LMFDB - Embedded newform 1150.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 60x^{2} - 45x + 108 $ x^4 - 2x^3 - 60x^2 - 45*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.12571$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -10.1257 q^{3} +4.00000 q^{4} +20.2514 q^{6} +24.6431 q^{7} -8.00000 q^{8} +75.5301 q^{9} -17.3867 q^{11} -40.5029 q^{12} -4.00699 q^{13} -49.2862 q^{14} +16.0000 q^{16} -48.2740 q^{17} -151.060 q^{18} +79.3172 q^{19} -249.529 q^{21} +34.7734 q^{22} +23.0000 q^{23} +81.0057 q^{24} +8.01398 q^{26} -491.402 q^{27} +98.5723 q^{28} -254.267 q^{29} -220.696 q^{31} -32.0000 q^{32} +176.053 q^{33} +96.5480 q^{34} +302.120 q^{36} +422.904 q^{37} -158.634 q^{38} +40.5736 q^{39} -170.251 q^{41} +499.058 q^{42} +228.920 q^{43} -69.5468 q^{44} -46.0000 q^{46} -580.087 q^{47} -162.011 q^{48} +264.282 q^{49} +488.809 q^{51} -16.0280 q^{52} +260.354 q^{53} +982.803 q^{54} -197.145 q^{56} -803.144 q^{57} +508.535 q^{58} +353.130 q^{59} -80.6108 q^{61} +441.392 q^{62} +1861.29 q^{63} +64.0000 q^{64} -352.106 q^{66} +820.011 q^{67} -193.096 q^{68} -232.891 q^{69} +614.845 q^{71} -604.241 q^{72} -511.586 q^{73} -845.808 q^{74} +317.269 q^{76} -428.462 q^{77} -81.1472 q^{78} +160.464 q^{79} +2936.48 q^{81} +340.502 q^{82} +32.5646 q^{83} -998.115 q^{84} -457.840 q^{86} +2574.64 q^{87} +139.094 q^{88} -25.0375 q^{89} -98.7445 q^{91} +92.0000 q^{92} +2234.70 q^{93} +1160.17 q^{94} +324.023 q^{96} +249.798 q^{97} -528.563 q^{98} -1313.22 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 14 q^{3} + 16 q^{4} + 28 q^{6} - 8 q^{7} - 32 q^{8} + 64 q^{9} + 21 q^{11} - 56 q^{12} - 70 q^{13} + 16 q^{14} + 64 q^{16} - 56 q^{17} - 128 q^{18} + 173 q^{19} - 120 q^{21} - 42 q^{22}+ \cdots - 2745 q^{99}+O(q^{100}) $ 4 * q - 8 * q^2 - 14 * q^3 + 16 * q^4 + 28 * q^6 - 8 * q^7 - 32 * q^8 + 64 * q^9 + 21 * q^11 - 56 * q^12 - 70 * q^13 + 16 * q^14 + 64 * q^16 - 56 * q^17 - 128 * q^18 + 173 * q^19 - 120 * q^21 - 42 * q^22 + 92 * q^23 + 112 * q^24 + 140 * q^26 - 389 * q^27 - 32 * q^28 - 118 * q^29 + 17 * q^31 - 128 * q^32 + 89 * q^33 + 112 * q^34 + 256 * q^36 + 343 * q^37 - 346 * q^38 - 221 * q^39 + 139 * q^41 + 240 * q^42 + 50 * q^43 + 84 * q^44 - 184 * q^46 - 367 * q^47 - 224 * q^48 - 124 * q^49 + 439 * q^51 - 280 * q^52 + 353 * q^53 + 778 * q^54 + 64 * q^56 + 238 * q^57 + 236 * q^58 - 453 * q^59 - 327 * q^61 - 34 * q^62 + 1723 * q^63 + 256 * q^64 - 178 * q^66 + 455 * q^67 - 224 * q^68 - 322 * q^69 + 195 * q^71 - 512 * q^72 + 633 * q^73 - 686 * q^74 + 692 * q^76 + 2 * q^77 + 442 * q^78 - 1140 * q^79 + 1456 * q^81 - 278 * q^82 + 1199 * q^83 - 480 * q^84 - 100 * q^86 + 1775 * q^87 - 168 * q^88 - 2170 * q^89 + 557 * q^91 + 368 * q^92 + 3241 * q^93 + 734 * q^94 + 448 * q^96 + 703 * q^97 + 248 * q^98 - 2745 * q^99

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
$3$	−10.1257	−1.94869	−0.974347	−	0.225050i	$-0.927745\pi$
$3$	−10.1257	−1.94869	−0.974347	+	0.225050i	$0.927745\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	20.2514	1.37794
$7$	24.6431	1.33060	0.665301	−	0.746575i	$-0.268303\pi$
$7$	24.6431	1.33060	0.665301	+	0.746575i	$0.268303\pi$
$8$	−8.00000	−0.353553
$9$	75.5301	2.79741
$10$	0	0
$11$	−17.3867	−0.476572	−0.238286	−	0.971195i	$-0.576586\pi$
$11$	−17.3867	−0.476572	−0.238286	+	0.971195i	$0.576586\pi$
$12$	−40.5029	−0.974347
$13$	−4.00699	−0.0854876	−0.0427438	−	0.999086i	$-0.513610\pi$
$13$	−4.00699	−0.0854876	−0.0427438	+	0.999086i	$0.513610\pi$
$14$	−49.2862	−0.940877
$15$	0	0
$16$	16.0000	0.250000
$17$	−48.2740	−0.688716	−0.344358	−	0.938838i	$-0.611903\pi$
$17$	−48.2740	−0.688716	−0.344358	+	0.938838i	$0.611903\pi$
$18$	−151.060	−1.97807
$19$	79.3172	0.957717	0.478859	−	0.877892i	$-0.341051\pi$
$19$	79.3172	0.957717	0.478859	+	0.877892i	$0.341051\pi$
$20$	0	0
$21$	−249.529	−2.59294
$22$	34.7734	0.336987
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	81.0057	0.688968
$25$	0	0
$26$	8.01398	0.0604488
$27$	−491.402	−3.50260
$28$	98.5723	0.665301
$29$	−254.267	−1.62815	−0.814074	−	0.580761i	$-0.802755\pi$
$29$	−254.267	−1.62815	−0.814074	+	0.580761i	$0.802755\pi$
$30$	0	0
$31$	−220.696	−1.27865	−0.639325	−	0.768937i	$-0.720786\pi$
$31$	−220.696	−1.27865	−0.639325	+	0.768937i	$0.720786\pi$
$32$	−32.0000	−0.176777
$33$	176.053	0.928693
$34$	96.5480	0.486996
$35$	0	0
$36$	302.120	1.39870
$37$	422.904	1.87905	0.939527	−	0.342476i	$-0.111265\pi$
$37$	422.904	1.87905	0.939527	+	0.342476i	$0.111265\pi$
$38$	−158.634	−0.677208
$39$	40.5736	0.166589
$40$	0	0
