LMFDB - Embedded newform 1160.4.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,4,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, 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("1160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1160.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:	$68.4422156067$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 179 x^{9} + 370 x^{8} + 10353 x^{7} - 19394 x^{6} - 210392 x^{5} + 267796 x^{4} + \cdots + 567808 $ x^11 - 2x^10 - 179x^9 + 370x^8 + 10353x^7 - 19394x^6 - 210392x^5 + 267796x^4 + 1107520x^3 - 894176x^2 - 757696x + 567808
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.848128$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.848128 q^{3} -5.00000 q^{5} +28.0939 q^{7} -26.2807 q^{9} -69.7247 q^{11} -38.8297 q^{13} -4.24064 q^{15} -13.6867 q^{17} -21.6449 q^{19} +23.8273 q^{21} +111.788 q^{23} +25.0000 q^{25} -45.1888 q^{27} -29.0000 q^{29} +255.526 q^{31} -59.1355 q^{33} -140.470 q^{35} -188.168 q^{37} -32.9325 q^{39} +469.212 q^{41} +437.992 q^{43} +131.403 q^{45} +309.302 q^{47} +446.270 q^{49} -11.6081 q^{51} -497.585 q^{53} +348.624 q^{55} -18.3576 q^{57} +887.837 q^{59} -415.947 q^{61} -738.328 q^{63} +194.148 q^{65} -295.924 q^{67} +94.8106 q^{69} -438.592 q^{71} -20.8671 q^{73} +21.2032 q^{75} -1958.84 q^{77} +176.943 q^{79} +671.252 q^{81} +709.660 q^{83} +68.4337 q^{85} -24.5957 q^{87} -13.1512 q^{89} -1090.88 q^{91} +216.718 q^{93} +108.225 q^{95} +493.631 q^{97} +1832.41 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) $ 11 * q + 2 * q^3 - 55 * q^5 + 56 * q^7 + 65 * q^9 - 8 * q^11 + 16 * q^13 - 10 * q^15 + 64 * q^17 + 160 * q^19 - 48 * q^21 + 140 * q^23 + 275 * q^25 - 136 * q^27 - 319 * q^29 - 144 * q^31 - 424 * q^33 - 280 * q^35 - 50 * q^37 + 572 * q^39 + 62 * q^41 + 326 * q^43 - 325 * q^45 + 1074 * q^47 + 1513 * q^49 - 144 * q^51 + 576 * q^53 + 40 * q^55 - 896 * q^57 - 468 * q^59 + 660 * q^61 + 2552 * q^63 - 80 * q^65 + 892 * q^67 - 510 * q^69 + 920 * q^71 - 676 * q^73 + 50 * q^75 + 1708 * q^77 + 2308 * q^79 + 1287 * q^81 - 1548 * q^83 - 320 * q^85 - 58 * q^87 + 1854 * q^89 + 3100 * q^91 + 5424 * q^93 - 800 * q^95 + 4724 * q^97 + 6884 * q^99

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
$3$	0.848128	0.163222	0.0816111	−	0.996664i	$-0.473993\pi$
$3$	0.848128	0.163222	0.0816111	+	0.996664i	$0.473993\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	28.0939	1.51693	0.758465	−	0.651713i	$-0.225949\pi$
$7$	28.0939	1.51693	0.758465	+	0.651713i	$0.225949\pi$
$8$	0	0
$9$	−26.2807	−0.973359
$10$	0	0
$11$	−69.7247	−1.91116	−0.955582	−	0.294727i	$-0.904771\pi$
$11$	−69.7247	−1.91116	−0.955582	+	0.294727i	$0.904771\pi$
$12$	0	0
$13$	−38.8297	−0.828416	−0.414208	−	0.910182i	$-0.635941\pi$
$13$	−38.8297	−0.828416	−0.414208	+	0.910182i	$0.635941\pi$
$14$	0	0
$15$	−4.24064	−0.0729952
$16$	0	0
$17$	−13.6867	−0.195266	−0.0976330	−	0.995222i	$-0.531127\pi$
$17$	−13.6867	−0.195266	−0.0976330	+	0.995222i	$0.531127\pi$
$18$	0	0
$19$	−21.6449	−0.261352	−0.130676	−	0.991425i	$-0.541715\pi$
$19$	−21.6449	−0.261352	−0.130676	+	0.991425i	$0.541715\pi$
$20$	0	0
$21$	23.8273	0.247597
$22$	0	0
$23$	111.788	1.01345	0.506727	−	0.862107i	$-0.330855\pi$
$23$	111.788	1.01345	0.506727	+	0.862107i	$0.330855\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−45.1888	−0.322096
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	255.526	1.48044	0.740222	−	0.672363i	$-0.234720\pi$
$31$	255.526	1.48044	0.740222	+	0.672363i	$0.234720\pi$
$32$	0	0
$33$	−59.1355	−0.311944
$34$	0	0
$35$	−140.470	−0.678392
$36$	0	0
$37$	−188.168	−0.836069	−0.418035	−	0.908431i	$-0.637281\pi$
$37$	−188.168	−0.836069	−0.418035	+	0.908431i	$0.637281\pi$
$38$	0	0
$39$	−32.9325	−0.135216
$40$	0	0
$41$	469.212	1.78728	0.893642	−	0.448780i	$-0.148141\pi$
$41$	469.212	1.78728	0.893642	+	0.448780i	$0.148141\pi$
$42$	0	0
$43$	437.992	1.55333	0.776665	−	0.629914i	$-0.216910\pi$
$43$	437.992	1.55333	0.776665	+	0.629914i	$0.216910\pi$
$44$	0	0
$45$	131.403	0.435299
$46$	0	0
$47$	309.302	0.959922	0.479961	−	0.877290i	$-0.340651\pi$
$47$	309.302	0.959922	0.479961	+	0.877290i	$0.340651\pi$
$48$	0	0
$49$	446.270	1.30108
$50$	0	0
$51$	−11.6081	−0.0318717
$52$	0	0
$53$	−497.585	−1.28960	−0.644798	−	0.764353i	$-0.723058\pi$
$53$	−497.585	−1.28960	−0.644798	+	0.764353i	$0.723058\pi$
$54$	0	0
$55$	348.624	0.854698
$56$	0	0
$57$	−18.3576	−0.0426584
$58$	0	0
$59$	887.837	1.95909	0.979547	−	0.201217i	$-0.0644898\pi$
$59$	887.837	1.95909	0.979547	+	0.201217i	$0.0644898\pi$
$60$	0	0
$61$	−415.947	−0.873057	−0.436529	−	0.899690i	$-0.643792\pi$
