LMFDB - Embedded newform 315.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$18.5856016518$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.70156 q^{2} +14.1047 q^{4} +5.00000 q^{5} +7.00000 q^{7} -28.7016 q^{8} +O(q^{10})$	\(q-4.70156 q^{2} +14.1047 q^{4} +5.00000 q^{5} +7.00000 q^{7} -28.7016 q^{8} -23.5078 q^{10} -24.5969 q^{11} -35.0156 q^{13} -32.9109 q^{14} +22.1047 q^{16} +18.4187 q^{17} -67.4031 q^{19} +70.5234 q^{20} +115.644 q^{22} +145.675 q^{23} +25.0000 q^{25} +164.628 q^{26} +98.7328 q^{28} -214.419 q^{29} -88.6594 q^{31} +125.686 q^{32} -86.5969 q^{34} +35.0000 q^{35} +162.125 q^{37} +316.900 q^{38} -143.508 q^{40} +337.769 q^{41} +122.156 q^{43} -346.931 q^{44} -684.900 q^{46} -354.219 q^{47} +49.0000 q^{49} -117.539 q^{50} -493.884 q^{52} -676.691 q^{53} -122.984 q^{55} -200.911 q^{56} +1008.10 q^{58} -501.319 q^{59} -708.931 q^{61} +416.837 q^{62} -767.758 q^{64} -175.078 q^{65} -907.956 q^{67} +259.791 q^{68} -164.555 q^{70} -430.334 q^{71} +41.3406 q^{73} -762.241 q^{74} -950.700 q^{76} -172.178 q^{77} +890.388 q^{79} +110.523 q^{80} -1588.04 q^{82} +1057.15 q^{83} +92.0937 q^{85} -574.325 q^{86} +705.969 q^{88} -1473.72 q^{89} -245.109 q^{91} +2054.70 q^{92} +1665.38 q^{94} -337.016 q^{95} +555.034 q^{97} -230.377 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 9 * q^4 + 10 * q^5 + 14 * q^7 - 51 * q^8	$ 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8} - 15 q^{10} - 62 q^{11} - 6 q^{13} - 21 q^{14} + 25 q^{16} - 40 q^{17} - 122 q^{19} + 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} + 214 q^{26} + 63 q^{28} - 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{34} + 70 q^{35} - 188 q^{37} + 224 q^{38} - 255 q^{40} - 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} + 188 q^{47} + 98 q^{49} - 75 q^{50} - 642 q^{52} - 982 q^{53} - 310 q^{55} - 357 q^{56} + 774 q^{58} - 516 q^{59} - 880 q^{61} + 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} + 558 q^{68} - 105 q^{70} - 310 q^{71} + 326 q^{73} - 1358 q^{74} - 672 q^{76} - 434 q^{77} + 1832 q^{79} + 125 q^{80} - 2190 q^{82} + 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} - 796 q^{89} - 42 q^{91} + 2880 q^{92} + 2588 q^{94} - 610 q^{95} - 670 q^{97} - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 9 * q^4 + 10 * q^5 + 14 * q^7 - 51 * q^8 - 15 * q^10 - 62 * q^11 - 6 * q^13 - 21 * q^14 + 25 * q^16 - 40 * q^17 - 122 * q^19 + 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 + 214 * q^26 + 63 * q^28 - 352 * q^29 + 66 * q^31 + 309 * q^32 - 186 * q^34 + 70 * q^35 - 188 * q^37 + 224 * q^38 - 255 * q^40 - 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 + 188 * q^47 + 98 * q^49 - 75 * q^50 - 642 * q^52 - 982 * q^53 - 310 * q^55 - 357 * q^56 + 774 * q^58 - 516 * q^59 - 880 * q^61 + 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 + 558 * q^68 - 105 * q^70 - 310 * q^71 + 326 * q^73 - 1358 * q^74 - 672 * q^76 - 434 * q^77 + 1832 * q^79 + 125 * q^80 - 2190 * q^82 + 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 - 796 * q^89 - 42 * q^91 + 2880 * q^92 + 2588 * q^94 - 610 * q^95 - 670 * q^97 - 147 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−4.70156	−1.66225	−0.831127	−	0.556083i	$-0.812304\pi$
$2$	−4.70156	−1.66225	−0.831127	+	0.556083i	$0.812304\pi$
$3$	0	0
$3$	0	0
$4$	14.1047	1.76309
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−28.7016	−1.26844
$9$	0	0
$10$	−23.5078	−0.743382
$11$	−24.5969	−0.674203	−0.337102	−	0.941468i	$-0.609447\pi$
$11$	−24.5969	−0.674203	−0.337102	+	0.941468i	$0.609447\pi$
$12$	0	0
$13$	−35.0156	−0.747045	−0.373523	−	0.927621i	$-0.621850\pi$
$13$	−35.0156	−0.747045	−0.373523	+	0.927621i	$0.621850\pi$
$14$	−32.9109	−0.628273
$15$	0	0
$16$	22.1047	0.345386
$17$	18.4187	0.262777	0.131388	−	0.991331i	$-0.458057\pi$
$17$	18.4187	0.262777	0.131388	+	0.991331i	$0.458057\pi$
$18$	0	0
$19$	−67.4031	−0.813860	−0.406930	−	0.913459i	$-0.633401\pi$
$19$	−67.4031	−0.813860	−0.406930	+	0.913459i	$0.633401\pi$
$20$	70.5234	0.788476
$21$	0	0
$22$	115.644	1.12070
$23$	145.675	1.32067	0.660333	−	0.750973i	$-0.270415\pi$
$23$	145.675	1.32067	0.660333	+	0.750973i	$0.270415\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	164.628	1.24178
$27$	0	0
$28$	98.7328	0.666384
$29$	−214.419	−1.37298	−0.686492	−	0.727137i	$-0.740851\pi$
$29$	−214.419	−1.37298	−0.686492	+	0.727137i	$0.740851\pi$
$30$	0	0
$31$	−88.6594	−0.513667	−0.256834	−	0.966456i	$-0.582679\pi$
$31$	−88.6594	−0.513667	−0.256834	+	0.966456i	$0.582679\pi$
$32$	125.686	0.694323
$33$	0	0
$34$	−86.5969	−0.436801
$35$	35.0000	0.169031
$36$	0	0
$37$	162.125	0.720356	0.360178	−	0.932884i	$-0.382716\pi$
$37$	162.125	0.720356	0.360178	+	0.932884i	$0.382716\pi$
$38$	316.900	1.35284
$39$	0	0
$40$	−143.508	−0.567264
$41$	337.769	1.28660	0.643300	−	0.765614i	$-0.277565\pi$
$41$	337.769	1.28660	0.643300	+	0.765614i	$0.277565\pi$
$42$	0	0
$43$	122.156	0.433224	0.216612	−	0.976258i	$-0.430499\pi$
$43$	122.156	0.433224	0.216612	+	0.976258i	$0.430499\pi$
$44$	−346.931	−1.18868
$45$	0	0
$46$	−684.900	−2.19528
$47$	−354.219	−1.09932	−0.549661	−	0.835388i	$-0.685243\pi$
$47$	−354.219	−1.09932	−0.549661	+	0.835388i	$0.685243\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−117.539	−0.332451
$51$	0	0
