LMFDB - Embedded newform 2116.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2116,4,Mod(1,2116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2116.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2116 = 2^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2116.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:	$124.848041572$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 92)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2116.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.94474 q^{3} +12.4605 q^{5} -26.2182 q^{7} +71.8979 q^{9} -14.4815 q^{11} -75.8882 q^{13} -123.917 q^{15} -47.8555 q^{17} +0.645912 q^{19} +260.733 q^{21} +30.2652 q^{25} -446.498 q^{27} -47.5434 q^{29} -253.352 q^{31} +144.015 q^{33} -326.693 q^{35} -111.121 q^{37} +754.688 q^{39} +399.220 q^{41} +100.756 q^{43} +895.887 q^{45} -343.912 q^{47} +344.395 q^{49} +475.910 q^{51} -318.014 q^{53} -180.447 q^{55} -6.42342 q^{57} +159.340 q^{59} -38.3680 q^{61} -1885.03 q^{63} -945.608 q^{65} -497.979 q^{67} -481.080 q^{71} -83.5123 q^{73} -300.980 q^{75} +379.679 q^{77} -1275.70 q^{79} +2499.06 q^{81} +153.064 q^{83} -596.305 q^{85} +472.807 q^{87} +673.680 q^{89} +1989.65 q^{91} +2519.52 q^{93} +8.04841 q^{95} -1664.32 q^{97} -1041.19 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 2 q^{3} + 50 q^{5} - 2 q^{7} + 272 q^{9} + 92 q^{11} - 28 q^{13} - 64 q^{15} + 316 q^{17} + 222 q^{19} + 512 q^{21} + 786 q^{25} - 212 q^{27} + 218 q^{29} + 350 q^{31} + 576 q^{33} + 150 q^{35} + 314 q^{37}+ \cdots + 930 q^{99}+O(q^{100}) $ 30 * q - 2 * q^3 + 50 * q^5 - 2 * q^7 + 272 * q^9 + 92 * q^11 - 28 * q^13 - 64 * q^15 + 316 * q^17 + 222 * q^19 + 512 * q^21 + 786 * q^25 - 212 * q^27 + 218 * q^29 + 350 * q^31 + 576 * q^33 + 150 * q^35 + 314 * q^37 - 144 * q^39 - 90 * q^41 + 224 * q^43 + 1406 * q^45 - 734 * q^47 + 1002 * q^49 + 1126 * q^51 + 1866 * q^53 - 306 * q^55 + 1246 * q^57 - 1070 * q^59 + 1656 * q^61 + 1026 * q^63 + 2524 * q^65 - 46 * q^67 - 1064 * q^71 - 2570 * q^73 + 3138 * q^75 + 3628 * q^77 + 410 * q^79 + 4590 * q^81 + 2264 * q^83 - 2322 * q^85 + 1584 * q^87 + 5736 * q^89 + 2350 * q^91 + 2022 * q^93 + 3182 * q^95 + 3144 * q^97 + 930 * 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
$3$	−9.94474	−1.91387	−0.956933	−	0.290308i	$-0.906242\pi$
$3$	−9.94474	−1.91387	−0.956933	+	0.290308i	$0.906242\pi$
$4$	0	0
$5$	12.4605	1.11451	0.557253	−	0.830343i	$-0.311856\pi$
$5$	12.4605	1.11451	0.557253	+	0.830343i	$0.311856\pi$
$6$	0	0
$7$	−26.2182	−1.41565	−0.707826	−	0.706387i	$-0.750324\pi$
$7$	−26.2182	−1.41565	−0.707826	+	0.706387i	$0.750324\pi$
$8$	0	0
$9$	71.8979	2.66288
$10$	0	0
$11$	−14.4815	−0.396939	−0.198470	−	0.980107i	$-0.563597\pi$
$11$	−14.4815	−0.396939	−0.198470	+	0.980107i	$0.563597\pi$
$12$	0	0
$13$	−75.8882	−1.61905	−0.809523	−	0.587089i	$-0.800274\pi$
$13$	−75.8882	−1.61905	−0.809523	+	0.587089i	$0.800274\pi$
$14$	0	0
$15$	−123.917	−2.13301
$16$	0	0
$17$	−47.8555	−0.682745	−0.341372	−	0.939928i	$-0.610892\pi$
$17$	−47.8555	−0.682745	−0.341372	+	0.939928i	$0.610892\pi$
$18$	0	0
$19$	0.645912	0.00779907	0.00389953	−	0.999992i	$-0.498759\pi$
$19$	0.645912	0.00779907	0.00389953	+	0.999992i	$0.498759\pi$
$20$	0	0
$21$	260.733	2.70937
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	30.2652	0.242122
$26$	0	0
$27$	−446.498	−3.18254
$28$	0	0
$29$	−47.5434	−0.304434	−0.152217	−	0.988347i	$-0.548641\pi$
$29$	−47.5434	−0.304434	−0.152217	+	0.988347i	$0.548641\pi$
$30$	0	0
$31$	−253.352	−1.46785	−0.733924	−	0.679231i	$-0.762313\pi$
$31$	−253.352	−1.46785	−0.733924	+	0.679231i	$0.762313\pi$
$32$	0	0
$33$	144.015	0.759689
$34$	0	0
$35$	−326.693	−1.57775
$36$	0	0
$37$	−111.121	−0.493734	−0.246867	−	0.969049i	$-0.579401\pi$
$37$	−111.121	−0.493734	−0.246867	+	0.969049i	$0.579401\pi$
$38$	0	0
$39$	754.688	3.09864
$40$	0	0
$41$	399.220	1.52068	0.760339	−	0.649527i	$-0.225033\pi$
$41$	399.220	1.52068	0.760339	+	0.649527i	$0.225033\pi$
$42$	0	0
$43$	100.756	0.357331	0.178665	−	0.983910i	$-0.442822\pi$
$43$	100.756	0.357331	0.178665	+	0.983910i	$0.442822\pi$
$44$	0	0
$45$	895.887	2.96780
$46$	0	0
$47$	−343.912	−1.06734	−0.533668	−	0.845694i	$-0.679187\pi$
$47$	−343.912	−1.06734	−0.533668	+	0.845694i	$0.679187\pi$
$48$	0	0
$49$	344.395	1.00407
$50$	0	0
$51$	475.910	1.30668
$52$	0	0
$53$	−318.014	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$53$	−318.014	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$54$	0	0
$55$	−180.447	−0.442391
$56$	0	0
$57$	−6.42342	−0.0149264
$58$	0	0
$59$	159.340	0.351598	0.175799	−	0.984426i	$-0.443749\pi$
$59$	159.340	0.351598	0.175799	+	0.984426i	$0.443749\pi$
$60$	0	0
$61$	−38.3680	−0.0805332	−0.0402666	−	0.999189i	$-0.512821\pi$
