LMFDB - Embedded newform 2070.4.a.bg.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2070,4,Mod(1,2070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,-8,0,16,-20,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$122.133953712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 60x^{2} - 45x + 108 $ x^4 - 2x^3 - 60x^2 - 45*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.92711$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +24.6272 q^{7} -8.00000 q^{8} +10.0000 q^{10} +24.6573 q^{11} +15.2681 q^{13} -49.2544 q^{14} +16.0000 q^{16} +78.7383 q^{17} -16.3731 q^{19} -20.0000 q^{20} -49.3146 q^{22} +23.0000 q^{23} +25.0000 q^{25} -30.5362 q^{26} +98.5088 q^{28} -36.5449 q^{29} +186.438 q^{31} -32.0000 q^{32} -157.477 q^{34} -123.136 q^{35} +327.841 q^{37} +32.7462 q^{38} +40.0000 q^{40} +313.067 q^{41} -394.105 q^{43} +98.6293 q^{44} -46.0000 q^{46} +252.906 q^{47} +263.499 q^{49} -50.0000 q^{50} +61.0723 q^{52} +8.12126 q^{53} -123.287 q^{55} -197.018 q^{56} +73.0898 q^{58} -173.659 q^{59} +420.666 q^{61} -372.875 q^{62} +64.0000 q^{64} -76.3404 q^{65} +142.945 q^{67} +314.953 q^{68} +246.272 q^{70} -658.533 q^{71} -144.794 q^{73} -655.683 q^{74} -65.4924 q^{76} +607.241 q^{77} -521.592 q^{79} -80.0000 q^{80} -626.134 q^{82} +987.368 q^{83} -393.691 q^{85} +788.210 q^{86} -197.259 q^{88} -176.234 q^{89} +376.010 q^{91} +92.0000 q^{92} -505.813 q^{94} +81.8655 q^{95} +769.092 q^{97} -526.998 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 8 * q^7 - 32 * q^8 + 40 * q^10 - 21 * q^11 + 70 * q^13 - 16 * q^14 + 64 * q^16 - 56 * q^17 + 173 * q^19 - 80 * q^20 + 42 * q^22 + 92 * q^23 + 100 * q^25 - 140 * q^26 + 32 * q^28 + 118 * q^29 + 17 * q^31 - 128 * q^32 + 112 * q^34 - 40 * q^35 - 343 * q^37 - 346 * q^38 + 160 * q^40 - 139 * q^41 - 50 * q^43 - 84 * q^44 - 184 * q^46 - 367 * q^47 - 124 * q^49 - 200 * q^50 + 280 * q^52 + 353 * q^53 + 105 * q^55 - 64 * q^56 - 236 * q^58 + 453 * q^59 - 327 * q^61 - 34 * q^62 + 256 * q^64 - 350 * q^65 - 455 * q^67 - 224 * q^68 + 80 * q^70 - 195 * q^71 - 633 * q^73 + 686 * q^74 + 692 * q^76 + 2 * q^77 - 1140 * q^79 - 320 * q^80 + 278 * q^82 + 1199 * q^83 + 280 * q^85 + 100 * q^86 + 168 * q^88 + 2170 * q^89 + 557 * q^91 + 368 * q^92 + 734 * q^94 - 865 * q^95 - 703 * q^97 + 248 * q^98

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
$7$	24.6272	1.32974	0.664872	−	0.746957i	$-0.268486\pi$
$7$	24.6272	1.32974	0.664872	+	0.746957i	$0.268486\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	24.6573	0.675860	0.337930	−	0.941171i	$-0.390273\pi$
$11$	24.6573	0.675860	0.337930	+	0.941171i	$0.390273\pi$
$12$	0	0
$13$	15.2681	0.325739	0.162869	−	0.986648i	$-0.447925\pi$
$13$	15.2681	0.325739	0.162869	+	0.986648i	$0.447925\pi$
$14$	−49.2544	−0.940271
$15$	0	0
$16$	16.0000	0.250000
$17$	78.7383	1.12334	0.561672	−	0.827360i	$-0.310158\pi$
$17$	78.7383	1.12334	0.561672	+	0.827360i	$0.310158\pi$
$18$	0	0
$19$	−16.3731	−0.197697	−0.0988486	−	0.995102i	$-0.531516\pi$
$19$	−16.3731	−0.197697	−0.0988486	+	0.995102i	$0.531516\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−49.3146	−0.477905
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	−30.5362	−0.230332
$27$	0	0
$28$	98.5088	0.664872
$29$	−36.5449	−0.234008	−0.117004	−	0.993131i	$-0.537329\pi$
$29$	−36.5449	−0.234008	−0.117004	+	0.993131i	$0.537329\pi$
$30$	0	0
$31$	186.438	1.08017	0.540083	−	0.841612i	$-0.318393\pi$
$31$	186.438	1.08017	0.540083	+	0.841612i	$0.318393\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−157.477	−0.794324
$35$	−123.136	−0.594680
$36$	0	0
$37$	327.841	1.45667	0.728335	−	0.685221i	$-0.240294\pi$
$37$	327.841	1.45667	0.728335	+	0.685221i	$0.240294\pi$
$38$	32.7462	0.139793
$39$	0	0
$40$	40.0000	0.158114
$41$	313.067	1.19251	0.596254	−	0.802796i	$-0.296655\pi$
$41$	313.067	1.19251	0.596254	+	0.802796i	$0.296655\pi$
$42$	0	0
$43$	−394.105	−1.39768	−0.698842	−	0.715276i	$-0.746301\pi$
$43$	−394.105	−1.39768	−0.698842	+	0.715276i	$0.746301\pi$
$44$	98.6293	0.337930
$45$	0	0
$46$	−46.0000	−0.147442
$47$	252.906	0.784897	0.392449	−	0.919774i	$-0.371628\pi$
$47$	252.906	0.784897	0.392449	+	0.919774i	$0.371628\pi$
$48$	0	0
$49$	263.499	0.768219
$50$	−50.0000	−0.141421
$51$	0	0
$52$	61.0723	0.162869
$53$	8.12126	0.0210480	0.0105240	−	0.999945i	$-0.496650\pi$
$53$	8.12126	0.0210480	0.0105240	+	0.999945i	$0.496650\pi$
$54$	0	0
$55$	−123.287	−0.302254
$56$	−197.018	−0.470135
$57$	0	0
$58$	73.0898	0.165468
$59$	−173.659	−0.383193	−0.191597	−	0.981474i	$-0.561367\pi$
$59$	−173.659	−0.383193	−0.191597	+	0.981474i	$0.561367\pi$
$60$	0	0
$61$	420.666	0.882964	0.441482	−	0.897270i	$-0.354453\pi$
$61$	420.666	0.882964	0.441482	+	0.897270i	$0.354453\pi$
$62$	−372.875	−0.763793
$63$	0	0
$64$	64.0000	0.125000
$65$	−76.3404	−0.145675
$66$	0	0
$67$	142.945	0.260649	0.130324	−	0.991471i	$-0.458398\pi$
$67$	142.945	0.260649	0.130324	+	0.991471i	$0.458398\pi$
$68$	314.953	0.561672
$69$	0	0
$70$	246.272	0.420502
$71$	−658.533	−1.10075	−0.550376	−	0.834917i	$-0.685516\pi$