$41$	−170.251	−0.648505	−0.324252	−	0.945971i	$-0.605113\pi$
$41$	−170.251	−0.648505	−0.324252	+	0.945971i	$0.605113\pi$
$42$	499.058	1.83348
$43$	228.920	0.811860	0.405930	−	0.913904i	$-0.366948\pi$
$43$	228.920	0.811860	0.405930	+	0.913904i	$0.366948\pi$
$44$	−69.5468	−0.238286
$45$	0	0
$46$	−46.0000	−0.147442
$47$	−580.087	−1.80031	−0.900154	−	0.435572i	$-0.856546\pi$
$47$	−580.087	−1.80031	−0.900154	+	0.435572i	$0.856546\pi$
$48$	−162.011	−0.487174
$49$	264.282	0.770500
$50$	0	0
$51$	488.809	1.34210
$52$	−16.0280	−0.0427438
$53$	260.354	0.674762	0.337381	−	0.941368i	$-0.390459\pi$
$53$	260.354	0.674762	0.337381	+	0.941368i	$0.390459\pi$
$54$	982.803	2.47671
$55$	0	0
$56$	−197.145	−0.470439
$57$	−803.144	−1.86630
$58$	508.535	1.15127
$59$	353.130	0.779215	0.389607	−	0.920981i	$-0.372611\pi$
$59$	353.130	0.779215	0.389607	+	0.920981i	$0.372611\pi$
$60$	0	0
$61$	−80.6108	−0.169199	−0.0845996	−	0.996415i	$-0.526961\pi$
$61$	−80.6108	−0.169199	−0.0845996	+	0.996415i	$0.526961\pi$
$62$	441.392	0.904142
$63$	1861.29	3.72224
$64$	64.0000	0.125000
$65$	0	0
$66$	−352.106	−0.656685
$67$	820.011	1.49523	0.747614	−	0.664133i	$-0.231199\pi$
$67$	820.011	1.49523	0.747614	+	0.664133i	$0.231199\pi$
$68$	−193.096	−0.344358
$69$	−232.891	−0.406331
$70$	0	0
$71$	614.845	1.02773	0.513864	−	0.857872i	$-0.328214\pi$
$71$	614.845	1.02773	0.513864	+	0.857872i	$0.328214\pi$
$72$	−604.241	−0.989034
$73$	−511.586	−0.820228	−0.410114	−	0.912034i	$-0.634511\pi$
$73$	−511.586	−0.820228	−0.410114	+	0.912034i	$0.634511\pi$
$74$	−845.808	−1.32869
$75$	0	0
$76$	317.269	0.478859
$77$	−428.462	−0.634127
$78$	−81.1472	−0.117796
$79$	160.464	0.228526	0.114263	−	0.993451i	$-0.463549\pi$
$79$	160.464	0.228526	0.114263	+	0.993451i	$0.463549\pi$
$80$	0	0
$81$	2936.48	4.02809
$82$	340.502	0.458562
$83$	32.5646	0.0430654	0.0215327	−	0.999768i	$-0.493145\pi$
$83$	32.5646	0.0430654	0.0215327	+	0.999768i	$0.493145\pi$
$84$	−998.115	−1.29647
$85$	0	0
$86$	−457.840	−0.574072
$87$	2574.64	3.17276
$88$	139.094	0.168494
$89$	−25.0375	−0.0298199	−0.0149100	−	0.999889i	$-0.504746\pi$
$89$	−25.0375	−0.0298199	−0.0149100	+	0.999889i	$0.504746\pi$
$90$	0	0
$91$	−98.7445	−0.113750
$92$	92.0000	0.104257
$93$	2234.70	2.49170
$94$	1160.17	1.27301
$95$	0	0
$96$	324.023	0.344484
$97$	249.798	0.261476	0.130738	−	0.991417i	$-0.458265\pi$
$97$	249.798	0.261476	0.130738	+	0.991417i	$0.458265\pi$
$98$	−528.563	−0.544826
$99$	−1313.22	−1.33317
$100$	0	0
$101$	620.493	0.611300	0.305650	−	0.952144i	$-0.401126\pi$
$101$	620.493	0.611300	0.305650	+	0.952144i	$0.401126\pi$
$102$	−977.618	−0.949006
$103$	−1473.24	−1.40935	−0.704673	−	0.709532i	$-0.748906\pi$
$103$	−1473.24	−1.40935	−0.704673	+	0.709532i	$0.748906\pi$
$104$	32.0559	0.0302244
$105$	0	0
$106$	−520.708	−0.477129
$107$	940.141	0.849410	0.424705	−	0.905332i	$-0.360378\pi$
$107$	940.141	0.849410	0.424705	+	0.905332i	$0.360378\pi$
$108$	−1965.61	−1.75130
$109$	−636.264	−0.559111	−0.279555	−	0.960130i	$-0.590187\pi$
$109$	−636.264	−0.559111	−0.279555	+	0.960130i	$0.590187\pi$
$110$	0	0
$111$	−4282.20	−3.66170
$112$	394.289	0.332650
$113$	−832.451	−0.693013	−0.346506	−	0.938048i	$-0.612632\pi$
$113$	−832.451	−0.693013	−0.346506	+	0.938048i	$0.612632\pi$
$114$	1606.29	1.31967
$115$	0	0
$116$	−1017.07	−0.814074
$117$	−302.648	−0.239144
$118$	−706.261	−0.550988
$119$	−1189.62	−0.916406
$120$	0	0
$121$	−1028.70	−0.772879
$122$	161.222	0.119642
$123$	1723.91	1.26374
$124$	−882.783	−0.639325
$125$	0	0
$126$	−3722.59	−2.63202
$127$	−1614.60	−1.12813	−0.564067	−	0.825729i	$-0.690764\pi$
$127$	−1614.60	−1.12813	−0.564067	+	0.825729i	$0.690764\pi$
$128$	−128.000	−0.0883883
$129$	−2317.98	−1.58207
$130$	0	0
$131$	−1974.56	−1.31693	−0.658464	−	0.752612i	$-0.728794\pi$
$131$	−1974.56	−1.31693	−0.658464	+	0.752612i	$0.728794\pi$
$132$	704.211	0.464346
$133$	1954.62	1.27434
$134$	−1640.02	−1.05729
$135$	0	0
$136$	386.192	0.243498
$137$	−753.805	−0.470087	−0.235044	−	0.971985i	$-0.575523\pi$
$137$	−753.805	−0.470087	−0.235044	+	0.971985i	$0.575523\pi$
$138$	465.783	0.287319
$139$	1014.10	0.618810	0.309405	−	0.950930i	$-0.399870\pi$
$139$	1014.10	0.618810	0.309405	+	0.950930i	$0.399870\pi$
$140$	0	0
$141$	5873.80	3.50825
$142$	−1229.69	−0.726714
$143$	69.6683	0.0407410
$144$	1208.48	0.699352
$145$	0	0
$146$	1023.17	0.579989
$147$	−2676.04	−1.50147
$148$	1691.62	0.939527
$149$	−2771.80	−1.52399	−0.761995	−	0.647583i	$-0.775780\pi$
$149$	−2771.80	−1.52399	−0.761995	+	0.647583i	$0.775780\pi$
$150$	0	0
