LMFDB - Embedded newform 1148.4.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 278 x^{13} + 106 x^{12} + 30089 x^{11} + 6907 x^{10} - 1613884 x^{9} + \cdots - 6369125904 $ x^15 - x^14 - 278x^13 + 106x^12 + 30089x^11 + 6907x^10 - 1613884x^9 - 1342154x^8 + 45234374x^7 + 64400510x^6 - 632496875x^5 - 1277681173x^4 + 3431570979x^3 + 9451873305x^2 + 973796742*x - 6369125904
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.78398$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.78398 q^{3} -6.11125 q^{5} +7.00000 q^{7} +6.45437 q^{9} +O(q^{10})$	$q-5.78398 q^{3} -6.11125 q^{5} +7.00000 q^{7} +6.45437 q^{9} +64.7694 q^{11} -83.7744 q^{13} +35.3473 q^{15} -87.3046 q^{17} +130.970 q^{19} -40.4878 q^{21} -48.6690 q^{23} -87.6527 q^{25} +118.835 q^{27} +18.8616 q^{29} -2.57883 q^{31} -374.625 q^{33} -42.7787 q^{35} +434.839 q^{37} +484.549 q^{39} +41.0000 q^{41} +195.702 q^{43} -39.4443 q^{45} +189.617 q^{47} +49.0000 q^{49} +504.968 q^{51} -60.3919 q^{53} -395.822 q^{55} -757.530 q^{57} +286.148 q^{59} -921.385 q^{61} +45.1806 q^{63} +511.966 q^{65} -473.140 q^{67} +281.500 q^{69} +1139.15 q^{71} +1121.40 q^{73} +506.981 q^{75} +453.386 q^{77} -99.9330 q^{79} -861.609 q^{81} -481.637 q^{83} +533.540 q^{85} -109.095 q^{87} -807.624 q^{89} -586.421 q^{91} +14.9159 q^{93} -800.393 q^{95} -1512.59 q^{97} +418.046 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 14 q^{3} - 24 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) $ 15 * q - 14 * q^3 - 24 * q^5 + 105 * q^7 + 165 * q^9	$ 15 q - 14 q^{3} - 24 q^{5} + 105 q^{7} + 165 q^{9} - 64 q^{11} - 112 q^{13} - 228 q^{15} - 128 q^{17} - 240 q^{19} - 98 q^{21} - 302 q^{23} + 325 q^{25} - 410 q^{27} - 330 q^{29} - 432 q^{31} - 40 q^{33} - 168 q^{35} - 318 q^{37} - 454 q^{39} + 615 q^{41} - 882 q^{43} - 606 q^{45} + 520 q^{47} + 735 q^{49} + 1194 q^{51} - 304 q^{53} - 400 q^{55} + 26 q^{57} - 782 q^{59} - 710 q^{61} + 1155 q^{63} + 1344 q^{65} - 102 q^{67} - 1040 q^{69} + 1392 q^{71} - 366 q^{73} - 1134 q^{75} - 448 q^{77} - 1482 q^{79} + 951 q^{81} - 1920 q^{83} - 2496 q^{85} - 4764 q^{87} - 1404 q^{89} - 784 q^{91} - 3734 q^{93} - 226 q^{95} - 7366 q^{97} - 3724 q^{99}+O(q^{100}) $ 15 * q - 14 * q^3 - 24 * q^5 + 105 * q^7 + 165 * q^9 - 64 * q^11 - 112 * q^13 - 228 * q^15 - 128 * q^17 - 240 * q^19 - 98 * q^21 - 302 * q^23 + 325 * q^25 - 410 * q^27 - 330 * q^29 - 432 * q^31 - 40 * q^33 - 168 * q^35 - 318 * q^37 - 454 * q^39 + 615 * q^41 - 882 * q^43 - 606 * q^45 + 520 * q^47 + 735 * q^49 + 1194 * q^51 - 304 * q^53 - 400 * q^55 + 26 * q^57 - 782 * q^59 - 710 * q^61 + 1155 * q^63 + 1344 * q^65 - 102 * q^67 - 1040 * q^69 + 1392 * q^71 - 366 * q^73 - 1134 * q^75 - 448 * q^77 - 1482 * q^79 + 951 * q^81 - 1920 * q^83 - 2496 * q^85 - 4764 * q^87 - 1404 * q^89 - 784 * q^91 - 3734 * q^93 - 226 * q^95 - 7366 * q^97 - 3724 * 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
$3$	−5.78398	−1.11313	−0.556563	−	0.830805i	$-0.687880\pi$
$3$	−5.78398	−1.11313	−0.556563	+	0.830805i	$0.687880\pi$
$4$	0	0
$5$	−6.11125	−0.546606	−0.273303	−	0.961928i	$-0.588116\pi$
$5$	−6.11125	−0.546606	−0.273303	+	0.961928i	$0.588116\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	6.45437	0.239051
$10$	0	0
$11$	64.7694	1.77534	0.887669	−	0.460482i	$-0.152324\pi$
$11$	64.7694	1.77534	0.887669	+	0.460482i	$0.152324\pi$
$12$	0	0
$13$	−83.7744	−1.78730	−0.893648	−	0.448769i	$-0.851863\pi$
$13$	−83.7744	−1.78730	−0.893648	+	0.448769i	$0.851863\pi$
$14$	0	0
$15$	35.3473	0.608442
$16$	0	0
$17$	−87.3046	−1.24556	−0.622779	−	0.782398i	$-0.713996\pi$
$17$	−87.3046	−1.24556	−0.622779	+	0.782398i	$0.713996\pi$
$18$	0	0
$19$	130.970	1.58140	0.790702	−	0.612201i	$-0.209716\pi$
$19$	130.970	1.58140	0.790702	+	0.612201i	$0.209716\pi$
$20$	0	0
$21$	−40.4878	−0.420722
$22$	0	0
$23$	−48.6690	−0.441225	−0.220613	−	0.975362i	$-0.570806\pi$
$23$	−48.6690	−0.441225	−0.220613	+	0.975362i	$0.570806\pi$
$24$	0	0
$25$	−87.6527	−0.701221
$26$	0	0
$27$	118.835	0.847033
$28$	0	0
$29$	18.8616	0.120776	0.0603881	−	0.998175i	$-0.480766\pi$
$29$	18.8616	0.120776	0.0603881	+	0.998175i	$0.480766\pi$
$30$	0	0
$31$	−2.57883	−0.0149410	−0.00747051	−	0.999972i	$-0.502378\pi$
$31$	−2.57883	−0.0149410	−0.00747051	+	0.999972i	$0.502378\pi$
$32$	0	0
$33$	−374.625	−1.97618
$34$	0	0
$35$	−42.7787	−0.206598
$36$	0	0
$37$	434.839	1.93208	0.966041	−	0.258387i	$-0.0831910\pi$
$37$	434.839	1.93208	0.966041	+	0.258387i	$0.0831910\pi$
$38$	0	0
$39$	484.549	1.98949
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	195.702	0.694055	0.347027	−	0.937855i	$-0.387191\pi$
$43$	195.702	0.694055	0.347027	+	0.937855i	$0.387191\pi$
$44$	0	0
$45$	−39.4443	−0.130667
$46$	0	0
$47$	189.617	0.588479	0.294239	−	0.955732i	$-0.404934\pi$
$47$	189.617	0.588479	0.294239	+	0.955732i	$0.404934\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	504.968	1.38646