$71$	−658.533	−1.10075	−0.550376	+	0.834917i	$0.685516\pi$
$72$	0	0
$73$	−144.794	−0.232149	−0.116075	−	0.993241i	$-0.537031\pi$
$73$	−144.794	−0.232149	−0.116075	+	0.993241i	$0.537031\pi$
$74$	−655.683	−1.03002
$75$	0	0
$76$	−65.4924	−0.0988486
$77$	607.241	0.898721
$78$	0	0
$79$	−521.592	−0.742831	−0.371415	−	0.928467i	$-0.621127\pi$
$79$	−521.592	−0.742831	−0.371415	+	0.928467i	$0.621127\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	−626.134	−0.843230
$83$	987.368	1.30576	0.652878	−	0.757463i	$-0.273562\pi$
$83$	987.368	1.30576	0.652878	+	0.757463i	$0.273562\pi$
$84$	0	0
$85$	−393.691	−0.502375
$86$	788.210	0.988312
$87$	0	0
$88$	−197.259	−0.238953
$89$	−176.234	−0.209896	−0.104948	−	0.994478i	$-0.533468\pi$
$89$	−176.234	−0.209896	−0.104948	+	0.994478i	$0.533468\pi$
$90$	0	0
$91$	376.010	0.433149
$92$	92.0000	0.104257
$93$	0	0
$94$	−505.813	−0.555006
$95$	81.8655	0.0884129
$96$	0	0
$97$	769.092	0.805046	0.402523	−	0.915410i	$-0.368133\pi$
$97$	769.092	0.805046	0.402523	+	0.915410i	$0.368133\pi$
$98$	−526.998	−0.543213
$99$	0	0
$100$	100.000	0.100000
$101$	33.6036	0.0331058	0.0165529	−	0.999863i	$-0.494731\pi$
$101$	33.6036	0.0331058	0.0165529	+	0.999863i	$0.494731\pi$
$102$	0	0
$103$	−1868.28	−1.78725	−0.893626	−	0.448813i	$-0.851847\pi$
$103$	−1868.28	−1.78725	−0.893626	+	0.448813i	$0.851847\pi$
$104$	−122.145	−0.115166
$105$	0	0
$106$	−16.2425	−0.0148832
$107$	1438.84	1.29998	0.649992	−	0.759941i	$-0.274772\pi$
$107$	1438.84	1.29998	0.649992	+	0.759941i	$0.274772\pi$
$108$	0	0
$109$	−1864.49	−1.63840	−0.819202	−	0.573504i	$-0.805584\pi$
$109$	−1864.49	−1.63840	−0.819202	+	0.573504i	$0.805584\pi$
$110$	246.573	0.213726
$111$	0	0
$112$	394.035	0.332436
$113$	397.768	0.331141	0.165570	−	0.986198i	$-0.447053\pi$
$113$	397.768	0.331141	0.165570	+	0.986198i	$0.447053\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	−146.180	−0.117004
$117$	0	0
$118$	347.317	0.270959
$119$	1939.10	1.49376
$120$	0	0
$121$	−723.016	−0.543213
$122$	−841.332	−0.624350
$123$	0	0
$124$	745.750	0.540083
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1370.48	0.957564	0.478782	−	0.877934i	$-0.341078\pi$
$127$	1370.48	0.957564	0.478782	+	0.877934i	$0.341078\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	152.681	0.103008
$131$	−1891.14	−1.26130	−0.630649	−	0.776068i	$-0.717211\pi$
$131$	−1891.14	−1.26130	−0.630649	+	0.776068i	$0.717211\pi$
$132$	0	0
$133$	−403.224	−0.262887
$134$	−285.889	−0.184307
$135$	0	0
$136$	−629.906	−0.397162
$137$	−2226.21	−1.38831	−0.694155	−	0.719826i	$-0.744222\pi$
$137$	−2226.21	−1.38831	−0.694155	+	0.719826i	$0.744222\pi$
$138$	0	0
$139$	−699.554	−0.426873	−0.213437	−	0.976957i	$-0.568466\pi$
$139$	−699.554	−0.426873	−0.213437	+	0.976957i	$0.568466\pi$
$140$	−492.544	−0.297340
$141$	0	0
$142$	1317.07	0.778350
$143$	376.470	0.220154
$144$	0	0
$145$	182.725	0.104651
$146$	289.589	0.164154
$147$	0	0
$148$	1311.37	0.728335
$149$	1174.11	0.645549	0.322775	−	0.946476i	$-0.395384\pi$
$149$	1174.11	0.645549	0.322775	+	0.946476i	$0.395384\pi$
$150$	0	0
$151$	2733.01	1.47291	0.736455	−	0.676487i	$-0.236498\pi$
$151$	2733.01	1.47291	0.736455	+	0.676487i	$0.236498\pi$
$152$	130.985	0.0698965
$153$	0	0
$154$	−1214.48	−0.635492
$155$	−932.188	−0.483065
$156$	0	0
$157$	−4.77232	−0.00242594	−0.00121297	−	0.999999i	$-0.500386\pi$
$157$	−4.77232	−0.00242594	−0.00121297	+	0.999999i	$0.500386\pi$
$158$	1043.18	0.525261
$159$	0	0
$160$	160.000	0.0790569
$161$	566.426	0.277271
$162$	0	0
$163$	1523.51	0.732090	0.366045	−	0.930597i	$-0.380712\pi$
$163$	1523.51	0.732090	0.366045	+	0.930597i	$0.380712\pi$
$164$	1252.27	0.596254
$165$	0	0
$166$	−1974.74	−0.923308
$167$	−3603.81	−1.66989	−0.834944	−	0.550335i	$-0.814500\pi$
$167$	−3603.81	−1.66989	−0.834944	+	0.550335i	$0.814500\pi$
$168$	0	0
$169$	−1963.89	−0.893894
$170$	787.383	0.355232
$171$	0	0
$172$	−1576.42	−0.698842
$173$	−349.993	−0.153812	−0.0769061	−	0.997038i	$-0.524504\pi$
$173$	−349.993	−0.153812	−0.0769061	+	0.997038i	$0.524504\pi$
$174$	0	0
$175$	615.680	0.265949
$176$	394.517	0.168965
$177$	0	0
$178$	352.467	0.148419
$179$	1283.17	0.535804	0.267902	−	0.963446i	$-0.413670\pi$
$179$	1283.17	0.535804	0.267902	+	0.963446i	$0.413670\pi$
$180$	0	0
$181$	3076.40	1.26336	0.631678	−	0.775231i	$-0.282367\pi$
$181$	3076.40	1.26336	0.631678	+	0.775231i	$0.282367\pi$
$182$	−752.020	−0.306283
$183$	0	0
$184$	−184.000	−0.0737210
$185$	−1639.21	−0.651443
$186$	0	0
$187$	1941.48	0.759223
$188$	1011.63	0.392449
$189$	0	0
$190$	−163.731	−0.0625174
$191$	1787.86	0.677306	0.338653	−	0.940911i	$-0.390029\pi$
$191$	1787.86	0.677306	0.338653	+	0.940911i	$0.390029\pi$
$192$	0	0
$193$	−881.506	−0.328768	−0.164384	−	0.986396i	$-0.552564\pi$
$193$	−881.506	−0.328768	−0.164384	+	0.986396i	$0.552564\pi$
$194$	−1538.18	−0.569253
$195$	0	0
$196$	1054.00	0.384109
$197$	−2217.28	−0.801904	−0.400952	−	0.916099i	$-0.631321\pi$
$197$	−2217.28	−0.801904	−0.400952	+	0.916099i	$0.631321\pi$
$198$	0	0
$199$	−211.313	−0.0752742	−0.0376371	−	0.999291i	$-0.511983\pi$
