LMFDB - Embedded newform 2070.4.a.bn.1.5

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 614x^{3} - 950x^{2} + 96373x + 445660 $ x^5 - 2x^4 - 614x^3 - 950x^2 + 96373x + 445660
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-15.4995$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +32.3809 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +32.3809 q^{7} +8.00000 q^{8} +10.0000 q^{10} +34.1176 q^{11} -74.7007 q^{13} +64.7619 q^{14} +16.0000 q^{16} -49.6826 q^{17} +12.1176 q^{19} +20.0000 q^{20} +68.2351 q^{22} -23.0000 q^{23} +25.0000 q^{25} -149.401 q^{26} +129.524 q^{28} -5.10359 q^{29} +143.018 q^{31} +32.0000 q^{32} -99.3653 q^{34} +161.905 q^{35} +315.348 q^{37} +24.2351 q^{38} +40.0000 q^{40} +263.380 q^{41} +190.104 q^{43} +136.470 q^{44} -46.0000 q^{46} +405.220 q^{47} +705.525 q^{49} +50.0000 q^{50} -298.803 q^{52} +634.972 q^{53} +170.588 q^{55} +259.047 q^{56} -10.2072 q^{58} -480.680 q^{59} -571.081 q^{61} +286.036 q^{62} +64.0000 q^{64} -373.504 q^{65} -992.582 q^{67} -198.731 q^{68} +323.809 q^{70} +681.170 q^{71} -820.067 q^{73} +630.697 q^{74} +48.4702 q^{76} +1104.76 q^{77} +855.443 q^{79} +80.0000 q^{80} +526.760 q^{82} -396.418 q^{83} -248.413 q^{85} +380.207 q^{86} +272.940 q^{88} -314.598 q^{89} -2418.88 q^{91} -92.0000 q^{92} +810.440 q^{94} +60.5878 q^{95} +1306.36 q^{97} +1411.05 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{2} + 20 q^{4} + 25 q^{5} + 12 q^{7} + 40 q^{8}+O(q^{10}) $ 5 * q + 10 * q^2 + 20 * q^4 + 25 * q^5 + 12 * q^7 + 40 * q^8	$ 5 q + 10 q^{2} + 20 q^{4} + 25 q^{5} + 12 q^{7} + 40 q^{8} + 50 q^{10} + 82 q^{11} + 16 q^{13} + 24 q^{14} + 80 q^{16} - 10 q^{17} - 28 q^{19} + 100 q^{20} + 164 q^{22} - 115 q^{23} + 125 q^{25} + 32 q^{26} + 48 q^{28} + 396 q^{29} + 564 q^{31} + 160 q^{32} - 20 q^{34} + 60 q^{35} + 290 q^{37} - 56 q^{38} + 200 q^{40} - 130 q^{41} + 468 q^{43} + 328 q^{44} - 230 q^{46} + 976 q^{47} + 1293 q^{49} + 250 q^{50} + 64 q^{52} + 598 q^{53} + 410 q^{55} + 96 q^{56} + 792 q^{58} - 400 q^{59} - 670 q^{61} + 1128 q^{62} + 320 q^{64} + 80 q^{65} - 10 q^{67} - 40 q^{68} + 120 q^{70} + 82 q^{71} + 1410 q^{73} + 580 q^{74} - 112 q^{76} + 316 q^{77} + 1140 q^{79} + 400 q^{80} - 260 q^{82} + 2640 q^{83} - 50 q^{85} + 936 q^{86} + 656 q^{88} - 856 q^{89} - 920 q^{91} - 460 q^{92} + 1952 q^{94} - 140 q^{95} + 946 q^{97} + 2586 q^{98}+O(q^{100}) $ 5 * q + 10 * q^2 + 20 * q^4 + 25 * q^5 + 12 * q^7 + 40 * q^8 + 50 * q^10 + 82 * q^11 + 16 * q^13 + 24 * q^14 + 80 * q^16 - 10 * q^17 - 28 * q^19 + 100 * q^20 + 164 * q^22 - 115 * q^23 + 125 * q^25 + 32 * q^26 + 48 * q^28 + 396 * q^29 + 564 * q^31 + 160 * q^32 - 20 * q^34 + 60 * q^35 + 290 * q^37 - 56 * q^38 + 200 * q^40 - 130 * q^41 + 468 * q^43 + 328 * q^44 - 230 * q^46 + 976 * q^47 + 1293 * q^49 + 250 * q^50 + 64 * q^52 + 598 * q^53 + 410 * q^55 + 96 * q^56 + 792 * q^58 - 400 * q^59 - 670 * q^61 + 1128 * q^62 + 320 * q^64 + 80 * q^65 - 10 * q^67 - 40 * q^68 + 120 * q^70 + 82 * q^71 + 1410 * q^73 + 580 * q^74 - 112 * q^76 + 316 * q^77 + 1140 * q^79 + 400 * q^80 - 260 * q^82 + 2640 * q^83 - 50 * q^85 + 936 * q^86 + 656 * q^88 - 856 * q^89 - 920 * q^91 - 460 * q^92 + 1952 * q^94 - 140 * q^95 + 946 * q^97 + 2586 * 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
$7$	32.3809	1.74841	0.874203	−	0.485561i	$-0.161385\pi$
$7$	32.3809	1.74841	0.874203	+	0.485561i	$0.161385\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	34.1176	0.935166	0.467583	−	0.883949i	$-0.345125\pi$
$11$	34.1176	0.935166	0.467583	+	0.883949i	$0.345125\pi$
$12$	0	0
$13$	−74.7007	−1.59371	−0.796856	−	0.604169i	$-0.793505\pi$
$13$	−74.7007	−1.59371	−0.796856	+	0.604169i	$0.793505\pi$
$14$	64.7619	1.23631
$15$	0	0
$16$	16.0000	0.250000
$17$	−49.6826	−0.708812	−0.354406	−	0.935092i	$-0.615317\pi$
$17$	−49.6826	−0.708812	−0.354406	+	0.935092i	$0.615317\pi$
$18$	0	0
$19$	12.1176	0.146314	0.0731568	−	0.997320i	$-0.476693\pi$
$19$	12.1176	0.146314	0.0731568	+	0.997320i	$0.476693\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	68.2351	0.661262
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	−149.401	−1.12692
$27$	0	0
$28$	129.524	0.874203
$29$	−5.10359	−0.0326798	−0.0163399	−	0.999866i	$-0.505201\pi$
$29$	−5.10359	−0.0326798	−0.0163399	+	0.999866i	$0.505201\pi$
$30$	0	0
$31$	143.018	0.828607	0.414303	−	0.910139i	$-0.364025\pi$
$31$	143.018	0.828607	0.414303	+	0.910139i	$0.364025\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−99.3653	−0.501206
$35$	161.905	0.781911
$36$	0	0
$37$	315.348	1.40116	0.700580	−	0.713574i	$-0.252925\pi$
$37$	315.348	1.40116	0.700580	+	0.713574i	$0.252925\pi$
$38$	24.2351	0.103459
$39$	0	0
$40$	40.0000	0.158114
$41$	263.380	1.00324	0.501622	−	0.865087i	$-0.332737\pi$
$41$	263.380	1.00324	0.501622	+	0.865087i	$0.332737\pi$
$42$	0	0
$43$	190.104	0.674198	0.337099	−	0.941469i	$-0.390554\pi$
$43$	190.104	0.674198	0.337099	+	0.941469i	$0.390554\pi$
$44$	136.470	0.467583
$45$	0	0
$46$	−46.0000	−0.147442
$47$	405.220	1.25760	0.628802	−	0.777565i	$-0.283546\pi$
$47$	405.220	1.25760	0.628802	+	0.777565i	$0.283546\pi$
$48$	0	0
$49$	705.525	2.05692
$50$	50.0000	0.141421
$51$	0	0
$52$	−298.803	−0.796856
$53$	634.972	1.64566	0.822831	−	0.568286i	$-0.192393\pi$
