LMFDB - Embedded newform 2070.4.a.bi.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)

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);

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")

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,-8,0,16,20,0,26] 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} - 84x^{2} - 11x + 1242 $ x^4 - 84x^2 - 11x + 1242
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.26018$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +27.7921 q^{7} -8.00000 q^{8} -10.0000 q^{10} +10.8182 q^{11} +36.9683 q^{13} -55.5842 q^{14} +16.0000 q^{16} -118.996 q^{17} -19.3321 q^{19} +20.0000 q^{20} -21.6365 q^{22} -23.0000 q^{23} +25.0000 q^{25} -73.9366 q^{26} +111.168 q^{28} -234.499 q^{29} +165.319 q^{31} -32.0000 q^{32} +237.992 q^{34} +138.960 q^{35} +202.301 q^{37} +38.6643 q^{38} -40.0000 q^{40} +295.846 q^{41} -65.9221 q^{43} +43.2729 q^{44} +46.0000 q^{46} +110.279 q^{47} +429.400 q^{49} -50.0000 q^{50} +147.873 q^{52} +688.135 q^{53} +54.0911 q^{55} -222.337 q^{56} +468.998 q^{58} -10.5847 q^{59} +110.579 q^{61} -330.639 q^{62} +64.0000 q^{64} +184.842 q^{65} -643.290 q^{67} -475.984 q^{68} -277.921 q^{70} -143.216 q^{71} -158.213 q^{73} -404.601 q^{74} -77.3285 q^{76} +300.661 q^{77} +1123.34 q^{79} +80.0000 q^{80} -591.692 q^{82} +824.600 q^{83} -594.981 q^{85} +131.844 q^{86} -86.5458 q^{88} +879.672 q^{89} +1027.43 q^{91} -92.0000 q^{92} -220.557 q^{94} -96.6607 q^{95} -938.437 q^{97} -858.801 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} + 20 q^{5} + 26 q^{7} - 32 q^{8} - 40 q^{10} - 93 q^{11} + 32 q^{13} - 52 q^{14} + 64 q^{16} - 108 q^{17} + 185 q^{19} + 80 q^{20} + 186 q^{22} - 92 q^{23} + 100 q^{25} - 64 q^{26}+ \cdots - 1200 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 + 20 * q^5 + 26 * q^7 - 32 * q^8 - 40 * q^10 - 93 * q^11 + 32 * q^13 - 52 * q^14 + 64 * q^16 - 108 * q^17 + 185 * q^19 + 80 * q^20 + 186 * q^22 - 92 * q^23 + 100 * q^25 - 64 * q^26 + 104 * q^28 - 294 * q^29 - 211 * q^31 - 128 * q^32 + 216 * q^34 + 130 * q^35 + 5 * q^37 - 370 * q^38 - 160 * q^40 + 369 * q^41 - 100 * q^43 - 372 * q^44 + 184 * q^46 + 363 * q^47 + 600 * q^49 - 200 * q^50 + 128 * q^52 - 21 * q^53 - 465 * q^55 - 208 * q^56 + 588 * q^58 + 33 * q^59 - 307 * q^61 + 422 * q^62 + 256 * q^64 + 160 * q^65 + 725 * q^67 - 432 * q^68 - 260 * q^70 + 1257 * q^71 + 509 * q^73 - 10 * q^74 + 740 * q^76 + 1962 * q^77 + 1202 * q^79 + 320 * q^80 - 738 * q^82 + 1377 * q^83 - 540 * q^85 + 200 * q^86 + 744 * q^88 + 984 * q^89 - 995 * q^91 - 368 * q^92 - 726 * q^94 + 925 * q^95 + 137 * q^97 - 1200 * 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
$7$	27.7921	1.50063	0.750316	−	0.661079i	$-0.229901\pi$
$7$	27.7921	1.50063	0.750316	+	0.661079i	$0.229901\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	10.8182	0.296529	0.148264	−	0.988948i	$-0.452631\pi$
$11$	10.8182	0.296529	0.148264	+	0.988948i	$0.452631\pi$
$12$	0	0
$13$	36.9683	0.788705	0.394352	−	0.918959i	$-0.370969\pi$
$13$	36.9683	0.788705	0.394352	+	0.918959i	$0.370969\pi$
$14$	−55.5842	−1.06111
$15$	0	0
$16$	16.0000	0.250000
$17$	−118.996	−1.69769	−0.848847	−	0.528639i	$-0.822703\pi$
$17$	−118.996	−1.69769	−0.848847	+	0.528639i	$0.822703\pi$
$18$	0	0
$19$	−19.3321	−0.233426	−0.116713	−	0.993166i	$-0.537236\pi$
$19$	−19.3321	−0.233426	−0.116713	+	0.993166i	$0.537236\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−21.6365	−0.209678
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	−73.9366	−0.557699
$27$	0	0
$28$	111.168	0.750316
$29$	−234.499	−1.50156	−0.750782	−	0.660550i	$-0.770323\pi$
$29$	−234.499	−1.50156	−0.750782	+	0.660550i	$0.770323\pi$
$30$	0	0
$31$	165.319	0.957814	0.478907	−	0.877866i	$-0.341033\pi$
$31$	165.319	0.957814	0.478907	+	0.877866i	$0.341033\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	237.992	1.20045
$35$	138.960	0.671103
$36$	0	0
$37$	202.301	0.898865	0.449432	−	0.893314i	$-0.351626\pi$
$37$	202.301	0.898865	0.449432	+	0.893314i	$0.351626\pi$
$38$	38.6643	0.165057
$39$	0	0
$40$	−40.0000	−0.158114
$41$	295.846	1.12691	0.563456	−	0.826146i	$-0.309471\pi$
$41$	295.846	1.12691	0.563456	+	0.826146i	$0.309471\pi$
$42$	0	0
$43$	−65.9221	−0.233791	−0.116896	−	0.993144i	$-0.537294\pi$
$43$	−65.9221	−0.233791	−0.116896	+	0.993144i	$0.537294\pi$
$44$	43.2729	0.148264
$45$	0	0
$46$	46.0000	0.147442
$47$	110.279	0.342251	0.171126	−	0.985249i	$-0.445260\pi$
$47$	110.279	0.342251	0.171126	+	0.985249i	$0.445260\pi$
$48$	0	0
$49$	429.400	1.25190
$50$	−50.0000	−0.141421
$51$	0	0
$52$	147.873	0.394352
$53$	688.135	1.78345	0.891723	−	0.452582i	$-0.149497\pi$
$53$	688.135	1.78345	0.891723	+	0.452582i	$0.149497\pi$
$54$	0	0
$55$	54.0911	0.132612
$56$	−222.337	−0.530553
$57$	0	0
$58$	468.998	1.06177
$59$	−10.5847	−0.0233561	−0.0116781	−	0.999932i	$-0.503717\pi$
$59$	−10.5847	−0.0233561	−0.0116781	+	0.999932i	$0.503717\pi$
$60$	0	0
$61$	110.579	0.232101	0.116050	−	0.993243i	$-0.462977\pi$
$61$	110.579	0.232101	0.116050	+	0.993243i	$0.462977\pi$
$62$	−330.639	−0.677276
$63$	0	0
$64$	64.0000	0.125000
$65$	184.842	0.352720