$199$	−211.313	−0.0752742	−0.0376371	+	0.999291i	$0.511983\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−67.2072	−0.0234093
$203$	−899.999	−0.311170
$204$	0	0
$205$	−1565.33	−0.533306
$206$	3736.56	1.26378
$207$	0	0
$208$	244.289	0.0814347
$209$	−403.717	−0.133616
$210$	0	0
$211$	3939.97	1.28549	0.642745	−	0.766080i	$-0.277795\pi$
$211$	3939.97	1.28549	0.642745	+	0.766080i	$0.277795\pi$
$212$	32.4851	0.0105240
$213$	0	0
$214$	−2877.69	−0.919227
$215$	1970.52	0.625063
$216$	0	0
$217$	4591.43	1.43635
$218$	3728.99	1.15853
$219$	0	0
$220$	−493.146	−0.151127
$221$	1202.18	0.365916
$222$	0	0
$223$	4154.48	1.24755	0.623777	−	0.781603i	$-0.285597\pi$
$223$	4154.48	1.24755	0.623777	+	0.781603i	$0.285597\pi$
$224$	−788.070	−0.235068
$225$	0	0
$226$	−795.537	−0.234152
$227$	−6133.31	−1.79331	−0.896656	−	0.442727i	$-0.854011\pi$
$227$	−6133.31	−1.79331	−0.896656	+	0.442727i	$0.854011\pi$
$228$	0	0
$229$	2244.20	0.647603	0.323802	−	0.946125i	$-0.395039\pi$
$229$	2244.20	0.647603	0.323802	+	0.946125i	$0.395039\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	292.359	0.0827342
$233$	−2161.23	−0.607670	−0.303835	−	0.952725i	$-0.598267\pi$
$233$	−2161.23	−0.607670	−0.303835	+	0.952725i	$0.598267\pi$
$234$	0	0
$235$	−1264.53	−0.351017
$236$	−694.634	−0.191597
$237$	0	0
$238$	−3878.21	−1.05625
$239$	−3448.18	−0.933240	−0.466620	−	0.884458i	$-0.654528\pi$
$239$	−3448.18	−0.933240	−0.466620	+	0.884458i	$0.654528\pi$
$240$	0	0
$241$	1654.43	0.442204	0.221102	−	0.975251i	$-0.429035\pi$
$241$	1654.43	0.442204	0.221102	+	0.975251i	$0.429035\pi$
$242$	1446.03	0.384110
$243$	0	0
$244$	1682.66	0.441482
$245$	−1317.50	−0.343558
$246$	0	0
$247$	−249.986	−0.0643976
$248$	−1491.50	−0.381897
$249$	0	0
$250$	250.000	0.0632456
$251$	6157.96	1.54855	0.774277	−	0.632846i	$-0.218113\pi$
$251$	6157.96	1.54855	0.774277	+	0.632846i	$0.218113\pi$
$252$	0	0
$253$	567.118	0.140927
$254$	−2740.96	−0.677100
$255$	0	0
$256$	256.000	0.0625000
$257$	448.443	0.108845	0.0544224	−	0.998518i	$-0.482668\pi$
$257$	448.443	0.108845	0.0544224	+	0.998518i	$0.482668\pi$
$258$	0	0
$259$	8073.82	1.93700
$260$	−305.362	−0.0728374
$261$	0	0
$262$	3782.29	0.891873
$263$	−6298.76	−1.47680	−0.738399	−	0.674364i	$-0.764418\pi$
$263$	−6298.76	−1.47680	−0.738399	+	0.674364i	$0.764418\pi$
$264$	0	0
$265$	−40.6063	−0.00941293
$266$	806.447	0.185889
$267$	0	0
$268$	571.779	0.130324
$269$	2106.69	0.477500	0.238750	−	0.971081i	$-0.423262\pi$
$269$	2106.69	0.477500	0.238750	+	0.971081i	$0.423262\pi$
$270$	0	0
$271$	−3116.92	−0.698668	−0.349334	−	0.936998i	$-0.613592\pi$
$271$	−3116.92	−0.698668	−0.349334	+	0.936998i	$0.613592\pi$
$272$	1259.81	0.280836
$273$	0	0
$274$	4452.43	0.981683
$275$	616.433	0.135172
$276$	0	0
$277$	1177.86	0.255490	0.127745	−	0.991807i	$-0.459226\pi$
$277$	1177.86	0.255490	0.127745	+	0.991807i	$0.459226\pi$
$278$	1399.11	0.301845
$279$	0	0
$280$	985.088	0.210251
$281$	5240.30	1.11249	0.556246	−	0.831018i	$-0.312241\pi$
$281$	5240.30	1.11249	0.556246	+	0.831018i	$0.312241\pi$
$282$	0	0
$283$	7162.44	1.50446	0.752232	−	0.658899i	$-0.228978\pi$
$283$	7162.44	1.50446	0.752232	+	0.658899i	$0.228978\pi$
$284$	−2634.13	−0.550376
$285$	0	0
$286$	−752.940	−0.155672
$287$	7709.96	1.58573
$288$	0	0
$289$	1286.72	0.261901
$290$	−365.449	−0.0739997
$291$	0	0
$292$	−579.177	−0.116075
$293$	7564.70	1.50831	0.754154	−	0.656698i	$-0.228047\pi$
$293$	7564.70	1.50831	0.754154	+	0.656698i	$0.228047\pi$
$294$	0	0
$295$	868.293	0.171369
$296$	−2622.73	−0.515011
$297$	0	0
$298$	−2348.22	−0.456472
$299$	351.166	0.0679212
$300$	0	0
$301$	−9705.70	−1.85856
$302$	−5466.02	−1.04150
$303$	0	0
$304$	−261.970	−0.0494243
$305$	−2103.33	−0.394873
$306$	0	0
$307$	5675.81	1.05516	0.527582	−	0.849504i	$-0.323099\pi$
$307$	5675.81	1.05516	0.527582	+	0.849504i	$0.323099\pi$
$308$	2428.96	0.449361
$309$	0	0
$310$	1864.38	0.341579
$311$	−4934.29	−0.899671	−0.449835	−	0.893111i	$-0.648517\pi$
$311$	−4934.29	−0.899671	−0.449835	+	0.893111i	$0.648517\pi$
$312$	0	0
$313$	−4458.67	−0.805173	−0.402587	−	0.915382i	$-0.631889\pi$
$313$	−4458.67	−0.805173	−0.402587	+	0.915382i	$0.631889\pi$
$314$	9.54464	0.00171540
$315$	0	0
$316$	−2086.37	−0.371415
$317$	−4132.87	−0.732256	−0.366128	−	0.930564i	$-0.619317\pi$
$317$	−4132.87	−0.732256	−0.366128	+	0.930564i	$0.619317\pi$
$318$	0	0
$319$	−901.100	−0.158156
$320$	−320.000	−0.0559017
$321$	0	0
$322$	−1132.85	−0.196060
$323$	−1289.19	−0.222082
$324$	0	0
$325$	381.702	0.0651477
$326$	−3047.02	−0.517665
$327$	0	0
$328$	−2504.53	−0.421615
$329$	6228.37	1.04371
$330$	0	0
$331$	10319.9	1.71370	0.856849	−	0.515567i	$-0.172419\pi$
$331$	10319.9	1.71370	0.856849	+	0.515567i	$0.172419\pi$
$332$	3949.47	0.652878
$333$	0	0
$334$	7207.62	1.18079
$335$	−714.723	−0.116566
$336$	0	0
$337$	−9817.59	−1.58694	−0.793469	−	0.608611i	$-0.791727\pi$
$337$	−9817.59	−1.58694	−0.793469	+	0.608611i	$0.791727\pi$
$338$	3927.77	0.632079
$339$	0	0
$340$	−1574.77	−0.251187
$341$	4597.05	0.730042
$342$	0	0
$343$	−1957.89	−0.308210
$344$	3152.84	0.494156
$345$	0	0
$346$	699.987	0.108762