$53$	634.972	1.64566	0.822831	+	0.568286i	$0.192393\pi$
$54$	0	0
$55$	170.588	0.418219
$56$	259.047	0.618155
$57$	0	0
$58$	−10.2072	−0.0231081
$59$	−480.680	−1.06066	−0.530332	−	0.847790i	$-0.677933\pi$
$59$	−480.680	−1.06066	−0.530332	+	0.847790i	$0.677933\pi$
$60$	0	0
$61$	−571.081	−1.19868	−0.599339	−	0.800495i	$-0.704570\pi$
$61$	−571.081	−1.19868	−0.599339	+	0.800495i	$0.704570\pi$
$62$	286.036	0.585913
$63$	0	0
$64$	64.0000	0.125000
$65$	−373.504	−0.712730
$66$	0	0
$67$	−992.582	−1.80990	−0.904949	−	0.425520i	$-0.860091\pi$
$67$	−992.582	−1.80990	−0.904949	+	0.425520i	$0.860091\pi$
$68$	−198.731	−0.354406
$69$	0	0
$70$	323.809	0.552894
$71$	681.170	1.13859	0.569295	−	0.822133i	$-0.307216\pi$
$71$	681.170	1.13859	0.569295	+	0.822133i	$0.307216\pi$
$72$	0	0
$73$	−820.067	−1.31482	−0.657408	−	0.753535i	$-0.728347\pi$
$73$	−820.067	−1.31482	−0.657408	+	0.753535i	$0.728347\pi$
$74$	630.697	0.990770
$75$	0	0
$76$	48.4702	0.0731568
$77$	1104.76	1.63505
$78$	0	0
$79$	855.443	1.21829	0.609145	−	0.793059i	$-0.291513\pi$
$79$	855.443	1.21829	0.609145	+	0.793059i	$0.291513\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	526.760	0.709401
$83$	−396.418	−0.524247	−0.262123	−	0.965034i	$-0.584423\pi$
$83$	−396.418	−0.524247	−0.262123	+	0.965034i	$0.584423\pi$
$84$	0	0
$85$	−248.413	−0.316990
$86$	380.207	0.476730
$87$	0	0
$88$	272.940	0.330631
$89$	−314.598	−0.374688	−0.187344	−	0.982294i	$-0.559988\pi$
$89$	−314.598	−0.374688	−0.187344	+	0.982294i	$0.559988\pi$
$90$	0	0
$91$	−2418.88	−2.78646
$92$	−92.0000	−0.104257
$93$	0	0
$94$	810.440	0.889260
$95$	60.5878	0.0654334
$96$	0	0
$97$	1306.36	1.36743	0.683716	−	0.729749i	$-0.260363\pi$
$97$	1306.36	1.36743	0.683716	+	0.729749i	$0.260363\pi$
$98$	1411.05	1.45446
$99$	0	0
$100$	100.000	0.100000
$101$	−1113.86	−1.09735	−0.548677	−	0.836035i	$-0.684868\pi$
$101$	−1113.86	−1.09735	−0.548677	+	0.836035i	$0.684868\pi$
$102$	0	0
$103$	172.349	0.164874	0.0824369	−	0.996596i	$-0.473730\pi$
$103$	172.349	0.164874	0.0824369	+	0.996596i	$0.473730\pi$
$104$	−597.606	−0.563462
$105$	0	0
$106$	1269.94	1.16366
$107$	453.769	0.409977	0.204988	−	0.978764i	$-0.434284\pi$
$107$	453.769	0.409977	0.204988	+	0.978764i	$0.434284\pi$
$108$	0	0
$109$	811.588	0.713175	0.356587	−	0.934262i	$-0.383940\pi$
$109$	811.588	0.713175	0.356587	+	0.934262i	$0.383940\pi$
$110$	341.176	0.295726
$111$	0	0
$112$	518.095	0.437101
$113$	7.71533	0.00642299	0.00321149	−	0.999995i	$-0.498978\pi$
$113$	7.71533	0.00642299	0.00321149	+	0.999995i	$0.498978\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	−20.4144	−0.0163399
$117$	0	0
$118$	−961.360	−0.750003
$119$	−1608.77	−1.23929
$120$	0	0
$121$	−166.993	−0.125464
$122$	−1142.16	−0.847593
$123$	0	0
$124$	572.072	0.414303
$125$	125.000	0.0894427
$126$	0	0
$127$	1542.29	1.07761	0.538803	−	0.842432i	$-0.318877\pi$
$127$	1542.29	1.07761	0.538803	+	0.842432i	$0.318877\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−747.007	−0.503976
$131$	−876.212	−0.584389	−0.292195	−	0.956359i	$-0.594386\pi$
$131$	−876.212	−0.584389	−0.292195	+	0.956359i	$0.594386\pi$
$132$	0	0
$133$	392.378	0.255815
$134$	−1985.16	−1.27979
$135$	0	0
$136$	−397.461	−0.250603
$137$	109.107	0.0680410	0.0340205	−	0.999421i	$-0.489169\pi$
$137$	109.107	0.0680410	0.0340205	+	0.999421i	$0.489169\pi$
$138$	0	0
$139$	−2600.98	−1.58714	−0.793568	−	0.608481i	$-0.791779\pi$
$139$	−2600.98	−1.58714	−0.793568	+	0.608481i	$0.791779\pi$
$140$	647.619	0.390955
$141$	0	0
$142$	1362.34	0.805105
$143$	−2548.61	−1.49039
$144$	0	0
$145$	−25.5179	−0.0146148
$146$	−1640.13	−0.929715
$147$	0	0
$148$	1261.39	0.700580
$149$	3079.78	1.69332	0.846661	−	0.532132i	$-0.178609\pi$
$149$	3079.78	1.69332	0.846661	+	0.532132i	$0.178609\pi$
$150$	0	0
$151$	−311.553	−0.167906	−0.0839530	−	0.996470i	$-0.526755\pi$
$151$	−311.553	−0.167906	−0.0839530	+	0.996470i	$0.526755\pi$
$152$	96.9404	0.0517296
$153$	0	0
$154$	2209.52	1.15616
$155$	715.091	0.370564
$156$	0	0
$157$	2586.96	1.31505	0.657523	−	0.753435i	$-0.271604\pi$
$157$	2586.96	1.31505	0.657523	+	0.753435i	$0.271604\pi$
$158$	1710.89	0.861461
$159$	0	0
$160$	160.000	0.0790569
$161$	−744.761	−0.364568
$162$	0	0
$163$	−688.681	−0.330930	−0.165465	−	0.986216i	$-0.552913\pi$
$163$	−688.681	−0.330930	−0.165465	+	0.986216i	$0.552913\pi$
$164$	1053.52	0.501622
$165$	0	0
$166$	−792.835	−0.370699
$167$	2095.79	0.971122	0.485561	−	0.874203i	$-0.338615\pi$
$167$	2095.79	0.971122	0.485561	+	0.874203i	$0.338615\pi$
$168$	0	0
$169$	3383.20	1.53992
$170$	−496.826	−0.224146
$171$	0	0
$172$	760.414	0.337099
$173$	2918.90	1.28278	0.641388	−	0.767217i	$-0.278359\pi$
$173$	2918.90	1.28278	0.641388	+	0.767217i	$0.278359\pi$
$174$	0	0
$175$	809.523	0.349681
$176$	545.881	0.233792
$177$	0	0
$178$	−629.195	−0.264945
$179$	−3018.21	−1.26029	−0.630145	−	0.776478i	$-0.717004\pi$
$179$	−3018.21	−1.26029	−0.630145	+	0.776478i	$0.717004\pi$
$180$	0	0
$181$	−3072.49	−1.26175	−0.630874	−	0.775886i	$-0.717303\pi$
$181$	−3072.49	−1.26175	−0.630874	+	0.775886i	$0.717303\pi$
$182$	−4837.76	−1.97032
$183$	0	0
$184$	−184.000	−0.0737210
$185$	1576.74	0.626618
$186$	0	0
$187$	−1695.05	−0.662857
$188$	1620.88	0.628802
$189$	0	0
$190$	121.176	0.0462684
