LMFDB - Embedded newform 1840.4.a.p.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 34x^{3} - 9x^{2} + 260x + 60 $ x^5 - 34x^3 - 9x^2 + 260*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.67392$ 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)

$f(q)$	$=$	$q+9.09294 q^{3} -5.00000 q^{5} -6.08885 q^{7} +55.6816 q^{9} +O(q^{10})$	$q+9.09294 q^{3} -5.00000 q^{5} -6.08885 q^{7} +55.6816 q^{9} -23.3883 q^{11} +37.3062 q^{13} -45.4647 q^{15} -91.4681 q^{17} +4.18673 q^{19} -55.3656 q^{21} -23.0000 q^{23} +25.0000 q^{25} +260.800 q^{27} -251.116 q^{29} -305.603 q^{31} -212.668 q^{33} +30.4442 q^{35} +262.641 q^{37} +339.223 q^{39} -288.844 q^{41} +377.593 q^{43} -278.408 q^{45} -463.271 q^{47} -305.926 q^{49} -831.715 q^{51} -581.906 q^{53} +116.941 q^{55} +38.0697 q^{57} +58.8318 q^{59} -45.6685 q^{61} -339.037 q^{63} -186.531 q^{65} +772.445 q^{67} -209.138 q^{69} +16.4499 q^{71} +664.642 q^{73} +227.324 q^{75} +142.408 q^{77} +1138.34 q^{79} +868.040 q^{81} +187.943 q^{83} +457.341 q^{85} -2283.38 q^{87} -730.250 q^{89} -227.152 q^{91} -2778.83 q^{93} -20.9336 q^{95} -212.290 q^{97} -1302.30 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9}+O(q^{10}) $ 5 * q + 6 * q^3 - 25 * q^5 + 15 * q^7 + 79 * q^9	$ 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9} + 153 q^{11} + 28 q^{13} - 30 q^{15} - 341 q^{17} - 3 q^{19} - 212 q^{21} - 115 q^{23} + 125 q^{25} + 243 q^{27} - 583 q^{29} - 662 q^{31} - 457 q^{33} - 75 q^{35} - 172 q^{37} - 83 q^{39} + 344 q^{41} + 230 q^{43} - 395 q^{45} + 337 q^{47} - 4 q^{49} + 205 q^{51} - 942 q^{53} - 765 q^{55} - 890 q^{57} + 1166 q^{59} + 499 q^{61} + 1228 q^{63} - 140 q^{65} + 972 q^{67} - 138 q^{69} + 14 q^{71} - 229 q^{73} + 150 q^{75} + 312 q^{77} + 88 q^{79} + 897 q^{81} + 72 q^{83} + 1705 q^{85} - 1157 q^{87} - 90 q^{89} + 1309 q^{91} - 3071 q^{93} + 15 q^{95} - 1765 q^{97} + 91 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 - 25 * q^5 + 15 * q^7 + 79 * q^9 + 153 * q^11 + 28 * q^13 - 30 * q^15 - 341 * q^17 - 3 * q^19 - 212 * q^21 - 115 * q^23 + 125 * q^25 + 243 * q^27 - 583 * q^29 - 662 * q^31 - 457 * q^33 - 75 * q^35 - 172 * q^37 - 83 * q^39 + 344 * q^41 + 230 * q^43 - 395 * q^45 + 337 * q^47 - 4 * q^49 + 205 * q^51 - 942 * q^53 - 765 * q^55 - 890 * q^57 + 1166 * q^59 + 499 * q^61 + 1228 * q^63 - 140 * q^65 + 972 * q^67 - 138 * q^69 + 14 * q^71 - 229 * q^73 + 150 * q^75 + 312 * q^77 + 88 * q^79 + 897 * q^81 + 72 * q^83 + 1705 * q^85 - 1157 * q^87 - 90 * q^89 + 1309 * q^91 - 3071 * q^93 + 15 * q^95 - 1765 * q^97 + 91 * 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
$3$	9.09294	1.74994	0.874969	−	0.484179i	$-0.160882\pi$
$3$	9.09294	1.74994	0.874969	+	0.484179i	$0.160882\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−6.08885	−0.328767	−0.164383	−	0.986397i	$-0.552563\pi$
$7$	−6.08885	−0.328767	−0.164383	+	0.986397i	$0.552563\pi$
$8$	0	0
$9$	55.6816	2.06228
$10$	0	0
$11$	−23.3883	−0.641076	−0.320538	−	0.947236i	$-0.603864\pi$
$11$	−23.3883	−0.641076	−0.320538	+	0.947236i	$0.603864\pi$
$12$	0	0
$13$	37.3062	0.795913	0.397957	−	0.917404i	$-0.369719\pi$
$13$	37.3062	0.795913	0.397957	+	0.917404i	$0.369719\pi$
$14$	0	0
$15$	−45.4647	−0.782596
$16$	0	0
$17$	−91.4681	−1.30496	−0.652479	−	0.757807i	$-0.726271\pi$
$17$	−91.4681	−1.30496	−0.652479	+	0.757807i	$0.726271\pi$
$18$	0	0
$19$	4.18673	0.0505527	0.0252764	−	0.999681i	$-0.491953\pi$
$19$	4.18673	0.0505527	0.0252764	+	0.999681i	$0.491953\pi$
$20$	0	0
$21$	−55.3656	−0.575322
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	260.800	1.85893
$28$	0	0
$29$	−251.116	−1.60797	−0.803983	−	0.594653i	$-0.797289\pi$
$29$	−251.116	−1.60797	−0.803983	+	0.594653i	$0.797289\pi$
$30$	0	0
$31$	−305.603	−1.77058	−0.885289	−	0.465042i	$-0.846039\pi$
$31$	−305.603	−1.77058	−0.885289	+	0.465042i	$0.846039\pi$
$32$	0	0
$33$	−212.668	−1.12184
$34$	0	0
$35$	30.4442	0.147029
$36$	0	0
$37$	262.641	1.16697	0.583486	−	0.812123i	$-0.301688\pi$
$37$	262.641	1.16697	0.583486	+	0.812123i	$0.301688\pi$
$38$	0	0
$39$	339.223	1.39280
$40$	0	0
$41$	−288.844	−1.10024	−0.550120	−	0.835085i	$-0.685418\pi$
$41$	−288.844	−1.10024	−0.550120	+	0.835085i	$0.685418\pi$
$42$	0	0
$43$	377.593	1.33913	0.669563	−	0.742755i	$-0.266481\pi$
$43$	377.593	1.33913	0.669563	+	0.742755i	$0.266481\pi$
$44$	0	0
$45$	−278.408	−0.922281
$46$	0	0
$47$	−463.271	−1.43777	−0.718883	−	0.695131i	$-0.755346\pi$
$47$	−463.271	−1.43777	−0.718883	+	0.695131i	$0.755346\pi$
$48$	0	0
$49$	−305.926	−0.891912
$50$	0	0
$51$	−831.715	−2.28359
$52$	0	0
$53$	−581.906	−1.50813	−0.754066	−	0.656799i	$-0.771910\pi$
$53$	−581.906	−1.50813	−0.754066	+	0.656799i	$0.771910\pi$
$54$	0	0
$55$	116.941	0.286698
$56$	0	0
$57$	38.0697	0.0884641
$58$	0	0
$59$	58.8318	0.129818	0.0649089	−	0.997891i	$-0.479324\pi$