$151$	3108.62	1.67534	0.837668	−	0.546180i	$-0.183918\pi$
$151$	3108.62	1.67534	0.837668	+	0.546180i	$0.183918\pi$
$152$	−634.538	−0.338604
$153$	−3646.14	−1.92662
$154$	856.924	0.448396
$155$	0	0
$156$	162.294	0.0832946
$157$	−3712.87	−1.88739	−0.943693	−	0.330822i	$-0.892674\pi$
$157$	−3712.87	−1.88739	−0.943693	+	0.330822i	$0.892674\pi$
$158$	−320.928	−0.161593
$159$	−2636.27	−1.31490
$160$	0	0
$161$	566.791	0.277450
$162$	−5872.96	−2.84829
$163$	−915.791	−0.440063	−0.220032	−	0.975493i	$-0.570616\pi$
$163$	−915.791	−0.440063	−0.220032	+	0.975493i	$0.570616\pi$
$164$	−681.003	−0.324252
$165$	0	0
$166$	−65.1292	−0.0304518
$167$	−1432.32	−0.663688	−0.331844	−	0.943334i	$-0.607671\pi$
$167$	−1432.32	−0.663688	−0.331844	+	0.943334i	$0.607671\pi$
$168$	1996.23	0.916741
$169$	−2180.94	−0.992692
$170$	0	0
$171$	5990.84	2.67913
$172$	915.680	0.405930
$173$	3479.54	1.52916	0.764580	−	0.644529i	$-0.222946\pi$
$173$	3479.54	1.52916	0.764580	+	0.644529i	$0.222946\pi$
$174$	−5149.28	−2.24348
$175$	0	0
$176$	−278.187	−0.119143
$177$	−3575.70	−1.51845
$178$	50.0751	0.0210859
$179$	3642.71	1.52105	0.760527	−	0.649307i	$-0.224941\pi$
$179$	3642.71	1.52105	0.760527	+	0.649307i	$0.224941\pi$
$180$	0	0
$181$	2409.15	0.989342	0.494671	−	0.869080i	$-0.335289\pi$
$181$	2409.15	0.989342	0.494671	+	0.869080i	$0.335289\pi$
$182$	197.489	0.0804333
$183$	816.241	0.329717
$184$	−184.000	−0.0737210
$185$	0	0
$186$	−4469.41	−1.76190
$187$	839.326	0.328223
$188$	−2320.35	−0.900154
$189$	−12109.6	−4.66057
$190$	0	0
$191$	−189.608	−0.0718299	−0.0359150	−	0.999355i	$-0.511435\pi$
$191$	−189.608	−0.0718299	−0.0359150	+	0.999355i	$0.511435\pi$
$192$	−648.046	−0.243587
$193$	1855.45	0.692012	0.346006	−	0.938232i	$-0.387538\pi$
$193$	1855.45	0.692012	0.346006	+	0.938232i	$0.387538\pi$
$194$	−499.597	−0.184891
$195$	0	0
$196$	1057.13	0.385250
$197$	−2429.85	−0.878779	−0.439390	−	0.898297i	$-0.644805\pi$
$197$	−2429.85	−0.878779	−0.439390	+	0.898297i	$0.644805\pi$
$198$	2626.44	0.942691
$199$	4333.49	1.54368	0.771842	−	0.635815i	$-0.219336\pi$
$199$	4333.49	1.54368	0.771842	+	0.635815i	$0.219336\pi$
$200$	0	0
$201$	−8303.20	−2.91374
$202$	−1240.99	−0.432255
$203$	−6265.94	−2.16642
$204$	1955.24	0.671048
$205$	0	0
$206$	2946.48	0.996558
$207$	1737.19	0.583300
$208$	−64.1118	−0.0213719
$209$	−1379.07	−0.456421
$210$	0	0
$211$	−816.788	−0.266493	−0.133247	−	0.991083i	$-0.542540\pi$
$211$	−816.788	−0.266493	−0.133247	+	0.991083i	$0.542540\pi$
$212$	1041.42	0.337381
$213$	−6225.75	−2.00273
$214$	−1880.28	−0.600624
$215$	0	0
$216$	3931.21	1.23836
$217$	−5438.63	−1.70137
$218$	1272.53	0.395351
$219$	5180.18	1.59837
$220$	0	0
$221$	193.433	0.0588767
$222$	8564.41	2.58921
$223$	4513.80	1.35546	0.677728	−	0.735313i	$-0.262965\pi$
$223$	4513.80	1.35546	0.677728	+	0.735313i	$0.262965\pi$
$224$	−788.579	−0.235219
$225$	0	0
$226$	1664.90	0.490034
$227$	−2792.85	−0.816599	−0.408300	−	0.912848i	$-0.633878\pi$
$227$	−2792.85	−0.816599	−0.408300	+	0.912848i	$0.633878\pi$
$228$	−3212.57	−0.933149
$229$	1404.50	0.405292	0.202646	−	0.979252i	$-0.435046\pi$
$229$	1404.50	0.405292	0.202646	+	0.979252i	$0.435046\pi$
$230$	0	0
$231$	4338.48	1.23572
$232$	2034.14	0.575637
$233$	−1073.79	−0.301916	−0.150958	−	0.988540i	$-0.548236\pi$
$233$	−1073.79	−0.301916	−0.150958	+	0.988540i	$0.548236\pi$
$234$	605.296	0.169100
$235$	0	0
$236$	1412.52	0.389607
$237$	−1624.81	−0.445328
$238$	2379.24	0.647997
$239$	−2573.18	−0.696423	−0.348212	−	0.937416i	$-0.613211\pi$
$239$	−2573.18	−0.696423	−0.348212	+	0.937416i	$0.613211\pi$
$240$	0	0
$241$	−696.127	−0.186064	−0.0930321	−	0.995663i	$-0.529656\pi$
$241$	−696.127	−0.186064	−0.0930321	+	0.995663i	$0.529656\pi$
$242$	2057.40	0.546508
$243$	−16466.1	−4.34692
$244$	−322.443	−0.0845996
$245$	0	0
$246$	−3447.82	−0.893598
$247$	−317.823	−0.0818729
$248$	1765.57	0.452071
$249$	−329.740	−0.0839213
$250$	0	0
$251$	6467.81	1.62647	0.813236	−	0.581934i	$-0.197704\pi$
$251$	6467.81	1.62647	0.813236	+	0.581934i	$0.197704\pi$
$252$	7445.18	1.86112
$253$	−399.894	−0.0993721
$254$	3229.21	0.797711
$255$	0	0
$256$	256.000	0.0625000
$257$	−5332.72	−1.29434	−0.647172	−	0.762344i	$-0.724048\pi$
$257$	−5332.72	−1.29434	−0.647172	+	0.762344i	$0.724048\pi$
$258$	4635.96	1.11869
$259$	10421.7	2.50027
$260$	0	0
$261$	−19204.8	−4.55460
$262$	3949.11	0.931209
$263$	6872.85	1.61140	0.805699	−	0.592325i	$-0.201790\pi$
$263$	6872.85	1.61140	0.805699	+	0.592325i	$0.201790\pi$
$264$	−1408.42	−0.328342
$265$	0	0
$266$	−3909.24	−0.901094
