LMFDB - Embedded newform 207.4.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [207,4,Mod(1,207)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("207.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$207 = 3^{2} \cdot 23$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	207.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.2133953712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2009704.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 27x^{2} - 6x + 112$ x^4 - 27x^2 - 6x + 112
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.09731$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.09731 q^{2} -6.79592 q^{4} -3.76407 q^{5} +27.3318 q^{7} -16.2356 q^{8} -4.13033 q^{10} -23.0798 q^{11} -10.1463 q^{13} +29.9913 q^{14} +36.5519 q^{16} -137.293 q^{17} -110.571 q^{19} +25.5803 q^{20} -25.3256 q^{22} -23.0000 q^{23} -110.832 q^{25} -11.1336 q^{26} -185.745 q^{28} +0.290564 q^{29} -294.053 q^{31} +169.994 q^{32} -150.652 q^{34} -102.879 q^{35} +230.689 q^{37} -121.330 q^{38} +61.1121 q^{40} +470.221 q^{41} +130.676 q^{43} +156.849 q^{44} -25.2380 q^{46} -230.727 q^{47} +404.026 q^{49} -121.616 q^{50} +68.9538 q^{52} -135.626 q^{53} +86.8740 q^{55} -443.749 q^{56} +0.318838 q^{58} -245.965 q^{59} -358.608 q^{61} -322.666 q^{62} -105.880 q^{64} +38.1915 q^{65} +357.125 q^{67} +933.031 q^{68} -112.889 q^{70} +340.997 q^{71} -725.178 q^{73} +253.136 q^{74} +751.432 q^{76} -630.813 q^{77} -40.8802 q^{79} -137.584 q^{80} +515.977 q^{82} -459.288 q^{83} +516.779 q^{85} +143.391 q^{86} +374.716 q^{88} +322.262 q^{89} -277.318 q^{91} +156.306 q^{92} -253.178 q^{94} +416.197 q^{95} +1198.96 q^{97} +443.340 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + 26 q^{4} - 4 q^{5} - 14 q^{7} - 84 q^{8} - 100 q^{10} - 70 q^{11} - 12 q^{13} + 18 q^{14} + 130 q^{16} - 178 q^{17} + 96 q^{19} + 296 q^{20} - 326 q^{22} - 92 q^{23} - 80 q^{25} + 464 q^{26}+ \cdots + 2784 q^{98}+O(q^{100})$ 4 * q - 4 * q^2 + 26 * q^4 - 4 * q^5 - 14 * q^7 - 84 * q^8 - 100 * q^10 - 70 * q^11 - 12 * q^13 + 18 * q^14 + 130 * q^16 - 178 * q^17 + 96 * q^19 + 296 * q^20 - 326 * q^22 - 92 * q^23 - 80 * q^25 + 464 * q^26 - 774 * q^28 + 24 * q^29 - 400 * q^31 - 636 * q^32 - 18 * q^34 - 224 * q^35 - 358 * q^37 + 136 * q^38 - 1268 * q^40 - 152 * q^41 - 196 * q^43 - 70 * q^44 + 92 * q^46 - 260 * q^47 + 476 * q^49 - 876 * q^50 - 876 * q^52 - 968 * q^53 + 684 * q^55 + 1234 * q^56 + 904 * q^58 - 1236 * q^59 - 482 * q^61 + 1304 * q^62 + 1658 * q^64 - 1424 * q^65 + 620 * q^67 - 838 * q^68 + 2636 * q^70 + 88 * q^71 - 948 * q^73 - 710 * q^74 + 3344 * q^76 + 892 * q^77 + 254 * q^79 + 5144 * q^80 + 1568 * q^82 + 550 * q^83 - 1208 * q^85 + 1236 * q^86 + 202 * q^88 - 202 * q^89 - 356 * q^91 - 598 * q^92 - 1328 * q^94 + 460 * q^95 - 1800 * q^97 + 2784 * 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.09731	0.387956	0.193978	−	0.981006i	$-0.437861\pi$
$2$	1.09731	0.387956	0.193978	+	0.981006i	$0.437861\pi$
$3$	0	0
$3$	0	0
$4$	−6.79592	−0.849490
$5$	−3.76407	−0.336668	−0.168334	−	0.985730i	$-0.553839\pi$
$5$	−3.76407	−0.336668	−0.168334	+	0.985730i	$0.553839\pi$
$6$	0	0
$7$	27.3318	1.47578	0.737889	−	0.674922i	$-0.235823\pi$
$7$	27.3318	1.47578	0.737889	+	0.674922i	$0.235823\pi$
$8$	−16.2356	−0.717521
$9$	0	0
$10$	−4.13033	−0.130613
$11$	−23.0798	−0.632621	−0.316310	−	0.948656i	$-0.602444\pi$
$11$	−23.0798	−0.632621	−0.316310	+	0.948656i	$0.602444\pi$
$12$	0	0
$13$	−10.1463	−0.216469	−0.108234	−	0.994125i	$-0.534520\pi$
$13$	−10.1463	−0.216469	−0.108234	+	0.994125i	$0.534520\pi$
$14$	29.9913	0.572537
$15$	0	0
$16$	36.5519	0.571123
$17$	−137.293	−1.95873	−0.979364	−	0.202103i	$-0.935222\pi$
$17$	−137.293	−1.95873	−0.979364	+	0.202103i	$0.935222\pi$
$18$	0	0
$19$	−110.571	−1.33509	−0.667546	−	0.744569i	$-0.732655\pi$
$19$	−110.571	−1.33509	−0.667546	+	0.744569i	$0.732655\pi$
$20$	25.5803	0.285997
$21$	0	0
$22$	−25.3256	−0.245429
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−110.832	−0.886654
$26$	−11.1336	−0.0839803
$27$	0	0
$28$	−185.745	−1.25366
$29$	0.290564	0.00186057	0.000930284	−	1.00000i	$-0.499704\pi$
$29$	0.290564	0.00186057	0.000930284		1.00000i	$0.499704\pi$
$30$	0	0
$31$	−294.053	−1.70366	−0.851829	−	0.523820i	$-0.824507\pi$
$31$	−294.053	−1.70366	−0.851829	+	0.523820i	$0.824507\pi$
$32$	169.994	0.939092
$33$	0	0
$34$	−150.652	−0.759901
$35$	−102.879	−0.496848
$36$	0	0
$37$	230.689	1.02500	0.512501	−	0.858687i	$-0.328719\pi$
$37$	230.689	1.02500	0.512501	+	0.858687i	$0.328719\pi$
$38$	−121.330	−0.517957
$39$	0	0
$40$	61.1121	0.241567
$41$	470.221	1.79113	0.895564	−	0.444933i	$-0.146772\pi$
$41$	470.221	1.79113	0.895564	+	0.444933i	$0.146772\pi$
$42$	0	0
$43$	130.676	0.463438	0.231719	−	0.972783i	$-0.425565\pi$
$43$	130.676	0.463438	0.231719	+	0.972783i	$0.425565\pi$
$44$	156.849	0.537405
$45$	0	0
$46$	−25.2380	−0.0808944