$59$	58.8318	0.129818	0.0649089	+	0.997891i	$0.479324\pi$
$60$	0	0
$61$	−45.6685	−0.0958566	−0.0479283	−	0.998851i	$-0.515262\pi$
$61$	−45.6685	−0.0958566	−0.0479283	+	0.998851i	$0.515262\pi$
$62$	0	0
$63$	−339.037	−0.678010
$64$	0	0
$65$	−186.531	−0.355943
$66$	0	0
$67$	772.445	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$67$	772.445	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$68$	0	0
$69$	−209.138	−0.364887
$70$	0	0
$71$	16.4499	0.0274964	0.0137482	−	0.999905i	$-0.495624\pi$
$71$	16.4499	0.0274964	0.0137482	+	0.999905i	$0.495624\pi$
$72$	0	0
$73$	664.642	1.06562	0.532811	−	0.846234i	$-0.321135\pi$
$73$	664.642	1.06562	0.532811	+	0.846234i	$0.321135\pi$
$74$	0	0
$75$	227.324	0.349988
$76$	0	0
$77$	142.408	0.210764
$78$	0	0
$79$	1138.34	1.62118	0.810590	−	0.585614i	$-0.199146\pi$
$79$	1138.34	1.62118	0.810590	+	0.585614i	$0.199146\pi$
$80$	0	0
$81$	868.040	1.19073
$82$	0	0
$83$	187.943	0.248548	0.124274	−	0.992248i	$-0.460340\pi$
$83$	187.943	0.248548	0.124274	+	0.992248i	$0.460340\pi$
$84$	0	0
$85$	457.341	0.583595
$86$	0	0
$87$	−2283.38	−2.81384
$88$	0	0
$89$	−730.250	−0.869734	−0.434867	−	0.900495i	$-0.643205\pi$
$89$	−730.250	−0.869734	−0.434867	+	0.900495i	$0.643205\pi$
$90$	0	0
$91$	−227.152	−0.261670
$92$	0	0
$93$	−2778.83	−3.09840
$94$	0	0
$95$	−20.9336	−0.0226079
$96$	0	0
$97$	−212.290	−0.222214	−0.111107	−	0.993808i	$-0.535440\pi$
$97$	−212.290	−0.222214	−0.111107	+	0.993808i	$0.535440\pi$
$98$	0	0
$99$	−1302.30	−1.32208
$100$	0	0
$101$	306.073	0.301539	0.150770	−	0.988569i	$-0.451825\pi$
$101$	306.073	0.301539	0.150770	+	0.988569i	$0.451825\pi$
$102$	0	0
$103$	−1142.06	−1.09253	−0.546263	−	0.837614i	$-0.683950\pi$
$103$	−1142.06	−1.09253	−0.546263	+	0.837614i	$0.683950\pi$
$104$	0	0
$105$	276.828	0.257292
$106$	0	0
$107$	1042.14	0.941567	0.470783	−	0.882249i	$-0.343971\pi$
$107$	1042.14	0.941567	0.470783	+	0.882249i	$0.343971\pi$
$108$	0	0
$109$	−1350.29	−1.18655	−0.593276	−	0.804999i	$-0.702166\pi$
$109$	−1350.29	−1.18655	−0.593276	+	0.804999i	$0.702166\pi$
$110$	0	0
$111$	2388.18	2.04213
$112$	0	0
$113$	−979.622	−0.815532	−0.407766	−	0.913086i	$-0.633692\pi$
$113$	−979.622	−0.815532	−0.407766	+	0.913086i	$0.633692\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	2077.27	1.64140
$118$	0	0
$119$	556.936	0.429027
$120$	0	0
$121$	−783.988	−0.589022
$122$	0	0
$123$	−2626.44	−1.92535
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	827.970	0.578508	0.289254	−	0.957252i	$-0.406593\pi$
$127$	827.970	0.578508	0.289254	+	0.957252i	$0.406593\pi$
$128$	0	0
$129$	3433.43	2.34339
$130$	0	0
$131$	−1908.48	−1.27286	−0.636430	−	0.771335i	$-0.719589\pi$
$131$	−1908.48	−1.27286	−0.636430	+	0.771335i	$0.719589\pi$
$132$	0	0
$133$	−25.4924	−0.0166201
$134$	0	0
$135$	−1304.00	−0.831338
$136$	0	0
$137$	861.188	0.537053	0.268526	−	0.963272i	$-0.413463\pi$
$137$	861.188	0.537053	0.268526	+	0.963272i	$0.413463\pi$
$138$	0	0
$139$	−544.460	−0.332234	−0.166117	−	0.986106i	$-0.553123\pi$
$139$	−544.460	−0.332234	−0.166117	+	0.986106i	$0.553123\pi$
$140$	0	0
$141$	−4212.50	−2.51600
$142$	0	0
$143$	−872.527	−0.510240
$144$	0	0
$145$	1255.58	0.719104
$146$	0	0
$147$	−2781.77	−1.56079
$148$	0	0
$149$	2040.97	1.12217	0.561084	−	0.827759i	$-0.310385\pi$
$149$	2040.97	1.12217	0.561084	+	0.827759i	$0.310385\pi$
$150$	0	0
$151$	−1429.68	−0.770503	−0.385251	−	0.922812i	$-0.625885\pi$
$151$	−1429.68	−0.770503	−0.385251	+	0.922812i	$0.625885\pi$
$152$	0	0
$153$	−5093.09	−2.69119
$154$	0	0
$155$	1528.01	0.791826
$156$	0	0
$157$	11.2494	0.00571847	0.00285923	−	0.999996i	$-0.499090\pi$
$157$	11.2494	0.00571847	0.00285923	+	0.999996i	$0.499090\pi$
$158$	0	0
$159$	−5291.24	−2.63914
$160$	0	0
$161$	140.044	0.0685527
$162$	0	0
$163$	499.193	0.239876	0.119938	−	0.992781i	$-0.461730\pi$
$163$	499.193	0.239876	0.119938	+	0.992781i	$0.461730\pi$
$164$	0	0
$165$	1063.34	0.501703
$166$	0	0
$167$	−1282.80	−0.594408	−0.297204	−	0.954814i	$-0.596054\pi$
$167$	−1282.80	−0.594408	−0.297204	+	0.954814i	$0.596054\pi$
$168$	0	0
$169$	−805.249	−0.366522
$170$	0	0
$171$	233.124	0.104254
$172$	0	0
$173$	−4501.79	−1.97841	−0.989204	−	0.146543i	$-0.953185\pi$
$173$	−4501.79	−1.97841	−0.989204	+	0.146543i	$0.953185\pi$
$174$	0	0
$175$	−152.221	−0.0657534
$176$	0	0
$177$	534.954	0.227173
$178$	0	0
$179$	24.3630	0.0101730	0.00508652	−	0.999987i	$-0.498381\pi$
$179$	24.3630	0.0101730	0.00508652	+	0.999987i	$0.498381\pi$
$180$	0	0
$181$	−1664.65	−0.683603	−0.341802	−	0.939772i	$-0.611037\pi$
$181$	−1664.65	−0.683603	−0.341802	+	0.939772i	$0.611037\pi$
$182$	0	0
$183$	−415.261	−0.167743
$184$	0	0
$185$	−1313.21	−0.521886
$186$	0	0
$187$	2139.28	0.836576
$188$	0	0
$189$	−1587.97	−0.611154
$190$	0	0
$191$	−4095.76	−1.55162	−0.775809	−	0.630968i	$-0.782658\pi$
$191$	−4095.76	−1.55162	−0.775809	+	0.630968i	$0.782658\pi$