$267$	253.523	0.0581099
$268$	3280.04	0.747614
$269$	−1926.13	−0.436572	−0.218286	−	0.975885i	$-0.570047\pi$
$269$	−1926.13	−0.436572	−0.218286	+	0.975885i	$0.570047\pi$
$270$	0	0
$271$	−3653.06	−0.818847	−0.409423	−	0.912344i	$-0.634270\pi$
$271$	−3653.06	−0.818847	−0.409423	+	0.912344i	$0.634270\pi$
$272$	−772.384	−0.172179
$273$	999.859	0.221664
$274$	1507.61	0.332402
$275$	0	0
$276$	−931.566	−0.203165
$277$	1047.37	0.227185	0.113592	−	0.993527i	$-0.463764\pi$
$277$	1047.37	0.227185	0.113592	+	0.993527i	$0.463764\pi$
$278$	−2028.19	−0.437565
$279$	−16669.2	−3.57691
$280$	0	0
$281$	−2758.90	−0.585701	−0.292851	−	0.956158i	$-0.594604\pi$
$281$	−2758.90	−0.585701	−0.292851	+	0.956158i	$0.594604\pi$
$282$	−11747.6	−2.48071
$283$	−3355.58	−0.704837	−0.352418	−	0.935843i	$-0.614641\pi$
$283$	−3355.58	−0.704837	−0.352418	+	0.935843i	$0.614641\pi$
$284$	2459.38	0.513864
$285$	0	0
$286$	−139.337	−0.0288082
$287$	−4195.50	−0.862902
$288$	−2416.96	−0.494517
$289$	−2582.62	−0.525670
$290$	0	0
$291$	−2529.39	−0.509537
$292$	−2046.35	−0.410114
$293$	−419.625	−0.0836681	−0.0418341	−	0.999125i	$-0.513320\pi$
$293$	−419.625	−0.0836681	−0.0418341	+	0.999125i	$0.513320\pi$
$294$	5352.08	1.06170
$295$	0	0
$296$	−3383.23	−0.664346
$297$	8543.85	1.66924
$298$	5543.59	1.07762
$299$	−92.1607	−0.0178254
$300$	0	0
$301$	5641.30	1.08026
$302$	−6217.24	−1.18464
$303$	−6282.93	−1.19124
$304$	1269.08	0.239429
$305$	0	0
$306$	7292.28	1.36233
$307$	4133.41	0.768425	0.384212	−	0.923245i	$-0.374473\pi$
$307$	4133.41	0.768425	0.384212	+	0.923245i	$0.374473\pi$
$308$	−1713.85	−0.317064
$309$	14917.6	2.74639
$310$	0	0
$311$	−6991.03	−1.27468	−0.637339	−	0.770584i	$-0.719965\pi$
$311$	−6991.03	−1.27468	−0.637339	+	0.770584i	$0.719965\pi$
$312$	−324.589	−0.0588982
$313$	−9380.53	−1.69399	−0.846996	−	0.531600i	$-0.821591\pi$
$313$	−9380.53	−1.69399	−0.846996	+	0.531600i	$0.821591\pi$
$314$	7425.75	1.33458
$315$	0	0
$316$	641.855	0.114263
$317$	7995.35	1.41660	0.708302	−	0.705910i	$-0.249462\pi$
$317$	7995.35	1.41660	0.708302	+	0.705910i	$0.249462\pi$
$318$	5272.54	0.929778
$319$	4420.87	0.775929
$320$	0	0
$321$	−9519.60	−1.65524
$322$	−1133.58	−0.196186
$323$	−3828.96	−0.659595
$324$	11745.9	2.01405
$325$	0	0
$326$	1831.58	0.311172
$327$	6442.63	1.08954
$328$	1362.01	0.229281
$329$	−14295.1	−2.39549
$330$	0	0
$331$	−4798.86	−0.796886	−0.398443	−	0.917193i	$-0.630449\pi$
$331$	−4798.86	−0.796886	−0.398443	+	0.917193i	$0.630449\pi$
$332$	130.258	0.0215327
$333$	31942.0	5.25648
$334$	2864.63	0.469298
$335$	0	0
$336$	−3992.46	−0.648234
$337$	−6895.51	−1.11461	−0.557303	−	0.830309i	$-0.688164\pi$
$337$	−6895.51	−1.11461	−0.557303	+	0.830309i	$0.688164\pi$
$338$	4361.89	0.701939
$339$	8429.16	1.35047
$340$	0	0
$341$	3837.17	0.609368
$342$	−11981.7	−1.89443
$343$	−1939.86	−0.305372
$344$	−1831.36	−0.287036
$345$	0	0
$346$	−6959.08	−1.08128
$347$	8744.17	1.35277	0.676386	−	0.736548i	$-0.263545\pi$
$347$	8744.17	1.35277	0.676386	+	0.736548i	$0.263545\pi$
$348$	10298.6	1.58638
$349$	−6387.08	−0.979635	−0.489818	−	0.871825i	$-0.662937\pi$
$349$	−6387.08	−0.979635	−0.489818	+	0.871825i	$0.662937\pi$
$350$	0	0
$351$	1969.04	0.299429
$352$	556.375	0.0842468
$353$	589.384	0.0888661	0.0444331	−	0.999012i	$-0.485852\pi$
$353$	589.384	0.0888661	0.0444331	+	0.999012i	$0.485852\pi$
$354$	7151.39	1.07371
$355$	0	0
$356$	−100.150	−0.0149100
$357$	12045.8	1.78580
$358$	−7285.41	−1.07555
$359$	7214.13	1.06058	0.530289	−	0.847817i	$-0.322083\pi$
$359$	7214.13	1.06058	0.530289	+	0.847817i	$0.322083\pi$
$360$	0	0
$361$	−567.774	−0.0827780
$362$	−4818.30	−0.699570
$363$	10416.3	1.50611
$364$	−394.978	−0.0568749
$365$	0	0
$366$	−1632.48	−0.233145
$367$	−12356.4	−1.75750	−0.878748	−	0.477286i	$-0.841621\pi$
$367$	−12356.4	−1.75750	−0.878748	+	0.477286i	$0.841621\pi$
$368$	368.000	0.0521286
$369$	−12859.1	−1.81413
$370$	0	0
$371$	6415.92	0.897839
$372$	8938.81	1.24585
$373$	−7145.98	−0.991969	−0.495985	−	0.868331i	$-0.665193\pi$
$373$	−7145.98	−0.991969	−0.495985	+	0.868331i	$0.665193\pi$
$374$	−1678.65	−0.232088
$375$	0	0
$376$	4640.70	0.636505
$377$	1018.85	0.139186
$378$	24219.3	3.29552
$379$	2170.69	0.294197	0.147099	−	0.989122i	$-0.453007\pi$
$379$	2170.69	0.294197	0.147099	+	0.989122i	$0.453007\pi$
$380$	0	0
$381$	16349.0	2.19839
$382$	379.215	0.0507914
$383$	7967.21	1.06294	0.531469	−	0.847078i	$-0.321640\pi$
$383$	7967.21	1.06294	0.531469	+	0.847078i	$0.321640\pi$
$384$	1296.09	0.172242
$385$	0	0
$386$	−3710.90	−0.489326
$387$	17290.3	2.27111