$191$	3721.59	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$191$	3721.59	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$192$	0	0
$193$	355.539	0.132603	0.0663013	−	0.997800i	$-0.478880\pi$
$193$	355.539	0.132603	0.0663013	+	0.997800i	$0.478880\pi$
$194$	2612.72	0.966920
$195$	0	0
$196$	2822.10	1.02846
$197$	−3177.92	−1.14933	−0.574664	−	0.818389i	$-0.694867\pi$
$197$	−3177.92	−1.14933	−0.574664	+	0.818389i	$0.694867\pi$
$198$	0	0
$199$	−1923.52	−0.685199	−0.342599	−	0.939482i	$-0.611307\pi$
$199$	−1923.52	−0.685199	−0.342599	+	0.939482i	$0.611307\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−2227.71	−0.775946
$203$	−165.259	−0.0571375
$204$	0	0
$205$	1316.90	0.448665
$206$	344.697	0.116583
$207$	0	0
$208$	−1195.21	−0.398428
$209$	413.421	0.136827
$210$	0	0
$211$	−3356.82	−1.09523	−0.547614	−	0.836731i	$-0.684464\pi$
$211$	−3356.82	−1.09523	−0.547614	+	0.836731i	$0.684464\pi$
$212$	2539.89	0.822831
$213$	0	0
$214$	907.538	0.289897
$215$	950.518	0.301511
$216$	0	0
$217$	4631.06	1.44874
$218$	1623.18	0.504291
$219$	0	0
$220$	682.351	0.209110
$221$	3711.33	1.12964
$222$	0	0
$223$	−5129.78	−1.54043	−0.770215	−	0.637785i	$-0.779851\pi$
$223$	−5129.78	−1.54043	−0.770215	+	0.637785i	$0.779851\pi$
$224$	1036.19	0.309077
$225$	0	0
$226$	15.4307	0.00454174
$227$	3344.55	0.977909	0.488954	−	0.872309i	$-0.337378\pi$
$227$	3344.55	0.977909	0.488954	+	0.872309i	$0.337378\pi$
$228$	0	0
$229$	4161.52	1.20088	0.600438	−	0.799671i	$-0.294993\pi$
$229$	4161.52	1.20088	0.600438	+	0.799671i	$0.294993\pi$
$230$	−230.000	−0.0659380
$231$	0	0
$232$	−40.8287	−0.0115540
$233$	2252.72	0.633392	0.316696	−	0.948527i	$-0.397426\pi$
$233$	2252.72	0.633392	0.316696	+	0.948527i	$0.397426\pi$
$234$	0	0
$235$	2026.10	0.562418
$236$	−1922.72	−0.530332
$237$	0	0
$238$	−3217.54	−0.876311
$239$	6728.46	1.82104	0.910518	−	0.413469i	$-0.135683\pi$
$239$	6728.46	1.82104	0.910518	+	0.413469i	$0.135683\pi$
$240$	0	0
$241$	5173.39	1.38277	0.691385	−	0.722487i	$-0.257001\pi$
$241$	5173.39	1.38277	0.691385	+	0.722487i	$0.257001\pi$
$242$	−333.985	−0.0887165
$243$	0	0
$244$	−2284.32	−0.599339
$245$	3527.62	0.919884
$246$	0	0
$247$	−905.190	−0.233182
$248$	1144.14	0.292957
$249$	0	0
$250$	250.000	0.0632456
$251$	2541.04	0.639000	0.319500	−	0.947586i	$-0.396485\pi$
$251$	2541.04	0.639000	0.319500	+	0.947586i	$0.396485\pi$
$252$	0	0
$253$	−784.704	−0.194996
$254$	3084.57	0.761982
$255$	0	0
$256$	256.000	0.0625000
$257$	5064.47	1.22923	0.614617	−	0.788826i	$-0.289311\pi$
$257$	5064.47	1.22923	0.614617	+	0.788826i	$0.289311\pi$
$258$	0	0
$259$	10211.3	2.44980
$260$	−1494.01	−0.356365
$261$	0	0
$262$	−1752.42	−0.413226
$263$	−1891.78	−0.443545	−0.221773	−	0.975098i	$-0.571184\pi$
$263$	−1891.78	−0.443545	−0.221773	+	0.975098i	$0.571184\pi$
$264$	0	0
$265$	3174.86	0.735962
$266$	784.755	0.180889
$267$	0	0
$268$	−3970.33	−0.904949
$269$	−454.300	−0.102971	−0.0514854	−	0.998674i	$-0.516396\pi$
$269$	−454.300	−0.102971	−0.0514854	+	0.998674i	$0.516396\pi$
$270$	0	0
$271$	−7709.69	−1.72816	−0.864078	−	0.503359i	$-0.832097\pi$
$271$	−7709.69	−1.72816	−0.864078	+	0.503359i	$0.832097\pi$
$272$	−794.922	−0.177203
$273$	0	0
$274$	218.214	0.0481123
$275$	852.939	0.187033
$276$	0	0
$277$	−5660.50	−1.22782	−0.613911	−	0.789375i	$-0.710405\pi$
$277$	−5660.50	−1.22782	−0.613911	+	0.789375i	$0.710405\pi$
$278$	−5201.95	−1.12228
$279$	0	0
$280$	1295.24	0.276447
$281$	−4529.90	−0.961676	−0.480838	−	0.876809i	$-0.659667\pi$
$281$	−4529.90	−0.961676	−0.480838	+	0.876809i	$0.659667\pi$
$282$	0	0
$283$	−678.391	−0.142495	−0.0712476	−	0.997459i	$-0.522698\pi$
$283$	−678.391	−0.142495	−0.0712476	+	0.997459i	$0.522698\pi$
$284$	2724.68	0.569295
$285$	0	0
$286$	−5097.21	−1.05386
$287$	8528.49	1.75408
$288$	0	0
$289$	−2444.64	−0.497585
$290$	−51.0359	−0.0103342
$291$	0	0
$292$	−3280.27	−0.657408
$293$	446.249	0.0889765	0.0444883	−	0.999010i	$-0.485834\pi$
$293$	446.249	0.0889765	0.0444883	+	0.999010i	$0.485834\pi$
$294$	0	0
$295$	−2403.40	−0.474344
$296$	2522.79	0.495385
$297$	0	0
$298$	6159.55	1.19736
$299$	1718.12	0.332312
$300$	0	0
$301$	6155.73	1.17877
$302$	−623.105	−0.118727
$303$	0	0
$304$	193.881	0.0365784
$305$	−2855.40	−0.536065
$306$	0	0
$307$	−5828.74	−1.08360	−0.541798	−	0.840509i	$-0.682256\pi$
$307$	−5828.74	−1.08360	−0.541798	+	0.840509i	$0.682256\pi$
$308$	4419.03	0.817525
$309$	0	0
$310$	1430.18	0.262028
$311$	−2052.55	−0.374243	−0.187122	−	0.982337i	$-0.559916\pi$
$311$	−2052.55	−0.374243	−0.187122	+	0.982337i	$0.559916\pi$
$312$	0	0
$313$	−154.090	−0.0278265	−0.0139132	−	0.999903i	$-0.504429\pi$
$313$	−154.090	−0.0278265	−0.0139132	+	0.999903i	$0.504429\pi$
$314$	5173.93	0.929878
$315$	0	0
$316$	3421.77	0.609145
$317$	−1847.57	−0.327349	−0.163674	−	0.986514i	$-0.552335\pi$
$317$	−1847.57	−0.327349	−0.163674	+	0.986514i	$0.552335\pi$
$318$	0	0
$319$	−174.122	−0.0305610
$320$	320.000	0.0559017
$321$	0	0
$322$	−1489.52	−0.257788
$323$	−602.032	−0.103709
$324$	0	0
$325$	−1867.52	−0.318742
$326$	−1377.36	−0.234003
$327$	0	0
$328$	2107.04	0.354701
$329$	13121.4	2.19880
$330$	0	0
$331$	523.803	0.0869814	0.0434907	−	0.999054i	$-0.486152\pi$
$331$	523.803	0.0869814	0.0434907	+	0.999054i	$0.486152\pi$
$332$	−1585.67	−0.262123
$333$	0	0