$192$	0	0
$193$	−3191.15	−1.19018	−0.595088	−	0.803661i	$-0.702883\pi$
$193$	−3191.15	−1.19018	−0.595088	+	0.803661i	$0.702883\pi$
$194$	0	0
$195$	−1696.11	−0.622878
$196$	0	0
$197$	−3411.93	−1.23396	−0.616979	−	0.786979i	$-0.711644\pi$
$197$	−3411.93	−1.23396	−0.616979	+	0.786979i	$0.711644\pi$
$198$	0	0
$199$	1497.33	0.533380	0.266690	−	0.963782i	$-0.414070\pi$
$199$	1497.33	0.533380	0.266690	+	0.963782i	$0.414070\pi$
$200$	0	0
$201$	7023.80	2.46478
$202$	0	0
$203$	1529.00	0.528646
$204$	0	0
$205$	1444.22	0.492043
$206$	0	0
$207$	−1280.68	−0.430016
$208$	0	0
$209$	−97.9204	−0.0324081
$210$	0	0
$211$	1750.46	0.571122	0.285561	−	0.958360i	$-0.407820\pi$
$211$	1750.46	0.571122	0.285561	+	0.958360i	$0.407820\pi$
$212$	0	0
$213$	149.578	0.0481170
$214$	0	0
$215$	−1887.97	−0.598875
$216$	0	0
$217$	1860.77	0.582107
$218$	0	0
$219$	6043.55	1.86477
$220$	0	0
$221$	−3412.33	−1.03863
$222$	0	0
$223$	3344.95	1.00446	0.502229	−	0.864734i	$-0.332513\pi$
$223$	3344.95	1.00446	0.502229	+	0.864734i	$0.332513\pi$
$224$	0	0
$225$	1392.04	0.412457
$226$	0	0
$227$	4819.05	1.40904	0.704519	−	0.709685i	$-0.251163\pi$
$227$	4819.05	1.40904	0.704519	+	0.709685i	$0.251163\pi$
$228$	0	0
$229$	−5778.01	−1.66734	−0.833672	−	0.552260i	$-0.813765\pi$
$229$	−5778.01	−1.66734	−0.833672	+	0.552260i	$0.813765\pi$
$230$	0	0
$231$	1294.91	0.368825
$232$	0	0
$233$	725.579	0.204010	0.102005	−	0.994784i	$-0.467474\pi$
$233$	725.579	0.204010	0.102005	+	0.994784i	$0.467474\pi$
$234$	0	0
$235$	2316.35	0.642988
$236$	0	0
$237$	10350.9	2.83697
$238$	0	0
$239$	2967.85	0.803239	0.401619	−	0.915807i	$-0.368447\pi$
$239$	2967.85	0.803239	0.401619	+	0.915807i	$0.368447\pi$
$240$	0	0
$241$	4738.02	1.26640	0.633201	−	0.773987i	$-0.281741\pi$
$241$	4738.02	1.26640	0.633201	+	0.773987i	$0.281741\pi$
$242$	0	0
$243$	851.427	0.224770
$244$	0	0
$245$	1529.63	0.398875
$246$	0	0
$247$	156.191	0.0402356
$248$	0	0
$249$	1708.96	0.434943
$250$	0	0
$251$	3320.80	0.835087	0.417543	−	0.908657i	$-0.362891\pi$
$251$	3320.80	0.835087	0.417543	+	0.908657i	$0.362891\pi$
$252$	0	0
$253$	537.930	0.133673
$254$	0	0
$255$	4158.57	1.02125
$256$	0	0
$257$	−2422.68	−0.588025	−0.294012	−	0.955802i	$-0.594991\pi$
$257$	−2422.68	−0.588025	−0.294012	+	0.955802i	$0.594991\pi$
$258$	0	0
$259$	−1599.18	−0.383662
$260$	0	0
$261$	−13982.5	−3.31608
$262$	0	0
$263$	−6954.54	−1.63055	−0.815277	−	0.579072i	$-0.803415\pi$
$263$	−6954.54	−1.63055	−0.815277	+	0.579072i	$0.803415\pi$
$264$	0	0
$265$	2909.53	0.674457
$266$	0	0
$267$	−6640.12	−1.52198
$268$	0	0
$269$	5717.90	1.29601	0.648005	−	0.761636i	$-0.275604\pi$
$269$	5717.90	1.29601	0.648005	+	0.761636i	$0.275604\pi$
$270$	0	0
$271$	−6459.11	−1.44783	−0.723917	−	0.689887i	$-0.757660\pi$
$271$	−6459.11	−1.44783	−0.723917	+	0.689887i	$0.757660\pi$
$272$	0	0
$273$	−2065.48	−0.457906
$274$	0	0
$275$	−584.707	−0.128215
$276$	0	0
$277$	−599.177	−0.129968	−0.0649838	−	0.997886i	$-0.520700\pi$
$277$	−599.177	−0.129968	−0.0649838	+	0.997886i	$0.520700\pi$
$278$	0	0
$279$	−17016.5	−3.65143
$280$	0	0
$281$	−290.857	−0.0617476	−0.0308738	−	0.999523i	$-0.509829\pi$
$281$	−290.857	−0.0617476	−0.0308738	+	0.999523i	$0.509829\pi$
$282$	0	0
$283$	8928.37	1.87539	0.937697	−	0.347455i	$-0.112954\pi$
$283$	8928.37	1.87539	0.937697	+	0.347455i	$0.112954\pi$
$284$	0	0
$285$	−190.348	−0.0395624
$286$	0	0
$287$	1758.73	0.361723
$288$	0	0
$289$	3453.42	0.702915
$290$	0	0
$291$	−1930.34	−0.388861
$292$	0	0
$293$	6131.90	1.22263	0.611313	−	0.791389i	$-0.290642\pi$
$293$	6131.90	1.22263	0.611313	+	0.791389i	$0.290642\pi$
$294$	0	0
$295$	−294.159	−0.0580563
$296$	0	0
$297$	−6099.67	−1.19171
$298$	0	0
$299$	−858.042	−0.165959
$300$	0	0
$301$	−2299.11	−0.440260
$302$	0	0
$303$	2783.11	0.527675
$304$	0	0
$305$	228.343	0.0428684
$306$	0	0
$307$	4677.89	0.869646	0.434823	−	0.900516i	$-0.356811\pi$
$307$	4677.89	0.869646	0.434823	+	0.900516i	$0.356811\pi$
$308$	0	0
$309$	−10384.7	−1.91185
$310$	0	0
$311$	2876.66	0.524504	0.262252	−	0.964999i	$-0.415535\pi$
$311$	2876.66	0.524504	0.262252	+	0.964999i	$0.415535\pi$
$312$	0	0
$313$	9384.77	1.69476	0.847378	−	0.530991i	$-0.178180\pi$
$313$	9384.77	1.69476	0.847378	+	0.530991i	$0.178180\pi$
$314$	0	0
$315$	1695.19	0.303215
$316$	0	0
$317$	3897.38	0.690532	0.345266	−	0.938505i	$-0.387789\pi$
$317$	3897.38	0.690532	0.345266	+	0.938505i	$0.387789\pi$
$318$	0	0
$319$	5873.16	1.03083
$320$	0	0
$321$	9476.14	1.64768
$322$	0	0
$323$	−382.952	−0.0659691
$324$	0	0
$325$	932.654	0.159183
$326$	0	0
$327$	−12278.1	−2.07639
$328$	0	0
$329$	2820.79	0.472690
$330$	0	0
$331$	3220.53	0.534792	0.267396	−	0.963587i	$-0.413837\pi$
$331$	3220.53	0.534792	0.267396	+	0.963587i	$0.413837\pi$
$332$	0	0
$333$	14624.3	2.40663
$334$	0	0
$335$	−3862.22	−0.629898
$336$	0	0