$47$	−230.727	−0.716064	−0.358032	−	0.933709i	$-0.616552\pi$
$47$	−230.727	−0.716064	−0.358032	+	0.933709i	$0.616552\pi$
$48$	0	0
$49$	404.026	1.17792
$50$	−121.616	−0.343983
$51$	0	0
$52$	68.9538	0.183888
$53$	−135.626	−0.351504	−0.175752	−	0.984434i	$-0.556236\pi$
$53$	−135.626	−0.351504	−0.175752	+	0.984434i	$0.556236\pi$
$54$	0	0
$55$	86.8740	0.212984
$56$	−443.749	−1.05890
$57$	0	0
$58$	0.318838	0.000721818 0
$59$	−245.965	−0.542744	−0.271372	−	0.962475i	$-0.587477\pi$
$59$	−245.965	−0.542744	−0.271372	+	0.962475i	$0.587477\pi$
$60$	0	0
$61$	−358.608	−0.752707	−0.376353	−	0.926476i	$-0.622822\pi$
$61$	−358.608	−0.752707	−0.376353	+	0.926476i	$0.622822\pi$
$62$	−322.666	−0.660945
$63$	0	0
$64$	−105.880	−0.206797
$65$	38.1915	0.0728781
$66$	0	0
$67$	357.125	0.651191	0.325596	−	0.945509i	$-0.394435\pi$
$67$	357.125	0.651191	0.325596	+	0.945509i	$0.394435\pi$
$68$	933.031	1.66392
$69$	0	0
$70$	−112.889	−0.192755
$71$	340.997	0.569985	0.284992	−	0.958530i	$-0.408009\pi$
$71$	340.997	0.569985	0.284992	+	0.958530i	$0.408009\pi$
$72$	0	0
$73$	−725.178	−1.16268	−0.581340	−	0.813661i	$-0.697471\pi$
$73$	−725.178	−1.16268	−0.581340	+	0.813661i	$0.697471\pi$
$74$	253.136	0.397656
$75$	0	0
$76$	751.432	1.13415
$77$	−630.813	−0.933608
$78$	0	0
$79$	−40.8802	−0.0582200	−0.0291100	−	0.999576i	$-0.509267\pi$
$79$	−40.8802	−0.0582200	−0.0291100	+	0.999576i	$0.509267\pi$
$80$	−137.584	−0.192279
$81$	0	0
$82$	515.977	0.694879
$83$	−459.288	−0.607391	−0.303696	−	0.952769i	$-0.598221\pi$
$83$	−459.288	−0.607391	−0.303696	+	0.952769i	$0.598221\pi$
$84$	0	0
$85$	516.779	0.659442
$86$	143.391	0.179794
$87$	0	0
$88$	374.716	0.453919
$89$	322.262	0.383817	0.191908	−	0.981413i	$-0.438532\pi$
$89$	322.262	0.383817	0.191908	+	0.981413i	$0.438532\pi$
$90$	0	0
$91$	−277.318	−0.319459
$92$	156.306	0.177131
$93$	0	0
$94$	−253.178	−0.277802
$95$	416.197	0.449483
$96$	0	0
$97$	1198.96	1.25501	0.627507	−	0.778611i	$-0.284076\pi$
$97$	1198.96	1.25501	0.627507	+	0.778611i	$0.284076\pi$
$98$	443.340	0.456981
$99$	0	0
$100$	753.204	0.753204
$101$	458.744	0.451947	0.225974	−	0.974133i	$-0.427444\pi$
$101$	458.744	0.451947	0.225974	+	0.974133i	$0.427444\pi$
$102$	0	0
$103$	−1104.37	−1.05647	−0.528237	−	0.849097i	$-0.677147\pi$
$103$	−1104.37	−1.05647	−0.528237	+	0.849097i	$0.677147\pi$
$104$	164.733	0.155321
$105$	0	0
$106$	−148.823	−0.136368
$107$	1923.58	1.73794	0.868970	−	0.494864i	$-0.164782\pi$
$107$	1923.58	1.73794	0.868970	+	0.494864i	$0.164782\pi$
$108$	0	0
$109$	−1466.58	−1.28874	−0.644371	−	0.764713i	$-0.722881\pi$
$109$	−1466.58	−1.28874	−0.644371	+	0.764713i	$0.722881\pi$
$110$	95.3274	0.0826283
$111$	0	0
$112$	999.028	0.842851
$113$	1085.34	0.903539	0.451769	−	0.892135i	$-0.350793\pi$
$113$	1085.34	0.903539	0.451769	+	0.892135i	$0.350793\pi$
$114$	0	0
$115$	86.5736	0.0702002
$116$	−1.97465	−0.00158053
$117$	0	0
$118$	−269.898	−0.210561
$119$	−3752.46	−2.89065
$120$	0	0
$121$	−798.321	−0.599791
$122$	−393.503	−0.292017
$123$	0	0
$124$	1998.36	1.44724
$125$	887.687	0.635177
$126$	0	0
$127$	773.227	0.540258	0.270129	−	0.962824i	$-0.412934\pi$
$127$	773.227	0.540258	0.270129	+	0.962824i	$0.412934\pi$
$128$	−1476.13	−1.01932
$129$	0	0
$130$	41.9078	0.0282735
$131$	−1335.38	−0.890629	−0.445314	−	0.895374i	$-0.646908\pi$
$131$	−1335.38	−0.890629	−0.445314	+	0.895374i	$0.646908\pi$
$132$	0	0
$133$	−3022.10	−1.97030
$134$	391.876	0.252634
$135$	0	0
$136$	2229.04	1.40543
$137$	2132.45	1.32984	0.664918	−	0.746916i	$-0.268466\pi$
$137$	2132.45	1.32984	0.664918	+	0.746916i	$0.268466\pi$
$138$	0	0
$139$	1690.32	1.03145	0.515723	−	0.856755i	$-0.327523\pi$
$139$	1690.32	1.03145	0.515723	+	0.856755i	$0.327523\pi$
$140$	699.155	0.422067
$141$	0	0
$142$	374.178	0.221129
$143$	234.176	0.136943
$144$	0	0
$145$	−1.09370	−0.000626394 0
$146$	−795.742	−0.451069
$147$	0	0
$148$	−1567.75	−0.870729
$149$	−2653.60	−1.45900	−0.729501	−	0.683979i	$-0.760248\pi$
$149$	−2653.60	−1.45900	−0.729501	+	0.683979i	$0.760248\pi$
$150$	0	0
$151$	−488.924	−0.263497	−0.131749	−	0.991283i	$-0.542059\pi$
$151$	−488.924	−0.263497	−0.131749	+	0.991283i	$0.542059\pi$
$152$	1795.19	0.957956
$153$	0	0
$154$	−692.194	−0.362199
$155$	1106.83	0.573568
$156$	0	0
$157$	1903.73	0.967735	0.483868	−	0.875141i	$-0.339232\pi$
$157$	1903.73	0.967735	0.483868	+	0.875141i	$0.339232\pi$
$158$	−44.8581	−0.0225868
$159$	0	0
$160$	−639.868	−0.316163
$161$	−628.631	−0.307721
$162$	0	0
$163$	810.227	0.389337	0.194668	−	0.980869i	$-0.437637\pi$
$163$	810.227	0.389337	0.194668	+	0.980869i	$0.437637\pi$
$164$	−3195.59	−1.52155
$165$	0	0
$166$	−503.980	−0.235641
$167$	−483.976	−0.224258	−0.112129	−	0.993694i	$-0.535767\pi$
$167$	−483.976	−0.224258	−0.112129	+	0.993694i	$0.535767\pi$
$168$	0	0
$169$	−2094.05	−0.953141
$170$	567.065	0.255835
$171$	0	0
$172$	−888.061	−0.393686
$173$	1210.96	0.532183	0.266091	−	0.963948i	$-0.414268\pi$
