LMFDB - Embedded newform 1160.4.a.d.1.2

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 153x^{7} + 229x^{6} + 7393x^{5} - 8331x^{4} - 115371x^{3} + 125775x^{2} + 306882x + 29241 $ x^9 - 2x^8 - 153x^7 + 229x^6 + 7393x^5 - 8331x^4 - 115371x^3 + 125775x^2 + 306882x + 29241
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.79757$ 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)

$f(q)$	$=$	$q-7.79757 q^{3} +5.00000 q^{5} -15.5907 q^{7} +33.8021 q^{9} +0.904566 q^{11} +44.0785 q^{13} -38.9879 q^{15} -24.7763 q^{17} -144.622 q^{19} +121.569 q^{21} +20.9321 q^{23} +25.0000 q^{25} -53.0402 q^{27} +29.0000 q^{29} +86.8967 q^{31} -7.05342 q^{33} -77.9534 q^{35} +258.033 q^{37} -343.705 q^{39} +261.855 q^{41} +167.197 q^{43} +169.011 q^{45} -108.300 q^{47} -99.9307 q^{49} +193.195 q^{51} -59.0204 q^{53} +4.52283 q^{55} +1127.70 q^{57} +594.350 q^{59} -265.219 q^{61} -526.998 q^{63} +220.392 q^{65} +155.916 q^{67} -163.220 q^{69} -485.415 q^{71} +654.740 q^{73} -194.939 q^{75} -14.1028 q^{77} +10.4630 q^{79} -499.073 q^{81} -151.760 q^{83} -123.882 q^{85} -226.130 q^{87} +42.2972 q^{89} -687.214 q^{91} -677.583 q^{93} -723.111 q^{95} -120.546 q^{97} +30.5763 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 7 q^{3} + 45 q^{5} - 39 q^{7} + 72 q^{9} - 108 q^{11} - 19 q^{13} - 35 q^{15} + 91 q^{17} - 36 q^{19} - 200 q^{21} - 209 q^{23} + 225 q^{25} - 316 q^{27} + 261 q^{29} - 599 q^{31} - 192 q^{33} - 195 q^{35}+ \cdots - 2946 q^{99}+O(q^{100}) $ 9 * q - 7 * q^3 + 45 * q^5 - 39 * q^7 + 72 * q^9 - 108 * q^11 - 19 * q^13 - 35 * q^15 + 91 * q^17 - 36 * q^19 - 200 * q^21 - 209 * q^23 + 225 * q^25 - 316 * q^27 + 261 * q^29 - 599 * q^31 - 192 * q^33 - 195 * q^35 + 204 * q^37 - 221 * q^39 - 328 * q^41 - 717 * q^43 + 360 * q^45 - 646 * q^47 + 200 * q^49 - 1038 * q^51 + 71 * q^53 - 540 * q^55 + 494 * q^57 - 1591 * q^59 - 439 * q^61 - 354 * q^63 - 95 * q^65 - 1222 * q^67 + 873 * q^69 - 1240 * q^71 - 69 * q^73 - 175 * q^75 + 102 * q^77 - 1013 * q^79 - 171 * q^81 - 2728 * q^83 + 455 * q^85 - 203 * q^87 - 312 * q^89 - 1652 * q^91 - 702 * q^93 - 180 * q^95 + 1145 * q^97 - 2946 * 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
$3$	−7.79757	−1.50064	−0.750322	−	0.661073i	$-0.770101\pi$
$3$	−7.79757	−1.50064	−0.750322	+	0.661073i	$0.770101\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−15.5907	−0.841818	−0.420909	−	0.907103i	$-0.638289\pi$
$7$	−15.5907	−0.841818	−0.420909	+	0.907103i	$0.638289\pi$
$8$	0	0
$9$	33.8021	1.25193
$10$	0	0
$11$	0.904566	0.0247943	0.0123971	−	0.999923i	$-0.496054\pi$
$11$	0.904566	0.0247943	0.0123971	+	0.999923i	$0.496054\pi$
$12$	0	0
$13$	44.0785	0.940398	0.470199	−	0.882560i	$-0.344182\pi$
$13$	44.0785	0.940398	0.470199	+	0.882560i	$0.344182\pi$
$14$	0	0
$15$	−38.9879	−0.671108
$16$	0	0
$17$	−24.7763	−0.353479	−0.176739	−	0.984258i	$-0.556555\pi$
$17$	−24.7763	−0.353479	−0.176739	+	0.984258i	$0.556555\pi$
$18$	0	0
$19$	−144.622	−1.74624	−0.873122	−	0.487503i	$-0.837908\pi$
$19$	−144.622	−1.74624	−0.873122	+	0.487503i	$0.837908\pi$
$20$	0	0
$21$	121.569	1.26327
$22$	0	0
$23$	20.9321	0.189767	0.0948837	−	0.995488i	$-0.469752\pi$
$23$	20.9321	0.189767	0.0948837	+	0.995488i	$0.469752\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−53.0402	−0.378059
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	86.8967	0.503455	0.251727	−	0.967798i	$-0.419001\pi$
$31$	86.8967	0.503455	0.251727	+	0.967798i	$0.419001\pi$
$32$	0	0
$33$	−7.05342	−0.0372074
$34$	0	0
$35$	−77.9534	−0.376472
$36$	0	0
$37$	258.033	1.14650	0.573248	−	0.819382i	$-0.305683\pi$
$37$	258.033	1.14650	0.573248	+	0.819382i	$0.305683\pi$
$38$	0	0
$39$	−343.705	−1.41120
$40$	0	0
$41$	261.855	0.997436	0.498718	−	0.866764i	$-0.333804\pi$
$41$	261.855	0.997436	0.498718	+	0.866764i	$0.333804\pi$
$42$	0	0
$43$	167.197	0.592959	0.296480	−	0.955039i	$-0.404187\pi$
$43$	167.197	0.592959	0.296480	+	0.955039i	$0.404187\pi$
$44$	0	0
$45$	169.011	0.559881
$46$	0	0
$47$	−108.300	−0.336111	−0.168055	−	0.985778i	$-0.553749\pi$
$47$	−108.300	−0.336111	−0.168055	+	0.985778i	$0.553749\pi$
$48$	0	0
$49$	−99.9307	−0.291343
$50$	0	0
$51$	193.195	0.530446
$52$	0	0
$53$	−59.0204	−0.152964	−0.0764819	−	0.997071i	$-0.524369\pi$
$53$	−59.0204	−0.152964	−0.0764819	+	0.997071i	$0.524369\pi$
$54$	0	0
$55$	4.52283	0.0110883
$56$	0	0
$57$	1127.70	2.62049
$58$	0	0
$59$	594.350	1.31149	0.655744	−	0.754984i	$-0.272355\pi$
$59$	594.350	1.31149	0.655744	+	0.754984i	$0.272355\pi$
$60$	0	0
$61$	−265.219	−0.556686	−0.278343	−	0.960482i	$-0.589785\pi$
$61$	−265.219	−0.556686	−0.278343	+	0.960482i	$0.589785\pi$
$62$	0	0
$63$	−526.998	−1.05390
$64$	0	0
$65$	220.392	0.420559
$66$	0	0