$337$	−728.120	−0.117695	−0.0588475	−	0.998267i	$-0.518743\pi$
$337$	−728.120	−0.117695	−0.0588475	+	0.998267i	$0.518743\pi$
$338$	0	0
$339$	−8907.65	−1.42713
$340$	0	0
$341$	7147.52	1.13507
$342$	0	0
$343$	3951.21	0.621998
$344$	0	0
$345$	1045.69	0.163183
$346$	0	0
$347$	3086.63	0.477518	0.238759	−	0.971079i	$-0.423259\pi$
$347$	3086.63	0.477518	0.238759	+	0.971079i	$0.423259\pi$
$348$	0	0
$349$	−1999.39	−0.306662	−0.153331	−	0.988175i	$-0.549000\pi$
$349$	−1999.39	−0.306662	−0.153331	+	0.988175i	$0.549000\pi$
$350$	0	0
$351$	9729.47	1.47955
$352$	0	0
$353$	7313.70	1.10275	0.551373	−	0.834259i	$-0.314104\pi$
$353$	7313.70	1.10275	0.551373	+	0.834259i	$0.314104\pi$
$354$	0	0
$355$	−82.2495	−0.0122968
$356$	0	0
$357$	5064.19	0.750771
$358$	0	0
$359$	−3934.11	−0.578369	−0.289184	−	0.957273i	$-0.593384\pi$
$359$	−3934.11	−0.578369	−0.289184	+	0.957273i	$0.593384\pi$
$360$	0	0
$361$	−6841.47	−0.997444
$362$	0	0
$363$	−7128.76	−1.03075
$364$	0	0
$365$	−3323.21	−0.476561
$366$	0	0
$367$	3052.00	0.434096	0.217048	−	0.976161i	$-0.430357\pi$
$367$	3052.00	0.434096	0.217048	+	0.976161i	$0.430357\pi$
$368$	0	0
$369$	−16083.3	−2.26901
$370$	0	0
$371$	3543.14	0.495824
$372$	0	0
$373$	−10909.5	−1.51440	−0.757202	−	0.653180i	$-0.773434\pi$
$373$	−10909.5	−1.51440	−0.757202	+	0.653180i	$0.773434\pi$
$374$	0	0
$375$	−1136.62	−0.156519
$376$	0	0
$377$	−9368.16	−1.27980
$378$	0	0
$379$	−3050.04	−0.413378	−0.206689	−	0.978407i	$-0.566269\pi$
$379$	−3050.04	−0.413378	−0.206689	+	0.978407i	$0.566269\pi$
$380$	0	0
$381$	7528.69	1.01235
$382$	0	0
$383$	−2257.74	−0.301215	−0.150607	−	0.988594i	$-0.548123\pi$
$383$	−2257.74	−0.301215	−0.150607	+	0.988594i	$0.548123\pi$
$384$	0	0
$385$	−712.039	−0.0942567
$386$	0	0
$387$	21025.0	2.76166
$388$	0	0
$389$	5012.00	0.653261	0.326631	−	0.945152i	$-0.394087\pi$
$389$	5012.00	0.653261	0.326631	+	0.945152i	$0.394087\pi$
$390$	0	0
$391$	2103.77	0.272102
$392$	0	0
$393$	−17353.7	−2.22742
$394$	0	0
$395$	−5691.70	−0.725014
$396$	0	0
$397$	13110.3	1.65740	0.828702	−	0.559690i	$-0.189080\pi$
$397$	13110.3	1.65740	0.828702	+	0.559690i	$0.189080\pi$
$398$	0	0
$399$	−231.801	−0.0290841
$400$	0	0
$401$	−4328.91	−0.539091	−0.269546	−	0.962988i	$-0.586874\pi$
$401$	−4328.91	−0.539091	−0.269546	+	0.962988i	$0.586874\pi$
$402$	0	0
$403$	−11400.9	−1.40923
$404$	0	0
$405$	−4340.20	−0.532509
$406$	0	0
$407$	−6142.73	−0.748118
$408$	0	0
$409$	−9470.32	−1.14493	−0.572466	−	0.819928i	$-0.694013\pi$
$409$	−9470.32	−1.14493	−0.572466	+	0.819928i	$0.694013\pi$
$410$	0	0
$411$	7830.73	0.939809
$412$	0	0
$413$	−358.218	−0.0426798
$414$	0	0
$415$	−939.716	−0.111154
$416$	0	0
$417$	−4950.74	−0.581388
$418$	0	0
$419$	6279.53	0.732160	0.366080	−	0.930583i	$-0.380700\pi$
$419$	6279.53	0.732160	0.366080	+	0.930583i	$0.380700\pi$
$420$	0	0
$421$	−2252.13	−0.260717	−0.130359	−	0.991467i	$-0.541613\pi$
$421$	−2252.13	−0.260717	−0.130359	+	0.991467i	$0.541613\pi$
$422$	0	0
$423$	−25795.7	−2.96508
$424$	0	0
$425$	−2286.70	−0.260992
$426$	0	0
$427$	278.069	0.0315145
$428$	0	0
$429$	−7933.84	−0.892889
$430$	0	0
$431$	−1497.33	−0.167341	−0.0836706	−	0.996493i	$-0.526664\pi$
$431$	−1497.33	−0.167341	−0.0836706	+	0.996493i	$0.526664\pi$
$432$	0	0
$433$	−12114.1	−1.34449	−0.672247	−	0.740327i	$-0.734671\pi$
$433$	−12114.1	−1.34449	−0.672247	+	0.740327i	$0.734671\pi$
$434$	0	0
$435$	11416.9	1.25839
$436$	0	0
$437$	−96.2948	−0.0105410
$438$	0	0
$439$	−6642.77	−0.722191	−0.361096	−	0.932529i	$-0.617597\pi$
$439$	−6642.77	−0.722191	−0.361096	+	0.932529i	$0.617597\pi$
$440$	0	0
$441$	−17034.5	−1.83938
$442$	0	0
$443$	5341.45	0.572867	0.286434	−	0.958100i	$-0.407530\pi$
$443$	5341.45	0.572867	0.286434	+	0.958100i	$0.407530\pi$
$444$	0	0
$445$	3651.25	0.388957
$446$	0	0
$447$	18558.4	1.96372
$448$	0	0
$449$	−5383.77	−0.565870	−0.282935	−	0.959139i	$-0.591308\pi$
$449$	−5383.77	−0.565870	−0.282935	+	0.959139i	$0.591308\pi$
$450$	0	0
$451$	6755.57	0.705337
$452$	0	0
$453$	−13000.0	−1.34833
$454$	0	0
$455$	1135.76	0.117022
$456$	0	0
$457$	3966.41	0.405998	0.202999	−	0.979179i	$-0.434931\pi$
$457$	3966.41	0.405998	0.202999	+	0.979179i	$0.434931\pi$
$458$	0	0
$459$	−23854.9	−2.42582
$460$	0	0
$461$	−1750.10	−0.176812	−0.0884059	−	0.996085i	$-0.528177\pi$
$461$	−1750.10	−0.176812	−0.0884059	+	0.996085i	$0.528177\pi$
$462$	0	0
$463$	−5398.87	−0.541915	−0.270958	−	0.962591i	$-0.587340\pi$
$463$	−5398.87	−0.541915	−0.270958	+	0.962591i	$0.587340\pi$
$464$	0	0
$465$	13894.1	1.38565
$466$	0	0
$467$	3496.86	0.346500	0.173250	−	0.984878i	$-0.444573\pi$
$467$	3496.86	0.346500	0.173250	+	0.984878i	$0.444573\pi$
$468$	0	0
$469$	−4703.30	−0.463067
$470$	0	0
$471$	102.290	0.0100070
$472$	0	0
$473$	−8831.25	−0.858481
$474$	0	0
$475$	104.668	0.0101105
$476$	0	0
$477$	−32401.5	−3.11019
$478$	0	0
$479$	−10708.6	−1.02148	−0.510741	−	0.859735i	$-0.670629\pi$