$52$	−493.884	−1.31710
$53$	−676.691	−1.75378	−0.876892	−	0.480687i	$-0.840387\pi$
$53$	−676.691	−1.75378	−0.876892	+	0.480687i	$0.840387\pi$
$54$	0	0
$55$	−122.984	−0.301513
$56$	−200.911	−0.479426
$57$	0	0
$58$	1008.10	2.28225
$59$	−501.319	−1.10621	−0.553103	−	0.833113i	$-0.686556\pi$
$59$	−501.319	−1.10621	−0.553103	+	0.833113i	$0.686556\pi$
$60$	0	0
$61$	−708.931	−1.48802	−0.744011	−	0.668167i	$-0.767079\pi$
$61$	−708.931	−1.48802	−0.744011	+	0.668167i	$0.767079\pi$
$62$	416.837	0.853845
$63$	0	0
$64$	−767.758	−1.49953
$65$	−175.078	−0.334089
$66$	0	0
$67$	−907.956	−1.65559	−0.827795	−	0.561031i	$-0.810405\pi$
$67$	−907.956	−1.65559	−0.827795	+	0.561031i	$0.810405\pi$
$68$	259.791	0.463298
$69$	0	0
$70$	−164.555	−0.280972
$71$	−430.334	−0.719314	−0.359657	−	0.933085i	$-0.617106\pi$
$71$	−430.334	−0.719314	−0.359657	+	0.933085i	$0.617106\pi$
$72$	0	0
$73$	41.3406	0.0662816	0.0331408	−	0.999451i	$-0.489449\pi$
$73$	41.3406	0.0662816	0.0331408	+	0.999451i	$0.489449\pi$
$74$	−762.241	−1.19741
$75$	0	0
$76$	−950.700	−1.43490
$77$	−172.178	−0.254825
$78$	0	0
$79$	890.388	1.26806	0.634028	−	0.773310i	$-0.281400\pi$
$79$	890.388	1.26806	0.634028	+	0.773310i	$0.281400\pi$
$80$	110.523	0.154461
$81$	0	0
$82$	−1588.04	−2.13866
$83$	1057.15	1.39804	0.699020	−	0.715102i	$-0.253620\pi$
$83$	1057.15	1.39804	0.699020	+	0.715102i	$0.253620\pi$
$84$	0	0
$85$	92.0937	0.117517
$86$	−574.325	−0.720129
$87$	0	0
$88$	705.969	0.855188
$89$	−1473.72	−1.75522	−0.877610	−	0.479376i	$-0.840863\pi$
$89$	−1473.72	−1.75522	−0.877610	+	0.479376i	$0.840863\pi$
$90$	0	0
$91$	−245.109	−0.282356
$92$	2054.70	2.32845
$93$	0	0
$94$	1665.38	1.82735
$95$	−337.016	−0.363969
$96$	0	0
$97$	555.034	0.580981	0.290491	−	0.956878i	$-0.406181\pi$
$97$	555.034	0.580981	0.290491	+	0.956878i	$0.406181\pi$
$98$	−230.377	−0.237465
$99$	0	0
$100$	352.617	0.352617
$101$	−1890.14	−1.86214	−0.931071	−	0.364838i	$-0.881124\pi$
$101$	−1890.14	−1.86214	−0.931071	+	0.364838i	$0.881124\pi$
$102$	0	0
$103$	662.700	0.633959	0.316979	−	0.948432i	$-0.397331\pi$
$103$	662.700	0.633959	0.316979	+	0.948432i	$0.397331\pi$
$104$	1005.00	0.947583
$105$	0	0
$106$	3181.50	2.91523
$107$	−1614.53	−1.45872	−0.729358	−	0.684132i	$-0.760181\pi$
$107$	−1614.53	−1.45872	−0.729358	+	0.684132i	$0.760181\pi$
$108$	0	0
$109$	217.206	0.190868	0.0954339	−	0.995436i	$-0.469576\pi$
$109$	217.206	0.190868	0.0954339	+	0.995436i	$0.469576\pi$
$110$	578.219	0.501191
$111$	0	0
$112$	154.733	0.130544
$113$	1658.20	1.38044	0.690221	−	0.723598i	$-0.257513\pi$
$113$	1658.20	1.38044	0.690221	+	0.723598i	$0.257513\pi$
$114$	0	0
$115$	728.375	0.590620
$116$	−3024.31	−2.42069
$117$	0	0
$118$	2356.98	1.83879
$119$	128.931	0.0993202
$120$	0	0
$121$	−725.994	−0.545450
$122$	3333.08	2.47347
$123$	0	0
$124$	−1250.51	−0.905640
$125$	125.000	0.0894427
$126$	0	0
$127$	−1108.81	−0.774734	−0.387367	−	0.921926i	$-0.626615\pi$
$127$	−1108.81	−0.774734	−0.387367	+	0.921926i	$0.626615\pi$
$128$	2604.17	1.79827
$129$	0	0
$130$	823.141	0.555340
$131$	−185.488	−0.123711	−0.0618554	−	0.998085i	$-0.519702\pi$
$131$	−185.488	−0.123711	−0.0618554	+	0.998085i	$0.519702\pi$
$132$	0	0
$133$	−471.822	−0.307610
$134$	4268.81	2.75201
$135$	0	0
$136$	−528.647	−0.333317
$137$	37.9907	0.0236917	0.0118458	−	0.999930i	$-0.496229\pi$
$137$	37.9907	0.0236917	0.0118458	+	0.999930i	$0.496229\pi$
$138$	0	0
$139$	183.609	0.112040	0.0560199	−	0.998430i	$-0.482159\pi$
$139$	183.609	0.112040	0.0560199	+	0.998430i	$0.482159\pi$
$140$	493.664	0.298016
$141$	0	0
$142$	2023.24	1.19568
$143$	861.275	0.503660
$144$	0	0
$145$	−1072.09	−0.614018
$146$	−194.366	−0.110177
$147$	0	0
$148$	2286.72	1.27005
$149$	1383.34	0.760587	0.380293	−	0.924866i	$-0.375823\pi$
$149$	1383.34	0.760587	0.380293	+	0.924866i	$0.375823\pi$
$150$	0	0
$151$	765.256	0.412422	0.206211	−	0.978508i	$-0.433887\pi$
$151$	765.256	0.412422	0.206211	+	0.978508i	$0.433887\pi$
$152$	1934.57	1.03233
$153$	0	0
$154$	809.506	0.423584
$155$	−443.297	−0.229719
$156$	0	0
$157$	−2366.76	−1.20311	−0.601554	−	0.798832i	$-0.705452\pi$
$157$	−2366.76	−1.20311	−0.601554	+	0.798832i	$0.705452\pi$
$158$	−4186.21	−2.10783
$159$	0	0
$160$	628.430	0.310511
$161$	1019.72	0.499165
$162$	0	0
$163$	−3137.69	−1.50775	−0.753875	−	0.657018i	$-0.771817\pi$
$163$	−3137.69	−1.50775	−0.753875	+	0.657018i	$0.771817\pi$
$164$	4764.12	2.26839
$165$	0	0
$166$	−4970.26	−2.32390
$167$	−146.469	−0.0678688	−0.0339344	−	0.999424i	$-0.510804\pi$
$167$	−146.469	−0.0678688	−0.0339344	+	0.999424i	$0.510804\pi$
$168$	0	0
$169$	−970.906	−0.441924
$170$	−432.984	−0.195343
$171$	0	0
$172$	1722.98	0.763812
$173$	1424.12	0.625860	0.312930	−	0.949776i	$-0.398689\pi$
$173$	1424.12	0.625860	0.312930	+	0.949776i	$0.398689\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	−543.706	−0.232860
$177$	0	0
$178$	6928.81	2.91762
$179$	−1244.70	−0.519737	−0.259869	−	0.965644i	$-0.583679\pi$
$179$	−1244.70	−0.519737	−0.259869	+	0.965644i	$0.583679\pi$
$180$	0	0
$181$	−3879.09	−1.59299	−0.796493	−	0.604648i	$-0.793314\pi$
$181$	−3879.09	−1.59299	−0.796493	+	0.604648i	$0.793314\pi$
$182$	1152.40	0.469348
$183$	0	0
$184$	−4181.10	−1.67519
$185$	810.625	0.322153
$186$	0	0
$187$	−453.044	−0.177165
$188$	−4996.14	−1.93820
$189$	0	0
$190$	1584.50	0.605009