$347$	5827.53	0.901552	0.450776	−	0.892637i	$-0.351147\pi$
$347$	5827.53	0.901552	0.450776	+	0.892637i	$0.351147\pi$
$348$	0	0
$349$	6483.51	0.994426	0.497213	−	0.867629i	$-0.334357\pi$
$349$	6483.51	0.994426	0.497213	+	0.867629i	$0.334357\pi$
$350$	−1231.36	−0.188054
$351$	0	0
$352$	−789.034	−0.119476
$353$	11535.2	1.73925	0.869627	−	0.493709i	$-0.164359\pi$
$353$	11535.2	1.73925	0.869627	+	0.493709i	$0.164359\pi$
$354$	0	0
$355$	3292.66	0.492272
$356$	−704.934	−0.104948
$357$	0	0
$358$	−2566.35	−0.378870
$359$	−5922.53	−0.870695	−0.435347	−	0.900263i	$-0.643374\pi$
$359$	−5922.53	−0.870695	−0.435347	+	0.900263i	$0.643374\pi$
$360$	0	0
$361$	−6590.92	−0.960916
$362$	−6152.81	−0.893327
$363$	0	0
$364$	1504.04	0.216575
$365$	723.971	0.103820
$366$	0	0
$367$	−4042.72	−0.575009	−0.287504	−	0.957779i	$-0.592826\pi$
$367$	−4042.72	−0.575009	−0.287504	+	0.957779i	$0.592826\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	3278.41	0.460639
$371$	200.004	0.0279884
$372$	0	0
$373$	−9078.68	−1.26026	−0.630129	−	0.776491i	$-0.716998\pi$
$373$	−9078.68	−1.26026	−0.630129	+	0.776491i	$0.716998\pi$
$374$	−3882.95	−0.536852
$375$	0	0
$376$	−2023.25	−0.277503
$377$	−557.971	−0.0762253
$378$	0	0
$379$	6561.42	0.889280	0.444640	−	0.895709i	$-0.353332\pi$
$379$	6561.42	0.889280	0.444640	+	0.895709i	$0.353332\pi$
$380$	327.462	0.0442064
$381$	0	0
$382$	−3575.73	−0.478927
$383$	−587.548	−0.0783872	−0.0391936	−	0.999232i	$-0.512479\pi$
$383$	−587.548	−0.0783872	−0.0391936	+	0.999232i	$0.512479\pi$
$384$	0	0
$385$	−3036.20	−0.401920
$386$	1763.01	0.232474
$387$	0	0
$388$	3076.37	0.402523
$389$	6470.36	0.843343	0.421672	−	0.906749i	$-0.361444\pi$
$389$	6470.36	0.843343	0.421672	+	0.906749i	$0.361444\pi$
$390$	0	0
$391$	1810.98	0.234233
$392$	−2107.99	−0.271606
$393$	0	0
$394$	4434.57	0.567031
$395$	2607.96	0.332204
$396$	0	0
$397$	9118.76	1.15279	0.576395	−	0.817171i	$-0.304459\pi$
$397$	9118.76	1.15279	0.576395	+	0.817171i	$0.304459\pi$
$398$	422.626	0.0532269
$399$	0	0
$400$	400.000	0.0500000
$401$	3199.75	0.398474	0.199237	−	0.979951i	$-0.436154\pi$
$401$	3199.75	0.398474	0.199237	+	0.979951i	$0.436154\pi$
$402$	0	0
$403$	2846.54	0.351852
$404$	134.414	0.0165529
$405$	0	0
$406$	1800.00	0.220031
$407$	8083.69	0.984505
$408$	0	0
$409$	6146.22	0.743059	0.371529	−	0.928421i	$-0.378833\pi$
$409$	6146.22	0.743059	0.371529	+	0.928421i	$0.378833\pi$
$410$	3130.67	0.377104
$411$	0	0
$412$	−7473.11	−0.893626
$413$	−4276.72	−0.509549
$414$	0	0
$415$	−4936.84	−0.583951
$416$	−488.578	−0.0575830
$417$	0	0
$418$	807.434	0.0944806
$419$	9933.76	1.15822	0.579112	−	0.815248i	$-0.303399\pi$
$419$	9933.76	1.15822	0.579112	+	0.815248i	$0.303399\pi$
$420$	0	0
$421$	3653.89	0.422993	0.211496	−	0.977379i	$-0.432166\pi$
$421$	3653.89	0.422993	0.211496	+	0.977379i	$0.432166\pi$
$422$	−7879.93	−0.908979
$423$	0	0
$424$	−64.9701	−0.00744158
$425$	1968.46	0.224669
$426$	0	0
$427$	10359.8	1.17412
$428$	5755.37	0.649992
$429$	0	0
$430$	−3941.05	−0.441987
$431$	3056.87	0.341634	0.170817	−	0.985303i	$-0.445359\pi$
$431$	3056.87	0.341634	0.170817	+	0.985303i	$0.445359\pi$
$432$	0	0
$433$	1703.05	0.189014	0.0945072	−	0.995524i	$-0.469872\pi$
$433$	1703.05	0.189014	0.0945072	+	0.995524i	$0.469872\pi$
$434$	−9182.87	−1.01565
$435$	0	0
$436$	−7457.98	−0.819202
$437$	−376.581	−0.0412227
$438$	0	0
$439$	−758.630	−0.0824771	−0.0412386	−	0.999149i	$-0.513130\pi$
$439$	−758.630	−0.0824771	−0.0412386	+	0.999149i	$0.513130\pi$
$440$	986.293	0.106863
$441$	0	0
$442$	−2404.36	−0.258742
$443$	−5971.42	−0.640430	−0.320215	−	0.947345i	$-0.603755\pi$
$443$	−5971.42	−0.640430	−0.320215	+	0.947345i	$0.603755\pi$
$444$	0	0
$445$	881.168	0.0938682
$446$	−8308.95	−0.882153
$447$	0	0
$448$	1576.14	0.166218
$449$	3515.62	0.369515	0.184757	−	0.982784i	$-0.440850\pi$
$449$	3515.62	0.369515	0.184757	+	0.982784i	$0.440850\pi$
$450$	0	0
$451$	7719.39	0.805969
$452$	1591.07	0.165570
$453$	0	0
$454$	12266.6	1.26806
$455$	−1880.05	−0.193710
$456$	0	0
$457$	12155.6	1.24423	0.622117	−	0.782925i	$-0.286273\pi$
$457$	12155.6	1.24423	0.622117	+	0.782925i	$0.286273\pi$
$458$	−4488.41	−0.457925
$459$	0	0
$460$	−460.000	−0.0466252
$461$	8854.82	0.894599	0.447299	−	0.894384i	$-0.352386\pi$
$461$	8854.82	0.894599	0.447299	+	0.894384i	$0.352386\pi$
$462$	0	0
$463$	17908.6	1.79759	0.898793	−	0.438374i	$-0.144445\pi$
$463$	17908.6	1.79759	0.898793	+	0.438374i	$0.144445\pi$
$464$	−584.719	−0.0585019
$465$	0	0
$466$	4322.46	0.429687
$467$	−8596.74	−0.851841	−0.425920	−	0.904761i	$-0.640050\pi$
$467$	−8596.74	−0.851841	−0.425920	+	0.904761i	$0.640050\pi$
$468$	0	0
$469$	3520.33	0.346596
$470$	2529.06	0.248206
$471$	0	0
$472$	1389.27	0.135479
$473$	−9717.57	−0.944639
$474$	0	0
$475$	−409.328	−0.0395394
$476$	7756.42	0.746880
$477$	0	0
$478$	6896.37	0.659901
$479$	12385.2	1.18140	0.590702	−	0.806890i	$-0.298851\pi$
$479$	12385.2	1.18140	0.590702	+	0.806890i	$0.298851\pi$
$480$	0	0
$481$	5005.51	0.474494
$482$	−3308.86	−0.312685
$483$	0	0
$484$	−2892.07	−0.271606
$485$	−3845.46	−0.360027