$66$	0	0
$67$	−643.290	−1.17299	−0.586496	−	0.809952i	$-0.699493\pi$
$67$	−643.290	−1.17299	−0.586496	+	0.809952i	$0.699493\pi$
$68$	−475.984	−0.848847
$69$	0	0
$70$	−277.921	−0.474541
$71$	−143.216	−0.239389	−0.119695	−	0.992811i	$-0.538192\pi$
$71$	−143.216	−0.239389	−0.119695	+	0.992811i	$0.538192\pi$
$72$	0	0
$73$	−158.213	−0.253664	−0.126832	−	0.991924i	$-0.540481\pi$
$73$	−158.213	−0.253664	−0.126832	+	0.991924i	$0.540481\pi$
$74$	−404.601	−0.635593
$75$	0	0
$76$	−77.3285	−0.116713
$77$	300.661	0.444981
$78$	0	0
$79$	1123.34	1.59982	0.799912	−	0.600118i	$-0.204880\pi$
$79$	1123.34	1.59982	0.799912	+	0.600118i	$0.204880\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−591.692	−0.796847
$83$	824.600	1.09050	0.545251	−	0.838273i	$-0.316434\pi$
$83$	824.600	1.09050	0.545251	+	0.838273i	$0.316434\pi$
$84$	0	0
$85$	−594.981	−0.759232
$86$	131.844	0.165316
$87$	0	0
$88$	−86.5458	−0.104839
$89$	879.672	1.04770	0.523849	−	0.851811i	$-0.324496\pi$
$89$	879.672	1.04770	0.523849	+	0.851811i	$0.324496\pi$
$90$	0	0
$91$	1027.43	1.18356
$92$	−92.0000	−0.104257
$93$	0	0
$94$	−220.557	−0.242008
$95$	−96.6607	−0.104391
$96$	0	0
$97$	−938.437	−0.982308	−0.491154	−	0.871073i	$-0.663425\pi$
$97$	−938.437	−0.982308	−0.491154	+	0.871073i	$0.663425\pi$
$98$	−858.801	−0.885224
$99$	0	0
$100$	100.000	0.100000
$101$	688.428	0.678229	0.339114	−	0.940745i	$-0.389873\pi$
$101$	688.428	0.678229	0.339114	+	0.940745i	$0.389873\pi$
$102$	0	0
$103$	2041.55	1.95301	0.976504	−	0.215500i	$-0.0691381\pi$
$103$	2041.55	1.95301	0.976504	+	0.215500i	$0.0691381\pi$
$104$	−295.746	−0.278849
$105$	0	0
$106$	−1376.27	−1.26109
$107$	−287.420	−0.259682	−0.129841	−	0.991535i	$-0.541447\pi$
$107$	−287.420	−0.259682	−0.129841	+	0.991535i	$0.541447\pi$
$108$	0	0
$109$	−1211.29	−1.06441	−0.532204	−	0.846616i	$-0.678636\pi$
$109$	−1211.29	−1.06441	−0.532204	+	0.846616i	$0.678636\pi$
$110$	−108.182	−0.0937707
$111$	0	0
$112$	444.673	0.375158
$113$	−741.316	−0.617143	−0.308571	−	0.951201i	$-0.599851\pi$
$113$	−741.316	−0.617143	−0.308571	+	0.951201i	$0.599851\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	−937.996	−0.750782
$117$	0	0
$118$	21.1694	0.0165153
$119$	−3307.15	−2.54761
$120$	0	0
$121$	−1213.97	−0.912071
$122$	−221.157	−0.164120
$123$	0	0
$124$	661.277	0.478907
$125$	125.000	0.0894427
$126$	0	0
$127$	1579.99	1.10395	0.551976	−	0.833860i	$-0.313874\pi$
$127$	1579.99	1.10395	0.551976	+	0.833860i	$0.313874\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−369.683	−0.249410
$131$	2348.86	1.56657	0.783285	−	0.621662i	$-0.213542\pi$
$131$	2348.86	1.56657	0.783285	+	0.621662i	$0.213542\pi$
$132$	0	0
$133$	−537.280	−0.350287
$134$	1286.58	0.829430
$135$	0	0
$136$	951.969	0.600225
$137$	617.567	0.385127	0.192563	−	0.981285i	$-0.438320\pi$
$137$	617.567	0.385127	0.192563	+	0.981285i	$0.438320\pi$
$138$	0	0
$139$	1509.75	0.921260	0.460630	−	0.887592i	$-0.347623\pi$
$139$	1509.75	0.921260	0.460630	+	0.887592i	$0.347623\pi$
$140$	555.842	0.335551
$141$	0	0
$142$	286.432	0.169274
$143$	399.932	0.233874
$144$	0	0
$145$	−1172.49	−0.671520
$146$	316.427	0.179368
$147$	0	0
$148$	809.202	0.449432
$149$	−1025.46	−0.563818	−0.281909	−	0.959441i	$-0.590968\pi$
$149$	−1025.46	−0.563818	−0.281909	+	0.959441i	$0.590968\pi$
$150$	0	0
$151$	−1516.67	−0.817383	−0.408692	−	0.912672i	$-0.634015\pi$
$151$	−1516.67	−0.817383	−0.408692	+	0.912672i	$0.634015\pi$
$152$	154.657	0.0825286
$153$	0	0
$154$	−601.322	−0.314649
$155$	826.596	0.428347
$156$	0	0
$157$	−2757.31	−1.40164	−0.700819	−	0.713339i	$-0.747182\pi$
$157$	−2757.31	−1.40164	−0.700819	+	0.713339i	$0.747182\pi$
$158$	−2246.69	−1.13125
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−639.218	−0.312903
$162$	0	0
$163$	1263.29	0.607047	0.303524	−	0.952824i	$-0.401837\pi$
$163$	1263.29	0.607047	0.303524	+	0.952824i	$0.401837\pi$
$164$	1183.38	0.563456
$165$	0	0
$166$	−1649.20	−0.771101
$167$	349.646	0.162014	0.0810072	−	0.996714i	$-0.474186\pi$
$167$	349.646	0.162014	0.0810072	+	0.996714i	$0.474186\pi$
$168$	0	0
$169$	−830.344	−0.377945
$170$	1189.96	0.536858
$171$	0	0
$172$	−263.689	−0.116896
$173$	−2313.83	−1.01686	−0.508432	−	0.861102i	$-0.669775\pi$
$173$	−2313.83	−1.01686	−0.508432	+	0.861102i	$0.669775\pi$
$174$	0	0
$175$	694.802	0.300126
$176$	173.092	0.0741322
$177$	0	0
$178$	−1759.34	−0.740834
$179$	2347.69	0.980306	0.490153	−	0.871636i	$-0.336941\pi$
$179$	2347.69	0.980306	0.490153	+	0.871636i	$0.336941\pi$
$180$	0	0
$181$	4396.31	1.80539	0.902695	−	0.430282i	$-0.141586\pi$
$181$	4396.31	1.80539	0.902695	+	0.430282i	$0.141586\pi$
$182$	−2054.85	−0.836900
$183$	0	0
$184$	184.000	0.0737210
$185$	1011.50	0.401985
$186$	0	0
$187$	−1287.33	−0.503415
$188$	441.115	0.171126
$189$	0	0
$190$	193.321	0.0738158
$191$	−4153.14	−1.57335	−0.786677	−	0.617365i	$-0.788200\pi$
$191$	−4153.14	−1.57335	−0.786677	+	0.617365i	$0.788200\pi$
$192$	0	0
$193$	−1020.52	−0.380616	−0.190308	−	0.981724i	$-0.560949\pi$
$193$	−1020.52	−0.380616	−0.190308	+	0.981724i	$0.560949\pi$
$194$	1876.87	0.694596
$195$	0	0
$196$	1717.60	0.625948