$191$	−1574.90	−0.596628	−0.298314	−	0.954468i	$-0.596424\pi$
$191$	−1574.90	−0.596628	−0.298314	+	0.954468i	$0.596424\pi$
$192$	0	0
$193$	−4775.67	−1.78114	−0.890572	−	0.454843i	$-0.849695\pi$
$193$	−4775.67	−1.78114	−0.890572	+	0.454843i	$0.849695\pi$
$194$	−2609.53	−0.965738
$195$	0	0
$196$	691.130	0.251869
$197$	2803.58	1.01394	0.506971	−	0.861963i	$-0.330765\pi$
$197$	2803.58	1.01394	0.506971	+	0.861963i	$0.330765\pi$
$198$	0	0
$199$	4102.92	1.46155	0.730774	−	0.682620i	$-0.239159\pi$
$199$	4102.92	1.46155	0.730774	+	0.682620i	$0.239159\pi$
$200$	−717.539	−0.253688
$201$	0	0
$202$	8886.63	3.09535
$203$	−1500.93	−0.518940
$204$	0	0
$205$	1688.84	0.575385
$206$	−3115.72	−1.05380
$207$	0	0
$208$	−774.009	−0.258019
$209$	1657.91	0.548707
$210$	0	0
$211$	−823.512	−0.268687	−0.134343	−	0.990935i	$-0.542893\pi$
$211$	−823.512	−0.268687	−0.134343	+	0.990935i	$0.542893\pi$
$212$	−9544.51	−3.09207
$213$	0	0
$214$	7590.82	2.42476
$215$	610.781	0.193744
$216$	0	0
$217$	−620.616	−0.194148
$218$	−1021.21	−0.317271
$219$	0	0
$220$	−1734.66	−0.531593
$221$	−644.944	−0.196306
$222$	0	0
$223$	817.194	0.245396	0.122698	−	0.992444i	$-0.460845\pi$
$223$	817.194	0.245396	0.122698	+	0.992444i	$0.460845\pi$
$224$	879.802	0.262430
$225$	0	0
$226$	−7796.12	−2.29465
$227$	−3655.85	−1.06893	−0.534465	−	0.845190i	$-0.679487\pi$
$227$	−3655.85	−1.06893	−0.534465	+	0.845190i	$0.679487\pi$
$228$	0	0
$229$	939.393	0.271078	0.135539	−	0.990772i	$-0.456723\pi$
$229$	939.393	0.271078	0.135539	+	0.990772i	$0.456723\pi$
$230$	−3424.50	−0.981760
$231$	0	0
$232$	6154.15	1.74155
$233$	7.64701	0.00215010	0.00107505	−	0.999999i	$-0.499658\pi$
$233$	7.64701	0.00215010	0.00107505	+	0.999999i	$0.499658\pi$
$234$	0	0
$235$	−1771.09	−0.491631
$236$	−7070.94	−1.95034
$237$	0	0
$238$	−606.178	−0.165095
$239$	889.115	0.240636	0.120318	−	0.992735i	$-0.461609\pi$
$239$	889.115	0.240636	0.120318	+	0.992735i	$0.461609\pi$
$240$	0	0
$241$	2140.23	0.572051	0.286026	−	0.958222i	$-0.407666\pi$
$241$	2140.23	0.572051	0.286026	+	0.958222i	$0.407666\pi$
$242$	3413.30	0.906676
$243$	0	0
$244$	−9999.25	−2.62351
$245$	245.000	0.0638877
$246$	0	0
$247$	2360.16	0.607990
$248$	2544.66	0.651557
$249$	0	0
$250$	−587.695	−0.148676
$251$	6749.81	1.69739	0.848693	−	0.528886i	$-0.177390\pi$
$251$	6749.81	1.69739	0.848693	+	0.528886i	$0.177390\pi$
$252$	0	0
$253$	−3583.15	−0.890398
$254$	5213.15	1.28780
$255$	0	0
$256$	−6101.62	−1.48965
$257$	−3068.64	−0.744811	−0.372405	−	0.928070i	$-0.621467\pi$
$257$	−3068.64	−0.744811	−0.372405	+	0.928070i	$0.621467\pi$
$258$	0	0
$259$	1134.87	0.272269
$260$	−2469.42	−0.589027
$261$	0	0
$262$	872.081	0.205639
$263$	4674.12	1.09589	0.547944	−	0.836515i	$-0.315411\pi$
$263$	4674.12	1.09589	0.547944	+	0.836515i	$0.315411\pi$
$264$	0	0
$265$	−3383.45	−0.784316
$266$	2218.30	0.511326
$267$	0	0
$268$	−12806.4	−2.91895
$269$	−2417.38	−0.547919	−0.273960	−	0.961741i	$-0.588333\pi$
$269$	−2417.38	−0.547919	−0.273960	+	0.961741i	$0.588333\pi$
$270$	0	0
$271$	7724.30	1.73143	0.865715	−	0.500537i	$-0.166864\pi$
$271$	7724.30	1.73143	0.865715	+	0.500537i	$0.166864\pi$
$272$	407.141	0.0907593
$273$	0	0
$274$	−178.616	−0.0393816
$275$	−614.922	−0.134841
$276$	0	0
$277$	−4576.17	−0.992620	−0.496310	−	0.868145i	$-0.665312\pi$
$277$	−4576.17	−0.992620	−0.496310	+	0.868145i	$0.665312\pi$
$278$	−863.250	−0.186239
$279$	0	0
$280$	−1004.55	−0.214406
$281$	1358.56	0.288415	0.144208	−	0.989547i	$-0.453937\pi$
$281$	1358.56	0.288415	0.144208	+	0.989547i	$0.453937\pi$
$282$	0	0
$283$	3885.04	0.816048	0.408024	−	0.912971i	$-0.366218\pi$
$283$	3885.04	0.816048	0.408024	+	0.912971i	$0.366218\pi$
$284$	−6069.73	−1.26821
$285$	0	0
$286$	−4049.34	−0.837211
$287$	2364.38	0.486289
$288$	0	0
$289$	−4573.75	−0.930948
$290$	5040.52	1.02065
$291$	0	0
$292$	583.097	0.116860
$293$	4033.91	0.804312	0.402156	−	0.915571i	$-0.368261\pi$
$293$	4033.91	0.804312	0.402156	+	0.915571i	$0.368261\pi$
$294$	0	0
$295$	−2506.59	−0.494710
$296$	−4653.24	−0.913730
$297$	0	0
$298$	−6503.85	−1.26429
$299$	−5100.90	−0.986598
$300$	0	0
$301$	855.093	0.163743
$302$	−3597.90	−0.685549
$303$	0	0
$304$	−1489.92	−0.281096
$305$	−3544.66	−0.665464
$306$	0	0
$307$	−4620.36	−0.858950	−0.429475	−	0.903079i	$-0.641301\pi$
$307$	−4620.36	−0.858950	−0.429475	+	0.903079i	$0.641301\pi$
$308$	−2428.52	−0.449278
$309$	0	0
$310$	2084.19	0.381851
$311$	−6675.89	−1.21722	−0.608609	−	0.793470i	$-0.708272\pi$
$311$	−6675.89	−1.21722	−0.608609	+	0.793470i	$0.708272\pi$
$312$	0	0
$313$	2836.78	0.512283	0.256141	−	0.966639i	$-0.417549\pi$
$313$	2836.78	0.512283	0.256141	+	0.966639i	$0.417549\pi$
$314$	11127.5	1.99987
$315$	0	0
$316$	12558.6	2.23569
$317$	−4010.63	−0.710597	−0.355299	−	0.934753i	$-0.615621\pi$
$317$	−4010.63	−0.710597	−0.355299	+	0.934753i	$0.615621\pi$
$318$	0	0
$319$	5274.03	0.925671
$320$	−3838.79	−0.670609
$321$	0	0
$322$	−4794.30	−0.829739
$323$	−1241.48	−0.213863
$324$	0	0
$325$	−875.391	−0.149409
$326$	14752.1	2.50626
$327$	0	0
$328$	−9694.49	−1.63198
$329$	−2479.53	−0.415504
$330$	0	0
$331$	11087.5	1.84117	0.920583	−	0.390546i	$-0.127714\pi$
$331$	11087.5	1.84117	0.920583	+	0.390546i	$0.127714\pi$
$332$	14910.8	2.46486
$333$	0	0
$334$	688.631	0.112815
$335$	−4539.78	−0.740402
$336$	0	0