$388$	999.193	0.130738
$389$	−568.951	−0.0741567	−0.0370783	−	0.999312i	$-0.511805\pi$
$389$	−568.951	−0.0741567	−0.0370783	+	0.999312i	$0.511805\pi$
$390$	0	0
$391$	−1110.30	−0.143607
$392$	−2114.25	−0.272413
$393$	19993.8	2.56629
$394$	4859.70	0.621391
$395$	0	0
$396$	−5252.88	−0.666583
$397$	−8564.88	−1.08277	−0.541384	−	0.840775i	$-0.682100\pi$
$397$	−8564.88	−1.08277	−0.541384	+	0.840775i	$0.682100\pi$
$398$	−8666.98	−1.09155
$399$	−19791.9	−2.48330
$400$	0	0
$401$	−12455.6	−1.55113	−0.775563	−	0.631270i	$-0.782534\pi$
$401$	−12455.6	−1.55113	−0.775563	+	0.631270i	$0.782534\pi$
$402$	16606.4	2.06033
$403$	884.325	0.109309
$404$	2481.97	0.305650
$405$	0	0
$406$	12531.9	1.53189
$407$	−7352.91	−0.895504
$408$	−3910.47	−0.474503
$409$	11838.9	1.43129	0.715645	−	0.698465i	$-0.246133\pi$
$409$	11838.9	1.43129	0.715645	+	0.698465i	$0.246133\pi$
$410$	0	0
$411$	7632.82	0.916056
$412$	−5892.96	−0.704673
$413$	8702.22	1.03682
$414$	−3474.38	−0.412456
$415$	0	0
$416$	128.224	0.0151122
$417$	−10268.5	−1.20587
$418$	2758.13	0.322738
$419$	−1531.77	−0.178596	−0.0892981	−	0.996005i	$-0.528462\pi$
$419$	−1531.77	−0.178596	−0.0892981	+	0.996005i	$0.528462\pi$
$420$	0	0
$421$	−9985.06	−1.15592	−0.577959	−	0.816066i	$-0.696151\pi$
$421$	−9985.06	−1.15592	−0.577959	+	0.816066i	$0.696151\pi$
$422$	1633.58	0.188439
$423$	−43814.0	−5.03620
$424$	−2082.83	−0.238564
$425$	0	0
$426$	12451.5	1.41614
$427$	−1986.50	−0.225137
$428$	3760.56	0.424705
$429$	−705.441	−0.0793917
$430$	0	0
$431$	−13786.7	−1.54079	−0.770396	−	0.637566i	$-0.779941\pi$
$431$	−13786.7	−1.54079	−0.770396	+	0.637566i	$0.779941\pi$
$432$	−7862.42	−0.875651
$433$	2621.92	0.290996	0.145498	−	0.989359i	$-0.453522\pi$
$433$	2621.92	0.290996	0.145498	+	0.989359i	$0.453522\pi$
$434$	10877.3	1.20305
$435$	0	0
$436$	−2545.06	−0.279555
$437$	1824.30	0.199698
$438$	−10360.4	−1.13022
$439$	12062.7	1.31143	0.655717	−	0.755007i	$-0.272367\pi$
$439$	12062.7	1.31143	0.655717	+	0.755007i	$0.272367\pi$
$440$	0	0
$441$	19961.2	2.15541
$442$	−386.867	−0.0416321
$443$	−3659.26	−0.392453	−0.196227	−	0.980559i	$-0.562869\pi$
$443$	−3659.26	−0.392453	−0.196227	+	0.980559i	$0.562869\pi$
$444$	−17128.8	−1.83085
$445$	0	0
$446$	−9027.60	−0.958452
$447$	28066.4	2.96979
$448$	1577.16	0.166325
$449$	−10529.6	−1.10674	−0.553368	−	0.832937i	$-0.686658\pi$
$449$	−10529.6	−1.10674	−0.553368	+	0.832937i	$0.686658\pi$
$450$	0	0
$451$	2960.10	0.309059
$452$	−3329.81	−0.346506
$453$	−31477.0	−3.26472
$454$	5585.70	0.577423
$455$	0	0
$456$	6425.15	0.659836
$457$	−6443.23	−0.659522	−0.329761	−	0.944064i	$-0.606968\pi$
$457$	−6443.23	−0.659522	−0.329761	+	0.944064i	$0.606968\pi$
$458$	−2809.00	−0.286585
$459$	23721.9	2.41230
$460$	0	0
$461$	−3263.86	−0.329747	−0.164873	−	0.986315i	$-0.552722\pi$
$461$	−3263.86	−0.329747	−0.164873	+	0.986315i	$0.552722\pi$
$462$	−8676.97	−0.873786
$463$	−9518.12	−0.955388	−0.477694	−	0.878526i	$-0.658527\pi$
$463$	−9518.12	−0.955388	−0.477694	+	0.878526i	$0.658527\pi$
$464$	−4068.28	−0.407037
$465$	0	0
$466$	2147.59	0.213487
$467$	−19092.6	−1.89187	−0.945934	−	0.324360i	$-0.894851\pi$
$467$	−19092.6	−1.89187	−0.945934	+	0.324360i	$0.894851\pi$
$468$	−1210.59	−0.119572
$469$	20207.6	1.98955
$470$	0	0
$471$	37595.5	3.67794
$472$	−2825.04	−0.275494
$473$	−3980.17	−0.386910
$474$	3249.62	0.314895
$475$	0	0
$476$	−4758.48	−0.458203
$477$	19664.6	1.88758
$478$	5146.36	0.492446
$479$	−6324.81	−0.603315	−0.301658	−	0.953416i	$-0.597540\pi$
$479$	−6324.81	−0.603315	−0.301658	+	0.953416i	$0.597540\pi$
$480$	0	0
$481$	−1694.57	−0.160636
$482$	1392.25	0.131567
$483$	−5739.16	−0.540664
$484$	−4114.81	−0.386440
$485$	0	0
$486$	32932.2	3.07373
$487$	7873.07	0.732573	0.366286	−	0.930502i	$-0.380629\pi$
$487$	7873.07	0.732573	0.366286	+	0.930502i	$0.380629\pi$
$488$	644.886	0.0598209
$489$	9273.04	0.857549
$490$	0	0
$491$	−3556.82	−0.326918	−0.163459	−	0.986550i	$-0.552265\pi$
$491$	−3556.82	−0.326918	−0.163459	+	0.986550i	$0.552265\pi$
$492$	6895.64	0.631869
$493$	12274.5	1.12133
$494$	635.646	0.0578929
$495$	0	0
$496$	−3531.13	−0.319662
$497$	15151.7	1.36750
$498$	659.479	0.0593413
$499$	−1933.37	−0.173446	−0.0867231	−	0.996232i	$-0.527640\pi$
$499$	−1933.37	−0.173446	−0.0867231	+	0.996232i	$0.527640\pi$
$500$	0	0
$501$	14503.2	1.29332
$502$	−12935.6	−1.15009
$503$	−2114.36	−0.187425	−0.0937123	−	0.995599i	$-0.529873\pi$
$503$	−2114.36	−0.187425	−0.0937123	+	0.995599i	$0.529873\pi$
$504$	−14890.4	−1.31601
$505$	0	0
$506$	799.789	0.0702667