$52$	0	0
$53$	−60.3919	−0.156518	−0.0782591	−	0.996933i	$-0.524936\pi$
$53$	−60.3919	−0.156518	−0.0782591	+	0.996933i	$0.524936\pi$
$54$	0	0
$55$	−395.822	−0.970411
$56$	0	0
$57$	−757.530	−1.76030
$58$	0	0
$59$	286.148	0.631413	0.315706	−	0.948857i	$-0.397759\pi$
$59$	286.148	0.631413	0.315706	+	0.948857i	$0.397759\pi$
$60$	0	0
$61$	−921.385	−1.93395	−0.966977	−	0.254863i	$-0.917969\pi$
$61$	−921.385	−1.93395	−0.966977	+	0.254863i	$0.917969\pi$
$62$	0	0
$63$	45.1806	0.0903527
$64$	0	0
$65$	511.966	0.976947
$66$	0	0
$67$	−473.140	−0.862734	−0.431367	−	0.902176i	$-0.641969\pi$
$67$	−473.140	−0.862734	−0.431367	+	0.902176i	$0.641969\pi$
$68$	0	0
$69$	281.500	0.491139
$70$	0	0
$71$	1139.15	1.90411	0.952056	−	0.305923i	$-0.0989649\pi$
$71$	1139.15	1.90411	0.952056	+	0.305923i	$0.0989649\pi$
$72$	0	0
$73$	1121.40	1.79794	0.898972	−	0.438007i	$-0.144315\pi$
$73$	1121.40	1.79794	0.898972	+	0.438007i	$0.144315\pi$
$74$	0	0
$75$	506.981	0.780548
$76$	0	0
$77$	453.386	0.671015
$78$	0	0
$79$	−99.9330	−0.142321	−0.0711604	−	0.997465i	$-0.522670\pi$
$79$	−99.9330	−0.142321	−0.0711604	+	0.997465i	$0.522670\pi$
$80$	0	0
$81$	−861.609	−1.18191
$82$	0	0
$83$	−481.637	−0.636946	−0.318473	−	0.947932i	$-0.603170\pi$
$83$	−481.637	−0.636946	−0.318473	+	0.947932i	$0.603170\pi$
$84$	0	0
$85$	533.540	0.680830
$86$	0	0
$87$	−109.095	−0.134439
$88$	0	0
$89$	−807.624	−0.961887	−0.480944	−	0.876752i	$-0.659706\pi$
$89$	−807.624	−0.961887	−0.480944	+	0.876752i	$0.659706\pi$
$90$	0	0
$91$	−586.421	−0.675534
$92$	0	0
$93$	14.9159	0.0166313
$94$	0	0
$95$	−800.393	−0.864406
$96$	0	0
$97$	−1512.59	−1.58330	−0.791650	−	0.610975i	$-0.790777\pi$
$97$	−1512.59	−1.58330	−0.791650	+	0.610975i	$0.790777\pi$
$98$	0	0
$99$	418.046	0.424396
$100$	0	0
$101$	−1111.02	−1.09456	−0.547280	−	0.836950i	$-0.684337\pi$
$101$	−1111.02	−1.09456	−0.547280	+	0.836950i	$0.684337\pi$
$102$	0	0
$103$	−524.090	−0.501360	−0.250680	−	0.968070i	$-0.580654\pi$
$103$	−524.090	−0.501360	−0.250680	+	0.968070i	$0.580654\pi$
$104$	0	0
$105$	247.431	0.229970
$106$	0	0
$107$	−901.155	−0.814187	−0.407093	−	0.913387i	$-0.633458\pi$
$107$	−901.155	−0.814187	−0.407093	+	0.913387i	$0.633458\pi$
$108$	0	0
$109$	280.501	0.246488	0.123244	−	0.992376i	$-0.460670\pi$
$109$	280.501	0.246488	0.123244	+	0.992376i	$0.460670\pi$
$110$	0	0
$111$	−2515.10	−2.15065
$112$	0	0
$113$	−807.124	−0.671928	−0.335964	−	0.941875i	$-0.609062\pi$
$113$	−807.124	−0.671928	−0.335964	+	0.941875i	$0.609062\pi$
$114$	0	0
$115$	297.428	0.241177
$116$	0	0
$117$	−540.711	−0.427254
$118$	0	0
$119$	−611.133	−0.470777
$120$	0	0
$121$	2864.08	2.15182
$122$	0	0
$123$	−237.143	−0.173841
$124$	0	0
$125$	1299.57	0.929899
$126$	0	0
$127$	−2547.94	−1.78026	−0.890130	−	0.455706i	$-0.849387\pi$
$127$	−2547.94	−1.78026	−0.890130	+	0.455706i	$0.849387\pi$
$128$	0	0
$129$	−1131.94	−0.772571
$130$	0	0
$131$	−817.395	−0.545162	−0.272581	−	0.962133i	$-0.587877\pi$
$131$	−817.395	−0.545162	−0.272581	+	0.962133i	$0.587877\pi$
$132$	0	0
$133$	916.793	0.597715
$134$	0	0
$135$	−726.232	−0.462994
$136$	0	0
$137$	−1340.95	−0.836239	−0.418119	−	0.908392i	$-0.637311\pi$
$137$	−1340.95	−0.836239	−0.418119	+	0.908392i	$0.637311\pi$
$138$	0	0
$139$	1225.72	0.747944	0.373972	−	0.927440i	$-0.377996\pi$
$139$	1225.72	0.747944	0.373972	+	0.927440i	$0.377996\pi$
$140$	0	0
$141$	−1096.74	−0.655052
$142$	0	0
$143$	−5426.02	−3.17305
$144$	0	0
$145$	−115.268	−0.0660171
$146$	0	0
$147$	−283.415	−0.159018
$148$	0	0
$149$	3570.70	1.96324	0.981622	−	0.190834i	$-0.0611191\pi$
$149$	3570.70	1.96324	0.981622	+	0.190834i	$0.0611191\pi$
$150$	0	0
$151$	−126.421	−0.0681322	−0.0340661	−	0.999420i	$-0.510846\pi$
$151$	−126.421	−0.0681322	−0.0340661	+	0.999420i	$0.510846\pi$
$152$	0	0
$153$	−563.497	−0.297752
$154$	0	0
$155$	15.7599	0.00816686
$156$	0	0
$157$	−2138.03	−1.08684	−0.543418	−	0.839462i	$-0.682870\pi$
$157$	−2138.03	−1.08684	−0.543418	+	0.839462i	$0.682870\pi$
$158$	0	0
$159$	349.305	0.174225
$160$	0	0
$161$	−340.683	−0.166767
$162$	0	0
$163$	1114.97	0.535776	0.267888	−	0.963450i	$-0.413674\pi$
$163$	1114.97	0.535776	0.267888	+	0.963450i	$0.413674\pi$
$164$	0	0
$165$	2289.42	1.08019
$166$	0	0
$167$	−2379.91	−1.10277	−0.551385	−	0.834251i	$-0.685900\pi$
$167$	−2379.91	−1.10277	−0.551385	+	0.834251i	$0.685900\pi$
$168$	0	0
$169$	4821.15	2.19443
$170$	0	0
$171$	845.332	0.378036
$172$	0	0
$173$	−3519.04	−1.54652	−0.773260	−	0.634089i	$-0.781375\pi$
$173$	−3519.04	−1.54652	−0.773260	+	0.634089i	$0.781375\pi$
$174$	0	0
$175$	−613.569	−0.265037
$176$	0	0
$177$	−1655.08	−0.702842
$178$	0	0
$179$	−1140.89	−0.476393	−0.238197	−	0.971217i	$-0.576556\pi$
$179$	−1140.89	−0.476393	−0.238197	+	0.971217i	$0.576556\pi$
$180$	0	0
$181$	−2563.99	−1.05293	−0.526463	−	0.850198i	$-0.676482\pi$