$197$	398.643	0.144173	0.0720866	−	0.997398i	$-0.477034\pi$
$197$	398.643	0.144173	0.0720866	+	0.997398i	$0.477034\pi$
$198$	0	0
$199$	4131.40	1.47169	0.735847	−	0.677148i	$-0.236784\pi$
$199$	4131.40	1.47169	0.735847	+	0.677148i	$0.236784\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−1376.86	−0.479580
$203$	−6517.22	−2.25329
$204$	0	0
$205$	1479.23	0.503970
$206$	−4083.10	−1.38098
$207$	0	0
$208$	591.493	0.197176
$209$	−209.139	−0.0692176
$210$	0	0
$211$	826.315	0.269601	0.134801	−	0.990873i	$-0.456961\pi$
$211$	826.315	0.269601	0.134801	+	0.990873i	$0.456961\pi$
$212$	2752.54	0.891723
$213$	0	0
$214$	574.840	0.183623
$215$	−329.611	−0.104555
$216$	0	0
$217$	4594.57	1.43733
$218$	2422.58	0.752650
$219$	0	0
$220$	216.365	0.0663059
$221$	−4399.08	−1.33898
$222$	0	0
$223$	1110.09	0.333351	0.166676	−	0.986012i	$-0.446697\pi$
$223$	1110.09	0.333351	0.166676	+	0.986012i	$0.446697\pi$
$224$	−889.347	−0.265277
$225$	0	0
$226$	1482.63	0.436386
$227$	4917.90	1.43794	0.718971	−	0.695040i	$-0.244614\pi$
$227$	4917.90	1.43794	0.718971	+	0.695040i	$0.244614\pi$
$228$	0	0
$229$	−1390.62	−0.401288	−0.200644	−	0.979664i	$-0.564304\pi$
$229$	−1390.62	−0.401288	−0.200644	+	0.979664i	$0.564304\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	1875.99	0.530883
$233$	3409.59	0.958668	0.479334	−	0.877633i	$-0.340878\pi$
$233$	3409.59	0.958668	0.479334	+	0.877633i	$0.340878\pi$
$234$	0	0
$235$	551.394	0.153059
$236$	−42.3388	−0.0116781
$237$	0	0
$238$	6614.30	1.80143
$239$	−2056.35	−0.556544	−0.278272	−	0.960502i	$-0.589762\pi$
$239$	−2056.35	−0.556544	−0.278272	+	0.960502i	$0.589762\pi$
$240$	0	0
$241$	−1957.03	−0.523085	−0.261543	−	0.965192i	$-0.584231\pi$
$241$	−1957.03	−0.523085	−0.261543	+	0.965192i	$0.584231\pi$
$242$	2427.93	0.644931
$243$	0	0
$244$	442.315	0.116050
$245$	2147.00	0.559865
$246$	0	0
$247$	−714.676	−0.184104
$248$	−1322.55	−0.338638
$249$	0	0
$250$	−250.000	−0.0632456
$251$	1954.13	0.491410	0.245705	−	0.969345i	$-0.420981\pi$
$251$	1954.13	0.491410	0.245705	+	0.969345i	$0.420981\pi$
$252$	0	0
$253$	−248.819	−0.0618306
$254$	−3159.99	−0.780612
$255$	0	0
$256$	256.000	0.0625000
$257$	−1958.40	−0.475338	−0.237669	−	0.971346i	$-0.576383\pi$
$257$	−1958.40	−0.475338	−0.237669	+	0.971346i	$0.576383\pi$
$258$	0	0
$259$	5622.35	1.34887
$260$	739.366	0.176360
$261$	0	0
$262$	−4697.72	−1.10773
$263$	−7205.23	−1.68933	−0.844665	−	0.535295i	$-0.820200\pi$
$263$	−7205.23	−1.68933	−0.844665	+	0.535295i	$0.820200\pi$
$264$	0	0
$265$	3440.67	0.797581
$266$	1074.56	0.247690
$267$	0	0
$268$	−2573.16	−0.586496
$269$	3764.30	0.853210	0.426605	−	0.904438i	$-0.359709\pi$
$269$	3764.30	0.853210	0.426605	+	0.904438i	$0.359709\pi$
$270$	0	0
$271$	−5208.88	−1.16759	−0.583795	−	0.811901i	$-0.698433\pi$
$271$	−5208.88	−1.16759	−0.583795	+	0.811901i	$0.698433\pi$
$272$	−1903.94	−0.424423
$273$	0	0
$274$	−1235.13	−0.272326
$275$	270.456	0.0593058
$276$	0	0
$277$	1550.21	0.336256	0.168128	−	0.985765i	$-0.446228\pi$
$277$	1550.21	0.336256	0.168128	+	0.985765i	$0.446228\pi$
$278$	−3019.50	−0.651430
$279$	0	0
$280$	−1111.68	−0.237271
$281$	−7997.53	−1.69784	−0.848920	−	0.528522i	$-0.822746\pi$
$281$	−7997.53	−1.69784	−0.848920	+	0.528522i	$0.822746\pi$
$282$	0	0
$283$	−5808.41	−1.22005	−0.610025	−	0.792382i	$-0.708841\pi$
$283$	−5808.41	−1.22005	−0.610025	+	0.792382i	$0.708841\pi$
$284$	−572.864	−0.119695
$285$	0	0
$286$	−799.863	−0.165374
$287$	8222.18	1.69108
$288$	0	0
$289$	9247.08	1.88216
$290$	2344.99	0.474836
$291$	0	0
$292$	−632.854	−0.126832
$293$	8854.66	1.76551	0.882756	−	0.469832i	$-0.155686\pi$
$293$	8854.66	1.76551	0.882756	+	0.469832i	$0.155686\pi$
$294$	0	0
$295$	−52.9235	−0.0104452
$296$	−1618.40	−0.317797
$297$	0	0
$298$	2050.92	0.398680
$299$	−850.271	−0.164456
$300$	0	0
$301$	−1832.11	−0.350835
$302$	3033.34	0.577977
$303$	0	0
$304$	−309.314	−0.0583565
$305$	552.893	0.103799
$306$	0	0
$307$	−1501.03	−0.279050	−0.139525	−	0.990219i	$-0.544557\pi$
$307$	−1501.03	−0.279050	−0.139525	+	0.990219i	$0.544557\pi$
$308$	1202.64	0.222490
$309$	0	0
$310$	−1653.19	−0.302887
$311$	7440.48	1.35663	0.678313	−	0.734773i	$-0.262711\pi$
$311$	7440.48	1.35663	0.678313	+	0.734773i	$0.262711\pi$
$312$	0	0
$313$	−2528.63	−0.456634	−0.228317	−	0.973587i	$-0.573322\pi$
$313$	−2528.63	−0.456634	−0.228317	+	0.973587i	$0.573322\pi$
$314$	5514.61	0.991107
$315$	0	0
$316$	4493.38	0.799912
$317$	9636.72	1.70742	0.853710	−	0.520749i	$-0.174347\pi$
$317$	9636.72	1.70742	0.853710	+	0.520749i	$0.174347\pi$
$318$	0	0
$319$	−2536.86	−0.445257
$320$	320.000	0.0559017
$321$	0	0
$322$	1278.44	0.221256
$323$	2300.45	0.396286
$324$	0	0
$325$	924.208	0.157741
$326$	−2526.58	−0.429247
$327$	0	0
$328$	−2366.77	−0.398424
$329$	3064.88	0.513593
$330$	0	0
$331$	2875.82	0.477551	0.238776	−	0.971075i	$-0.423254\pi$
$331$	2875.82	0.477551	0.238776	+	0.971075i	$0.423254\pi$
$332$	3298.40	0.545251
$333$	0	0
$334$	−699.291	−0.114561
$335$	−3216.45	−0.524578
$336$	0	0
$337$	9911.19	1.60207	0.801034	−	0.598619i	$-0.204283\pi$
$337$	9911.19	1.60207	0.801034	+	0.598619i	$0.204283\pi$
$338$	1660.69	0.267247
$339$	0	0