$334$	4191.59	0.686687
$335$	−4962.91	−0.809411
$336$	0	0
$337$	4047.97	0.654323	0.327161	−	0.944968i	$-0.393908\pi$
$337$	4047.97	0.654323	0.327161	+	0.944968i	$0.393908\pi$
$338$	6766.40	1.08889
$339$	0	0
$340$	−993.653	−0.158495
$341$	4879.43	0.774885
$342$	0	0
$343$	11738.9	1.84793
$344$	1520.83	0.238365
$345$	0	0
$346$	5837.81	0.907059
$347$	−11925.8	−1.84499	−0.922493	−	0.386015i	$-0.873851\pi$
$347$	−11925.8	−1.84499	−0.922493	+	0.386015i	$0.873851\pi$
$348$	0	0
$349$	−1790.26	−0.274586	−0.137293	−	0.990530i	$-0.543840\pi$
$349$	−1790.26	−0.274586	−0.137293	+	0.990530i	$0.543840\pi$
$350$	1619.05	0.247262
$351$	0	0
$352$	1091.76	0.165316
$353$	−9731.82	−1.46734	−0.733672	−	0.679503i	$-0.762195\pi$
$353$	−9731.82	−1.46734	−0.733672	+	0.679503i	$0.762195\pi$
$354$	0	0
$355$	3405.85	0.509193
$356$	−1258.39	−0.187344
$357$	0	0
$358$	−6036.43	−0.891159
$359$	−11983.0	−1.76167	−0.880836	−	0.473421i	$-0.843019\pi$
$359$	−11983.0	−1.76167	−0.880836	+	0.473421i	$0.843019\pi$
$360$	0	0
$361$	−6712.16	−0.978592
$362$	−6144.98	−0.892190
$363$	0	0
$364$	−9675.52	−1.39323
$365$	−4100.33	−0.588004
$366$	0	0
$367$	316.169	0.0449698	0.0224849	−	0.999747i	$-0.492842\pi$
$367$	316.169	0.0449698	0.0224849	+	0.999747i	$0.492842\pi$
$368$	−368.000	−0.0521286
$369$	0	0
$370$	3153.48	0.443086
$371$	20561.0	2.87728
$372$	0	0
$373$	−2419.78	−0.335901	−0.167951	−	0.985795i	$-0.553715\pi$
$373$	−2419.78	−0.335901	−0.167951	+	0.985795i	$0.553715\pi$
$374$	−3390.10	−0.468711
$375$	0	0
$376$	3241.76	0.444630
$377$	381.242	0.0520821
$378$	0	0
$379$	−4705.17	−0.637700	−0.318850	−	0.947805i	$-0.603297\pi$
$379$	−4705.17	−0.637700	−0.318850	+	0.947805i	$0.603297\pi$
$380$	242.351	0.0327167
$381$	0	0
$382$	7443.18	0.996928
$383$	−8075.26	−1.07735	−0.538677	−	0.842512i	$-0.681076\pi$
$383$	−8075.26	−1.07735	−0.538677	+	0.842512i	$0.681076\pi$
$384$	0	0
$385$	5523.79	0.731217
$386$	711.079	0.0937641
$387$	0	0
$388$	5225.44	0.683716
$389$	15154.8	1.97526	0.987631	−	0.156795i	$-0.0501161\pi$
$389$	15154.8	1.97526	0.987631	+	0.156795i	$0.0501161\pi$
$390$	0	0
$391$	1142.70	0.147798
$392$	5644.20	0.727232
$393$	0	0
$394$	−6355.85	−0.812698
$395$	4277.22	0.544836
$396$	0	0
$397$	−483.481	−0.0611215	−0.0305607	−	0.999533i	$-0.509729\pi$
$397$	−483.481	−0.0611215	−0.0305607	+	0.999533i	$0.509729\pi$
$398$	−3847.04	−0.484509
$399$	0	0
$400$	400.000	0.0500000
$401$	4657.80	0.580049	0.290024	−	0.957019i	$-0.406337\pi$
$401$	4657.80	0.580049	0.290024	+	0.957019i	$0.406337\pi$
$402$	0	0
$403$	−10683.6	−1.32056
$404$	−4455.42	−0.548677
$405$	0	0
$406$	−330.518	−0.0404023
$407$	10758.9	1.31032
$408$	0	0
$409$	−3580.34	−0.432852	−0.216426	−	0.976299i	$-0.569440\pi$
$409$	−3580.34	−0.432852	−0.216426	+	0.976299i	$0.569440\pi$
$410$	2633.80	0.317254
$411$	0	0
$412$	689.394	0.0824369
$413$	−15564.9	−1.85447
$414$	0	0
$415$	−1982.09	−0.234450
$416$	−2390.42	−0.281731
$417$	0	0
$418$	826.842	0.0967516
$419$	12895.5	1.50355	0.751775	−	0.659420i	$-0.229198\pi$
$419$	12895.5	1.50355	0.751775	+	0.659420i	$0.229198\pi$
$420$	0	0
$421$	5033.44	0.582696	0.291348	−	0.956617i	$-0.405896\pi$
$421$	5033.44	0.582696	0.291348	+	0.956617i	$0.405896\pi$
$422$	−6713.65	−0.774444
$423$	0	0
$424$	5079.77	0.581829
$425$	−1242.07	−0.141762
$426$	0	0
$427$	−18492.1	−2.09578
$428$	1815.08	0.204988
$429$	0	0
$430$	1901.04	0.213200
$431$	9425.93	1.05344	0.526718	−	0.850040i	$-0.323422\pi$
$431$	9425.93	1.05344	0.526718	+	0.850040i	$0.323422\pi$
$432$	0	0
$433$	−14328.2	−1.59023	−0.795117	−	0.606457i	$-0.792590\pi$
$433$	−14328.2	−1.59023	−0.795117	+	0.606457i	$0.792590\pi$
$434$	9262.12	1.02441
$435$	0	0
$436$	3246.35	0.356587
$437$	−278.704	−0.0305085
$438$	0	0
$439$	−668.891	−0.0727207	−0.0363604	−	0.999339i	$-0.511576\pi$
$439$	−668.891	−0.0727207	−0.0363604	+	0.999339i	$0.511576\pi$
$440$	1364.70	0.147863
$441$	0	0
$442$	7422.66	0.798778
$443$	5530.10	0.593099	0.296550	−	0.955017i	$-0.404164\pi$
$443$	5530.10	0.593099	0.296550	+	0.955017i	$0.404164\pi$
$444$	0	0
$445$	−1572.99	−0.167566
$446$	−10259.6	−1.08925
$447$	0	0
$448$	2072.38	0.218551
$449$	−2446.62	−0.257156	−0.128578	−	0.991699i	$-0.541041\pi$
$449$	−2446.62	−0.257156	−0.128578	+	0.991699i	$0.541041\pi$
$450$	0	0
$451$	8985.88	0.938200
$452$	30.8613	0.00321149
$453$	0	0
$454$	6689.09	0.691486
$455$	−12094.4	−1.24614
$456$	0	0
$457$	9101.79	0.931650	0.465825	−	0.884877i	$-0.345758\pi$
$457$	9101.79	0.931650	0.465825	+	0.884877i	$0.345758\pi$
$458$	8323.03	0.849148
$459$	0	0
$460$	−460.000	−0.0466252
$461$	−3992.26	−0.403336	−0.201668	−	0.979454i	$-0.564636\pi$
$461$	−3992.26	−0.403336	−0.201668	+	0.979454i	$0.564636\pi$
$462$	0	0
$463$	−14685.8	−1.47410	−0.737050	−	0.675838i	$-0.763782\pi$
$463$	−14685.8	−1.47410	−0.737050	+	0.675838i	$0.763782\pi$
$464$	−81.6574	−0.00816994
$465$	0	0
$466$	4505.43	0.447876
$467$	621.331	0.0615669	0.0307835	−	0.999526i	$-0.490200\pi$
$467$	621.331	0.0615669	0.0307835	+	0.999526i	$0.490200\pi$
$468$	0	0
$469$	−32140.7	−3.16444
$470$	4052.20	0.397689
$471$	0	0
$472$	−3845.44	−0.375002
$473$	6485.87	0.630487
$474$	0	0
$475$	302.939	0.0292627
$476$	−6435.08	−0.619646
$477$	0	0
$478$	13456.9	1.28767
$479$	−7598.19	−0.724781	−0.362391	−	0.932026i	$-0.618039\pi$