$181$	−2563.99	−1.05293	−0.526463	+	0.850198i	$0.676482\pi$
$182$	0	0
$183$	5329.27	2.15274
$184$	0	0
$185$	−2657.41	−1.05609
$186$	0	0
$187$	−5654.67	−2.21129
$188$	0	0
$189$	831.848	0.320148
$190$	0	0
$191$	−712.672	−0.269985	−0.134992	−	0.990847i	$-0.543101\pi$
$191$	−712.672	−0.269985	−0.134992	+	0.990847i	$0.543101\pi$
$192$	0	0
$193$	1818.13	0.678094	0.339047	−	0.940769i	$-0.389895\pi$
$193$	1818.13	0.678094	0.339047	+	0.940769i	$0.389895\pi$
$194$	0	0
$195$	−2961.20	−1.08747
$196$	0	0
$197$	−3744.37	−1.35419	−0.677095	−	0.735896i	$-0.736761\pi$
$197$	−3744.37	−1.35419	−0.677095	+	0.735896i	$0.736761\pi$
$198$	0	0
$199$	1022.89	0.364374	0.182187	−	0.983264i	$-0.441682\pi$
$199$	1022.89	0.364374	0.182187	+	0.983264i	$0.441682\pi$
$200$	0	0
$201$	2736.63	0.960332
$202$	0	0
$203$	132.031	0.0456491
$204$	0	0
$205$	−250.561	−0.0853656
$206$	0	0
$207$	−314.128	−0.105475
$208$	0	0
$209$	8482.88	2.80753
$210$	0	0
$211$	−3053.44	−0.996244	−0.498122	−	0.867107i	$-0.665977\pi$
$211$	−3053.44	−0.996244	−0.498122	+	0.867107i	$0.665977\pi$
$212$	0	0
$213$	−6588.80	−2.11952
$214$	0	0
$215$	−1195.99	−0.379375
$216$	0	0
$217$	−18.0518	−0.00564718
$218$	0	0
$219$	−6486.15	−2.00134
$220$	0	0
$221$	7313.90	2.22618
$222$	0	0
$223$	5675.36	1.70426	0.852131	−	0.523329i	$-0.175310\pi$
$223$	5675.36	1.70426	0.852131	+	0.523329i	$0.175310\pi$
$224$	0	0
$225$	−565.743	−0.167628
$226$	0	0
$227$	5802.17	1.69649	0.848245	−	0.529603i	$-0.177659\pi$
$227$	5802.17	1.69649	0.848245	+	0.529603i	$0.177659\pi$
$228$	0	0
$229$	−4873.68	−1.40638	−0.703191	−	0.711001i	$-0.748242\pi$
$229$	−4873.68	−1.40638	−0.703191	+	0.711001i	$0.748242\pi$
$230$	0	0
$231$	−2622.37	−0.746924
$232$	0	0
$233$	−3217.10	−0.904546	−0.452273	−	0.891880i	$-0.649387\pi$
$233$	−3217.10	−0.904546	−0.452273	+	0.891880i	$0.649387\pi$
$234$	0	0
$235$	−1158.80	−0.321666
$236$	0	0
$237$	578.010	0.158421
$238$	0	0
$239$	−4315.01	−1.16784	−0.583922	−	0.811810i	$-0.698483\pi$
$239$	−4315.01	−1.16784	−0.583922	+	0.811810i	$0.698483\pi$
$240$	0	0
$241$	−1553.68	−0.415275	−0.207638	−	0.978206i	$-0.566577\pi$
$241$	−1553.68	−0.415275	−0.207638	+	0.978206i	$0.566577\pi$
$242$	0	0
$243$	1774.97	0.468578
$244$	0	0
$245$	−299.451	−0.0780866
$246$	0	0
$247$	−10972.0	−2.82644
$248$	0	0
$249$	2785.78	0.709001
$250$	0	0
$251$	−3589.56	−0.902674	−0.451337	−	0.892354i	$-0.649053\pi$
$251$	−3589.56	−0.902674	−0.451337	+	0.892354i	$0.649053\pi$
$252$	0	0
$253$	−3152.26	−0.783324
$254$	0	0
$255$	−3085.98	−0.757850
$256$	0	0
$257$	1202.26	0.291810	0.145905	−	0.989299i	$-0.453391\pi$
$257$	1202.26	0.291810	0.145905	+	0.989299i	$0.453391\pi$
$258$	0	0
$259$	3043.87	0.730259
$260$	0	0
$261$	121.740	0.0288717
$262$	0	0
$263$	76.5966	0.0179587	0.00897937	−	0.999960i	$-0.497142\pi$
$263$	76.5966	0.0179587	0.00897937	+	0.999960i	$0.497142\pi$
$264$	0	0
$265$	369.070	0.0855538
$266$	0	0
$267$	4671.28	1.07070
$268$	0	0
$269$	−5175.70	−1.17311	−0.586557	−	0.809908i	$-0.699517\pi$
$269$	−5175.70	−1.17311	−0.586557	+	0.809908i	$0.699517\pi$
$270$	0	0
$271$	−3001.38	−0.672772	−0.336386	−	0.941724i	$-0.609205\pi$
$271$	−3001.38	−0.672772	−0.336386	+	0.941724i	$0.609205\pi$
$272$	0	0
$273$	3391.84	0.751955
$274$	0	0
$275$	−5677.21	−1.24490
$276$	0	0
$277$	−7602.76	−1.64912	−0.824559	−	0.565776i	$-0.808577\pi$
$277$	−7602.76	−1.64912	−0.824559	+	0.565776i	$0.808577\pi$
$278$	0	0
$279$	−16.6447	−0.00357166
$280$	0	0
$281$	2915.76	0.619002	0.309501	−	0.950899i	$-0.399838\pi$
$281$	2915.76	0.619002	0.309501	+	0.950899i	$0.399838\pi$
$282$	0	0
$283$	5756.80	1.20921	0.604605	−	0.796526i	$-0.293331\pi$
$283$	5756.80	1.20921	0.604605	+	0.796526i	$0.293331\pi$
$284$	0	0
$285$	4629.45	0.962193
$286$	0	0
$287$	287.000	0.0590281
$288$	0	0
$289$	2709.10	0.551415
$290$	0	0
$291$	8748.77	1.76241
$292$	0	0
$293$	−1309.93	−0.261185	−0.130593	−	0.991436i	$-0.541688\pi$
$293$	−1309.93	−0.261185	−0.130593	+	0.991436i	$0.541688\pi$
$294$	0	0
$295$	−1748.72	−0.345134
$296$	0	0
$297$	7696.90	1.50377
$298$	0	0
$299$	4077.21	0.788600
$300$	0	0
$301$	1369.92	0.262328
$302$	0	0
$303$	6426.11	1.21838
$304$	0	0
$305$	5630.81	1.05711
$306$	0	0
$307$	4996.64	0.928903	0.464452	−	0.885599i	$-0.346251\pi$
$307$	4996.64	0.928903	0.464452	+	0.885599i	$0.346251\pi$
$308$	0	0
$309$	3031.32	0.558077
$310$	0	0
$311$	9462.76	1.72535	0.862675	−	0.505758i	$-0.168787\pi$
$311$	9462.76	1.72535	0.862675	+	0.505758i	$0.168787\pi$
$312$	0	0
$313$	−7648.38	−1.38119	−0.690594	−	0.723243i	$-0.742651\pi$
$313$	−7648.38	−1.38119	−0.690594	+	0.723243i	$0.742651\pi$
$314$	0	0
$315$	−276.110	−0.0493874
$316$	0	0
$317$	−2770.19	−0.490819	−0.245409	−	0.969420i	$-0.578922\pi$
$317$	−2770.19	−0.490819	−0.245409	+	0.969420i	$0.578922\pi$
$318$	0	0
$319$	1221.66	0.214419
$320$	0	0
$321$	5212.26	0.906293
$322$	0	0
$323$	−11434.3	−1.96973