$67$	155.916	0.284301	0.142150	−	0.989845i	$-0.454598\pi$
$67$	155.916	0.284301	0.142150	+	0.989845i	$0.454598\pi$
$68$	0	0
$69$	−163.220	−0.284773
$70$	0	0
$71$	−485.415	−0.811382	−0.405691	−	0.914010i	$-0.632969\pi$
$71$	−485.415	−0.811382	−0.405691	+	0.914010i	$0.632969\pi$
$72$	0	0
$73$	654.740	1.04975	0.524874	−	0.851180i	$-0.324113\pi$
$73$	654.740	1.04975	0.524874	+	0.851180i	$0.324113\pi$
$74$	0	0
$75$	−194.939	−0.300129
$76$	0	0
$77$	−14.1028	−0.0208723
$78$	0	0
$79$	10.4630	0.0149011	0.00745053	−	0.999972i	$-0.497628\pi$
$79$	10.4630	0.0149011	0.00745053	+	0.999972i	$0.497628\pi$
$80$	0	0
$81$	−499.073	−0.684600
$82$	0	0
$83$	−151.760	−0.200697	−0.100348	−	0.994952i	$-0.531996\pi$
$83$	−151.760	−0.200697	−0.100348	+	0.994952i	$0.531996\pi$
$84$	0	0
$85$	−123.882	−0.158080
$86$	0	0
$87$	−226.130	−0.278663
$88$	0	0
$89$	42.2972	0.0503763	0.0251882	−	0.999683i	$-0.491982\pi$
$89$	42.2972	0.0503763	0.0251882	+	0.999683i	$0.491982\pi$
$90$	0	0
$91$	−687.214	−0.791643
$92$	0	0
$93$	−677.583	−0.755506
$94$	0	0
$95$	−723.111	−0.780944
$96$	0	0
$97$	−120.546	−0.126182	−0.0630908	−	0.998008i	$-0.520096\pi$
$97$	−120.546	−0.126182	−0.0630908	+	0.998008i	$0.520096\pi$
$98$	0	0
$99$	30.5763	0.0310407
$100$	0	0
$101$	170.001	0.167482	0.0837412	−	0.996488i	$-0.473313\pi$
$101$	170.001	0.167482	0.0837412	+	0.996488i	$0.473313\pi$
$102$	0	0
$103$	541.303	0.517827	0.258913	−	0.965901i	$-0.416636\pi$
$103$	541.303	0.517827	0.258913	+	0.965901i	$0.416636\pi$
$104$	0	0
$105$	607.847	0.564951
$106$	0	0
$107$	−1603.37	−1.44863	−0.724317	−	0.689467i	$-0.757845\pi$
$107$	−1603.37	−1.44863	−0.724317	+	0.689467i	$0.757845\pi$
$108$	0	0
$109$	−693.395	−0.609313	−0.304657	−	0.952462i	$-0.598542\pi$
$109$	−693.395	−0.609313	−0.304657	+	0.952462i	$0.598542\pi$
$110$	0	0
$111$	−2012.03	−1.72048
$112$	0	0
$113$	272.427	0.226794	0.113397	−	0.993550i	$-0.463827\pi$
$113$	272.427	0.226794	0.113397	+	0.993550i	$0.463827\pi$
$114$	0	0
$115$	104.661	0.0848666
$116$	0	0
$117$	1489.95	1.17731
$118$	0	0
$119$	386.280	0.297565
$120$	0	0
$121$	−1330.18	−0.999385
$122$	0	0
$123$	−2041.83	−1.49680
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	990.503	0.692070	0.346035	−	0.938222i	$-0.387528\pi$
$127$	990.503	0.692070	0.346035	+	0.938222i	$0.387528\pi$
$128$	0	0
$129$	−1303.73	−0.889820
$130$	0	0
$131$	−1461.36	−0.974652	−0.487326	−	0.873220i	$-0.662028\pi$
$131$	−1461.36	−0.974652	−0.487326	+	0.873220i	$0.662028\pi$
$132$	0	0
$133$	2254.76	1.47002
$134$	0	0
$135$	−265.201	−0.169073
$136$	0	0
$137$	537.074	0.334929	0.167465	−	0.985878i	$-0.446442\pi$
$137$	537.074	0.334929	0.167465	+	0.985878i	$0.446442\pi$
$138$	0	0
$139$	−1442.25	−0.880071	−0.440035	−	0.897980i	$-0.645034\pi$
$139$	−1442.25	−0.880071	−0.440035	+	0.897980i	$0.645034\pi$
$140$	0	0
$141$	844.478	0.504382
$142$	0	0
$143$	39.8719	0.0233165
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	779.217	0.437202
$148$	0	0
$149$	−761.634	−0.418761	−0.209381	−	0.977834i	$-0.567145\pi$
$149$	−761.634	−0.418761	−0.209381	+	0.977834i	$0.567145\pi$
$150$	0	0
$151$	−307.916	−0.165946	−0.0829729	−	0.996552i	$-0.526441\pi$
$151$	−307.916	−0.165946	−0.0829729	+	0.996552i	$0.526441\pi$
$152$	0	0
$153$	−837.492	−0.442531
$154$	0	0
$155$	434.483	0.225152
$156$	0	0
$157$	−1031.05	−0.524120	−0.262060	−	0.965052i	$-0.584402\pi$
$157$	−1031.05	−0.524120	−0.262060	+	0.965052i	$0.584402\pi$
$158$	0	0
$159$	460.216	0.229544
$160$	0	0
$161$	−326.346	−0.159750
$162$	0	0
$163$	2486.39	1.19478	0.597389	−	0.801951i	$-0.296205\pi$
$163$	2486.39	1.19478	0.597389	+	0.801951i	$0.296205\pi$
$164$	0	0
$165$	−35.2671	−0.0166396
$166$	0	0
$167$	−1662.88	−0.770524	−0.385262	−	0.922807i	$-0.625889\pi$
$167$	−1662.88	−0.770524	−0.385262	+	0.922807i	$0.625889\pi$
$168$	0	0
$169$	−254.087	−0.115652
$170$	0	0
$171$	−4888.54	−2.18618
$172$	0	0
$173$	−1627.97	−0.715446	−0.357723	−	0.933828i	$-0.616447\pi$
$173$	−1627.97	−0.715446	−0.357723	+	0.933828i	$0.616447\pi$
$174$	0	0
$175$	−389.767	−0.168364
$176$	0	0
$177$	−4634.49	−1.96807
$178$	0	0
$179$	−1329.47	−0.555134	−0.277567	−	0.960706i	$-0.589528\pi$
$179$	−1329.47	−0.555134	−0.277567	+	0.960706i	$0.589528\pi$
$180$	0	0
$181$	−2434.68	−0.999826	−0.499913	−	0.866076i	$-0.666635\pi$
$181$	−2434.68	−0.999826	−0.499913	+	0.866076i	$0.666635\pi$
$182$	0	0
$183$	2068.07	0.835388
$184$	0	0
$185$	1290.17	0.512729
$186$	0	0
$187$	−22.4118	−0.00876425
$188$	0	0
$189$	826.933	0.318257
$190$	0	0
$191$	−4951.01	−1.87561	−0.937807	−	0.347156i	$-0.887148\pi$
$191$	−4951.01	−1.87561	−0.937807	+	0.347156i	$0.887148\pi$
$192$	0	0
$193$	−1492.91	−0.556799	−0.278400	−	0.960465i	$-0.589804\pi$
