LMFDB - Embedded newform 2070.4.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,4,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, 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("2070.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2070.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:	$122.133953712$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 486x - 3340 $ x^3 - 486*x - 3340
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$24.9023$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +8.08465 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +8.08465 q^{7} -8.00000 q^{8} -10.0000 q^{10} -38.9023 q^{11} -50.7069 q^{13} -16.1693 q^{14} +16.0000 q^{16} -64.8176 q^{17} +64.5376 q^{19} +20.0000 q^{20} +77.8046 q^{22} +23.0000 q^{23} +25.0000 q^{25} +101.414 q^{26} +32.3386 q^{28} +144.036 q^{29} +248.600 q^{31} -32.0000 q^{32} +129.635 q^{34} +40.4233 q^{35} +290.769 q^{37} -129.075 q^{38} -40.0000 q^{40} +123.307 q^{41} +96.9581 q^{43} -155.609 q^{44} -46.0000 q^{46} -342.629 q^{47} -277.638 q^{49} -50.0000 q^{50} -202.828 q^{52} -39.6254 q^{53} -194.511 q^{55} -64.6772 q^{56} -288.072 q^{58} -388.739 q^{59} -178.147 q^{61} -497.199 q^{62} +64.0000 q^{64} -253.534 q^{65} -27.8576 q^{67} -259.271 q^{68} -80.8465 q^{70} -165.064 q^{71} +391.901 q^{73} -581.538 q^{74} +258.150 q^{76} -314.511 q^{77} +785.383 q^{79} +80.0000 q^{80} -246.613 q^{82} -746.890 q^{83} -324.088 q^{85} -193.916 q^{86} +311.218 q^{88} -15.5166 q^{89} -409.947 q^{91} +92.0000 q^{92} +685.257 q^{94} +322.688 q^{95} -1180.85 q^{97} +555.277 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 12 q^{4} + 15 q^{5} - 2 q^{7} - 24 q^{8}+O(q^{10}) $ 3 * q - 6 * q^2 + 12 * q^4 + 15 * q^5 - 2 * q^7 - 24 * q^8	$ 3 q - 6 q^{2} + 12 q^{4} + 15 q^{5} - 2 q^{7} - 24 q^{8} - 30 q^{10} - 42 q^{11} + 72 q^{13} + 4 q^{14} + 48 q^{16} - 146 q^{17} + 22 q^{19} + 60 q^{20} + 84 q^{22} + 69 q^{23} + 75 q^{25} - 144 q^{26} - 8 q^{28} - 214 q^{29} + 130 q^{31} - 96 q^{32} + 292 q^{34} - 10 q^{35} + 204 q^{37} - 44 q^{38} - 120 q^{40} - 470 q^{41} + 404 q^{43} - 168 q^{44} - 138 q^{46} - 198 q^{47} + 845 q^{49} - 150 q^{50} + 288 q^{52} - 870 q^{53} - 210 q^{55} + 16 q^{56} + 428 q^{58} - 512 q^{59} - 64 q^{61} - 260 q^{62} + 192 q^{64} + 360 q^{65} - 538 q^{67} - 584 q^{68} + 20 q^{70} - 600 q^{71} + 552 q^{73} - 408 q^{74} + 88 q^{76} - 570 q^{77} + 208 q^{79} + 240 q^{80} + 940 q^{82} - 900 q^{83} - 730 q^{85} - 808 q^{86} + 336 q^{88} - 6 q^{89} - 1842 q^{91} + 276 q^{92} + 396 q^{94} + 110 q^{95} - 518 q^{97} - 1690 q^{98}+O(q^{100}) $ 3 * q - 6 * q^2 + 12 * q^4 + 15 * q^5 - 2 * q^7 - 24 * q^8 - 30 * q^10 - 42 * q^11 + 72 * q^13 + 4 * q^14 + 48 * q^16 - 146 * q^17 + 22 * q^19 + 60 * q^20 + 84 * q^22 + 69 * q^23 + 75 * q^25 - 144 * q^26 - 8 * q^28 - 214 * q^29 + 130 * q^31 - 96 * q^32 + 292 * q^34 - 10 * q^35 + 204 * q^37 - 44 * q^38 - 120 * q^40 - 470 * q^41 + 404 * q^43 - 168 * q^44 - 138 * q^46 - 198 * q^47 + 845 * q^49 - 150 * q^50 + 288 * q^52 - 870 * q^53 - 210 * q^55 + 16 * q^56 + 428 * q^58 - 512 * q^59 - 64 * q^61 - 260 * q^62 + 192 * q^64 + 360 * q^65 - 538 * q^67 - 584 * q^68 + 20 * q^70 - 600 * q^71 + 552 * q^73 - 408 * q^74 + 88 * q^76 - 570 * q^77 + 208 * q^79 + 240 * q^80 + 940 * q^82 - 900 * q^83 - 730 * q^85 - 808 * q^86 + 336 * q^88 - 6 * q^89 - 1842 * q^91 + 276 * q^92 + 396 * q^94 + 110 * q^95 - 518 * q^97 - 1690 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	8.08465	0.436530	0.218265	−	0.975890i	$-0.429960\pi$
$7$	8.08465	0.436530	0.218265	+	0.975890i	$0.429960\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	−38.9023	−1.06632	−0.533158	−	0.846015i	$-0.678995\pi$
$11$	−38.9023	−1.06632	−0.533158	+	0.846015i	$0.678995\pi$
$12$	0	0
$13$	−50.7069	−1.08181	−0.540906	−	0.841083i	$-0.681919\pi$
$13$	−50.7069	−1.08181	−0.540906	+	0.841083i	$0.681919\pi$
$14$	−16.1693	−0.308673
$15$	0	0
$16$	16.0000	0.250000
$17$	−64.8176	−0.924740	−0.462370	−	0.886687i	$-0.653001\pi$
$17$	−64.8176	−0.924740	−0.462370	+	0.886687i	$0.653001\pi$
$18$	0	0
$19$	64.5376	0.779260	0.389630	−	0.920972i	$-0.372603\pi$
$19$	64.5376	0.779260	0.389630	+	0.920972i	$0.372603\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	77.8046	0.754000
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	101.414	0.764957
$27$	0	0
$28$	32.3386	0.218265
$29$	144.036	0.922304	0.461152	−	0.887321i	$-0.347436\pi$
$29$	144.036	0.922304	0.461152	+	0.887321i	$0.347436\pi$
$30$	0	0
$31$	248.600	1.44032	0.720158	−	0.693810i	$-0.244069\pi$
$31$	248.600	1.44032	0.720158	+	0.693810i	$0.244069\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	129.635	0.653890
$35$	40.4233	0.195222
$36$	0	0
$37$	290.769	1.29195	0.645975	−	0.763359i	$-0.276451\pi$
$37$	290.769	1.29195	0.645975	+	0.763359i	$0.276451\pi$
$38$	−129.075	−0.551020
$39$	0	0
$40$	−40.0000	−0.158114
$41$	123.307	0.469689	0.234844	−	0.972033i	$-0.424542\pi$
$41$	123.307	0.469689	0.234844	+	0.972033i	$0.424542\pi$
$42$	0	0
$43$	96.9581	0.343860	0.171930	−	0.985109i	$-0.445000\pi$
$43$	96.9581	0.343860	0.171930	+	0.985109i	$0.445000\pi$
$44$	−155.609	−0.533158
$45$	0	0
$46$	−46.0000	−0.147442
$47$	−342.629	−1.06335	−0.531676	−	0.846948i	$-0.678437\pi$
$47$	−342.629	−1.06335	−0.531676	+	0.846948i	$0.678437\pi$