$479$	−10708.6	−1.02148	−0.510741	+	0.859735i	$0.670629\pi$
$480$	0	0
$481$	9798.15	0.928809
$482$	0	0
$483$	1273.41	0.119963
$484$	0	0
$485$	1061.45	0.0993773
$486$	0	0
$487$	7441.58	0.692423	0.346212	−	0.938156i	$-0.387468\pi$
$487$	7441.58	0.692423	0.346212	+	0.938156i	$0.387468\pi$
$488$	0	0
$489$	4539.13	0.419768
$490$	0	0
$491$	12195.4	1.12091	0.560457	−	0.828184i	$-0.310626\pi$
$491$	12195.4	1.12091	0.560457	+	0.828184i	$0.310626\pi$
$492$	0	0
$493$	22969.1	2.09833
$494$	0	0
$495$	6511.49	0.591252
$496$	0	0
$497$	−100.161	−0.00903991
$498$	0	0
$499$	−18698.5	−1.67747	−0.838737	−	0.544537i	$-0.816705\pi$
$499$	−18698.5	−1.67747	−0.838737	+	0.544537i	$0.816705\pi$
$500$	0	0
$501$	−11664.4	−1.04018
$502$	0	0
$503$	−677.534	−0.0600592	−0.0300296	−	0.999549i	$-0.509560\pi$
$503$	−677.534	−0.0600592	−0.0300296	+	0.999549i	$0.509560\pi$
$504$	0	0
$505$	−1530.37	−0.134852
$506$	0	0
$507$	−7322.09	−0.641391
$508$	0	0
$509$	−2236.19	−0.194730	−0.0973649	−	0.995249i	$-0.531041\pi$
$509$	−2236.19	−0.194730	−0.0973649	+	0.995249i	$0.531041\pi$
$510$	0	0
$511$	−4046.91	−0.350342
$512$	0	0
$513$	1091.90	0.0939739
$514$	0	0
$515$	5710.28	0.488593
$516$	0	0
$517$	10835.1	0.921717
$518$	0	0
$519$	−40934.5	−3.46209
$520$	0	0
$521$	6093.61	0.512411	0.256205	−	0.966622i	$-0.417528\pi$
$521$	6093.61	0.512411	0.256205	+	0.966622i	$0.417528\pi$
$522$	0	0
$523$	18419.2	1.53999	0.769996	−	0.638048i	$-0.220258\pi$
$523$	18419.2	1.53999	0.769996	+	0.638048i	$0.220258\pi$
$524$	0	0
$525$	−1384.14	−0.115064
$526$	0	0
$527$	27952.9	2.31053
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	3275.85	0.267721
$532$	0	0
$533$	−10775.7	−0.875696
$534$	0	0
$535$	−5210.71	−0.421082
$536$	0	0
$537$	221.531	0.0178022
$538$	0	0
$539$	7155.08	0.571783
$540$	0	0
$541$	7550.87	0.600069	0.300035	−	0.953928i	$-0.403002\pi$
$541$	7550.87	0.600069	0.300035	+	0.953928i	$0.403002\pi$
$542$	0	0
$543$	−15136.5	−1.19626
$544$	0	0
$545$	6751.44	0.530642
$546$	0	0
$547$	−11201.2	−0.875552	−0.437776	−	0.899084i	$-0.644234\pi$
$547$	−11201.2	−0.875552	−0.437776	+	0.899084i	$0.644234\pi$
$548$	0	0
$549$	−2542.90	−0.197683
$550$	0	0
$551$	−1051.35	−0.0812870
$552$	0	0
$553$	−6931.18	−0.532991
$554$	0	0
$555$	−11940.9	−0.913268
$556$	0	0
$557$	−4414.29	−0.335798	−0.167899	−	0.985804i	$-0.553698\pi$
$557$	−4414.29	−0.335798	−0.167899	+	0.985804i	$0.553698\pi$
$558$	0	0
$559$	14086.6	1.06583
$560$	0	0
$561$	19452.4	1.46396
$562$	0	0
$563$	19600.8	1.46728	0.733638	−	0.679540i	$-0.237821\pi$
$563$	19600.8	1.46728	0.733638	+	0.679540i	$0.237821\pi$
$564$	0	0
$565$	4898.11	0.364717
$566$	0	0
$567$	−5285.37	−0.391472
$568$	0	0
$569$	14529.6	1.07049	0.535247	−	0.844695i	$-0.320218\pi$
$569$	14529.6	1.07049	0.535247	+	0.844695i	$0.320218\pi$
$570$	0	0
$571$	−9398.85	−0.688844	−0.344422	−	0.938815i	$-0.611925\pi$
$571$	−9398.85	−0.688844	−0.344422	+	0.938815i	$0.611925\pi$
$572$	0	0
$573$	−37242.6	−2.71524
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−19166.1	−1.38284	−0.691418	−	0.722455i	$-0.743014\pi$
$577$	−19166.1	−1.38284	−0.691418	+	0.722455i	$0.743014\pi$
$578$	0	0
$579$	−29016.9	−2.08273
$580$	0	0
$581$	−1144.36	−0.0817143
$582$	0	0
$583$	13609.8	0.966827
$584$	0	0
$585$	−10386.3	−0.734055
$586$	0	0
$587$	2935.53	0.206409	0.103205	−	0.994660i	$-0.467090\pi$
$587$	2935.53	0.206409	0.103205	+	0.994660i	$0.467090\pi$
$588$	0	0
$589$	−1279.48	−0.0895075
$590$	0	0
$591$	−31024.5	−2.15935
$592$	0	0
$593$	3401.07	0.235523	0.117761	−	0.993042i	$-0.462428\pi$
$593$	3401.07	0.235523	0.117761	+	0.993042i	$0.462428\pi$
$594$	0	0
$595$	−2784.68	−0.191867
$596$	0	0
$597$	13615.1	0.933382
$598$	0	0
$599$	12678.1	0.864798	0.432399	−	0.901682i	$-0.357667\pi$
$599$	12678.1	0.864798	0.432399	+	0.901682i	$0.357667\pi$
$600$	0	0
$601$	18186.9	1.23438	0.617188	−	0.786816i	$-0.288272\pi$
$601$	18186.9	1.23438	0.617188	+	0.786816i	$0.288272\pi$
$602$	0	0
$603$	43011.0	2.90471
$604$	0	0
$605$	3919.94	0.263419
$606$	0	0
$607$	−11286.6	−0.754710	−0.377355	−	0.926069i	$-0.623166\pi$
$607$	−11286.6	−0.754710	−0.377355	+	0.926069i	$0.623166\pi$
$608$	0	0
$609$	13903.2	0.925097
$610$	0	0
$611$	−17282.9	−1.14434
$612$	0	0
$613$	−1874.43	−0.123503	−0.0617516	−	0.998092i	$-0.519669\pi$
$613$	−1874.43	−0.123503	−0.0617516	+	0.998092i	$0.519669\pi$
$614$	0	0
$615$	13132.2	0.861044
$616$	0	0
$617$	9829.98	0.641394	0.320697	−	0.947182i	$-0.396083\pi$
$617$	9829.98	0.641394	0.320697	+	0.947182i	$0.396083\pi$
$618$	0	0
$619$	−10785.0	−0.700298	−0.350149	−	0.936694i	$-0.613869\pi$
$619$	−10785.0	−0.700298	−0.350149	+	0.936694i	$0.613869\pi$
$620$	0	0
$621$	−5998.41	−0.387613
$622$	0	0
$623$	4446.38	0.285940
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−890.385	−0.0567122
$628$	0	0
$629$	−24023.3	−1.52285