$61$	−38.3680	−0.0805332	−0.0402666	+	0.999189i	$0.512821\pi$
$62$	0	0
$63$	−1885.03	−3.76971
$64$	0	0
$65$	−945.608	−1.80443
$66$	0	0
$67$	−497.979	−0.908027	−0.454013	−	0.890995i	$-0.650008\pi$
$67$	−497.979	−0.908027	−0.454013	+	0.890995i	$0.650008\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−481.080	−0.804136	−0.402068	−	0.915610i	$-0.631709\pi$
$71$	−481.080	−0.804136	−0.402068	+	0.915610i	$0.631709\pi$
$72$	0	0
$73$	−83.5123	−0.133895	−0.0669477	−	0.997756i	$-0.521326\pi$
$73$	−83.5123	−0.133895	−0.0669477	+	0.997756i	$0.521326\pi$
$74$	0	0
$75$	−300.980	−0.463389
$76$	0	0
$77$	379.679	0.561928
$78$	0	0
$79$	−1275.70	−1.81681	−0.908405	−	0.418091i	$-0.862699\pi$
$79$	−1275.70	−1.81681	−0.908405	+	0.418091i	$0.862699\pi$
$80$	0	0
$81$	2499.06	3.42807
$82$	0	0
$83$	153.064	0.202421	0.101211	−	0.994865i	$-0.467728\pi$
$83$	153.064	0.202421	0.101211	+	0.994865i	$0.467728\pi$
$84$	0	0
$85$	−596.305	−0.760922
$86$	0	0
$87$	472.807	0.582646
$88$	0	0
$89$	673.680	0.802358	0.401179	−	0.916000i	$-0.368600\pi$
$89$	673.680	0.802358	0.401179	+	0.916000i	$0.368600\pi$
$90$	0	0
$91$	1989.65	2.29200
$92$	0	0
$93$	2519.52	2.80927
$94$	0	0
$95$	8.04841	0.00869210
$96$	0	0
$97$	−1664.32	−1.74213	−0.871063	−	0.491171i	$-0.836569\pi$
$97$	−1664.32	−1.74213	−0.871063	+	0.491171i	$0.836569\pi$
$98$	0	0
$99$	−1041.19	−1.05700
$100$	0	0
$101$	−497.369	−0.490000	−0.245000	−	0.969523i	$-0.578788\pi$
$101$	−497.369	−0.490000	−0.245000	+	0.969523i	$0.578788\pi$
$102$	0	0
$103$	−733.531	−0.701718	−0.350859	−	0.936428i	$-0.614110\pi$
$103$	−733.531	−0.701718	−0.350859	+	0.936428i	$0.614110\pi$
$104$	0	0
$105$	3248.88	3.01960
$106$	0	0
$107$	−1540.68	−1.39200	−0.695998	−	0.718044i	$-0.745038\pi$
$107$	−1540.68	−1.39200	−0.695998	+	0.718044i	$0.745038\pi$
$108$	0	0
$109$	1136.06	0.998301	0.499150	−	0.866515i	$-0.333646\pi$
$109$	1136.06	0.998301	0.499150	+	0.866515i	$0.333646\pi$
$110$	0	0
$111$	1105.07	0.944940
$112$	0	0
$113$	−179.174	−0.149162	−0.0745808	−	0.997215i	$-0.523762\pi$
$113$	−179.174	−0.149162	−0.0745808	+	0.997215i	$0.523762\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5456.20	−4.31133
$118$	0	0
$119$	1254.69	0.966528
$120$	0	0
$121$	−1121.29	−0.842439
$122$	0	0
$123$	−3970.14	−2.91037
$124$	0	0
$125$	−1180.45	−0.844659
$126$	0	0
$127$	97.4070	0.0680589	0.0340294	−	0.999421i	$-0.489166\pi$
$127$	97.4070	0.0680589	0.0340294	+	0.999421i	$0.489166\pi$
$128$	0	0
$129$	−1002.00	−0.683883
$130$	0	0
$131$	−2113.20	−1.40940	−0.704700	−	0.709505i	$-0.748918\pi$
$131$	−2113.20	−1.40940	−0.704700	+	0.709505i	$0.748918\pi$
$132$	0	0
$133$	−16.9347	−0.0110408
$134$	0	0
$135$	−5563.60	−3.54695
$136$	0	0
$137$	1094.21	0.682370	0.341185	−	0.939996i	$-0.389172\pi$
$137$	1094.21	0.682370	0.341185	+	0.939996i	$0.389172\pi$
$138$	0	0
$139$	−80.5304	−0.0491403	−0.0245701	−	0.999698i	$-0.507822\pi$
$139$	−80.5304	−0.0491403	−0.0245701	+	0.999698i	$0.507822\pi$
$140$	0	0
$141$	3420.12	2.04274
$142$	0	0
$143$	1098.97	0.642663
$144$	0	0
$145$	−592.416	−0.339293
$146$	0	0
$147$	−3424.92	−1.92165
$148$	0	0
$149$	−2652.59	−1.45845	−0.729224	−	0.684275i	$-0.760119\pi$
$149$	−2652.59	−1.45845	−0.729224	+	0.684275i	$0.760119\pi$
$150$	0	0
$151$	−149.188	−0.0804022	−0.0402011	−	0.999192i	$-0.512800\pi$
$151$	−149.188	−0.0804022	−0.0402011	+	0.999192i	$0.512800\pi$
$152$	0	0
$153$	−3440.71	−1.81807
$154$	0	0
$155$	−3156.90	−1.63593
$156$	0	0
$157$	−84.4369	−0.0429223	−0.0214611	−	0.999770i	$-0.506832\pi$
$157$	−84.4369	−0.0429223	−0.0214611	+	0.999770i	$0.506832\pi$
$158$	0	0
$159$	3162.56	1.57741
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2461.06	1.18261	0.591304	−	0.806449i	$-0.298613\pi$
$163$	2461.06	1.18261	0.591304	+	0.806449i	$0.298613\pi$
$164$	0	0
$165$	1794.50	0.846677
$166$	0	0
$167$	1061.13	0.491694	0.245847	−	0.969309i	$-0.420934\pi$
$167$	1061.13	0.491694	0.245847	+	0.969309i	$0.420934\pi$
$168$	0	0
$169$	3562.01	1.62131
$170$	0	0
$171$	46.4397	0.0207680
$172$	0	0
$173$	−789.507	−0.346966	−0.173483	−	0.984837i	$-0.555502\pi$
$173$	−789.507	−0.346966	−0.173483	+	0.984837i	$0.555502\pi$
$174$	0	0
$175$	−793.500	−0.342760
$176$	0	0
$177$	−1584.59	−0.672912
$178$	0	0
$179$	1973.57	0.824088	0.412044	−	0.911164i	$-0.364815\pi$
$179$	1973.57	0.824088	0.412044	+	0.911164i	$0.364815\pi$
$180$	0	0
$181$	3726.29	1.53024	0.765120	−	0.643888i	$-0.222680\pi$
$181$	3726.29	1.53024	0.765120	+	0.643888i	$0.222680\pi$
$182$	0	0
$183$	381.560	0.154130
$184$	0	0
$185$	−1384.63	−0.550269
$186$	0	0
$187$	693.018	0.271008
$188$	0	0
$189$	11706.4	4.50536
$190$	0	0
$191$	−1822.11	−0.690277	−0.345139	−	0.938552i	$-0.612168\pi$
$191$	−1822.11	−0.690277	−0.345139	+	0.938552i	$0.612168\pi$