$61$	−415.947	−0.873057	−0.436529	+	0.899690i	$0.643792\pi$
$62$	0	0
$63$	−738.328	−1.47652
$64$	0	0
$65$	194.148	0.370479
$66$	0	0
$67$	−295.924	−0.539594	−0.269797	−	0.962917i	$-0.586957\pi$
$67$	−295.924	−0.539594	−0.269797	+	0.962917i	$0.586957\pi$
$68$	0	0
$69$	94.8106	0.165418
$70$	0	0
$71$	−438.592	−0.733117	−0.366559	−	0.930395i	$-0.619464\pi$
$71$	−438.592	−0.733117	−0.366559	+	0.930395i	$0.619464\pi$
$72$	0	0
$73$	−20.8671	−0.0334563	−0.0167281	−	0.999860i	$-0.505325\pi$
$73$	−20.8671	−0.0334563	−0.0167281	+	0.999860i	$0.505325\pi$
$74$	0	0
$75$	21.2032	0.0326444
$76$	0	0
$77$	−1958.84	−2.89910
$78$	0	0
$79$	176.943	0.251996	0.125998	−	0.992030i	$-0.459787\pi$
$79$	176.943	0.251996	0.125998	+	0.992030i	$0.459787\pi$
$80$	0	0
$81$	671.252	0.920785
$82$	0	0
$83$	709.660	0.938497	0.469249	−	0.883066i	$-0.344525\pi$
$83$	709.660	0.938497	0.469249	+	0.883066i	$0.344525\pi$
$84$	0	0
$85$	68.4337	0.0873256
$86$	0	0
$87$	−24.5957	−0.0303096
$88$	0	0
$89$	−13.1512	−0.0156632	−0.00783162	−	0.999969i	$-0.502493\pi$
$89$	−13.1512	−0.0156632	−0.00783162	+	0.999969i	$0.502493\pi$
$90$	0	0
$91$	−1090.88	−1.25665
$92$	0	0
$93$	216.718	0.241641
$94$	0	0
$95$	108.225	0.116880
$96$	0	0
$97$	493.631	0.516707	0.258354	−	0.966050i	$-0.416820\pi$
$97$	493.631	0.516707	0.258354	+	0.966050i	$0.416820\pi$
$98$	0	0
$99$	1832.41	1.86025
$100$	0	0
$101$	−1187.45	−1.16986	−0.584930	−	0.811084i	$-0.698878\pi$
$101$	−1187.45	−1.16986	−0.584930	+	0.811084i	$0.698878\pi$
$102$	0	0
$103$	1048.91	1.00342	0.501712	−	0.865035i	$-0.332704\pi$
$103$	1048.91	1.00342	0.501712	+	0.865035i	$0.332704\pi$
$104$	0	0
$105$	−119.136	−0.110729
$106$	0	0
$107$	−1328.40	−1.20020	−0.600099	−	0.799925i	$-0.704872\pi$
$107$	−1328.40	−1.20020	−0.600099	+	0.799925i	$0.704872\pi$
$108$	0	0
$109$	2217.54	1.94864	0.974321	−	0.225161i	$-0.0722909\pi$
$109$	2217.54	1.94864	0.974321	+	0.225161i	$0.0722909\pi$
$110$	0	0
$111$	−159.590	−0.136465
$112$	0	0
$113$	−493.891	−0.411162	−0.205581	−	0.978640i	$-0.565908\pi$
$113$	−493.891	−0.411162	−0.205581	+	0.978640i	$0.565908\pi$
$114$	0	0
$115$	−558.941	−0.453230
$116$	0	0
$117$	1020.47	0.806346
$118$	0	0
$119$	−384.514	−0.296205
$120$	0	0
$121$	3530.54	2.65254
$122$	0	0
$123$	397.952	0.291725
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2351.55	−1.64304	−0.821522	−	0.570177i	$-0.806875\pi$
$127$	−2351.55	−1.64304	−0.821522	+	0.570177i	$0.806875\pi$
$128$	0	0
$129$	371.473	0.253538
$130$	0	0
$131$	1678.33	1.11936	0.559682	−	0.828707i	$-0.310923\pi$
$131$	1678.33	1.11936	0.559682	+	0.828707i	$0.310923\pi$
$132$	0	0
$133$	−608.091	−0.396452
$134$	0	0
$135$	225.944	0.144046
$136$	0	0
$137$	1220.12	0.760889	0.380445	−	0.924804i	$-0.375771\pi$
$137$	1220.12	0.760889	0.380445	+	0.924804i	$0.375771\pi$
$138$	0	0
$139$	−1171.97	−0.715143	−0.357572	−	0.933886i	$-0.616395\pi$
$139$	−1171.97	−0.715143	−0.357572	+	0.933886i	$0.616395\pi$
$140$	0	0
$141$	262.328	0.156681
$142$	0	0
$143$	2707.39	1.58324
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	378.494	0.212365
$148$	0	0
$149$	2139.45	1.17631	0.588157	−	0.808746i	$-0.299854\pi$
$149$	2139.45	1.17631	0.588157	+	0.808746i	$0.299854\pi$
$150$	0	0
$151$	2623.32	1.41379	0.706897	−	0.707316i	$-0.250094\pi$
$151$	2623.32	1.41379	0.706897	+	0.707316i	$0.250094\pi$
$152$	0	0
$153$	359.697	0.190064
$154$	0	0
$155$	−1277.63	−0.662075
$156$	0	0
$157$	2750.34	1.39810	0.699048	−	0.715074i	$-0.253607\pi$
$157$	2750.34	1.39810	0.699048	+	0.715074i	$0.253607\pi$
$158$	0	0
$159$	−422.016	−0.210491
$160$	0	0
$161$	3140.57	1.53734
$162$	0	0
$163$	2207.46	1.06075	0.530374	−	0.847764i	$-0.322051\pi$
$163$	2207.46	1.06075	0.530374	+	0.847764i	$0.322051\pi$
$164$	0	0
$165$	295.677	0.139506
$166$	0	0
$167$	252.313	0.116914	0.0584568	−	0.998290i	$-0.481382\pi$
$167$	252.313	0.116914	0.0584568	+	0.998290i	$0.481382\pi$
$168$	0	0
$169$	−689.257	−0.313727
$170$	0	0
$171$	568.843	0.254389
$172$	0	0
$173$	4.51450	0.00198400	0.000991998	−	1.00000i	$-0.499684\pi$
$173$	4.51450	0.00198400	0.000991998		1.00000i	$0.499684\pi$
$174$	0	0
$175$	702.349	0.303386
$176$	0	0
$177$	752.999	0.319768
$178$	0	0
$179$	−3397.76	−1.41878	−0.709388	−	0.704818i	$-0.751028\pi$
$179$	−3397.76	−1.41878	−0.709388	+	0.704818i	$0.751028\pi$
$180$	0	0
$181$	2429.13	0.997546	0.498773	−	0.866733i	$-0.333784\pi$
$181$	2429.13	0.997546	0.498773	+	0.866733i	$0.333784\pi$
$182$	0	0
$183$	−352.776	−0.142502
$184$	0	0
$185$	940.838	0.373902
$186$	0	0
$187$	954.304	0.373185
$188$	0	0
$189$	−1269.53	−0.488597
$190$	0	0
$191$	−4256.62	−1.61256	−0.806278	−	0.591537i	$-0.798521\pi$
$191$	−4256.62	−1.61256	−0.806278	+	0.591537i	$0.798521\pi$
$192$	0	0