$479$	−7598.19	−0.724781	−0.362391	+	0.932026i	$0.618039\pi$
$480$	0	0
$481$	−23556.7	−2.23305
$482$	10346.8	0.977766
$483$	0	0
$484$	−667.971	−0.0627320
$485$	6531.80	0.611534
$486$	0	0
$487$	−11896.2	−1.10692	−0.553458	−	0.832877i	$-0.686692\pi$
$487$	−11896.2	−1.10692	−0.553458	+	0.832877i	$0.686692\pi$
$488$	−4568.64	−0.423797
$489$	0	0
$490$	7055.25	0.650456
$491$	19118.9	1.75728	0.878639	−	0.477487i	$-0.158452\pi$
$491$	19118.9	1.75728	0.878639	+	0.477487i	$0.158452\pi$
$492$	0	0
$493$	253.560	0.0231638
$494$	−1810.38	−0.164884
$495$	0	0
$496$	2288.29	0.207152
$497$	22056.9	1.99072
$498$	0	0
$499$	−17258.3	−1.54828	−0.774138	−	0.633017i	$-0.781816\pi$
$499$	−17258.3	−1.54828	−0.774138	+	0.633017i	$0.781816\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	5082.08	0.451841
$503$	516.744	0.0458061	0.0229030	−	0.999738i	$-0.492709\pi$
$503$	516.744	0.0458061	0.0229030	+	0.999738i	$0.492709\pi$
$504$	0	0
$505$	−5569.28	−0.490752
$506$	−1569.41	−0.137883
$507$	0	0
$508$	6169.15	0.538803
$509$	8671.35	0.755109	0.377555	−	0.925987i	$-0.376765\pi$
$509$	8671.35	0.755109	0.377555	+	0.925987i	$0.376765\pi$
$510$	0	0
$511$	−26554.5	−2.29883
$512$	512.000	0.0441942
$513$	0	0
$514$	10128.9	0.869200
$515$	861.743	0.0737338
$516$	0	0
$517$	13825.1	1.17607
$518$	20422.5	1.73227
$519$	0	0
$520$	−2988.03	−0.251988
$521$	−22726.5	−1.91107	−0.955533	−	0.294884i	$-0.904719\pi$
$521$	−22726.5	−1.91107	−0.955533	+	0.294884i	$0.904719\pi$
$522$	0	0
$523$	570.680	0.0477134	0.0238567	−	0.999715i	$-0.492405\pi$
$523$	570.680	0.0477134	0.0238567	+	0.999715i	$0.492405\pi$
$524$	−3504.85	−0.292195
$525$	0	0
$526$	−3783.57	−0.313634
$527$	−7105.52	−0.587327
$528$	0	0
$529$	529.000	0.0434783
$530$	6349.72	0.520404
$531$	0	0
$532$	1569.51	0.127908
$533$	−19674.7	−1.59888
$534$	0	0
$535$	2268.85	0.183347
$536$	−7940.65	−0.639895
$537$	0	0
$538$	−908.600	−0.0728114
$539$	24070.8	1.92357
$540$	0	0
$541$	2955.50	0.234874	0.117437	−	0.993080i	$-0.462532\pi$
$541$	2955.50	0.234874	0.117437	+	0.993080i	$0.462532\pi$
$542$	−15419.4	−1.22199
$543$	0	0
$544$	−1589.84	−0.125301
$545$	4057.94	0.318941
$546$	0	0
$547$	4518.52	0.353196	0.176598	−	0.984283i	$-0.443491\pi$
$547$	4518.52	0.353196	0.176598	+	0.984283i	$0.443491\pi$
$548$	436.427	0.0340205
$549$	0	0
$550$	1705.88	0.132252
$551$	−61.8430	−0.00478149
$552$	0	0
$553$	27700.0	2.13006
$554$	−11321.0	−0.868201
$555$	0	0
$556$	−10403.9	−0.793568
$557$	18458.7	1.40417	0.702083	−	0.712095i	$-0.252253\pi$
$557$	18458.7	1.40417	0.702083	+	0.712095i	$0.252253\pi$
$558$	0	0
$559$	−14200.9	−1.07448
$560$	2590.47	0.195478
$561$	0	0
$562$	−9059.79	−0.680008
$563$	−7293.91	−0.546007	−0.273003	−	0.962013i	$-0.588017\pi$
$563$	−7293.91	−0.546007	−0.273003	+	0.962013i	$0.588017\pi$
$564$	0	0
$565$	38.5767	0.00287245
$566$	−1356.78	−0.100759
$567$	0	0
$568$	5449.36	0.402553
$569$	−9904.21	−0.729712	−0.364856	−	0.931064i	$-0.618882\pi$
$569$	−9904.21	−0.729712	−0.364856	+	0.931064i	$0.618882\pi$
$570$	0	0
$571$	−8829.05	−0.647082	−0.323541	−	0.946214i	$-0.604873\pi$
$571$	−8829.05	−0.647082	−0.323541	+	0.946214i	$0.604873\pi$
$572$	−10194.4	−0.745193
$573$	0	0
$574$	17057.0	1.24032
$575$	−575.000	−0.0417029
$576$	0	0
$577$	3782.21	0.272886	0.136443	−	0.990648i	$-0.456433\pi$
$577$	3782.21	0.272886	0.136443	+	0.990648i	$0.456433\pi$
$578$	−4889.27	−0.351846
$579$	0	0
$580$	−102.072	−0.00730741
$581$	−12836.4	−0.916597
$582$	0	0
$583$	21663.7	1.53897
$584$	−6560.53	−0.464858
$585$	0	0
$586$	892.497	0.0629159
$587$	20938.6	1.47228	0.736140	−	0.676830i	$-0.236647\pi$
$587$	20938.6	1.47228	0.736140	+	0.676830i	$0.236647\pi$
$588$	0	0
$589$	1733.03	0.121236
$590$	−4806.80	−0.335412
$591$	0	0
$592$	5045.57	0.350290
$593$	7394.78	0.512086	0.256043	−	0.966665i	$-0.417581\pi$
$593$	7394.78	0.512086	0.256043	+	0.966665i	$0.417581\pi$
$594$	0	0
$595$	−8043.85	−0.554228
$596$	12319.1	0.846661
$597$	0	0
$598$	3436.23	0.234980
$599$	19894.9	1.35707	0.678535	−	0.734568i	$-0.262615\pi$
$599$	19894.9	1.35707	0.678535	+	0.734568i	$0.262615\pi$
$600$	0	0
$601$	−23799.9	−1.61534	−0.807669	−	0.589636i	$-0.799271\pi$
$601$	−23799.9	−1.61534	−0.807669	+	0.589636i	$0.799271\pi$
$602$	12311.5	0.833518
$603$	0	0
$604$	−1246.21	−0.0839530
$605$	−834.963	−0.0561092
$606$	0	0
$607$	19447.5	1.30041	0.650205	−	0.759759i	$-0.274683\pi$
$607$	19447.5	1.30041	0.650205	+	0.759759i	$0.274683\pi$
$608$	387.762	0.0258648
$609$	0	0
$610$	−5710.81	−0.379055
$611$	−30270.2	−2.00426
$612$	0	0
$613$	−14307.7	−0.942711	−0.471355	−	0.881943i	$-0.656235\pi$
$613$	−14307.7	−0.942711	−0.471355	+	0.881943i	$0.656235\pi$
$614$	−11657.5	−0.766218
$615$	0	0
$616$	8838.06	0.578078
$617$	−3241.23	−0.211486	−0.105743	−	0.994393i	$-0.533722\pi$
$617$	−3241.23	−0.211486	−0.105743	+	0.994393i	$0.533722\pi$
$618$	0	0
$619$	−7191.88	−0.466988	−0.233494	−	0.972358i	$-0.575016\pi$
$619$	−7191.88	−0.466988	−0.233494	+	0.972358i	$0.575016\pi$
$620$	2860.36	0.185282
$621$	0	0
$622$	−4105.11	−0.264630
$623$	−10187.0	−0.655107
$624$	0	0
$625$	625.000	0.0400000
$626$	−308.180	−0.0196763
$627$	0	0
$628$	10347.9	0.657523
$629$	−15667.3	−0.993160
$630$	0	0
$631$	25605.5	1.61544	0.807718	−	0.589569i	$-0.200702\pi$