$193$	−1492.91	−0.556799	−0.278400	+	0.960465i	$0.589804\pi$
$194$	0	0
$195$	−1718.53	−0.631109
$196$	0	0
$197$	178.455	0.0645399	0.0322700	−	0.999479i	$-0.489726\pi$
$197$	178.455	0.0645399	0.0322700	+	0.999479i	$0.489726\pi$
$198$	0	0
$199$	−2665.50	−0.949507	−0.474754	−	0.880119i	$-0.657463\pi$
$199$	−2665.50	−0.949507	−0.474754	+	0.880119i	$0.657463\pi$
$200$	0	0
$201$	−1215.76	−0.426634
$202$	0	0
$203$	−452.130	−0.156322
$204$	0	0
$205$	1309.27	0.446067
$206$	0	0
$207$	707.551	0.237576
$208$	0	0
$209$	−130.820	−0.0432968
$210$	0	0
$211$	−1857.37	−0.606002	−0.303001	−	0.952990i	$-0.597989\pi$
$211$	−1857.37	−0.606002	−0.303001	+	0.952990i	$0.597989\pi$
$212$	0	0
$213$	3785.06	1.21760
$214$	0	0
$215$	835.983	0.265179
$216$	0	0
$217$	−1354.78	−0.423817
$218$	0	0
$219$	−5105.39	−1.57530
$220$	0	0
$221$	−1092.10	−0.332411
$222$	0	0
$223$	−55.6672	−0.0167164	−0.00835818	−	0.999965i	$-0.502661\pi$
$223$	−55.6672	−0.0167164	−0.00835818	+	0.999965i	$0.502661\pi$
$224$	0	0
$225$	845.054	0.250386
$226$	0	0
$227$	−5179.30	−1.51437	−0.757185	−	0.653200i	$-0.773426\pi$
$227$	−5179.30	−1.51437	−0.757185	+	0.653200i	$0.773426\pi$
$228$	0	0
$229$	−4430.18	−1.27840	−0.639202	−	0.769039i	$-0.720735\pi$
$229$	−4430.18	−1.27840	−0.639202	+	0.769039i	$0.720735\pi$
$230$	0	0
$231$	109.968	0.0313218
$232$	0	0
$233$	2518.58	0.708144	0.354072	−	0.935218i	$-0.384797\pi$
$233$	2518.58	0.708144	0.354072	+	0.935218i	$0.384797\pi$
$234$	0	0
$235$	−541.501	−0.150313
$236$	0	0
$237$	−81.5863	−0.0223612
$238$	0	0
$239$	−4559.50	−1.23401	−0.617007	−	0.786957i	$-0.711655\pi$
$239$	−4559.50	−1.23401	−0.617007	+	0.786957i	$0.711655\pi$
$240$	0	0
$241$	811.460	0.216891	0.108445	−	0.994102i	$-0.465413\pi$
$241$	811.460	0.216891	0.108445	+	0.994102i	$0.465413\pi$
$242$	0	0
$243$	5323.64	1.40540
$244$	0	0
$245$	−499.653	−0.130293
$246$	0	0
$247$	−6374.73	−1.64216
$248$	0	0
$249$	1183.36	0.301174
$250$	0	0
$251$	822.590	0.206858	0.103429	−	0.994637i	$-0.467019\pi$
$251$	822.590	0.206858	0.103429	+	0.994637i	$0.467019\pi$
$252$	0	0
$253$	18.9345	0.00470515
$254$	0	0
$255$	965.975	0.237222
$256$	0	0
$257$	2667.62	0.647477	0.323739	−	0.946147i	$-0.395060\pi$
$257$	2667.62	0.647477	0.323739	+	0.946147i	$0.395060\pi$
$258$	0	0
$259$	−4022.91	−0.965141
$260$	0	0
$261$	980.262	0.232478
$262$	0	0
$263$	−363.692	−0.0852707	−0.0426353	−	0.999091i	$-0.513575\pi$
$263$	−363.692	−0.0852707	−0.0426353	+	0.999091i	$0.513575\pi$
$264$	0	0
$265$	−295.102	−0.0684075
$266$	0	0
$267$	−329.816	−0.0755969
$268$	0	0
$269$	121.990	0.0276499	0.0138250	−	0.999904i	$-0.495599\pi$
$269$	121.990	0.0276499	0.0138250	+	0.999904i	$0.495599\pi$
$270$	0	0
$271$	7132.79	1.59884	0.799420	−	0.600772i	$-0.205140\pi$
$271$	7132.79	1.59884	0.799420	+	0.600772i	$0.205140\pi$
$272$	0	0
$273$	5358.60	1.18797
$274$	0	0
$275$	22.6142	0.00495885
$276$	0	0
$277$	−4082.12	−0.885453	−0.442727	−	0.896657i	$-0.645989\pi$
$277$	−4082.12	−0.885453	−0.442727	+	0.896657i	$0.645989\pi$
$278$	0	0
$279$	2937.29	0.630291
$280$	0	0
$281$	4713.45	1.00064	0.500322	−	0.865839i	$-0.333215\pi$
$281$	4713.45	1.00064	0.500322	+	0.865839i	$0.333215\pi$
$282$	0	0
$283$	4088.40	0.858763	0.429382	−	0.903123i	$-0.358732\pi$
$283$	4088.40	0.858763	0.429382	+	0.903123i	$0.358732\pi$
$284$	0	0
$285$	5638.51	1.17192
$286$	0	0
$287$	−4082.50	−0.839659
$288$	0	0
$289$	−4299.13	−0.875053
$290$	0	0
$291$	939.969	0.189354
$292$	0	0
$293$	−3699.22	−0.737580	−0.368790	−	0.929513i	$-0.620228\pi$
$293$	−3699.22	−0.737580	−0.368790	+	0.929513i	$0.620228\pi$
$294$	0	0
$295$	2971.75	0.586515
$296$	0	0
$297$	−47.9784	−0.00937369
$298$	0	0
$299$	922.657	0.178457
$300$	0	0
$301$	−2606.71	−0.499163
$302$	0	0
$303$	−1325.59	−0.251331
$304$	0	0
$305$	−1326.10	−0.248958
$306$	0	0
$307$	1956.32	0.363691	0.181845	−	0.983327i	$-0.441793\pi$
$307$	1956.32	0.363691	0.181845	+	0.983327i	$0.441793\pi$
$308$	0	0
$309$	−4220.85	−0.777073
$310$	0	0
$311$	1605.41	0.292715	0.146358	−	0.989232i	$-0.453245\pi$
$311$	1605.41	0.292715	0.146358	+	0.989232i	$0.453245\pi$
$312$	0	0
$313$	−1691.08	−0.305385	−0.152693	−	0.988274i	$-0.548794\pi$
$313$	−1691.08	−0.305385	−0.152693	+	0.988274i	$0.548794\pi$
$314$	0	0
$315$	−2634.99	−0.471317
$316$	0	0
$317$	−7790.78	−1.38036	−0.690180	−	0.723638i	$-0.742468\pi$
$317$	−7790.78	−1.38036	−0.690180	+	0.723638i	$0.742468\pi$
$318$	0	0
$319$	26.2324	0.00460418
$320$	0	0
$321$	12502.4	2.17388
$322$	0	0
$323$	3583.21	0.617260
$324$	0	0
$325$	1101.96	0.188080
$326$	0	0
$327$	5406.80	0.914362
$328$	0	0
$329$	1688.47	0.282944
$330$	0	0
$331$	−7105.62	−1.17994	−0.589970	−	0.807425i	$-0.700861\pi$
$331$	−7105.62	−1.17994	−0.589970	+	0.807425i	$0.700861\pi$
$332$	0	0