$507$	22083.6	1.93445
$508$	−6458.42	−0.564067
$509$	−316.452	−0.0275570	−0.0137785	−	0.999905i	$-0.504386\pi$
$509$	−316.452	−0.0275570	−0.0137785	+	0.999905i	$0.504386\pi$
$510$	0	0
$511$	−12607.1	−1.09140
$512$	−512.000	−0.0441942
$513$	−38976.6	−3.35450
$514$	10665.4	0.915239
$515$	0	0
$516$	−9271.92	−0.791034
$517$	10085.8	0.857976
$518$	−20843.3	−1.76796
$519$	−35232.8	−2.97987
$520$	0	0
$521$	−309.041	−0.0259872	−0.0129936	−	0.999916i	$-0.504136\pi$
$521$	−309.041	−0.0259872	−0.0129936	+	0.999916i	$0.504136\pi$
$522$	38409.7	3.22059
$523$	5892.46	0.492656	0.246328	−	0.969186i	$-0.420776\pi$
$523$	5892.46	0.492656	0.246328	+	0.969186i	$0.420776\pi$
$524$	−7898.22	−0.658464
$525$	0	0
$526$	−13745.7	−1.13943
$527$	10653.9	0.880626
$528$	2816.84	0.232173
$529$	529.000	0.0434783
$530$	0	0
$531$	26672.0	2.17978
$532$	7818.49	0.637170
$533$	682.193	0.0554391
$534$	−507.046	−0.0410899
$535$	0	0
$536$	−6560.09	−0.528643
$537$	−36885.0	−2.96407
$538$	3852.25	0.308703
$539$	−4594.99	−0.367199
$540$	0	0
$541$	1560.55	0.124017	0.0620085	−	0.998076i	$-0.480249\pi$
$541$	1560.55	0.124017	0.0620085	+	0.998076i	$0.480249\pi$
$542$	7306.12	0.579012
$543$	−24394.4	−1.92792
$544$	1544.77	0.121749
$545$	0	0
$546$	−1999.72	−0.156740
$547$	−17756.3	−1.38795	−0.693973	−	0.720001i	$-0.744141\pi$
$547$	−17756.3	−1.38795	−0.693973	+	0.720001i	$0.744141\pi$
$548$	−3015.22	−0.235044
$549$	−6088.54	−0.473319
$550$	0	0
$551$	−20167.8	−1.55930
$552$	1863.13	0.143660
$553$	3954.32	0.304078
$554$	−2094.73	−0.160644
$555$	0	0
$556$	4056.39	0.309405
$557$	−1212.77	−0.0922559	−0.0461280	−	0.998936i	$-0.514688\pi$
$557$	−1212.77	−0.0922559	−0.0461280	+	0.998936i	$0.514688\pi$
$558$	33338.3	2.52926
$559$	−917.280	−0.0694040
$560$	0	0
$561$	−8498.78	−0.639605
$562$	5517.80	0.414153
$563$	−12558.2	−0.940078	−0.470039	−	0.882646i	$-0.655760\pi$
$563$	−12558.2	−0.940078	−0.470039	+	0.882646i	$0.655760\pi$
$564$	23495.2	1.75412
$565$	0	0
$566$	6711.17	0.498395
$567$	72363.9	5.35978
$568$	−4918.76	−0.363357
$569$	9776.72	0.720319	0.360159	−	0.932891i	$-0.382722\pi$
$569$	9776.72	0.720319	0.360159	+	0.932891i	$0.382722\pi$
$570$	0	0
$571$	−18733.9	−1.37301	−0.686505	−	0.727125i	$-0.740856\pi$
$571$	−18733.9	−1.37301	−0.686505	+	0.727125i	$0.740856\pi$
$572$	278.673	0.0203705
$573$	1919.91	0.139975
$574$	8391.01	0.610164
$575$	0	0
$576$	4833.92	0.349676
$577$	−5113.58	−0.368944	−0.184472	−	0.982838i	$-0.559058\pi$
$577$	−5113.58	−0.368944	−0.184472	+	0.982838i	$0.559058\pi$
$578$	5165.24	0.371705
$579$	−18787.8	−1.34852
$580$	0	0
$581$	802.492	0.0573029
$582$	5058.77	0.360297
$583$	−4526.70	−0.321572
$584$	4092.69	0.289995
$585$	0	0
$586$	839.250	0.0591623
$587$	5379.95	0.378287	0.189143	−	0.981949i	$-0.439429\pi$
$587$	5379.95	0.378287	0.189143	+	0.981949i	$0.439429\pi$
$588$	−10704.2	−0.750735
$589$	−17505.0	−1.22458
$590$	0	0
$591$	24603.9	1.71247
$592$	6766.46	0.469763
$593$	15060.3	1.04292	0.521461	−	0.853275i	$-0.325387\pi$
$593$	15060.3	1.04292	0.521461	+	0.853275i	$0.325387\pi$
$594$	−17087.7	−1.18033
$595$	0	0
$596$	−11087.2	−0.761995
$597$	−43879.7	−3.00817
$598$	184.321	0.0126045
$599$	−3772.04	−0.257298	−0.128649	−	0.991690i	$-0.541064\pi$
$599$	−3772.04	−0.257298	−0.128649	+	0.991690i	$0.541064\pi$
$600$	0	0
$601$	−14663.9	−0.995261	−0.497631	−	0.867389i	$-0.665796\pi$
$601$	−14663.9	−0.995261	−0.497631	+	0.867389i	$0.665796\pi$
$602$	−11282.6	−0.763861
$603$	61935.5	4.18277
$604$	12434.5	0.837668
$605$	0	0
$606$	12565.9	0.842332
$607$	24514.5	1.63923	0.819614	−	0.572915i	$-0.194188\pi$
$607$	24514.5	1.63923	0.819614	+	0.572915i	$0.194188\pi$
$608$	−2538.15	−0.169302
$609$	63447.1	4.22168
$610$	0	0
$611$	2324.40	0.153904
$612$	−14584.6	−0.963310
$613$	−14451.2	−0.952167	−0.476084	−	0.879400i	$-0.657944\pi$
$613$	−14451.2	−0.952167	−0.476084	+	0.879400i	$0.657944\pi$
$614$	−8266.83	−0.543358
$615$	0	0
$616$	3427.70	0.224198
$617$	−3292.19	−0.214811	−0.107406	−	0.994215i	$-0.534254\pi$
$617$	−3292.19	−0.214811	−0.107406	+	0.994215i	$0.534254\pi$
$618$	−29835.2	−1.94199
$619$	−12595.5	−0.817858	−0.408929	−	0.912566i	$-0.634098\pi$
$619$	−12595.5	−0.817858	−0.408929	+	0.912566i	$0.634098\pi$
$620$	0	0
$621$	−11302.2	−0.730343
$622$	13982.1	0.901333
$623$	−617.002	−0.0396785
$624$	649.178	0.0416473
$625$	0	0
$626$	18761.1	1.19783
$627$	13964.0	0.889425
$628$	−14851.5	−0.943693
$629$	−20415.3	−1.29413
$630$	0	0
$631$	19889.3	1.25480	0.627401	−	0.778697i	$-0.284119\pi$
$631$	19889.3	1.25480	0.627401	+	0.778697i	$0.284119\pi$