$192$	0	0
$193$	−1946.08	−0.725813	−0.362907	−	0.931825i	$-0.618216\pi$
$193$	−1946.08	−0.725813	−0.362907	+	0.931825i	$0.618216\pi$
$194$	0	0
$195$	9403.83	3.45345
$196$	0	0
$197$	−3026.22	−1.09446	−0.547232	−	0.836981i	$-0.684318\pi$
$197$	−3026.22	−1.09446	−0.547232	+	0.836981i	$0.684318\pi$
$198$	0	0
$199$	−3640.06	−1.29667	−0.648333	−	0.761356i	$-0.724534\pi$
$199$	−3640.06	−1.29667	−0.648333	+	0.761356i	$0.724534\pi$
$200$	0	0
$201$	4952.27	1.73784
$202$	0	0
$203$	1246.50	0.430972
$204$	0	0
$205$	4974.51	1.69480
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.35376	−0.00309576
$210$	0	0
$211$	−1266.37	−0.413177	−0.206588	−	0.978428i	$-0.566236\pi$
$211$	−1266.37	−0.413177	−0.206588	+	0.978428i	$0.566236\pi$
$212$	0	0
$213$	4784.21	1.53901
$214$	0	0
$215$	1255.48	0.398247
$216$	0	0
$217$	6642.43	2.07796
$218$	0	0
$219$	830.508	0.256258
$220$	0	0
$221$	3631.66	1.10539
$222$	0	0
$223$	3078.13	0.924336	0.462168	−	0.886792i	$-0.347072\pi$
$223$	3078.13	0.924336	0.462168	+	0.886792i	$0.347072\pi$
$224$	0	0
$225$	2176.01	0.644742
$226$	0	0
$227$	3207.90	0.937956	0.468978	−	0.883210i	$-0.344622\pi$
$227$	3207.90	0.937956	0.468978	+	0.883210i	$0.344622\pi$
$228$	0	0
$229$	1985.81	0.573040	0.286520	−	0.958074i	$-0.407501\pi$
$229$	1985.81	0.573040	0.286520	+	0.958074i	$0.407501\pi$
$230$	0	0
$231$	−3775.81	−1.07545
$232$	0	0
$233$	−810.586	−0.227911	−0.113955	−	0.993486i	$-0.536352\pi$
$233$	−810.586	−0.227911	−0.113955	+	0.993486i	$0.536352\pi$
$234$	0	0
$235$	−4285.34	−1.18955
$236$	0	0
$237$	12686.6	3.47713
$238$	0	0
$239$	−2970.27	−0.803895	−0.401948	−	0.915663i	$-0.631667\pi$
$239$	−2970.27	−0.803895	−0.401948	+	0.915663i	$0.631667\pi$
$240$	0	0
$241$	6822.51	1.82356	0.911778	−	0.410685i	$-0.134710\pi$
$241$	6822.51	1.82356	0.911778	+	0.410685i	$0.134710\pi$
$242$	0	0
$243$	−12797.1	−3.37832
$244$	0	0
$245$	4291.35	1.11904
$246$	0	0
$247$	−49.0170	−0.0126270
$248$	0	0
$249$	−1522.18	−0.387407
$250$	0	0
$251$	3162.52	0.795284	0.397642	−	0.917541i	$-0.369829\pi$
$251$	3162.52	0.795284	0.397642	+	0.917541i	$0.369829\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	5930.10	1.45630
$256$	0	0
$257$	2840.10	0.689341	0.344671	−	0.938724i	$-0.387991\pi$
$257$	2840.10	0.689341	0.344671	+	0.938724i	$0.387991\pi$
$258$	0	0
$259$	2913.39	0.698955
$260$	0	0
$261$	−3418.27	−0.810672
$262$	0	0
$263$	−3199.05	−0.750044	−0.375022	−	0.927016i	$-0.622365\pi$
$263$	−3199.05	−0.750044	−0.375022	+	0.927016i	$0.622365\pi$
$264$	0	0
$265$	−3962.62	−0.918574
$266$	0	0
$267$	−6699.57	−1.53561
$268$	0	0
$269$	−2030.42	−0.460211	−0.230106	−	0.973166i	$-0.573907\pi$
$269$	−2030.42	−0.460211	−0.230106	+	0.973166i	$0.573907\pi$
$270$	0	0
$271$	2839.12	0.636399	0.318200	−	0.948024i	$-0.396922\pi$
$271$	2839.12	0.636399	0.318200	+	0.948024i	$0.396922\pi$
$272$	0	0
$273$	−19786.6	−4.38659
$274$	0	0
$275$	−438.285	−0.0961076
$276$	0	0
$277$	4849.52	1.05191	0.525955	−	0.850512i	$-0.323708\pi$
$277$	4849.52	1.05191	0.525955	+	0.850512i	$0.323708\pi$
$278$	0	0
$279$	−18215.5	−3.90871
$280$	0	0
$281$	6494.90	1.37884	0.689418	−	0.724364i	$-0.257866\pi$
$281$	6494.90	1.37884	0.689418	+	0.724364i	$0.257866\pi$
$282$	0	0
$283$	3429.96	0.720458	0.360229	−	0.932864i	$-0.382698\pi$
$283$	3429.96	0.720458	0.360229	+	0.932864i	$0.382698\pi$
$284$	0	0
$285$	−80.0394	−0.0166355
$286$	0	0
$287$	−10466.9	−2.15275
$288$	0	0
$289$	−2622.85	−0.533860
$290$	0	0
$291$	16551.2	3.33420
$292$	0	0
$293$	3867.37	0.771106	0.385553	−	0.922686i	$-0.374011\pi$
$293$	3867.37	0.771106	0.385553	+	0.922686i	$0.374011\pi$
$294$	0	0
$295$	1985.46	0.391858
$296$	0	0
$297$	6465.95	1.26327
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−2641.66	−0.505856
$302$	0	0
$303$	4946.20	0.937795
$304$	0	0
$305$	−478.087	−0.0897546
$306$	0	0
$307$	693.762	0.128974	0.0644871	−	0.997919i	$-0.479459\pi$
$307$	693.762	0.128974	0.0644871	+	0.997919i	$0.479459\pi$
$308$	0	0
$309$	7294.78	1.34299
$310$	0	0
$311$	4639.29	0.845885	0.422943	−	0.906157i	$-0.360997\pi$
$311$	4639.29	0.845885	0.422943	+	0.906157i	$0.360997\pi$
$312$	0	0
$313$	−823.427	−0.148699	−0.0743496	−	0.997232i	$-0.523688\pi$
$313$	−823.427	−0.148699	−0.0743496	+	0.997232i	$0.523688\pi$
$314$	0	0
$315$	−23488.6	−4.20137
$316$	0	0
$317$	−3673.18	−0.650808	−0.325404	−	0.945575i	$-0.605500\pi$
$317$	−3673.18	−0.650808	−0.325404	+	0.945575i	$0.605500\pi$
$318$	0	0
$319$	688.499	0.120842
$320$	0	0
$321$	15321.7	2.66409
$322$	0	0
$323$	−30.9104	−0.00532477
$324$	0	0
$325$	−2296.77	−0.392006
$326$	0	0
$327$	−11297.8	−1.91061
$328$	0	0
$329$	9016.77	1.51097
$330$	0	0
$331$	9634.53	1.59988	0.799942	−	0.600078i	$-0.204864\pi$