$631$	25605.5	1.61544	0.807718	+	0.589569i	$0.200702\pi$
$632$	6843.55	0.430730
$633$	0	0
$634$	−3695.13	−0.231471
$635$	7711.43	0.481920
$636$	0	0
$637$	−52703.2	−3.27814
$638$	−348.244	−0.0216099
$639$	0	0
$640$	640.000	0.0395285
$641$	21469.0	1.32289	0.661445	−	0.749993i	$-0.269943\pi$
$641$	21469.0	1.32289	0.661445	+	0.749993i	$0.269943\pi$
$642$	0	0
$643$	22846.5	1.40121	0.700604	−	0.713551i	$-0.252914\pi$
$643$	22846.5	1.40121	0.700604	+	0.713551i	$0.252914\pi$
$644$	−2979.05	−0.182284
$645$	0	0
$646$	−1204.06	−0.0733332
$647$	−1735.24	−0.105439	−0.0527196	−	0.998609i	$-0.516789\pi$
$647$	−1735.24	−0.105439	−0.0527196	+	0.998609i	$0.516789\pi$
$648$	0	0
$649$	−16399.6	−0.991898
$650$	−3735.04	−0.225385
$651$	0	0
$652$	−2754.72	−0.165465
$653$	7309.34	0.438035	0.219017	−	0.975721i	$-0.429715\pi$
$653$	7309.34	0.438035	0.219017	+	0.975721i	$0.429715\pi$
$654$	0	0
$655$	−4381.06	−0.261347
$656$	4214.08	0.250811
$657$	0	0
$658$	26242.8	1.55479
$659$	24502.7	1.44839	0.724197	−	0.689593i	$-0.242211\pi$
$659$	24502.7	1.44839	0.724197	+	0.689593i	$0.242211\pi$
$660$	0	0
$661$	−15207.1	−0.894837	−0.447419	−	0.894325i	$-0.647657\pi$
$661$	−15207.1	−0.894837	−0.447419	+	0.894325i	$0.647657\pi$
$662$	1047.61	0.0615051
$663$	0	0
$664$	−3171.34	−0.185349
$665$	1961.89	0.114404
$666$	0	0
$667$	117.383	0.00681420
$668$	8383.18	0.485561
$669$	0	0
$670$	−9925.82	−0.572340
$671$	−19483.9	−1.12096
$672$	0	0
$673$	8762.36	0.501879	0.250939	−	0.968003i	$-0.419261\pi$
$673$	8762.36	0.501879	0.250939	+	0.968003i	$0.419261\pi$
$674$	8095.93	0.462676
$675$	0	0
$676$	13532.8	0.769959
$677$	−25564.9	−1.45131	−0.725657	−	0.688057i	$-0.758464\pi$
$677$	−25564.9	−1.45131	−0.725657	+	0.688057i	$0.758464\pi$
$678$	0	0
$679$	42301.2	2.39082
$680$	−1987.31	−0.112073
$681$	0	0
$682$	9758.85	0.547926
$683$	5792.33	0.324506	0.162253	−	0.986749i	$-0.448124\pi$
$683$	5792.33	0.324506	0.162253	+	0.986749i	$0.448124\pi$
$684$	0	0
$685$	545.534	0.0304289
$686$	23477.8	1.30668
$687$	0	0
$688$	3041.66	0.168550
$689$	−47432.9	−2.62271
$690$	0	0
$691$	−15005.1	−0.826081	−0.413041	−	0.910713i	$-0.635533\pi$
$691$	−15005.1	−0.826081	−0.413041	+	0.910713i	$0.635533\pi$
$692$	11675.6	0.641388
$693$	0	0
$694$	−23851.6	−1.30460
$695$	−13004.9	−0.709789
$696$	0	0
$697$	−13085.4	−0.711112
$698$	−3580.53	−0.194162
$699$	0	0
$700$	3238.09	0.174841
$701$	10883.2	0.586379	0.293189	−	0.956054i	$-0.405283\pi$
$701$	10883.2	0.586379	0.293189	+	0.956054i	$0.405283\pi$
$702$	0	0
$703$	3821.25	0.205009
$704$	2183.52	0.116896
$705$	0	0
$706$	−19463.6	−1.03757
$707$	−36067.7	−1.91862
$708$	0	0
$709$	−4394.69	−0.232787	−0.116393	−	0.993203i	$-0.537133\pi$
$709$	−4394.69	−0.232787	−0.116393	+	0.993203i	$0.537133\pi$
$710$	6811.70	0.360054
$711$	0	0
$712$	−2516.78	−0.132472
$713$	−3289.42	−0.172776
$714$	0	0
$715$	−12743.0	−0.666521
$716$	−12072.9	−0.630145
$717$	0	0
$718$	−23966.1	−1.24569
$719$	30107.1	1.56162	0.780811	−	0.624767i	$-0.214806\pi$
$719$	30107.1	1.56162	0.780811	+	0.624767i	$0.214806\pi$
$720$	0	0
$721$	5580.81	0.288266
$722$	−13424.3	−0.691969
$723$	0	0
$724$	−12290.0	−0.630874
$725$	−127.590	−0.00653595
$726$	0	0
$727$	24606.5	1.25530	0.627652	−	0.778494i	$-0.284016\pi$
$727$	24606.5	1.25530	0.627652	+	0.778494i	$0.284016\pi$
$728$	−19351.0	−0.985161
$729$	0	0
$730$	−8200.67	−0.415781
$731$	−9444.85	−0.477880
$732$	0	0
$733$	−13942.3	−0.702550	−0.351275	−	0.936272i	$-0.614252\pi$
$733$	−13942.3	−0.702550	−0.351275	+	0.936272i	$0.614252\pi$
$734$	632.339	0.0317984
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−33864.5	−1.69256
$738$	0	0
$739$	−34676.3	−1.72610	−0.863051	−	0.505117i	$-0.831449\pi$
$739$	−34676.3	−1.72610	−0.863051	+	0.505117i	$0.831449\pi$
$740$	6306.97	0.313309
$741$	0	0
$742$	41122.0	2.03455
$743$	−18059.8	−0.891721	−0.445861	−	0.895102i	$-0.647102\pi$
$743$	−18059.8	−0.891721	−0.445861	+	0.895102i	$0.647102\pi$
$744$	0	0
$745$	15398.9	0.757277
$746$	−4839.55	−0.237518
$747$	0	0
$748$	−6780.20	−0.331429
$749$	14693.5	0.716806
$750$	0	0
$751$	3262.88	0.158541	0.0792704	−	0.996853i	$-0.474741\pi$
$751$	3262.88	0.158541	0.0792704	+	0.996853i	$0.474741\pi$
$752$	6483.52	0.314401
$753$	0	0
$754$	762.484	0.0368276
$755$	−1557.76	−0.0750898
$756$	0	0
$757$	−1667.17	−0.0800452	−0.0400226	−	0.999199i	$-0.512743\pi$
$757$	−1667.17	−0.0800452	−0.0400226	+	0.999199i	$0.512743\pi$
$758$	−9410.34	−0.450922
$759$	0	0
$760$	484.702	0.0231342
$761$	−31816.2	−1.51555	−0.757777	−	0.652514i	$-0.773714\pi$
$761$	−31816.2	−1.51555	−0.757777	+	0.652514i	$0.773714\pi$
$762$	0	0
$763$	26280.0	1.24692
$764$	14886.4	0.704934
$765$	0	0
$766$	−16150.5	−0.761804
$767$	35907.2	1.69039
$768$	0	0
$769$	−12994.4	−0.609349	−0.304675	−	0.952456i	$-0.598548\pi$
$769$	−12994.4	−0.609349	−0.304675	+	0.952456i	$0.598548\pi$
$770$	11047.6	0.517048
$771$	0	0
$772$	1422.16	0.0663013
$773$	−6350.96	−0.295508	−0.147754	−	0.989024i	$-0.547204\pi$
$773$	−6350.96	−0.295508	−0.147754	+	0.989024i	$0.547204\pi$
$774$	0	0
$775$	3575.45	0.165721
$776$	10450.9	0.483460
$777$	0	0
$778$	30309.5	1.39672
$779$	3191.52	0.146788
$780$	0	0
$781$	23239.8	1.06477
$782$	2285.40	0.104509
$783$	0	0
$784$	11288.4	0.514231
$785$	12934.8	0.588106
$786$	0	0