$337$	12118.7	1.95890	0.979450	−	0.201689i	$-0.0646431\pi$
$337$	12118.7	1.95890	0.979450	+	0.201689i	$0.0646431\pi$
$338$	4564.78	0.734589
$339$	0	0
$340$	1298.95	0.207193
$341$	2180.74	0.346316
$342$	0	0
$343$	343.000	0.0539949
$344$	−3506.07	−0.549520
$345$	0	0
$346$	−6695.58	−1.04034
$347$	6361.22	0.984116	0.492058	−	0.870562i	$-0.336245\pi$
$347$	6361.22	0.984116	0.492058	+	0.870562i	$0.336245\pi$
$348$	0	0
$349$	−3115.18	−0.477799	−0.238899	−	0.971044i	$-0.576787\pi$
$349$	−3115.18	−0.477799	−0.238899	+	0.971044i	$0.576787\pi$
$350$	−822.773	−0.125655
$351$	0	0
$352$	−3091.48	−0.468115
$353$	11927.4	1.79839	0.899194	−	0.437550i	$-0.144154\pi$
$353$	11927.4	1.79839	0.899194	+	0.437550i	$0.144154\pi$
$354$	0	0
$355$	−2151.67	−0.321687
$356$	−20786.4	−3.09460
$357$	0	0
$358$	5852.02	0.863935
$359$	6143.95	0.903245	0.451623	−	0.892209i	$-0.350845\pi$
$359$	6143.95	0.903245	0.451623	+	0.892209i	$0.350845\pi$
$360$	0	0
$361$	−2315.82	−0.337632
$362$	18237.8	2.64794
$363$	0	0
$364$	−3457.19	−0.497819
$365$	206.703	0.0296420
$366$	0	0
$367$	−1927.67	−0.274178	−0.137089	−	0.990559i	$-0.543775\pi$
$367$	−1927.67	−0.274178	−0.137089	+	0.990559i	$0.543775\pi$
$368$	3220.10	0.456139
$369$	0	0
$370$	−3811.20	−0.535500
$371$	−4736.83	−0.662868
$372$	0	0
$373$	10452.0	1.45090	0.725449	−	0.688276i	$-0.241632\pi$
$373$	10452.0	1.45090	0.725449	+	0.688276i	$0.241632\pi$
$374$	2130.01	0.294493
$375$	0	0
$376$	10166.6	1.39443
$377$	7508.01	1.02568
$378$	0	0
$379$	7066.43	0.957726	0.478863	−	0.877890i	$-0.341049\pi$
$379$	7066.43	0.957726	0.478863	+	0.877890i	$0.341049\pi$
$380$	−4753.50	−0.641709
$381$	0	0
$382$	7404.51	0.991747
$383$	−7168.04	−0.956318	−0.478159	−	0.878273i	$-0.658696\pi$
$383$	−7168.04	−0.956318	−0.478159	+	0.878273i	$0.658696\pi$
$384$	0	0
$385$	−860.891	−0.113961
$386$	22453.1	2.96071
$387$	0	0
$388$	7828.58	1.02432
$389$	7414.06	0.966344	0.483172	−	0.875525i	$-0.339485\pi$
$389$	7414.06	0.966344	0.483172	+	0.875525i	$0.339485\pi$
$390$	0	0
$391$	2683.15	0.347040
$392$	−1406.38	−0.181206
$393$	0	0
$394$	−13181.2	−1.68543
$395$	4451.94	0.567092
$396$	0	0
$397$	−8936.01	−1.12969	−0.564843	−	0.825198i	$-0.691063\pi$
$397$	−8936.01	−1.12969	−0.564843	+	0.825198i	$0.691063\pi$
$398$	−19290.1	−2.42946
$399$	0	0
$400$	552.617	0.0690771
$401$	−1782.91	−0.222031	−0.111015	−	0.993819i	$-0.535410\pi$
$401$	−1782.91	−0.222031	−0.111015	+	0.993819i	$0.535410\pi$
$402$	0	0
$403$	3104.46	0.383733
$404$	−26659.9	−3.28312
$405$	0	0
$406$	7056.72	0.862609
$407$	−3987.77	−0.485667
$408$	0	0
$409$	−8759.92	−1.05905	−0.529524	−	0.848295i	$-0.677629\pi$
$409$	−8759.92	−1.05905	−0.529524	+	0.848295i	$0.677629\pi$
$410$	−7940.20	−0.956436
$411$	0	0
$412$	9347.17	1.11772
$413$	−3509.23	−0.418106
$414$	0	0
$415$	5285.75	0.625222
$416$	−4400.97	−0.518691
$417$	0	0
$418$	−7794.75	−0.912090
$419$	3212.74	0.374588	0.187294	−	0.982304i	$-0.440028\pi$
$419$	3212.74	0.374588	0.187294	+	0.982304i	$0.440028\pi$
$420$	0	0
$421$	15757.8	1.82420	0.912101	−	0.409965i	$-0.134459\pi$
$421$	15757.8	1.82420	0.912101	+	0.409965i	$0.134459\pi$
$422$	3871.79	0.446626
$423$	0	0
$424$	19422.1	2.22457
$425$	460.469	0.0525553
$426$	0	0
$427$	−4962.52	−0.562419
$428$	−22772.5	−2.57184
$429$	0	0
$430$	−2871.63	−0.322051
$431$	405.917	0.0453650	0.0226825	−	0.999743i	$-0.492779\pi$
$431$	405.917	0.0453650	0.0226825	+	0.999743i	$0.492779\pi$
$432$	0	0
$433$	−7845.25	−0.870713	−0.435357	−	0.900258i	$-0.643378\pi$
$433$	−7845.25	−0.870713	−0.435357	+	0.900258i	$0.643378\pi$
$434$	2917.86	0.322723
$435$	0	0
$436$	3063.63	0.336516
$437$	−9818.95	−1.07484
$438$	0	0
$439$	423.029	0.0459911	0.0229955	−	0.999736i	$-0.492680\pi$
$439$	423.029	0.0459911	0.0229955	+	0.999736i	$0.492680\pi$
$440$	3529.84	0.382452
$441$	0	0
$442$	3032.24	0.326310
$443$	16058.7	1.72229	0.861143	−	0.508362i	$-0.169749\pi$
$443$	16058.7	1.72229	0.861143	+	0.508362i	$0.169749\pi$
$444$	0	0
$445$	−7368.62	−0.784958
$446$	−3842.09	−0.407911
$447$	0	0
$448$	−5374.30	−0.566768
$449$	−2186.75	−0.229842	−0.114921	−	0.993375i	$-0.536662\pi$
$449$	−2186.75	−0.229842	−0.114921	+	0.993375i	$0.536662\pi$
$450$	0	0
$451$	−8308.05	−0.867430
$452$	23388.3	2.43384
$453$	0	0
$454$	17188.2	1.77683
$455$	−1225.55	−0.126274
$456$	0	0
$457$	−5799.22	−0.593602	−0.296801	−	0.954939i	$-0.595920\pi$
$457$	−5799.22	−0.593602	−0.296801	+	0.954939i	$0.595920\pi$
$458$	−4416.62	−0.450600
$459$	0	0
$460$	10273.5	1.04131
$461$	−9873.35	−0.997500	−0.498750	−	0.866746i	$-0.666207\pi$
$461$	−9873.35	−0.997500	−0.498750	+	0.866746i	$0.666207\pi$
$462$	0	0
$463$	−6181.84	−0.620506	−0.310253	−	0.950654i	$-0.600414\pi$
$463$	−6181.84	−0.620506	−0.310253	+	0.950654i	$0.600414\pi$
$464$	−4739.66	−0.474209
$465$	0	0
$466$	−35.9529	−0.00357400
$467$	−6145.50	−0.608950	−0.304475	−	0.952520i	$-0.598481\pi$
$467$	−6145.50	−0.608950	−0.304475	+	0.952520i	$0.598481\pi$
$468$	0	0
$469$	−6355.69	−0.625754
$470$	8326.91	0.817216
$471$	0	0
$472$	14388.6	1.40316
$473$	−3004.66	−0.292081
$474$	0	0
$475$	−1685.08	−0.162772
$476$	1818.53	0.175110
$477$	0	0
$478$	−4180.23	−0.399999
$479$	−10879.4	−1.03777	−0.518887	−	0.854843i	$-0.673653\pi$
$479$	−10879.4	−1.03777	−0.518887	+	0.854843i	$0.673653\pi$
$480$	0	0
$481$	−5676.91	−0.538139