$331$	9634.53	1.59988	0.799942	+	0.600078i	$0.204864\pi$
$332$	0	0
$333$	−7989.35	−1.31476
$334$	0	0
$335$	−6205.09	−1.01200
$336$	0	0
$337$	7767.46	1.25555	0.627775	−	0.778395i	$-0.283966\pi$
$337$	7767.46	1.25555	0.627775	+	0.778395i	$0.283966\pi$
$338$	0	0
$339$	1781.84	0.285475
$340$	0	0
$341$	3668.91	0.582647
$342$	0	0
$343$	−36.5848	−0.00575916
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11132.6	−1.72228	−0.861141	−	0.508367i	$-0.830249\pi$
$347$	−11132.6	−1.72228	−0.861141	+	0.508367i	$0.830249\pi$
$348$	0	0
$349$	4483.75	0.687706	0.343853	−	0.939023i	$-0.388268\pi$
$349$	4483.75	0.687706	0.343853	+	0.939023i	$0.388268\pi$
$350$	0	0
$351$	33883.9	5.15267
$352$	0	0
$353$	8677.65	1.30840	0.654199	−	0.756322i	$-0.273006\pi$
$353$	8677.65	1.30840	0.654199	+	0.756322i	$0.273006\pi$
$354$	0	0
$355$	−5994.52	−0.896214
$356$	0	0
$357$	−12477.5	−1.84981
$358$	0	0
$359$	−10327.2	−1.51824	−0.759122	−	0.650948i	$-0.774371\pi$
$359$	−10327.2	−1.51824	−0.759122	+	0.650948i	$0.774371\pi$
$360$	0	0
$361$	−6858.58	−0.999939
$362$	0	0
$363$	11150.9	1.61232
$364$	0	0
$365$	−1040.61	−0.149227
$366$	0	0
$367$	−11612.9	−1.65174	−0.825868	−	0.563863i	$-0.809315\pi$
$367$	−11612.9	−1.65174	−0.825868	+	0.563863i	$0.809315\pi$
$368$	0	0
$369$	28703.1	4.04939
$370$	0	0
$371$	8337.76	1.16678
$372$	0	0
$373$	−3949.83	−0.548296	−0.274148	−	0.961688i	$-0.588396\pi$
$373$	−3949.83	−0.548296	−0.274148	+	0.961688i	$0.588396\pi$
$374$	0	0
$375$	11739.2	1.61656
$376$	0	0
$377$	3607.98	0.492892
$378$	0	0
$379$	13201.3	1.78920	0.894599	−	0.446870i	$-0.147461\pi$
$379$	13201.3	1.78920	0.894599	+	0.446870i	$0.147461\pi$
$380$	0	0
$381$	−968.688	−0.130256
$382$	0	0
$383$	6019.63	0.803104	0.401552	−	0.915836i	$-0.368471\pi$
$383$	6019.63	0.803104	0.401552	+	0.915836i	$0.368471\pi$
$384$	0	0
$385$	4731.01	0.626271
$386$	0	0
$387$	7244.18	0.951530
$388$	0	0
$389$	−12961.4	−1.68938	−0.844691	−	0.535254i	$-0.820216\pi$
$389$	−12961.4	−1.68938	−0.844691	+	0.535254i	$0.820216\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	21015.3	2.69740
$394$	0	0
$395$	−15896.0	−2.02484
$396$	0	0
$397$	5025.40	0.635309	0.317655	−	0.948207i	$-0.397105\pi$
$397$	5025.40	0.635309	0.317655	+	0.948207i	$0.397105\pi$
$398$	0	0
$399$	168.411	0.0211305
$400$	0	0
$401$	2357.46	0.293581	0.146791	−	0.989168i	$-0.453106\pi$
$401$	2357.46	0.293581	0.146791	+	0.989168i	$0.453106\pi$
$402$	0	0
$403$	19226.4	2.37651
$404$	0	0
$405$	31139.7	3.82060
$406$	0	0
$407$	1609.19	0.195982
$408$	0	0
$409$	1505.90	0.182058	0.0910292	−	0.995848i	$-0.470984\pi$
$409$	1505.90	0.182058	0.0910292	+	0.995848i	$0.470984\pi$
$410$	0	0
$411$	−10881.6	−1.30596
$412$	0	0
$413$	−4177.61	−0.497740
$414$	0	0
$415$	1907.26	0.225599
$416$	0	0
$417$	800.854	0.0940479
$418$	0	0
$419$	13376.0	1.55957	0.779784	−	0.626049i	$-0.215329\pi$
$419$	13376.0	1.55957	0.779784	+	0.626049i	$0.215329\pi$
$420$	0	0
$421$	−10134.8	−1.17325	−0.586626	−	0.809858i	$-0.699544\pi$
$421$	−10134.8	−1.17325	−0.586626	+	0.809858i	$0.699544\pi$
$422$	0	0
$423$	−24726.6	−2.84219
$424$	0	0
$425$	−1448.36	−0.165307
$426$	0	0
$427$	1005.94	0.114007
$428$	0	0
$429$	−10929.0	−1.22997
$430$	0	0
$431$	692.544	0.0773983	0.0386992	−	0.999251i	$-0.487679\pi$
$431$	692.544	0.0773983	0.0386992	+	0.999251i	$0.487679\pi$
$432$	0	0
$433$	−3912.69	−0.434254	−0.217127	−	0.976143i	$-0.569669\pi$
$433$	−3912.69	−0.434254	−0.217127	+	0.976143i	$0.569669\pi$
$434$	0	0
$435$	5891.43	0.649362
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−3027.77	−0.329174	−0.164587	−	0.986363i	$-0.552629\pi$
$439$	−3027.77	−0.329174	−0.164587	+	0.986363i	$0.552629\pi$
$440$	0	0
$441$	24761.3	2.67372
$442$	0	0
$443$	4266.35	0.457563	0.228781	−	0.973478i	$-0.426526\pi$
$443$	4266.35	0.457563	0.228781	+	0.973478i	$0.426526\pi$
$444$	0	0
$445$	8394.42	0.894233
$446$	0	0
$447$	26379.3	2.79128
$448$	0	0
$449$	−10438.0	−1.09711	−0.548553	−	0.836116i	$-0.684821\pi$
$449$	−10438.0	−1.09711	−0.548553	+	0.836116i	$0.684821\pi$
$450$	0	0
$451$	−5781.31	−0.603616
$452$	0	0
$453$	1483.63	0.153879
$454$	0	0
$455$	24792.2	2.55445
$456$	0	0
$457$	2123.45	0.217354	0.108677	−	0.994077i	$-0.465339\pi$
$457$	2123.45	0.217354	0.108677	+	0.994077i	$0.465339\pi$
$458$	0	0
$459$	21367.4	2.17286
$460$	0	0
$461$	−15784.0	−1.59465	−0.797324	−	0.603552i	$-0.793752\pi$
$461$	−15784.0	−1.59465	−0.797324	+	0.603552i	$0.793752\pi$
$462$	0	0
$463$	10711.3	1.07516	0.537579	−	0.843214i	$-0.319339\pi$
$463$	10711.3	1.07516	0.537579	+	0.843214i	$0.319339\pi$
$464$	0	0
$465$	31394.6	3.13094
$466$	0	0
$467$	7492.65	0.742438	0.371219	−	0.928545i	$-0.378940\pi$
$467$	7492.65	0.742438	0.371219	+	0.928545i	$0.378940\pi$
$468$	0	0
$469$	13056.1	1.28545