$173$	1210.96	0.532183	0.266091	+	0.963948i	$0.414268\pi$
$174$	0	0
$175$	−3029.23	−1.30850
$176$	−843.612	−0.361305
$177$	0	0
$178$	353.620	0.148904
$179$	929.904	0.388292	0.194146	−	0.980973i	$-0.437806\pi$
$179$	929.904	0.388292	0.194146	+	0.980973i	$0.437806\pi$
$180$	0	0
$181$	−3049.96	−1.25249	−0.626247	−	0.779625i	$-0.715410\pi$
$181$	−3049.96	−1.25249	−0.626247	+	0.779625i	$0.715410\pi$
$182$	−304.302	−0.123936
$183$	0	0
$184$	373.420	0.149613
$185$	−868.330	−0.345086
$186$	0	0
$187$	3168.69	1.23913
$188$	1568.00	0.608290
$189$	0	0
$190$	456.695	0.174380
$191$	−4434.70	−1.68002	−0.840010	−	0.542571i	$-0.817451\pi$
$191$	−4434.70	−1.68002	−0.840010	+	0.542571i	$0.817451\pi$
$192$	0	0
$193$	−3028.10	−1.12936	−0.564682	−	0.825309i	$-0.691001\pi$
$193$	−3028.10	−1.12936	−0.564682	+	0.825309i	$0.691001\pi$
$194$	1315.63	0.486890
$195$	0	0
$196$	−2745.73	−1.00063
$197$	187.415	0.0677807	0.0338903	−	0.999426i	$-0.489210\pi$
$197$	187.415	0.0677807	0.0338903	+	0.999426i	$0.489210\pi$
$198$	0	0
$199$	−2092.79	−0.745497	−0.372748	−	0.927932i	$-0.621585\pi$
$199$	−2092.79	−0.745497	−0.372748	+	0.927932i	$0.621585\pi$
$200$	1799.43	0.636193
$201$	0	0
$202$	503.382	0.175336
$203$	7.94164	0.00274578
$204$	0	0
$205$	−1769.95	−0.603016
$206$	−1211.83	−0.409866
$207$	0	0
$208$	−370.868	−0.123630
$209$	2551.96	0.844607
$210$	0	0
$211$	4521.06	1.47508	0.737541	−	0.675302i	$-0.235987\pi$
$211$	4521.06	1.47508	0.737541	+	0.675302i	$0.235987\pi$
$212$	921.705	0.298599
$213$	0	0
$214$	2110.76	0.674245
$215$	−491.872	−0.156025
$216$	0	0
$217$	−8036.98	−2.51422
$218$	−1609.29	−0.499975
$219$	0	0
$220$	−590.389	−0.180927
$221$	1393.02	0.424003
$222$	0	0
$223$	5184.60	1.55689	0.778445	−	0.627713i	$-0.216009\pi$
$223$	5184.60	1.55689	0.778445	+	0.627713i	$0.216009\pi$
$224$	4646.23	1.38589
$225$	0	0
$226$	1190.95	0.350533
$227$	−1499.77	−0.438517	−0.219258	−	0.975667i	$-0.570364\pi$
$227$	−1499.77	−0.438517	−0.219258	+	0.975667i	$0.570364\pi$
$228$	0	0
$229$	1152.48	0.332569	0.166284	−	0.986078i	$-0.446823\pi$
$229$	1152.48	0.332569	0.166284	+	0.986078i	$0.446823\pi$
$230$	94.9976	0.0272346
$231$	0	0
$232$	−4.71750	−0.00133500
$233$	−5721.06	−1.60858	−0.804290	−	0.594238i	$-0.797454\pi$
$233$	−5721.06	−1.60858	−0.804290	+	0.594238i	$0.797454\pi$
$234$	0	0
$235$	868.473	0.241076
$236$	1671.56	0.461055
$237$	0	0
$238$	−4117.59	−1.12144
$239$	2653.57	0.718182	0.359091	−	0.933303i	$-0.383087\pi$
$239$	2653.57	0.718182	0.359091	+	0.933303i	$0.383087\pi$
$240$	0	0
$241$	−660.897	−0.176648	−0.0883239	−	0.996092i	$-0.528151\pi$
$241$	−660.897	−0.176648	−0.0883239	+	0.996092i	$0.528151\pi$
$242$	−876.003	−0.232692
$243$	0	0
$244$	2437.07	0.639417
$245$	−1520.78	−0.396568
$246$	0	0
$247$	1121.89	0.289005
$248$	4774.13	1.22241
$249$	0	0
$250$	974.064	0.246421
$251$	−7162.32	−1.80112	−0.900561	−	0.434730i	$-0.856844\pi$
$251$	−7162.32	−1.80112	−0.900561	+	0.434730i	$0.856844\pi$
$252$	0	0
$253$	530.836	0.131911
$254$	848.466	0.209597
$255$	0	0
$256$	−772.728	−0.188654
$257$	−6388.77	−1.55066	−0.775332	−	0.631554i	$-0.782418\pi$
$257$	−6388.77	−1.55066	−0.775332	+	0.631554i	$0.782418\pi$
$258$	0	0
$259$	6305.14	1.51267
$260$	−259.547	−0.0619092
$261$	0	0
$262$	−1465.32	−0.345525
$263$	−1166.14	−0.273411	−0.136705	−	0.990612i	$-0.543651\pi$
$263$	−1166.14	−0.273411	−0.136705	+	0.990612i	$0.543651\pi$
$264$	0	0
$265$	510.506	0.118340
$266$	−3316.17	−0.764389
$267$	0	0
$268$	−2427.00	−0.553180
$269$	7527.35	1.70614	0.853068	−	0.521801i	$-0.174739\pi$
$269$	7527.35	1.70614	0.853068	+	0.521801i	$0.174739\pi$
$270$	0	0
$271$	−2105.48	−0.471951	−0.235975	−	0.971759i	$-0.575828\pi$
$271$	−2105.48	−0.471951	−0.235975	+	0.971759i	$0.575828\pi$
$272$	−5018.31	−1.11868
$273$	0	0
$274$	2339.95	0.515918
$275$	2557.98	0.560916
$276$	0	0
$277$	−2012.19	−0.436466	−0.218233	−	0.975897i	$-0.570029\pi$
$277$	−2012.19	−0.436466	−0.218233	+	0.975897i	$0.570029\pi$
$278$	1854.80	0.400156
$279$	0	0
$280$	1670.30	0.356499
$281$	−7474.36	−1.58677	−0.793386	−	0.608719i	$-0.791684\pi$
$281$	−7474.36	−1.58677	−0.793386	+	0.608719i	$0.791684\pi$
$282$	0	0
$283$	529.468	0.111214	0.0556071	−	0.998453i	$-0.482291\pi$
$283$	529.468	0.111214	0.0556071	+	0.998453i	$0.482291\pi$
$284$	−2317.39	−0.484196
$285$	0	0
$286$	256.963	0.0531277
$287$	12852.0	2.64331
$288$	0	0
$289$	13936.3	2.83662
$290$	−1.20013	−0.000243013 0
$291$	0	0
$292$	4928.25	0.987686
$293$	6974.64	1.39066	0.695329	−	0.718692i	$-0.255259\pi$
$293$	6974.64	1.39066	0.695329	+	0.718692i	$0.255259\pi$
$294$	0	0
$295$	925.828	0.182725
$296$	−3745.39	−0.735460
$297$	0	0
$298$	−2911.81	−0.566029
$299$	233.366	0.0451368
$300$	0	0
$301$	3571.60	0.683932
$302$	−536.499	−0.102225
$303$	0	0
$304$	−4041.58	−0.762502
$305$	1349.83	0.253413
$306$	0	0
$307$	7715.11	1.43428	0.717141	−	0.696928i	$-0.245450\pi$
$307$	7715.11	1.43428	0.717141	+	0.696928i	$0.245450\pi$