$482$	−10062.4	−0.950894
$483$	0	0
$484$	−10239.9	−0.961675
$485$	2775.17	0.259823
$486$	0	0
$487$	8087.51	0.752526	0.376263	−	0.926513i	$-0.377209\pi$
$487$	8087.51	0.752526	0.376263	+	0.926513i	$0.377209\pi$
$488$	20347.4	1.88747
$489$	0	0
$490$	−1151.88	−0.106197
$491$	6959.90	0.639707	0.319853	−	0.947467i	$-0.396366\pi$
$491$	6959.90	0.639707	0.319853	+	0.947467i	$0.396366\pi$
$492$	0	0
$493$	−3949.32	−0.360788
$494$	−11096.4	−1.01063
$495$	0	0
$496$	−1959.79	−0.177413
$497$	−3012.34	−0.271875
$498$	0	0
$499$	18632.0	1.67151	0.835756	−	0.549101i	$-0.185030\pi$
$499$	18632.0	1.67151	0.835756	+	0.549101i	$0.185030\pi$
$500$	1763.09	0.157695
$501$	0	0
$502$	−31734.6	−2.82149
$503$	−4627.62	−0.410209	−0.205105	−	0.978740i	$-0.565753\pi$
$503$	−4627.62	−0.410209	−0.205105	+	0.978740i	$0.565753\pi$
$504$	0	0
$505$	−9450.72	−0.832775
$506$	16846.4	1.48007
$507$	0	0
$508$	−15639.4	−1.36592
$509$	11351.8	0.988528	0.494264	−	0.869312i	$-0.335438\pi$
$509$	11351.8	0.988528	0.494264	+	0.869312i	$0.335438\pi$
$510$	0	0
$511$	289.384	0.0250521
$512$	7853.76	0.677911
$513$	0	0
$514$	14427.4	1.23806
$515$	3313.50	0.283515
$516$	0	0
$517$	8712.67	0.741166
$518$	−5335.68	−0.452580
$519$	0	0
$520$	5025.02	0.423772
$521$	−19096.1	−1.60579	−0.802893	−	0.596123i	$-0.796707\pi$
$521$	−19096.1	−1.60579	−0.802893	+	0.596123i	$0.796707\pi$
$522$	0	0
$523$	−3145.11	−0.262956	−0.131478	−	0.991319i	$-0.541972\pi$
$523$	−3145.11	−0.262956	−0.131478	+	0.991319i	$0.541972\pi$
$524$	−2616.24	−0.218113
$525$	0	0
$526$	−21975.7	−1.82164
$527$	−1632.99	−0.134980
$528$	0	0
$529$	9054.20	0.744160
$530$	15907.5	1.30373
$531$	0	0
$532$	−6654.90	−0.542343
$533$	−11827.2	−0.961148
$534$	0	0
$535$	−8072.66	−0.652358
$536$	26059.8	2.10002
$537$	0	0
$538$	11365.5	0.910781
$539$	−1205.25	−0.0963148
$540$	0	0
$541$	8776.12	0.697440	0.348720	−	0.937227i	$-0.386616\pi$
$541$	8776.12	0.697440	0.348720	+	0.937227i	$0.386616\pi$
$542$	−36316.3	−2.87808
$543$	0	0
$544$	2314.98	0.182452
$545$	1086.03	0.0853587
$546$	0	0
$547$	−13695.1	−1.07049	−0.535247	−	0.844696i	$-0.679781\pi$
$547$	−13695.1	−1.07049	−0.535247	+	0.844696i	$0.679781\pi$
$548$	535.847	0.0417705
$549$	0	0
$550$	2891.09	0.224139
$551$	14452.5	1.11742
$552$	0	0
$553$	6232.71	0.479280
$554$	21515.2	1.64999
$555$	0	0
$556$	2589.75	0.197536
$557$	−7850.44	−0.597188	−0.298594	−	0.954380i	$-0.596518\pi$
$557$	−7850.44	−0.597188	−0.298594	+	0.954380i	$0.596518\pi$
$558$	0	0
$559$	−4277.38	−0.323638
$560$	773.664	0.0583808
$561$	0	0
$562$	−6387.33	−0.479419
$563$	4948.81	0.370457	0.185229	−	0.982695i	$-0.440697\pi$
$563$	4948.81	0.370457	0.185229	+	0.982695i	$0.440697\pi$
$564$	0	0
$565$	8290.98	0.617353
$566$	−18265.7	−1.35648
$567$	0	0
$568$	12351.3	0.912408
$569$	8115.76	0.597945	0.298972	−	0.954262i	$-0.403356\pi$
$569$	8115.76	0.597945	0.298972	+	0.954262i	$0.403356\pi$
$570$	0	0
$571$	5656.42	0.414560	0.207280	−	0.978282i	$-0.433539\pi$
$571$	5656.42	0.414560	0.207280	+	0.978282i	$0.433539\pi$
$572$	12148.0	0.887996
$573$	0	0
$574$	−11116.3	−0.808336
$575$	3641.87	0.264133
$576$	0	0
$577$	−9536.77	−0.688078	−0.344039	−	0.938955i	$-0.611795\pi$
$577$	−9536.77	−0.688078	−0.344039	+	0.938955i	$0.611795\pi$
$578$	21503.8	1.54747
$579$	0	0
$580$	−15121.5	−1.08257
$581$	7400.05	0.528409
$582$	0	0
$583$	16644.5	1.18241
$584$	−1186.54	−0.0840743
$585$	0	0
$586$	−18965.7	−1.33697
$587$	13089.6	0.920383	0.460191	−	0.887820i	$-0.347781\pi$
$587$	13089.6	0.920383	0.460191	+	0.887820i	$0.347781\pi$
$588$	0	0
$589$	5975.92	0.418053
$590$	11784.9	0.822334
$591$	0	0
$592$	3583.72	0.248801
$593$	−4281.96	−0.296524	−0.148262	−	0.988948i	$-0.547368\pi$
$593$	−4281.96	−0.296524	−0.148262	+	0.988948i	$0.547368\pi$
$594$	0	0
$595$	644.656	0.0444173
$596$	19511.5	1.34098
$597$	0	0
$598$	23982.2	1.63997
$599$	−3699.92	−0.252378	−0.126189	−	0.992006i	$-0.540275\pi$
$599$	−3699.92	−0.252378	−0.126189	+	0.992006i	$0.540275\pi$
$600$	0	0
$601$	−17286.1	−1.17323	−0.586616	−	0.809865i	$-0.699540\pi$
$601$	−17286.1	−1.17323	−0.586616	+	0.809865i	$0.699540\pi$
$602$	−4020.28	−0.272183
$603$	0	0
$604$	10793.7	0.727135
$605$	−3629.97	−0.243933
$606$	0	0
$607$	14456.7	0.966689	0.483344	−	0.875430i	$-0.339422\pi$
$607$	14456.7	0.966689	0.483344	+	0.875430i	$0.339422\pi$
$608$	−8471.63	−0.565082
$609$	0	0
$610$	16665.4	1.10617
$611$	12403.2	0.821243
$612$	0	0
$613$	17981.9	1.18480	0.592400	−	0.805644i	$-0.298181\pi$
$613$	17981.9	1.18480	0.592400	+	0.805644i	$0.298181\pi$
$614$	21722.9	1.42779
$615$	0	0
$616$	4941.78	0.323231
$617$	−19614.7	−1.27983	−0.639916	−	0.768445i	$-0.721031\pi$
$617$	−19614.7	−1.27983	−0.639916	+	0.768445i	$0.721031\pi$
$618$	0	0
$619$	−10462.9	−0.679385	−0.339692	−	0.940537i	$-0.610323\pi$
$619$	−10462.9	−0.679385	−0.339692	+	0.940537i	$0.610323\pi$
$620$	−6252.56	−0.405014
$621$	0	0
$622$	31387.1	2.02332
$623$	−10316.1	−0.663411
$624$	0	0
$625$	625.000	0.0400000
$626$	−13337.3	−0.851544
$627$	0	0
$628$	−33382.4	−2.12118
$629$	2986.14	0.189293
$630$	0	0
$631$	24481.9	1.54454	0.772272	−	0.635292i	$-0.219120\pi$
$631$	24481.9	1.54454	0.772272	+	0.635292i	$0.219120\pi$
$632$	−25555.5	−1.60846
$633$	0	0
$634$	18856.2	1.18119
$635$	−5544.06	−0.346471
$636$	0	0
$637$	−1715.77	−0.106721
$638$	−24796.2	−1.53870
$639$	0	0