$787$	3395.76	0.153807	0.0769033	−	0.997039i	$-0.475497\pi$
$787$	3395.76	0.153807	0.0769033	+	0.997039i	$0.475497\pi$
$788$	−12711.7	−0.574664
$789$	0	0
$790$	8554.43	0.385257
$791$	249.830	0.0112300
$792$	0	0
$793$	42660.1	1.91035
$794$	−966.962	−0.0432194
$795$	0	0
$796$	−7694.07	−0.342599
$797$	9113.03	0.405019	0.202509	−	0.979280i	$-0.435090\pi$
$797$	9113.03	0.405019	0.202509	+	0.979280i	$0.435090\pi$
$798$	0	0
$799$	−20132.4	−0.891405
$800$	800.000	0.0353553
$801$	0	0
$802$	9315.61	0.410156
$803$	−27978.7	−1.22957
$804$	0	0
$805$	−3723.81	−0.163040
$806$	−21367.1	−0.933777
$807$	0	0
$808$	−8910.84	−0.387973
$809$	−34900.4	−1.51673	−0.758363	−	0.651832i	$-0.774001\pi$
$809$	−34900.4	−1.51673	−0.758363	+	0.651832i	$0.774001\pi$
$810$	0	0
$811$	−32772.3	−1.41898	−0.709490	−	0.704716i	$-0.751074\pi$
$811$	−32772.3	−1.41898	−0.709490	+	0.704716i	$0.751074\pi$
$812$	−661.036	−0.0285687
$813$	0	0
$814$	21517.8	0.926535
$815$	−3443.41	−0.147997
$816$	0	0
$817$	2303.59	0.0986443
$818$	−7160.68	−0.306072
$819$	0	0
$820$	5267.60	0.224332
$821$	−18437.6	−0.783773	−0.391887	−	0.920013i	$-0.628177\pi$
$821$	−18437.6	−0.783773	−0.391887	+	0.920013i	$0.628177\pi$
$822$	0	0
$823$	40837.4	1.72965	0.864825	−	0.502073i	$-0.167429\pi$
$823$	40837.4	1.72965	0.864825	+	0.502073i	$0.167429\pi$
$824$	1378.79	0.0582917
$825$	0	0
$826$	−31129.7	−1.31131
$827$	−14236.9	−0.598627	−0.299314	−	0.954155i	$-0.596758\pi$
$827$	−14236.9	−0.598627	−0.299314	+	0.954155i	$0.596758\pi$
$828$	0	0
$829$	−31917.9	−1.33722	−0.668609	−	0.743614i	$-0.733110\pi$
$829$	−31917.9	−1.33722	−0.668609	+	0.743614i	$0.733110\pi$
$830$	−3964.18	−0.165781
$831$	0	0
$832$	−4780.85	−0.199214
$833$	−35052.3	−1.45797
$834$	0	0
$835$	10479.0	0.434299
$836$	1653.68	0.0684137
$837$	0	0
$838$	25791.0	1.06317
$839$	12956.2	0.533132	0.266566	−	0.963817i	$-0.414111\pi$
$839$	12956.2	0.533132	0.266566	+	0.963817i	$0.414111\pi$
$840$	0	0
$841$	−24363.0	−0.998932
$842$	10066.9	0.412028
$843$	0	0
$844$	−13427.3	−0.547614
$845$	16916.0	0.688672
$846$	0	0
$847$	−5407.38	−0.219362
$848$	10159.5	0.411415
$849$	0	0
$850$	−2484.13	−0.100241
$851$	−7253.01	−0.292162
$852$	0	0
$853$	38863.4	1.55998	0.779988	−	0.625795i	$-0.215225\pi$
$853$	38863.4	1.55998	0.779988	+	0.625795i	$0.215225\pi$
$854$	−36984.2	−1.48194
$855$	0	0
$856$	3630.15	0.144949
$857$	10618.6	0.423249	0.211624	−	0.977351i	$-0.432125\pi$
$857$	10618.6	0.423249	0.211624	+	0.977351i	$0.432125\pi$
$858$	0	0
$859$	−12830.2	−0.509616	−0.254808	−	0.966992i	$-0.582012\pi$
$859$	−12830.2	−0.509616	−0.254808	+	0.966992i	$0.582012\pi$
$860$	3802.07	0.150755
$861$	0	0
$862$	18851.9	0.744892
$863$	−14812.7	−0.584277	−0.292139	−	0.956376i	$-0.594367\pi$
$863$	−14812.7	−0.584277	−0.292139	+	0.956376i	$0.594367\pi$
$864$	0	0
$865$	14594.5	0.573674
$866$	−28656.5	−1.12446
$867$	0	0
$868$	18524.2	0.724370
$869$	29185.6	1.13930
$870$	0	0
$871$	74146.6	2.88446
$872$	6492.70	0.252145
$873$	0	0
$874$	−557.407	−0.0215728
$875$	4047.62	0.156382
$876$	0	0
$877$	−46280.5	−1.78196	−0.890981	−	0.454041i	$-0.849982\pi$
$877$	−46280.5	−1.78196	−0.890981	+	0.454041i	$0.849982\pi$
$878$	−1337.78	−0.0514213
$879$	0	0
$880$	2729.40	0.104555
$881$	498.905	0.0190789	0.00953946	−	0.999954i	$-0.496963\pi$
$881$	498.905	0.0190789	0.00953946	+	0.999954i	$0.496963\pi$
$882$	0	0
$883$	29474.2	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$883$	29474.2	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$884$	14845.3	0.564821
$885$	0	0
$886$	11060.2	0.419385
$887$	−40034.8	−1.51549	−0.757744	−	0.652552i	$-0.773699\pi$
$887$	−40034.8	−1.51549	−0.757744	+	0.652552i	$0.773699\pi$
$888$	0	0
$889$	49940.7	1.88409
$890$	−3145.98	−0.118487
$891$	0	0
$892$	−20519.1	−0.770215
$893$	4910.27	0.184004
$894$	0	0
$895$	−15091.1	−0.563619
$896$	4144.76	0.154539
$897$	0	0
$898$	−4893.24	−0.181837
$899$	−729.906	−0.0270787
$900$	0	0
$901$	−31547.1	−1.16647
$902$	17971.8	0.663408
$903$	0	0
$904$	61.7227	0.00227087
$905$	−15362.4	−0.564271
$906$	0	0
$907$	19536.6	0.715217	0.357608	−	0.933872i	$-0.383592\pi$
$907$	19536.6	0.715217	0.357608	+	0.933872i	$0.383592\pi$
$908$	13378.2	0.488954
$909$	0	0
$910$	−24188.8	−0.881155
$911$	−35050.3	−1.27472	−0.637359	−	0.770567i	$-0.719973\pi$
$911$	−35050.3	−1.27472	−0.637359	+	0.770567i	$0.719973\pi$
$912$	0	0
$913$	−13524.8	−0.490258
$914$	18203.6	0.658776
$915$	0	0
$916$	16646.1	0.600438
$917$	−28372.6	−1.02175
$918$	0	0
$919$	−12097.0	−0.434215	−0.217107	−	0.976148i	$-0.569662\pi$
$919$	−12097.0	−0.434215	−0.217107	+	0.976148i	$0.569662\pi$
$920$	−920.000	−0.0329690
$921$	0	0
$922$	−7984.51	−0.285202
$923$	−50883.9	−1.81459
$924$	0	0
$925$	7883.71	0.280232
$926$	−29371.7	−1.04235
$927$	0	0
$928$	−163.315	−0.00577702
$929$	−10791.1	−0.381101	−0.190551	−	0.981677i	$-0.561027\pi$
$929$	−10791.1	−0.381101	−0.190551	+	0.981677i	$0.561027\pi$
$930$	0	0
$931$	8549.23	0.300956
$932$	9010.87	0.316696
$933$	0	0
$934$	1242.66	0.0435344
$935$	−8475.25	−0.296439
$936$	0	0
$937$	22186.8	0.773543	0.386772	−	0.922176i	$-0.373590\pi$
$937$	22186.8	0.773543	0.386772	+	0.922176i	$0.373590\pi$
$938$	−64281.4	−2.23759
$939$	0	0
$940$	8104.40	0.281209
$941$	1899.06	0.0657891	0.0328945	−	0.999459i	$-0.489527\pi$