$486$	0	0
$487$	4331.97	0.403081	0.201541	−	0.979480i	$-0.435405\pi$
$487$	4331.97	0.403081	0.201541	+	0.979480i	$0.435405\pi$
$488$	−3365.33	−0.312175
$489$	0	0
$490$	2634.99	0.242932
$491$	11366.0	1.04468	0.522341	−	0.852737i	$-0.325059\pi$
$491$	11366.0	1.04468	0.522341	+	0.852737i	$0.325059\pi$
$492$	0	0
$493$	−2877.48	−0.262871
$494$	499.972	0.0455360
$495$	0	0
$496$	2983.00	0.270042
$497$	−16217.8	−1.46372
$498$	0	0
$499$	12620.5	1.13221	0.566105	−	0.824333i	$-0.308450\pi$
$499$	12620.5	1.13221	0.566105	+	0.824333i	$0.308450\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	−12315.9	−1.09499
$503$	11183.5	0.991348	0.495674	−	0.868509i	$-0.334921\pi$
$503$	11183.5	0.991348	0.495674	+	0.868509i	$0.334921\pi$
$504$	0	0
$505$	−168.018	−0.0148054
$506$	−1134.24	−0.0996502
$507$	0	0
$508$	5481.93	0.478782
$509$	5066.54	0.441199	0.220599	−	0.975364i	$-0.429199\pi$
$509$	5066.54	0.441199	0.220599	+	0.975364i	$0.429199\pi$
$510$	0	0
$511$	−3565.88	−0.308699
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−896.886	−0.0769649
$515$	9341.39	0.799283
$516$	0	0
$517$	6235.99	0.530481
$518$	−16147.6	−1.36966
$519$	0	0
$520$	610.723	0.0515038
$521$	−20873.3	−1.75523	−0.877616	−	0.479364i	$-0.840867\pi$
$521$	−20873.3	−1.75523	−0.877616	+	0.479364i	$0.840867\pi$
$522$	0	0
$523$	8937.71	0.747264	0.373632	−	0.927577i	$-0.378112\pi$
$523$	8937.71	0.747264	0.373632	+	0.927577i	$0.378112\pi$
$524$	−7564.58	−0.630649
$525$	0	0
$526$	12597.5	1.04425
$527$	14679.8	1.21340
$528$	0	0
$529$	529.000	0.0434783
$530$	81.2126	0.00665595
$531$	0	0
$532$	−1612.89	−0.131443
$533$	4779.93	0.388446
$534$	0	0
$535$	−7194.22	−0.581370
$536$	−1143.56	−0.0921533
$537$	0	0
$538$	−4213.39	−0.337643
$539$	6497.18	0.519209
$540$	0	0
$541$	−14320.8	−1.13808	−0.569038	−	0.822311i	$-0.692684\pi$
$541$	−14320.8	−1.13808	−0.569038	+	0.822311i	$0.692684\pi$
$542$	6233.83	0.494033
$543$	0	0
$544$	−2519.63	−0.198581
$545$	9322.47	0.732717
$546$	0	0
$547$	−17084.3	−1.33541	−0.667706	−	0.744425i	$-0.732724\pi$
$547$	−17084.3	−1.33541	−0.667706	+	0.744425i	$0.732724\pi$
$548$	−8904.86	−0.694155
$549$	0	0
$550$	−1232.87	−0.0955811
$551$	598.354	0.0462627
$552$	0	0
$553$	−12845.3	−0.987775
$554$	−2355.72	−0.180659
$555$	0	0
$556$	−2798.22	−0.213437
$557$	21376.9	1.62616	0.813079	−	0.582153i	$-0.197789\pi$
$557$	21376.9	1.62616	0.813079	+	0.582153i	$0.197789\pi$
$558$	0	0
$559$	−6017.22	−0.455280
$560$	−1970.18	−0.148670
$561$	0	0
$562$	−10480.6	−0.786651
$563$	−1867.78	−0.139818	−0.0699091	−	0.997553i	$-0.522271\pi$
$563$	−1867.78	−0.139818	−0.0699091	+	0.997553i	$0.522271\pi$
$564$	0	0
$565$	−1988.84	−0.148091
$566$	−14324.9	−1.06382
$567$	0	0
$568$	5268.26	0.389175
$569$	−9340.49	−0.688179	−0.344089	−	0.938937i	$-0.611812\pi$
$569$	−9340.49	−0.688179	−0.344089	+	0.938937i	$0.611812\pi$
$570$	0	0
$571$	−5452.88	−0.399643	−0.199821	−	0.979832i	$-0.564036\pi$
$571$	−5452.88	−0.399643	−0.199821	+	0.979832i	$0.564036\pi$
$572$	1505.88	0.110077
$573$	0	0
$574$	−15419.9	−1.12128
$575$	575.000	0.0417029
$576$	0	0
$577$	3343.59	0.241240	0.120620	−	0.992699i	$-0.461512\pi$
$577$	3343.59	0.241240	0.120620	+	0.992699i	$0.461512\pi$
$578$	−2573.44	−0.185192
$579$	0	0
$580$	730.898	0.0523257
$581$	24316.1	1.73632
$582$	0	0
$583$	200.249	0.0142255
$584$	1158.35	0.0820771
$585$	0	0
$586$	−15129.4	−1.06653
$587$	19990.7	1.40563	0.702813	−	0.711375i	$-0.251927\pi$
$587$	19990.7	1.40563	0.702813	+	0.711375i	$0.251927\pi$
$588$	0	0
$589$	−3052.56	−0.213546
$590$	−1736.59	−0.121176
$591$	0	0
$592$	5245.46	0.364167
$593$	−15181.6	−1.05132	−0.525662	−	0.850694i	$-0.676182\pi$
$593$	−15181.6	−1.05132	−0.525662	+	0.850694i	$0.676182\pi$
$594$	0	0
$595$	−9695.52	−0.668030
$596$	4696.44	0.322775
$597$	0	0
$598$	−702.332	−0.0480275
$599$	5524.97	0.376869	0.188434	−	0.982086i	$-0.439659\pi$
$599$	5524.97	0.376869	0.188434	+	0.982086i	$0.439659\pi$
$600$	0	0
$601$	22916.0	1.55535	0.777673	−	0.628669i	$-0.216400\pi$
$601$	22916.0	1.55535	0.777673	+	0.628669i	$0.216400\pi$
$602$	19411.4	1.31420
$603$	0	0
$604$	10932.0	0.736455
$605$	3615.08	0.242932
$606$	0	0
$607$	−10630.6	−0.710844	−0.355422	−	0.934706i	$-0.615663\pi$
$607$	−10630.6	−0.710844	−0.355422	+	0.934706i	$0.615663\pi$
$608$	523.939	0.0349483
$609$	0	0
$610$	4206.66	0.279218
$611$	3861.39	0.255671
$612$	0	0
$613$	−25267.8	−1.66486	−0.832429	−	0.554132i	$-0.813050\pi$
$613$	−25267.8	−1.66486	−0.832429	+	0.554132i	$0.813050\pi$
$614$	−11351.6	−0.746114
$615$	0	0
$616$	−4857.93	−0.317746
$617$	−21199.3	−1.38323	−0.691615	−	0.722266i	$-0.743101\pi$
$617$	−21199.3	−1.38323	−0.691615	+	0.722266i	$0.743101\pi$
$618$	0	0
$619$	−8059.26	−0.523310	−0.261655	−	0.965161i	$-0.584268\pi$
$619$	−8059.26	−0.523310	−0.261655	+	0.965161i	$0.584268\pi$
$620$	−3728.75	−0.241533
$621$	0	0
$622$	9868.57	0.636163
$623$	−4340.14	−0.279107
$624$	0	0
$625$	625.000	0.0400000
$626$	8917.35	0.569343
$627$	0	0
$628$	−19.0893	−0.00121297
$629$	25813.7	1.63634
$630$	0	0
$631$	10528.9	0.664262	0.332131	−	0.943233i	$-0.392232\pi$