$193$	2983.93	1.11289	0.556445	−	0.830884i	$-0.312165\pi$
$193$	2983.93	1.11289	0.556445	+	0.830884i	$0.312165\pi$
$194$	0	0
$195$	164.663	0.0604704
$196$	0	0
$197$	−3711.23	−1.34221	−0.671103	−	0.741364i	$-0.734179\pi$
$197$	−3711.23	−1.34221	−0.671103	+	0.741364i	$0.734179\pi$
$198$	0	0
$199$	1399.38	0.498490	0.249245	−	0.968440i	$-0.419818\pi$
$199$	1399.38	0.498490	0.249245	+	0.968440i	$0.419818\pi$
$200$	0	0
$201$	−250.981	−0.0880738
$202$	0	0
$203$	−814.725	−0.281687
$204$	0	0
$205$	−2346.06	−0.799298
$206$	0	0
$207$	−2937.87	−0.986454
$208$	0	0
$209$	1509.18	0.499486
$210$	0	0
$211$	−4315.78	−1.40811	−0.704053	−	0.710148i	$-0.748628\pi$
$211$	−4315.78	−1.40811	−0.704053	+	0.710148i	$0.748628\pi$
$212$	0	0
$213$	−371.982	−0.119661
$214$	0	0
$215$	−2189.96	−0.694670
$216$	0	0
$217$	7178.72	2.24573
$218$	0	0
$219$	−17.6980	−0.00546081
$220$	0	0
$221$	531.451	0.161761
$222$	0	0
$223$	3871.79	1.16266	0.581332	−	0.813666i	$-0.302532\pi$
$223$	3871.79	1.16266	0.581332	+	0.813666i	$0.302532\pi$
$224$	0	0
$225$	−657.017	−0.194672
$226$	0	0
$227$	−2029.62	−0.593440	−0.296720	−	0.954964i	$-0.595893\pi$
$227$	−2029.62	−0.593440	−0.296720	+	0.954964i	$0.595893\pi$
$228$	0	0
$229$	5622.78	1.62255	0.811274	−	0.584666i	$-0.198775\pi$
$229$	5622.78	1.62255	0.811274	+	0.584666i	$0.198775\pi$
$230$	0	0
$231$	−1661.35	−0.473198
$232$	0	0
$233$	−4229.06	−1.18908	−0.594539	−	0.804067i	$-0.702665\pi$
$233$	−4229.06	−1.18908	−0.594539	+	0.804067i	$0.702665\pi$
$234$	0	0
$235$	−1546.51	−0.429290
$236$	0	0
$237$	150.071	0.0411314
$238$	0	0
$239$	−944.259	−0.255561	−0.127780	−	0.991802i	$-0.540785\pi$
$239$	−944.259	−0.255561	−0.127780	+	0.991802i	$0.540785\pi$
$240$	0	0
$241$	1288.24	0.344328	0.172164	−	0.985068i	$-0.444924\pi$
$241$	1288.24	0.344328	0.172164	+	0.985068i	$0.444924\pi$
$242$	0	0
$243$	1789.41	0.472389
$244$	0	0
$245$	−2231.35	−0.581860
$246$	0	0
$247$	840.464	0.216508
$248$	0	0
$249$	601.882	0.153184
$250$	0	0
$251$	4513.45	1.13501	0.567503	−	0.823371i	$-0.307910\pi$
$251$	4513.45	1.13501	0.567503	+	0.823371i	$0.307910\pi$
$252$	0	0
$253$	−7794.40	−1.93688
$254$	0	0
$255$	58.0405	0.0142535
$256$	0	0
$257$	−1762.18	−0.427711	−0.213855	−	0.976865i	$-0.568602\pi$
$257$	−1762.18	−0.427711	−0.213855	+	0.976865i	$0.568602\pi$
$258$	0	0
$259$	−5286.37	−1.26826
$260$	0	0
$261$	762.140	0.180748
$262$	0	0
$263$	223.502	0.0524019	0.0262010	−	0.999657i	$-0.491659\pi$
$263$	223.502	0.0524019	0.0262010	+	0.999657i	$0.491659\pi$
$264$	0	0
$265$	2487.92	0.576725
$266$	0	0
$267$	−11.1539	−0.00255659
$268$	0	0
$269$	−7893.90	−1.78922	−0.894609	−	0.446850i	$-0.852546\pi$
$269$	−7893.90	−1.78922	−0.894609	+	0.446850i	$0.852546\pi$
$270$	0	0
$271$	5146.81	1.15368	0.576838	−	0.816858i	$-0.304286\pi$
$271$	5146.81	1.15368	0.576838	+	0.816858i	$0.304286\pi$
$272$	0	0
$273$	−925.204	−0.205113
$274$	0	0
$275$	−1743.12	−0.382233
$276$	0	0
$277$	−0.499889	−0.000108431 0	−5.42156e−5	−	1.00000i	$-0.500017\pi$
$277$	−0.499889	−0.000108431 0	−5.42156e−5		1.00000i	$0.500017\pi$
$278$	0	0
$279$	−6715.39	−1.44100
$280$	0	0
$281$	−2273.99	−0.482758	−0.241379	−	0.970431i	$-0.577600\pi$
$281$	−2273.99	−0.482758	−0.241379	+	0.970431i	$0.577600\pi$
$282$	0	0
$283$	6710.84	1.40960	0.704802	−	0.709404i	$-0.251036\pi$
$283$	6710.84	1.40960	0.704802	+	0.709404i	$0.251036\pi$
$284$	0	0
$285$	91.7882	0.0190774
$286$	0	0
$287$	13182.0	2.71119
$288$	0	0
$289$	−4725.67	−0.961871
$290$	0	0
$291$	418.662	0.0843381
$292$	0	0
$293$	8246.25	1.64420	0.822101	−	0.569342i	$-0.192802\pi$
$293$	8246.25	1.64420	0.822101	+	0.569342i	$0.192802\pi$
$294$	0	0
$295$	−4439.18	−0.876133
$296$	0	0
$297$	3150.78	0.615578
$298$	0	0
$299$	−4340.69	−0.839561
$300$	0	0
$301$	12304.9	2.35629
$302$	0	0
$303$	−1007.11	−0.190947
$304$	0	0
$305$	2079.73	0.390443
$306$	0	0
$307$	−3348.40	−0.622486	−0.311243	−	0.950330i	$-0.600745\pi$
$307$	−3348.40	−0.622486	−0.311243	+	0.950330i	$0.600745\pi$
$308$	0	0
$309$	889.613	0.163781
$310$	0	0
$311$	−3183.02	−0.580361	−0.290181	−	0.956972i	$-0.593715\pi$
$311$	−3183.02	−0.580361	−0.290181	+	0.956972i	$0.593715\pi$
$312$	0	0
$313$	194.603	0.0351425	0.0175713	−	0.999846i	$-0.494407\pi$
$313$	194.603	0.0351425	0.0175713	+	0.999846i	$0.494407\pi$
$314$	0	0
$315$	3691.64	0.660319
$316$	0	0
$317$	7850.74	1.39098	0.695491	−	0.718535i	$-0.255187\pi$
$317$	7850.74	1.39098	0.695491	+	0.718535i	$0.255187\pi$
$318$	0	0
$319$	2022.02	0.354894
$320$	0	0
$321$	−1126.65	−0.195899
$322$	0	0
$323$	296.248	0.0510331
$324$	0	0
$325$	−970.742	−0.165683
$326$	0	0
$327$	1880.76	0.318062
$328$	0	0
$329$	8689.52	1.45614
$330$	0	0
$331$	10688.8	1.77495	0.887476	−	0.460853i	$-0.152456\pi$