$340$	−2379.92	−0.379616
$341$	1788.46	0.284019
$342$	0	0
$343$	2401.24	0.378003
$344$	527.377	0.0826578
$345$	0	0
$346$	4627.67	0.719031
$347$	−4497.70	−0.695820	−0.347910	−	0.937528i	$-0.613108\pi$
$347$	−4497.70	−0.695820	−0.347910	+	0.937528i	$0.613108\pi$
$348$	0	0
$349$	888.112	0.136216	0.0681082	−	0.997678i	$-0.478304\pi$
$349$	888.112	0.136216	0.0681082	+	0.997678i	$0.478304\pi$
$350$	−1389.60	−0.212221
$351$	0	0
$352$	−346.183	−0.0524194
$353$	11722.2	1.76745	0.883727	−	0.468002i	$-0.155026\pi$
$353$	11722.2	1.76745	0.883727	+	0.468002i	$0.155026\pi$
$354$	0	0
$355$	−716.081	−0.107058
$356$	3518.69	0.523849
$357$	0	0
$358$	−4695.39	−0.693181
$359$	1257.49	0.184869	0.0924343	−	0.995719i	$-0.470535\pi$
$359$	1257.49	0.184869	0.0924343	+	0.995719i	$0.470535\pi$
$360$	0	0
$361$	−6485.27	−0.945512
$362$	−8792.63	−1.27660
$363$	0	0
$364$	4109.71	0.591778
$365$	−791.067	−0.113442
$366$	0	0
$367$	10122.2	1.43972	0.719859	−	0.694120i	$-0.244206\pi$
$367$	10122.2	1.43972	0.719859	+	0.694120i	$0.244206\pi$
$368$	−368.000	−0.0521286
$369$	0	0
$370$	−2023.01	−0.284246
$371$	19124.7	2.67629
$372$	0	0
$373$	6530.08	0.906473	0.453237	−	0.891390i	$-0.350269\pi$
$373$	6530.08	0.906473	0.453237	+	0.891390i	$0.350269\pi$
$374$	2574.65	0.355968
$375$	0	0
$376$	−882.230	−0.121004
$377$	−8669.03	−1.18429
$378$	0	0
$379$	277.850	0.0376575	0.0188288	−	0.999823i	$-0.494006\pi$
$379$	277.850	0.0376575	0.0188288	+	0.999823i	$0.494006\pi$
$380$	−386.643	−0.0521957
$381$	0	0
$382$	8306.28	1.11253
$383$	10679.6	1.42482	0.712408	−	0.701766i	$-0.247605\pi$
$383$	10679.6	1.42482	0.712408	+	0.701766i	$0.247605\pi$
$384$	0	0
$385$	1503.31	0.199001
$386$	2041.05	0.269136
$387$	0	0
$388$	−3753.75	−0.491154
$389$	−1546.05	−0.201511	−0.100756	−	0.994911i	$-0.532126\pi$
$389$	−1546.05	−0.201511	−0.100756	+	0.994911i	$0.532126\pi$
$390$	0	0
$391$	2736.91	0.353994
$392$	−3435.20	−0.442612
$393$	0	0
$394$	−797.285	−0.101946
$395$	5616.72	0.715463
$396$	0	0
$397$	−12901.8	−1.63103	−0.815517	−	0.578733i	$-0.803547\pi$
$397$	−12901.8	−1.63103	−0.815517	+	0.578733i	$0.803547\pi$
$398$	−8262.80	−1.04064
$399$	0	0
$400$	400.000	0.0500000
$401$	8534.31	1.06280	0.531400	−	0.847121i	$-0.321666\pi$
$401$	8534.31	1.06280	0.531400	+	0.847121i	$0.321666\pi$
$402$	0	0
$403$	6111.57	0.755432
$404$	2753.71	0.339114
$405$	0	0
$406$	13034.4	1.59332
$407$	2188.53	0.266539
$408$	0	0
$409$	650.968	0.0787000	0.0393500	−	0.999225i	$-0.487471\pi$
$409$	650.968	0.0787000	0.0393500	+	0.999225i	$0.487471\pi$
$410$	−2958.46	−0.356361
$411$	0	0
$412$	8166.20	0.976504
$413$	−294.171	−0.0350489
$414$	0	0
$415$	4123.00	0.487687
$416$	−1182.99	−0.139425
$417$	0	0
$418$	418.279	0.0489442
$419$	−13266.2	−1.54677	−0.773384	−	0.633938i	$-0.781437\pi$
$419$	−13266.2	−1.54677	−0.773384	+	0.633938i	$0.781437\pi$
$420$	0	0
$421$	−8274.69	−0.957918	−0.478959	−	0.877837i	$-0.658986\pi$
$421$	−8274.69	−0.957918	−0.478959	+	0.877837i	$0.658986\pi$
$422$	−1652.63	−0.190637
$423$	0	0
$424$	−5505.08	−0.630543
$425$	−2974.90	−0.339539
$426$	0	0
$427$	3073.21	0.348298
$428$	−1149.68	−0.129841
$429$	0	0
$430$	659.221	0.0739314
$431$	−724.084	−0.0809232	−0.0404616	−	0.999181i	$-0.512883\pi$
$431$	−724.084	−0.0809232	−0.0404616	+	0.999181i	$0.512883\pi$
$432$	0	0
$433$	−11996.5	−1.33145	−0.665724	−	0.746198i	$-0.731877\pi$
$433$	−11996.5	−1.33145	−0.665724	+	0.746198i	$0.731877\pi$
$434$	−9189.14	−1.01634
$435$	0	0
$436$	−4845.16	−0.532204
$437$	444.639	0.0486727
$438$	0	0
$439$	−7765.40	−0.844242	−0.422121	−	0.906539i	$-0.638714\pi$
$439$	−7765.40	−0.844242	−0.422121	+	0.906539i	$0.638714\pi$
$440$	−432.729	−0.0468853
$441$	0	0
$442$	8798.17	0.946801
$443$	9061.54	0.971844	0.485922	−	0.874002i	$-0.338484\pi$
$443$	9061.54	0.971844	0.485922	+	0.874002i	$0.338484\pi$
$444$	0	0
$445$	4398.36	0.468544
$446$	−2220.19	−0.235715
$447$	0	0
$448$	1778.69	0.187579
$449$	10944.3	1.15032	0.575159	−	0.818041i	$-0.304940\pi$
$449$	10944.3	1.15032	0.575159	+	0.818041i	$0.304940\pi$
$450$	0	0
$451$	3200.53	0.334162
$452$	−2965.26	−0.308571
$453$	0	0
$454$	−9835.81	−1.01678
$455$	5137.13	0.529302
$456$	0	0
$457$	10934.9	1.11928	0.559641	−	0.828735i	$-0.310939\pi$
$457$	10934.9	1.11928	0.559641	+	0.828735i	$0.310939\pi$
$458$	2781.25	0.283754
$459$	0	0
$460$	−460.000	−0.0466252
$461$	−2652.25	−0.267956	−0.133978	−	0.990984i	$-0.542775\pi$
$461$	−2652.25	−0.267956	−0.133978	+	0.990984i	$0.542775\pi$
$462$	0	0
$463$	707.699	0.0710358	0.0355179	−	0.999369i	$-0.488692\pi$
$463$	707.699	0.0710358	0.0355179	+	0.999369i	$0.488692\pi$
$464$	−3751.98	−0.375391
$465$	0	0
$466$	−6819.18	−0.677880
$467$	10671.1	1.05738	0.528691	−	0.848814i	$-0.322683\pi$
$467$	10671.1	1.05738	0.528691	+	0.848814i	$0.322683\pi$
$468$	0	0
$469$	−17878.4	−1.76023
$470$	−1102.79	−0.108229
$471$	0	0
$472$	84.6776	0.00825763
$473$	−713.161	−0.0693259
$474$	0	0
$475$	−483.303	−0.0466852
$476$	−13228.6	−1.27381
$477$	0	0
$478$	4112.69	0.393536
$479$	−1951.17	−0.186119	−0.0930597	−	0.995661i	$-0.529665\pi$
$479$	−1951.17	−0.186119	−0.0930597	+	0.995661i	$0.529665\pi$
$480$	0	0