$470$	0	0
$471$	839.703	0.0821475
$472$	0	0
$473$	−1459.10	−0.141839
$474$	0	0
$475$	19.5487	0.00188832
$476$	0	0
$477$	−22864.5	−2.19475
$478$	0	0
$479$	470.436	0.0448743	0.0224371	−	0.999748i	$-0.492857\pi$
$479$	470.436	0.0448743	0.0224371	+	0.999748i	$0.492857\pi$
$480$	0	0
$481$	8432.75	0.799377
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20738.4	−1.94161
$486$	0	0
$487$	4683.09	0.435751	0.217876	−	0.975977i	$-0.430087\pi$
$487$	4683.09	0.435751	0.217876	+	0.975977i	$0.430087\pi$
$488$	0	0
$489$	−24474.6	−2.26335
$490$	0	0
$491$	−16056.4	−1.47580	−0.737899	−	0.674911i	$-0.764182\pi$
$491$	−16056.4	−1.47580	−0.737899	+	0.674911i	$0.764182\pi$
$492$	0	0
$493$	2275.21	0.207851
$494$	0	0
$495$	−12973.8	−1.17804
$496$	0	0
$497$	12613.1	1.13838
$498$	0	0
$499$	−1182.03	−0.106042	−0.0530211	−	0.998593i	$-0.516885\pi$
$499$	−1182.03	−0.106042	−0.0530211	+	0.998593i	$0.516885\pi$
$500$	0	0
$501$	−10552.7	−0.941037
$502$	0	0
$503$	−4567.73	−0.404901	−0.202450	−	0.979293i	$-0.564890\pi$
$503$	−4567.73	−0.404901	−0.202450	+	0.979293i	$0.564890\pi$
$504$	0	0
$505$	−6197.49	−0.546108
$506$	0	0
$507$	−35423.3	−3.10297
$508$	0	0
$509$	−9219.65	−0.802856	−0.401428	−	0.915890i	$-0.631486\pi$
$509$	−9219.65	−0.802856	−0.401428	+	0.915890i	$0.631486\pi$
$510$	0	0
$511$	2189.54	0.189549
$512$	0	0
$513$	−288.398	−0.0248208
$514$	0	0
$515$	−9140.20	−0.782069
$516$	0	0
$517$	4980.36	0.423667
$518$	0	0
$519$	7851.44	0.664046
$520$	0	0
$521$	−12111.9	−1.01849	−0.509243	−	0.860623i	$-0.670074\pi$
$521$	−12111.9	−1.01849	−0.509243	+	0.860623i	$0.670074\pi$
$522$	0	0
$523$	9483.09	0.792862	0.396431	−	0.918065i	$-0.370249\pi$
$523$	9483.09	0.792862	0.396431	+	0.918065i	$0.370249\pi$
$524$	0	0
$525$	7891.16	0.655997
$526$	0	0
$527$	12124.3	1.00217
$528$	0	0
$529$	0	0
$530$	0	0
$531$	11456.2	0.936265
$532$	0	0
$533$	−30296.1	−2.46205
$534$	0	0
$535$	−19197.8	−1.55139
$536$	0	0
$537$	−19626.7	−1.57719
$538$	0	0
$539$	−4987.36	−0.398554
$540$	0	0
$541$	9704.41	0.771211	0.385605	−	0.922664i	$-0.373993\pi$
$541$	9704.41	0.771211	0.385605	+	0.922664i	$0.373993\pi$
$542$	0	0
$543$	−37057.0	−2.92867
$544$	0	0
$545$	14155.9	1.11261
$546$	0	0
$547$	5983.20	0.467684	0.233842	−	0.972275i	$-0.424870\pi$
$547$	5983.20	0.467684	0.233842	+	0.972275i	$0.424870\pi$
$548$	0	0
$549$	−2758.58	−0.214450
$550$	0	0
$551$	−30.7088	−0.00237430
$552$	0	0
$553$	33446.7	2.57197
$554$	0	0
$555$	13769.7	1.05314
$556$	0	0
$557$	−2004.07	−0.152451	−0.0762255	−	0.997091i	$-0.524287\pi$
$557$	−2004.07	−0.152451	−0.0762255	+	0.997091i	$0.524287\pi$
$558$	0	0
$559$	−7646.22	−0.578534
$560$	0	0
$561$	−6891.89	−0.518673
$562$	0	0
$563$	22091.0	1.65369	0.826844	−	0.562431i	$-0.190134\pi$
$563$	22091.0	1.65369	0.826844	+	0.562431i	$0.190134\pi$
$564$	0	0
$565$	−2232.60	−0.166241
$566$	0	0
$567$	−65520.9	−4.85295
$568$	0	0
$569$	21872.6	1.61151	0.805754	−	0.592251i	$-0.201761\pi$
$569$	21872.6	1.61151	0.805754	+	0.592251i	$0.201761\pi$
$570$	0	0
$571$	−3662.90	−0.268455	−0.134227	−	0.990951i	$-0.542855\pi$
$571$	−3662.90	−0.268455	−0.134227	+	0.990951i	$0.542855\pi$
$572$	0	0
$573$	18120.4	1.32110
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−12531.7	−0.904159	−0.452079	−	0.891978i	$-0.649318\pi$
$577$	−12531.7	−0.904159	−0.452079	+	0.891978i	$0.649318\pi$
$578$	0	0
$579$	19353.3	1.38911
$580$	0	0
$581$	−4013.07	−0.286558
$582$	0	0
$583$	4605.31	0.327157
$584$	0	0
$585$	−67987.2	−4.80500
$586$	0	0
$587$	10668.6	0.750156	0.375078	−	0.926993i	$-0.377616\pi$
$587$	10668.6	0.750156	0.375078	+	0.926993i	$0.377616\pi$
$588$	0	0
$589$	−163.643	−0.0114479
$590$	0	0
$591$	30095.0	2.09466
$592$	0	0
$593$	−19712.7	−1.36510	−0.682549	−	0.730840i	$-0.739129\pi$
$593$	−19712.7	−1.36510	−0.682549	+	0.730840i	$0.739129\pi$
$594$	0	0
$595$	15634.1	1.07720
$596$	0	0
$597$	36199.4	2.48165
$598$	0	0
$599$	6365.07	0.434173	0.217086	−	0.976152i	$-0.430345\pi$
$599$	6365.07	0.434173	0.217086	+	0.976152i	$0.430345\pi$
$600$	0	0
$601$	4056.20	0.275301	0.137651	−	0.990481i	$-0.456045\pi$
$601$	4056.20	0.275301	0.137651	+	0.990481i	$0.456045\pi$
$602$	0	0
$603$	−35803.6	−2.41797
$604$	0	0
$605$	−13971.8	−0.938903
$606$	0	0
$607$	5331.59	0.356512	0.178256	−	0.983984i	$-0.442955\pi$
$607$	5331.59	0.356512	0.178256	+	0.983984i	$0.442955\pi$
$608$	0	0
$609$	−12396.1	−0.824823
$610$	0	0
$611$	26098.9	1.72806
$612$	0	0
$613$	2298.06	0.151416	0.0757079	−	0.997130i	$-0.475878\pi$
$613$	2298.06	0.151416	0.0757079	+	0.997130i	$0.475878\pi$
$614$	0	0
$615$	−49470.2	−3.24363
$616$	0	0
$617$	−9356.41	−0.610494	−0.305247	−	0.952273i	$-0.598739\pi$
$617$	−9356.41	−0.610494	−0.305247	+	0.952273i	$0.598739\pi$
$618$	0	0