$632$	−1283.71	−0.0807963
$633$	8270.56	0.519313
$634$	−15990.7	−1.00169
$635$	0	0
$636$	−10545.1	−0.657452
$637$	−1058.97	−0.0658682
$638$	−8841.75	−0.548665
$639$	46439.3	2.87498
$640$	0	0
$641$	−19276.0	−1.18776	−0.593880	−	0.804553i	$-0.702405\pi$
$641$	−19276.0	−1.18776	−0.593880	+	0.804553i	$0.702405\pi$
$642$	19039.2	1.17043
$643$	10219.2	0.626758	0.313379	−	0.949628i	$-0.398539\pi$
$643$	10219.2	0.626758	0.313379	+	0.949628i	$0.398539\pi$
$644$	2267.16	0.138725
$645$	0	0
$646$	7657.93	0.466404
$647$	−20818.4	−1.26500	−0.632500	−	0.774560i	$-0.717971\pi$
$647$	−20818.4	−1.26500	−0.632500	+	0.774560i	$0.717971\pi$
$648$	−23491.8	−1.42415
$649$	−6139.77	−0.371352
$650$	0	0
$651$	55070.0	3.31546
$652$	−3663.17	−0.220032
$653$	15135.3	0.907031	0.453516	−	0.891248i	$-0.350170\pi$
$653$	15135.3	0.907031	0.453516	+	0.891248i	$0.350170\pi$
$654$	−12885.3	−0.770418
$655$	0	0
$656$	−2724.01	−0.162126
$657$	−38640.2	−2.29451
$658$	28590.3	1.69387
$659$	−13207.9	−0.780737	−0.390369	−	0.920659i	$-0.627652\pi$
$659$	−13207.9	−0.780737	−0.390369	+	0.920659i	$0.627652\pi$
$660$	0	0
$661$	6671.34	0.392564	0.196282	−	0.980547i	$-0.437113\pi$
$661$	6671.34	0.392564	0.196282	+	0.980547i	$0.437113\pi$
$662$	9597.72	0.563484
$663$	−1958.65	−0.114733
$664$	−260.517	−0.0152259
$665$	0	0
$666$	−63883.9	−3.71689
$667$	−5848.15	−0.339492
$668$	−5729.26	−0.331844
$669$	−45705.5	−2.64137
$670$	0	0
$671$	1401.56	0.0806355
$672$	7984.92	0.458371
$673$	9534.41	0.546099	0.273049	−	0.962000i	$-0.411968\pi$
$673$	9534.41	0.546099	0.273049	+	0.962000i	$0.411968\pi$
$674$	13791.0	0.788146
$675$	0	0
$676$	−8723.78	−0.496346
$677$	−17748.4	−1.00757	−0.503787	−	0.863828i	$-0.668060\pi$
$677$	−17748.4	−1.00757	−0.503787	+	0.863828i	$0.668060\pi$
$678$	−16858.3	−0.954927
$679$	6155.80	0.347920
$680$	0	0
$681$	28279.6	1.59130
$682$	−7674.35	−0.430888
$683$	−9583.57	−0.536904	−0.268452	−	0.963293i	$-0.586512\pi$
$683$	−9583.57	−0.536904	−0.268452	+	0.963293i	$0.586512\pi$
$684$	23963.3	1.33956
$685$	0	0
$686$	3879.73	0.215931
$687$	−14221.5	−0.789790
$688$	3662.72	0.202965
$689$	−1043.24	−0.0576837
$690$	0	0
$691$	9410.74	0.518092	0.259046	−	0.965865i	$-0.416592\pi$
$691$	9410.74	0.518092	0.259046	+	0.965865i	$0.416592\pi$
$692$	13918.2	0.764580
$693$	−32361.8	−1.77391
$694$	−17488.3	−0.956554
$695$	0	0
$696$	−20597.1	−1.12174
$697$	8218.69	0.446636
$698$	12774.2	0.692707
$699$	10872.9	0.588343
$700$	0	0
$701$	−19517.9	−1.05161	−0.525806	−	0.850605i	$-0.676236\pi$
$701$	−19517.9	−1.05161	−0.525806	+	0.850605i	$0.676236\pi$
$702$	−3938.08	−0.211728
$703$	33543.6	1.79960
$704$	−1112.75	−0.0595715
$705$	0	0
$706$	−1178.77	−0.0628378
$707$	15290.9	0.813397
$708$	−14302.8	−0.759225
$709$	−28430.7	−1.50598	−0.752990	−	0.658033i	$-0.771389\pi$
$709$	−28430.7	−1.50598	−0.752990	+	0.658033i	$0.771389\pi$
$710$	0	0
$711$	12119.8	0.639282
$712$	200.300	0.0105429
$713$	−5076.00	−0.266617
$714$	−24091.5	−1.26275
$715$	0	0
$716$	14570.8	0.760527
$717$	26055.3	1.35712
$718$	−14428.3	−0.749942
$719$	18716.0	0.970779	0.485390	−	0.874298i	$-0.338678\pi$
$719$	18716.0	0.970779	0.485390	+	0.874298i	$0.338678\pi$
$720$	0	0
$721$	−36305.2	−1.87528
$722$	1135.55	0.0585329
$723$	7048.78	0.362582
$724$	9636.61	0.494671
$725$	0	0
$726$	−20832.7	−1.06498
$727$	−18419.5	−0.939670	−0.469835	−	0.882754i	$-0.655687\pi$
$727$	−18419.5	−0.939670	−0.469835	+	0.882754i	$0.655687\pi$
$728$	789.956	0.0402167
$729$	87446.1	4.44272
$730$	0	0
$731$	−11050.9	−0.559141
$732$	3264.97	0.164859
$733$	21548.4	1.08582	0.542912	−	0.839790i	$-0.317322\pi$
$733$	21548.4	1.08582	0.542912	+	0.839790i	$0.317322\pi$
$734$	24712.9	1.24274
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−14257.3	−0.712584
$738$	25718.1	1.28279
$739$	12066.4	0.600636	0.300318	−	0.953839i	$-0.402907\pi$
$739$	12066.4	0.600636	0.300318	+	0.953839i	$0.402907\pi$
$740$	0	0
$741$	3218.19	0.159545
$742$	−12831.8	−0.634868
$743$	21951.0	1.08385	0.541926	−	0.840426i	$-0.317695\pi$
$743$	21951.0	1.08385	0.541926	+	0.840426i	$0.317695\pi$
$744$	−17877.6	−0.880948
$745$	0	0
$746$	14292.0	0.701428
$747$	2459.61	0.120472
$748$	3357.31	0.164111
$749$	23168.0	1.13023
$750$	0	0
$751$	−3112.43	−0.151230	−0.0756152	−	0.997137i	$-0.524092\pi$
$751$	−3112.43	−0.151230	−0.0756152	+	0.997137i	$0.524092\pi$
$752$	−9281.40	−0.450077
$753$	−65491.2	−3.16950
$754$	−2037.69	−0.0984197
$755$	0	0
$756$	−48438.6	−2.33028
$757$	7684.64	0.368960	0.184480	−	0.982836i	$-0.440940\pi$