$308$	4286.95	0.793090
$309$	0	0
$310$	1214.54	0.222519
$311$	7546.75	1.37600	0.688002	−	0.725709i	$-0.258488\pi$
$311$	7546.75	1.37600	0.688002	+	0.725709i	$0.258488\pi$
$312$	0	0
$313$	−634.350	−0.114555	−0.0572773	−	0.998358i	$-0.518242\pi$
$313$	−634.350	−0.114555	−0.0572773	+	0.998358i	$0.518242\pi$
$314$	2088.98	0.375439
$315$	0	0
$316$	277.819	0.0494573
$317$	−5791.00	−1.02604	−0.513020	−	0.858377i	$-0.671473\pi$
$317$	−5791.00	−1.02604	−0.513020	+	0.858377i	$0.671473\pi$
$318$	0	0
$319$	−6.70618	−0.00117703
$320$	398.540	0.0696220
$321$	0	0
$322$	−689.800	−0.119382
$323$	15180.6	2.61508
$324$	0	0
$325$	1124.54	0.191933
$326$	889.067	0.151046
$327$	0	0
$328$	−7634.35	−1.28517
$329$	−6306.18	−1.05675
$330$	0	0
$331$	1071.45	0.177922	0.0889609	−	0.996035i	$-0.471645\pi$
$331$	1071.45	0.177922	0.0889609	+	0.996035i	$0.471645\pi$
$332$	3121.29	0.515973
$333$	0	0
$334$	−531.069	−0.0870024
$335$	−1344.24	−0.219236
$336$	0	0
$337$	788.327	0.127427	0.0637135	−	0.997968i	$-0.479706\pi$
$337$	788.327	0.127427	0.0637135	+	0.997968i	$0.479706\pi$
$338$	−2297.81	−0.369777
$339$	0	0
$340$	−3511.99	−0.560190
$341$	6786.68	1.07777
$342$	0	0
$343$	1667.95	0.262568
$344$	−2121.60	−0.332527
$345$	0	0
$346$	1328.79	0.206464
$347$	2677.65	0.414247	0.207123	−	0.978315i	$-0.433590\pi$
$347$	2677.65	0.414247	0.207123	+	0.978315i	$0.433590\pi$
$348$	0	0
$349$	−9605.93	−1.47333	−0.736667	−	0.676255i	$-0.763602\pi$
$349$	−9605.93	−1.47333	−0.736667	+	0.676255i	$0.763602\pi$
$350$	−3323.99	−0.507642
$351$	0	0
$352$	−3923.43	−0.594089
$353$	−6876.00	−1.03675	−0.518375	−	0.855153i	$-0.673463\pi$
$353$	−6876.00	−1.03675	−0.518375	+	0.855153i	$0.673463\pi$
$354$	0	0
$355$	−1283.54	−0.191896
$356$	−2190.07	−0.326048
$357$	0	0
$358$	1020.39	0.150640
$359$	−6330.89	−0.930728	−0.465364	−	0.885119i	$-0.654077\pi$
$359$	−6330.89	−0.930728	−0.465364	+	0.885119i	$0.654077\pi$
$360$	0	0
$361$	5366.96	0.782469
$362$	−3346.73	−0.485913
$363$	0	0
$364$	1884.63	0.271378
$365$	2729.62	0.391438
$366$	0	0
$367$	−9216.63	−1.31091	−0.655455	−	0.755234i	$-0.727523\pi$
$367$	−9216.63	−1.31091	−0.655455	+	0.755234i	$0.727523\pi$
$368$	−840.694	−0.119087
$369$	0	0
$370$	−952.823	−0.133878
$371$	−3706.91	−0.518741
$372$	0	0
$373$	−1852.00	−0.257086	−0.128543	−	0.991704i	$-0.541030\pi$
$373$	−1852.00	−0.257086	−0.128543	+	0.991704i	$0.541030\pi$
$374$	3477.02	0.480729
$375$	0	0
$376$	3746.01	0.513791
$377$	−2.94817	−0.000402754 0
$378$	0	0
$379$	53.5867	0.00726270	0.00363135	−	0.999993i	$-0.498844\pi$
$379$	53.5867	0.00726270	0.00363135	+	0.999993i	$0.498844\pi$
$380$	−2828.44	−0.381832
$381$	0	0
$382$	−4866.22	−0.651774
$383$	7178.87	0.957762	0.478881	−	0.877880i	$-0.341042\pi$
$383$	7178.87	0.957762	0.478881	+	0.877880i	$0.341042\pi$
$384$	0	0
$385$	2374.42	0.314316
$386$	−3322.75	−0.438144
$387$	0	0
$388$	−8148.06	−1.06612
$389$	4878.46	0.635856	0.317928	−	0.948115i	$-0.397013\pi$
$389$	4878.46	0.635856	0.317928	+	0.948115i	$0.397013\pi$
$390$	0	0
$391$	3157.73	0.408423
$392$	−6559.62	−0.845181
$393$	0	0
$394$	205.652	0.0262959
$395$	153.876	0.0196008
$396$	0	0
$397$	5122.65	0.647603	0.323802	−	0.946125i	$-0.395039\pi$
$397$	5122.65	0.647603	0.323802	+	0.946125i	$0.395039\pi$
$398$	−2296.43	−0.289220
$399$	0	0
$400$	−4051.11	−0.506389
$401$	−3824.52	−0.476278	−0.238139	−	0.971231i	$-0.576537\pi$
$401$	−3824.52	−0.476278	−0.238139	+	0.971231i	$0.576537\pi$
$402$	0	0
$403$	2983.56	0.368788
$404$	−3117.58	−0.383925
$405$	0	0
$406$	8.71441	0.00106524
$407$	−5324.27	−0.648438
$408$	0	0
$409$	−7367.31	−0.890685	−0.445342	−	0.895360i	$-0.646918\pi$
$409$	−7367.31	−0.890685	−0.445342	+	0.895360i	$0.646918\pi$
$410$	−1942.17	−0.233944
$411$	0	0
$412$	7505.22	0.897465
$413$	−6722.65	−0.800969
$414$	0	0
$415$	1728.79	0.204489
$416$	−1724.82	−0.203284
$417$	0	0
$418$	2800.28	0.327670
$419$	−2125.12	−0.247778	−0.123889	−	0.992296i	$-0.539537\pi$
$419$	−2125.12	−0.247778	−0.123889	+	0.992296i	$0.539537\pi$
$420$	0	0
$421$	126.465	0.0146402	0.00732012	−	0.999973i	$-0.497670\pi$
$421$	126.465	0.0146402	0.00732012	+	0.999973i	$0.497670\pi$
$422$	4960.98	0.572267
$423$	0	0
$424$	2201.98	0.252211
$425$	15216.4	1.73672
$426$	0	0
$427$	−9801.41	−1.11083
$428$	−13072.5	−1.47636
$429$	0	0
$430$	−539.734	−0.0605309
$431$	10175.9	1.13726	0.568628	−	0.822595i	$-0.307474\pi$
$431$	10175.9	1.13726	0.568628	+	0.822595i	$0.307474\pi$
$432$	0	0
$433$	−13812.1	−1.53295	−0.766474	−	0.642275i	$-0.777991\pi$
$433$	−13812.1	−1.53295	−0.766474	+	0.642275i	$0.777991\pi$
$434$	−8819.02	−0.975407
$435$	0	0
$436$	9966.76	1.09477
$437$	2543.13	0.278386
$438$	0	0
$439$	4417.32	0.480245	0.240122	−	0.970743i	$-0.422812\pi$
$439$	4417.32	0.480245	0.240122	+	0.970743i	$0.422812\pi$
$440$	−1410.46	−0.152820
$441$	0	0
$442$	1528.57	0.164495
$443$	−8426.02	−0.903685	−0.451842	−	0.892098i	$-0.649233\pi$