$48$	0	0
$49$	−277.638	−0.809441
$50$	−50.0000	−0.141421
$51$	0	0
$52$	−202.828	−0.540906
$53$	−39.6254	−0.102698	−0.0513488	−	0.998681i	$-0.516352\pi$
$53$	−39.6254	−0.102698	−0.0513488	+	0.998681i	$0.516352\pi$
$54$	0	0
$55$	−194.511	−0.476871
$56$	−64.6772	−0.154337
$57$	0	0
$58$	−288.072	−0.652167
$59$	−388.739	−0.857789	−0.428894	−	0.903355i	$-0.641097\pi$
$59$	−388.739	−0.857789	−0.428894	+	0.903355i	$0.641097\pi$
$60$	0	0
$61$	−178.147	−0.373924	−0.186962	−	0.982367i	$-0.559864\pi$
$61$	−178.147	−0.373924	−0.186962	+	0.982367i	$0.559864\pi$
$62$	−497.199	−1.01846
$63$	0	0
$64$	64.0000	0.125000
$65$	−253.534	−0.483801
$66$	0	0
$67$	−27.8576	−0.0507963	−0.0253981	−	0.999677i	$-0.508085\pi$
$67$	−27.8576	−0.0507963	−0.0253981	+	0.999677i	$0.508085\pi$
$68$	−259.271	−0.462370
$69$	0	0
$70$	−80.8465	−0.138043
$71$	−165.064	−0.275908	−0.137954	−	0.990439i	$-0.544053\pi$
$71$	−165.064	−0.275908	−0.137954	+	0.990439i	$0.544053\pi$
$72$	0	0
$73$	391.901	0.628336	0.314168	−	0.949367i	$-0.398275\pi$
$73$	391.901	0.628336	0.314168	+	0.949367i	$0.398275\pi$
$74$	−581.538	−0.913546
$75$	0	0
$76$	258.150	0.389630
$77$	−314.511	−0.465479
$78$	0	0
$79$	785.383	1.11851	0.559256	−	0.828995i	$-0.311087\pi$
$79$	785.383	1.11851	0.559256	+	0.828995i	$0.311087\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−246.613	−0.332120
$83$	−746.890	−0.987732	−0.493866	−	0.869538i	$-0.664417\pi$
$83$	−746.890	−0.987732	−0.493866	+	0.869538i	$0.664417\pi$
$84$	0	0
$85$	−324.088	−0.413556
$86$	−193.916	−0.243146
$87$	0	0
$88$	311.218	0.377000
$89$	−15.5166	−0.0184804	−0.00924021	−	0.999957i	$-0.502941\pi$
$89$	−15.5166	−0.0184804	−0.00924021	+	0.999957i	$0.502941\pi$
$90$	0	0
$91$	−409.947	−0.472244
$92$	92.0000	0.104257
$93$	0	0
$94$	685.257	0.751903
$95$	322.688	0.348496
$96$	0	0
$97$	−1180.85	−1.23606	−0.618029	−	0.786155i	$-0.712069\pi$
$97$	−1180.85	−1.23606	−0.618029	+	0.786155i	$0.712069\pi$
$98$	555.277	0.572362
$99$	0	0
$100$	100.000	0.100000
$101$	850.356	0.837758	0.418879	−	0.908042i	$-0.362423\pi$
$101$	850.356	0.837758	0.418879	+	0.908042i	$0.362423\pi$
$102$	0	0
$103$	−850.829	−0.813929	−0.406964	−	0.913444i	$-0.633413\pi$
$103$	−850.829	−0.813929	−0.406964	+	0.913444i	$0.633413\pi$
$104$	405.655	0.382478
$105$	0	0
$106$	79.2508	0.0726181
$107$	−983.125	−0.888246	−0.444123	−	0.895966i	$-0.646485\pi$
$107$	−983.125	−0.888246	−0.444123	+	0.895966i	$0.646485\pi$
$108$	0	0
$109$	−1374.23	−1.20759	−0.603797	−	0.797138i	$-0.706346\pi$
$109$	−1374.23	−1.20759	−0.603797	+	0.797138i	$0.706346\pi$
$110$	389.023	0.337199
$111$	0	0
$112$	129.354	0.109133
$113$	468.595	0.390103	0.195052	−	0.980793i	$-0.437513\pi$
$113$	468.595	0.390103	0.195052	+	0.980793i	$0.437513\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	576.144	0.461152
$117$	0	0
$118$	777.479	0.606548
$119$	−524.028	−0.403677
$120$	0	0
$121$	182.388	0.137031
$122$	356.293	0.264404
$123$	0	0
$124$	994.399	0.720158
$125$	125.000	0.0894427
$126$	0	0
$127$	−349.258	−0.244029	−0.122014	−	0.992528i	$-0.538935\pi$
$127$	−349.258	−0.244029	−0.122014	+	0.992528i	$0.538935\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	507.069	0.342099
$131$	−1236.33	−0.824569	−0.412285	−	0.911055i	$-0.635269\pi$
$131$	−1236.33	−0.824569	−0.412285	+	0.911055i	$0.635269\pi$
$132$	0	0
$133$	521.764	0.340170
$134$	55.7152	0.0359184
$135$	0	0
$136$	518.541	0.326945
$137$	525.555	0.327746	0.163873	−	0.986481i	$-0.447601\pi$
$137$	525.555	0.327746	0.163873	+	0.986481i	$0.447601\pi$
$138$	0	0
$139$	−694.253	−0.423639	−0.211819	−	0.977309i	$-0.567939\pi$
$139$	−694.253	−0.423639	−0.211819	+	0.977309i	$0.567939\pi$
$140$	161.693	0.0976111
$141$	0	0
$142$	330.127	0.195096
$143$	1972.61	1.15355
$144$	0	0
$145$	720.180	0.412467
$146$	−783.801	−0.444300
$147$	0	0
$148$	1163.08	0.645975
$149$	1205.93	0.663046	0.331523	−	0.943447i	$-0.392438\pi$
$149$	1205.93	0.663046	0.331523	+	0.943447i	$0.392438\pi$
$150$	0	0
$151$	−1345.87	−0.725334	−0.362667	−	0.931919i	$-0.618134\pi$
$151$	−1345.87	−0.725334	−0.362667	+	0.931919i	$0.618134\pi$
$152$	−516.301	−0.275510
$153$	0	0
$154$	629.023	0.329144
$155$	1243.00	0.644129
$156$	0	0
$157$	−809.506	−0.411501	−0.205750	−	0.978604i	$-0.565964\pi$
$157$	−809.506	−0.411501	−0.205750	+	0.978604i	$0.565964\pi$
$158$	−1570.77	−0.790908
$159$	0	0
$160$	−160.000	−0.0790569
$161$	185.947	0.0910228
$162$	0	0
$163$	304.481	0.146312	0.0731558	−	0.997321i	$-0.476693\pi$
$163$	304.481	0.146312	0.0731558	+	0.997321i	$0.476693\pi$
$164$	493.226	0.234844
$165$	0	0
$166$	1493.78	0.698432
$167$	350.891	0.162591	0.0812957	−	0.996690i	$-0.474094\pi$
$167$	350.891	0.162591	0.0812957	+	0.996690i	$0.474094\pi$
$168$	0	0
$169$	374.188	0.170318
$170$	648.176	0.292429
$171$	0	0
$172$	387.832	0.171930
$173$	2157.18	0.948017	0.474009	−	0.880520i	$-0.342807\pi$
$173$	2157.18	0.948017	0.474009	+	0.880520i	$0.342807\pi$
$174$	0	0
$175$	202.116	0.0873060
$176$	−622.437	−0.266579
$177$	0	0
$178$	31.0332	0.0130676
$179$	−3659.97	−1.52826	−0.764131	−	0.645061i	$-0.776832\pi$
$179$	−3659.97	−1.52826	−0.764131	+	0.645061i	$0.776832\pi$
$180$	0	0
$181$	888.887	0.365030	0.182515	−	0.983203i	$-0.441576\pi$
$181$	888.887	0.365030	0.182515	+	0.983203i	$0.441576\pi$