$481$	7478.71	0.708939
$482$	3914.06	0.369877
$483$	0	0
$484$	−4855.86	−0.456035
$485$	−4692.18	−0.439301
$486$	0	0
$487$	17071.2	1.58844	0.794220	−	0.607630i	$-0.207880\pi$
$487$	17071.2	1.58844	0.794220	+	0.607630i	$0.207880\pi$
$488$	−884.629	−0.0820600
$489$	0	0
$490$	−4294.00	−0.395884
$491$	−16979.9	−1.56067	−0.780337	−	0.625359i	$-0.784953\pi$
$491$	−16979.9	−1.56067	−0.780337	+	0.625359i	$0.784953\pi$
$492$	0	0
$493$	27904.5	2.54920
$494$	1429.35	0.130181
$495$	0	0
$496$	2645.11	0.239453
$497$	−3980.28	−0.359235
$498$	0	0
$499$	−2788.83	−0.250191	−0.125095	−	0.992145i	$-0.539924\pi$
$499$	−2788.83	−0.250191	−0.125095	+	0.992145i	$0.539924\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−3908.27	−0.347479
$503$	15525.1	1.37620	0.688102	−	0.725614i	$-0.258444\pi$
$503$	15525.1	1.37620	0.688102	+	0.725614i	$0.258444\pi$
$504$	0	0
$505$	3442.14	0.303313
$506$	497.638	0.0437208
$507$	0	0
$508$	6319.98	0.551976
$509$	−9039.20	−0.787143	−0.393571	−	0.919294i	$-0.628761\pi$
$509$	−9039.20	−0.787143	−0.393571	+	0.919294i	$0.628761\pi$
$510$	0	0
$511$	−4397.08	−0.380657
$512$	−512.000	−0.0441942
$513$	0	0
$514$	3916.81	0.336115
$515$	10207.7	0.873412
$516$	0	0
$517$	1193.02	0.101487
$518$	−11244.7	−0.953792
$519$	0	0
$520$	−1478.73	−0.124705
$521$	−1093.03	−0.0919128	−0.0459564	−	0.998943i	$-0.514634\pi$
$521$	−1093.03	−0.0919128	−0.0459564	+	0.998943i	$0.514634\pi$
$522$	0	0
$523$	−4660.98	−0.389695	−0.194848	−	0.980834i	$-0.562421\pi$
$523$	−4660.98	−0.389695	−0.194848	+	0.980834i	$0.562421\pi$
$524$	9395.44	0.783285
$525$	0	0
$526$	14410.5	1.19454
$527$	−19672.4	−1.62607
$528$	0	0
$529$	529.000	0.0434783
$530$	−6881.35	−0.563975
$531$	0	0
$532$	−2149.12	−0.175143
$533$	10936.9	0.888801
$534$	0	0
$535$	−1437.10	−0.116133
$536$	5146.32	0.414715
$537$	0	0
$538$	−7528.60	−0.603311
$539$	4645.35	0.371223
$540$	0	0
$541$	−8072.40	−0.641515	−0.320757	−	0.947161i	$-0.603937\pi$
$541$	−8072.40	−0.641515	−0.320757	+	0.947161i	$0.603937\pi$
$542$	10417.8	0.825611
$543$	0	0
$544$	3807.88	0.300113
$545$	−6056.45	−0.476018
$546$	0	0
$547$	−2877.27	−0.224905	−0.112452	−	0.993657i	$-0.535871\pi$
$547$	−2877.27	−0.224905	−0.112452	+	0.993657i	$0.535871\pi$
$548$	2470.27	0.192563
$549$	0	0
$550$	−540.911	−0.0419355
$551$	4533.36	0.350504
$552$	0	0
$553$	31220.1	2.40075
$554$	−3100.42	−0.237769
$555$	0	0
$556$	6038.99	0.460630
$557$	−15928.7	−1.21171	−0.605855	−	0.795575i	$-0.707169\pi$
$557$	−15928.7	−1.21171	−0.605855	+	0.795575i	$0.707169\pi$
$558$	0	0
$559$	−2437.03	−0.184392
$560$	2223.37	0.167776
$561$	0	0
$562$	15995.1	1.20055
$563$	−7988.51	−0.598003	−0.299001	−	0.954253i	$-0.596654\pi$
$563$	−7988.51	−0.598003	−0.299001	+	0.954253i	$0.596654\pi$
$564$	0	0
$565$	−3706.58	−0.275995
$566$	11616.8	0.862706
$567$	0	0
$568$	1145.73	0.0846368
$569$	6429.49	0.473705	0.236853	−	0.971546i	$-0.423884\pi$
$569$	6429.49	0.473705	0.236853	+	0.971546i	$0.423884\pi$
$570$	0	0
$571$	2886.51	0.211553	0.105776	−	0.994390i	$-0.466267\pi$
$571$	2886.51	0.211553	0.105776	+	0.994390i	$0.466267\pi$
$572$	1599.73	0.116937
$573$	0	0
$574$	−16444.4	−1.19577
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−9966.59	−0.719089	−0.359545	−	0.933128i	$-0.617068\pi$
$577$	−9966.59	−0.719089	−0.359545	+	0.933128i	$0.617068\pi$
$578$	−18494.2	−1.33089
$579$	0	0
$580$	−4689.98	−0.335760
$581$	22917.4	1.63644
$582$	0	0
$583$	7444.40	0.528843
$584$	1265.71	0.0896838
$585$	0	0
$586$	−17709.3	−1.24841
$587$	3522.91	0.247710	0.123855	−	0.992300i	$-0.460474\pi$
$587$	3522.91	0.247710	0.123855	+	0.992300i	$0.460474\pi$
$588$	0	0
$589$	−3195.97	−0.223579
$590$	105.847	0.00738585
$591$	0	0
$592$	3236.81	0.224716
$593$	28713.0	1.98837	0.994184	−	0.107695i	$-0.0343469\pi$
$593$	28713.0	1.98837	0.994184	+	0.107695i	$0.0343469\pi$
$594$	0	0
$595$	−16535.8	−1.13933
$596$	−4101.84	−0.281909
$597$	0	0
$598$	1700.54	0.116288
$599$	−7736.44	−0.527717	−0.263858	−	0.964561i	$-0.584995\pi$
$599$	−7736.44	−0.527717	−0.263858	+	0.964561i	$0.584995\pi$
$600$	0	0
$601$	−3863.89	−0.262249	−0.131124	−	0.991366i	$-0.541859\pi$
$601$	−3863.89	−0.262249	−0.131124	+	0.991366i	$0.541859\pi$
$602$	3664.23	0.248078
$603$	0	0
$604$	−6066.68	−0.408692
$605$	−6069.83	−0.407890
$606$	0	0
$607$	24954.6	1.66866	0.834330	−	0.551265i	$-0.185855\pi$
$607$	24954.6	1.66866	0.834330	+	0.551265i	$0.185855\pi$
$608$	618.628	0.0412643
$609$	0	0
$610$	−1105.79	−0.0733967
$611$	4076.82	0.269935
$612$	0	0
$613$	12712.0	0.837574	0.418787	−	0.908085i	$-0.362455\pi$
$613$	12712.0	0.837574	0.418787	+	0.908085i	$0.362455\pi$
$614$	3002.06	0.197318
$615$	0	0
$616$	−2405.29	−0.157324
$617$	−11017.6	−0.718884	−0.359442	−	0.933167i	$-0.617033\pi$
$617$	−11017.6	−0.718884	−0.359442	+	0.933167i	$0.617033\pi$
$618$	0	0
$619$	6859.65	0.445416	0.222708	−	0.974885i	$-0.428510\pi$
$619$	6859.65	0.445416	0.222708	+	0.974885i	$0.428510\pi$
$620$	3306.39	0.214174
$621$	0	0
$622$	−14881.0	−0.959280
$623$	24447.9	1.57221
$624$	0	0
$625$	625.000	0.0400000
$626$	5057.25	0.322889
$627$	0	0
$628$	−11029.2	−0.700819
$629$	−24073.0	−1.52600
$630$	0	0