$443$	−8426.02	−0.903685	−0.451842	+	0.892098i	$0.649233\pi$
$444$	0	0
$445$	−1213.02	−0.129219
$446$	5689.09	0.604005
$447$	0	0
$448$	−2893.89	−0.305186
$449$	5829.80	0.612751	0.306375	−	0.951911i	$-0.400884\pi$
$449$	5829.80	0.612751	0.306375	+	0.951911i	$0.400884\pi$
$450$	0	0
$451$	−10852.6	−1.13311
$452$	−7375.86	−0.767547
$453$	0	0
$454$	−1645.71	−0.170125
$455$	1043.84	0.107552
$456$	0	0
$457$	−989.909	−0.101326	−0.0506630	−	0.998716i	$-0.516133\pi$
$457$	−989.909	−0.101326	−0.0506630	+	0.998716i	$0.516133\pi$
$458$	1264.63	0.129022
$459$	0	0
$460$	−588.347	−0.0596344
$461$	−18559.3	−1.87504	−0.937519	−	0.347935i	$-0.886883\pi$
$461$	−18559.3	−1.87504	−0.937519	+	0.347935i	$0.886883\pi$
$462$	0	0
$463$	−7161.29	−0.718819	−0.359410	−	0.933180i	$-0.617022\pi$
$463$	−7161.29	−0.718819	−0.359410	+	0.933180i	$0.617022\pi$
$464$	10.6207	0.00106261
$465$	0	0
$466$	−6277.75	−0.624058
$467$	−15859.9	−1.57154	−0.785771	−	0.618517i	$-0.787734\pi$
$467$	−15859.9	−1.57154	−0.785771	+	0.618517i	$0.787734\pi$
$468$	0	0
$469$	9760.87	0.961013
$470$	952.980	0.0935270
$471$	0	0
$472$	3993.40	0.389430
$473$	−3015.97	−0.293181
$474$	0	0
$475$	12254.8	1.18376
$476$	25501.4	2.45558
$477$	0	0
$478$	2911.78	0.278623
$479$	−2275.07	−0.217015	−0.108508	−	0.994096i	$-0.534607\pi$
$479$	−2275.07	−0.217015	−0.108508	+	0.994096i	$0.534607\pi$
$480$	0	0
$481$	−2340.65	−0.221881
$482$	−725.206	−0.0685316
$483$	0	0
$484$	5425.33	0.509516
$485$	−4512.98	−0.422523
$486$	0	0
$487$	−6758.12	−0.628829	−0.314415	−	0.949286i	$-0.601808\pi$
$487$	−6758.12	−0.628829	−0.314415	+	0.949286i	$0.601808\pi$
$488$	5822.24	0.540083
$489$	0	0
$490$	−1668.76	−0.153851
$491$	4825.23	0.443502	0.221751	−	0.975103i	$-0.428823\pi$
$491$	4825.23	0.443502	0.221751	+	0.975103i	$0.428823\pi$
$492$	0	0
$493$	−39.8924	−0.00364435
$494$	1231.06	0.112121
$495$	0	0
$496$	−10748.2	−0.972999
$497$	9320.06	0.841171
$498$	0	0
$499$	4729.06	0.424252	0.212126	−	0.977242i	$-0.431961\pi$
$499$	4729.06	0.424252	0.212126	+	0.977242i	$0.431961\pi$
$500$	−6032.65	−0.539577
$501$	0	0
$502$	−7859.25	−0.698756
$503$	−3782.85	−0.335326	−0.167663	−	0.985844i	$-0.553622\pi$
$503$	−3782.85	−0.335326	−0.167663	+	0.985844i	$0.553622\pi$
$504$	0	0
$505$	−1726.74	−0.152156
$506$	582.489	0.0511755
$507$	0	0
$508$	−5254.79	−0.458944
$509$	20133.0	1.75320	0.876602	−	0.481216i	$-0.159805\pi$
$509$	20133.0	1.75320	0.876602	+	0.481216i	$0.159805\pi$
$510$	0	0
$511$	−19820.4	−1.71586
$512$	10961.1	0.946130
$513$	0	0
$514$	−7010.44	−0.601590
$515$	4156.93	0.355682
$516$	0	0
$517$	5325.14	0.452997
$518$	6918.67	0.586851
$519$	0	0
$520$	−620.064	−0.0522916
$521$	−10307.1	−0.866719	−0.433360	−	0.901221i	$-0.642672\pi$
$521$	−10307.1	−0.866719	−0.433360	+	0.901221i	$0.642672\pi$
$522$	0	0
$523$	11637.8	0.973016	0.486508	−	0.873676i	$-0.338270\pi$
$523$	11637.8	0.973016	0.486508	+	0.873676i	$0.338270\pi$
$524$	9075.11	0.756580
$525$	0	0
$526$	−1279.61	−0.106071
$527$	40371.3	3.33700
$528$	0	0
$529$	529.000	0.0434783
$530$	560.182	0.0459108
$531$	0	0
$532$	20538.0	1.67375
$533$	−4771.03	−0.387723
$534$	0	0
$535$	−7240.49	−0.585110
$536$	−5798.16	−0.467243
$537$	0	0
$538$	8259.80	0.661906
$539$	−9324.85	−0.745176
$540$	0	0
$541$	12440.2	0.988622	0.494311	−	0.869285i	$-0.335420\pi$
$541$	12440.2	0.988622	0.494311	+	0.869285i	$0.335420\pi$
$542$	−2310.35	−0.183096
$543$	0	0
$544$	−23338.9	−1.83943
$545$	5520.31	0.433879
$546$	0	0
$547$	24086.8	1.88277	0.941386	−	0.337331i	$-0.109524\pi$
$547$	24086.8	1.88277	0.941386	+	0.337331i	$0.109524\pi$
$548$	−14492.0	−1.12968
$549$	0	0
$550$	2806.88	0.217611
$551$	−32.1280	−0.00248403
$552$	0	0
$553$	−1117.33	−0.0859198
$554$	−2207.99	−0.169330
$555$	0	0
$556$	−11487.3	−0.876204
$557$	−23520.8	−1.78924	−0.894620	−	0.446828i	$-0.852554\pi$
$557$	−23520.8	−1.78924	−0.894620	+	0.446828i	$0.852554\pi$
$558$	0	0
$559$	−1325.88	−0.100320
$560$	−3760.41	−0.283761
$561$	0	0
$562$	−8201.66	−0.615598
$563$	−17852.3	−1.33639	−0.668193	−	0.743988i	$-0.732932\pi$
$563$	−17852.3	−1.33639	−0.668193	+	0.743988i	$0.732932\pi$
$564$	0	0
$565$	−4085.28	−0.304193
$566$	580.989	0.0431462
$567$	0	0
$568$	−5536.31	−0.408976
$569$	690.384	0.0508654	0.0254327	−	0.999677i	$-0.491904\pi$
$569$	690.384	0.0508654	0.0254327	+	0.999677i	$0.491904\pi$
$570$	0	0
$571$	18770.3	1.37568	0.687839	−	0.725864i	$-0.258560\pi$
$571$	18770.3	1.37568	0.687839	+	0.725864i	$0.258560\pi$
$572$	−1591.44	−0.116331
$573$	0	0
$574$	14102.6	1.02549
$575$	2549.13	0.184880
$576$	0	0
$577$	−20267.5	−1.46230	−0.731149	−	0.682218i	$-0.761016\pi$
$577$	−20267.5	−1.46230	−0.731149	+	0.682218i	$0.761016\pi$
$578$	15292.4	1.10048
$579$	0	0
$580$	7.43272	0.000532116 0
$581$	−12553.2	−0.896374
$582$	0	0
$583$	3130.23	0.222369
$584$	11773.7	0.834248
$585$	0	0
$586$	7653.31	0.539514
$587$	14385.9	1.01153	0.505765	−	0.862671i	$-0.331210\pi$