$630$	0	0
$631$	−18254.8	−1.15168	−0.575841	−	0.817562i	$-0.695325\pi$
$631$	−18254.8	−1.15168	−0.575841	+	0.817562i	$0.695325\pi$
$632$	0	0
$633$	15916.9	0.999429
$634$	0	0
$635$	−4139.85	−0.258717
$636$	0	0
$637$	−11412.9	−0.709885
$638$	0	0
$639$	915.957	0.0567053
$640$	0	0
$641$	24421.9	1.50485	0.752423	−	0.658681i	$-0.228885\pi$
$641$	24421.9	1.50485	0.752423	+	0.658681i	$0.228885\pi$
$642$	0	0
$643$	−6756.53	−0.414388	−0.207194	−	0.978300i	$-0.566433\pi$
$643$	−6756.53	−0.414388	−0.207194	+	0.978300i	$0.566433\pi$
$644$	0	0
$645$	−17167.2	−1.04799
$646$	0	0
$647$	18630.8	1.13208	0.566038	−	0.824379i	$-0.308476\pi$
$647$	18630.8	1.13208	0.566038	+	0.824379i	$0.308476\pi$
$648$	0	0
$649$	−1375.97	−0.0832230
$650$	0	0
$651$	16919.9	1.01865
$652$	0	0
$653$	17406.2	1.04312	0.521560	−	0.853215i	$-0.325350\pi$
$653$	17406.2	1.04312	0.521560	+	0.853215i	$0.325350\pi$
$654$	0	0
$655$	9542.39	0.569240
$656$	0	0
$657$	37008.4	2.19762
$658$	0	0
$659$	−9406.78	−0.556048	−0.278024	−	0.960574i	$-0.589680\pi$
$659$	−9406.78	−0.556048	−0.278024	+	0.960574i	$0.589680\pi$
$660$	0	0
$661$	−19184.8	−1.12890	−0.564448	−	0.825468i	$-0.690911\pi$
$661$	−19184.8	−1.12890	−0.564448	+	0.825468i	$0.690911\pi$
$662$	0	0
$663$	−31028.1	−1.81754
$664$	0	0
$665$	127.462	0.00743272
$666$	0	0
$667$	5775.66	0.335284
$668$	0	0
$669$	30415.4	1.75774
$670$	0	0
$671$	1068.11	0.0614513
$672$	0	0
$673$	22916.4	1.31257	0.656286	−	0.754512i	$-0.272126\pi$
$673$	22916.4	1.31257	0.656286	+	0.754512i	$0.272126\pi$
$674$	0	0
$675$	6520.01	0.371786
$676$	0	0
$677$	3942.02	0.223788	0.111894	−	0.993720i	$-0.464308\pi$
$677$	3942.02	0.223788	0.111894	+	0.993720i	$0.464308\pi$
$678$	0	0
$679$	1292.60	0.0730568
$680$	0	0
$681$	43819.4	2.46573
$682$	0	0
$683$	−30363.4	−1.70106	−0.850529	−	0.525928i	$-0.823718\pi$
$683$	−30363.4	−1.70106	−0.850529	+	0.525928i	$0.823718\pi$
$684$	0	0
$685$	−4305.94	−0.240177
$686$	0	0
$687$	−52539.1	−2.91775
$688$	0	0
$689$	−21708.7	−1.20034
$690$	0	0
$691$	7051.96	0.388233	0.194117	−	0.980978i	$-0.437816\pi$
$691$	7051.96	0.388233	0.194117	+	0.980978i	$0.437816\pi$
$692$	0	0
$693$	7929.49	0.434656
$694$	0	0
$695$	2722.30	0.148579
$696$	0	0
$697$	26420.0	1.43577
$698$	0	0
$699$	6597.65	0.357005
$700$	0	0
$701$	−3759.62	−0.202566	−0.101283	−	0.994858i	$-0.532295\pi$
$701$	−3759.62	−0.202566	−0.101283	+	0.994858i	$0.532295\pi$
$702$	0	0
$703$	1099.61	0.0589936
$704$	0	0
$705$	21062.5	1.12519
$706$	0	0
$707$	−1863.63	−0.0991361
$708$	0	0
$709$	18003.2	0.953630	0.476815	−	0.879004i	$-0.341791\pi$
$709$	18003.2	0.953630	0.476815	+	0.879004i	$0.341791\pi$
$710$	0	0
$711$	63384.6	3.34333
$712$	0	0
$713$	7028.87	0.369191
$714$	0	0
$715$	4362.64	0.228186
$716$	0	0
$717$	26986.5	1.40562
$718$	0	0
$719$	17960.3	0.931579	0.465790	−	0.884896i	$-0.345770\pi$
$719$	17960.3	0.931579	0.465790	+	0.884896i	$0.345770\pi$
$720$	0	0
$721$	6953.81	0.359187
$722$	0	0
$723$	43082.6	2.21613
$724$	0	0
$725$	−6277.89	−0.321593
$726$	0	0
$727$	24624.4	1.25621	0.628106	−	0.778128i	$-0.283830\pi$
$727$	24624.4	1.25621	0.628106	+	0.778128i	$0.283830\pi$
$728$	0	0
$729$	−15695.1	−0.797394
$730$	0	0
$731$	−34537.7	−1.74750
$732$	0	0
$733$	−14879.3	−0.749767	−0.374883	−	0.927072i	$-0.622317\pi$
$733$	−14879.3	−0.749767	−0.374883	+	0.927072i	$0.622317\pi$
$734$	0	0
$735$	13908.8	0.698007
$736$	0	0
$737$	−18066.2	−0.902952
$738$	0	0
$739$	−11114.8	−0.553267	−0.276633	−	0.960976i	$-0.589219\pi$
$739$	−11114.8	−0.553267	−0.276633	+	0.960976i	$0.589219\pi$
$740$	0	0
$741$	1420.23	0.0704097
$742$	0	0
$743$	19788.5	0.977077	0.488539	−	0.872542i	$-0.337530\pi$
$743$	19788.5	0.977077	0.488539	+	0.872542i	$0.337530\pi$
$744$	0	0
$745$	−10204.9	−0.501849
$746$	0	0
$747$	10465.0	0.512576
$748$	0	0
$749$	−6345.45	−0.309556
$750$	0	0
$751$	25874.9	1.25724	0.628621	−	0.777712i	$-0.283620\pi$
$751$	25874.9	1.25724	0.628621	+	0.777712i	$0.283620\pi$
$752$	0	0
$753$	30195.8	1.46135
$754$	0	0
$755$	7148.41	0.344579
$756$	0	0
$757$	24849.3	1.19308	0.596542	−	0.802582i	$-0.296541\pi$
$757$	24849.3	1.19308	0.596542	+	0.802582i	$0.296541\pi$
$758$	0	0
$759$	4891.37	0.233920
$760$	0	0
$761$	−22285.7	−1.06157	−0.530786	−	0.847506i	$-0.678103\pi$
$761$	−22285.7	−1.06157	−0.530786	+	0.847506i	$0.678103\pi$
$762$	0	0
$763$	8221.70	0.390099
$764$	0	0
$765$	25465.5	1.20354
$766$	0	0
$767$	2194.79	0.103324
$768$	0	0
$769$	−18158.9	−0.851530	−0.425765	−	0.904834i	$-0.639995\pi$
$769$	−18158.9	−0.851530	−0.425765	+	0.904834i	$0.639995\pi$
$770$	0	0
$771$	−22029.3	−1.02901
$772$	0	0
$773$	23789.9	1.10694	0.553469	−	0.832870i	$-0.313304\pi$
$773$	23789.9	1.10694	0.553469	+	0.832870i	$0.313304\pi$
$774$	0	0
$775$	−7640.07	−0.354115
$776$	0	0
$777$	−14541.3	−0.671385
$778$	0	0
$779$	−1209.31	−0.0556201
$780$	0	0
$781$	−384.735	−0.0176273
$782$	0	0