$631$	−14104.0	−0.889815	−0.444907	−	0.895577i	$-0.646763\pi$
$631$	−14104.0	−0.889815	−0.444907	+	0.895577i	$0.646763\pi$
$632$	−8986.75	−0.565623
$633$	0	0
$634$	−19273.4	−1.20733
$635$	7899.97	0.493702
$636$	0	0
$637$	15874.2	0.987376
$638$	5073.73	0.314844
$639$	0	0
$640$	−640.000	−0.0395285
$641$	144.931	0.00893047	0.00446524	−	0.999990i	$-0.498579\pi$
$641$	144.931	0.00893047	0.00446524	+	0.999990i	$0.498579\pi$
$642$	0	0
$643$	30891.4	1.89461	0.947307	−	0.320328i	$-0.103793\pi$
$643$	30891.4	1.89461	0.947307	+	0.320328i	$0.103793\pi$
$644$	−2556.87	−0.156452
$645$	0	0
$646$	−4600.90	−0.280217
$647$	−22784.1	−1.38444	−0.692222	−	0.721684i	$-0.743368\pi$
$647$	−22784.1	−1.38444	−0.692222	+	0.721684i	$0.743368\pi$
$648$	0	0
$649$	−114.508	−0.00692576
$650$	−1848.42	−0.111540
$651$	0	0
$652$	5053.17	0.303524
$653$	32125.7	1.92523	0.962617	−	0.270868i	$-0.0873106\pi$
$653$	32125.7	1.92523	0.962617	+	0.270868i	$0.0873106\pi$
$654$	0	0
$655$	11744.3	0.700592
$656$	4733.54	0.281728
$657$	0	0
$658$	−6129.75	−0.363165
$659$	−15521.7	−0.917508	−0.458754	−	0.888563i	$-0.651704\pi$
$659$	−15521.7	−0.917508	−0.458754	+	0.888563i	$0.651704\pi$
$660$	0	0
$661$	19428.4	1.14323	0.571617	−	0.820521i	$-0.306316\pi$
$661$	19428.4	1.14323	0.571617	+	0.820521i	$0.306316\pi$
$662$	−5751.64	−0.337680
$663$	0	0
$664$	−6596.80	−0.385550
$665$	−2686.40	−0.156653
$666$	0	0
$667$	5393.48	0.313098
$668$	1398.58	0.0810072
$669$	0	0
$670$	6432.90	0.370932
$671$	1196.27	0.0688246
$672$	0	0
$673$	18992.7	1.08784	0.543919	−	0.839138i	$-0.316940\pi$
$673$	18992.7	1.08784	0.543919	+	0.839138i	$0.316940\pi$
$674$	−19822.4	−1.13283
$675$	0	0
$676$	−3321.38	−0.188972
$677$	−19104.3	−1.08455	−0.542273	−	0.840202i	$-0.682436\pi$
$677$	−19104.3	−1.08455	−0.542273	+	0.840202i	$0.682436\pi$
$678$	0	0
$679$	−26081.1	−1.47408
$680$	4759.84	0.268429
$681$	0	0
$682$	−3576.92	−0.200832
$683$	24737.2	1.38586	0.692931	−	0.721004i	$-0.256319\pi$
$683$	24737.2	1.38586	0.692931	+	0.721004i	$0.256319\pi$
$684$	0	0
$685$	3087.84	0.172234
$686$	−4802.49	−0.267288
$687$	0	0
$688$	−1054.75	−0.0584479
$689$	25439.2	1.40661
$690$	0	0
$691$	12957.9	0.713373	0.356687	−	0.934224i	$-0.383906\pi$
$691$	12957.9	0.713373	0.356687	+	0.934224i	$0.383906\pi$
$692$	−9255.33	−0.508432
$693$	0	0
$694$	8995.41	0.492019
$695$	7548.74	0.412000
$696$	0	0
$697$	−35204.5	−1.91315
$698$	−1776.22	−0.0963195
$699$	0	0
$700$	2779.21	0.150063
$701$	−20730.7	−1.11696	−0.558478	−	0.829519i	$-0.688615\pi$
$701$	−20730.7	−1.11696	−0.558478	+	0.829519i	$0.688615\pi$
$702$	0	0
$703$	−3910.90	−0.209819
$704$	692.367	0.0370661
$705$	0	0
$706$	−23444.5	−1.24978
$707$	19132.8	1.01777
$708$	0	0
$709$	−22191.2	−1.17547	−0.587735	−	0.809053i	$-0.699980\pi$
$709$	−22191.2	−1.17547	−0.587735	+	0.809053i	$0.699980\pi$
$710$	1432.16	0.0757015
$711$	0	0
$712$	−7037.38	−0.370417
$713$	−3802.34	−0.199718
$714$	0	0
$715$	1999.66	0.104592
$716$	9390.77	0.490153
$717$	0	0
$718$	−2514.98	−0.130722
$719$	773.962	0.0401445	0.0200723	−	0.999799i	$-0.493610\pi$
$719$	773.962	0.0401445	0.0200723	+	0.999799i	$0.493610\pi$
$720$	0	0
$721$	56738.9	2.93075
$722$	12970.5	0.668578
$723$	0	0
$724$	17585.3	0.902695
$725$	−5862.47	−0.300313
$726$	0	0
$727$	−32484.7	−1.65721	−0.828604	−	0.559835i	$-0.810864\pi$
$727$	−32484.7	−1.65721	−0.828604	+	0.559835i	$0.810864\pi$
$728$	−8219.41	−0.418450
$729$	0	0
$730$	1582.13	0.0802157
$731$	7844.48	0.396906
$732$	0	0
$733$	−13701.5	−0.690416	−0.345208	−	0.938526i	$-0.612192\pi$
$733$	−13701.5	−0.690416	−0.345208	+	0.938526i	$0.612192\pi$
$734$	−20244.5	−1.01803
$735$	0	0
$736$	736.000	0.0368605
$737$	−6959.26	−0.347826
$738$	0	0
$739$	−31474.2	−1.56671	−0.783354	−	0.621576i	$-0.786493\pi$
$739$	−31474.2	−1.56671	−0.783354	+	0.621576i	$0.786493\pi$
$740$	4046.01	0.200992
$741$	0	0
$742$	−38249.4	−1.89243
$743$	−28664.2	−1.41533	−0.707664	−	0.706549i	$-0.750251\pi$
$743$	−28664.2	−1.41533	−0.707664	+	0.706549i	$0.750251\pi$
$744$	0	0
$745$	−5127.30	−0.252147
$746$	−13060.2	−0.640974
$747$	0	0
$748$	−5149.31	−0.251708
$749$	−7988.00	−0.389687
$750$	0	0
$751$	−25904.0	−1.25866	−0.629328	−	0.777140i	$-0.716670\pi$
$751$	−25904.0	−1.25866	−0.629328	+	0.777140i	$0.716670\pi$
$752$	1764.46	0.0855628
$753$	0	0
$754$	17338.1	0.837420
$755$	−7583.35	−0.365545
$756$	0	0
$757$	2569.70	0.123378	0.0616892	−	0.998095i	$-0.480351\pi$
$757$	2569.70	0.123378	0.0616892	+	0.998095i	$0.480351\pi$
$758$	−555.700	−0.0266279
$759$	0	0
$760$	773.285	0.0369079
$761$	23219.4	1.10605	0.553024	−	0.833165i	$-0.313474\pi$
$761$	23219.4	1.10605	0.553024	+	0.833165i	$0.313474\pi$
$762$	0	0
$763$	−33664.3	−1.59728
$764$	−16612.6	−0.786677
$765$	0	0
$766$	−21359.3	−1.00750
$767$	−391.298	−0.0184211
$768$	0	0
$769$	34771.1	1.63053	0.815265	−	0.579088i	$-0.196591\pi$
$769$	34771.1	1.63053	0.815265	+	0.579088i	$0.196591\pi$
$770$	−3006.61	−0.140715
$771$	0	0
$772$	−4082.10	−0.190308
$773$	36377.2	1.69262	0.846312	−	0.532687i	$-0.178818\pi$
$773$	36377.2	1.69262	0.846312	+	0.532687i	$0.178818\pi$
$774$	0	0
$775$	4132.98	0.191563
$776$	7507.50	0.347298
$777$	0	0