$587$	14385.9	1.01153	0.505765	+	0.862671i	$0.331210\pi$
$588$	0	0
$589$	32513.7	2.27454
$590$	1015.92	0.0708892
$591$	0	0
$592$	8432.13	0.585402
$593$	−22603.3	−1.56527	−0.782636	−	0.622480i	$-0.786125\pi$
$593$	−22603.3	−1.56527	−0.782636	+	0.622480i	$0.786125\pi$
$594$	0	0
$595$	14124.5	0.973190
$596$	18033.7	1.23941
$597$	0	0
$598$	256.074	0.0175111
$599$	−1664.25	−0.113521	−0.0567606	−	0.998388i	$-0.518077\pi$
$599$	−1664.25	−0.113521	−0.0567606	+	0.998388i	$0.518077\pi$
$600$	0	0
$601$	−15033.5	−1.02035	−0.510174	−	0.860071i	$-0.670419\pi$
$601$	−15033.5	−1.02035	−0.510174	+	0.860071i	$0.670419\pi$
$602$	3919.13	0.265336
$603$	0	0
$604$	3322.69	0.223838
$605$	3004.94	0.201931
$606$	0	0
$607$	5597.56	0.374296	0.187148	−	0.982332i	$-0.440076\pi$
$607$	5597.56	0.374296	0.187148	+	0.982332i	$0.440076\pi$
$608$	−18796.4	−1.25377
$609$	0	0
$610$	1481.17	0.0983130
$611$	2341.04	0.155005
$612$	0	0
$613$	−567.595	−0.0373980	−0.0186990	−	0.999825i	$-0.505952\pi$
$613$	−567.595	−0.0373980	−0.0186990	+	0.999825i	$0.505952\pi$
$614$	8465.84	0.556439
$615$	0	0
$616$	10241.7	0.669883
$617$	−13840.0	−0.903044	−0.451522	−	0.892260i	$-0.649119\pi$
$617$	−13840.0	−0.903044	−0.451522	+	0.892260i	$0.649119\pi$
$618$	0	0
$619$	563.304	0.0365769	0.0182885	−	0.999833i	$-0.494178\pi$
$619$	563.304	0.0365769	0.0182885	+	0.999833i	$0.494178\pi$
$620$	−7521.96	−0.487240
$621$	0	0
$622$	8281.10	0.533829
$623$	8807.99	0.566428
$624$	0	0
$625$	10512.7	0.672810
$626$	−696.076	−0.0444422
$627$	0	0
$628$	−12937.6	−0.822082
$629$	−31671.9	−2.00770
$630$	0	0
$631$	2274.68	0.143508	0.0717540	−	0.997422i	$-0.477140\pi$
$631$	2274.68	0.143508	0.0717540	+	0.997422i	$0.477140\pi$
$632$	663.716	0.0417741
$633$	0	0
$634$	−6354.49	−0.398059
$635$	−2910.48	−0.181888
$636$	0	0
$637$	−4099.39	−0.254982
$638$	−7.35872	−0.000456637 0
$639$	0	0
$640$	5556.26	0.343173
$641$	5401.53	0.332836	0.166418	−	0.986055i	$-0.446780\pi$
$641$	5401.53	0.332836	0.166418	+	0.986055i	$0.446780\pi$
$642$	0	0
$643$	6504.37	0.398922	0.199461	−	0.979906i	$-0.436081\pi$
$643$	6504.37	0.398922	0.199461	+	0.979906i	$0.436081\pi$
$644$	4272.13	0.261406
$645$	0	0
$646$	16657.8	1.01454
$647$	25795.0	1.56740	0.783699	−	0.621141i	$-0.213331\pi$
$647$	25795.0	1.56740	0.783699	+	0.621141i	$0.213331\pi$
$648$	0	0
$649$	5676.82	0.343351
$650$	1233.96	0.0744615
$651$	0	0
$652$	−5506.24	−0.330738
$653$	−2258.17	−0.135328	−0.0676639	−	0.997708i	$-0.521555\pi$
$653$	−2258.17	−0.135328	−0.0676639	+	0.997708i	$0.521555\pi$
$654$	0	0
$655$	5026.45	0.299847
$656$	17187.5	1.02296
$657$	0	0
$658$	−6919.81	−0.409973
$659$	14039.1	0.829874	0.414937	−	0.909850i	$-0.363804\pi$
$659$	14039.1	0.829874	0.414937	+	0.909850i	$0.363804\pi$
$660$	0	0
$661$	−16239.0	−0.955555	−0.477778	−	0.878481i	$-0.658558\pi$
$661$	−16239.0	−0.955555	−0.477778	+	0.878481i	$0.658558\pi$
$662$	1175.71	0.0690258
$663$	0	0
$664$	7456.85	0.435816
$665$	11375.4	0.663337
$666$	0	0
$667$	−6.68298	−0.000387955 0
$668$	3289.06	0.190505
$669$	0	0
$670$	−1475.05	−0.0850538
$671$	8276.62	0.476178
$672$	0	0
$673$	23650.9	1.35464	0.677322	−	0.735687i	$-0.263140\pi$
$673$	23650.9	1.35464	0.677322	+	0.735687i	$0.263140\pi$
$674$	865.036	0.0494361
$675$	0	0
$676$	14231.0	0.809684
$677$	−8737.86	−0.496046	−0.248023	−	0.968754i	$-0.579781\pi$
$677$	−8737.86	−0.496046	−0.248023	+	0.968754i	$0.579781\pi$
$678$	0	0
$679$	32769.8	1.85212
$680$	−8390.24	−0.473164
$681$	0	0
$682$	7447.07	0.418127
$683$	2416.73	0.135393	0.0676967	−	0.997706i	$-0.478435\pi$
$683$	2416.73	0.135393	0.0676967	+	0.997706i	$0.478435\pi$
$684$	0	0
$685$	−8026.69	−0.447714
$686$	1830.25	0.101865
$687$	0	0
$688$	4776.44	0.264681
$689$	1376.11	0.0760895
$690$	0	0
$691$	−29291.4	−1.61259	−0.806293	−	0.591517i	$-0.798529\pi$
$691$	−29291.4	−1.61259	−0.806293	+	0.591517i	$0.798529\pi$
$692$	−8229.59	−0.452084
$693$	0	0
$694$	2938.20	0.160710
$695$	−6362.48	−0.347256
$696$	0	0
$697$	−64558.0	−3.50833
$698$	−10540.6	−0.571589
$699$	0	0
$700$	20586.4	1.11156
$701$	1756.30	0.0946285	0.0473142	−	0.998880i	$-0.484934\pi$
$701$	1756.30	0.0946285	0.0473142	+	0.998880i	$0.484934\pi$
$702$	0	0
$703$	−25507.5	−1.36847
$704$	2443.69	0.130824
$705$	0	0
$706$	−7545.08	−0.402213
$707$	12538.3	0.666974
$708$	0	0
$709$	−29509.3	−1.56311	−0.781554	−	0.623837i	$-0.785573\pi$
$709$	−29509.3	−1.56311	−0.781554	+	0.623837i	$0.785573\pi$
$710$	−1408.43	−0.0744472
$711$	0	0
$712$	−5232.13	−0.275397
$713$	6763.21	0.355237
$714$	0	0
$715$	−881.454	−0.0461042
$716$	−6319.55	−0.329850
$717$	0	0
$718$	−6946.92	−0.361082
$719$	−26137.7	−1.35573	−0.677866	−	0.735185i	$-0.737095\pi$
$719$	−26137.7	−1.35573	−0.677866	+	0.735185i	$0.737095\pi$
$720$	0	0
$721$	−30184.4	−1.55912
$722$	5889.19	0.303564
$723$	0	0
$724$	20727.3	1.06398
$725$	−32.2038	−0.00164968
$726$	0	0
$727$	−31695.0	−1.61692	−0.808462	−	0.588549i	$-0.799700\pi$