$333$	8722.07	1.43534
$334$	0	0
$335$	779.579	0.127143
$336$	0	0
$337$	−3455.62	−0.558574	−0.279287	−	0.960208i	$-0.590098\pi$
$337$	−3455.62	−0.558574	−0.279287	+	0.960208i	$0.590098\pi$
$338$	0	0
$339$	−2124.27	−0.340337
$340$	0	0
$341$	78.6038	0.0124828
$342$	0	0
$343$	6905.59	1.08708
$344$	0	0
$345$	−816.099	−0.127354
$346$	0	0
$347$	−5644.82	−0.873284	−0.436642	−	0.899635i	$-0.643832\pi$
$347$	−5644.82	−0.873284	−0.436642	+	0.899635i	$0.643832\pi$
$348$	0	0
$349$	2584.08	0.396340	0.198170	−	0.980168i	$-0.436500\pi$
$349$	2584.08	0.396340	0.198170	+	0.980168i	$0.436500\pi$
$350$	0	0
$351$	−2337.93	−0.355526
$352$	0	0
$353$	1766.21	0.266305	0.133153	−	0.991096i	$-0.457490\pi$
$353$	1766.21	0.266305	0.133153	+	0.991096i	$0.457490\pi$
$354$	0	0
$355$	−2427.07	−0.362861
$356$	0	0
$357$	−3012.04	−0.446538
$358$	0	0
$359$	8470.53	1.24528	0.622642	−	0.782507i	$-0.286059\pi$
$359$	8470.53	1.24528	0.622642	+	0.782507i	$0.286059\pi$
$360$	0	0
$361$	14056.6	2.04937
$362$	0	0
$363$	10372.2	1.49972
$364$	0	0
$365$	3273.70	0.469461
$366$	0	0
$367$	11962.7	1.70150	0.850748	−	0.525573i	$-0.176149\pi$
$367$	11962.7	1.70150	0.850748	+	0.525573i	$0.176149\pi$
$368$	0	0
$369$	8851.26	1.24872
$370$	0	0
$371$	920.169	0.128768
$372$	0	0
$373$	10471.0	1.45353	0.726764	−	0.686887i	$-0.241023\pi$
$373$	10471.0	1.45353	0.726764	+	0.686887i	$0.241023\pi$
$374$	0	0
$375$	−974.697	−0.134222
$376$	0	0
$377$	1278.28	0.174627
$378$	0	0
$379$	11884.4	1.61071	0.805354	−	0.592795i	$-0.201975\pi$
$379$	11884.4	1.61071	0.805354	+	0.592795i	$0.201975\pi$
$380$	0	0
$381$	−7723.52	−1.03855
$382$	0	0
$383$	1154.80	0.154067	0.0770333	−	0.997029i	$-0.475455\pi$
$383$	1154.80	0.154067	0.0770333	+	0.997029i	$0.475455\pi$
$384$	0	0
$385$	−70.5140	−0.00933436
$386$	0	0
$387$	5651.60	0.742344
$388$	0	0
$389$	7269.51	0.947504	0.473752	−	0.880658i	$-0.342899\pi$
$389$	7269.51	0.947504	0.473752	+	0.880658i	$0.342899\pi$
$390$	0	0
$391$	−518.621	−0.0670787
$392$	0	0
$393$	11395.0	1.46260
$394$	0	0
$395$	52.3152	0.00666396
$396$	0	0
$397$	−7778.26	−0.983324	−0.491662	−	0.870786i	$-0.663610\pi$
$397$	−7778.26	−0.983324	−0.491662	+	0.870786i	$0.663610\pi$
$398$	0	0
$399$	−17581.7	−2.20597
$400$	0	0
$401$	−7533.41	−0.938156	−0.469078	−	0.883157i	$-0.655414\pi$
$401$	−7533.41	−0.938156	−0.469078	+	0.883157i	$0.655414\pi$
$402$	0	0
$403$	3830.27	0.473448
$404$	0	0
$405$	−2495.37	−0.306162
$406$	0	0
$407$	233.408	0.0284266
$408$	0	0
$409$	11081.6	1.33973	0.669867	−	0.742482i	$-0.266351\pi$
$409$	11081.6	1.33973	0.669867	+	0.742482i	$0.266351\pi$
$410$	0	0
$411$	−4187.87	−0.502609
$412$	0	0
$413$	−9266.32	−1.10403
$414$	0	0
$415$	−758.800	−0.0897543
$416$	0	0
$417$	11246.0	1.32067
$418$	0	0
$419$	6405.75	0.746877	0.373438	−	0.927655i	$-0.378179\pi$
$419$	6405.75	0.746877	0.373438	+	0.927655i	$0.378179\pi$
$420$	0	0
$421$	−2929.76	−0.339163	−0.169581	−	0.985516i	$-0.554242\pi$
$421$	−2929.76	−0.339163	−0.169581	+	0.985516i	$0.554242\pi$
$422$	0	0
$423$	−3660.78	−0.420787
$424$	0	0
$425$	−619.408	−0.0706957
$426$	0	0
$427$	4134.95	0.468628
$428$	0	0
$429$	−310.904	−0.0349897
$430$	0	0
$431$	2013.91	0.225074	0.112537	−	0.993648i	$-0.464102\pi$
$431$	2013.91	0.225074	0.112537	+	0.993648i	$0.464102\pi$
$432$	0	0
$433$	5405.69	0.599956	0.299978	−	0.953946i	$-0.403021\pi$
$433$	5405.69	0.599956	0.299978	+	0.953946i	$0.403021\pi$
$434$	0	0
$435$	−1130.65	−0.124622
$436$	0	0
$437$	−3027.25	−0.331380
$438$	0	0
$439$	−4144.69	−0.450604	−0.225302	−	0.974289i	$-0.572337\pi$
$439$	−4144.69	−0.450604	−0.225302	+	0.974289i	$0.572337\pi$
$440$	0	0
$441$	−3377.87	−0.364741
$442$	0	0
$443$	13110.1	1.40605	0.703027	−	0.711163i	$-0.251831\pi$
$443$	13110.1	1.40605	0.703027	+	0.711163i	$0.251831\pi$
$444$	0	0
$445$	211.486	0.0225290
$446$	0	0
$447$	5938.89	0.628412
$448$	0	0
$449$	−2901.88	−0.305007	−0.152503	−	0.988303i	$-0.548733\pi$
$449$	−2901.88	−0.305007	−0.152503	+	0.988303i	$0.548733\pi$
$450$	0	0
$451$	236.865	0.0247307
$452$	0	0
$453$	2400.99	0.249025
$454$	0	0
$455$	−3436.07	−0.354034
$456$	0	0
$457$	1766.65	0.180832	0.0904160	−	0.995904i	$-0.471180\pi$
$457$	1766.65	0.180832	0.0904160	+	0.995904i	$0.471180\pi$
$458$	0	0
$459$	1314.14	0.133636
$460$	0	0
$461$	−15729.5	−1.58915	−0.794574	−	0.607167i	$-0.792306\pi$
$461$	−15729.5	−1.58915	−0.794574	+	0.607167i	$0.792306\pi$
$462$	0	0
$463$	−16697.6	−1.67603	−0.838016	−	0.545646i	$-0.816284\pi$
$463$	−16697.6	−1.67603	−0.838016	+	0.545646i	$0.816284\pi$
$464$	0	0
$465$	−3387.92	−0.337873
$466$	0	0
$467$	5640.48	0.558908	0.279454	−	0.960159i	$-0.409847\pi$
$467$	5640.48	0.558908	0.279454	+	0.960159i	$0.409847\pi$
$468$	0	0
$469$	−2430.83	−0.239329
$470$	0	0