$783$	−65491.1	−2.98909
$784$	0	0
$785$	−56.2470	−0.00255738
$786$	0	0
$787$	−18652.8	−0.844855	−0.422427	−	0.906397i	$-0.638822\pi$
$787$	−18652.8	−0.844855	−0.422427	+	0.906397i	$0.638822\pi$
$788$	0	0
$789$	−63237.3	−2.85337
$790$	0	0
$791$	5964.77	0.268120
$792$	0	0
$793$	−1703.72	−0.0762935
$794$	0	0
$795$	26456.2	1.18026
$796$	0	0
$797$	−7173.54	−0.318820	−0.159410	−	0.987212i	$-0.550959\pi$
$797$	−7173.54	−0.318820	−0.159410	+	0.987212i	$0.550959\pi$
$798$	0	0
$799$	42374.5	1.87622
$800$	0	0
$801$	−40661.5	−1.79364
$802$	0	0
$803$	−15544.8	−0.683145
$804$	0	0
$805$	−700.218	−0.0306577
$806$	0	0
$807$	51992.5	2.26794
$808$	0	0
$809$	−32204.8	−1.39958	−0.699790	−	0.714348i	$-0.746723\pi$
$809$	−32204.8	−1.39958	−0.699790	+	0.714348i	$0.746723\pi$
$810$	0	0
$811$	−5718.25	−0.247589	−0.123795	−	0.992308i	$-0.539506\pi$
$811$	−5718.25	−0.247589	−0.123795	+	0.992308i	$0.539506\pi$
$812$	0	0
$813$	−58732.4	−2.53362
$814$	0	0
$815$	−2495.97	−0.107276
$816$	0	0
$817$	1580.88	0.0676964
$818$	0	0
$819$	−12648.2	−0.539637
$820$	0	0
$821$	−45293.4	−1.92540	−0.962699	−	0.270576i	$-0.912786\pi$
$821$	−45293.4	−1.92540	−0.962699	+	0.270576i	$0.912786\pi$
$822$	0	0
$823$	27520.0	1.16560	0.582799	−	0.812617i	$-0.301958\pi$
$823$	27520.0	1.16560	0.582799	+	0.812617i	$0.301958\pi$
$824$	0	0
$825$	−5316.71	−0.224368
$826$	0	0
$827$	−29720.5	−1.24968	−0.624839	−	0.780754i	$-0.714835\pi$
$827$	−29720.5	−1.24968	−0.624839	+	0.780754i	$0.714835\pi$
$828$	0	0
$829$	46805.2	1.96093	0.980467	−	0.196686i	$-0.0630180\pi$
$829$	46805.2	1.96093	0.980467	+	0.196686i	$0.0630180\pi$
$830$	0	0
$831$	−5448.28	−0.227435
$832$	0	0
$833$	27982.5	1.16391
$834$	0	0
$835$	6414.00	0.265827
$836$	0	0
$837$	−79701.4	−3.29138
$838$	0	0
$839$	−17095.1	−0.703441	−0.351720	−	0.936105i	$-0.614403\pi$
$839$	−17095.1	−0.703441	−0.351720	+	0.936105i	$0.614403\pi$
$840$	0	0
$841$	38670.0	1.58555
$842$	0	0
$843$	−2644.75	−0.108054
$844$	0	0
$845$	4026.25	0.163914
$846$	0	0
$847$	4773.59	0.193651
$848$	0	0
$849$	81185.1	3.28182
$850$	0	0
$851$	−6040.75	−0.243331
$852$	0	0
$853$	6229.79	0.250063	0.125032	−	0.992153i	$-0.460097\pi$
$853$	6229.79	0.250063	0.125032	+	0.992153i	$0.460097\pi$
$854$	0	0
$855$	−1165.62	−0.0466238
$856$	0	0
$857$	41110.7	1.63864	0.819320	−	0.573336i	$-0.194351\pi$
$857$	41110.7	1.63864	0.819320	+	0.573336i	$0.194351\pi$
$858$	0	0
$859$	16800.0	0.667297	0.333649	−	0.942698i	$-0.391720\pi$
$859$	16800.0	0.667297	0.333649	+	0.942698i	$0.391720\pi$
$860$	0	0
$861$	15992.0	0.632992
$862$	0	0
$863$	23136.9	0.912619	0.456309	−	0.889821i	$-0.349171\pi$
$863$	23136.9	0.912619	0.456309	+	0.889821i	$0.349171\pi$
$864$	0	0
$865$	22508.9	0.884771
$866$	0	0
$867$	31401.8	1.23006
$868$	0	0
$869$	−26623.8	−1.03930
$870$	0	0
$871$	28817.0	1.12104
$872$	0	0
$873$	−11820.7	−0.458269
$874$	0	0
$875$	761.106	0.0294058
$876$	0	0
$877$	−8550.22	−0.329214	−0.164607	−	0.986359i	$-0.552636\pi$
$877$	−8550.22	−0.329214	−0.164607	+	0.986359i	$0.552636\pi$
$878$	0	0
$879$	55757.0	2.13952
$880$	0	0
$881$	−5655.10	−0.216260	−0.108130	−	0.994137i	$-0.534486\pi$
$881$	−5655.10	−0.216260	−0.108130	+	0.994137i	$0.534486\pi$
$882$	0	0
$883$	−45667.4	−1.74046	−0.870232	−	0.492642i	$-0.836031\pi$
$883$	−45667.4	−1.74046	−0.870232	+	0.492642i	$0.836031\pi$
$884$	0	0
$885$	−2674.77	−0.101595
$886$	0	0
$887$	−15597.2	−0.590419	−0.295210	−	0.955432i	$-0.595390\pi$
$887$	−15597.2	−0.590419	−0.295210	+	0.955432i	$0.595390\pi$
$888$	0	0
$889$	−5041.39	−0.190194
$890$	0	0
$891$	−20302.0	−0.763346
$892$	0	0
$893$	−1939.59	−0.0726830
$894$	0	0
$895$	−121.815	−0.00454952
$896$	0	0
$897$	−7802.13	−0.290419
$898$	0	0
$899$	76741.6	2.84703
$900$	0	0
$901$	53225.9	1.96805
$902$	0	0
$903$	−20905.7	−0.770428
$904$	0	0
$905$	8323.23	0.305717
$906$	0	0
$907$	−32954.8	−1.20645	−0.603223	−	0.797573i	$-0.706117\pi$
$907$	−32954.8	−1.20645	−0.603223	+	0.797573i	$0.706117\pi$
$908$	0	0
$909$	17042.7	0.621859
$910$	0	0
$911$	53565.9	1.94810	0.974049	−	0.226339i	$-0.0726756\pi$
$911$	53565.9	1.94810	0.974049	+	0.226339i	$0.0726756\pi$
$912$	0	0
$913$	−4395.67	−0.159338
$914$	0	0
$915$	2076.31	0.0750170
$916$	0	0
$917$	11620.4	0.418474
$918$	0	0
$919$	−24574.2	−0.882075	−0.441037	−	0.897489i	$-0.645389\pi$
$919$	−24574.2	−0.882075	−0.441037	+	0.897489i	$0.645389\pi$
$920$	0	0
$921$	42535.8	1.52183
$922$	0	0
$923$	613.683	0.0218847
$924$	0	0
$925$	6566.04	0.233395
$926$	0	0
$927$	−63591.6	−2.25310
$928$	0	0
$929$	28749.5	1.01533	0.507665	−	0.861555i	$-0.330509\pi$
$929$	28749.5	1.01533	0.507665	+	0.861555i	$0.330509\pi$
$930$	0	0
$931$	−1280.83	−0.0450886
$932$	0	0
$933$	26157.3	0.917849
$934$	0	0
$935$	−10696.4	−0.374128
$936$	0	0
$937$	−829.435	−0.0289183	−0.0144592	−	0.999895i	$-0.504603\pi$
$937$	−829.435	−0.0289183	−0.0144592	+	0.999895i	$0.504603\pi$