$324$	0	0
$325$	7343.05	1.25329
$326$	0	0
$327$	−1622.41	−0.274372
$328$	0	0
$329$	1327.32	0.222424
$330$	0	0
$331$	1.01805	0.000169054 0	8.45269e−5	−	1.00000i	$-0.499973\pi$
$331$	1.01805	0.000169054 0	8.45269e−5		1.00000i	$0.499973\pi$
$332$	0	0
$333$	2806.61	0.461866
$334$	0	0
$335$	2891.47	0.471576
$336$	0	0
$337$	−1976.68	−0.319515	−0.159757	−	0.987156i	$-0.551071\pi$
$337$	−1976.68	−0.319515	−0.159757	+	0.987156i	$0.551071\pi$
$338$	0	0
$339$	4668.39	0.747941
$340$	0	0
$341$	−167.029	−0.0265254
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−1720.32	−0.268460
$346$	0	0
$347$	−5085.19	−0.786707	−0.393353	−	0.919387i	$-0.628685\pi$
$347$	−5085.19	−0.786707	−0.393353	+	0.919387i	$0.628685\pi$
$348$	0	0
$349$	−10785.3	−1.65422	−0.827110	−	0.562041i	$-0.810016\pi$
$349$	−10785.3	−1.65422	−0.827110	+	0.562041i	$0.810016\pi$
$350$	0	0
$351$	−9955.37	−1.51390
$352$	0	0
$353$	9020.52	1.36010	0.680048	−	0.733167i	$-0.261959\pi$
$353$	9020.52	1.36010	0.680048	+	0.733167i	$0.261959\pi$
$354$	0	0
$355$	−6961.61	−1.04080
$356$	0	0
$357$	3534.78	0.524034
$358$	0	0
$359$	10528.9	1.54790	0.773950	−	0.633247i	$-0.218278\pi$
$359$	10528.9	1.54790	0.773950	+	0.633247i	$0.218278\pi$
$360$	0	0
$361$	10294.3	1.50084
$362$	0	0
$363$	−16565.8	−2.39525
$364$	0	0
$365$	−6853.15	−0.982767
$366$	0	0
$367$	4987.94	0.709451	0.354726	−	0.934970i	$-0.384574\pi$
$367$	4987.94	0.709451	0.354726	+	0.934970i	$0.384574\pi$
$368$	0	0
$369$	264.629	0.0373335
$370$	0	0
$371$	−422.743	−0.0591583
$372$	0	0
$373$	−6578.78	−0.913234	−0.456617	−	0.889663i	$-0.650939\pi$
$373$	−6578.78	−0.913234	−0.456617	+	0.889663i	$0.650939\pi$
$374$	0	0
$375$	−7516.70	−1.03509
$376$	0	0
$377$	−1580.12	−0.215863
$378$	0	0
$379$	−2012.29	−0.272730	−0.136365	−	0.990659i	$-0.543542\pi$
$379$	−2012.29	−0.272730	−0.136365	+	0.990659i	$0.543542\pi$
$380$	0	0
$381$	14737.2	1.98166
$382$	0	0
$383$	−8255.15	−1.10135	−0.550677	−	0.834718i	$-0.685630\pi$
$383$	−8255.15	−1.10135	−0.550677	+	0.834718i	$0.685630\pi$
$384$	0	0
$385$	−2770.75	−0.366781
$386$	0	0
$387$	1263.14	0.165914
$388$	0	0
$389$	4847.96	0.631880	0.315940	−	0.948779i	$-0.397680\pi$
$389$	4847.96	0.631880	0.315940	+	0.948779i	$0.397680\pi$
$390$	0	0
$391$	4249.03	0.549572
$392$	0	0
$393$	4727.79	0.606834
$394$	0	0
$395$	610.715	0.0777934
$396$	0	0
$397$	−9809.55	−1.24012	−0.620059	−	0.784555i	$-0.712891\pi$
$397$	−9809.55	−1.24012	−0.620059	+	0.784555i	$0.712891\pi$
$398$	0	0
$399$	−5302.71	−0.665332
$400$	0	0
$401$	−5716.85	−0.711935	−0.355967	−	0.934498i	$-0.615849\pi$
$401$	−5716.85	−0.711935	−0.355967	+	0.934498i	$0.615849\pi$
$402$	0	0
$403$	216.040	0.0267040
$404$	0	0
$405$	5265.51	0.646037
$406$	0	0
$407$	28164.3	3.43010
$408$	0	0
$409$	3741.51	0.452337	0.226169	−	0.974088i	$-0.427380\pi$
$409$	3741.51	0.452337	0.226169	+	0.974088i	$0.427380\pi$
$410$	0	0
$411$	7756.00	0.930840
$412$	0	0
$413$	2003.04	0.238652
$414$	0	0
$415$	2943.40	0.348159
$416$	0	0
$417$	−7089.53	−0.832556
$418$	0	0
$419$	−9447.82	−1.10157	−0.550783	−	0.834648i	$-0.685671\pi$
$419$	−9447.82	−1.10157	−0.550783	+	0.834648i	$0.685671\pi$
$420$	0	0
$421$	8670.08	1.00369	0.501845	−	0.864957i	$-0.332655\pi$
$421$	8670.08	1.00369	0.501845	+	0.864957i	$0.332655\pi$
$422$	0	0
$423$	1223.86	0.140676
$424$	0	0
$425$	7652.49	0.873412
$426$	0	0
$427$	−6449.69	−0.730966
$428$	0	0
$429$	31384.0	3.53201
$430$	0	0
$431$	15960.6	1.78374	0.891871	−	0.452289i	$-0.149393\pi$
$431$	15960.6	1.78374	0.891871	+	0.452289i	$0.149393\pi$
$432$	0	0
$433$	8550.20	0.948952	0.474476	−	0.880268i	$-0.342637\pi$
$433$	8550.20	0.948952	0.474476	+	0.880268i	$0.342637\pi$
$434$	0	0
$435$	666.707	0.0734854
$436$	0	0
$437$	−6374.19	−0.697755
$438$	0	0
$439$	−14622.0	−1.58968	−0.794839	−	0.606821i	$-0.792445\pi$
$439$	−14622.0	−1.58968	−0.794839	+	0.606821i	$0.792445\pi$
$440$	0	0
$441$	316.264	0.0341501
$442$	0	0
$443$	691.155	0.0741259	0.0370629	−	0.999313i	$-0.488200\pi$
$443$	691.155	0.0741259	0.0370629	+	0.999313i	$0.488200\pi$
$444$	0	0
$445$	4935.59	0.525774
$446$	0	0
$447$	−20652.9	−2.18534
$448$	0	0
$449$	1659.10	0.174383	0.0871915	−	0.996192i	$-0.472211\pi$
$449$	1659.10	0.174383	0.0871915	+	0.996192i	$0.472211\pi$
$450$	0	0
$451$	2655.55	0.277261
$452$	0	0
$453$	731.213	0.0758397
$454$	0	0
$455$	3583.76	0.369251
$456$	0	0
$457$	−9655.48	−0.988325	−0.494162	−	0.869370i	$-0.664525\pi$
$457$	−9655.48	−0.988325	−0.494162	+	0.869370i	$0.664525\pi$
$458$	0	0
$459$	−10374.9	−1.05503
$460$	0	0
$461$	1480.43	0.149567	0.0747837	−	0.997200i	$-0.476173\pi$
$461$	1480.43	0.149567	0.0747837	+	0.997200i	$0.476173\pi$
$462$	0	0
$463$	−13645.0	−1.36963	−0.684814	−	0.728718i	$-0.740117\pi$
$463$	−13645.0	−1.36963	−0.684814	+	0.728718i	$0.740117\pi$
$464$	0	0
$465$	−91.1547	−0.00909075
$466$	0	0
$467$	−3768.76	−0.373442	−0.186721	−	0.982413i	$-0.559786\pi$