$640$	13020.9	0.804211
$641$	1109.39	0.0683595	0.0341797	−	0.999416i	$-0.489118\pi$
$641$	1109.39	0.0683595	0.0341797	+	0.999416i	$0.489118\pi$
$642$	0	0
$643$	30112.5	1.84684	0.923422	−	0.383787i	$-0.125380\pi$
$643$	30112.5	1.84684	0.923422	+	0.383787i	$0.125380\pi$
$644$	14382.9	0.880071
$645$	0	0
$646$	5836.90	0.355495
$647$	4260.27	0.258869	0.129435	−	0.991588i	$-0.458684\pi$
$647$	4260.27	0.258869	0.129435	+	0.991588i	$0.458684\pi$
$648$	0	0
$649$	12330.9	0.745808
$650$	4115.70	0.248356
$651$	0	0
$652$	−44256.2	−2.65829
$653$	10576.8	0.633844	0.316922	−	0.948452i	$-0.397351\pi$
$653$	10576.8	0.633844	0.316922	+	0.948452i	$0.397351\pi$
$654$	0	0
$655$	−927.438	−0.0553252
$656$	7466.27	0.444373
$657$	0	0
$658$	11657.7	0.690674
$659$	−3394.70	−0.200666	−0.100333	−	0.994954i	$-0.531991\pi$
$659$	−3394.70	−0.200666	−0.100333	+	0.994954i	$0.531991\pi$
$660$	0	0
$661$	−33174.4	−1.95210	−0.976048	−	0.217554i	$-0.930192\pi$
$661$	−33174.4	−1.95210	−0.976048	+	0.217554i	$0.930192\pi$
$662$	−52128.7	−3.06048
$663$	0	0
$664$	−30341.9	−1.77333
$665$	−2359.11	−0.137567
$666$	0	0
$667$	−31235.4	−1.81326
$668$	−2065.89	−0.119658
$669$	0	0
$670$	21344.1	1.23074
$671$	17437.5	1.00323
$672$	0	0
$673$	753.881	0.0431797	0.0215899	−	0.999767i	$-0.493127\pi$
$673$	753.881	0.0431797	0.0215899	+	0.999767i	$0.493127\pi$
$674$	−56976.9	−3.25619
$675$	0	0
$676$	−13694.3	−0.779149
$677$	−15668.8	−0.889511	−0.444756	−	0.895652i	$-0.646709\pi$
$677$	−15668.8	−0.889511	−0.444756	+	0.895652i	$0.646709\pi$
$678$	0	0
$679$	3885.24	0.219590
$680$	−2643.23	−0.149064
$681$	0	0
$682$	−10252.9	−0.575665
$683$	11557.4	0.647485	0.323742	−	0.946145i	$-0.395059\pi$
$683$	11557.4	0.647485	0.323742	+	0.946145i	$0.395059\pi$
$684$	0	0
$685$	189.953	0.0105952
$686$	−1612.64	−0.0897532
$687$	0	0
$688$	2700.22	0.149630
$689$	23694.7	1.31016
$690$	0	0
$691$	−18503.1	−1.01866	−0.509328	−	0.860572i	$-0.670106\pi$
$691$	−18503.1	−1.01866	−0.509328	+	0.860572i	$0.670106\pi$
$692$	20086.8	1.10344
$693$	0	0
$694$	−29907.7	−1.63585
$695$	918.046	0.0501057
$696$	0	0
$697$	6221.28	0.338088
$698$	14646.2	0.794223
$699$	0	0
$700$	2468.32	0.133277
$701$	−22580.4	−1.21662	−0.608311	−	0.793699i	$-0.708153\pi$
$701$	−22580.4	−1.21662	−0.608311	+	0.793699i	$0.708153\pi$
$702$	0	0
$703$	−10927.7	−0.586269
$704$	18884.4	1.01099
$705$	0	0
$706$	−56077.4	−2.98938
$707$	−13231.0	−0.703823
$708$	0	0
$709$	−27426.6	−1.45279	−0.726394	−	0.687278i	$-0.758805\pi$
$709$	−27426.6	−1.45279	−0.726394	+	0.687278i	$0.758805\pi$
$710$	10116.2	0.534725
$711$	0	0
$712$	42298.2	2.22639
$713$	−12915.5	−0.678383
$714$	0	0
$715$	4306.37	0.225244
$716$	−17556.1	−0.916342
$717$	0	0
$718$	−28886.1	−1.50142
$719$	−19383.0	−1.00538	−0.502688	−	0.864468i	$-0.667656\pi$
$719$	−19383.0	−1.00538	−0.502688	+	0.864468i	$0.667656\pi$
$720$	0	0
$721$	4638.90	0.239614
$722$	10888.0	0.561230
$723$	0	0
$724$	−54713.3	−2.80857
$725$	−5360.47	−0.274597
$726$	0	0
$727$	−12317.3	−0.628368	−0.314184	−	0.949362i	$-0.601731\pi$
$727$	−12317.3	−0.628368	−0.314184	+	0.949362i	$0.601731\pi$
$728$	7035.02	0.358153
$729$	0	0
$730$	−971.828	−0.0492726
$731$	2249.96	0.113841
$732$	0	0
$733$	1234.02	0.0621822	0.0310911	−	0.999517i	$-0.490102\pi$
$733$	1234.02	0.0621822	0.0310911	+	0.999517i	$0.490102\pi$
$734$	9063.05	0.455754
$735$	0	0
$736$	18309.3	0.916970
$737$	22332.9	1.11620
$738$	0	0
$739$	−15257.3	−0.759473	−0.379736	−	0.925095i	$-0.623985\pi$
$739$	−15257.3	−0.759473	−0.379736	+	0.925095i	$0.623985\pi$
$740$	11433.6	0.567984
$741$	0	0
$742$	22270.5	1.10186
$743$	35565.1	1.75606	0.878032	−	0.478602i	$-0.158856\pi$
$743$	35565.1	1.75606	0.878032	+	0.478602i	$0.158856\pi$
$744$	0	0
$745$	6916.69	0.340145
$746$	−49140.8	−2.41176
$747$	0	0
$748$	−6390.04	−0.312357
$749$	−11301.7	−0.551343
$750$	0	0
$751$	14266.7	0.693209	0.346605	−	0.938011i	$-0.387335\pi$
$751$	14266.7	0.693209	0.346605	+	0.938011i	$0.387335\pi$
$752$	−7829.89	−0.379690
$753$	0	0
$754$	−35299.4	−1.70494
$755$	3826.28	0.184441
$756$	0	0
$757$	−15927.9	−0.764744	−0.382372	−	0.924009i	$-0.624893\pi$
$757$	−15927.9	−0.764744	−0.382372	+	0.924009i	$0.624893\pi$
$758$	−33223.3	−1.59198
$759$	0	0
$760$	9672.87	0.461674
$761$	2566.48	0.122253	0.0611266	−	0.998130i	$-0.480531\pi$
$761$	2566.48	0.122253	0.0611266	+	0.998130i	$0.480531\pi$
$762$	0	0
$763$	1520.44	0.0721413
$764$	−22213.5	−1.05191
$765$	0	0
$766$	33701.0	1.58964
$767$	17554.0	0.826386
$768$	0	0
$769$	14433.1	0.676816	0.338408	−	0.940999i	$-0.390112\pi$
$769$	14433.1	0.676816	0.338408	+	0.940999i	$0.390112\pi$
$770$	4047.53	0.189432
$771$	0	0
$772$	−67359.4	−3.14031
$773$	29443.2	1.36999	0.684993	−	0.728550i	$-0.259805\pi$
$773$	29443.2	1.36999	0.684993	+	0.728550i	$0.259805\pi$
$774$	0	0
$775$	−2216.48	−0.102733
$776$	−15930.4	−0.736941
$777$	0	0
$778$	−34857.7	−1.60631
$779$	−22766.7	−1.04711
$780$	0	0
$781$	10584.9	0.484964
$782$	−12615.0	−0.576869
$783$	0	0
$784$	1083.13	0.0493408
$785$	−11833.8	−0.538046
$786$	0	0
$787$	26390.6	1.19533	0.597664	−	0.801747i	$-0.296096\pi$
$787$	26390.6	1.19533	0.597664	+	0.801747i	$0.296096\pi$
$788$	39543.6	1.78767
$789$	0	0
$790$	−20931.1	−0.942650
$791$	11607.4	0.521758
$792$	0	0
$793$	24823.7	1.11162
$794$	42013.2	1.87783
$795$	0	0
$796$	57870.3	2.57683