$727$	−31695.0	−1.61692	−0.808462	+	0.588549i	$0.799700\pi$
$728$	4502.43	0.229219
$729$	0	0
$730$	2995.23	0.151861
$731$	−17940.8	−0.907750
$732$	0	0
$733$	−19079.0	−0.961388	−0.480694	−	0.876888i	$-0.659615\pi$
$733$	−19079.0	−0.961388	−0.480694	+	0.876888i	$0.659615\pi$
$734$	−10113.5	−0.508576
$735$	0	0
$736$	−3909.86	−0.195814
$737$	−8242.39	−0.411957
$738$	0	0
$739$	−17190.5	−0.855699	−0.427849	−	0.903850i	$-0.640729\pi$
$739$	−17190.5	−0.855699	−0.427849	+	0.903850i	$0.640729\pi$
$740$	5901.10	0.293147
$741$	0	0
$742$	−4067.61	−0.201249
$743$	−15471.5	−0.763922	−0.381961	−	0.924178i	$-0.624751\pi$
$743$	−15471.5	−0.763922	−0.381961	+	0.924178i	$0.624751\pi$
$744$	0	0
$745$	9988.33	0.491200
$746$	−2032.21	−0.0997379
$747$	0	0
$748$	−21534.2	−1.05263
$749$	52574.9	2.56481
$750$	0	0
$751$	16443.6	0.798984	0.399492	−	0.916737i	$-0.369187\pi$
$751$	16443.6	0.798984	0.399492	+	0.916737i	$0.369187\pi$
$752$	−8433.52	−0.408961
$753$	0	0
$754$	−3.23504	−0.000156251 0
$755$	1840.34	0.0887111
$756$	0	0
$757$	6296.77	0.302325	0.151162	−	0.988509i	$-0.451698\pi$
$757$	6296.77	0.302325	0.151162	+	0.988509i	$0.451698\pi$
$758$	58.8010	0.00281761
$759$	0	0
$760$	−6757.23	−0.322514
$761$	35631.9	1.69731	0.848656	−	0.528946i	$-0.177412\pi$
$761$	35631.9	1.69731	0.848656	+	0.528946i	$0.177412\pi$
$762$	0	0
$763$	−40084.2	−1.90190
$764$	30137.9	1.42716
$765$	0	0
$766$	7877.41	0.371570
$767$	2495.64	0.117487
$768$	0	0
$769$	14119.9	0.662130	0.331065	−	0.943608i	$-0.392592\pi$
$769$	14119.9	0.662130	0.331065	+	0.943608i	$0.392592\pi$
$770$	2605.47	0.121941
$771$	0	0
$772$	20578.7	0.959384
$773$	−30634.9	−1.42544	−0.712718	−	0.701451i	$-0.752536\pi$
$773$	−30634.9	−1.42544	−0.712718	+	0.701451i	$0.752536\pi$
$774$	0	0
$775$	32590.4	1.51056
$776$	−19465.9	−0.900498
$777$	0	0
$778$	5353.16	0.246684
$779$	−51992.9	−2.39132
$780$	0	0
$781$	−7870.16	−0.360584
$782$	3465.00	0.158450
$783$	0	0
$784$	14767.9	0.672737
$785$	−7165.78	−0.325806
$786$	0	0
$787$	40207.8	1.82116	0.910579	−	0.413334i	$-0.135636\pi$
$787$	40207.8	1.82116	0.910579	+	0.413334i	$0.135636\pi$
$788$	−1273.66	−0.0575790
$789$	0	0
$790$	168.849	0.00760427
$791$	29664.2	1.33342
$792$	0	0
$793$	3638.57	0.162937
$794$	5621.11	0.251242
$795$	0	0
$796$	14222.4	0.633292
$797$	−7300.18	−0.324449	−0.162224	−	0.986754i	$-0.551867\pi$
$797$	−7300.18	−0.324449	−0.162224	+	0.986754i	$0.551867\pi$
$798$	0	0
$799$	31677.2	1.40258
$800$	−18840.7	−0.832650
$801$	0	0
$802$	−4196.67	−0.184775
$803$	16737.0	0.735536
$804$	0	0
$805$	2366.21	0.103600
$806$	3273.88	0.143074
$807$	0	0
$808$	−7448.00	−0.324282
$809$	−409.938	−0.0178154	−0.00890770	−	0.999960i	$-0.502835\pi$
$809$	−409.938	−0.0178154	−0.00890770	+	0.999960i	$0.502835\pi$
$810$	0	0
$811$	−7855.15	−0.340113	−0.170057	−	0.985434i	$-0.554395\pi$
$811$	−7855.15	−0.340113	−0.170057	+	0.985434i	$0.554395\pi$
$812$	−53.9708	−0.00233252
$813$	0	0
$814$	−5842.35	−0.251565
$815$	−3049.75	−0.131077
$816$	0	0
$817$	−14448.9	−0.618733
$818$	−8084.19	−0.345547
$819$	0	0
$820$	12028.4	0.512256
$821$	33376.2	1.41880	0.709402	−	0.704804i	$-0.248965\pi$
$821$	33376.2	1.41880	0.709402	+	0.704804i	$0.248965\pi$
$822$	0	0
$823$	−25837.5	−1.09434	−0.547168	−	0.837023i	$-0.684294\pi$
$823$	−25837.5	−1.09434	−0.547168	+	0.837023i	$0.684294\pi$
$824$	17930.2	0.758043
$825$	0	0
$826$	−7376.80	−0.310741
$827$	−33346.5	−1.40214	−0.701071	−	0.713092i	$-0.747294\pi$
$827$	−33346.5	−1.40214	−0.701071	+	0.713092i	$0.747294\pi$
$828$	0	0
$829$	2183.56	0.0914815	0.0457407	−	0.998953i	$-0.485435\pi$
$829$	2183.56	0.0914815	0.0457407	+	0.998953i	$0.485435\pi$
$830$	1897.01	0.0793329
$831$	0	0
$832$	1074.30	0.0447651
$833$	−55469.8	−2.30722
$834$	0	0
$835$	1821.72	0.0755007
$836$	−17342.9	−0.717485
$837$	0	0
$838$	−2331.91	−0.0961271
$839$	−3359.93	−0.138257	−0.0691285	−	0.997608i	$-0.522022\pi$
$839$	−3359.93	−0.138257	−0.0691285	+	0.997608i	$0.522022\pi$
$840$	0	0
$841$	−24388.9	−0.999997
$842$	138.771	0.00567977
$843$	0	0
$844$	−30724.8	−1.25307
$845$	7882.15	0.320893
$846$	0	0
$847$	−21819.5	−0.885158
$848$	−4957.40	−0.200752
$849$	0	0
$850$	16697.0	0.673769
$851$	−5305.85	−0.213728
$852$	0	0
$853$	28869.9	1.15883	0.579417	−	0.815031i	$-0.303280\pi$
$853$	28869.9	1.15883	0.579417	+	0.815031i	$0.303280\pi$
$854$	−10755.1	−0.430952
$855$	0	0
$856$	−31230.6	−1.24701
$857$	10042.2	0.400275	0.200138	−	0.979768i	$-0.435861\pi$
$857$	10042.2	0.400275	0.200138	+	0.979768i	$0.435861\pi$
$858$	0	0
$859$	3285.78	0.130511	0.0652557	−	0.997869i	$-0.479214\pi$
$859$	3285.78	0.130511	0.0652557	+	0.997869i	$0.479214\pi$
$860$	3342.72	0.132542
$861$	0	0
$862$	11166.1	0.441206
$863$	−25059.3	−0.988446	−0.494223	−	0.869335i	$-0.664547\pi$
$863$	−25059.3	−0.988446	−0.494223	+	0.869335i	$0.664547\pi$
$864$	0	0
$865$	−4558.14	−0.179169
$866$	−15156.1	−0.594716
$867$	0	0
$868$	54618.7	2.13581