$778$	3092.10	0.142490
$779$	−5719.34	−0.263051
$780$	0	0
$781$	−1549.34	−0.0709858
$782$	−5473.82	−0.250311
$783$	0	0
$784$	6870.40	0.312974
$785$	−13786.5	−0.626831
$786$	0	0
$787$	−1814.43	−0.0821824	−0.0410912	−	0.999155i	$-0.513083\pi$
$787$	−1814.43	−0.0821824	−0.0410912	+	0.999155i	$0.513083\pi$
$788$	1594.57	0.0720866
$789$	0	0
$790$	−11233.4	−0.505909
$791$	−20602.7	−0.926104
$792$	0	0
$793$	4087.91	0.183059
$794$	25803.5	1.15332
$795$	0	0
$796$	16525.6	0.735847
$797$	7958.23	0.353695	0.176848	−	0.984238i	$-0.443410\pi$
$797$	7958.23	0.353695	0.176848	+	0.984238i	$0.443410\pi$
$798$	0	0
$799$	−13122.7	−0.581038
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−17068.6	−0.751513
$803$	−1711.59	−0.0752188
$804$	0	0
$805$	−3196.09	−0.139935
$806$	−12223.1	−0.534171
$807$	0	0
$808$	−5507.42	−0.239790
$809$	−20506.9	−0.891203	−0.445601	−	0.895232i	$-0.647010\pi$
$809$	−20506.9	−0.891203	−0.445601	+	0.895232i	$0.647010\pi$
$810$	0	0
$811$	1962.00	0.0849506	0.0424753	−	0.999098i	$-0.486476\pi$
$811$	1962.00	0.0849506	0.0424753	+	0.999098i	$0.486476\pi$
$812$	−26068.9	−1.12665
$813$	0	0
$814$	−4377.07	−0.188472
$815$	6316.46	0.271480
$816$	0	0
$817$	1274.42	0.0545730
$818$	−1301.94	−0.0556493
$819$	0	0
$820$	5916.92	0.251985
$821$	−25168.1	−1.06988	−0.534941	−	0.844889i	$-0.679666\pi$
$821$	−25168.1	−1.06988	−0.534941	+	0.844889i	$0.679666\pi$
$822$	0	0
$823$	3337.33	0.141351	0.0706756	−	0.997499i	$-0.477484\pi$
$823$	3337.33	0.141351	0.0706756	+	0.997499i	$0.477484\pi$
$824$	−16332.4	−0.690492
$825$	0	0
$826$	588.342	0.0247833
$827$	13087.4	0.550294	0.275147	−	0.961402i	$-0.411273\pi$
$827$	13087.4	0.550294	0.275147	+	0.961402i	$0.411273\pi$
$828$	0	0
$829$	−38486.8	−1.61243	−0.806213	−	0.591626i	$-0.798486\pi$
$829$	−38486.8	−1.61243	−0.806213	+	0.591626i	$0.798486\pi$
$830$	−8246.00	−0.344847
$831$	0	0
$832$	2365.97	0.0985881
$833$	−51097.0	−2.12534
$834$	0	0
$835$	1748.23	0.0724550
$836$	−836.558	−0.0346088
$837$	0	0
$838$	26532.4	1.09373
$839$	−8192.40	−0.337107	−0.168554	−	0.985692i	$-0.553910\pi$
$839$	−8192.40	−0.337107	−0.168554	+	0.985692i	$0.553910\pi$
$840$	0	0
$841$	30600.7	1.25469
$842$	16549.4	0.677350
$843$	0	0
$844$	3305.26	0.134801
$845$	−4151.72	−0.169022
$846$	0	0
$847$	−33738.7	−1.36868
$848$	11010.2	0.445861
$849$	0	0
$850$	5949.81	0.240090
$851$	−4652.91	−0.187426
$852$	0	0
$853$	−6774.83	−0.271941	−0.135971	−	0.990713i	$-0.543415\pi$
$853$	−6774.83	−0.271941	−0.135971	+	0.990713i	$0.543415\pi$
$854$	−6146.42	−0.246284
$855$	0	0
$856$	2299.36	0.0918113
$857$	−19272.3	−0.768177	−0.384089	−	0.923296i	$-0.625484\pi$
$857$	−19272.3	−0.768177	−0.384089	+	0.923296i	$0.625484\pi$
$858$	0	0
$859$	−5991.11	−0.237968	−0.118984	−	0.992896i	$-0.537964\pi$
$859$	−5991.11	−0.237968	−0.118984	+	0.992896i	$0.537964\pi$
$860$	−1318.44	−0.0522774
$861$	0	0
$862$	1448.17	0.0572214
$863$	−20711.4	−0.816945	−0.408472	−	0.912771i	$-0.633938\pi$
$863$	−20711.4	−0.816945	−0.408472	+	0.912771i	$0.633938\pi$
$864$	0	0
$865$	−11569.2	−0.454755
$866$	23993.1	0.941476
$867$	0	0
$868$	18378.3	0.718663
$869$	12152.6	0.474394
$870$	0	0
$871$	−23781.4	−0.925144
$872$	9690.31	0.376325
$873$	0	0
$874$	−889.278	−0.0344168
$875$	3474.01	0.134221
$876$	0	0
$877$	−45757.4	−1.76182	−0.880912	−	0.473281i	$-0.843070\pi$
$877$	−45757.4	−1.76182	−0.880912	+	0.473281i	$0.843070\pi$
$878$	15530.8	0.596969
$879$	0	0
$880$	865.458	0.0331529
$881$	−37019.6	−1.41569	−0.707845	−	0.706367i	$-0.750333\pi$
$881$	−37019.6	−1.41569	−0.707845	+	0.706367i	$0.750333\pi$
$882$	0	0
$883$	−9311.25	−0.354868	−0.177434	−	0.984133i	$-0.556780\pi$
$883$	−9311.25	−0.354868	−0.177434	+	0.984133i	$0.556780\pi$
$884$	−17596.3	−0.669490
$885$	0	0
$886$	−18123.1	−0.687197
$887$	−33041.8	−1.25077	−0.625387	−	0.780315i	$-0.715059\pi$
$887$	−33041.8	−1.25077	−0.625387	+	0.780315i	$0.715059\pi$
$888$	0	0
$889$	43911.4	1.65662
$890$	−8796.72	−0.331311
$891$	0	0
$892$	4440.37	0.166676
$893$	−2131.92	−0.0798903
$894$	0	0
$895$	11738.5	0.438406
$896$	−3557.39	−0.132638
$897$	0	0
$898$	−21888.6	−0.813398
$899$	−38767.2	−1.43822
$900$	0	0
$901$	−81885.4	−3.02774
$902$	−6401.06	−0.236288
$903$	0	0
$904$	5930.53	0.218193
$905$	21981.6	0.807395
$906$	0	0
$907$	−41359.1	−1.51412	−0.757060	−	0.653346i	$-0.773365\pi$
$907$	−41359.1	−1.51412	−0.757060	+	0.653346i	$0.773365\pi$
$908$	19671.6	0.718971
$909$	0	0
$910$	−10274.3	−0.374273
$911$	16960.3	0.616818	0.308409	−	0.951254i	$-0.400203\pi$
$911$	16960.3	0.616818	0.308409	+	0.951254i	$0.400203\pi$
$912$	0	0
$913$	8920.71	0.323365
$914$	−21869.8	−0.791452
$915$	0	0
$916$	−5562.50	−0.200644
$917$	65279.7	2.35085
$918$	0	0
$919$	3085.38	0.110748	0.0553739	−	0.998466i	$-0.482365\pi$
$919$	3085.38	0.110748	0.0553739	+	0.998466i	$0.482365\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	5304.51	0.189474
$923$	−5294.46	−0.188807
$924$	0	0
$925$	5057.51	0.179773
$926$	−1415.40	−0.0502299
$927$	0	0
$928$	7503.97	0.265442
$929$	42292.0	1.49360	0.746801	−	0.665048i	$-0.231589\pi$
$929$	42292.0	1.49360	0.746801	+	0.665048i	$0.231589\pi$
$930$	0	0
$931$	−8301.22	−0.292225