$471$	8039.70	0.786517
$472$	0	0
$473$	151.240	0.0147020
$474$	0	0
$475$	−3615.56	−0.349249
$476$	0	0
$477$	−1995.02	−0.191500
$478$	0	0
$479$	−9564.45	−0.912340	−0.456170	−	0.889893i	$-0.650779\pi$
$479$	−9564.45	−0.912340	−0.456170	+	0.889893i	$0.650779\pi$
$480$	0	0
$481$	11373.7	1.07816
$482$	0	0
$483$	2544.71	0.239727
$484$	0	0
$485$	−602.732	−0.0564302
$486$	0	0
$487$	906.403	0.0843389	0.0421694	−	0.999110i	$-0.486573\pi$
$487$	906.403	0.0843389	0.0421694	+	0.999110i	$0.486573\pi$
$488$	0	0
$489$	−19387.8	−1.79294
$490$	0	0
$491$	−20026.0	−1.84065	−0.920326	−	0.391153i	$-0.872076\pi$
$491$	−20026.0	−1.84065	−0.920326	+	0.391153i	$0.872076\pi$
$492$	0	0
$493$	−718.513	−0.0656393
$494$	0	0
$495$	152.881	0.0138818
$496$	0	0
$497$	7567.95	0.683036
$498$	0	0
$499$	−19999.0	−1.79414	−0.897070	−	0.441888i	$-0.854309\pi$
$499$	−19999.0	−1.79414	−0.897070	+	0.441888i	$0.854309\pi$
$500$	0	0
$501$	12966.4	1.15628
$502$	0	0
$503$	−19754.4	−1.75111	−0.875554	−	0.483121i	$-0.839503\pi$
$503$	−19754.4	−1.75111	−0.875554	+	0.483121i	$0.839503\pi$
$504$	0	0
$505$	850.005	0.0749004
$506$	0	0
$507$	1981.26	0.173552
$508$	0	0
$509$	−6235.90	−0.543028	−0.271514	−	0.962434i	$-0.587524\pi$
$509$	−6235.90	−0.543028	−0.271514	+	0.962434i	$0.587524\pi$
$510$	0	0
$511$	−10207.8	−0.883696
$512$	0	0
$513$	7670.79	0.660183
$514$	0	0
$515$	2706.51	0.231579
$516$	0	0
$517$	−97.9647	−0.00833362
$518$	0	0
$519$	12694.2	1.07363
$520$	0	0
$521$	15678.4	1.31839	0.659197	−	0.751970i	$-0.270896\pi$
$521$	15678.4	1.31839	0.659197	+	0.751970i	$0.270896\pi$
$522$	0	0
$523$	−8483.72	−0.709307	−0.354653	−	0.934998i	$-0.615401\pi$
$523$	−8483.72	−0.709307	−0.354653	+	0.934998i	$0.615401\pi$
$524$	0	0
$525$	3039.24	0.252654
$526$	0	0
$527$	−2152.98	−0.177961
$528$	0	0
$529$	−11728.8	−0.963988
$530$	0	0
$531$	20090.3	1.64189
$532$	0	0
$533$	11542.2	0.937987
$534$	0	0
$535$	−8016.86	−0.647849
$536$	0	0
$537$	10366.6	0.833058
$538$	0	0
$539$	−90.3939	−0.00722364
$540$	0	0
$541$	−20404.7	−1.62156	−0.810781	−	0.585350i	$-0.800957\pi$
$541$	−20404.7	−1.62156	−0.810781	+	0.585350i	$0.800957\pi$
$542$	0	0
$543$	18984.6	1.50038
$544$	0	0
$545$	−3466.97	−0.272493
$546$	0	0
$547$	−7199.63	−0.562768	−0.281384	−	0.959595i	$-0.590793\pi$
$547$	−7199.63	−0.562768	−0.281384	+	0.959595i	$0.590793\pi$
$548$	0	0
$549$	−8964.98	−0.696933
$550$	0	0
$551$	−4194.05	−0.324269
$552$	0	0
$553$	−163.126	−0.0125440
$554$	0	0
$555$	−10060.2	−0.769424
$556$	0	0
$557$	−10162.3	−0.773050	−0.386525	−	0.922279i	$-0.626325\pi$
$557$	−10162.3	−0.773050	−0.386525	+	0.922279i	$0.626325\pi$
$558$	0	0
$559$	7369.77	0.557617
$560$	0	0
$561$	174.758	0.0131520
$562$	0	0
$563$	1311.95	0.0982101	0.0491051	−	0.998794i	$-0.484363\pi$
$563$	1311.95	0.0982101	0.0491051	+	0.998794i	$0.484363\pi$
$564$	0	0
$565$	1362.13	0.101425
$566$	0	0
$567$	7780.89	0.576308
$568$	0	0
$569$	4043.91	0.297943	0.148971	−	0.988842i	$-0.452404\pi$
$569$	4043.91	0.297943	0.148971	+	0.988842i	$0.452404\pi$
$570$	0	0
$571$	−11300.2	−0.828193	−0.414096	−	0.910233i	$-0.635902\pi$
$571$	−11300.2	−0.828193	−0.414096	+	0.910233i	$0.635902\pi$
$572$	0	0
$573$	38605.9	2.81463
$574$	0	0
$575$	523.303	0.0379535
$576$	0	0
$577$	−27410.8	−1.97769	−0.988845	−	0.148946i	$-0.952412\pi$
$577$	−27410.8	−1.97769	−0.988845	+	0.148946i	$0.952412\pi$
$578$	0	0
$579$	11641.1	0.835557
$580$	0	0
$581$	2366.04	0.168950
$582$	0	0
$583$	−53.3879	−0.00379263
$584$	0	0
$585$	7449.74	0.526511
$586$	0	0
$587$	−7279.48	−0.511850	−0.255925	−	0.966697i	$-0.582380\pi$
$587$	−7279.48	−0.511850	−0.255925	+	0.966697i	$0.582380\pi$
$588$	0	0
$589$	−12567.2	−0.879155
$590$	0	0
$591$	−1391.51	−0.0968514
$592$	0	0
$593$	14007.0	0.969983	0.484992	−	0.874519i	$-0.338823\pi$
$593$	14007.0	0.969983	0.484992	+	0.874519i	$0.338823\pi$
$594$	0	0
$595$	1931.40	0.133075
$596$	0	0
$597$	20784.4	1.42487
$598$	0	0
$599$	15810.3	1.07845	0.539225	−	0.842162i	$-0.318717\pi$
$599$	15810.3	1.07845	0.539225	+	0.842162i	$0.318717\pi$
$600$	0	0
$601$	20133.1	1.36646	0.683232	−	0.730202i	$-0.260574\pi$
$601$	20133.1	1.36646	0.683232	+	0.730202i	$0.260574\pi$
$602$	0	0
$603$	5270.29	0.355925
$604$	0	0
$605$	−6650.91	−0.446939
$606$	0	0
$607$	10325.2	0.690421	0.345210	−	0.938525i	$-0.387808\pi$
$607$	10325.2	0.690421	0.345210	+	0.938525i	$0.387808\pi$
$608$	0	0
$609$	3525.51	0.234583
$610$	0	0
$611$	−4773.71	−0.316078
$612$	0	0
$613$	−6423.80	−0.423254	−0.211627	−	0.977350i	$-0.567876\pi$
$613$	−6423.80	−0.423254	−0.211627	+	0.977350i	$0.567876\pi$
$614$	0	0
$615$	−10209.2	−0.669387
$616$	0	0
$617$	−17619.9	−1.14967	−0.574837	−	0.818268i	$-0.694935\pi$
$617$	−17619.9	−1.14967	−0.574837	+	0.818268i	$0.694935\pi$