$869$	943.508	0.0368312
$870$	0	0
$871$	−3623.52	−0.140962
$872$	23810.9	0.924699
$873$	0	0
$874$	2790.60	0.108001
$875$	24262.1	0.937380
$876$	0	0
$877$	−11283.6	−0.434459	−0.217230	−	0.976121i	$-0.569702\pi$
$877$	−11283.6	−0.434459	−0.217230	+	0.976121i	$0.569702\pi$
$878$	4847.15	0.186314
$879$	0	0
$880$	3175.41	0.121640
$881$	7757.23	0.296649	0.148324	−	0.988939i	$-0.452612\pi$
$881$	7757.23	0.296649	0.148324	+	0.988939i	$0.452612\pi$
$882$	0	0
$883$	−19730.0	−0.751947	−0.375973	−	0.926630i	$-0.622692\pi$
$883$	−19730.0	−0.751947	−0.375973	+	0.926630i	$0.622692\pi$
$884$	−9466.85	−0.360186
$885$	0	0
$886$	−9245.92	−0.350590
$887$	20784.1	0.786765	0.393383	−	0.919375i	$-0.371305\pi$
$887$	20784.1	0.786765	0.393383	+	0.919375i	$0.371305\pi$
$888$	0	0
$889$	21133.7	0.797301
$890$	−1331.05	−0.0501313
$891$	0	0
$892$	−35234.1	−1.32256
$893$	25511.7	0.956011
$894$	0	0
$895$	−3500.22	−0.130726
$896$	−40345.3	−1.50429
$897$	0	0
$898$	6397.07	0.237720
$899$	−85.4412	−0.00316977
$900$	0	0
$901$	18620.5	0.688500
$902$	−11908.7	−0.439595
$903$	0	0
$904$	−17621.1	−0.648308
$905$	11480.2	0.421675
$906$	0	0
$907$	28539.4	1.04480	0.522401	−	0.852700i	$-0.325036\pi$
$907$	28539.4	1.04480	0.522401	+	0.852700i	$0.325036\pi$
$908$	10192.3	0.372515
$909$	0	0
$910$	1145.41	0.0417254
$911$	−20277.9	−0.737472	−0.368736	−	0.929534i	$-0.620209\pi$
$911$	−20277.9	−0.737472	−0.368736	+	0.929534i	$0.620209\pi$
$912$	0	0
$913$	10600.3	0.384248
$914$	−1086.23	−0.0393101
$915$	0	0
$916$	−7832.19	−0.282514
$917$	−36498.2	−1.31437
$918$	0	0
$919$	−20829.3	−0.747654	−0.373827	−	0.927498i	$-0.621955\pi$
$919$	−20829.3	−0.747654	−0.373827	+	0.927498i	$0.621955\pi$
$920$	−1405.58	−0.0503701
$921$	0	0
$922$	−20365.2	−0.727432
$923$	−3459.88	−0.123384
$924$	0	0
$925$	−25567.7	−0.908822
$926$	−7858.12	−0.278870
$927$	0	0
$928$	49.3941	0.00174724
$929$	16551.7	0.584547	0.292273	−	0.956335i	$-0.405588\pi$
$929$	16551.7	0.584547	0.292273	+	0.956335i	$0.405588\pi$
$930$	0	0
$931$	−44673.6	−1.57263
$932$	38879.9	1.36647
$933$	0	0
$934$	−17403.2	−0.609690
$935$	−11927.2	−0.417177
$936$	0	0
$937$	−35576.9	−1.24039	−0.620195	−	0.784447i	$-0.712947\pi$
$937$	−35576.9	−1.24039	−0.620195	+	0.784447i	$0.712947\pi$
$938$	10710.7	0.372831
$939$	0	0
$940$	−5902.07	−0.204792
$941$	7372.81	0.255416	0.127708	−	0.991812i	$-0.459238\pi$
$941$	7372.81	0.255416	0.127708	+	0.991812i	$0.459238\pi$
$942$	0	0
$943$	−10815.1	−0.373476
$944$	−8990.48	−0.309974
$945$	0	0
$946$	−3309.44	−0.113741
$947$	3054.05	0.104798	0.0523988	−	0.998626i	$-0.483313\pi$
$947$	3054.05	0.104798	0.0523988	+	0.998626i	$0.483313\pi$
$948$	0	0
$949$	7357.91	0.251684
$950$	13447.2	0.459249
$951$	0	0
$952$	60923.5	2.07410
$953$	−18569.4	−0.631186	−0.315593	−	0.948895i	$-0.602203\pi$
$953$	−18569.4	−0.631186	−0.315593	+	0.948895i	$0.602203\pi$
$954$	0	0
$955$	16692.5	0.565610
$956$	−18033.5	−0.610088
$957$	0	0
$958$	−2496.44	−0.0841925
$959$	58283.7	1.96254
$960$	0	0
$961$	56676.0	1.90245
$962$	−2568.41	−0.0860799
$963$	0	0
$964$	4491.41	0.150061
$965$	11398.0	0.380221
$966$	0	0
$967$	48741.1	1.62090	0.810449	−	0.585809i	$-0.199223\pi$
$967$	48741.1	1.62090	0.810449	+	0.585809i	$0.199223\pi$
$968$	12961.3	0.430362
$969$	0	0
$970$	−4952.12	−0.163920
$971$	13441.8	0.444252	0.222126	−	0.975018i	$-0.428700\pi$
$971$	13441.8	0.444252	0.222126	+	0.975018i	$0.428700\pi$
$972$	0	0
$973$	46199.5	1.52219
$974$	−7415.73	−0.243958
$975$	0	0
$976$	−13107.8	−0.429888
$977$	5394.00	0.176632	0.0883159	−	0.996093i	$-0.471852\pi$
$977$	5394.00	0.176632	0.0883159	+	0.996093i	$0.471852\pi$
$978$	0	0
$979$	−7437.75	−0.242810
$980$	10335.1	0.336881
$981$	0	0
$982$	5294.75	0.172059
$983$	42903.5	1.39207	0.696037	−	0.718006i	$-0.254945\pi$
$983$	42903.5	1.39207	0.696037	+	0.718006i	$0.254945\pi$
$984$	0	0
$985$	−705.444	−0.0228196
$986$	−43.7741	−0.00141385
$987$	0	0
$988$	−7624.29	−0.245507
$989$	−3005.54	−0.0966336
$990$	0	0
$991$	52311.3	1.67681	0.838407	−	0.545045i	$-0.183487\pi$
$991$	52311.3	1.67681	0.838407	+	0.545045i	$0.183487\pi$
$992$	−49987.1	−1.59989
$993$	0	0
$994$	10227.0	0.326337
$995$	7877.40	0.250985
$996$	0	0
$997$	−28709.3	−0.911969	−0.455984	−	0.889988i	$-0.650713\pi$
$997$	−28709.3	−0.911969	−0.455984	+	0.889988i	$0.650713\pi$
$998$	5189.23	0.164591
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.d.1.3		4
3.2	odd	2		69.4.a.d.1.2	✓	4
12.11	even	2		1104.4.a.t.1.2		4
15.14	odd	2		1725.4.a.p.1.3		4
69.68	even	2		1587.4.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.d.1.2	✓	4	3.2	odd	2
207.4.a.d.1.3		4	1.1	even	1	trivial
1104.4.a.t.1.2		4	12.11	even	2
1587.4.a.g.1.2		4	69.68	even	2
1725.4.a.p.1.3		4	15.14	odd	2

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	207.4.a.d.1.3

Level	$207$
Weight	$4$
Character	207.1
Self dual	yes
Analytic conductor	$12.213$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$