$619$	10068.1	0.653752	0.326876	−	0.945067i	$-0.394004\pi$
$619$	10068.1	0.653752	0.326876	+	0.945067i	$0.394004\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17662.7	−1.13586
$624$	0	0
$625$	−18492.2	−1.18350
$626$	0	0
$627$	93.0207	0.00592486
$628$	0	0
$629$	5317.74	0.337094
$630$	0	0
$631$	23713.2	1.49605	0.748024	−	0.663672i	$-0.231003\pi$
$631$	23713.2	1.49605	0.748024	+	0.663672i	$0.231003\pi$
$632$	0	0
$633$	12593.7	0.790765
$634$	0	0
$635$	1213.74	0.0758519
$636$	0	0
$637$	−26135.5	−1.62563
$638$	0	0
$639$	−34588.6	−2.14132
$640$	0	0
$641$	29042.1	1.78954	0.894770	−	0.446527i	$-0.147339\pi$
$641$	29042.1	1.78954	0.894770	+	0.446527i	$0.147339\pi$
$642$	0	0
$643$	−9383.74	−0.575519	−0.287759	−	0.957703i	$-0.592910\pi$
$643$	−9383.74	−0.575519	−0.287759	+	0.957703i	$0.592910\pi$
$644$	0	0
$645$	−12485.4	−0.762191
$646$	0	0
$647$	−1459.50	−0.0886844	−0.0443422	−	0.999016i	$-0.514119\pi$
$647$	−1459.50	−0.0886844	−0.0443422	+	0.999016i	$0.514119\pi$
$648$	0	0
$649$	−2307.48	−0.139563
$650$	0	0
$651$	−66057.3	−3.97694
$652$	0	0
$653$	−30851.8	−1.84889	−0.924445	−	0.381316i	$-0.875471\pi$
$653$	−30851.8	−1.84889	−0.924445	+	0.381316i	$0.875471\pi$
$654$	0	0
$655$	−26331.7	−1.57078
$656$	0	0
$657$	−6004.35	−0.356548
$658$	0	0
$659$	20099.3	1.18810	0.594049	−	0.804429i	$-0.297528\pi$
$659$	20099.3	1.18810	0.594049	+	0.804429i	$0.297528\pi$
$660$	0	0
$661$	−4147.58	−0.244058	−0.122029	−	0.992527i	$-0.538940\pi$
$661$	−4147.58	−0.244058	−0.122029	+	0.992527i	$0.538940\pi$
$662$	0	0
$663$	−36116.0	−2.11558
$664$	0	0
$665$	−211.015	−0.0123050
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−30611.2	−1.76905
$670$	0	0
$671$	555.626	0.0319668
$672$	0	0
$673$	−23389.5	−1.33967	−0.669836	−	0.742509i	$-0.733636\pi$
$673$	−23389.5	−1.33967	−0.669836	+	0.742509i	$0.733636\pi$
$674$	0	0
$675$	−13513.4	−0.770562
$676$	0	0
$677$	−2511.53	−0.142579	−0.0712895	−	0.997456i	$-0.522711\pi$
$677$	−2511.53	−0.142579	−0.0712895	+	0.997456i	$0.522711\pi$
$678$	0	0
$679$	43635.5	2.46624
$680$	0	0
$681$	−31901.8	−1.79512
$682$	0	0
$683$	−21785.7	−1.22051	−0.610253	−	0.792206i	$-0.708932\pi$
$683$	−21785.7	−1.22051	−0.610253	+	0.792206i	$0.708932\pi$
$684$	0	0
$685$	13634.5	0.760505
$686$	0	0
$687$	−19748.4	−1.09672
$688$	0	0
$689$	24133.5	1.33442
$690$	0	0
$691$	−23045.5	−1.26873	−0.634364	−	0.773035i	$-0.718738\pi$
$691$	−23045.5	−1.26873	−0.634364	+	0.773035i	$0.718738\pi$
$692$	0	0
$693$	27298.1	1.49635
$694$	0	0
$695$	−1003.45	−0.0547671
$696$	0	0
$697$	−19104.9	−1.03823
$698$	0	0
$699$	8061.06	0.436191
$700$	0	0
$701$	−21575.1	−1.16246	−0.581228	−	0.813741i	$-0.697428\pi$
$701$	−21575.1	−1.16246	−0.581228	+	0.813741i	$0.697428\pi$
$702$	0	0
$703$	−71.7742	−0.00385066
$704$	0	0
$705$	42616.6	2.27664
$706$	0	0
$707$	13040.1	0.693669
$708$	0	0
$709$	−17788.8	−0.942272	−0.471136	−	0.882061i	$-0.656156\pi$
$709$	−17788.8	−0.942272	−0.471136	+	0.882061i	$0.656156\pi$
$710$	0	0
$711$	−91720.5	−4.83796
$712$	0	0
$713$	0	0
$714$	0	0
$715$	13693.8	0.716251
$716$	0	0
$717$	29538.6	1.53855
$718$	0	0
$719$	−473.961	−0.0245838	−0.0122919	−	0.999924i	$-0.503913\pi$
$719$	−473.961	−0.0245838	−0.0122919	+	0.999924i	$0.503913\pi$
$720$	0	0
$721$	19231.9	0.993388
$722$	0	0
$723$	−67848.1	−3.49004
$724$	0	0
$725$	−1438.91	−0.0737101
$726$	0	0
$727$	−3899.59	−0.198938	−0.0994688	−	0.995041i	$-0.531714\pi$
$727$	−3899.59	−0.198938	−0.0994688	+	0.995041i	$0.531714\pi$
$728$	0	0
$729$	59789.0	3.03759
$730$	0	0
$731$	−4821.75	−0.243966
$732$	0	0
$733$	17905.4	0.902251	0.451126	−	0.892460i	$-0.351023\pi$
$733$	17905.4	0.902251	0.451126	+	0.892460i	$0.351023\pi$
$734$	0	0
$735$	−42676.4	−2.14169
$736$	0	0
$737$	7211.47	0.360431
$738$	0	0
$739$	−13312.1	−0.662643	−0.331322	−	0.943518i	$-0.607494\pi$
$739$	−13312.1	−0.662643	−0.331322	+	0.943518i	$0.607494\pi$
$740$	0	0
$741$	487.462	0.0241665
$742$	0	0
$743$	33895.7	1.67364	0.836819	−	0.547480i	$-0.184413\pi$
$743$	33895.7	1.67364	0.836819	+	0.547480i	$0.184413\pi$
$744$	0	0
$745$	−33052.7	−1.62545
$746$	0	0
$747$	11005.0	0.539024
$748$	0	0
$749$	40394.0	1.97058
$750$	0	0
$751$	−26911.0	−1.30759	−0.653793	−	0.756674i	$-0.726823\pi$
$751$	−26911.0	−1.30759	−0.653793	+	0.756674i	$0.726823\pi$
$752$	0	0
$753$	−31450.4	−1.52207
$754$	0	0
$755$	−1858.96	−0.0896086
$756$	0	0
$757$	7322.40	0.351568	0.175784	−	0.984429i	$-0.443754\pi$
$757$	7322.40	0.351568	0.175784	+	0.984429i	$0.443754\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12135.2	−0.578056	−0.289028	−	0.957321i	$-0.593332\pi$
$761$	−12135.2	−0.578056	−0.289028	+	0.957321i	$0.593332\pi$
$762$	0	0
$763$	−29785.5	−1.41325
$764$	0	0
$765$	−42873.1	−2.02625
$766$	0	0
$767$	−12092.0	−0.569253
$768$	0	0