$331$	10688.8	1.77495	0.887476	+	0.460853i	$0.152456\pi$
$332$	0	0
$333$	4945.17	0.813795
$334$	0	0
$335$	1479.62	0.241314
$336$	0	0
$337$	8950.21	1.44673	0.723367	−	0.690464i	$-0.242594\pi$
$337$	8950.21	1.44673	0.723367	+	0.690464i	$0.242594\pi$
$338$	0	0
$339$	−418.882	−0.0671108
$340$	0	0
$341$	−17816.5	−2.82937
$342$	0	0
$343$	2901.26	0.456715
$344$	0	0
$345$	−474.053	−0.0739773
$346$	0	0
$347$	8407.47	1.30068	0.650341	−	0.759642i	$-0.274626\pi$
$347$	8407.47	1.30068	0.650341	+	0.759642i	$0.274626\pi$
$348$	0	0
$349$	3592.04	0.550939	0.275469	−	0.961310i	$-0.411167\pi$
$349$	3592.04	0.550939	0.275469	+	0.961310i	$0.411167\pi$
$350$	0	0
$351$	1754.67	0.266830
$352$	0	0
$353$	−2530.91	−0.381606	−0.190803	−	0.981628i	$-0.561109\pi$
$353$	−2530.91	−0.381606	−0.190803	+	0.981628i	$0.561109\pi$
$354$	0	0
$355$	2192.96	0.327860
$356$	0	0
$357$	−326.117	−0.0483472
$358$	0	0
$359$	−2373.47	−0.348932	−0.174466	−	0.984663i	$-0.555820\pi$
$359$	−2373.47	−0.348932	−0.174466	+	0.984663i	$0.555820\pi$
$360$	0	0
$361$	−6390.50	−0.931695
$362$	0	0
$363$	2994.35	0.432954
$364$	0	0
$365$	104.335	0.0149621
$366$	0	0
$367$	−12847.3	−1.82731	−0.913656	−	0.406488i	$-0.866753\pi$
$367$	−12847.3	−1.82731	−0.913656	+	0.406488i	$0.866753\pi$
$368$	0	0
$369$	−12331.2	−1.73967
$370$	0	0
$371$	−13979.1	−1.95623
$372$	0	0
$373$	3451.72	0.479151	0.239575	−	0.970878i	$-0.422992\pi$
$373$	3451.72	0.479151	0.239575	+	0.970878i	$0.422992\pi$
$374$	0	0
$375$	−106.016	−0.0145990
$376$	0	0
$377$	1126.06	0.153833
$378$	0	0
$379$	−242.865	−0.0329160	−0.0164580	−	0.999865i	$-0.505239\pi$
$379$	−242.865	−0.0329160	−0.0164580	+	0.999865i	$0.505239\pi$
$380$	0	0
$381$	−1994.42	−0.268181
$382$	0	0
$383$	−6607.94	−0.881593	−0.440797	−	0.897607i	$-0.645304\pi$
$383$	−6607.94	−0.881593	−0.440797	+	0.897607i	$0.645304\pi$
$384$	0	0
$385$	9794.21	1.29652
$386$	0	0
$387$	−11510.7	−1.51195
$388$	0	0
$389$	−198.661	−0.0258933	−0.0129467	−	0.999916i	$-0.504121\pi$
$389$	−198.661	−0.0258933	−0.0129467	+	0.999916i	$0.504121\pi$
$390$	0	0
$391$	−1530.01	−0.197893
$392$	0	0
$393$	1423.44	0.182705
$394$	0	0
$395$	−884.717	−0.112696
$396$	0	0
$397$	171.465	0.0216766	0.0108383	−	0.999941i	$-0.496550\pi$
$397$	171.465	0.0216766	0.0108383	+	0.999941i	$0.496550\pi$
$398$	0	0
$399$	−515.739	−0.0647098
$400$	0	0
$401$	−2380.44	−0.296443	−0.148222	−	0.988954i	$-0.547355\pi$
$401$	−2380.44	−0.296443	−0.148222	+	0.988954i	$0.547355\pi$
$402$	0	0
$403$	−9921.97	−1.22642
$404$	0	0
$405$	−3356.26	−0.411788
$406$	0	0
$407$	13119.9	1.59787
$408$	0	0
$409$	−7.26717	−0.000878578 0	−0.000439289	−	1.00000i	$-0.500140\pi$
$409$	−7.26717	−0.000878578 0	−0.000439289		1.00000i	$0.500140\pi$
$410$	0	0
$411$	1034.82	0.124194
$412$	0	0
$413$	24942.8	2.97181
$414$	0	0
$415$	−3548.30	−0.419709
$416$	0	0
$417$	−993.977	−0.116727
$418$	0	0
$419$	−15010.8	−1.75018	−0.875090	−	0.483959i	$-0.839198\pi$
$419$	−15010.8	−1.75018	−0.875090	+	0.483959i	$0.839198\pi$
$420$	0	0
$421$	6873.68	0.795731	0.397865	−	0.917444i	$-0.369751\pi$
$421$	6873.68	0.795731	0.397865	+	0.917444i	$0.369751\pi$
$422$	0	0
$423$	−8128.67	−0.934348
$424$	0	0
$425$	−342.168	−0.0390532
$426$	0	0
$427$	−11685.6	−1.32437
$428$	0	0
$429$	2296.21	0.258420
$430$	0	0
$431$	14365.1	1.60544	0.802719	−	0.596357i	$-0.203386\pi$
$431$	14365.1	1.60544	0.802719	+	0.596357i	$0.203386\pi$
$432$	0	0
$433$	9970.97	1.10664	0.553319	−	0.832970i	$-0.313361\pi$
$433$	9970.97	1.10664	0.553319	+	0.832970i	$0.313361\pi$
$434$	0	0
$435$	122.979	0.0135549
$436$	0	0
$437$	−2419.64	−0.264868
$438$	0	0
$439$	1278.00	0.138943	0.0694713	−	0.997584i	$-0.477869\pi$
$439$	1278.00	0.138943	0.0694713	+	0.997584i	$0.477869\pi$
$440$	0	0
$441$	−11728.3	−1.26642
$442$	0	0
$443$	−11527.2	−1.23628	−0.618142	−	0.786066i	$-0.712114\pi$
$443$	−11527.2	−1.23628	−0.618142	+	0.786066i	$0.712114\pi$
$444$	0	0
$445$	65.7562	0.00700481
$446$	0	0
$447$	1814.53	0.192001
$448$	0	0
$449$	−12869.5	−1.35267	−0.676334	−	0.736595i	$-0.736432\pi$
$449$	−12869.5	−1.35267	−0.676334	+	0.736595i	$0.736432\pi$
$450$	0	0
$451$	−32715.7	−3.41579
$452$	0	0
$453$	2224.91	0.230763
$454$	0	0
$455$	5454.39	0.561991
$456$	0	0
$457$	−3157.97	−0.323246	−0.161623	−	0.986853i	$-0.551673\pi$
$457$	−3157.97	−0.323246	−0.161623	+	0.986853i	$0.551673\pi$
$458$	0	0
$459$	618.487	0.0628944
$460$	0	0
$461$	9396.45	0.949320	0.474660	−	0.880169i	$-0.342571\pi$
$461$	9396.45	0.949320	0.474660	+	0.880169i	$0.342571\pi$
$462$	0	0
$463$	−5949.10	−0.597145	−0.298572	−	0.954387i	$-0.596510\pi$
$463$	−5949.10	−0.597145	−0.298572	+	0.954387i	$0.596510\pi$
$464$	0	0
$465$	−1083.59	−0.108065