$797$	3738.33	0.166146	0.0830730	−	0.996543i	$-0.473527\pi$
$797$	3738.33	0.166146	0.0830730	+	0.996543i	$0.473527\pi$
$798$	0	0
$799$	−6524.26	−0.288876
$800$	3142.15	0.138865
$801$	0	0
$802$	8382.48	0.369072
$803$	−1016.85	−0.0446873
$804$	0	0
$805$	5098.62	0.223233
$806$	−14595.8	−0.637861
$807$	0	0
$808$	54250.1	2.36202
$809$	−43204.1	−1.87760	−0.938798	−	0.344468i	$-0.888059\pi$
$809$	−43204.1	−1.87760	−0.938798	+	0.344468i	$0.888059\pi$
$810$	0	0
$811$	−30192.4	−1.30727	−0.653637	−	0.756809i	$-0.726758\pi$
$811$	−30192.4	−1.30727	−0.653637	+	0.756809i	$0.726758\pi$
$812$	−21170.2	−0.914935
$813$	0	0
$814$	18748.7	0.807301
$815$	−15688.5	−0.674286
$816$	0	0
$817$	−8233.71	−0.352584
$818$	41185.3	1.76041
$819$	0	0
$820$	23820.6	1.01445
$821$	40274.7	1.71206	0.856028	−	0.516929i	$-0.172925\pi$
$821$	40274.7	1.71206	0.856028	+	0.516929i	$0.172925\pi$
$822$	0	0
$823$	25184.2	1.06667	0.533334	−	0.845905i	$-0.320939\pi$
$823$	25184.2	1.06667	0.533334	+	0.845905i	$0.320939\pi$
$824$	−19020.5	−0.804140
$825$	0	0
$826$	16498.9	0.694999
$827$	38941.7	1.63741	0.818703	−	0.574218i	$-0.194694\pi$
$827$	38941.7	1.63741	0.818703	+	0.574218i	$0.194694\pi$
$828$	0	0
$829$	−8327.05	−0.348867	−0.174433	−	0.984669i	$-0.555809\pi$
$829$	−8327.05	−0.348867	−0.174433	+	0.984669i	$0.555809\pi$
$830$	−24851.3	−1.03928
$831$	0	0
$832$	26883.5	1.12021
$833$	902.519	0.0375395
$834$	0	0
$835$	−732.343	−0.0303518
$836$	23384.2	0.967418
$837$	0	0
$838$	−15104.9	−0.622660
$839$	−8784.41	−0.361468	−0.180734	−	0.983532i	$-0.557847\pi$
$839$	−8784.41	−0.361468	−0.180734	+	0.983532i	$0.557847\pi$
$840$	0	0
$841$	21586.4	0.885087
$842$	−74086.4	−3.03229
$843$	0	0
$844$	−11615.4	−0.473718
$845$	−4854.53	−0.197634
$846$	0	0
$847$	−5081.96	−0.206161
$848$	−14958.0	−0.605732
$849$	0	0
$850$	−2164.92	−0.0873602
$851$	23617.6	0.951350
$852$	0	0
$853$	−9076.15	−0.364316	−0.182158	−	0.983269i	$-0.558308\pi$
$853$	−9076.15	−0.364316	−0.182158	+	0.983269i	$0.558308\pi$
$854$	23331.6	0.934884
$855$	0	0
$856$	46339.6	1.85030
$857$	−36396.7	−1.45074	−0.725372	−	0.688357i	$-0.758332\pi$
$857$	−36396.7	−1.45074	−0.725372	+	0.688357i	$0.758332\pi$
$858$	0	0
$859$	8915.27	0.354115	0.177058	−	0.984200i	$-0.443342\pi$
$859$	8915.27	0.354115	0.177058	+	0.984200i	$0.443342\pi$
$860$	8614.88	0.341587
$861$	0	0
$862$	−1908.44	−0.0754081
$863$	6148.26	0.242514	0.121257	−	0.992621i	$-0.461308\pi$
$863$	6148.26	0.242514	0.121257	+	0.992621i	$0.461308\pi$
$864$	0	0
$865$	7120.59	0.279893
$866$	36884.9	1.44735
$867$	0	0
$868$	−8753.59	−0.342300
$869$	−21900.8	−0.854928
$870$	0	0
$871$	31792.6	1.23680
$872$	−6234.16	−0.242105
$873$	0	0
$874$	46164.4	1.78665
$875$	875.000	0.0338062
$876$	0	0
$877$	−14287.0	−0.550101	−0.275050	−	0.961430i	$-0.588694\pi$
$877$	−14287.0	−0.550101	−0.275050	+	0.961430i	$0.588694\pi$
$878$	−1988.90	−0.0764488
$879$	0	0
$880$	−2718.53	−0.104138
$881$	13315.9	0.509221	0.254610	−	0.967044i	$-0.418053\pi$
$881$	13315.9	0.509221	0.254610	+	0.967044i	$0.418053\pi$
$882$	0	0
$883$	−5271.78	−0.200917	−0.100458	−	0.994941i	$-0.532031\pi$
$883$	−5271.78	−0.200917	−0.100458	+	0.994941i	$0.532031\pi$
$884$	−9096.73	−0.346104
$885$	0	0
$886$	−75501.1	−2.86288
$887$	−2606.07	−0.0986507	−0.0493253	−	0.998783i	$-0.515707\pi$
$887$	−2606.07	−0.0986507	−0.0493253	+	0.998783i	$0.515707\pi$
$888$	0	0
$889$	−7761.69	−0.292822
$890$	34644.0	1.30480
$891$	0	0
$892$	11526.3	0.432654
$893$	23875.4	0.894694
$894$	0	0
$895$	−6223.48	−0.232434
$896$	18229.2	0.679682
$897$	0	0
$898$	10281.1	0.382056
$899$	19010.2	0.705258
$900$	0	0
$901$	−12463.8	−0.460854
$902$	39060.8	1.44189
$903$	0	0
$904$	−47592.8	−1.75101
$905$	−19395.4	−0.712405
$906$	0	0
$907$	18610.6	0.681317	0.340659	−	0.940187i	$-0.389350\pi$
$907$	18610.6	0.681317	0.340659	+	0.940187i	$0.389350\pi$
$908$	−51564.6	−1.88462
$909$	0	0
$910$	5761.98	0.209899
$911$	−41091.7	−1.49443	−0.747216	−	0.664581i	$-0.768610\pi$
$911$	−41091.7	−1.49443	−0.747216	+	0.664581i	$0.768610\pi$
$912$	0	0
$913$	−26002.6	−0.942563
$914$	27265.4	0.986716
$915$	0	0
$916$	13249.8	0.477934
$917$	−1298.41	−0.0467583
$918$	0	0
$919$	38891.3	1.39598	0.697990	−	0.716107i	$-0.254078\pi$
$919$	38891.3	1.39598	0.697990	+	0.716107i	$0.254078\pi$
$920$	−20905.5	−0.749167
$921$	0	0
$922$	46420.2	1.65810
$923$	15068.4	0.537360
$924$	0	0
$925$	4053.12	0.144071
$926$	29064.3	1.03144
$927$	0	0
$928$	−26949.4	−0.953295
$929$	−18699.4	−0.660396	−0.330198	−	0.943912i	$-0.607115\pi$
$929$	−18699.4	−0.660396	−0.330198	+	0.943912i	$0.607115\pi$
$930$	0	0
$931$	−3302.75	−0.116266
$932$	107.859	0.00379080
$933$	0	0
$934$	28893.4	1.01223
$935$	−2265.22	−0.0792305
$936$	0	0
$937$	21509.6	0.749933	0.374967	−	0.927038i	$-0.377654\pi$
$937$	21509.6	0.749933	0.374967	+	0.927038i	$0.377654\pi$
$938$	29881.7	1.04016
$939$	0	0
$940$	−24980.7	−0.866788
$941$	11241.7	0.389448	0.194724	−	0.980858i	$-0.437619\pi$
$941$	11241.7	0.389448	0.194724	+	0.980858i	$0.437619\pi$
$942$	0	0
$943$	49204.5	1.69917
$944$	−11081.5	−0.382068
$945$	0	0
$946$	14126.6	0.485513
$947$	36556.3	1.25441	0.627203	−	0.778856i	$-0.284200\pi$
$947$	36556.3	1.25441	0.627203	+	0.778856i	$0.284200\pi$
$948$	0	0
$949$	−1447.57	−0.0495153
$950$	7922.50	0.270568
$951$	0	0
$952$	−3700.53	−0.125982
