0
$779$	11243.6	0.517129
$780$	0	0
$781$	−45422.7	−2.08112
$782$	0	0
$783$	21854.2	0.997451
$784$	0	0
$785$	−15035.4	−0.683612
$786$	0	0
$787$	9121.27	0.413136	0.206568	−	0.978432i	$-0.433771\pi$
$787$	9121.27	0.413136	0.206568	+	0.978432i	$0.433771\pi$
$788$	0	0
$789$	−37327.8	−1.68429
$790$	0	0
$791$	38986.0	1.75244
$792$	0	0
$793$	6943.16	0.310919
$794$	0	0
$795$	14246.6	0.635564
$796$	0	0
$797$	−22575.1	−1.00333	−0.501664	−	0.865062i	$-0.667279\pi$
$797$	−22575.1	−1.00333	−0.501664	+	0.865062i	$0.667279\pi$
$798$	0	0
$799$	−747.294	−0.0330881
$800$	0	0
$801$	−13706.7	−0.604620
$802$	0	0
$803$	31904.6	1.40210
$804$	0	0
$805$	−2441.03	−0.106876
$806$	0	0
$807$	44666.2	1.94836
$808$	0	0
$809$	29450.8	1.27989	0.639947	−	0.768419i	$-0.278956\pi$
$809$	29450.8	1.27989	0.639947	+	0.768419i	$0.278956\pi$
$810$	0	0
$811$	19531.9	0.845692	0.422846	−	0.906201i	$-0.361031\pi$
$811$	19531.9	0.845692	0.422846	+	0.906201i	$0.361031\pi$
$812$	0	0
$813$	35703.3	1.54019
$814$	0	0
$815$	8733.21	0.375351
$816$	0	0
$817$	250.254	0.0107164
$818$	0	0
$819$	19078.7	0.813999
$820$	0	0
$821$	−12231.0	−0.519933	−0.259966	−	0.965618i	$-0.583711\pi$
$821$	−12231.0	−0.519933	−0.259966	+	0.965618i	$0.583711\pi$
$822$	0	0
$823$	5666.78	0.240014	0.120007	−	0.992773i	$-0.461708\pi$
$823$	5666.78	0.240014	0.120007	+	0.992773i	$0.461708\pi$
$824$	0	0
$825$	13380.9	0.564684
$826$	0	0
$827$	−26084.1	−1.09677	−0.548387	−	0.836225i	$-0.684758\pi$
$827$	−26084.1	−1.09677	−0.548387	+	0.836225i	$0.684758\pi$
$828$	0	0
$829$	−22124.1	−0.926900	−0.463450	−	0.886123i	$-0.653389\pi$
$829$	−22124.1	−0.926900	−0.463450	+	0.886123i	$0.653389\pi$
$830$	0	0
$831$	63081.4	2.63330
$832$	0	0
$833$	−337.649	−0.0140442
$834$	0	0
$835$	−2810.25	−0.116470
$836$	0	0
$837$	−57362.0	−2.36884
$838$	0	0
$839$	24133.7	0.993073	0.496537	−	0.868016i	$-0.334605\pi$
$839$	24133.7	0.993073	0.496537	+	0.868016i	$0.334605\pi$
$840$	0	0
$841$	−20489.9	−0.840129
$842$	0	0
$843$	−29361.7	−1.19961
$844$	0	0
$845$	−9990.78	−0.406738
$846$	0	0
$847$	−38761.6	−1.57245
$848$	0	0
$849$	22757.5	0.919946
$850$	0	0
$851$	1834.27	0.0738873
$852$	0	0
$853$	−6078.12	−0.243975	−0.121988	−	0.992532i	$-0.538927\pi$
$853$	−6078.12	−0.243975	−0.121988	+	0.992532i	$0.538927\pi$
$854$	0	0
$855$	10927.2	0.437078
$856$	0	0
$857$	28506.9	1.13626	0.568132	−	0.822937i	$-0.307666\pi$
$857$	28506.9	1.13626	0.568132	+	0.822937i	$0.307666\pi$
$858$	0	0
$859$	5837.32	0.231859	0.115929	−	0.993257i	$-0.463015\pi$
$859$	5837.32	0.231859	0.115929	+	0.993257i	$0.463015\pi$
$860$	0	0
$861$	−66307.5	−2.62457
$862$	0	0
$863$	−11975.1	−0.472349	−0.236174	−	0.971711i	$-0.575894\pi$
$863$	−11975.1	−0.472349	−0.236174	+	0.971711i	$0.575894\pi$
$864$	0	0
$865$	−6691.92	−0.263043
$866$	0	0
$867$	−46706.4	−1.82957
$868$	0	0
$869$	37924.6	1.48044
$870$	0	0
$871$	−13981.4	−0.543904
$872$	0	0
$873$	−96576.4	−3.74412
$874$	0	0
$875$	−2653.29	−0.102512
$876$	0	0
$877$	−30561.3	−1.17672	−0.588359	−	0.808600i	$-0.700226\pi$
$877$	−30561.3	−1.17672	−0.588359	+	0.808600i	$0.700226\pi$
$878$	0	0
$879$	17194.2	0.659778
$880$	0	0
$881$	5188.32	0.198410	0.0992049	−	0.995067i	$-0.468370\pi$
$881$	5188.32	0.198410	0.0992049	+	0.995067i	$0.468370\pi$
$882$	0	0
$883$	−19996.6	−0.762105	−0.381052	−	0.924553i	$-0.624438\pi$
$883$	−19996.6	−0.762105	−0.381052	+	0.924553i	$0.624438\pi$
$884$	0	0
$885$	−158.475	−0.00601929
$886$	0	0
$887$	−31944.8	−1.20924	−0.604622	−	0.796512i	$-0.706676\pi$
$887$	−31944.8	−1.20924	−0.604622	+	0.796512i	$0.706676\pi$