$941$	1899.06	0.0657891	0.0328945	+	0.999459i	$0.489527\pi$
$942$	0	0
$943$	−6057.74	−0.209191
$944$	−7690.88	−0.265166
$945$	0	0
$946$	12971.7	0.445822
$947$	−44333.4	−1.52127	−0.760635	−	0.649180i	$-0.775112\pi$
$947$	−44333.4	−1.52127	−0.760635	+	0.649180i	$0.775112\pi$
$948$	0	0
$949$	61259.6	2.09544
$950$	605.878	0.0206919
$951$	0	0
$952$	−12870.2	−0.438156
$953$	52107.8	1.77118	0.885592	−	0.464464i	$-0.153753\pi$
$953$	52107.8	1.77118	0.885592	+	0.464464i	$0.153753\pi$
$954$	0	0
$955$	18608.0	0.630512
$956$	26913.8	0.910518
$957$	0	0
$958$	−15196.4	−0.512498
$959$	3532.98	0.118963
$960$	0	0
$961$	−9336.82	−0.313411
$962$	−47113.5	−1.57900
$963$	0	0
$964$	20693.6	0.691385
$965$	1777.70	0.0593017
$966$	0	0
$967$	16426.7	0.546274	0.273137	−	0.961975i	$-0.411939\pi$
$967$	16426.7	0.546274	0.273137	+	0.961975i	$0.411939\pi$
$968$	−1335.94	−0.0443582
$969$	0	0
$970$	13063.6	0.432420
$971$	−18590.7	−0.614421	−0.307211	−	0.951642i	$-0.599396\pi$
$971$	−18590.7	−0.614421	−0.307211	+	0.951642i	$0.599396\pi$
$972$	0	0
$973$	−84222.1	−2.77496
$974$	−23792.4	−0.782707
$975$	0	0
$976$	−9137.29	−0.299670
$977$	−58200.0	−1.90582	−0.952908	−	0.303258i	$-0.901926\pi$
$977$	−58200.0	−1.90582	−0.952908	+	0.303258i	$0.901926\pi$
$978$	0	0
$979$	−10733.3	−0.350396
$980$	14110.5	0.459942
$981$	0	0
$982$	38237.8	1.24258
$983$	−28037.4	−0.909720	−0.454860	−	0.890563i	$-0.650311\pi$
$983$	−28037.4	−0.909720	−0.454860	+	0.890563i	$0.650311\pi$
$984$	0	0
$985$	−15889.6	−0.513995
$986$	507.120	0.0163793
$987$	0	0
$988$	−3620.76	−0.116591
$989$	−4372.38	−0.140580
$990$	0	0
$991$	39782.6	1.27521	0.637607	−	0.770362i	$-0.279924\pi$
$991$	39782.6	1.27521	0.637607	+	0.770362i	$0.279924\pi$
$992$	4576.58	0.146478
$993$	0	0
$994$	44113.8	1.40765
$995$	−9617.59	−0.306430
$996$	0	0
$997$	−36951.1	−1.17377	−0.586887	−	0.809669i	$-0.699647\pi$
$997$	−36951.1	−1.17377	−0.586887	+	0.809669i	$0.699647\pi$
$998$	−34516.7	−1.09480
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bn.1.5	yes	5
3.2	odd	2		2070.4.a.bk.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2070.4.a.bk.1.5	✓	5	3.2	odd	2
2070.4.a.bn.1.5	yes	5	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +32.3809 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +32.3809 q^{7} +8.00000 q^{8} +10.0000 q^{10} +34.1176 q^{11} -74.7007 q^{13} +64.7619 q^{14} +16.0000 q^{16} -49.6826 q^{17} +12.1176 q^{19} +20.0000 q^{20} +68.2351 q^{22} -23.0000 q^{23} +25.0000 q^{25} -149.401 q^{26} +129.524 q^{28} -5.10359 q^{29} +143.018 q^{31} +32.0000 q^{32} -99.3653 q^{34} +161.905 q^{35} +315.348 q^{37} +24.2351 q^{38} +40.0000 q^{40} +263.380 q^{41} +190.104 q^{43} +136.470 q^{44} -46.0000 q^{46} +405.220 q^{47} +705.525 q^{49} +50.0000 q^{50} -298.803 q^{52} +634.972 q^{53} +170.588 q^{55} +259.047 q^{56} -10.2072 q^{58} -480.680 q^{59} -571.081 q^{61} +286.036 q^{62} +64.0000 q^{64} -373.504 q^{65} -992.582 q^{67} -198.731 q^{68} +323.809 q^{70} +681.170 q^{71} -820.067 q^{73} +630.697 q^{74} +48.4702 q^{76} +1104.76 q^{77} +855.443 q^{79} +80.0000 q^{80} +526.760 q^{82} -396.418 q^{83} -248.413 q^{85} +380.207 q^{86} +272.940 q^{88} -314.598 q^{89} -2418.88 q^{91} -92.0000 q^{92} +810.440 q^{94} +60.5878 q^{95} +1306.36 q^{97} +1411.05 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{2} + 20 q^{4} + 25 q^{5} + 12 q^{7} + 40 q^{8}+O(q^{10}) \) 5 * q + 10 * q^2 + 20 * q^4 + 25 * q^5 + 12 * q^7 + 40 * q^8	\( 5 q + 10 q^{2} + 20 q^{4} + 25 q^{5} + 12 q^{7} + 40 q^{8} + 50 q^{10} + 82 q^{11} + 16 q^{13} + 24 q^{14} + 80 q^{16} - 10 q^{17} - 28 q^{19} + 100 q^{20} + 164 q^{22} - 115 q^{23} + 125 q^{25} + 32 q^{26} + 48 q^{28} + 396 q^{29} + 564 q^{31} + 160 q^{32} - 20 q^{34} + 60 q^{35} + 290 q^{37} - 56 q^{38} + 200 q^{40} - 130 q^{41} + 468 q^{43} + 328 q^{44} - 230 q^{46} + 976 q^{47} + 1293 q^{49} + 250 q^{50} + 64 q^{52} + 598 q^{53} + 410 q^{55} + 96 q^{56} + 792 q^{58} - 400 q^{59} - 670 q^{61} + 1128 q^{62} + 320 q^{64} + 80 q^{65} - 10 q^{67} - 40 q^{68} + 120 q^{70} + 82 q^{71} + 1410 q^{73} + 580 q^{74} - 112 q^{76} + 316 q^{77} + 1140 q^{79} + 400 q^{80} - 260 q^{82} + 2640 q^{83} - 50 q^{85} + 936 q^{86} + 656 q^{88} - 856 q^{89} - 920 q^{91} - 460 q^{92} + 1952 q^{94} - 140 q^{95} + 946 q^{97} + 2586 q^{98}+O(q^{100}) \) 5 * q + 10 * q^2 + 20 * q^4 + 25 * q^5 + 12 * q^7 + 40 * q^8 + 50 * q^10 + 82 * q^11 + 16 * q^13 + 24 * q^14 + 80 * q^16 - 10 * q^17 - 28 * q^19 + 100 * q^20 + 164 * q^22 - 115 * q^23 + 125 * q^25 + 32 * q^26 + 48 * q^28 + 396 * q^29 + 564 * q^31 + 160 * q^32 - 20 * q^34 + 60 * q^35 + 290 * q^37 - 56 * q^38 + 200 * q^40 - 130 * q^41 + 468 * q^43 + 328 * q^44 - 230 * q^46 + 976 * q^47 + 1293 * q^49 + 250 * q^50 + 64 * q^52 + 598 * q^53 + 410 * q^55 + 96 * q^56 + 792 * q^58 - 400 * q^59 - 670 * q^61 + 1128 * q^62 + 320 * q^64 + 80 * q^65 - 10 * q^67 - 40 * q^68 + 120 * q^70 + 82 * q^71 + 1410 * q^73 + 580 * q^74 - 112 * q^76 + 316 * q^77 + 1140 * q^79 + 400 * q^80 - 260 * q^82 + 2640 * q^83 - 50 * q^85 + 936 * q^86 + 656 * q^88 - 856 * q^89 - 920 * q^91 - 460 * q^92 + 1952 * q^94 - 140 * q^95 + 946 * q^97 + 2586 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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