$182$	819.895	0.333927
$183$	0	0
$184$	−184.000	−0.0737210
$185$	1453.84	0.577777
$186$	0	0
$187$	2521.55	0.986066
$188$	−1370.51	−0.531676
$189$	0	0
$190$	−645.376	−0.246424
$191$	−614.493	−0.232791	−0.116396	−	0.993203i	$-0.537134\pi$
$191$	−614.493	−0.232791	−0.116396	+	0.993203i	$0.537134\pi$
$192$	0	0
$193$	−2736.43	−1.02058	−0.510292	−	0.860001i	$-0.670463\pi$
$193$	−2736.43	−1.02058	−0.510292	+	0.860001i	$0.670463\pi$
$194$	2361.71	0.874025
$195$	0	0
$196$	−1110.55	−0.404721
$197$	−414.411	−0.149876	−0.0749379	−	0.997188i	$-0.523876\pi$
$197$	−414.411	−0.149876	−0.0749379	+	0.997188i	$0.523876\pi$
$198$	0	0
$199$	−3091.20	−1.10115	−0.550577	−	0.834784i	$-0.685592\pi$
$199$	−3091.20	−1.10115	−0.550577	+	0.834784i	$0.685592\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−1700.71	−0.592384
$203$	1164.48	0.402613
$204$	0	0
$205$	616.533	0.210051
$206$	1701.66	0.575535
$207$	0	0
$208$	−811.310	−0.270453
$209$	−2510.66	−0.830938
$210$	0	0
$211$	−675.753	−0.220478	−0.110239	−	0.993905i	$-0.535162\pi$
$211$	−675.753	−0.220478	−0.110239	+	0.993905i	$0.535162\pi$
$212$	−158.502	−0.0513488
$213$	0	0
$214$	1966.25	0.628085
$215$	484.790	0.153779
$216$	0	0
$217$	2009.84	0.628742
$218$	2748.47	0.853897
$219$	0	0
$220$	−778.046	−0.238436
$221$	3286.70	1.00040
$222$	0	0
$223$	−480.899	−0.144410	−0.0722048	−	0.997390i	$-0.523004\pi$
$223$	−480.899	−0.144410	−0.0722048	+	0.997390i	$0.523004\pi$
$224$	−258.709	−0.0771684
$225$	0	0
$226$	−937.189	−0.275845
$227$	−4296.54	−1.25626	−0.628131	−	0.778108i	$-0.716180\pi$
$227$	−4296.54	−1.25626	−0.628131	+	0.778108i	$0.716180\pi$
$228$	0	0
$229$	−1601.12	−0.462031	−0.231016	−	0.972950i	$-0.574205\pi$
$229$	−1601.12	−0.462031	−0.231016	+	0.972950i	$0.574205\pi$
$230$	−230.000	−0.0659380
$231$	0	0
$232$	−1152.29	−0.326084
$233$	−3573.49	−1.00475	−0.502376	−	0.864650i	$-0.667541\pi$
$233$	−3573.49	−1.00475	−0.502376	+	0.864650i	$0.667541\pi$
$234$	0	0
$235$	−1713.14	−0.475545
$236$	−1554.96	−0.428894
$237$	0	0
$238$	1048.06	0.285443
$239$	−55.9933	−0.0151544	−0.00757721	−	0.999971i	$-0.502412\pi$
$239$	−55.9933	−0.0151544	−0.00757721	+	0.999971i	$0.502412\pi$
$240$	0	0
$241$	−2783.05	−0.743866	−0.371933	−	0.928259i	$-0.621305\pi$
$241$	−2783.05	−0.743866	−0.371933	+	0.928259i	$0.621305\pi$
$242$	−364.777	−0.0968956
$243$	0	0
$244$	−712.587	−0.186962
$245$	−1388.19	−0.361993
$246$	0	0
$247$	−3272.50	−0.843013
$248$	−1988.80	−0.509229
$249$	0	0
$250$	−250.000	−0.0632456
$251$	4804.56	1.20821	0.604106	−	0.796904i	$-0.293530\pi$
$251$	4804.56	1.20821	0.604106	+	0.796904i	$0.293530\pi$
$252$	0	0
$253$	−894.753	−0.222342
$254$	698.517	0.172554
$255$	0	0
$256$	256.000	0.0625000
$257$	−2399.01	−0.582281	−0.291140	−	0.956680i	$-0.594035\pi$
$257$	−2399.01	−0.582281	−0.291140	+	0.956680i	$0.594035\pi$
$258$	0	0
$259$	2350.77	0.563975
$260$	−1014.14	−0.241901
$261$	0	0
$262$	2472.66	0.583059
$263$	4069.70	0.954176	0.477088	−	0.878855i	$-0.341692\pi$
$263$	4069.70	0.954176	0.477088	+	0.878855i	$0.341692\pi$
$264$	0	0
$265$	−198.127	−0.0459277
$266$	−1043.53	−0.240537
$267$	0	0
$268$	−111.430	−0.0253981
$269$	1457.95	0.330456	0.165228	−	0.986255i	$-0.447164\pi$
$269$	1457.95	0.330456	0.165228	+	0.986255i	$0.447164\pi$
$270$	0	0
$271$	−2323.28	−0.520771	−0.260386	−	0.965505i	$-0.583850\pi$
$271$	−2323.28	−0.520771	−0.260386	+	0.965505i	$0.583850\pi$
$272$	−1037.08	−0.231185
$273$	0	0
$274$	−1051.11	−0.231752
$275$	−972.557	−0.213263
$276$	0	0
$277$	−7552.79	−1.63828	−0.819139	−	0.573595i	$-0.805549\pi$
$277$	−7552.79	−1.63828	−0.819139	+	0.573595i	$0.805549\pi$
$278$	1388.51	0.299558
$279$	0	0
$280$	−323.386	−0.0690215
$281$	589.094	0.125062	0.0625309	−	0.998043i	$-0.480083\pi$
$281$	589.094	0.125062	0.0625309	+	0.998043i	$0.480083\pi$
$282$	0	0
$283$	−4990.59	−1.04827	−0.524133	−	0.851636i	$-0.675611\pi$
$283$	−4990.59	−1.04827	−0.524133	+	0.851636i	$0.675611\pi$
$284$	−660.255	−0.137954
$285$	0	0
$286$	−3945.23	−0.815686
$287$	996.890	0.205033
$288$	0	0
$289$	−711.673	−0.144855
$290$	−1440.36	−0.291658
$291$	0	0
$292$	1567.60	0.314168
$293$	5916.06	1.17959	0.589796	−	0.807553i	$-0.299208\pi$
$293$	5916.06	1.17959	0.589796	+	0.807553i	$0.299208\pi$
$294$	0	0
$295$	−1943.70	−0.383615
$296$	−2326.15	−0.456773
$297$	0	0
$298$	−2411.86	−0.468844
$299$	−1166.26	−0.225573
$300$	0	0
$301$	783.872	0.150105
$302$	2691.74	0.512889
$303$	0	0
$304$	1032.60	0.194815
$305$	−890.734	−0.167224
$306$	0	0
$307$	4262.05	0.792339	0.396170	−	0.918177i	$-0.370339\pi$
$307$	4262.05	0.792339	0.396170	+	0.918177i	$0.370339\pi$
$308$	−1258.05	−0.232740
$309$	0	0
$310$	−2486.00	−0.455468
$311$	10270.4	1.87260	0.936301	−	0.351198i	$-0.114226\pi$
$311$	10270.4	1.87260	0.936301	+	0.351198i	$0.114226\pi$
$312$	0	0
$313$	−1844.55	−0.333100	−0.166550	−	0.986033i	$-0.553263\pi$
$313$	−1844.55	−0.333100	−0.166550	+	0.986033i	$0.553263\pi$
$314$	1619.01	0.290975
$315$	0	0
$316$	3141.53	0.559256
$317$	−1829.65	−0.324175	−0.162087	−	0.986776i	$-0.551823\pi$
$317$	−1829.65	−0.324175	−0.162087	+	0.986776i	$0.551823\pi$
$318$	0	0
$319$	−5603.33	−0.983468
$320$	320.000	0.0559017
$321$	0	0
$322$	−371.894	−0.0643629
$323$	−4183.17	−0.720613
$324$	0	0
$325$	−1267.67	−0.216362
$326$	−608.962	−0.103458
$327$	0	0
$328$	−986.452	−0.166060