$769$	3754.71	0.176071	0.0880353	−	0.996117i	$-0.471941\pi$
$769$	3754.71	0.176071	0.0880353	+	0.996117i	$0.471941\pi$
$770$	0	0
$771$	−28244.1	−1.31931
$772$	0	0
$773$	−25609.6	−1.19161	−0.595804	−	0.803130i	$-0.703167\pi$
$773$	−25609.6	−1.19161	−0.595804	+	0.803130i	$0.703167\pi$
$774$	0	0
$775$	−7667.75	−0.355398
$776$	0	0
$777$	−28972.9	−1.33771
$778$	0	0
$779$	257.861	0.0118599
$780$	0	0
$781$	6966.75	0.319193
$782$	0	0
$783$	21228.0	0.968872
$784$	0	0
$785$	−1052.13	−0.0478371
$786$	0	0
$787$	−7226.43	−0.327312	−0.163656	−	0.986517i	$-0.552329\pi$
$787$	−7226.43	−0.327312	−0.163656	+	0.986517i	$0.552329\pi$
$788$	0	0
$789$	31813.7	1.43548
$790$	0	0
$791$	4697.62	0.211161
$792$	0	0
$793$	2911.68	0.130387
$794$	0	0
$795$	39407.3	1.75803
$796$	0	0
$797$	34935.9	1.55269	0.776344	−	0.630309i	$-0.217072\pi$
$797$	34935.9	1.55269	0.776344	+	0.630309i	$0.217072\pi$
$798$	0	0
$799$	16458.1	0.728718
$800$	0	0
$801$	48436.1	2.13659
$802$	0	0
$803$	1209.38	0.0531484
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20192.0	0.880783
$808$	0	0
$809$	−27288.3	−1.18592	−0.592958	−	0.805233i	$-0.702040\pi$
$809$	−27288.3	−1.18592	−0.592958	+	0.805233i	$0.702040\pi$
$810$	0	0
$811$	37405.1	1.61957	0.809785	−	0.586727i	$-0.199584\pi$
$811$	37405.1	1.61957	0.809785	+	0.586727i	$0.199584\pi$
$812$	0	0
$813$	−28234.3	−1.21798
$814$	0	0
$815$	30666.1	1.31802
$816$	0	0
$817$	65.0798	0.00278685
$818$	0	0
$819$	143052.	6.10334
$820$	0	0
$821$	15104.5	0.642084	0.321042	−	0.947065i	$-0.395967\pi$
$821$	15104.5	0.642084	0.321042	+	0.947065i	$0.395967\pi$
$822$	0	0
$823$	17898.2	0.758071	0.379035	−	0.925382i	$-0.376256\pi$
$823$	17898.2	0.758071	0.379035	+	0.925382i	$0.376256\pi$
$824$	0	0
$825$	4358.63	0.183937
$826$	0	0
$827$	10734.0	0.451341	0.225671	−	0.974204i	$-0.427543\pi$
$827$	10734.0	0.451341	0.225671	+	0.974204i	$0.427543\pi$
$828$	0	0
$829$	−44607.5	−1.86886	−0.934428	−	0.356151i	$-0.884089\pi$
$829$	−44607.5	−1.86886	−0.934428	+	0.356151i	$0.884089\pi$
$830$	0	0
$831$	−48227.2	−2.01322
$832$	0	0
$833$	−16481.2	−0.685522
$834$	0	0
$835$	13222.3	0.547996
$836$	0	0
$837$	113121.	4.67148
$838$	0	0
$839$	19778.0	0.813843	0.406921	−	0.913463i	$-0.366602\pi$
$839$	19778.0	0.813843	0.406921	+	0.913463i	$0.366602\pi$
$840$	0	0
$841$	−22128.6	−0.907320
$842$	0	0
$843$	−64590.0	−2.63891
$844$	0	0
$845$	44384.6	1.80696
$846$	0	0
$847$	29398.1	1.19260
$848$	0	0
$849$	−34110.0	−1.37886
$850$	0	0
$851$	0	0
$852$	0	0
$853$	32806.3	1.31684	0.658421	−	0.752650i	$-0.271225\pi$
$853$	32806.3	1.31684	0.658421	+	0.752650i	$0.271225\pi$
$854$	0	0
$855$	578.664	0.0231461
$856$	0	0
$857$	−32892.0	−1.31105	−0.655524	−	0.755174i	$-0.727552\pi$
$857$	−32892.0	−1.31105	−0.655524	+	0.755174i	$0.727552\pi$
$858$	0	0
$859$	19148.4	0.760574	0.380287	−	0.924868i	$-0.375825\pi$
$859$	19148.4	0.760574	0.380287	+	0.924868i	$0.375825\pi$
$860$	0	0
$861$	104090.	4.12007
$862$	0	0
$863$	−39969.2	−1.57656	−0.788278	−	0.615320i	$-0.789027\pi$
$863$	−39969.2	−1.57656	−0.788278	+	0.615320i	$0.789027\pi$
$864$	0	0
$865$	−9837.69	−0.386695
$866$	0	0
$867$	26083.6	1.02174
$868$	0	0
$869$	18474.1	0.721163
$870$	0	0
$871$	37790.7	1.47014
$872$	0	0
$873$	−119661.	−4.63908
$874$	0	0
$875$	30949.2	1.19574
$876$	0	0
$877$	17323.6	0.667020	0.333510	−	0.942747i	$-0.391767\pi$
$877$	17323.6	0.667020	0.333510	+	0.942747i	$0.391767\pi$
$878$	0	0
$879$	−38459.9	−1.47579
$880$	0	0
$881$	−6911.21	−0.264296	−0.132148	−	0.991230i	$-0.542187\pi$
$881$	−6911.21	−0.264296	−0.132148	+	0.991230i	$0.542187\pi$
$882$	0	0
$883$	44073.8	1.67973	0.839865	−	0.542795i	$-0.182634\pi$
$883$	44073.8	1.67973	0.839865	+	0.542795i	$0.182634\pi$
$884$	0	0
$885$	−19744.9	−0.749963
$886$	0	0
$887$	−21893.8	−0.828775	−0.414387	−	0.910101i	$-0.636004\pi$
$887$	−21893.8	−0.828775	−0.414387	+	0.910101i	$0.636004\pi$
$888$	0	0
$889$	−2553.84	−0.0963476
$890$	0	0
$891$	−36190.1	−1.36073
$892$	0	0
$893$	−222.137	−0.00832422
$894$	0	0
$895$	24591.8	0.918450
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12045.2	0.446863
$900$	0	0
$901$	15218.7	0.562717
$902$	0	0
$903$	26270.6	0.968140
$904$	0	0
$905$	46431.7	1.70546
$906$	0	0
$907$	−14618.3	−0.535163	−0.267581	−	0.963535i	$-0.586224\pi$
$907$	−14618.3	−0.535163	−0.267581	+	0.963535i	$0.586224\pi$
$908$	0	0
$909$	−35759.7	−1.30481
$910$	0	0
$911$	2071.66	0.0753427	0.0376714	−	0.999290i	$-0.488006\pi$
$911$	2071.66	0.0753427	0.0376714	+	0.999290i	$0.488006\pi$
$912$	0	0
$913$	−2216.59	−0.0803489
$914$	0	0
$915$	4754.45	0.171778
$916$	0	0
$917$	55404.4	1.99522
$918$	0	0
$919$	−7935.82	−0.284852	−0.142426	−	0.989805i	$-0.545490\pi$
$919$	−7935.82	−0.284852	−0.142426	+	0.989805i	$0.545490\pi$