$953$	36633.4	1.24520	0.622598	−	0.782542i	$-0.286077\pi$
$953$	36633.4	1.24520	0.622598	+	0.782542i	$0.286077\pi$
$954$	0	0
$955$	−7874.52	−0.266820
$956$	12540.7	0.424263
$957$	0	0
$958$	51150.3	1.72504
$959$	265.935	0.00895462
$960$	0	0
$961$	−21930.5	−0.736146
$962$	26690.3	0.894523
$963$	0	0
$964$	30187.3	1.00858
$965$	−23878.4	−0.796551
$966$	0	0
$967$	−35515.8	−1.18109	−0.590544	−	0.807006i	$-0.701087\pi$
$967$	−35515.8	−1.18109	−0.590544	+	0.807006i	$0.701087\pi$
$968$	20837.2	0.691871
$969$	0	0
$970$	−13047.6	−0.431891
$971$	39661.0	1.31080	0.655398	−	0.755283i	$-0.272501\pi$
$971$	39661.0	1.31080	0.655398	+	0.755283i	$0.272501\pi$
$972$	0	0
$973$	1285.26	0.0423471
$974$	−38023.9	−1.25089
$975$	0	0
$976$	−15670.7	−0.513942
$977$	−50325.3	−1.64795	−0.823977	−	0.566624i	$-0.808249\pi$
$977$	−50325.3	−1.64795	−0.823977	+	0.566624i	$0.808249\pi$
$978$	0	0
$979$	36249.0	1.18337
$980$	3455.65	0.112639
$981$	0	0
$982$	−32722.4	−1.06335
$983$	−51189.0	−1.66091	−0.830456	−	0.557084i	$-0.811920\pi$
$983$	−51189.0	−1.66091	−0.830456	+	0.557084i	$0.811920\pi$
$984$	0	0
$985$	14017.9	0.453449
$986$	18568.0	0.599721
$987$	0	0
$988$	33289.3	1.07194
$989$	17795.1	0.572145
$990$	0	0
$991$	−55137.3	−1.76740	−0.883700	−	0.468054i	$-0.844955\pi$
$991$	−55137.3	−1.76740	−0.883700	+	0.468054i	$0.844955\pi$
$992$	−11143.2	−0.356651
$993$	0	0
$994$	14162.7	0.451925
$995$	20514.6	0.653624
$996$	0	0
$997$	41606.5	1.32166	0.660828	−	0.750537i	$-0.270205\pi$
$997$	41606.5	1.32166	0.660828	+	0.750537i	$0.270205\pi$
$998$	−87599.6	−2.77848
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.g.1.1		2
3.2	odd	2		105.4.a.g.1.2	✓	2
5.4	even	2		1575.4.a.y.1.2		2
7.6	odd	2		2205.4.a.v.1.1		2
12.11	even	2		1680.4.a.y.1.2		2
15.2	even	4		525.4.d.j.274.4		4
15.8	even	4		525.4.d.j.274.1		4
15.14	odd	2		525.4.a.i.1.1		2
21.20	even	2		735.4.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.g.1.2	✓	2	3.2	odd	2
315.4.a.g.1.1		2	1.1	even	1	trivial
525.4.a.i.1.1		2	15.14	odd	2
525.4.d.j.274.1		4	15.8	even	4
525.4.d.j.274.4		4	15.2	even	4
735.4.a.q.1.2		2	21.20	even	2
1575.4.a.y.1.2		2	5.4	even	2
1680.4.a.y.1.2		2	12.11	even	2
2205.4.a.v.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-4.70156 q^{2} +14.1047 q^{4} +5.00000 q^{5} +7.00000 q^{7} -28.7016 q^{8} +O(q^{10})\)	\(q-4.70156 q^{2} +14.1047 q^{4} +5.00000 q^{5} +7.00000 q^{7} -28.7016 q^{8} -23.5078 q^{10} -24.5969 q^{11} -35.0156 q^{13} -32.9109 q^{14} +22.1047 q^{16} +18.4187 q^{17} -67.4031 q^{19} +70.5234 q^{20} +115.644 q^{22} +145.675 q^{23} +25.0000 q^{25} +164.628 q^{26} +98.7328 q^{28} -214.419 q^{29} -88.6594 q^{31} +125.686 q^{32} -86.5969 q^{34} +35.0000 q^{35} +162.125 q^{37} +316.900 q^{38} -143.508 q^{40} +337.769 q^{41} +122.156 q^{43} -346.931 q^{44} -684.900 q^{46} -354.219 q^{47} +49.0000 q^{49} -117.539 q^{50} -493.884 q^{52} -676.691 q^{53} -122.984 q^{55} -200.911 q^{56} +1008.10 q^{58} -501.319 q^{59} -708.931 q^{61} +416.837 q^{62} -767.758 q^{64} -175.078 q^{65} -907.956 q^{67} +259.791 q^{68} -164.555 q^{70} -430.334 q^{71} +41.3406 q^{73} -762.241 q^{74} -950.700 q^{76} -172.178 q^{77} +890.388 q^{79} +110.523 q^{80} -1588.04 q^{82} +1057.15 q^{83} +92.0937 q^{85} -574.325 q^{86} +705.969 q^{88} -1473.72 q^{89} -245.109 q^{91} +2054.70 q^{92} +1665.38 q^{94} -337.016 q^{95} +555.034 q^{97} -230.377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 9 * q^4 + 10 * q^5 + 14 * q^7 - 51 * q^8	\( 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8} - 15 q^{10} - 62 q^{11} - 6 q^{13} - 21 q^{14} + 25 q^{16} - 40 q^{17} - 122 q^{19} + 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} + 214 q^{26} + 63 q^{28} - 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{34} + 70 q^{35} - 188 q^{37} + 224 q^{38} - 255 q^{40} - 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} + 188 q^{47} + 98 q^{49} - 75 q^{50} - 642 q^{52} - 982 q^{53} - 310 q^{55} - 357 q^{56} + 774 q^{58} - 516 q^{59} - 880 q^{61} + 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} + 558 q^{68} - 105 q^{70} - 310 q^{71} + 326 q^{73} - 1358 q^{74} - 672 q^{76} - 434 q^{77} + 1832 q^{79} + 125 q^{80} - 2190 q^{82} + 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} - 796 q^{89} - 42 q^{91} + 2880 q^{92} + 2588 q^{94} - 610 q^{95} - 670 q^{97} - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 9 * q^4 + 10 * q^5 + 14 * q^7 - 51 * q^8 - 15 * q^10 - 62 * q^11 - 6 * q^13 - 21 * q^14 + 25 * q^16 - 40 * q^17 - 122 * q^19 + 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 + 214 * q^26 + 63 * q^28 - 352 * q^29 + 66 * q^31 + 309 * q^32 - 186 * q^34 + 70 * q^35 - 188 * q^37 + 224 * q^38 - 255 * q^40 - 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 + 188 * q^47 + 98 * q^49 - 75 * q^50 - 642 * q^52 - 982 * q^53 - 310 * q^55 - 357 * q^56 + 774 * q^58 - 516 * q^59 - 880 * q^61 + 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 + 558 * q^68 - 105 * q^70 - 310 * q^71 + 326 * q^73 - 1358 * q^74 - 672 * q^76 - 434 * q^77 + 1832 * q^79 + 125 * q^80 - 2190 * q^82 + 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 - 796 * q^89 - 42 * q^91 + 2880 * q^92 + 2588 * q^94 - 610 * q^95 - 670 * q^97 - 147 * q^98

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