$618$	0	0
$619$	−19401.6	−1.25980	−0.629901	−	0.776675i	$-0.716905\pi$
$619$	−19401.6	−1.25980	−0.629901	+	0.776675i	$0.716905\pi$
$620$	0	0
$621$	−1110.24	−0.0717433
$622$	0	0
$623$	−659.442	−0.0424077
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1020.08	0.0649731
$628$	0	0
$629$	−6393.11	−0.405262
$630$	0	0
$631$	3806.79	0.240168	0.120084	−	0.992764i	$-0.461684\pi$
$631$	3806.79	0.240168	0.120084	+	0.992764i	$0.461684\pi$
$632$	0	0
$633$	14483.0	0.909393
$634$	0	0
$635$	4952.51	0.309503
$636$	0	0
$637$	−4404.79	−0.273978
$638$	0	0
$639$	−16408.1	−1.01580
$640$	0	0
$641$	24251.8	1.49436	0.747182	−	0.664620i	$-0.231407\pi$
$641$	24251.8	1.49436	0.747182	+	0.664620i	$0.231407\pi$
$642$	0	0
$643$	−21994.4	−1.34895	−0.674476	−	0.738297i	$-0.735630\pi$
$643$	−21994.4	−1.34895	−0.674476	+	0.738297i	$0.735630\pi$
$644$	0	0
$645$	−6518.64	−0.397940
$646$	0	0
$647$	13553.0	0.823530	0.411765	−	0.911290i	$-0.364912\pi$
$647$	13553.0	0.823530	0.411765	+	0.911290i	$0.364912\pi$
$648$	0	0
$649$	537.629	0.0325174
$650$	0	0
$651$	10564.0	0.635999
$652$	0	0
$653$	1789.08	0.107216	0.0536080	−	0.998562i	$-0.482928\pi$
$653$	1789.08	0.107216	0.0536080	+	0.998562i	$0.482928\pi$
$654$	0	0
$655$	−7306.78	−0.435877
$656$	0	0
$657$	22131.6	1.31421
$658$	0	0
$659$	2820.26	0.166710	0.0833548	−	0.996520i	$-0.473437\pi$
$659$	2820.26	0.166710	0.0833548	+	0.996520i	$0.473437\pi$
$660$	0	0
$661$	10362.8	0.609782	0.304891	−	0.952387i	$-0.401380\pi$
$661$	10362.8	0.609782	0.304891	+	0.952387i	$0.401380\pi$
$662$	0	0
$663$	8515.75	0.498830
$664$	0	0
$665$	11273.8	0.657412
$666$	0	0
$667$	607.032	0.0352389
$668$	0	0
$669$	434.069	0.0250853
$670$	0	0
$671$	−239.908	−0.0138026
$672$	0	0
$673$	15137.2	0.867010	0.433505	−	0.901151i	$-0.357277\pi$
$673$	15137.2	0.867010	0.433505	+	0.901151i	$0.357277\pi$
$674$	0	0
$675$	−1326.00	−0.0756118
$676$	0	0
$677$	−2478.96	−0.140730	−0.0703650	−	0.997521i	$-0.522416\pi$
$677$	−2478.96	−0.140730	−0.0703650	+	0.997521i	$0.522416\pi$
$678$	0	0
$679$	1879.40	0.106222
$680$	0	0
$681$	40385.9	2.27253
$682$	0	0
$683$	16268.0	0.911389	0.455695	−	0.890136i	$-0.349391\pi$
$683$	16268.0	0.911389	0.455695	+	0.890136i	$0.349391\pi$
$684$	0	0
$685$	2685.37	0.149785
$686$	0	0
$687$	34544.6	1.91843
$688$	0	0
$689$	−2601.53	−0.143847
$690$	0	0
$691$	9048.46	0.498147	0.249074	−	0.968485i	$-0.419874\pi$
$691$	9048.46	0.498147	0.249074	+	0.968485i	$0.419874\pi$
$692$	0	0
$693$	−476.705	−0.0261306
$694$	0	0
$695$	−7211.24	−0.393580
$696$	0	0
$697$	−6487.80	−0.352572
$698$	0	0
$699$	−19638.8	−1.06267
$700$	0	0
$701$	−10797.5	−0.581764	−0.290882	−	0.956759i	$-0.593949\pi$
$701$	−10797.5	−0.581764	−0.290882	+	0.956759i	$0.593949\pi$
$702$	0	0
$703$	−37317.3	−2.00206
$704$	0	0
$705$	4222.39	0.225567
$706$	0	0
$707$	−2650.43	−0.140990
$708$	0	0
$709$	−15359.5	−0.813594	−0.406797	−	0.913519i	$-0.633354\pi$
$709$	−15359.5	−0.813594	−0.406797	+	0.913519i	$0.633354\pi$
$710$	0	0
$711$	353.673	0.0186551
$712$	0	0
$713$	1818.93	0.0955393
$714$	0	0
$715$	199.360	0.0104274
$716$	0	0
$717$	35553.0	1.85182
$718$	0	0
$719$	21305.1	1.10507	0.552537	−	0.833489i	$-0.313660\pi$
$719$	21305.1	1.10507	0.552537	+	0.833489i	$0.313660\pi$
$720$	0	0
$721$	−8439.28	−0.435916
$722$	0	0
$723$	−6327.41	−0.325476
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	16591.5	0.846417	0.423208	−	0.906032i	$-0.360904\pi$
$727$	16591.5	0.846417	0.423208	+	0.906032i	$0.360904\pi$
$728$	0	0
$729$	−28036.5	−1.42440
$730$	0	0
$731$	−4142.51	−0.209598
$732$	0	0
$733$	−10379.7	−0.523035	−0.261518	−	0.965199i	$-0.584223\pi$
$733$	−10379.7	−0.523035	−0.261518	+	0.965199i	$0.584223\pi$
$734$	0	0
$735$	3896.08	0.195523
$736$	0	0
$737$	141.036	0.00704903
$738$	0	0
$739$	−22623.7	−1.12615	−0.563076	−	0.826405i	$-0.690382\pi$
$739$	−22623.7	−1.12615	−0.563076	+	0.826405i	$0.690382\pi$
$740$	0	0
$741$	49707.4	2.46430
$742$	0	0
$743$	12857.3	0.634841	0.317420	−	0.948285i	$-0.397183\pi$
$743$	12857.3	0.634841	0.317420	+	0.948285i	$0.397183\pi$
$744$	0	0
$745$	−3808.17	−0.187276
$746$	0	0
$747$	−5129.81	−0.251259
$748$	0	0
$749$	24997.7	1.21949
$750$	0	0
$751$	−10496.1	−0.509997	−0.254999	−	0.966941i	$-0.582075\pi$
$751$	−10496.1	−0.509997	−0.254999	+	0.966941i	$0.582075\pi$
$752$	0	0
$753$	−6414.20	−0.310420
$754$	0	0
$755$	−1539.58	−0.0742132
$756$	0	0
$757$	−22022.8	−1.05738	−0.528688	−	0.848817i	$-0.677316\pi$
$757$	−22022.8	−1.05738	−0.528688	+	0.848817i	$0.677316\pi$
$758$	0	0
$759$	−147.643	−0.00706075
$760$	0	0
$761$	−2439.55	−0.116207	−0.0581034	−	0.998311i	$-0.518505\pi$
$761$	−2439.55	−0.116207	−0.0581034	+	0.998311i	$0.518505\pi$
$762$	0	0
$763$	10810.5	0.512931
$764$	0	0
$765$	−4187.46	−0.197906