$329$	−2770.03	−0.464185
$330$	0	0
$331$	722.273	0.119939	0.0599694	−	0.998200i	$-0.480900\pi$
$331$	722.273	0.119939	0.0599694	+	0.998200i	$0.480900\pi$
$332$	−2987.56	−0.493866
$333$	0	0
$334$	−701.782	−0.114970
$335$	−139.288	−0.0227168
$336$	0	0
$337$	−9733.11	−1.57328	−0.786642	−	0.617410i	$-0.788182\pi$
$337$	−9733.11	−1.57328	−0.786642	+	0.617410i	$0.788182\pi$
$338$	−748.375	−0.120433
$339$	0	0
$340$	−1296.35	−0.206778
$341$	−9671.10	−1.53583
$342$	0	0
$343$	−5017.65	−0.789876
$344$	−775.665	−0.121573
$345$	0	0
$346$	−4314.35	−0.670350
$347$	3773.98	0.583856	0.291928	−	0.956440i	$-0.405703\pi$
$347$	3773.98	0.583856	0.291928	+	0.956440i	$0.405703\pi$
$348$	0	0
$349$	7255.01	1.11276	0.556378	−	0.830929i	$-0.312191\pi$
$349$	7255.01	1.11276	0.556378	+	0.830929i	$0.312191\pi$
$350$	−404.233	−0.0617347
$351$	0	0
$352$	1244.87	0.188500
$353$	−2160.96	−0.325825	−0.162912	−	0.986641i	$-0.552089\pi$
$353$	−2160.96	−0.325825	−0.162912	+	0.986641i	$0.552089\pi$
$354$	0	0
$355$	−825.318	−0.123390
$356$	−62.0664	−0.00924021
$357$	0	0
$358$	7319.94	1.08064
$359$	−268.024	−0.0394033	−0.0197016	−	0.999806i	$-0.506272\pi$
$359$	−268.024	−0.0394033	−0.0197016	+	0.999806i	$0.506272\pi$
$360$	0	0
$361$	−2693.90	−0.392754
$362$	−1777.77	−0.258115
$363$	0	0
$364$	−1639.79	−0.236122
$365$	1959.50	0.281000
$366$	0	0
$367$	8594.01	1.22235	0.611177	−	0.791494i	$-0.290696\pi$
$367$	8594.01	1.22235	0.611177	+	0.791494i	$0.290696\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	−2907.69	−0.408550
$371$	−320.358	−0.0448306
$372$	0	0
$373$	−13669.9	−1.89759	−0.948795	−	0.315893i	$-0.897696\pi$
$373$	−13669.9	−1.89759	−0.948795	+	0.315893i	$0.897696\pi$
$374$	−5043.11	−0.697254
$375$	0	0
$376$	2741.03	0.375951
$377$	−7303.62	−0.997759
$378$	0	0
$379$	9688.36	1.31308	0.656540	−	0.754291i	$-0.272019\pi$
$379$	9688.36	1.31308	0.656540	+	0.754291i	$0.272019\pi$
$380$	1290.75	0.174248
$381$	0	0
$382$	1228.99	0.164608
$383$	−11860.7	−1.58238	−0.791189	−	0.611571i	$-0.790538\pi$
$383$	−11860.7	−1.58238	−0.791189	+	0.611571i	$0.790538\pi$
$384$	0	0
$385$	−1572.56	−0.208169
$386$	5472.87	0.721662
$387$	0	0
$388$	−4723.42	−0.618029
$389$	7485.02	0.975593	0.487796	−	0.872957i	$-0.337801\pi$
$389$	7485.02	0.975593	0.487796	+	0.872957i	$0.337801\pi$
$390$	0	0
$391$	−1490.81	−0.192822
$392$	2221.11	0.286181
$393$	0	0
$394$	828.821	0.105978
$395$	3926.91	0.500214
$396$	0	0
$397$	6915.96	0.874313	0.437156	−	0.899386i	$-0.355986\pi$
$397$	6915.96	0.874313	0.437156	+	0.899386i	$0.355986\pi$
$398$	6182.41	0.778633
$399$	0	0
$400$	400.000	0.0500000
$401$	2.40567	0.000299585 0	0.000149793	−	1.00000i	$-0.499952\pi$
$401$	2.40567	0.000299585 0	0.000149793		1.00000i	$0.499952\pi$
$402$	0	0
$403$	−12605.7	−1.55815
$404$	3401.42	0.418879
$405$	0	0
$406$	−2328.96	−0.284691
$407$	−11311.6	−1.37763
$408$	0	0
$409$	3297.76	0.398689	0.199345	−	0.979929i	$-0.436119\pi$
$409$	3297.76	0.398689	0.199345	+	0.979929i	$0.436119\pi$
$410$	−1233.07	−0.148529
$411$	0	0
$412$	−3403.32	−0.406964
$413$	−3142.82	−0.374451
$414$	0	0
$415$	−3734.45	−0.441727
$416$	1622.62	0.191239
$417$	0	0
$418$	5021.32	0.587562
$419$	−5326.05	−0.620989	−0.310495	−	0.950575i	$-0.600495\pi$
$419$	−5326.05	−0.620989	−0.310495	+	0.950575i	$0.600495\pi$
$420$	0	0
$421$	−17060.3	−1.97498	−0.987492	−	0.157672i	$-0.949601\pi$
$421$	−17060.3	−1.97498	−0.987492	+	0.157672i	$0.949601\pi$
$422$	1351.51	0.155901
$423$	0	0
$424$	317.003	0.0363091
$425$	−1620.44	−0.184948
$426$	0	0
$427$	−1440.25	−0.163229
$428$	−3932.50	−0.444123
$429$	0	0
$430$	−969.581	−0.108738
$431$	434.550	0.0485651	0.0242825	−	0.999705i	$-0.492270\pi$
$431$	434.550	0.0485651	0.0242825	+	0.999705i	$0.492270\pi$
$432$	0	0
$433$	−1494.21	−0.165836	−0.0829179	−	0.996556i	$-0.526424\pi$
$433$	−1494.21	−0.165836	−0.0829179	+	0.996556i	$0.526424\pi$
$434$	−4019.68	−0.444587
$435$	0	0
$436$	−5496.93	−0.603797
$437$	1484.36	0.162487
$438$	0	0
$439$	3066.59	0.333394	0.166697	−	0.986008i	$-0.446690\pi$
$439$	3066.59	0.333394	0.166697	+	0.986008i	$0.446690\pi$
$440$	1556.09	0.168599
$441$	0	0
$442$	−6573.40	−0.707386
$443$	1415.68	0.151831	0.0759155	−	0.997114i	$-0.475812\pi$
$443$	1415.68	0.151831	0.0759155	+	0.997114i	$0.475812\pi$
$444$	0	0
$445$	−77.5830	−0.00826469
$446$	961.797	0.102113
$447$	0	0
$448$	517.418	0.0545663
$449$	−6699.91	−0.704206	−0.352103	−	0.935961i	$-0.614533\pi$
$449$	−6699.91	−0.704206	−0.352103	+	0.935961i	$0.614533\pi$
$450$	0	0
$451$	−4796.91	−0.500837
$452$	1874.38	0.195052
$453$	0	0
$454$	8593.08	0.888311
$455$	−2049.74	−0.211194
$456$	0	0
$457$	−4675.83	−0.478613	−0.239307	−	0.970944i	$-0.576920\pi$
$457$	−4675.83	−0.478613	−0.239307	+	0.970944i	$0.576920\pi$
$458$	3202.25	0.326705
$459$	0	0
$460$	460.000	0.0466252
$461$	−9693.07	−0.979287	−0.489644	−	0.871923i	$-0.662873\pi$
$461$	−9693.07	−0.979287	−0.489644	+	0.871923i	$0.662873\pi$
$462$	0	0
$463$	−8891.84	−0.892525	−0.446262	−	0.894902i	$-0.647245\pi$
$463$	−8891.84	−0.892525	−0.446262	+	0.894902i	$0.647245\pi$
$464$	2304.58	0.230576
$465$	0	0
$466$	7146.98	0.710466
$467$	−14889.0	−1.47533	−0.737666	−	0.675166i	$-0.764072\pi$
$467$	−14889.0	−1.47533	−0.737666	+	0.675166i	$0.764072\pi$
$468$	0	0
$469$	−225.219	−0.0221741
$470$	3426.29	0.336261
$471$	0	0