$467$	−3768.76	−0.373442	−0.186721	+	0.982413i	$0.559786\pi$
$468$	0	0
$469$	−3311.98	−0.326083
$470$	0	0
$471$	12366.3	1.20979
$472$	0	0
$473$	12675.5	1.23218
$474$	0	0
$475$	−11479.9	−1.10891
$476$	0	0
$477$	−389.792	−0.0374158
$478$	0	0
$479$	2272.94	0.216813	0.108406	−	0.994107i	$-0.465425\pi$
$479$	2272.94	0.216813	0.108406	+	0.994107i	$0.465425\pi$
$480$	0	0
$481$	−36428.4	−3.45320
$482$	0	0
$483$	1970.50	0.185633
$484$	0	0
$485$	9243.80	0.865442
$486$	0	0
$487$	10301.9	0.958566	0.479283	−	0.877661i	$-0.340897\pi$
$487$	10301.9	0.958566	0.479283	+	0.877661i	$0.340897\pi$
$488$	0	0
$489$	−6448.98	−0.596386
$490$	0	0
$491$	15806.0	1.45278	0.726388	−	0.687285i	$-0.241198\pi$
$491$	15806.0	1.45278	0.726388	+	0.687285i	$0.241198\pi$
$492$	0	0
$493$	−1646.71	−0.150434
$494$	0	0
$495$	−2554.78	−0.231978
$496$	0	0
$497$	7974.04	0.719687
$498$	0	0
$499$	−6332.33	−0.568084	−0.284042	−	0.958812i	$-0.591676\pi$
$499$	−6332.33	−0.568084	−0.284042	+	0.958812i	$0.591676\pi$
$500$	0	0
$501$	13765.3	1.22752
$502$	0	0
$503$	8971.64	0.795280	0.397640	−	0.917542i	$-0.369829\pi$
$503$	8971.64	0.795280	0.397640	+	0.917542i	$0.369829\pi$
$504$	0	0
$505$	6789.71	0.598293
$506$	0	0
$507$	−27885.4	−2.44267
$508$	0	0
$509$	16728.5	1.45674	0.728368	−	0.685186i	$-0.240279\pi$
$509$	16728.5	1.45674	0.728368	+	0.685186i	$0.240279\pi$
$510$	0	0
$511$	7849.80	0.679559
$512$	0	0
$513$	15563.9	1.33950
$514$	0	0
$515$	3202.84	0.274047
$516$	0	0
$517$	12281.4	1.04475
$518$	0	0
$519$	20354.1	1.72147
$520$	0	0
$521$	−2457.36	−0.206639	−0.103319	−	0.994648i	$-0.532946\pi$
$521$	−2457.36	−0.206639	−0.103319	+	0.994648i	$0.532946\pi$
$522$	0	0
$523$	−1460.92	−0.122145	−0.0610723	−	0.998133i	$-0.519452\pi$
$523$	−1460.92	−0.122145	−0.0610723	+	0.998133i	$0.519452\pi$
$524$	0	0
$525$	3548.87	0.295019
$526$	0	0
$527$	225.144	0.0186099
$528$	0	0
$529$	−9798.33	−0.805320
$530$	0	0
$531$	1846.91	0.150940
$532$	0	0
$533$	−3434.75	−0.279129
$534$	0	0
$535$	5507.18	0.445040
$536$	0	0
$537$	6598.90	0.530286
$538$	0	0
$539$	3173.70	0.253620
$540$	0	0
$541$	−16929.1	−1.34536	−0.672680	−	0.739934i	$-0.734857\pi$
$541$	−16929.1	−1.34536	−0.672680	+	0.739934i	$0.734857\pi$
$542$	0	0
$543$	14830.0	1.17204
$544$	0	0
$545$	−1714.21	−0.134732
$546$	0	0
$547$	−5652.72	−0.441852	−0.220926	−	0.975291i	$-0.570908\pi$
$547$	−5652.72	−0.441852	−0.220926	+	0.975291i	$0.570908\pi$
$548$	0	0
$549$	−5946.96	−0.462313
$550$	0	0
$551$	2470.31	0.190996
$552$	0	0
$553$	−699.531	−0.0537922
$554$	0	0
$555$	15370.4	1.17556
$556$	0	0
$557$	−5605.10	−0.426384	−0.213192	−	0.977010i	$-0.568386\pi$
$557$	−5605.10	−0.426384	−0.213192	+	0.977010i	$0.568386\pi$
$558$	0	0
$559$	−16394.9	−1.24048
$560$	0	0
$561$	32706.5	2.46144
$562$	0	0
$563$	−26536.8	−1.98649	−0.993244	−	0.116045i	$-0.962978\pi$
$563$	−26536.8	−1.98649	−0.993244	+	0.116045i	$0.962978\pi$
$564$	0	0
$565$	4932.54	0.367280
$566$	0	0
$567$	−6031.26	−0.446718
$568$	0	0
$569$	10982.8	0.809180	0.404590	−	0.914498i	$-0.367414\pi$
$569$	10982.8	0.809180	0.404590	+	0.914498i	$0.367414\pi$
$570$	0	0
$571$	425.464	0.0311824	0.0155912	−	0.999878i	$-0.495037\pi$
$571$	425.464	0.0311824	0.0155912	+	0.999878i	$0.495037\pi$
$572$	0	0
$573$	4122.07	0.300527
$574$	0	0
$575$	4265.96	0.309397
$576$	0	0
$577$	12999.8	0.937932	0.468966	−	0.883216i	$-0.344627\pi$
$577$	12999.8	0.937932	0.468966	+	0.883216i	$0.344627\pi$
$578$	0	0
$579$	−10516.0	−0.754804
$580$	0	0
$581$	−3371.46	−0.240743
$582$	0	0
$583$	−3911.55	−0.277873
$584$	0	0
$585$	3304.42	0.233540
$586$	0	0
$587$	13680.8	0.961954	0.480977	−	0.876733i	$-0.340282\pi$
$587$	13680.8	0.961954	0.480977	+	0.876733i	$0.340282\pi$
$588$	0	0
$589$	−337.751	−0.0236278
$590$	0	0
$591$	21657.3	1.50738
$592$	0	0
$593$	2951.32	0.204378	0.102189	−	0.994765i	$-0.467415\pi$
$593$	2951.32	0.204378	0.102189	+	0.994765i	$0.467415\pi$
$594$	0	0
$595$	3734.78	0.257330
$596$	0	0
$597$	−5916.34	−0.405594
$598$	0	0
$599$	9138.32	0.623342	0.311671	−	0.950190i	$-0.399111\pi$
$599$	9138.32	0.623342	0.311671	+	0.950190i	$0.399111\pi$
$600$	0	0
$601$	−9123.53	−0.619229	−0.309615	−	0.950862i	$-0.600200\pi$
$601$	−9123.53	−0.619229	−0.309615	+	0.950862i	$0.600200\pi$
$602$	0	0
$603$	−3053.82	−0.206237
$604$	0	0
$605$	−17503.1	−1.17620
$606$	0	0
$607$	−10271.1	−0.686805	−0.343402	−	0.939188i	$-0.611579\pi$
$607$	−10271.1	−0.686805	−0.343402	+	0.939188i	$0.611579\pi$
$608$	0	0
$609$	−763.665	−0.0508133
$610$	0	0
$611$	−15885.1	−1.05179
$612$	0	0
$613$	−1717.99	−0.113196	−0.0565979	−	0.998397i	$-0.518025\pi$
$613$	−1717.99	−0.113196	−0.0565979	+	0.998397i	$0.518025\pi$
$614$	0	0
$615$	1449.24	0.0950227
$616$	0	0
$617$	−18301.9	−1.19417	−0.597087	−	0.802177i	$-0.703675\pi$
$617$	−18301.9	−1.19417	−0.597087	+	0.802177i	$0.703675\pi$
$618$	0	0
$619$	14528.1	0.943351	0.471675	−	0.881772i	$-0.343649\pi$