$932$	13638.4	0.479334
$933$	0	0
$934$	−21342.1	−0.747682
$935$	−6436.64	−0.225134
$936$	0	0
$937$	15746.4	0.549000	0.274500	−	0.961587i	$-0.411488\pi$
$937$	15746.4	0.549000	0.274500	+	0.961587i	$0.411488\pi$
$938$	35756.8	1.24467
$939$	0	0
$940$	2205.57	0.0765297
$941$	−52033.4	−1.80259	−0.901296	−	0.433204i	$-0.857383\pi$
$941$	−52033.4	−1.80259	−0.901296	+	0.433204i	$0.857383\pi$
$942$	0	0
$943$	−6804.46	−0.234977
$944$	−169.355	−0.00583903
$945$	0	0
$946$	1426.32	0.0490208
$947$	17293.2	0.593405	0.296702	−	0.954970i	$-0.404113\pi$
$947$	17293.2	0.593405	0.296702	+	0.954970i	$0.404113\pi$
$948$	0	0
$949$	−5848.88	−0.200066
$950$	966.607	0.0330114
$951$	0	0
$952$	26457.2	0.900717
$953$	−8989.54	−0.305561	−0.152781	−	0.988260i	$-0.548823\pi$
$953$	−8989.54	−0.305561	−0.152781	+	0.988260i	$0.548823\pi$
$954$	0	0
$955$	−20765.7	−0.703625
$956$	−8225.39	−0.278272
$957$	0	0
$958$	3902.34	0.131606
$959$	17163.5	0.577933
$960$	0	0
$961$	−2460.53	−0.0825931
$962$	−14957.4	−0.501296
$963$	0	0
$964$	−7828.13	−0.261543
$965$	−5102.62	−0.170217
$966$	0	0
$967$	23285.3	0.774358	0.387179	−	0.922005i	$-0.373450\pi$
$967$	23285.3	0.774358	0.387179	+	0.922005i	$0.373450\pi$
$968$	9711.73	0.322466
$969$	0	0
$970$	9384.37	0.310633
$971$	−44316.5	−1.46466	−0.732330	−	0.680950i	$-0.761567\pi$
$971$	−44316.5	−1.46466	−0.732330	+	0.680950i	$0.761567\pi$
$972$	0	0
$973$	41959.1	1.38247
$974$	−34142.4	−1.12320
$975$	0	0
$976$	1769.26	0.0580252
$977$	−761.665	−0.0249415	−0.0124707	−	0.999922i	$-0.503970\pi$
$977$	−761.665	−0.0249415	−0.0124707	+	0.999922i	$0.503970\pi$
$978$	0	0
$979$	9516.49	0.310673
$980$	8588.01	0.279932
$981$	0	0
$982$	33959.8	1.10356
$983$	17423.3	0.565328	0.282664	−	0.959219i	$-0.408782\pi$
$983$	17423.3	0.565328	0.282664	+	0.959219i	$0.408782\pi$
$984$	0	0
$985$	1993.21	0.0644762
$986$	−55808.9	−1.80255
$987$	0	0
$988$	−2858.70	−0.0920521
$989$	1516.21	0.0487489
$990$	0	0
$991$	−33873.1	−1.08579	−0.542894	−	0.839801i	$-0.682671\pi$
$991$	−33873.1	−1.08579	−0.542894	+	0.839801i	$0.682671\pi$
$992$	−5290.22	−0.169319
$993$	0	0
$994$	7960.55	0.254017
$995$	20657.0	0.658162
$996$	0	0
$997$	−31163.3	−0.989921	−0.494960	−	0.868915i	$-0.664817\pi$
$997$	−31163.3	−0.989921	−0.494960	+	0.868915i	$0.664817\pi$
$998$	5577.66	0.176912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bi.1.4		4
3.2	odd	2		230.4.a.i.1.2	✓	4
12.11	even	2		1840.4.a.l.1.3		4
15.2	even	4		1150.4.b.m.599.7		8
15.8	even	4		1150.4.b.m.599.2		8
15.14	odd	2		1150.4.a.o.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.2	✓	4	3.2	odd	2
1150.4.a.o.1.3		4	15.14	odd	2
1150.4.b.m.599.2		8	15.8	even	4
1150.4.b.m.599.7		8	15.2	even	4
1840.4.a.l.1.3		4	12.11	even	2
2070.4.a.bi.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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +27.7921 q^{7} -8.00000 q^{8} -10.0000 q^{10} +10.8182 q^{11} +36.9683 q^{13} -55.5842 q^{14} +16.0000 q^{16} -118.996 q^{17} -19.3321 q^{19} +20.0000 q^{20} -21.6365 q^{22} -23.0000 q^{23} +25.0000 q^{25} -73.9366 q^{26} +111.168 q^{28} -234.499 q^{29} +165.319 q^{31} -32.0000 q^{32} +237.992 q^{34} +138.960 q^{35} +202.301 q^{37} +38.6643 q^{38} -40.0000 q^{40} +295.846 q^{41} -65.9221 q^{43} +43.2729 q^{44} +46.0000 q^{46} +110.279 q^{47} +429.400 q^{49} -50.0000 q^{50} +147.873 q^{52} +688.135 q^{53} +54.0911 q^{55} -222.337 q^{56} +468.998 q^{58} -10.5847 q^{59} +110.579 q^{61} -330.639 q^{62} +64.0000 q^{64} +184.842 q^{65} -643.290 q^{67} -475.984 q^{68} -277.921 q^{70} -143.216 q^{71} -158.213 q^{73} -404.601 q^{74} -77.3285 q^{76} +300.661 q^{77} +1123.34 q^{79} +80.0000 q^{80} -591.692 q^{82} +824.600 q^{83} -594.981 q^{85} +131.844 q^{86} -86.5458 q^{88} +879.672 q^{89} +1027.43 q^{91} -92.0000 q^{92} -220.557 q^{94} -96.6607 q^{95} -938.437 q^{97} -858.801 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} + 20 q^{5} + 26 q^{7} - 32 q^{8} - 40 q^{10} - 93 q^{11} + 32 q^{13} - 52 q^{14} + 64 q^{16} - 108 q^{17} + 185 q^{19} + 80 q^{20} + 186 q^{22} - 92 q^{23} + 100 q^{25} - 64 q^{26}+ \cdots - 1200 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 + 20 * q^5 + 26 * q^7 - 32 * q^8 - 40 * q^10 - 93 * q^11 + 32 * q^13 - 52 * q^14 + 64 * q^16 - 108 * q^17 + 185 * q^19 + 80 * q^20 + 186 * q^22 - 92 * q^23 + 100 * q^25 - 64 * q^26 + 104 * q^28 - 294 * q^29 - 211 * q^31 - 128 * q^32 + 216 * q^34 + 130 * q^35 + 5 * q^37 - 370 * q^38 - 160 * q^40 + 369 * q^41 - 100 * q^43 - 372 * q^44 + 184 * q^46 + 363 * q^47 + 600 * q^49 - 200 * q^50 + 128 * q^52 - 21 * q^53 - 465 * q^55 - 208 * q^56 + 588 * q^58 + 33 * q^59 - 307 * q^61 + 422 * q^62 + 256 * q^64 + 160 * q^65 + 725 * q^67 - 432 * q^68 - 260 * q^70 + 1257 * q^71 + 509 * q^73 - 10 * q^74 + 740 * q^76 + 1962 * q^77 + 1202 * q^79 + 320 * q^80 - 738 * q^82 + 1377 * q^83 - 540 * q^85 + 200 * q^86 + 744 * q^88 + 984 * q^89 - 995 * q^91 - 368 * q^92 - 726 * q^94 + 925 * q^95 + 137 * q^97 - 1200 * q^98

Introduction

L-functions

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Database

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