$631$	10528.9	0.664262	0.332131	+	0.943233i	$0.392232\pi$
$632$	4172.73	0.262630
$633$	0	0
$634$	8265.75	0.517783
$635$	−6852.41	−0.428236
$636$	0	0
$637$	4023.12	0.250239
$638$	1802.20	0.111834
$639$	0	0
$640$	640.000	0.0395285
$641$	13680.3	0.842961	0.421481	−	0.906837i	$-0.361511\pi$
$641$	13680.3	0.842961	0.421481	+	0.906837i	$0.361511\pi$
$642$	0	0
$643$	−8361.27	−0.512809	−0.256405	−	0.966570i	$-0.582538\pi$
$643$	−8361.27	−0.512809	−0.256405	+	0.966570i	$0.582538\pi$
$644$	2265.70	0.138635
$645$	0	0
$646$	2578.38	0.157036
$647$	31252.2	1.89900	0.949499	−	0.313769i	$-0.101592\pi$
$647$	31252.2	1.89900	0.949499	+	0.313769i	$0.101592\pi$
$648$	0	0
$649$	−4281.95	−0.258985
$650$	−763.404	−0.0460664
$651$	0	0
$652$	6094.04	0.366045
$653$	−22983.2	−1.37734	−0.688670	−	0.725075i	$-0.741805\pi$
$653$	−22983.2	−1.37734	−0.688670	+	0.725075i	$0.741805\pi$
$654$	0	0
$655$	9455.72	0.564070
$656$	5009.07	0.298127
$657$	0	0
$658$	−12456.7	−0.738016
$659$	−5781.74	−0.341767	−0.170884	−	0.985291i	$-0.554662\pi$
$659$	−5781.74	−0.341767	−0.170884	+	0.985291i	$0.554662\pi$
$660$	0	0
$661$	18973.9	1.11649	0.558244	−	0.829677i	$-0.311475\pi$
$661$	18973.9	1.11649	0.558244	+	0.829677i	$0.311475\pi$
$662$	−20639.8	−1.21177
$663$	0	0
$664$	−7898.94	−0.461654
$665$	2016.12	0.117567
$666$	0	0
$667$	−840.533	−0.0487940
$668$	−14415.2	−0.834944
$669$	0	0
$670$	1429.45	0.0824244
$671$	10372.5	0.596760
$672$	0	0
$673$	13833.4	0.792328	0.396164	−	0.918180i	$-0.370341\pi$
$673$	13833.4	0.792328	0.396164	+	0.918180i	$0.370341\pi$
$674$	19635.2	1.12213
$675$	0	0
$676$	−7855.54	−0.446947
$677$	3041.39	0.172659	0.0863295	−	0.996267i	$-0.472486\pi$
$677$	3041.39	0.172659	0.0863295	+	0.996267i	$0.472486\pi$
$678$	0	0
$679$	18940.6	1.07050
$680$	3149.53	0.177616
$681$	0	0
$682$	−9194.10	−0.516218
$683$	−3325.43	−0.186302	−0.0931509	−	0.995652i	$-0.529694\pi$
$683$	−3325.43	−0.186302	−0.0931509	+	0.995652i	$0.529694\pi$
$684$	0	0
$685$	11131.1	0.620871
$686$	3915.77	0.217937
$687$	0	0
$688$	−6305.68	−0.349421
$689$	123.996	0.00685613
$690$	0	0
$691$	−29403.7	−1.61877	−0.809384	−	0.587280i	$-0.800199\pi$
$691$	−29403.7	−1.61877	−0.809384	+	0.587280i	$0.800199\pi$
$692$	−1399.97	−0.0769061
$693$	0	0
$694$	−11655.1	−0.637493
$695$	3497.77	0.190904
$696$	0	0
$697$	24650.3	1.33960
$698$	−12967.0	−0.703165
$699$	0	0
$700$	2462.72	0.132974
$701$	−19653.5	−1.05892	−0.529461	−	0.848334i	$-0.677606\pi$
$701$	−19653.5	−1.05892	−0.529461	+	0.848334i	$0.677606\pi$
$702$	0	0
$703$	−5367.78	−0.287980
$704$	1578.07	0.0844825
$705$	0	0
$706$	−23070.4	−1.22984
$707$	827.563	0.0440222
$708$	0	0
$709$	36228.5	1.91903	0.959515	−	0.281659i	$-0.0908846\pi$
$709$	36228.5	1.91903	0.959515	+	0.281659i	$0.0908846\pi$
$710$	−6585.33	−0.348089
$711$	0	0
$712$	1409.87	0.0742093
$713$	4288.06	0.225230
$714$	0	0
$715$	−1882.35	−0.0984558
$716$	5132.69	0.267902
$717$	0	0
$718$	11845.1	0.615674
$719$	3129.71	0.162334	0.0811672	−	0.996700i	$-0.474135\pi$
$719$	3129.71	0.162334	0.0811672	+	0.996700i	$0.474135\pi$
$720$	0	0
$721$	−46010.5	−2.37659
$722$	13181.8	0.679470
$723$	0	0
$724$	12305.6	0.631678
$725$	−913.623	−0.0468015
$726$	0	0
$727$	7184.40	0.366513	0.183256	−	0.983065i	$-0.441336\pi$
$727$	7184.40	0.366513	0.183256	+	0.983065i	$0.441336\pi$
$728$	−3008.08	−0.153141
$729$	0	0
$730$	−1447.94	−0.0734120
$731$	−31031.1	−1.57008
$732$	0	0
$733$	−37118.8	−1.87042	−0.935208	−	0.354099i	$-0.884788\pi$
$733$	−37118.8	−1.87042	−0.935208	+	0.354099i	$0.884788\pi$
$734$	8085.44	0.406593
$735$	0	0
$736$	−736.000	−0.0368605
$737$	3524.63	0.176162
$738$	0	0
$739$	−6407.47	−0.318948	−0.159474	−	0.987202i	$-0.550980\pi$
$739$	−6407.47	−0.318948	−0.159474	+	0.987202i	$0.550980\pi$
$740$	−6556.83	−0.325721
$741$	0	0
$742$	−400.008	−0.0197908
$743$	−23437.3	−1.15724	−0.578620	−	0.815597i	$-0.696409\pi$
$743$	−23437.3	−1.15724	−0.578620	+	0.815597i	$0.696409\pi$
$744$	0	0
$745$	−5870.55	−0.288698
$746$	18157.4	0.891137
$747$	0	0
$748$	7765.90	0.379612
$749$	35434.7	1.72865
$750$	0	0
$751$	−10463.7	−0.508423	−0.254211	−	0.967149i	$-0.581816\pi$
$751$	−10463.7	−0.508423	−0.254211	+	0.967149i	$0.581816\pi$
$752$	4046.50	0.196224
$753$	0	0
$754$	1115.94	0.0538994
$755$	−13665.1	−0.658705
$756$	0	0
$757$	−27823.5	−1.33588	−0.667941	−	0.744214i	$-0.732824\pi$
$757$	−27823.5	−1.33588	−0.667941	+	0.744214i	$0.732824\pi$
$758$	−13122.8	−0.628816
$759$	0	0
$760$	−654.924	−0.0312587
$761$	30228.5	1.43992	0.719962	−	0.694013i	$-0.244159\pi$
$761$	30228.5	1.43992	0.719962	+	0.694013i	$0.244159\pi$
$762$	0	0
$763$	−45917.3	−2.17866
$764$	7151.46	0.338653
$765$	0	0
$766$	1175.10	0.0554281
$767$	−2651.43	−0.124821
$768$	0	0
$769$	3522.42	0.165178	0.0825889	−	0.996584i	$-0.473681\pi$
$769$	3522.42	0.165178	0.0825889	+	0.996584i	$0.473681\pi$
$770$	6072.41	0.284201
$771$	0	0
$772$	−3526.02	−0.164384
$773$	508.724	0.0236708	0.0118354	−	0.999930i	$-0.496233\pi$
$773$	508.724	0.0236708	0.0118354	+	0.999930i	$0.496233\pi$
$774$	0	0
$775$	4660.94	0.216033
$776$	−6152.74	−0.284627
$777$	0	0
$778$	−12940.7	−0.596334