$938$	0	0
$939$	85335.1	2.96572
$940$	0	0
$941$	34374.9	1.19085	0.595425	−	0.803411i	$-0.296984\pi$
$941$	34374.9	1.19085	0.595425	+	0.803411i	$0.296984\pi$
$942$	0	0
$943$	6643.41	0.229416
$944$	0	0
$945$	7939.87	0.273317
$946$	0	0
$947$	−18356.8	−0.629901	−0.314951	−	0.949108i	$-0.601988\pi$
$947$	−18356.8	−0.629901	−0.314951	+	0.949108i	$0.601988\pi$
$948$	0	0
$949$	24795.3	0.848143
$950$	0	0
$951$	35438.7	1.20839
$952$	0	0
$953$	−35115.7	−1.19361	−0.596805	−	0.802386i	$-0.703563\pi$
$953$	−35115.7	−1.19361	−0.596805	+	0.802386i	$0.703563\pi$
$954$	0	0
$955$	20478.8	0.693905
$956$	0	0
$957$	53404.3	1.80388
$958$	0	0
$959$	−5243.64	−0.176565
$960$	0	0
$961$	63602.1	2.13494
$962$	0	0
$963$	58028.2	1.94178
$964$	0	0
$965$	15955.7	0.532263
$966$	0	0
$967$	−33244.8	−1.10557	−0.552783	−	0.833325i	$-0.686434\pi$
$967$	−33244.8	−1.10557	−0.552783	+	0.833325i	$0.686434\pi$
$968$	0	0
$969$	−3482.16	−0.115442
$970$	0	0
$971$	30522.7	1.00877	0.504387	−	0.863477i	$-0.331718\pi$
$971$	30522.7	1.00877	0.504387	+	0.863477i	$0.331718\pi$
$972$	0	0
$973$	3315.13	0.109227
$974$	0	0
$975$	8480.57	0.278560
$976$	0	0
$977$	−1961.10	−0.0642183	−0.0321092	−	0.999484i	$-0.510222\pi$
$977$	−1961.10	−0.0642183	−0.0321092	+	0.999484i	$0.510222\pi$
$978$	0	0
$979$	17079.3	0.557565
$980$	0	0
$981$	−75186.2	−2.44700
$982$	0	0
$983$	−45485.4	−1.47585	−0.737925	−	0.674883i	$-0.764194\pi$
$983$	−45485.4	−1.47585	−0.737925	+	0.674883i	$0.764194\pi$
$984$	0	0
$985$	17059.6	0.551843
$986$	0	0
$987$	25649.3	0.827178
$988$	0	0
$989$	−8684.64	−0.279227
$990$	0	0
$991$	−31271.6	−1.00240	−0.501198	−	0.865333i	$-0.667107\pi$
$991$	−31271.6	−1.00240	−0.501198	+	0.865333i	$0.667107\pi$
$992$	0	0
$993$	29284.1	0.935853
$994$	0	0
$995$	−7486.63	−0.238535
$996$	0	0
$997$	−15009.0	−0.476770	−0.238385	−	0.971171i	$-0.576618\pi$
$997$	−15009.0	−0.476770	−0.238385	+	0.971171i	$0.576618\pi$
$998$	0	0
$999$	68497.0	2.16932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.p.1.5		5
4.3	odd	2		115.4.a.d.1.2	✓	5
12.11	even	2		1035.4.a.m.1.4		5
20.3	even	4		575.4.b.h.24.9		10
20.7	even	4		575.4.b.h.24.2		10
20.19	odd	2		575.4.a.k.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.d.1.2	✓	5	4.3	odd	2
575.4.a.k.1.4		5	20.19	odd	2
575.4.b.h.24.2		10	20.7	even	4
575.4.b.h.24.9		10	20.3	even	4
1035.4.a.m.1.4		5	12.11	even	2
1840.4.a.p.1.5		5	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)

\(f(q)\)	\(=\)	\(q+9.09294 q^{3} -5.00000 q^{5} -6.08885 q^{7} +55.6816 q^{9} +O(q^{10})\)	\(q+9.09294 q^{3} -5.00000 q^{5} -6.08885 q^{7} +55.6816 q^{9} -23.3883 q^{11} +37.3062 q^{13} -45.4647 q^{15} -91.4681 q^{17} +4.18673 q^{19} -55.3656 q^{21} -23.0000 q^{23} +25.0000 q^{25} +260.800 q^{27} -251.116 q^{29} -305.603 q^{31} -212.668 q^{33} +30.4442 q^{35} +262.641 q^{37} +339.223 q^{39} -288.844 q^{41} +377.593 q^{43} -278.408 q^{45} -463.271 q^{47} -305.926 q^{49} -831.715 q^{51} -581.906 q^{53} +116.941 q^{55} +38.0697 q^{57} +58.8318 q^{59} -45.6685 q^{61} -339.037 q^{63} -186.531 q^{65} +772.445 q^{67} -209.138 q^{69} +16.4499 q^{71} +664.642 q^{73} +227.324 q^{75} +142.408 q^{77} +1138.34 q^{79} +868.040 q^{81} +187.943 q^{83} +457.341 q^{85} -2283.38 q^{87} -730.250 q^{89} -227.152 q^{91} -2778.83 q^{93} -20.9336 q^{95} -212.290 q^{97} -1302.30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9}+O(q^{10}) \) 5 * q + 6 * q^3 - 25 * q^5 + 15 * q^7 + 79 * q^9	\( 5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9} + 153 q^{11} + 28 q^{13} - 30 q^{15} - 341 q^{17} - 3 q^{19} - 212 q^{21} - 115 q^{23} + 125 q^{25} + 243 q^{27} - 583 q^{29} - 662 q^{31} - 457 q^{33} - 75 q^{35} - 172 q^{37} - 83 q^{39} + 344 q^{41} + 230 q^{43} - 395 q^{45} + 337 q^{47} - 4 q^{49} + 205 q^{51} - 942 q^{53} - 765 q^{55} - 890 q^{57} + 1166 q^{59} + 499 q^{61} + 1228 q^{63} - 140 q^{65} + 972 q^{67} - 138 q^{69} + 14 q^{71} - 229 q^{73} + 150 q^{75} + 312 q^{77} + 88 q^{79} + 897 q^{81} + 72 q^{83} + 1705 q^{85} - 1157 q^{87} - 90 q^{89} + 1309 q^{91} - 3071 q^{93} + 15 q^{95} - 1765 q^{97} + 91 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 - 25 * q^5 + 15 * q^7 + 79 * q^9 + 153 * q^11 + 28 * q^13 - 30 * q^15 - 341 * q^17 - 3 * q^19 - 212 * q^21 - 115 * q^23 + 125 * q^25 + 243 * q^27 - 583 * q^29 - 662 * q^31 - 457 * q^33 - 75 * q^35 - 172 * q^37 - 83 * q^39 + 344 * q^41 + 230 * q^43 - 395 * q^45 + 337 * q^47 - 4 * q^49 + 205 * q^51 - 942 * q^53 - 765 * q^55 - 890 * q^57 + 1166 * q^59 + 499 * q^61 + 1228 * q^63 - 140 * q^65 + 972 * q^67 - 138 * q^69 + 14 * q^71 - 229 * q^73 + 150 * q^75 + 312 * q^77 + 88 * q^79 + 897 * q^81 + 72 * q^83 + 1705 * q^85 - 1157 * q^87 - 90 * q^89 + 1309 * q^91 - 3071 * q^93 + 15 * q^95 - 1765 * q^97 + 91 * q^99

Introduction

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