$619$	14528.1	0.943351	0.471675	+	0.881772i	$0.343649\pi$
$620$	0	0
$621$	−5783.60	−0.373732
$622$	0	0
$623$	−5653.37	−0.363559
$624$	0	0
$625$	3014.58	0.192933
$626$	0	0
$627$	−49064.8	−3.12513
$628$	0	0
$629$	−37963.5	−2.40652
$630$	0	0
$631$	−7561.51	−0.477051	−0.238525	−	0.971136i	$-0.576664\pi$
$631$	−7561.51	−0.477051	−0.238525	+	0.971136i	$0.576664\pi$
$632$	0	0
$633$	17661.0	1.10895
$634$	0	0
$635$	15571.1	0.973102
$636$	0	0
$637$	−4104.95	−0.255328
$638$	0	0
$639$	7352.49	0.455180
$640$	0	0
$641$	−10481.9	−0.645879	−0.322940	−	0.946420i	$-0.604671\pi$
$641$	−10481.9	−0.645879	−0.322940	+	0.946420i	$0.604671\pi$
$642$	0	0
$643$	18386.4	1.12766	0.563832	−	0.825889i	$-0.309327\pi$
$643$	18386.4	1.12766	0.563832	+	0.825889i	$0.309327\pi$
$644$	0	0
$645$	6917.55	0.422292
$646$	0	0
$647$	3377.74	0.205243	0.102622	−	0.994720i	$-0.467277\pi$
$647$	3377.74	0.205243	0.102622	+	0.994720i	$0.467277\pi$
$648$	0	0
$649$	18533.7	1.12097
$650$	0	0
$651$	104.411	0.00628602
$652$	0	0
$653$	32281.2	1.93455	0.967275	−	0.253731i	$-0.0816578\pi$
$653$	32281.2	1.93455	0.967275	+	0.253731i	$0.0816578\pi$
$654$	0	0
$655$	4995.30	0.297989
$656$	0	0
$657$	7237.93	0.429800
$658$	0	0
$659$	13591.7	0.803423	0.401712	−	0.915766i	$-0.368415\pi$
$659$	13591.7	0.803423	0.401712	+	0.915766i	$0.368415\pi$
$660$	0	0
$661$	2060.65	0.121256	0.0606278	−	0.998160i	$-0.480690\pi$
$661$	2060.65	0.121256	0.0606278	+	0.998160i	$0.480690\pi$
$662$	0	0
$663$	−42303.4	−2.47802
$664$	0	0
$665$	−5602.75	−0.326715
$666$	0	0
$667$	−917.975	−0.0532895
$668$	0	0
$669$	−32826.2	−1.89706
$670$	0	0
$671$	−59677.5	−3.43342
$672$	0	0
$673$	−8892.23	−0.509317	−0.254658	−	0.967031i	$-0.581963\pi$
$673$	−8892.23	−0.509317	−0.254658	+	0.967031i	$0.581963\pi$
$674$	0	0
$675$	−10416.2	−0.593958
$676$	0	0
$677$	−11729.4	−0.665874	−0.332937	−	0.942949i	$-0.608040\pi$
$677$	−11729.4	−0.665874	−0.332937	+	0.942949i	$0.608040\pi$
$678$	0	0
$679$	−10588.1	−0.598431
$680$	0	0
$681$	−33559.6	−1.88841
$682$	0	0
$683$	23232.9	1.30158	0.650791	−	0.759257i	$-0.274437\pi$
$683$	23232.9	1.30158	0.650791	+	0.759257i	$0.274437\pi$
$684$	0	0
$685$	8194.85	0.457094
$686$	0	0
$687$	28189.2	1.56548
$688$	0	0
$689$	5059.29	0.279744
$690$	0	0
$691$	−15774.1	−0.868418	−0.434209	−	0.900812i	$-0.642972\pi$
$691$	−15774.1	−0.868418	−0.434209	+	0.900812i	$0.642972\pi$
$692$	0	0
$693$	2926.32	0.160407
$694$	0	0
$695$	−7490.67	−0.408831
$696$	0	0
$697$	−3579.49	−0.194523
$698$	0	0
$699$	18607.6	1.00687
$700$	0	0
$701$	4104.53	0.221150	0.110575	−	0.993868i	$-0.464731\pi$
$701$	4104.53	0.221150	0.110575	+	0.993868i	$0.464731\pi$
$702$	0	0
$703$	56951.0	3.05540
$704$	0	0
$705$	6702.45	0.358055
$706$	0	0
$707$	−7777.13	−0.413705
$708$	0	0
$709$	−7196.43	−0.381196	−0.190598	−	0.981668i	$-0.561043\pi$
$709$	−7196.43	−0.381196	−0.190598	+	0.981668i	$0.561043\pi$
$710$	0	0
$711$	−645.004	−0.0340219
$712$	0	0
$713$	125.509	0.00659236
$714$	0	0
$715$	33159.7	1.73441
$716$	0	0
$717$	24957.9	1.29996
$718$	0	0
$719$	−1087.74	−0.0564197	−0.0282099	−	0.999602i	$-0.508981\pi$
$719$	−1087.74	−0.0564197	−0.0282099	+	0.999602i	$0.508981\pi$
$720$	0	0
$721$	−3668.63	−0.189496
$722$	0	0
$723$	8986.45	0.462254
$724$	0	0
$725$	−1653.27	−0.0846909
$726$	0	0
$727$	−1705.32	−0.0869972	−0.0434986	−	0.999053i	$-0.513850\pi$
$727$	−1705.32	−0.0869972	−0.0434986	+	0.999053i	$0.513850\pi$
$728$	0	0
$729$	12997.1	0.660319
$730$	0	0
$731$	−17085.7	−0.864485
$732$	0	0
$733$	−16964.8	−0.854855	−0.427428	−	0.904050i	$-0.640580\pi$
$733$	−16964.8	−0.854855	−0.427428	+	0.904050i	$0.640580\pi$
$734$	0	0
$735$	1732.02	0.0869203
$736$	0	0
$737$	−30645.0	−1.53164
$738$	0	0
$739$	−4266.27	−0.212364	−0.106182	−	0.994347i	$-0.533863\pi$
$739$	−4266.27	−0.212364	−0.106182	+	0.994347i	$0.533863\pi$
$740$	0	0
$741$	63461.6	3.14618
$742$	0	0
$743$	4476.25	0.221020	0.110510	−	0.993875i	$-0.464752\pi$
$743$	4476.25	0.221020	0.110510	+	0.993875i	$0.464752\pi$
$744$	0	0
$745$	−21821.4	−1.07312
$746$	0	0
$747$	−3108.66	−0.152262
$748$	0	0
$749$	−6308.09	−0.307734
$750$	0	0
$751$	5206.88	0.252998	0.126499	−	0.991967i	$-0.459626\pi$
$751$	5206.88	0.252998	0.126499	+	0.991967i	$0.459626\pi$
$752$	0	0
$753$	20761.9	1.00479
$754$	0	0
$755$	772.587	0.0372415
$756$	0	0
$757$	−34675.8	−1.66488	−0.832439	−	0.554117i	$-0.813056\pi$
$757$	−34675.8	−1.66488	−0.832439	+	0.554117i	$0.813056\pi$
$758$	0	0
$759$	18232.6	0.871938
$760$	0	0
$761$	−10553.7	−0.502722	−0.251361	−	0.967893i	$-0.580878\pi$
$761$	−10553.7	−0.502722	−0.251361	+	0.967893i	$0.580878\pi$
$762$	0	0
$763$	1963.51	0.0931635
$764$	0	0
$765$	3443.67	0.162753
$766$	0	0
$767$	−23971.9	−1.12852
$768$	0	0
$769$	24497.6	1.14877	0.574386	−	0.818585i	$-0.305241\pi$
$769$	24497.6	1.14877	0.574386	+	0.818585i	$0.305241\pi$
$770$	0	0