$920$	0	0
$921$	−6899.28	−0.246839
$922$	0	0
$923$	36508.3	1.30193
$924$	0	0
$925$	−3363.10	−0.119544
$926$	0	0
$927$	−52739.3	−1.86859
$928$	0	0
$929$	15621.5	0.551695	0.275848	−	0.961201i	$-0.411041\pi$
$929$	15621.5	0.551695	0.275848	+	0.961201i	$0.411041\pi$
$930$	0	0
$931$	222.449	0.00783080
$932$	0	0
$933$	−46136.6	−1.61891
$934$	0	0
$935$	8635.39	0.302040
$936$	0	0
$937$	39243.7	1.36823	0.684117	−	0.729372i	$-0.260188\pi$
$937$	39243.7	1.36823	0.684117	+	0.729372i	$0.260188\pi$
$938$	0	0
$939$	8188.77	0.284590
$940$	0	0
$941$	35170.9	1.21843	0.609213	−	0.793007i	$-0.291486\pi$
$941$	35170.9	1.21843	0.609213	+	0.793007i	$0.291486\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	145868.	5.02125
$946$	0	0
$947$	23076.2	0.791842	0.395921	−	0.918284i	$-0.370425\pi$
$947$	23076.2	0.791842	0.395921	+	0.918284i	$0.370425\pi$
$948$	0	0
$949$	6337.59	0.216783
$950$	0	0
$951$	36528.8	1.24556
$952$	0	0
$953$	609.227	0.0207081	0.0103540	−	0.999946i	$-0.496704\pi$
$953$	609.227	0.0207081	0.0103540	+	0.999946i	$0.496704\pi$
$954$	0	0
$955$	−22704.4	−0.769318
$956$	0	0
$957$	−6846.94	−0.231275
$958$	0	0
$959$	−28688.3	−0.965998
$960$	0	0
$961$	34396.1	1.15458
$962$	0	0
$963$	−110772.	−3.70672
$964$	0	0
$965$	−24249.2	−0.808923
$966$	0	0
$967$	605.367	0.0201316	0.0100658	−	0.999949i	$-0.496796\pi$
$967$	605.367	0.0201316	0.0100658	+	0.999949i	$0.496796\pi$
$968$	0	0
$969$	307.396	0.0101909
$970$	0	0
$971$	41656.8	1.37675	0.688377	−	0.725353i	$-0.258323\pi$
$971$	41656.8	1.37675	0.688377	+	0.725353i	$0.258323\pi$
$972$	0	0
$973$	2111.36	0.0695655
$974$	0	0
$975$	22840.8	0.750247
$976$	0	0
$977$	−17087.1	−0.559534	−0.279767	−	0.960068i	$-0.590257\pi$
$977$	−17087.1	−0.559534	−0.279767	+	0.960068i	$0.590257\pi$
$978$	0	0
$979$	−9755.88	−0.318488
$980$	0	0
$981$	81680.3	2.65836
$982$	0	0
$983$	21344.4	0.692554	0.346277	−	0.938132i	$-0.387446\pi$
$983$	21344.4	0.692554	0.346277	+	0.938132i	$0.387446\pi$
$984$	0	0
$985$	−37708.4	−1.21979
$986$	0	0
$987$	−89669.5	−2.89180
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−38896.1	−1.24680	−0.623398	−	0.781905i	$-0.714248\pi$
$991$	−38896.1	−1.24680	−0.623398	+	0.781905i	$0.714248\pi$
$992$	0	0
$993$	−95812.9	−3.06196
$994$	0	0
$995$	−45357.1	−1.44514
$996$	0	0
$997$	−40415.7	−1.28383	−0.641916	−	0.766775i	$-0.721860\pi$
$997$	−40415.7	−1.28383	−0.641916	+	0.766775i	$0.721860\pi$
$998$	0	0
$999$	49615.2	1.57133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2116.4.a.j.1.1		30
23.15	odd	22		92.4.e.a.41.6	yes	60
23.20	odd	22		92.4.e.a.9.6	✓	60
23.22	odd	2		2116.4.a.i.1.1		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.4.e.a.9.6	✓	60	23.20	odd	22
92.4.e.a.41.6	yes	60	23.15	odd	22
2116.4.a.i.1.1		30	23.22	odd	2
2116.4.a.j.1.1		30	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:	\( N \)	\(=\)	\( 2116 = 2^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2116.a (trivial)

\(f(q)\)	\(=\)	\(q-9.94474 q^{3} +12.4605 q^{5} -26.2182 q^{7} +71.8979 q^{9} -14.4815 q^{11} -75.8882 q^{13} -123.917 q^{15} -47.8555 q^{17} +0.645912 q^{19} +260.733 q^{21} +30.2652 q^{25} -446.498 q^{27} -47.5434 q^{29} -253.352 q^{31} +144.015 q^{33} -326.693 q^{35} -111.121 q^{37} +754.688 q^{39} +399.220 q^{41} +100.756 q^{43} +895.887 q^{45} -343.912 q^{47} +344.395 q^{49} +475.910 q^{51} -318.014 q^{53} -180.447 q^{55} -6.42342 q^{57} +159.340 q^{59} -38.3680 q^{61} -1885.03 q^{63} -945.608 q^{65} -497.979 q^{67} -481.080 q^{71} -83.5123 q^{73} -300.980 q^{75} +379.679 q^{77} -1275.70 q^{79} +2499.06 q^{81} +153.064 q^{83} -596.305 q^{85} +472.807 q^{87} +673.680 q^{89} +1989.65 q^{91} +2519.52 q^{93} +8.04841 q^{95} -1664.32 q^{97} -1041.19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 2 q^{3} + 50 q^{5} - 2 q^{7} + 272 q^{9} + 92 q^{11} - 28 q^{13} - 64 q^{15} + 316 q^{17} + 222 q^{19} + 512 q^{21} + 786 q^{25} - 212 q^{27} + 218 q^{29} + 350 q^{31} + 576 q^{33} + 150 q^{35} + 314 q^{37}+ \cdots + 930 q^{99}+O(q^{100}) \) 30 * q - 2 * q^3 + 50 * q^5 - 2 * q^7 + 272 * q^9 + 92 * q^11 - 28 * q^13 - 64 * q^15 + 316 * q^17 + 222 * q^19 + 512 * q^21 + 786 * q^25 - 212 * q^27 + 218 * q^29 + 350 * q^31 + 576 * q^33 + 150 * q^35 + 314 * q^37 - 144 * q^39 - 90 * q^41 + 224 * q^43 + 1406 * q^45 - 734 * q^47 + 1002 * q^49 + 1126 * q^51 + 1866 * q^53 - 306 * q^55 + 1246 * q^57 - 1070 * q^59 + 1656 * q^61 + 1026 * q^63 + 2524 * q^65 - 46 * q^67 - 1064 * q^71 - 2570 * q^73 + 3138 * q^75 + 3628 * q^77 + 410 * q^79 + 4590 * q^81 + 2264 * q^83 - 2322 * q^85 + 1584 * q^87 + 5736 * q^89 + 2350 * q^91 + 2022 * q^93 + 3182 * q^95 + 3144 * q^97 + 930 * q^99

Introduction

L-functions

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