$466$	0	0
$467$	19451.3	1.92741	0.963704	−	0.266975i	$-0.0860240\pi$
$467$	19451.3	1.92741	0.963704	+	0.266975i	$0.0860240\pi$
$468$	0	0
$469$	−8313.66	−0.818527
$470$	0	0
$471$	2332.64	0.228201
$472$	0	0
$473$	−30538.9	−2.96867
$474$	0	0
$475$	−541.123	−0.0522703
$476$	0	0
$477$	13076.9	1.25524
$478$	0	0
$479$	940.447	0.0897080	0.0448540	−	0.998994i	$-0.485718\pi$
$479$	940.447	0.0897080	0.0448540	+	0.998994i	$0.485718\pi$
$480$	0	0
$481$	7306.49	0.692614
$482$	0	0
$483$	2663.60	0.250928
$484$	0	0
$485$	−2468.15	−0.231078
$486$	0	0
$487$	−20060.6	−1.86660	−0.933300	−	0.359098i	$-0.883084\pi$
$487$	−20060.6	−1.86660	−0.933300	+	0.359098i	$0.883084\pi$
$488$	0	0
$489$	1872.21	0.173138
$490$	0	0
$491$	8069.26	0.741672	0.370836	−	0.928698i	$-0.379071\pi$
$491$	8069.26	0.741672	0.370836	+	0.928698i	$0.379071\pi$
$492$	0	0
$493$	396.915	0.0362600
$494$	0	0
$495$	−9162.07	−0.831928
$496$	0	0
$497$	−12321.8	−1.11209
$498$	0	0
$499$	−6221.99	−0.558186	−0.279093	−	0.960264i	$-0.590034\pi$
$499$	−6221.99	−0.558186	−0.279093	+	0.960264i	$0.590034\pi$
$500$	0	0
$501$	213.994	0.0190829
$502$	0	0
$503$	−19034.6	−1.68729	−0.843647	−	0.536898i	$-0.819596\pi$
$503$	−19034.6	−1.68729	−0.843647	+	0.536898i	$0.819596\pi$
$504$	0	0
$505$	5937.26	0.523178
$506$	0	0
$507$	−584.578	−0.0512072
$508$	0	0
$509$	18205.3	1.58534	0.792669	−	0.609652i	$-0.208691\pi$
$509$	18205.3	1.58534	0.792669	+	0.609652i	$0.208691\pi$
$510$	0	0
$511$	−586.239	−0.0507509
$512$	0	0
$513$	978.108	0.0841803
$514$	0	0
$515$	−5244.57	−0.448744
$516$	0	0
$517$	−21566.0	−1.83457
$518$	0	0
$519$	3.82888	0.000323832 0
$520$	0	0
$521$	11929.0	1.00311	0.501553	−	0.865127i	$-0.332762\pi$
$521$	11929.0	1.00311	0.501553	+	0.865127i	$0.332762\pi$
$522$	0	0
$523$	−8069.89	−0.674707	−0.337353	−	0.941378i	$-0.609532\pi$
$523$	−8069.89	−0.674707	−0.337353	+	0.941378i	$0.609532\pi$
$524$	0	0
$525$	595.681	0.0495194
$526$	0	0
$527$	−3497.31	−0.289080
$528$	0	0
$529$	329.583	0.0270882
$530$	0	0
$531$	−23333.0	−1.90690
$532$	0	0
$533$	−18219.4	−1.48062
$534$	0	0
$535$	6642.00	0.536745
$536$	0	0
$537$	−2881.74	−0.231576
$538$	0	0
$539$	−31116.0	−2.48657
$540$	0	0
$541$	7077.69	0.562465	0.281232	−	0.959640i	$-0.409257\pi$
$541$	7077.69	0.562465	0.281232	+	0.959640i	$0.409257\pi$
$542$	0	0
$543$	2060.21	0.162822
$544$	0	0
$545$	−11087.7	−0.871460
$546$	0	0
$547$	−13642.5	−1.06639	−0.533193	−	0.845994i	$-0.679008\pi$
$547$	−13642.5	−1.06639	−0.533193	+	0.845994i	$0.679008\pi$
$548$	0	0
$549$	10931.4	0.849798
$550$	0	0
$551$	627.702	0.0485318
$552$	0	0
$553$	4971.04	0.382261
$554$	0	0
$555$	797.951	0.0610291
$556$	0	0
$557$	2306.16	0.175431	0.0877156	−	0.996146i	$-0.472043\pi$
$557$	2306.16	0.175431	0.0877156	+	0.996146i	$0.472043\pi$
$558$	0	0
$559$	−17007.1	−1.28680
$560$	0	0
$561$	809.371	0.0609121
$562$	0	0
$563$	13007.8	0.973738	0.486869	−	0.873475i	$-0.338139\pi$
$563$	13007.8	0.973738	0.486869	+	0.873475i	$0.338139\pi$
$564$	0	0
$565$	2469.45	0.183877
$566$	0	0
$567$	18858.1	1.39677
$568$	0	0
$569$	−10975.6	−0.808651	−0.404326	−	0.914615i	$-0.632494\pi$
$569$	−10975.6	−0.808651	−0.404326	+	0.914615i	$0.632494\pi$
$570$	0	0
$571$	25367.0	1.85915	0.929574	−	0.368634i	$-0.120175\pi$
$571$	25367.0	1.85915	0.929574	+	0.368634i	$0.120175\pi$
$572$	0	0
$573$	−3610.16	−0.263205
$574$	0	0
$575$	2794.70	0.202691
$576$	0	0
$577$	−18004.5	−1.29902	−0.649512	−	0.760351i	$-0.725027\pi$
$577$	−18004.5	−1.29902	−0.649512	+	0.760351i	$0.725027\pi$
$578$	0	0
$579$	2530.75	0.181649
$580$	0	0
$581$	19937.1	1.42364
$582$	0	0
$583$	34694.0	2.46463
$584$	0	0
$585$	−5102.35	−0.360609
$586$	0	0
$587$	16433.8	1.15553	0.577763	−	0.816205i	$-0.303926\pi$
$587$	16433.8	1.15553	0.577763	+	0.816205i	$0.303926\pi$
$588$	0	0
$589$	−5530.83	−0.386916
$590$	0	0
$591$	−3147.60	−0.219078
$592$	0	0
$593$	−3323.36	−0.230141	−0.115071	−	0.993357i	$-0.536709\pi$
$593$	−3323.36	−0.230141	−0.115071	+	0.993357i	$0.536709\pi$
$594$	0	0
$595$	1922.57	0.132467
$596$	0	0
$597$	1186.85	0.0813647
$598$	0	0
$599$	16193.6	1.10459	0.552296	−	0.833648i	$-0.313752\pi$
$599$	16193.6	1.10459	0.552296	+	0.833648i	$0.313752\pi$
$600$	0	0
$601$	17580.1	1.19319	0.596594	−	0.802543i	$-0.296520\pi$
$601$	17580.1	1.19319	0.596594	+	0.802543i	$0.296520\pi$
$602$	0	0
$603$	7777.07	0.525219
$604$	0	0
$605$	−17652.7	−1.18625
$606$	0	0
$607$	−7866.77	−0.526034	−0.263017	−	0.964791i	$-0.584717\pi$
$607$	−7866.77	−0.526034	−0.263017	+	0.964791i	$0.584717\pi$
$608$	0	0
$609$	−690.990	−0.0459776
$610$	0	0
$611$	−12010.1	−0.795215
$612$	0	0
$613$	2110.21	0.139038	0.0695191	−	0.997581i	$-0.477854\pi$
$613$	2110.21	0.139038	0.0695191	+	0.997581i	$0.477854\pi$