$472$	3109.91	0.303274
$473$	−3771.89	−0.366663
$474$	0	0
$475$	1613.44	0.155852
$476$	−2096.11	−0.201839
$477$	0	0
$478$	111.987	0.0107158
$479$	17786.4	1.69662	0.848311	−	0.529498i	$-0.177620\pi$
$479$	17786.4	1.69662	0.848311	+	0.529498i	$0.177620\pi$
$480$	0	0
$481$	−14744.0	−1.39765
$482$	5566.09	0.525993
$483$	0	0
$484$	729.554	0.0685156
$485$	−5904.27	−0.552782
$486$	0	0
$487$	20479.4	1.90556	0.952780	−	0.303660i	$-0.0982088\pi$
$487$	20479.4	1.90556	0.952780	+	0.303660i	$0.0982088\pi$
$488$	1425.17	0.132202
$489$	0	0
$490$	2776.38	0.255968
$491$	9025.39	0.829552	0.414776	−	0.909924i	$-0.363860\pi$
$491$	9025.39	0.829552	0.414776	+	0.909924i	$0.363860\pi$
$492$	0	0
$493$	−9336.07	−0.852892
$494$	6545.00	0.596100
$495$	0	0
$496$	3977.59	0.360079
$497$	−1334.48	−0.120442
$498$	0	0
$499$	−14066.8	−1.26196	−0.630978	−	0.775801i	$-0.717346\pi$
$499$	−14066.8	−1.26196	−0.630978	+	0.775801i	$0.717346\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−9609.13	−0.854335
$503$	7109.56	0.630218	0.315109	−	0.949055i	$-0.397959\pi$
$503$	7109.56	0.630218	0.315109	+	0.949055i	$0.397959\pi$
$504$	0	0
$505$	4251.78	0.374657
$506$	1789.51	0.157220
$507$	0	0
$508$	−1397.03	−0.122014
$509$	−21956.4	−1.91199	−0.955993	−	0.293390i	$-0.905217\pi$
$509$	−21956.4	−1.91199	−0.955993	+	0.293390i	$0.905217\pi$
$510$	0	0
$511$	3168.38	0.274287
$512$	−512.000	−0.0441942
$513$	0	0
$514$	4798.02	0.411735
$515$	−4254.15	−0.364000
$516$	0	0
$517$	13329.0	1.13387
$518$	−4701.53	−0.398790
$519$	0	0
$520$	2028.28	0.171050
$521$	1075.53	0.0904410	0.0452205	−	0.998977i	$-0.485601\pi$
$521$	1075.53	0.0904410	0.0452205	+	0.998977i	$0.485601\pi$
$522$	0	0
$523$	−16456.0	−1.37585	−0.687926	−	0.725781i	$-0.741478\pi$
$523$	−16456.0	−1.37585	−0.687926	+	0.725781i	$0.741478\pi$
$524$	−4945.32	−0.412285
$525$	0	0
$526$	−8139.40	−0.674704
$527$	−16113.6	−1.33192
$528$	0	0
$529$	529.000	0.0434783
$530$	396.254	0.0324758
$531$	0	0
$532$	2087.06	0.170085
$533$	−6252.49	−0.508115
$534$	0	0
$535$	−4915.63	−0.397236
$536$	222.861	0.0179592
$537$	0	0
$538$	−2915.89	−0.233668
$539$	10800.8	0.863121
$540$	0	0
$541$	19194.3	1.52538	0.762688	−	0.646766i	$-0.223879\pi$
$541$	19194.3	1.52538	0.762688	+	0.646766i	$0.223879\pi$
$542$	4646.56	0.368241
$543$	0	0
$544$	2074.16	0.163473
$545$	−6871.16	−0.540052
$546$	0	0
$547$	1034.95	0.0808985	0.0404492	−	0.999182i	$-0.487121\pi$
$547$	1034.95	0.0808985	0.0404492	+	0.999182i	$0.487121\pi$
$548$	2102.22	0.163873
$549$	0	0
$550$	1945.11	0.150800
$551$	9295.73	0.718714
$552$	0	0
$553$	6349.55	0.488264
$554$	15105.6	1.15844
$555$	0	0
$556$	−2777.01	−0.211819
$557$	−18850.4	−1.43397	−0.716983	−	0.697091i	$-0.754477\pi$
$557$	−18850.4	−1.43397	−0.716983	+	0.697091i	$0.754477\pi$
$558$	0	0
$559$	−4916.44	−0.371992
$560$	646.772	0.0488056
$561$	0	0
$562$	−1178.19	−0.0884321
$563$	−18344.9	−1.37326	−0.686629	−	0.727008i	$-0.740910\pi$
$563$	−18344.9	−1.37326	−0.686629	+	0.727008i	$0.740910\pi$
$564$	0	0
$565$	2342.97	0.174459
$566$	9981.17	0.741237
$567$	0	0
$568$	1320.51	0.0975481
$569$	6296.12	0.463879	0.231940	−	0.972730i	$-0.425493\pi$
$569$	6296.12	0.463879	0.231940	+	0.972730i	$0.425493\pi$
$570$	0	0
$571$	−11309.7	−0.828889	−0.414445	−	0.910075i	$-0.636024\pi$
$571$	−11309.7	−0.828889	−0.414445	+	0.910075i	$0.636024\pi$
$572$	7890.46	0.576777
$573$	0	0
$574$	−1993.78	−0.144981
$575$	575.000	0.0417029
$576$	0	0
$577$	20068.7	1.44796	0.723978	−	0.689823i	$-0.242312\pi$
$577$	20068.7	1.44796	0.723978	+	0.689823i	$0.242312\pi$
$578$	1423.35	0.102428
$579$	0	0
$580$	2880.72	0.206233
$581$	−6038.34	−0.431175
$582$	0	0
$583$	1541.52	0.109508
$584$	−3135.21	−0.222150
$585$	0	0
$586$	−11832.1	−0.834097
$587$	−13033.6	−0.916447	−0.458224	−	0.888837i	$-0.651514\pi$
$587$	−13033.6	−0.916447	−0.458224	+	0.888837i	$0.651514\pi$
$588$	0	0
$589$	16044.0	1.12238
$590$	3887.39	0.271257
$591$	0	0
$592$	4652.30	0.322987
$593$	−21552.2	−1.49248	−0.746242	−	0.665675i	$-0.768144\pi$
$593$	−21552.2	−1.49248	−0.746242	+	0.665675i	$0.768144\pi$
$594$	0	0
$595$	−2620.14	−0.180530
$596$	4823.73	0.331523
$597$	0	0
$598$	2332.52	0.159504
$599$	3370.90	0.229936	0.114968	−	0.993369i	$-0.463324\pi$
$599$	3370.90	0.229936	0.114968	+	0.993369i	$0.463324\pi$
$600$	0	0
$601$	−22601.3	−1.53398	−0.766992	−	0.641657i	$-0.778247\pi$
$601$	−22601.3	−1.53398	−0.766992	+	0.641657i	$0.778247\pi$
$602$	−1567.74	−0.106140
$603$	0	0
$604$	−5383.48	−0.362667
$605$	911.942	0.0612822
$606$	0	0
$607$	13198.8	0.882576	0.441288	−	0.897366i	$-0.354522\pi$
$607$	13198.8	0.882576	0.441288	+	0.897366i	$0.354522\pi$
$608$	−2065.20	−0.137755
$609$	0	0
$610$	1781.47	0.118245
$611$	17373.6	1.15035
$612$	0	0
$613$	−6164.31	−0.406157	−0.203078	−	0.979162i	$-0.565095\pi$
$613$	−6164.31	−0.406157	−0.203078	+	0.979162i	$0.565095\pi$
$614$	−8524.10	−0.560268
$615$	0	0
$616$	2516.09	0.164572
$617$	−19285.7	−1.25837	−0.629185	−	0.777256i	$-0.716611\pi$
$617$	−19285.7	−1.25837	−0.629185	+	0.777256i	$0.716611\pi$
$618$	0	0
$619$	26882.2	1.74554	0.872769	−	0.488133i	$-0.162322\pi$
$619$	26882.2	1.74554	0.872769	+	0.488133i	$0.162322\pi$
$620$	4971.99	0.322065
$621$	0	0
$622$	−20540.7	−1.32413
$623$	−125.446	−0.00806726
$624$	0	0
$625$	625.000	0.0400000
$626$	3689.10	0.235537