$771$	−6953.86	−0.324821
$772$	0	0
$773$	−6158.49	−0.286553	−0.143276	−	0.989683i	$-0.545764\pi$
$773$	−6158.49	−0.286553	−0.143276	+	0.989683i	$0.545764\pi$
$774$	0	0
$775$	226.041	0.0104770
$776$	0	0
$777$	−17605.7	−0.812870
$778$	0	0
$779$	5369.79	0.246974
$780$	0	0
$781$	73782.0	3.38044
$782$	0	0
$783$	2241.43	0.102301
$784$	0	0
$785$	13066.0	0.594072
$786$	0	0
$787$	−1916.92	−0.0868245	−0.0434122	−	0.999057i	$-0.513823\pi$
$787$	−1916.92	−0.0868245	−0.0434122	+	0.999057i	$0.513823\pi$
$788$	0	0
$789$	−443.033	−0.0199903
$790$	0	0
$791$	−5649.87	−0.253965
$792$	0	0
$793$	77188.5	3.45655
$794$	0	0
$795$	−2134.69	−0.0952322
$796$	0	0
$797$	−38720.9	−1.72091	−0.860454	−	0.509529i	$-0.829820\pi$
$797$	−38720.9	−1.72091	−0.860454	+	0.509529i	$0.829820\pi$
$798$	0	0
$799$	−16554.5	−0.732985
$800$	0	0
$801$	−5212.71	−0.229940
$802$	0	0
$803$	72632.4	3.19196
$804$	0	0
$805$	2082.00	0.0911562
$806$	0	0
$807$	29936.1	1.30582
$808$	0	0
$809$	11868.7	0.515797	0.257899	−	0.966172i	$-0.416970\pi$
$809$	11868.7	0.515797	0.257899	+	0.966172i	$0.416970\pi$
$810$	0	0
$811$	−35944.7	−1.55634	−0.778168	−	0.628057i	$-0.783851\pi$
$811$	−35944.7	−1.55634	−0.778168	+	0.628057i	$0.783851\pi$
$812$	0	0
$813$	17359.9	0.748880
$814$	0	0
$815$	−6813.88	−0.292858
$816$	0	0
$817$	25631.2	1.09758
$818$	0	0
$819$	−3784.98	−0.161487
$820$	0	0
$821$	−9476.43	−0.402838	−0.201419	−	0.979505i	$-0.564555\pi$
$821$	−9476.43	−0.402838	−0.201419	+	0.979505i	$0.564555\pi$
$822$	0	0
$823$	−18223.8	−0.771862	−0.385931	−	0.922528i	$-0.626120\pi$
$823$	−18223.8	−0.771862	−0.385931	+	0.922528i	$0.626120\pi$
$824$	0	0
$825$	32836.9	1.38574
$826$	0	0
$827$	31101.8	1.30776	0.653878	−	0.756600i	$-0.273141\pi$
$827$	31101.8	1.30776	0.653878	+	0.756600i	$0.273141\pi$
$828$	0	0
$829$	4153.79	0.174025	0.0870127	−	0.996207i	$-0.472268\pi$
$829$	4153.79	0.174025	0.0870127	+	0.996207i	$0.472268\pi$
$830$	0	0
$831$	43974.2	1.83568
$832$	0	0
$833$	−4277.93	−0.177937
$834$	0	0
$835$	14544.2	0.602781
$836$	0	0
$837$	−306.456	−0.0126555
$838$	0	0
$839$	34695.0	1.42766	0.713830	−	0.700319i	$-0.246959\pi$
$839$	34695.0	1.42766	0.713830	+	0.700319i	$0.246959\pi$
$840$	0	0
$841$	−24033.2	−0.985413
$842$	0	0
$843$	−16864.7	−0.689027
$844$	0	0
$845$	−29463.3	−1.19949
$846$	0	0
$847$	20048.5	0.813313
$848$	0	0
$849$	−33297.2	−1.34600
$850$	0	0
$851$	−21163.2	−0.852484
$852$	0	0
$853$	5802.63	0.232917	0.116459	−	0.993196i	$-0.462846\pi$
$853$	5802.63	0.232917	0.116459	+	0.993196i	$0.462846\pi$
$854$	0	0
$855$	−5166.03	−0.206637
$856$	0	0
$857$	−21740.5	−0.866561	−0.433280	−	0.901259i	$-0.642644\pi$
$857$	−21740.5	−0.866561	−0.433280	+	0.901259i	$0.642644\pi$
$858$	0	0
$859$	−39614.4	−1.57349	−0.786743	−	0.617280i	$-0.788234\pi$
$859$	−39614.4	−1.57349	−0.786743	+	0.617280i	$0.788234\pi$
$860$	0	0
$861$	−1660.00	−0.0657058
$862$	0	0
$863$	−25741.9	−1.01537	−0.507684	−	0.861543i	$-0.669498\pi$
$863$	−25741.9	−1.01537	−0.507684	+	0.861543i	$0.669498\pi$
$864$	0	0
$865$	21505.7	0.845337
$866$	0	0
$867$	−15669.4	−0.613795
$868$	0	0
$869$	−6472.60	−0.252667
$870$	0	0
$871$	39637.0	1.54196
$872$	0	0
$873$	−9762.80	−0.378489
$874$	0	0
$875$	9097.01	0.351469
$876$	0	0
$877$	15106.5	0.581653	0.290826	−	0.956776i	$-0.406070\pi$
$877$	15106.5	0.581653	0.290826	+	0.956776i	$0.406070\pi$
$878$	0	0
$879$	7576.63	0.290732
$880$	0	0
$881$	−31908.3	−1.22022	−0.610112	−	0.792315i	$-0.708876\pi$
$881$	−31908.3	−1.22022	−0.610112	+	0.792315i	$0.708876\pi$
$882$	0	0
$883$	34413.6	1.31156	0.655782	−	0.754951i	$-0.272339\pi$
$883$	34413.6	1.31156	0.655782	+	0.754951i	$0.272339\pi$
$884$	0	0
$885$	10114.6	0.384178
$886$	0	0
$887$	−5373.04	−0.203392	−0.101696	−	0.994816i	$-0.532427\pi$
$887$	−5373.04	−0.203392	−0.101696	+	0.994816i	$0.532427\pi$
$888$	0	0
$889$	−17835.6	−0.672875
$890$	0	0
$891$	−55805.9	−2.09828
$892$	0	0
$893$	24834.2	0.930623
$894$	0	0
$895$	6972.28	0.260400
$896$	0	0
$897$	−23582.5	−0.877812
$898$	0	0
$899$	−48.6409	−0.00180452
$900$	0	0
$901$	5272.49	0.194952
$902$	0	0
$903$	−7923.57	−0.292004
$904$	0	0
$905$	15669.2	0.575536
$906$	0	0
$907$	−34203.4	−1.25215	−0.626077	−	0.779761i	$-0.715341\pi$
$907$	−34203.4	−1.25215	−0.626077	+	0.779761i	$0.715341\pi$
$908$	0	0
$909$	−7170.93	−0.261655
$910$	0	0
$911$	6076.08	0.220976	0.110488	−	0.993877i	$-0.464759\pi$
$911$	6076.08	0.220976	0.110488	+	0.993877i	$0.464759\pi$
$912$	0	0
$913$	−31195.3	−1.13079
$914$	0	0
$915$	−32568.5	−1.17670
$916$	0	0
$917$	−5721.77	−0.206052
$918$	0	0
$919$	−15517.4	−0.556987	−0.278493	−	0.960438i	$-0.589835\pi$
$919$	−15517.4	−0.556987	−0.278493	+	0.960438i	$0.589835\pi$
$920$	0	0
$921$	−28900.4	−1.03399
$922$	0	0
$923$	−95431.5	−3.40321
$924$	0	0
$925$	−38114.8	−1.35482
$926$	0	0
$927$	−3382.67	−0.119851
$928$	0	0
$929$	−46894.6	−1.65615	−0.828074	−	0.560620i	$-0.810563\pi$