$614$	0	0
$615$	−1989.76	−0.130463
$616$	0	0
$617$	−1700.56	−0.110959	−0.0554796	−	0.998460i	$-0.517669\pi$
$617$	−1700.56	−0.110959	−0.0554796	+	0.998460i	$0.517669\pi$
$618$	0	0
$619$	−5302.18	−0.344285	−0.172143	−	0.985072i	$-0.555069\pi$
$619$	−5302.18	−0.344285	−0.172143	+	0.985072i	$0.555069\pi$
$620$	0	0
$621$	−5051.57	−0.326429
$622$	0	0
$623$	−369.470	−0.0237600
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1279.98	0.0815272
$628$	0	0
$629$	2575.40	0.163256
$630$	0	0
$631$	−7741.34	−0.488396	−0.244198	−	0.969725i	$-0.578525\pi$
$631$	−7741.34	−0.488396	−0.244198	+	0.969725i	$0.578525\pi$
$632$	0	0
$633$	−3660.33	−0.229834
$634$	0	0
$635$	11757.8	0.734792
$636$	0	0
$637$	−17328.5	−1.07783
$638$	0	0
$639$	11526.5	0.713586
$640$	0	0
$641$	7562.73	0.466006	0.233003	−	0.972476i	$-0.425145\pi$
$641$	7562.73	0.466006	0.233003	+	0.972476i	$0.425145\pi$
$642$	0	0
$643$	14758.0	0.905130	0.452565	−	0.891731i	$-0.350509\pi$
$643$	14758.0	0.905130	0.452565	+	0.891731i	$0.350509\pi$
$644$	0	0
$645$	−1857.37	−0.113386
$646$	0	0
$647$	8945.49	0.543560	0.271780	−	0.962359i	$-0.412388\pi$
$647$	8945.49	0.543560	0.271780	+	0.962359i	$0.412388\pi$
$648$	0	0
$649$	−61904.2	−3.74415
$650$	0	0
$651$	6088.47	0.366553
$652$	0	0
$653$	−7162.10	−0.429211	−0.214605	−	0.976701i	$-0.568847\pi$
$653$	−7162.10	−0.429211	−0.214605	+	0.976701i	$0.568847\pi$
$654$	0	0
$655$	−8391.67	−0.500595
$656$	0	0
$657$	548.401	0.0325650
$658$	0	0
$659$	−27324.8	−1.61521	−0.807604	−	0.589726i	$-0.799236\pi$
$659$	−27324.8	−1.61521	−0.807604	+	0.589726i	$0.799236\pi$
$660$	0	0
$661$	−27268.2	−1.60455	−0.802276	−	0.596954i	$-0.796378\pi$
$661$	−27268.2	−1.60455	−0.802276	+	0.596954i	$0.796378\pi$
$662$	0	0
$663$	450.739	0.0264031
$664$	0	0
$665$	3040.45	0.177299
$666$	0	0
$667$	−3241.86	−0.188194
$668$	0	0
$669$	3283.77	0.189773
$670$	0	0
$671$	29001.8	1.66856
$672$	0	0
$673$	23189.7	1.32823	0.664115	−	0.747631i	$-0.268809\pi$
$673$	23189.7	1.32823	0.664115	+	0.747631i	$0.268809\pi$
$674$	0	0
$675$	−1129.72	−0.0644192
$676$	0	0
$677$	13786.6	0.782659	0.391330	−	0.920251i	$-0.372015\pi$
$677$	13786.6	0.782659	0.391330	+	0.920251i	$0.372015\pi$
$678$	0	0
$679$	13868.0	0.783809
$680$	0	0
$681$	−1721.38	−0.0968626
$682$	0	0
$683$	−4434.99	−0.248463	−0.124231	−	0.992253i	$-0.539646\pi$
$683$	−4434.99	−0.248463	−0.124231	+	0.992253i	$0.539646\pi$
$684$	0	0
$685$	−6100.60	−0.340280
$686$	0	0
$687$	4768.83	0.264836
$688$	0	0
$689$	19321.1	1.06832
$690$	0	0
$691$	23981.2	1.32024	0.660122	−	0.751159i	$-0.270505\pi$
$691$	23981.2	1.32024	0.660122	+	0.751159i	$0.270505\pi$
$692$	0	0
$693$	51479.7	2.82187
$694$	0	0
$695$	5859.83	0.319822
$696$	0	0
$697$	−6421.99	−0.348996
$698$	0	0
$699$	−3586.78	−0.194084
$700$	0	0
$701$	−27081.4	−1.45913	−0.729566	−	0.683911i	$-0.760278\pi$
$701$	−27081.4	−1.45913	−0.729566	+	0.683911i	$0.760278\pi$
$702$	0	0
$703$	4072.87	0.218508
$704$	0	0
$705$	−1311.64	−0.0700697
$706$	0	0
$707$	−33360.2	−1.77460
$708$	0	0
$709$	−14637.2	−0.775334	−0.387667	−	0.921799i	$-0.626719\pi$
$709$	−14637.2	−0.775334	−0.387667	+	0.921799i	$0.626719\pi$
$710$	0	0
$711$	−4650.19	−0.245282
$712$	0	0
$713$	28564.7	1.50036
$714$	0	0
$715$	−13536.9	−0.708046
$716$	0	0
$717$	−800.852	−0.0417132
$718$	0	0
$719$	12770.9	0.662413	0.331206	−	0.943558i	$-0.392544\pi$
$719$	12770.9	0.662413	0.331206	+	0.943558i	$0.392544\pi$
$720$	0	0
$721$	29468.1	1.52212
$722$	0	0
$723$	1092.59	0.0562019
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	11908.3	0.607504	0.303752	−	0.952751i	$-0.401761\pi$
$727$	11908.3	0.607504	0.303752	+	0.952751i	$0.401761\pi$
$728$	0	0
$729$	−16606.2	−0.843681
$730$	0	0
$731$	−5994.68	−0.303312
$732$	0	0
$733$	−11483.9	−0.578674	−0.289337	−	0.957227i	$-0.593435\pi$
$733$	−11483.9	−0.578674	−0.289337	+	0.957227i	$0.593435\pi$
$734$	0	0
$735$	−1892.47	−0.0949725
$736$	0	0
$737$	20633.2	1.03125
$738$	0	0
$739$	16051.6	0.799009	0.399504	−	0.916731i	$-0.369182\pi$
$739$	16051.6	0.799009	0.399504	+	0.916731i	$0.369182\pi$
$740$	0	0
$741$	712.821	0.0353389
$742$	0	0
$743$	−12941.8	−0.639015	−0.319508	−	0.947584i	$-0.603517\pi$
$743$	−12941.8	−0.639015	−0.319508	+	0.947584i	$0.603517\pi$
$744$	0	0
$745$	−10697.3	−0.526064
$746$	0	0
$747$	−18650.3	−0.913494
$748$	0	0
$749$	−37320.0	−1.82062
$750$	0	0
$751$	−6968.78	−0.338608	−0.169304	−	0.985564i	$-0.554152\pi$
$751$	−6968.78	−0.338608	−0.169304	+	0.985564i	$0.554152\pi$
$752$	0	0
$753$	3827.98	0.185258
$754$	0	0
$755$	−13116.6	−0.632268
$756$	0	0
$757$	−4699.25	−0.225624	−0.112812	−	0.993616i	$-0.535986\pi$
$757$	−4699.25	−0.225624	−0.112812	+	0.993616i	$0.535986\pi$