$766$	0	0
$767$	26198.0	1.23332
$768$	0	0
$769$	−14473.6	−0.678713	−0.339357	−	0.940658i	$-0.610209\pi$
$769$	−14473.6	−0.678713	−0.339357	+	0.940658i	$0.610209\pi$
$770$	0	0
$771$	−20801.0	−0.971633
$772$	0	0
$773$	20766.8	0.966275	0.483137	−	0.875545i	$-0.339497\pi$
$773$	20766.8	0.966275	0.483137	+	0.875545i	$0.339497\pi$
$774$	0	0
$775$	2172.42	0.100691
$776$	0	0
$777$	31369.0	1.44833
$778$	0	0
$779$	−37870.1	−1.74177
$780$	0	0
$781$	−439.090	−0.0201176
$782$	0	0
$783$	−1538.17	−0.0702038
$784$	0	0
$785$	−5155.26	−0.234394
$786$	0	0
$787$	27806.3	1.25945	0.629725	−	0.776819i	$-0.283168\pi$
$787$	27806.3	1.25945	0.629725	+	0.776819i	$0.283168\pi$
$788$	0	0
$789$	2835.91	0.127961
$790$	0	0
$791$	−4247.32	−0.190919
$792$	0	0
$793$	−11690.5	−0.523506
$794$	0	0
$795$	2301.08	0.102655
$796$	0	0
$797$	−43285.6	−1.92378	−0.961890	−	0.273436i	$-0.911840\pi$
$797$	−43285.6	−1.92378	−0.961890	+	0.273436i	$0.911840\pi$
$798$	0	0
$799$	2683.28	0.118808
$800$	0	0
$801$	1429.74	0.0630677
$802$	0	0
$803$	592.256	0.0260277
$804$	0	0
$805$	−1631.73	−0.0714422
$806$	0	0
$807$	−951.222	−0.0414927
$808$	0	0
$809$	−4576.49	−0.198888	−0.0994442	−	0.995043i	$-0.531706\pi$
$809$	−4576.49	−0.198888	−0.0994442	+	0.995043i	$0.531706\pi$
$810$	0	0
$811$	−12862.6	−0.556925	−0.278462	−	0.960447i	$-0.589825\pi$
$811$	−12862.6	−0.556925	−0.278462	+	0.960447i	$0.589825\pi$
$812$	0	0
$813$	−55618.4	−2.39929
$814$	0	0
$815$	12431.9	0.534321
$816$	0	0
$817$	−24180.3	−1.03545
$818$	0	0
$819$	−23229.3	−0.991083
$820$	0	0
$821$	−11106.9	−0.472147	−0.236074	−	0.971735i	$-0.575861\pi$
$821$	−11106.9	−0.472147	−0.236074	+	0.971735i	$0.575861\pi$
$822$	0	0
$823$	−19234.1	−0.814650	−0.407325	−	0.913283i	$-0.633538\pi$
$823$	−19234.1	−0.814650	−0.407325	+	0.913283i	$0.633538\pi$
$824$	0	0
$825$	−176.336	−0.00744147
$826$	0	0
$827$	22986.1	0.966513	0.483257	−	0.875479i	$-0.339454\pi$
$827$	22986.1	0.966513	0.483257	+	0.875479i	$0.339454\pi$
$828$	0	0
$829$	−10960.1	−0.459180	−0.229590	−	0.973287i	$-0.573739\pi$
$829$	−10960.1	−0.459180	−0.229590	+	0.973287i	$0.573739\pi$
$830$	0	0
$831$	31830.6	1.32875
$832$	0	0
$833$	2475.91	0.102984
$834$	0	0
$835$	−8314.40	−0.344589
$836$	0	0
$837$	−4609.02	−0.190336
$838$	0	0
$839$	35112.8	1.44485	0.722425	−	0.691449i	$-0.243027\pi$
$839$	35112.8	1.44485	0.722425	+	0.691449i	$0.243027\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−36753.5	−1.50161
$844$	0	0
$845$	−1270.44	−0.0517211
$846$	0	0
$847$	20738.4	0.841300
$848$	0	0
$849$	−31879.6	−1.28870
$850$	0	0
$851$	5401.18	0.217568
$852$	0	0
$853$	−40183.5	−1.61296	−0.806481	−	0.591259i	$-0.798631\pi$
$853$	−40183.5	−1.61296	−0.806481	+	0.591259i	$0.798631\pi$
$854$	0	0
$855$	−24442.7	−0.977688
$856$	0	0
$857$	−3343.14	−0.133255	−0.0666274	−	0.997778i	$-0.521224\pi$
$857$	−3343.14	−0.133255	−0.0666274	+	0.997778i	$0.521224\pi$
$858$	0	0
$859$	−17877.4	−0.710091	−0.355046	−	0.934849i	$-0.615535\pi$
$859$	−17877.4	−0.710091	−0.355046	+	0.934849i	$0.615535\pi$
$860$	0	0
$861$	31833.6	1.26003
$862$	0	0
$863$	10257.9	0.404615	0.202307	−	0.979322i	$-0.435156\pi$
$863$	10257.9	0.404615	0.202307	+	0.979322i	$0.435156\pi$
$864$	0	0
$865$	−8139.84	−0.319957
$866$	0	0
$867$	33522.8	1.31314
$868$	0	0
$869$	9.46451	0.000369461 0
$870$	0	0
$871$	6872.53	0.267356
$872$	0	0
$873$	−4074.72	−0.157971
$874$	0	0
$875$	−1948.84	−0.0752945
$876$	0	0
$877$	−15682.7	−0.603838	−0.301919	−	0.953334i	$-0.597627\pi$
$877$	−15682.7	−0.603838	−0.301919	+	0.953334i	$0.597627\pi$
$878$	0	0
$879$	28844.9	1.10684
$880$	0	0
$881$	29798.4	1.13954	0.569770	−	0.821805i	$-0.307032\pi$
$881$	29798.4	1.13954	0.569770	+	0.821805i	$0.307032\pi$
$882$	0	0
$883$	38705.2	1.47512	0.737562	−	0.675279i	$-0.235977\pi$
$883$	38705.2	1.47512	0.737562	+	0.675279i	$0.235977\pi$
$884$	0	0
$885$	−23172.4	−0.880150
$886$	0	0
$887$	−14576.7	−0.551790	−0.275895	−	0.961188i	$-0.588974\pi$
$887$	−14576.7	−0.551790	−0.275895	+	0.961188i	$0.588974\pi$
$888$	0	0
$889$	−15442.6	−0.582597
$890$	0	0
$891$	−451.445	−0.0169741
$892$	0	0
$893$	15662.6	0.586931
$894$	0	0
$895$	−6647.33	−0.248263
$896$	0	0
$897$	−7194.48	−0.267800
$898$	0	0
$899$	2520.00	0.0934892
$900$	0	0
$901$	1462.31	0.0540694
$902$	0	0
$903$	20326.0	0.749066
$904$	0	0
$905$	−12173.4	−0.447136
$906$	0	0
$907$	−7195.29	−0.263413	−0.131707	−	0.991289i	$-0.542046\pi$
$907$	−7195.29	−0.263413	−0.131707	+	0.991289i	$0.542046\pi$
$908$	0	0
$909$	5746.40	0.209676
$910$	0	0
$911$	43512.8	1.58248	0.791242	−	0.611503i	$-0.209435\pi$
$911$	43512.8	1.58248	0.791242	+	0.611503i	$0.209435\pi$
$912$	0	0
$913$	−137.277	−0.00497613
$914$	0	0
$915$	10340.3	0.373597
$916$	0	0
$917$	22783.6	0.820479