$929$	−46894.6	−1.65615	−0.828074	+	0.560620i	$0.810563\pi$
$930$	0	0
$931$	6417.55	0.225915
$932$	0	0
$933$	−54732.4	−1.92053
$934$	0	0
$935$	34557.1	1.20870
$936$	0	0
$937$	27436.3	0.956570	0.478285	−	0.878205i	$-0.341259\pi$
$937$	27436.3	0.956570	0.478285	+	0.878205i	$0.341259\pi$
$938$	0	0
$939$	44238.0	1.53744
$940$	0	0
$941$	18392.0	0.637156	0.318578	−	0.947897i	$-0.396795\pi$
$941$	18392.0	0.637156	0.318578	+	0.947897i	$0.396795\pi$
$942$	0	0
$943$	−1995.43	−0.0689078
$944$	0	0
$945$	−5083.63	−0.174995
$946$	0	0
$947$	10975.3	0.376608	0.188304	−	0.982111i	$-0.439701\pi$
$947$	10975.3	0.376608	0.188304	+	0.982111i	$0.439701\pi$
$948$	0	0
$949$	−93944.6	−3.21346
$950$	0	0
$951$	16022.7	0.546344
$952$	0	0
$953$	−35500.6	−1.20669	−0.603347	−	0.797479i	$-0.706166\pi$
$953$	−35500.6	−1.20669	−0.603347	+	0.797479i	$0.706166\pi$
$954$	0	0
$955$	4355.31	0.147575
$956$	0	0
$957$	−7066.02	−0.238675
$958$	0	0
$959$	−9386.62	−0.316069
$960$	0	0
$961$	−29784.3	−0.999777
$962$	0	0
$963$	−5816.39	−0.194632
$964$	0	0
$965$	−11111.1	−0.370651
$966$	0	0
$967$	−25480.1	−0.847346	−0.423673	−	0.905815i	$-0.639259\pi$
$967$	−25480.1	−0.847346	−0.423673	+	0.905815i	$0.639259\pi$
$968$	0	0
$969$	66135.9	2.19256
$970$	0	0
$971$	−4842.39	−0.160041	−0.0800205	−	0.996793i	$-0.525499\pi$
$971$	−4842.39	−0.160041	−0.0800205	+	0.996793i	$0.525499\pi$
$972$	0	0
$973$	8580.04	0.282696
$974$	0	0
$975$	−42472.0	−1.39507
$976$	0	0
$977$	53060.4	1.73752	0.868758	−	0.495237i	$-0.164919\pi$
$977$	53060.4	1.73752	0.868758	+	0.495237i	$0.164919\pi$
$978$	0	0
$979$	−52309.3	−1.70767
$980$	0	0
$981$	1810.46	0.0589231
$982$	0	0
$983$	6066.51	0.196838	0.0984190	−	0.995145i	$-0.468621\pi$
$983$	6066.51	0.196838	0.0984190	+	0.995145i	$0.468621\pi$
$984$	0	0
$985$	22882.8	0.740209
$986$	0	0
$987$	−7677.19	−0.247586
$988$	0	0
$989$	−9524.63	−0.306234
$990$	0	0
$991$	−10024.7	−0.321336	−0.160668	−	0.987009i	$-0.551365\pi$
$991$	−10024.7	−0.321336	−0.160668	+	0.987009i	$0.551365\pi$
$992$	0	0
$993$	−5.88835	−0.000188178 0
$994$	0	0
$995$	−6251.10	−0.199169
$996$	0	0
$997$	37287.8	1.18447	0.592236	−	0.805765i	$-0.298245\pi$
$997$	37287.8	1.18447	0.592236	+	0.805765i	$0.298245\pi$
$998$	0	0
$999$	51674.3	1.63654

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.a.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.a.1.4	✓	15	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-5.78398 q^{3} -6.11125 q^{5} +7.00000 q^{7} +6.45437 q^{9} +O(q^{10})\)	\(q-5.78398 q^{3} -6.11125 q^{5} +7.00000 q^{7} +6.45437 q^{9} +64.7694 q^{11} -83.7744 q^{13} +35.3473 q^{15} -87.3046 q^{17} +130.970 q^{19} -40.4878 q^{21} -48.6690 q^{23} -87.6527 q^{25} +118.835 q^{27} +18.8616 q^{29} -2.57883 q^{31} -374.625 q^{33} -42.7787 q^{35} +434.839 q^{37} +484.549 q^{39} +41.0000 q^{41} +195.702 q^{43} -39.4443 q^{45} +189.617 q^{47} +49.0000 q^{49} +504.968 q^{51} -60.3919 q^{53} -395.822 q^{55} -757.530 q^{57} +286.148 q^{59} -921.385 q^{61} +45.1806 q^{63} +511.966 q^{65} -473.140 q^{67} +281.500 q^{69} +1139.15 q^{71} +1121.40 q^{73} +506.981 q^{75} +453.386 q^{77} -99.9330 q^{79} -861.609 q^{81} -481.637 q^{83} +533.540 q^{85} -109.095 q^{87} -807.624 q^{89} -586.421 q^{91} +14.9159 q^{93} -800.393 q^{95} -1512.59 q^{97} +418.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 14 q^{3} - 24 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) \) 15 * q - 14 * q^3 - 24 * q^5 + 105 * q^7 + 165 * q^9	\( 15 q - 14 q^{3} - 24 q^{5} + 105 q^{7} + 165 q^{9} - 64 q^{11} - 112 q^{13} - 228 q^{15} - 128 q^{17} - 240 q^{19} - 98 q^{21} - 302 q^{23} + 325 q^{25} - 410 q^{27} - 330 q^{29} - 432 q^{31} - 40 q^{33} - 168 q^{35} - 318 q^{37} - 454 q^{39} + 615 q^{41} - 882 q^{43} - 606 q^{45} + 520 q^{47} + 735 q^{49} + 1194 q^{51} - 304 q^{53} - 400 q^{55} + 26 q^{57} - 782 q^{59} - 710 q^{61} + 1155 q^{63} + 1344 q^{65} - 102 q^{67} - 1040 q^{69} + 1392 q^{71} - 366 q^{73} - 1134 q^{75} - 448 q^{77} - 1482 q^{79} + 951 q^{81} - 1920 q^{83} - 2496 q^{85} - 4764 q^{87} - 1404 q^{89} - 784 q^{91} - 3734 q^{93} - 226 q^{95} - 7366 q^{97} - 3724 q^{99}+O(q^{100}) \) 15 * q - 14 * q^3 - 24 * q^5 + 105 * q^7 + 165 * q^9 - 64 * q^11 - 112 * q^13 - 228 * q^15 - 128 * q^17 - 240 * q^19 - 98 * q^21 - 302 * q^23 + 325 * q^25 - 410 * q^27 - 330 * q^29 - 432 * q^31 - 40 * q^33 - 168 * q^35 - 318 * q^37 - 454 * q^39 + 615 * q^41 - 882 * q^43 - 606 * q^45 + 520 * q^47 + 735 * q^49 + 1194 * q^51 - 304 * q^53 - 400 * q^55 + 26 * q^57 - 782 * q^59 - 710 * q^61 + 1155 * q^63 + 1344 * q^65 - 102 * q^67 - 1040 * q^69 + 1392 * q^71 - 366 * q^73 - 1134 * q^75 - 448 * q^77 - 1482 * q^79 + 951 * q^81 - 1920 * q^83 - 2496 * q^85 - 4764 * q^87 - 1404 * q^89 - 784 * q^91 - 3734 * q^93 - 226 * q^95 - 7366 * q^97 - 3724 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more