$779$	−5125.87	−0.235755
$780$	0	0
$781$	−16237.7	−0.743955
$782$	−3621.96	−0.165628
$783$	0	0
$784$	4215.98	0.192055
$785$	23.8616	0.00108491
$786$	0	0
$787$	30797.9	1.39495	0.697476	−	0.716608i	$-0.254306\pi$
$787$	30797.9	1.39495	0.697476	+	0.716608i	$0.254306\pi$
$788$	−8869.14	−0.400952
$789$	0	0
$790$	−5215.92	−0.234904
$791$	9795.92	0.440332
$792$	0	0
$793$	6422.76	0.287615
$794$	−18237.5	−0.815145
$795$	0	0
$796$	−845.251	−0.0376371
$797$	−11682.3	−0.519209	−0.259604	−	0.965715i	$-0.583592\pi$
$797$	−11682.3	−0.519209	−0.259604	+	0.965715i	$0.583592\pi$
$798$	0	0
$799$	19913.4	0.881709
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−6399.51	−0.281764
$803$	−3570.24	−0.156900
$804$	0	0
$805$	−2832.13	−0.123999
$806$	−5693.09	−0.248797
$807$	0	0
$808$	−268.829	−0.0117047
$809$	6661.27	0.289490	0.144745	−	0.989469i	$-0.453764\pi$
$809$	6661.27	0.289490	0.144745	+	0.989469i	$0.453764\pi$
$810$	0	0
$811$	−19875.6	−0.860574	−0.430287	−	0.902692i	$-0.641588\pi$
$811$	−19875.6	−0.860574	−0.430287	+	0.902692i	$0.641588\pi$
$812$	−3600.00	−0.155585
$813$	0	0
$814$	−16167.4	−0.696150
$815$	−7617.56	−0.327400
$816$	0	0
$817$	6452.72	0.276318
$818$	−12292.4	−0.525422
$819$	0	0
$820$	−6261.34	−0.266653
$821$	17128.2	0.728111	0.364055	−	0.931377i	$-0.381392\pi$
$821$	17128.2	0.728111	0.364055	+	0.931377i	$0.381392\pi$
$822$	0	0
$823$	−8.41573	−0.000356445 0	−0.000178222	−	1.00000i	$-0.500057\pi$
$823$	−8.41573	−0.000356445 0	−0.000178222		1.00000i	$0.500057\pi$
$824$	14946.2	0.631889
$825$	0	0
$826$	8553.45	0.360306
$827$	−30880.0	−1.29843	−0.649215	−	0.760605i	$-0.724902\pi$
$827$	−30880.0	−1.29843	−0.649215	+	0.760605i	$0.724902\pi$
$828$	0	0
$829$	−10566.0	−0.442668	−0.221334	−	0.975198i	$-0.571041\pi$
$829$	−10566.0	−0.442668	−0.221334	+	0.975198i	$0.571041\pi$
$830$	9873.68	0.412916
$831$	0	0
$832$	977.157	0.0407173
$833$	20747.5	0.862974
$834$	0	0
$835$	18019.1	0.746797
$836$	−1614.87	−0.0668079
$837$	0	0
$838$	−19867.5	−0.818988
$839$	−32747.3	−1.34751	−0.673755	−	0.738955i	$-0.735320\pi$
$839$	−32747.3	−1.34751	−0.673755	+	0.738955i	$0.735320\pi$
$840$	0	0
$841$	−23053.5	−0.945240
$842$	−7307.79	−0.299101
$843$	0	0
$844$	15759.9	0.642745
$845$	9819.43	0.399762
$846$	0	0
$847$	−17805.9	−0.722334
$848$	129.940	0.00526199
$849$	0	0
$850$	−3936.91	−0.158865
$851$	7540.35	0.303737
$852$	0	0
$853$	−37669.3	−1.51204	−0.756021	−	0.654547i	$-0.772859\pi$
$853$	−37669.3	−1.51204	−0.756021	+	0.654547i	$0.772859\pi$
$854$	−20719.7	−0.830225
$855$	0	0
$856$	−11510.7	−0.459614
$857$	−18749.5	−0.747340	−0.373670	−	0.927562i	$-0.621901\pi$
$857$	−18749.5	−0.747340	−0.373670	+	0.927562i	$0.621901\pi$
$858$	0	0
$859$	−41715.6	−1.65695	−0.828475	−	0.560026i	$-0.810791\pi$
$859$	−41715.6	−1.65695	−0.828475	+	0.560026i	$0.810791\pi$
$860$	7882.10	0.312532
$861$	0	0
$862$	−6113.75	−0.241572
$863$	−46493.1	−1.83389	−0.916943	−	0.399018i	$-0.869351\pi$
$863$	−46493.1	−1.83389	−0.916943	+	0.399018i	$0.869351\pi$
$864$	0	0
$865$	1749.97	0.0687869
$866$	−3406.09	−0.133653
$867$	0	0
$868$	18365.7	0.718173
$869$	−12861.1	−0.502050
$870$	0	0
$871$	2182.49	0.0849034
$872$	14916.0	0.579264
$873$	0	0
$874$	753.163	0.0291489
$875$	−3078.40	−0.118936
$876$	0	0
$877$	48214.7	1.85644	0.928218	−	0.372038i	$-0.121341\pi$
$877$	48214.7	1.85644	0.928218	+	0.372038i	$0.121341\pi$
$878$	1517.26	0.0583201
$879$	0	0
$880$	−1972.59	−0.0755635
$881$	−8809.87	−0.336903	−0.168452	−	0.985710i	$-0.553877\pi$
$881$	−8809.87	−0.336903	−0.168452	+	0.985710i	$0.553877\pi$
$882$	0	0
$883$	−50886.4	−1.93937	−0.969684	−	0.244362i	$-0.921422\pi$
$883$	−50886.4	−1.93937	−0.969684	+	0.244362i	$0.921422\pi$
$884$	4808.73	0.182958
$885$	0	0
$886$	11942.8	0.452853
$887$	−42296.0	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$887$	−42296.0	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$888$	0	0
$889$	33751.1	1.27332
$890$	−1762.34	−0.0663748
$891$	0	0
$892$	16617.9	0.623777
$893$	−4140.86	−0.155172
$894$	0	0
$895$	−6415.87	−0.239619
$896$	−3152.28	−0.117534
$897$	0	0
$898$	−7031.23	−0.261287
$899$	−6813.34	−0.252767
$900$	0	0
$901$	639.455	0.0236441
$902$	−15438.8	−0.569906
$903$	0	0
$904$	−3182.15	−0.117076
$905$	−15382.0	−0.564990
$906$	0	0
$907$	−37554.9	−1.37485	−0.687426	−	0.726254i	$-0.741260\pi$
$907$	−37554.9	−1.37485	−0.687426	+	0.726254i	$0.741260\pi$
$908$	−24533.2	−0.896656
$909$	0	0
$910$	3760.10	0.136974
$911$	−23951.7	−0.871080	−0.435540	−	0.900169i	$-0.643442\pi$
$911$	−23951.7	−0.871080	−0.435540	+	0.900169i	$0.643442\pi$
$912$	0	0
$913$	24345.8	0.882508
$914$	−24311.2	−0.879806
$915$	0	0
$916$	8976.81	0.323802
$917$	−46573.6	−1.67720
$918$	0	0
$919$	−8909.66	−0.319807	−0.159904	−	0.987133i	$-0.551118\pi$
$919$	−8909.66	−0.319807	−0.159904	+	0.987133i	$0.551118\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	−17709.6	−0.632577
$923$	−10054.5	−0.358558
$924$	0	0
$925$	8196.03	0.291334
$926$	−35817.1	−1.27108
$927$	0	0
$928$	1169.44	0.0413671
$929$	25002.6	0.883003	0.441501	−	0.897261i	$-0.354446\pi$