$758$	0	0
$759$	−6610.64	−0.316141
$760$	0	0
$761$	18565.6	0.884366	0.442183	−	0.896925i	$-0.354204\pi$
$761$	18565.6	0.884366	0.442183	+	0.896925i	$0.354204\pi$
$762$	0	0
$763$	62299.5	2.95596
$764$	0	0
$765$	−1798.48	−0.0849991
$766$	0	0
$767$	−34474.4	−1.62294
$768$	0	0
$769$	15732.3	0.737738	0.368869	−	0.929481i	$-0.379745\pi$
$769$	15732.3	0.737738	0.368869	+	0.929481i	$0.379745\pi$
$770$	0	0
$771$	−1494.55	−0.0698119
$772$	0	0
$773$	11610.6	0.540240	0.270120	−	0.962827i	$-0.412937\pi$
$773$	11610.6	0.540240	0.270120	+	0.962827i	$0.412937\pi$
$774$	0	0
$775$	6388.14	0.296089
$776$	0	0
$777$	−4483.52	−0.207008
$778$	0	0
$779$	−10156.1	−0.467110
$780$	0	0
$781$	30580.7	1.40111
$782$	0	0
$783$	1310.48	0.0598117
$784$	0	0
$785$	−13751.7	−0.625248
$786$	0	0
$787$	−28362.6	−1.28465	−0.642325	−	0.766433i	$-0.722030\pi$
$787$	−28362.6	−1.28465	−0.642325	+	0.766433i	$0.722030\pi$
$788$	0	0
$789$	189.558	0.00855316
$790$	0	0
$791$	−13875.3	−0.623705
$792$	0	0
$793$	16151.1	0.723255
$794$	0	0
$795$	2110.08	0.0941343
$796$	0	0
$797$	−4018.42	−0.178594	−0.0892971	−	0.996005i	$-0.528462\pi$
$797$	−4018.42	−0.178594	−0.0892971	+	0.996005i	$0.528462\pi$
$798$	0	0
$799$	−4233.34	−0.187440
$800$	0	0
$801$	345.623	0.0152459
$802$	0	0
$803$	1454.95	0.0639404
$804$	0	0
$805$	−15702.8	−0.687519
$806$	0	0
$807$	−6695.03	−0.292040
$808$	0	0
$809$	−35737.0	−1.55308	−0.776542	−	0.630065i	$-0.783028\pi$
$809$	−35737.0	−1.55308	−0.776542	+	0.630065i	$0.783028\pi$
$810$	0	0
$811$	7743.79	0.335291	0.167646	−	0.985847i	$-0.446384\pi$
$811$	7743.79	0.335291	0.167646	+	0.985847i	$0.446384\pi$
$812$	0	0
$813$	4365.15	0.188306
$814$	0	0
$815$	−11037.3	−0.474381
$816$	0	0
$817$	−9480.30	−0.405965
$818$	0	0
$819$	28669.0	1.22317
$820$	0	0
$821$	−11089.2	−0.471396	−0.235698	−	0.971826i	$-0.575738\pi$
$821$	−11089.2	−0.471396	−0.235698	+	0.971826i	$0.575738\pi$
$822$	0	0
$823$	3498.71	0.148186	0.0740932	−	0.997251i	$-0.476394\pi$
$823$	3498.71	0.148186	0.0740932	+	0.997251i	$0.476394\pi$
$824$	0	0
$825$	−1478.39	−0.0623889
$826$	0	0
$827$	−5140.31	−0.216138	−0.108069	−	0.994143i	$-0.534467\pi$
$827$	−5140.31	−0.216138	−0.108069	+	0.994143i	$0.534467\pi$
$828$	0	0
$829$	3324.47	0.139281	0.0696403	−	0.997572i	$-0.477815\pi$
$829$	3324.47	0.139281	0.0696403	+	0.997572i	$0.477815\pi$
$830$	0	0
$831$	−0.423970	−1.76984e−5 0
$832$	0	0
$833$	−6107.98	−0.254056
$834$	0	0
$835$	−1261.57	−0.0522854
$836$	0	0
$837$	−11546.9	−0.476845
$838$	0	0
$839$	−18892.5	−0.777405	−0.388703	−	0.921363i	$-0.627077\pi$
$839$	−18892.5	−0.777405	−0.388703	+	0.921363i	$0.627077\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−1928.64	−0.0787969
$844$	0	0
$845$	3446.29	0.140303
$846$	0	0
$847$	99186.7	4.02373
$848$	0	0
$849$	5691.65	0.230079
$850$	0	0
$851$	−21034.9	−0.847318
$852$	0	0
$853$	14256.5	0.572253	0.286126	−	0.958192i	$-0.407632\pi$
$853$	14256.5	0.572253	0.286126	+	0.958192i	$0.407632\pi$
$854$	0	0
$855$	−2844.21	−0.113766
$856$	0	0
$857$	29636.4	1.18128	0.590642	−	0.806934i	$-0.298874\pi$
$857$	29636.4	1.18128	0.590642	+	0.806934i	$0.298874\pi$
$858$	0	0
$859$	−12986.6	−0.515828	−0.257914	−	0.966168i	$-0.583035\pi$
$859$	−12986.6	−0.515828	−0.257914	+	0.966168i	$0.583035\pi$
$860$	0	0
$861$	11180.0	0.442526
$862$	0	0
$863$	8960.37	0.353435	0.176718	−	0.984262i	$-0.443452\pi$
$863$	8960.37	0.353435	0.176718	+	0.984262i	$0.443452\pi$
$864$	0	0
$865$	−22.5725	−0.000887270 0
$866$	0	0
$867$	−4007.97	−0.156999
$868$	0	0
$869$	−12337.3	−0.481606
$870$	0	0
$871$	11490.6	0.447009
$872$	0	0
$873$	−12972.9	−0.502941
$874$	0	0
$875$	−3511.74	−0.135678
$876$	0	0
$877$	13151.6	0.506382	0.253191	−	0.967416i	$-0.418520\pi$
$877$	13151.6	0.506382	0.253191	+	0.967416i	$0.418520\pi$
$878$	0	0
$879$	6993.87	0.268370
$880$	0	0
$881$	38097.3	1.45690	0.728450	−	0.685099i	$-0.240241\pi$
$881$	38097.3	1.45690	0.728450	+	0.685099i	$0.240241\pi$
$882$	0	0
$883$	22274.0	0.848903	0.424452	−	0.905451i	$-0.360467\pi$
$883$	22274.0	0.848903	0.424452	+	0.905451i	$0.360467\pi$
$884$	0	0
$885$	−3764.99	−0.143004
$886$	0	0
$887$	−15504.2	−0.586898	−0.293449	−	0.955975i	$-0.594803\pi$
$887$	−15504.2	−0.586898	−0.293449	+	0.955975i	$0.594803\pi$
$888$	0	0
$889$	−66064.4	−2.49238
$890$	0	0
$891$	−46802.9	−1.75977
$892$	0	0
$893$	−6694.81	−0.250877
$894$	0	0
$895$	16988.8	0.634496
$896$	0	0
$897$	−3681.46	−0.137035
$898$	0	0
$899$	−7410.24	−0.274911
$900$	0	0
$901$	6810.31	0.251814
$902$	0	0
$903$	10436.1	0.384599
$904$	0	0
$905$	−12145.7	−0.446116
$906$	0	0
$907$	52847.6	1.93470	0.967352	−	0.253436i	$-0.0815607\pi$
$907$	52847.6	1.93470	0.967352	+	0.253436i	$0.0815607\pi$
$908$	0	0