$888$	0	0
$889$	−49120.2	−1.85313
$890$	0	0
$891$	90626.5	3.40752
$892$	0	0
$893$	8161.80	0.305850
$894$	0	0
$895$	6906.83	0.257955
$896$	0	0
$897$	−3089.48	−0.115000
$898$	0	0
$899$	−10234.2	−0.379676
$900$	0	0
$901$	−938.993	−0.0347197
$902$	0	0
$903$	−1475.84	−0.0543886
$904$	0	0
$905$	18776.2	0.689660
$906$	0	0
$907$	−9742.42	−0.356661	−0.178331	−	0.983971i	$-0.557070\pi$
$907$	−9742.42	−0.356661	−0.178331	+	0.983971i	$0.557070\pi$
$908$	0	0
$909$	−19307.4	−0.704495
$910$	0	0
$911$	24081.2	0.875792	0.437896	−	0.899026i	$-0.355724\pi$
$911$	24081.2	0.875792	0.437896	+	0.899026i	$0.355724\pi$
$912$	0	0
$913$	−44953.4	−1.62951
$914$	0	0
$915$	−23451.7	−0.847309
$916$	0	0
$917$	−46639.4	−1.67957
$918$	0	0
$919$	−3077.10	−0.110451	−0.0552253	−	0.998474i	$-0.517588\pi$
$919$	−3077.10	−0.110451	−0.0552253	+	0.998474i	$0.517588\pi$
$920$	0	0
$921$	11686.4	0.418111
$922$	0	0
$923$	11399.4	0.406519
$924$	0	0
$925$	1993.78	0.0708702
$926$	0	0
$927$	31383.6	1.11195
$928$	0	0
$929$	9824.54	0.346967	0.173484	−	0.984837i	$-0.444498\pi$
$929$	9824.54	0.346967	0.173484	+	0.984837i	$0.444498\pi$
$930$	0	0
$931$	3687.74	0.129818
$932$	0	0
$933$	−35548.7	−1.24739
$934$	0	0
$935$	−881.941	−0.0308476
$936$	0	0
$937$	−48654.2	−1.69633	−0.848165	−	0.529731i	$-0.822293\pi$
$937$	−48654.2	−1.69633	−0.848165	+	0.529731i	$0.822293\pi$
$938$	0	0
$939$	−35106.5	−1.22008
$940$	0	0
$941$	−32224.5	−1.11635	−0.558176	−	0.829723i	$-0.688499\pi$
$941$	−32224.5	−1.11635	−0.558176	+	0.829723i	$0.688499\pi$
$942$	0	0
$943$	7542.47	0.260463
$944$	0	0
$945$	−37144.8	−1.27865
$946$	0	0
$947$	18909.7	0.648874	0.324437	−	0.945907i	$-0.394825\pi$
$947$	18909.7	0.648874	0.324437	+	0.945907i	$0.394825\pi$
$948$	0	0
$949$	−8006.88	−0.273882
$950$	0	0
$951$	−91767.1	−3.12908
$952$	0	0
$953$	21759.8	0.739633	0.369817	−	0.929105i	$-0.379421\pi$
$953$	21759.8	0.739633	0.369817	+	0.929105i	$0.379421\pi$
$954$	0	0
$955$	−5318.90	−0.180226
$956$	0	0
$957$	33421.6	1.12891
$958$	0	0
$959$	−52764.9	−1.77671
$960$	0	0
$961$	−2928.76	−0.0983102
$962$	0	0
$963$	18436.3	0.616928
$964$	0	0
$965$	−25998.1	−0.867264
$966$	0	0
$967$	23020.8	0.765564	0.382782	−	0.923839i	$-0.374966\pi$
$967$	23020.8	0.765564	0.382782	+	0.923839i	$0.374966\pi$
$968$	0	0
$969$	−1025.29	−0.0339907
$970$	0	0
$971$	−45806.9	−1.51392	−0.756958	−	0.653464i	$-0.773315\pi$
$971$	−45806.9	−1.51392	−0.756958	+	0.653464i	$0.773315\pi$
$972$	0	0
$973$	−9802.10	−0.322961
$974$	0	0
$975$	−3358.13	−0.110304
$976$	0	0
$977$	22310.6	0.730582	0.365291	−	0.930893i	$-0.380969\pi$
$977$	22310.6	0.730582	0.365291	+	0.930893i	$0.380969\pi$
$978$	0	0
$979$	−12082.5	−0.394442
$980$	0	0
$981$	84460.9	2.74886
$982$	0	0
$983$	−14157.1	−0.459351	−0.229675	−	0.973267i	$-0.573766\pi$
$983$	−14157.1	−0.459351	−0.229675	+	0.973267i	$0.573766\pi$
$984$	0	0
$985$	−12796.6	−0.413942
$986$	0	0
$987$	−48133.1	−1.55227
$988$	0	0
$989$	167.877	0.00539754
$990$	0	0
$991$	−30904.6	−0.990634	−0.495317	−	0.868712i	$-0.664948\pi$
$991$	−30904.6	−0.990634	−0.495317	+	0.868712i	$0.664948\pi$
$992$	0	0
$993$	−78947.1	−2.52297
$994$	0	0
$995$	−7044.25	−0.224440
$996$	0	0
$997$	−2638.84	−0.0838243	−0.0419121	−	0.999121i	$-0.513345\pi$
$997$	−2638.84	−0.0838243	−0.0419121	+	0.999121i	$0.513345\pi$
$998$	0	0
$999$	27911.9	0.883977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.x.1.8		8
4.3	odd	2		920.4.a.c.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.c.1.1	✓	8	4.3	odd	2
1840.4.a.x.1.8		8	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)