$929$	25002.6	0.883003	0.441501	+	0.897261i	$0.354446\pi$
$930$	0	0
$931$	−4314.30	−0.151875
$932$	−8644.93	−0.303835
$933$	0	0
$934$	17193.5	0.602342
$935$	−9707.38	−0.339535
$936$	0	0
$937$	−18592.9	−0.648244	−0.324122	−	0.946015i	$-0.605069\pi$
$937$	−18592.9	−0.648244	−0.324122	+	0.946015i	$0.605069\pi$
$938$	−7040.65	−0.245080
$939$	0	0
$940$	−5058.13	−0.175508
$941$	21625.8	0.749183	0.374591	−	0.927190i	$-0.377783\pi$
$941$	21625.8	0.749183	0.374591	+	0.927190i	$0.377783\pi$
$942$	0	0
$943$	7200.54	0.248655
$944$	−2778.54	−0.0957984
$945$	0	0
$946$	19435.1	0.667961
$947$	−4477.58	−0.153645	−0.0768224	−	0.997045i	$-0.524477\pi$
$947$	−4477.58	−0.153645	−0.0768224	+	0.997045i	$0.524477\pi$
$948$	0	0
$949$	−2210.73	−0.0756200
$950$	818.655	0.0279586
$951$	0	0
$952$	−15512.8	−0.528124
$953$	−54260.8	−1.84436	−0.922182	−	0.386755i	$-0.873596\pi$
$953$	−54260.8	−1.84436	−0.922182	+	0.386755i	$0.873596\pi$
$954$	0	0
$955$	−8939.32	−0.302900
$956$	−13792.7	−0.466620
$957$	0	0
$958$	−24770.3	−0.835379
$959$	−54825.4	−1.84610
$960$	0	0
$961$	4967.95	0.166760
$962$	−10011.0	−0.335518
$963$	0	0
$964$	6617.72	0.221102
$965$	4407.53	0.147029
$966$	0	0
$967$	−26295.4	−0.874462	−0.437231	−	0.899349i	$-0.644041\pi$
$967$	−26295.4	−0.874462	−0.437231	+	0.899349i	$0.644041\pi$
$968$	5784.13	0.192055
$969$	0	0
$970$	7690.92	0.254578
$971$	−21209.6	−0.700978	−0.350489	−	0.936567i	$-0.613985\pi$
$971$	−21209.6	−0.700978	−0.350489	+	0.936567i	$0.613985\pi$
$972$	0	0
$973$	−17228.1	−0.567632
$974$	−8663.95	−0.285021
$975$	0	0
$976$	6730.66	0.220741
$977$	607.767	0.0199019	0.00995097	−	0.999950i	$-0.496832\pi$
$977$	607.767	0.0199019	0.00995097	+	0.999950i	$0.496832\pi$
$978$	0	0
$979$	−4345.45	−0.141860
$980$	−5269.98	−0.171779
$981$	0	0
$982$	−22731.9	−0.738702
$983$	−47859.5	−1.55288	−0.776440	−	0.630192i	$-0.782976\pi$
$983$	−47859.5	−1.55288	−0.776440	+	0.630192i	$0.782976\pi$
$984$	0	0
$985$	11086.4	0.358622
$986$	5754.97	0.185878
$987$	0	0
$988$	−999.943	−0.0321988
$989$	−9064.41	−0.291437
$990$	0	0
$991$	34510.6	1.10622	0.553111	−	0.833108i	$-0.313441\pi$
$991$	34510.6	1.10622	0.553111	+	0.833108i	$0.313441\pi$
$992$	−5966.00	−0.190948
$993$	0	0
$994$	32435.6	1.03501
$995$	1056.56	0.0336637
$996$	0	0
$997$	21519.8	0.683590	0.341795	−	0.939775i	$-0.388965\pi$
$997$	21519.8	0.683590	0.341795	+	0.939775i	$0.388965\pi$
$998$	−25241.1	−0.800593
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bg.1.4		4
3.2	odd	2		230.4.a.j.1.3	✓	4
12.11	even	2		1840.4.a.k.1.2		4
15.2	even	4		1150.4.b.o.599.6		8
15.8	even	4		1150.4.b.o.599.3		8
15.14	odd	2		1150.4.a.n.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.3	✓	4	3.2	odd	2
1150.4.a.n.1.2		4	15.14	odd	2
1150.4.b.o.599.3		8	15.8	even	4
1150.4.b.o.599.6		8	15.2	even	4
1840.4.a.k.1.2		4	12.11	even	2
2070.4.a.bg.1.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +24.6272 q^{7} -8.00000 q^{8} +10.0000 q^{10} +24.6573 q^{11} +15.2681 q^{13} -49.2544 q^{14} +16.0000 q^{16} +78.7383 q^{17} -16.3731 q^{19} -20.0000 q^{20} -49.3146 q^{22} +23.0000 q^{23} +25.0000 q^{25} -30.5362 q^{26} +98.5088 q^{28} -36.5449 q^{29} +186.438 q^{31} -32.0000 q^{32} -157.477 q^{34} -123.136 q^{35} +327.841 q^{37} +32.7462 q^{38} +40.0000 q^{40} +313.067 q^{41} -394.105 q^{43} +98.6293 q^{44} -46.0000 q^{46} +252.906 q^{47} +263.499 q^{49} -50.0000 q^{50} +61.0723 q^{52} +8.12126 q^{53} -123.287 q^{55} -197.018 q^{56} +73.0898 q^{58} -173.659 q^{59} +420.666 q^{61} -372.875 q^{62} +64.0000 q^{64} -76.3404 q^{65} +142.945 q^{67} +314.953 q^{68} +246.272 q^{70} -658.533 q^{71} -144.794 q^{73} -655.683 q^{74} -65.4924 q^{76} +607.241 q^{77} -521.592 q^{79} -80.0000 q^{80} -626.134 q^{82} +987.368 q^{83} -393.691 q^{85} +788.210 q^{86} -197.259 q^{88} -176.234 q^{89} +376.010 q^{91} +92.0000 q^{92} -505.813 q^{94} +81.8655 q^{95} +769.092 q^{97} -526.998 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 8 q^{7} - 32 q^{8} + 40 q^{10} - 21 q^{11} + 70 q^{13} - 16 q^{14} + 64 q^{16} - 56 q^{17} + 173 q^{19} - 80 q^{20} + 42 q^{22} + 92 q^{23} + 100 q^{25} - 140 q^{26}+ \cdots + 248 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 8 * q^7 - 32 * q^8 + 40 * q^10 - 21 * q^11 + 70 * q^13 - 16 * q^14 + 64 * q^16 - 56 * q^17 + 173 * q^19 - 80 * q^20 + 42 * q^22 + 92 * q^23 + 100 * q^25 - 140 * q^26 + 32 * q^28 + 118 * q^29 + 17 * q^31 - 128 * q^32 + 112 * q^34 - 40 * q^35 - 343 * q^37 - 346 * q^38 + 160 * q^40 - 139 * q^41 - 50 * q^43 - 84 * q^44 - 184 * q^46 - 367 * q^47 - 124 * q^49 - 200 * q^50 + 280 * q^52 + 353 * q^53 + 105 * q^55 - 64 * q^56 - 236 * q^58 + 453 * q^59 - 327 * q^61 - 34 * q^62 + 256 * q^64 - 350 * q^65 - 455 * q^67 - 224 * q^68 + 80 * q^70 - 195 * q^71 - 633 * q^73 + 686 * q^74 + 692 * q^76 + 2 * q^77 - 1140 * q^79 - 320 * q^80 + 278 * q^82 + 1199 * q^83 + 280 * q^85 + 100 * q^86 + 168 * q^88 + 2170 * q^89 + 557 * q^91 + 368 * q^92 + 734 * q^94 - 865 * q^95 - 703 * q^97 + 248 * q^98

Introduction

L-functions

Modular forms

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