$627$	0	0
$628$	−3238.03	−0.205750
$629$	−18847.0	−1.19472
$630$	0	0
$631$	−14665.0	−0.925202	−0.462601	−	0.886567i	$-0.653084\pi$
$631$	−14665.0	−0.925202	−0.462601	+	0.886567i	$0.653084\pi$
$632$	−6283.06	−0.395454
$633$	0	0
$634$	3659.30	0.229226
$635$	−1746.29	−0.109133
$636$	0	0
$637$	14078.2	0.875664
$638$	11206.7	0.695417
$639$	0	0
$640$	−640.000	−0.0395285
$641$	22943.2	1.41373	0.706865	−	0.707349i	$-0.250109\pi$
$641$	22943.2	1.41373	0.706865	+	0.707349i	$0.250109\pi$
$642$	0	0
$643$	4723.28	0.289686	0.144843	−	0.989455i	$-0.453732\pi$
$643$	4723.28	0.289686	0.144843	+	0.989455i	$0.453732\pi$
$644$	743.788	0.0455114
$645$	0	0
$646$	8366.35	0.509550
$647$	−10716.3	−0.651162	−0.325581	−	0.945514i	$-0.605560\pi$
$647$	−10716.3	−0.651162	−0.325581	+	0.945514i	$0.605560\pi$
$648$	0	0
$649$	15122.9	0.914675
$650$	2535.34	0.152991
$651$	0	0
$652$	1217.92	0.0731558
$653$	−30846.6	−1.84858	−0.924289	−	0.381694i	$-0.875341\pi$
$653$	−30846.6	−1.84858	−0.924289	+	0.381694i	$0.875341\pi$
$654$	0	0
$655$	−6181.65	−0.368759
$656$	1972.90	0.117422
$657$	0	0
$658$	5540.06	0.328228
$659$	10095.1	0.596734	0.298367	−	0.954451i	$-0.403558\pi$
$659$	10095.1	0.596734	0.298367	+	0.954451i	$0.403558\pi$
$660$	0	0
$661$	26454.4	1.55667	0.778334	−	0.627850i	$-0.216065\pi$
$661$	26454.4	1.55667	0.778334	+	0.627850i	$0.216065\pi$
$662$	−1444.55	−0.0848095
$663$	0	0
$664$	5975.12	0.349216
$665$	2608.82	0.152129
$666$	0	0
$667$	3312.83	0.192314
$668$	1403.56	0.0812957
$669$	0	0
$670$	278.576	0.0160632
$671$	6930.32	0.398721
$672$	0	0
$673$	19675.0	1.12692	0.563460	−	0.826143i	$-0.309470\pi$
$673$	19675.0	1.12692	0.563460	+	0.826143i	$0.309470\pi$
$674$	19466.2	1.11248
$675$	0	0
$676$	1496.75	0.0851588
$677$	−13785.6	−0.782606	−0.391303	−	0.920262i	$-0.627976\pi$
$677$	−13785.6	−0.782606	−0.391303	+	0.920262i	$0.627976\pi$
$678$	0	0
$679$	−9546.80	−0.539576
$680$	2592.71	0.146214
$681$	0	0
$682$	19342.2	1.08600
$683$	−24286.4	−1.36060	−0.680302	−	0.732932i	$-0.738151\pi$
$683$	−24286.4	−1.36060	−0.680302	+	0.732932i	$0.738151\pi$
$684$	0	0
$685$	2627.78	0.146573
$686$	10035.3	0.558526
$687$	0	0
$688$	1551.33	0.0859649
$689$	2009.28	0.111099
$690$	0	0
$691$	−14518.4	−0.799284	−0.399642	−	0.916671i	$-0.630866\pi$
$691$	−14518.4	−0.799284	−0.399642	+	0.916671i	$0.630866\pi$
$692$	8628.70	0.474009
$693$	0	0
$694$	−7547.96	−0.412848
$695$	−3471.27	−0.189457
$696$	0	0
$697$	−7992.44	−0.434340
$698$	−14510.0	−0.786838
$699$	0	0
$700$	808.465	0.0436530
$701$	26083.3	1.40535	0.702677	−	0.711509i	$-0.251988\pi$
$701$	26083.3	1.40535	0.702677	+	0.711509i	$0.251988\pi$
$702$	0	0
$703$	18765.5	1.00676
$704$	−2489.75	−0.133290
$705$	0	0
$706$	4321.91	0.230393
$707$	6874.83	0.365707
$708$	0	0
$709$	−1831.92	−0.0970371	−0.0485185	−	0.998822i	$-0.515450\pi$
$709$	−1831.92	−0.0970371	−0.0485185	+	0.998822i	$0.515450\pi$
$710$	1650.64	0.0872497
$711$	0	0
$712$	124.133	0.00653381
$713$	5717.79	0.300327
$714$	0	0
$715$	9863.07	0.515885
$716$	−14639.9	−0.764131
$717$	0	0
$718$	536.049	0.0278623
$719$	37039.1	1.92117	0.960587	−	0.277980i	$-0.0896650\pi$
$719$	37039.1	1.92117	0.960587	+	0.277980i	$0.0896650\pi$
$720$	0	0
$721$	−6878.66	−0.355305
$722$	5387.80	0.277719
$723$	0	0
$724$	3555.55	0.182515
$725$	3600.90	0.184461
$726$	0	0
$727$	21356.1	1.08948	0.544740	−	0.838605i	$-0.316628\pi$
$727$	21356.1	1.08948	0.544740	+	0.838605i	$0.316628\pi$
$728$	3279.58	0.166963
$729$	0	0
$730$	−3919.01	−0.198697
$731$	−6284.59	−0.317981
$732$	0	0
$733$	28255.7	1.42380	0.711901	−	0.702280i	$-0.247834\pi$
$733$	28255.7	1.42380	0.711901	+	0.702280i	$0.247834\pi$
$734$	−17188.0	−0.864334
$735$	0	0
$736$	−736.000	−0.0368605
$737$	1083.72	0.0541649
$738$	0	0
$739$	21382.6	1.06437	0.532186	−	0.846628i	$-0.321371\pi$
$739$	21382.6	1.06437	0.532186	+	0.846628i	$0.321371\pi$
$740$	5815.38	0.288889
$741$	0	0
$742$	640.715	0.0317000
$743$	−14472.0	−0.714573	−0.357286	−	0.933995i	$-0.616298\pi$
$743$	−14472.0	−0.714573	−0.357286	+	0.933995i	$0.616298\pi$
$744$	0	0
$745$	6029.66	0.296523
$746$	27339.8	1.34180
$747$	0	0
$748$	10086.2	0.493033
$749$	−7948.23	−0.387746
$750$	0	0
$751$	−10350.6	−0.502927	−0.251464	−	0.967867i	$-0.580912\pi$
$751$	−10350.6	−0.502927	−0.251464	+	0.967867i	$0.580912\pi$
$752$	−5482.06	−0.265838
$753$	0	0
$754$	14607.2	0.705522
$755$	−6729.35	−0.324379
$756$	0	0
$757$	−922.620	−0.0442975	−0.0221487	−	0.999755i	$-0.507051\pi$
$757$	−922.620	−0.0442975	−0.0221487	+	0.999755i	$0.507051\pi$
$758$	−19376.7	−0.928488
$759$	0	0
$760$	−2581.50	−0.123212
$761$	−23019.0	−1.09650	−0.548250	−	0.836315i	$-0.684706\pi$
$761$	−23019.0	−1.09650	−0.548250	+	0.836315i	$0.684706\pi$
$762$	0	0
$763$	−11110.2	−0.527151
$764$	−2457.97	−0.116396
$765$	0	0
$766$	23721.3	1.11891
$767$	19711.8	0.927966
$768$	0	0
$769$	31321.3	1.46876	0.734380	−	0.678738i	$-0.237473\pi$
$769$	31321.3	1.46876	0.734380	+	0.678738i	$0.237473\pi$
$770$	3145.11	0.147197
$771$	0	0
$772$	−10945.7	−0.510292
$773$	−26124.8	−1.21558	−0.607790	−	0.794097i	$-0.707944\pi$
$773$	−26124.8	−1.21558	−0.607790	+	0.794097i	$0.707944\pi$
$774$	0	0
$775$	6214.99	0.288063
$776$	9446.84	0.437012
$777$	0	0
$778$	−14970.0	−0.689848
$779$	7957.91	0.366010
$780$	0	0
$781$	6421.36	0.294205
$782$	2981.61	0.136346
$783$	0	0