\(f(q)\)	\(=\)	\(q+9.52581 q^{3} +5.00000 q^{5} -21.2263 q^{7} +63.7410 q^{9} +O(q^{10})\)	\(q+9.52581 q^{3} +5.00000 q^{5} -21.2263 q^{7} +63.7410 q^{9} +56.1882 q^{11} -14.1012 q^{13} +47.6290 q^{15} -3.13924 q^{17} +34.2862 q^{19} -202.198 q^{21} +23.0000 q^{23} +25.0000 q^{25} +349.988 q^{27} +62.4426 q^{29} -163.897 q^{31} +535.238 q^{33} -106.132 q^{35} +79.7510 q^{37} -134.325 q^{39} +327.933 q^{41} +7.29898 q^{43} +318.705 q^{45} +238.049 q^{47} +107.557 q^{49} -29.9038 q^{51} +299.115 q^{53} +280.941 q^{55} +326.604 q^{57} -3.32727 q^{59} -492.381 q^{61} -1352.99 q^{63} -70.5059 q^{65} +991.503 q^{67} +219.094 q^{69} -808.403 q^{71} +567.816 q^{73} +238.145 q^{75} -1192.67 q^{77} +674.957 q^{79} +1612.91 q^{81} -800.050 q^{83} -15.6962 q^{85} +594.817 q^{87} -215.037 q^{89} +299.316 q^{91} -1561.25 q^{93} +171.431 q^{95} -1515.14 q^{97} +3581.49 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9}+O(q^{10}) \) 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9	\( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9} + 35 q^{11} - 54 q^{13} + 60 q^{15} - 11 q^{17} + 107 q^{19} - 148 q^{21} + 184 q^{23} + 200 q^{25} + 405 q^{27} + 37 q^{29} + 290 q^{31} + 289 q^{33} + 155 q^{35} - 172 q^{37} + 311 q^{39} + 308 q^{41} + 538 q^{43} + 310 q^{45} + 1035 q^{47} + 585 q^{49} + 387 q^{51} + 46 q^{53} + 175 q^{55} + 286 q^{57} + 1256 q^{59} + 399 q^{61} + 1234 q^{63} - 270 q^{65} + 1598 q^{67} + 276 q^{69} + 750 q^{71} + 177 q^{73} + 300 q^{75} - 1228 q^{77} + 1292 q^{79} + 380 q^{81} + 2094 q^{83} - 55 q^{85} + 2245 q^{87} + 484 q^{89} + 679 q^{91} - 2111 q^{93} + 535 q^{95} - 2225 q^{97} + 2117 q^{99}+O(q^{100}) \) 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9 + 35 * q^11 - 54 * q^13 + 60 * q^15 - 11 * q^17 + 107 * q^19 - 148 * q^21 + 184 * q^23 + 200 * q^25 + 405 * q^27 + 37 * q^29 + 290 * q^31 + 289 * q^33 + 155 * q^35 - 172 * q^37 + 311 * q^39 + 308 * q^41 + 538 * q^43 + 310 * q^45 + 1035 * q^47 + 585 * q^49 + 387 * q^51 + 46 * q^53 + 175 * q^55 + 286 * q^57 + 1256 * q^59 + 399 * q^61 + 1234 * q^63 - 270 * q^65 + 1598 * q^67 + 276 * q^69 + 750 * q^71 + 177 * q^73 + 300 * q^75 - 1228 * q^77 + 1292 * q^79 + 380 * q^81 + 2094 * q^83 - 55 * q^85 + 2245 * q^87 + 484 * q^89 + 679 * q^91 - 2111 * q^93 + 535 * q^95 - 2225 * q^97 + 2117 * q^99

Introduction

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Modular forms

Varieties

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Database

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