$909$	31207.1	1.13869
$910$	0	0
$911$	−41917.5	−1.52447	−0.762234	−	0.647302i	$-0.775897\pi$
$911$	−41917.5	−1.52447	−0.762234	+	0.647302i	$0.775897\pi$
$912$	0	0
$913$	−49480.8	−1.79362
$914$	0	0
$915$	1763.88	0.0637290
$916$	0	0
$917$	47151.0	1.69800
$918$	0	0
$919$	−34978.2	−1.25552	−0.627760	−	0.778407i	$-0.716028\pi$
$919$	−34978.2	−1.25552	−0.627760	+	0.778407i	$0.716028\pi$
$920$	0	0
$921$	−2839.87	−0.101604
$922$	0	0
$923$	17030.4	0.607326
$924$	0	0
$925$	−4704.19	−0.167214
$926$	0	0
$927$	−27566.2	−0.976690
$928$	0	0
$929$	16263.0	0.574350	0.287175	−	0.957878i	$-0.407284\pi$
$929$	16263.0	0.574350	0.287175	+	0.957878i	$0.407284\pi$
$930$	0	0
$931$	−9659.47	−0.340039
$932$	0	0
$933$	−2699.61	−0.0947279
$934$	0	0
$935$	−4771.52	−0.166893
$936$	0	0
$937$	−20933.5	−0.729846	−0.364923	−	0.931038i	$-0.618905\pi$
$937$	−20933.5	−0.729846	−0.364923	+	0.931038i	$0.618905\pi$
$938$	0	0
$939$	165.048	0.00573604
$940$	0	0
$941$	3025.35	0.104807	0.0524036	−	0.998626i	$-0.483312\pi$
$941$	3025.35	0.104807	0.0524036	+	0.998626i	$0.483312\pi$
$942$	0	0
$943$	52452.4	1.81133
$944$	0	0
$945$	6347.66	0.218507
$946$	0	0
$947$	13738.1	0.471414	0.235707	−	0.971824i	$-0.424259\pi$
$947$	13738.1	0.471414	0.235707	+	0.971824i	$0.424259\pi$
$948$	0	0
$949$	810.262	0.0277157
$950$	0	0
$951$	6658.43	0.227039
$952$	0	0
$953$	38152.7	1.29684	0.648420	−	0.761283i	$-0.275430\pi$
$953$	38152.7	1.29684	0.648420	+	0.761283i	$0.275430\pi$
$954$	0	0
$955$	21283.1	0.721157
$956$	0	0
$957$	1714.93	0.0579266
$958$	0	0
$959$	34278.0	1.15422
$960$	0	0
$961$	35502.3	1.19171
$962$	0	0
$963$	34911.3	1.16822
$964$	0	0
$965$	−14919.6	−0.497700
$966$	0	0
$967$	36348.2	1.20877	0.604385	−	0.796693i	$-0.293419\pi$
$967$	36348.2	1.20877	0.604385	+	0.796693i	$0.293419\pi$
$968$	0	0
$969$	251.256	0.00832973
$970$	0	0
$971$	50375.8	1.66492	0.832460	−	0.554085i	$-0.186932\pi$
$971$	50375.8	1.66492	0.832460	+	0.554085i	$0.186932\pi$
$972$	0	0
$973$	−32925.2	−1.08482
$974$	0	0
$975$	−823.313	−0.0270432
$976$	0	0
$977$	−28397.1	−0.929890	−0.464945	−	0.885340i	$-0.653926\pi$
$977$	−28397.1	−0.929890	−0.464945	+	0.885340i	$0.653926\pi$
$978$	0	0
$979$	916.966	0.0299350
$980$	0	0
$981$	−58278.5	−1.89673
$982$	0	0
$983$	42348.9	1.37408	0.687039	−	0.726620i	$-0.258910\pi$
$983$	42348.9	1.37408	0.687039	+	0.726620i	$0.258910\pi$
$984$	0	0
$985$	18556.2	0.600253
$986$	0	0
$987$	7369.82	0.237674
$988$	0	0
$989$	48962.3	1.57423
$990$	0	0
$991$	−14053.9	−0.450490	−0.225245	−	0.974302i	$-0.572318\pi$
$991$	−14053.9	−0.450490	−0.225245	+	0.974302i	$0.572318\pi$
$992$	0	0
$993$	9065.46	0.289712
$994$	0	0
$995$	−6996.91	−0.222932
$996$	0	0
$997$	−26552.1	−0.843442	−0.421721	−	0.906726i	$-0.638574\pi$
$997$	−26552.1	−0.843442	−0.421721	+	0.906726i	$0.638574\pi$
$998$	0	0
$999$	8503.07	0.269295

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.g.1.7	✓	11
4.3	odd	2		2320.4.a.bb.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.g.1.7	✓	11	1.1	even	1	trivial
2320.4.a.bb.1.5		11	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q+0.848128 q^{3} -5.00000 q^{5} +28.0939 q^{7} -26.2807 q^{9} -69.7247 q^{11} -38.8297 q^{13} -4.24064 q^{15} -13.6867 q^{17} -21.6449 q^{19} +23.8273 q^{21} +111.788 q^{23} +25.0000 q^{25} -45.1888 q^{27} -29.0000 q^{29} +255.526 q^{31} -59.1355 q^{33} -140.470 q^{35} -188.168 q^{37} -32.9325 q^{39} +469.212 q^{41} +437.992 q^{43} +131.403 q^{45} +309.302 q^{47} +446.270 q^{49} -11.6081 q^{51} -497.585 q^{53} +348.624 q^{55} -18.3576 q^{57} +887.837 q^{59} -415.947 q^{61} -738.328 q^{63} +194.148 q^{65} -295.924 q^{67} +94.8106 q^{69} -438.592 q^{71} -20.8671 q^{73} +21.2032 q^{75} -1958.84 q^{77} +176.943 q^{79} +671.252 q^{81} +709.660 q^{83} +68.4337 q^{85} -24.5957 q^{87} -13.1512 q^{89} -1090.88 q^{91} +216.718 q^{93} +108.225 q^{95} +493.631 q^{97} +1832.41 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) \) 11 * q + 2 * q^3 - 55 * q^5 + 56 * q^7 + 65 * q^9 - 8 * q^11 + 16 * q^13 - 10 * q^15 + 64 * q^17 + 160 * q^19 - 48 * q^21 + 140 * q^23 + 275 * q^25 - 136 * q^27 - 319 * q^29 - 144 * q^31 - 424 * q^33 - 280 * q^35 - 50 * q^37 + 572 * q^39 + 62 * q^41 + 326 * q^43 - 325 * q^45 + 1074 * q^47 + 1513 * q^49 - 144 * q^51 + 576 * q^53 + 40 * q^55 - 896 * q^57 - 468 * q^59 + 660 * q^61 + 2552 * q^63 - 80 * q^65 + 892 * q^67 - 510 * q^69 + 920 * q^71 - 676 * q^73 + 50 * q^75 + 1708 * q^77 + 2308 * q^79 + 1287 * q^81 - 1548 * q^83 - 320 * q^85 - 58 * q^87 + 1854 * q^89 + 3100 * q^91 + 5424 * q^93 - 800 * q^95 + 4724 * q^97 + 6884 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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