$784$	−4442.21	−0.202360
$785$	−4047.53	−0.184029
$786$	0	0
$787$	−34670.9	−1.57037	−0.785187	−	0.619258i	$-0.787433\pi$
$787$	−34670.9	−1.57037	−0.785187	+	0.619258i	$0.787433\pi$
$788$	−1657.64	−0.0749379
$789$	0	0
$790$	−7853.83	−0.353705
$791$	3788.42	0.170292
$792$	0	0
$793$	9033.27	0.404515
$794$	−13831.9	−0.618232
$795$	0	0
$796$	−12364.8	−0.550577
$797$	37989.3	1.68839	0.844197	−	0.536033i	$-0.180078\pi$
$797$	37989.3	1.68839	0.844197	+	0.536033i	$0.180078\pi$
$798$	0	0
$799$	22208.4	0.983324
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−4.81135	−0.000211839 0
$803$	−15245.8	−0.670005
$804$	0	0
$805$	929.735	0.0407066
$806$	25211.4	1.10178
$807$	0	0
$808$	−6802.84	−0.296192
$809$	6179.77	0.268565	0.134283	−	0.990943i	$-0.457127\pi$
$809$	6179.77	0.268565	0.134283	+	0.990943i	$0.457127\pi$
$810$	0	0
$811$	7374.64	0.319308	0.159654	−	0.987173i	$-0.448962\pi$
$811$	7374.64	0.319308	0.159654	+	0.987173i	$0.448962\pi$
$812$	4657.92	0.201307
$813$	0	0
$814$	22623.2	0.974129
$815$	1522.40	0.0654325
$816$	0	0
$817$	6257.44	0.267956
$818$	−6595.52	−0.281916
$819$	0	0
$820$	2466.13	0.105026
$821$	7400.58	0.314594	0.157297	−	0.987551i	$-0.449722\pi$
$821$	7400.58	0.314594	0.157297	+	0.987551i	$0.449722\pi$
$822$	0	0
$823$	−23872.2	−1.01109	−0.505547	−	0.862799i	$-0.668709\pi$
$823$	−23872.2	−1.01109	−0.505547	+	0.862799i	$0.668709\pi$
$824$	6806.63	0.287767
$825$	0	0
$826$	6285.64	0.264777
$827$	9555.16	0.401772	0.200886	−	0.979615i	$-0.435618\pi$
$827$	9555.16	0.401772	0.200886	+	0.979615i	$0.435618\pi$
$828$	0	0
$829$	−34160.5	−1.43117	−0.715586	−	0.698525i	$-0.753840\pi$
$829$	−34160.5	−1.43117	−0.715586	+	0.698525i	$0.753840\pi$
$830$	7468.90	0.312348
$831$	0	0
$832$	−3245.24	−0.135227
$833$	17995.9	0.748523
$834$	0	0
$835$	1754.46	0.0727131
$836$	−10042.6	−0.415469
$837$	0	0
$838$	10652.1	0.439106
$839$	21028.4	0.865293	0.432646	−	0.901564i	$-0.357580\pi$
$839$	21028.4	0.865293	0.432646	+	0.901564i	$0.357580\pi$
$840$	0	0
$841$	−3642.63	−0.149356
$842$	34120.6	1.39652
$843$	0	0
$844$	−2703.01	−0.110239
$845$	1870.94	0.0761683
$846$	0	0
$847$	1474.55	0.0598182
$848$	−634.007	−0.0256744
$849$	0	0
$850$	3240.88	0.130778
$851$	6687.69	0.269390
$852$	0	0
$853$	−20932.3	−0.840221	−0.420111	−	0.907473i	$-0.638009\pi$
$853$	−20932.3	−0.840221	−0.420111	+	0.907473i	$0.638009\pi$
$854$	2880.51	0.115420
$855$	0	0
$856$	7865.00	0.314042
$857$	12884.3	0.513558	0.256779	−	0.966470i	$-0.417339\pi$
$857$	12884.3	0.513558	0.256779	+	0.966470i	$0.417339\pi$
$858$	0	0
$859$	−29336.7	−1.16526	−0.582628	−	0.812739i	$-0.697976\pi$
$859$	−29336.7	−1.16526	−0.582628	+	0.812739i	$0.697976\pi$
$860$	1939.16	0.0768894
$861$	0	0
$862$	−869.101	−0.0343407
$863$	24878.6	0.981316	0.490658	−	0.871352i	$-0.336756\pi$
$863$	24878.6	0.981316	0.490658	+	0.871352i	$0.336756\pi$
$864$	0	0
$865$	10785.9	0.423966
$866$	2988.41	0.117264
$867$	0	0
$868$	8039.37	0.314371
$869$	−30553.2	−1.19269
$870$	0	0
$871$	1412.57	0.0549520
$872$	10993.9	0.426949
$873$	0	0
$874$	−2968.73	−0.114896
$875$	1010.58	0.0390444
$876$	0	0
$877$	20456.0	0.787628	0.393814	−	0.919190i	$-0.371155\pi$
$877$	20456.0	0.787628	0.393814	+	0.919190i	$0.371155\pi$
$878$	−6133.17	−0.235745
$879$	0	0
$880$	−3112.18	−0.119218
$881$	−22079.5	−0.844354	−0.422177	−	0.906513i	$-0.638734\pi$
$881$	−22079.5	−0.844354	−0.422177	+	0.906513i	$0.638734\pi$
$882$	0	0
$883$	21295.5	0.811607	0.405804	−	0.913960i	$-0.366992\pi$
$883$	21295.5	0.811607	0.405804	+	0.913960i	$0.366992\pi$
$884$	13146.8	0.500198
$885$	0	0
$886$	−2831.37	−0.107361
$887$	23006.2	0.870883	0.435442	−	0.900217i	$-0.356592\pi$
$887$	23006.2	0.870883	0.435442	+	0.900217i	$0.356592\pi$
$888$	0	0
$889$	−2823.63	−0.106526
$890$	155.166	0.00584402
$891$	0	0
$892$	−1923.59	−0.0722048
$893$	−22112.4	−0.828627
$894$	0	0
$895$	−18299.9	−0.683460
$896$	−1034.84	−0.0385842
$897$	0	0
$898$	13399.8	0.497949
$899$	35807.3	1.32841
$900$	0	0
$901$	2568.43	0.0949686
$902$	9593.82	0.354145
$903$	0	0
$904$	−3748.76	−0.137922
$905$	4444.43	0.163246
$906$	0	0
$907$	−17413.1	−0.637479	−0.318740	−	0.947842i	$-0.603260\pi$
$907$	−17413.1	−0.637479	−0.318740	+	0.947842i	$0.603260\pi$
$908$	−17186.2	−0.628131
$909$	0	0
$910$	4099.47	0.149337
$911$	−29875.5	−1.08652	−0.543259	−	0.839565i	$-0.682810\pi$
$911$	−29875.5	−1.08652	−0.543259	+	0.839565i	$0.682810\pi$
$912$	0	0
$913$	29055.7	1.05324
$914$	9351.66	0.338431
$915$	0	0
$916$	−6404.49	−0.231016
$917$	−9995.29	−0.359949
$918$	0	0
$919$	11585.3	0.415847	0.207923	−	0.978145i	$-0.433330\pi$
$919$	11585.3	0.415847	0.207923	+	0.978145i	$0.433330\pi$
$920$	−920.000	−0.0329690
$921$	0	0
$922$	19386.1	0.692461
$923$	8369.86	0.298480
$924$	0	0
$925$	7269.22	0.258390
$926$	17783.7	0.631110
$927$	0	0
$928$	−4609.15	−0.163042
$929$	2136.91	0.0754680	0.0377340	−	0.999288i	$-0.487986\pi$
$929$	2136.91	0.0754680	0.0377340	+	0.999288i	$0.487986\pi$
$930$	0	0
$931$	−17918.1	−0.630765
$932$	−14294.0	−0.502376
$933$	0	0
$934$	29778.0	1.04322
$935$	12607.8	0.440982
$936$	0	0
$937$	23155.5	0.807319	0.403659	−	0.914909i	$-0.367738\pi$
$937$	23155.5	0.807319	0.403659	+	0.914909i	$0.367738\pi$
$938$	450.438	0.0156795
$939$	0	0
$940$	−6852.57	−0.237773
$941$	−2006.95	−0.0695269	−0.0347634	−	0.999396i	$-0.511068\pi$