$918$	0	0
$919$	62.2336	0.00223384	0.00111692	−	0.999999i	$-0.499644\pi$
$919$	62.2336	0.00223384	0.00111692	+	0.999999i	$0.499644\pi$
$920$	0	0
$921$	−15254.5	−0.545770
$922$	0	0
$923$	−21396.4	−0.763022
$924$	0	0
$925$	6450.83	0.229299
$926$	0	0
$927$	18297.2	0.648283
$928$	0	0
$929$	−10018.2	−0.353806	−0.176903	−	0.984228i	$-0.556608\pi$
$929$	−10018.2	−0.353806	−0.176903	+	0.984228i	$0.556608\pi$
$930$	0	0
$931$	14452.2	0.508756
$932$	0	0
$933$	−12518.3	−0.439261
$934$	0	0
$935$	−112.059	−0.00391949
$936$	0	0
$937$	−46257.7	−1.61278	−0.806389	−	0.591385i	$-0.798581\pi$
$937$	−46257.7	−1.61278	−0.806389	+	0.591385i	$0.798581\pi$
$938$	0	0
$939$	13186.3	0.458274
$940$	0	0
$941$	35906.8	1.24392	0.621960	−	0.783049i	$-0.286337\pi$
$941$	35906.8	1.24392	0.621960	+	0.783049i	$0.286337\pi$
$942$	0	0
$943$	5481.18	0.189281
$944$	0	0
$945$	4134.66	0.142329
$946$	0	0
$947$	−6769.82	−0.232302	−0.116151	−	0.993232i	$-0.537056\pi$
$947$	−6769.82	−0.232302	−0.116151	+	0.993232i	$0.537056\pi$
$948$	0	0
$949$	28860.0	0.987180
$950$	0	0
$951$	60749.2	2.07143
$952$	0	0
$953$	33394.5	1.13510	0.567551	−	0.823338i	$-0.307891\pi$
$953$	33394.5	1.13510	0.567551	+	0.823338i	$0.307891\pi$
$954$	0	0
$955$	−24755.0	−0.838801
$956$	0	0
$957$	−204.549	−0.00690923
$958$	0	0
$959$	−8373.35	−0.281949
$960$	0	0
$961$	−22240.0	−0.746533
$962$	0	0
$963$	−54197.4	−1.81359
$964$	0	0
$965$	−7464.56	−0.249008
$966$	0	0
$967$	−32335.3	−1.07532	−0.537660	−	0.843162i	$-0.680691\pi$
$967$	−32335.3	−1.07532	−0.537660	+	0.843162i	$0.680691\pi$
$968$	0	0
$969$	−27940.3	−0.926287
$970$	0	0
$971$	−32008.8	−1.05789	−0.528945	−	0.848656i	$-0.677412\pi$
$971$	−32008.8	−1.05789	−0.528945	+	0.848656i	$0.677412\pi$
$972$	0	0
$973$	22485.6	0.740859
$974$	0	0
$975$	−8592.63	−0.282240
$976$	0	0
$977$	−29266.9	−0.958375	−0.479188	−	0.877713i	$-0.659069\pi$
$977$	−29266.9	−0.958375	−0.479188	+	0.877713i	$0.659069\pi$
$978$	0	0
$979$	38.2606	0.00124904
$980$	0	0
$981$	−23438.2	−0.762819
$982$	0	0
$983$	−21313.8	−0.691563	−0.345781	−	0.938315i	$-0.612386\pi$
$983$	−21313.8	−0.691563	−0.345781	+	0.938315i	$0.612386\pi$
$984$	0	0
$985$	892.273	0.0288631
$986$	0	0
$987$	−13166.0	−0.424598
$988$	0	0
$989$	3499.78	0.112524
$990$	0	0
$991$	44567.4	1.42859	0.714293	−	0.699846i	$-0.246748\pi$
$991$	44567.4	1.42859	0.714293	+	0.699846i	$0.246748\pi$
$992$	0	0
$993$	55406.6	1.77067
$994$	0	0
$995$	−13327.5	−0.424633
$996$	0	0
$997$	32449.3	1.03077	0.515386	−	0.856958i	$-0.327649\pi$
$997$	32449.3	1.03077	0.515386	+	0.856958i	$0.327649\pi$
$998$	0	0
$999$	−13686.1	−0.433443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.d.1.2	✓	9
4.3	odd	2		2320.4.a.y.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.d.1.2	✓	9	1.1	even	1	trivial
2320.4.a.y.1.8		9	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)

\(f(q)\)	\(=\)	\(q-7.79757 q^{3} +5.00000 q^{5} -15.5907 q^{7} +33.8021 q^{9} +0.904566 q^{11} +44.0785 q^{13} -38.9879 q^{15} -24.7763 q^{17} -144.622 q^{19} +121.569 q^{21} +20.9321 q^{23} +25.0000 q^{25} -53.0402 q^{27} +29.0000 q^{29} +86.8967 q^{31} -7.05342 q^{33} -77.9534 q^{35} +258.033 q^{37} -343.705 q^{39} +261.855 q^{41} +167.197 q^{43} +169.011 q^{45} -108.300 q^{47} -99.9307 q^{49} +193.195 q^{51} -59.0204 q^{53} +4.52283 q^{55} +1127.70 q^{57} +594.350 q^{59} -265.219 q^{61} -526.998 q^{63} +220.392 q^{65} +155.916 q^{67} -163.220 q^{69} -485.415 q^{71} +654.740 q^{73} -194.939 q^{75} -14.1028 q^{77} +10.4630 q^{79} -499.073 q^{81} -151.760 q^{83} -123.882 q^{85} -226.130 q^{87} +42.2972 q^{89} -687.214 q^{91} -677.583 q^{93} -723.111 q^{95} -120.546 q^{97} +30.5763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 7 q^{3} + 45 q^{5} - 39 q^{7} + 72 q^{9} - 108 q^{11} - 19 q^{13} - 35 q^{15} + 91 q^{17} - 36 q^{19} - 200 q^{21} - 209 q^{23} + 225 q^{25} - 316 q^{27} + 261 q^{29} - 599 q^{31} - 192 q^{33} - 195 q^{35}+ \cdots - 2946 q^{99}+O(q^{100}) \) 9 * q - 7 * q^3 + 45 * q^5 - 39 * q^7 + 72 * q^9 - 108 * q^11 - 19 * q^13 - 35 * q^15 + 91 * q^17 - 36 * q^19 - 200 * q^21 - 209 * q^23 + 225 * q^25 - 316 * q^27 + 261 * q^29 - 599 * q^31 - 192 * q^33 - 195 * q^35 + 204 * q^37 - 221 * q^39 - 328 * q^41 - 717 * q^43 + 360 * q^45 - 646 * q^47 + 200 * q^49 - 1038 * q^51 + 71 * q^53 - 540 * q^55 + 494 * q^57 - 1591 * q^59 - 439 * q^61 - 354 * q^63 - 95 * q^65 - 1222 * q^67 + 873 * q^69 - 1240 * q^71 - 69 * q^73 - 175 * q^75 + 102 * q^77 - 1013 * q^79 - 171 * q^81 - 2728 * q^83 + 455 * q^85 - 203 * q^87 - 312 * q^89 - 1652 * q^91 - 702 * q^93 - 180 * q^95 + 1145 * q^97 - 2946 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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