$941$	−2006.95	−0.0695269	−0.0347634	+	0.999396i	$0.511068\pi$
$942$	0	0
$943$	2836.05	0.0979369
$944$	−6219.83	−0.214447
$945$	0	0
$946$	7543.78	0.259270
$947$	−7896.67	−0.270969	−0.135484	−	0.990779i	$-0.543259\pi$
$947$	−7896.67	−0.270969	−0.135484	+	0.990779i	$0.543259\pi$
$948$	0	0
$949$	−19872.1	−0.679741
$950$	−3226.88	−0.110204
$951$	0	0
$952$	4192.22	0.142721
$953$	−11490.1	−0.390557	−0.195279	−	0.980748i	$-0.562561\pi$
$953$	−11490.1	−0.390557	−0.195279	+	0.980748i	$0.562561\pi$
$954$	0	0
$955$	−3072.47	−0.104108
$956$	−223.973	−0.00757721
$957$	0	0
$958$	−35572.8	−1.19969
$959$	4248.93	0.143071
$960$	0	0
$961$	32010.8	1.07451
$962$	29488.0	0.988285
$963$	0	0
$964$	−11132.2	−0.371933
$965$	−13682.2	−0.456419
$966$	0	0
$967$	−20841.5	−0.693089	−0.346545	−	0.938034i	$-0.612645\pi$
$967$	−20841.5	−0.693089	−0.346545	+	0.938034i	$0.612645\pi$
$968$	−1459.11	−0.0484478
$969$	0	0
$970$	11808.5	0.390876
$971$	45975.1	1.51948	0.759738	−	0.650229i	$-0.225327\pi$
$971$	45975.1	1.51948	0.759738	+	0.650229i	$0.225327\pi$
$972$	0	0
$973$	−5612.79	−0.184931
$974$	−40958.7	−1.34743
$975$	0	0
$976$	−2850.35	−0.0934810
$977$	28254.7	0.925227	0.462613	−	0.886560i	$-0.346912\pi$
$977$	28254.7	0.925227	0.462613	+	0.886560i	$0.346912\pi$
$978$	0	0
$979$	603.631	0.0197060
$980$	−5552.77	−0.180997
$981$	0	0
$982$	−18050.8	−0.586582
$983$	−8056.82	−0.261417	−0.130708	−	0.991421i	$-0.541725\pi$
$983$	−8056.82	−0.261417	−0.130708	+	0.991421i	$0.541725\pi$
$984$	0	0
$985$	−2072.05	−0.0670265
$986$	18672.1	0.603085
$987$	0	0
$988$	−13090.0	−0.421506
$989$	2230.04	0.0716997
$990$	0	0
$991$	11919.8	0.382085	0.191043	−	0.981582i	$-0.438813\pi$
$991$	11919.8	0.382085	0.191043	+	0.981582i	$0.438813\pi$
$992$	−7955.19	−0.254614
$993$	0	0
$994$	2668.96	0.0851654
$995$	−15456.0	−0.492451
$996$	0	0
$997$	58667.6	1.86361	0.931807	−	0.362955i	$-0.118232\pi$
$997$	58667.6	1.86361	0.931807	+	0.362955i	$0.118232\pi$
$998$	28133.6	0.892338
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.v.1.2		3
3.2	odd	2		690.4.a.r.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.4.a.r.1.2	✓	3	3.2	odd	2
2070.4.a.v.1.2		3	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +8.08465 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +8.08465 q^{7} -8.00000 q^{8} -10.0000 q^{10} -38.9023 q^{11} -50.7069 q^{13} -16.1693 q^{14} +16.0000 q^{16} -64.8176 q^{17} +64.5376 q^{19} +20.0000 q^{20} +77.8046 q^{22} +23.0000 q^{23} +25.0000 q^{25} +101.414 q^{26} +32.3386 q^{28} +144.036 q^{29} +248.600 q^{31} -32.0000 q^{32} +129.635 q^{34} +40.4233 q^{35} +290.769 q^{37} -129.075 q^{38} -40.0000 q^{40} +123.307 q^{41} +96.9581 q^{43} -155.609 q^{44} -46.0000 q^{46} -342.629 q^{47} -277.638 q^{49} -50.0000 q^{50} -202.828 q^{52} -39.6254 q^{53} -194.511 q^{55} -64.6772 q^{56} -288.072 q^{58} -388.739 q^{59} -178.147 q^{61} -497.199 q^{62} +64.0000 q^{64} -253.534 q^{65} -27.8576 q^{67} -259.271 q^{68} -80.8465 q^{70} -165.064 q^{71} +391.901 q^{73} -581.538 q^{74} +258.150 q^{76} -314.511 q^{77} +785.383 q^{79} +80.0000 q^{80} -246.613 q^{82} -746.890 q^{83} -324.088 q^{85} -193.916 q^{86} +311.218 q^{88} -15.5166 q^{89} -409.947 q^{91} +92.0000 q^{92} +685.257 q^{94} +322.688 q^{95} -1180.85 q^{97} +555.277 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 12 q^{4} + 15 q^{5} - 2 q^{7} - 24 q^{8}+O(q^{10}) \) 3 * q - 6 * q^2 + 12 * q^4 + 15 * q^5 - 2 * q^7 - 24 * q^8	\( 3 q - 6 q^{2} + 12 q^{4} + 15 q^{5} - 2 q^{7} - 24 q^{8} - 30 q^{10} - 42 q^{11} + 72 q^{13} + 4 q^{14} + 48 q^{16} - 146 q^{17} + 22 q^{19} + 60 q^{20} + 84 q^{22} + 69 q^{23} + 75 q^{25} - 144 q^{26} - 8 q^{28} - 214 q^{29} + 130 q^{31} - 96 q^{32} + 292 q^{34} - 10 q^{35} + 204 q^{37} - 44 q^{38} - 120 q^{40} - 470 q^{41} + 404 q^{43} - 168 q^{44} - 138 q^{46} - 198 q^{47} + 845 q^{49} - 150 q^{50} + 288 q^{52} - 870 q^{53} - 210 q^{55} + 16 q^{56} + 428 q^{58} - 512 q^{59} - 64 q^{61} - 260 q^{62} + 192 q^{64} + 360 q^{65} - 538 q^{67} - 584 q^{68} + 20 q^{70} - 600 q^{71} + 552 q^{73} - 408 q^{74} + 88 q^{76} - 570 q^{77} + 208 q^{79} + 240 q^{80} + 940 q^{82} - 900 q^{83} - 730 q^{85} - 808 q^{86} + 336 q^{88} - 6 q^{89} - 1842 q^{91} + 276 q^{92} + 396 q^{94} + 110 q^{95} - 518 q^{97} - 1690 q^{98}+O(q^{100}) \) 3 * q - 6 * q^2 + 12 * q^4 + 15 * q^5 - 2 * q^7 - 24 * q^8 - 30 * q^10 - 42 * q^11 + 72 * q^13 + 4 * q^14 + 48 * q^16 - 146 * q^17 + 22 * q^19 + 60 * q^20 + 84 * q^22 + 69 * q^23 + 75 * q^25 - 144 * q^26 - 8 * q^28 - 214 * q^29 + 130 * q^31 - 96 * q^32 + 292 * q^34 - 10 * q^35 + 204 * q^37 - 44 * q^38 - 120 * q^40 - 470 * q^41 + 404 * q^43 - 168 * q^44 - 138 * q^46 - 198 * q^47 + 845 * q^49 - 150 * q^50 + 288 * q^52 - 870 * q^53 - 210 * q^55 + 16 * q^56 + 428 * q^58 - 512 * q^59 - 64 * q^61 - 260 * q^62 + 192 * q^64 + 360 * q^65 - 538 * q^67 - 584 * q^68 + 20 * q^70 - 600 * q^71 + 552 * q^73 - 408 * q^74 + 88 * q^76 - 570 * q^77 + 208 * q^79 + 240 * q^80 + 940 * q^82 - 900 * q^83 - 730 * q^85 - 808 * q^86 + 336 * q^88 - 6 * q^89 - 1842 * q^91 + 276 * q^92 + 396 * q^94 + 110 * q^95 - 518 * q^97 - 1690 * q^98

Introduction

L-functions

Modular forms

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Database

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