$757$	7684.64	0.368960	0.184480	+	0.982836i	$0.440940\pi$
$758$	−4341.38	−0.208029
$759$	4049.21	0.193646
$760$	0	0
$761$	−36484.6	−1.73793	−0.868965	−	0.494874i	$-0.835214\pi$
$761$	−36484.6	−1.73793	−0.868965	+	0.494874i	$0.835214\pi$
$762$	−32698.0	−1.55450
$763$	−15679.5	−0.743954
$764$	−758.430	−0.0359150
$765$	0	0
$766$	−15934.4	−0.751611
$767$	−1414.99	−0.0666132
$768$	−2592.18	−0.121793
$769$	2004.39	0.0939925	0.0469962	−	0.998895i	$-0.485035\pi$
$769$	2004.39	0.0939925	0.0469962	+	0.998895i	$0.485035\pi$
$770$	0	0
$771$	53997.6	2.52228
$772$	7421.81	0.346006
$773$	−7716.17	−0.359031	−0.179516	−	0.983755i	$-0.557453\pi$
$773$	−7716.17	−0.359031	−0.179516	+	0.983755i	$0.557453\pi$
$774$	−34580.7	−1.60591
$775$	0	0
$776$	−1998.39	−0.0924457
$777$	−105527.	−4.87226
$778$	1137.90	0.0524367
$779$	−13503.8	−0.621084
$780$	0	0
$781$	−10690.1	−0.489786
$782$	2220.61	0.101546
$783$	124947.	5.70275
$784$	4228.51	0.192625
$785$	0	0
$786$	−39987.6	−1.81464
$787$	57.0149	0.00258241	0.00129121	−	0.999999i	$-0.499589\pi$
$787$	57.0149	0.00258241	0.00129121	+	0.999999i	$0.499589\pi$
$788$	−9719.39	−0.439390
$789$	−69592.5	−3.14012
$790$	0	0
$791$	−20514.2	−0.922124
$792$	10505.8	0.471346
$793$	323.006	0.0144644
$794$	17129.8	0.765633
$795$	0	0
$796$	17334.0	0.771842
$797$	15184.7	0.674870	0.337435	−	0.941349i	$-0.390441\pi$
$797$	15184.7	0.674870	0.337435	+	0.941349i	$0.390441\pi$
$798$	39583.9	1.75596
$799$	28003.2	1.23990
$800$	0	0
$801$	−1891.09	−0.0834186
$802$	24911.1	1.09681
$803$	8894.80	0.390898
$804$	−33212.8	−1.45687
$805$	0	0
$806$	−1768.65	−0.0772929
$807$	19503.4	0.850746
$808$	−4963.94	−0.216127
$809$	−30286.4	−1.31621	−0.658105	−	0.752926i	$-0.728642\pi$
$809$	−30286.4	−1.31621	−0.658105	+	0.752926i	$0.728642\pi$
$810$	0	0
$811$	−43936.2	−1.90235	−0.951176	−	0.308650i	$-0.900123\pi$
$811$	−43936.2	−1.90235	−0.951176	+	0.308650i	$0.900123\pi$
$812$	−25063.7	−1.08321
$813$	36989.8	1.59568
$814$	14705.8	0.633217
$815$	0	0
$816$	7820.94	0.335524
$817$	18157.3	0.777532
$818$	−23677.9	−1.01207
$819$	−7458.18	−0.318205
$820$	0	0
$821$	−5245.69	−0.222991	−0.111496	−	0.993765i	$-0.535564\pi$
$821$	−5245.69	−0.222991	−0.111496	+	0.993765i	$0.535564\pi$
$822$	−15265.6	−0.647750
$823$	−10678.0	−0.452260	−0.226130	−	0.974097i	$-0.572607\pi$
$823$	−10678.0	−0.452260	−0.226130	+	0.974097i	$0.572607\pi$
$824$	11785.9	0.498279
$825$	0	0
$826$	−17404.4	−0.733145
$827$	−3393.69	−0.142697	−0.0713484	−	0.997451i	$-0.522730\pi$
$827$	−3393.69	−0.142697	−0.0713484	+	0.997451i	$0.522730\pi$
$828$	6948.77	0.291650
$829$	9601.74	0.402270	0.201135	−	0.979564i	$-0.435537\pi$
$829$	9601.74	0.402270	0.201135	+	0.979564i	$0.435537\pi$
$830$	0	0
$831$	−10605.3	−0.442713
$832$	−256.447	−0.0106859
$833$	−12757.9	−0.530656
$834$	20536.9	0.852680
$835$	0	0
$836$	−5516.26	−0.228210
$837$	108450.	4.47860
$838$	3063.54	0.126287
$839$	11992.8	0.493489	0.246744	−	0.969081i	$-0.420639\pi$
$839$	11992.8	0.493489	0.246744	+	0.969081i	$0.420639\pi$
$840$	0	0
$841$	40263.0	1.65087
$842$	19970.1	0.817358
$843$	27935.8	1.14135
$844$	−3267.15	−0.133247
$845$	0	0
$846$	87628.1	3.56113
$847$	−25350.4	−1.02839
$848$	4165.66	0.168690
$849$	33977.7	1.37351
$850$	0	0
$851$	9726.79	0.391810
$852$	−24903.0	−1.00136
$853$	15441.5	0.619821	0.309911	−	0.950766i	$-0.399701\pi$
$853$	15441.5	0.619821	0.309911	+	0.950766i	$0.399701\pi$
$854$	3973.00	0.159196
$855$	0	0
$856$	−7521.13	−0.300312
$857$	44572.4	1.77662	0.888310	−	0.459244i	$-0.151880\pi$
$857$	44572.4	1.77662	0.888310	+	0.459244i	$0.151880\pi$
$858$	1410.88	0.0561384
$859$	2519.56	0.100077	0.0500386	−	0.998747i	$-0.484066\pi$
$859$	2519.56	0.100077	0.0500386	+	0.998747i	$0.484066\pi$
$860$	0	0
$861$	42482.5	1.68153
$862$	27573.4	1.08950
$863$	−28980.1	−1.14310	−0.571548	−	0.820568i	$-0.693657\pi$
$863$	−28980.1	−1.14310	−0.571548	+	0.820568i	$0.693657\pi$
$864$	15724.8	0.619178
$865$	0	0
$866$	−5243.83	−0.205765
$867$	26150.9	1.02437
$868$	−21754.5	−0.850687
$869$	−2789.94	−0.108909
$870$	0	0
$871$	−3285.77	−0.127823
$872$	5090.11	0.197675
$873$	18867.3	0.731455
$874$	−3648.59	−0.141208
$875$	0	0
$876$	20720.7	0.799187
$877$	7955.28	0.306307	0.153153	−	0.988202i	$-0.451057\pi$
$877$	7955.28	0.306307	0.153153	+	0.988202i	$0.451057\pi$
$878$	−24125.3	−0.927323
$879$	4249.00	0.163044
$880$	0	0
$881$	29722.0	1.13662	0.568309	−	0.822815i	$-0.307598\pi$
$881$	29722.0	1.13662	0.568309	+	0.822815i	$0.307598\pi$
$882$	−39922.4	−1.52410
$883$	19379.7	0.738596	0.369298	−	0.929311i	$-0.379598\pi$
$883$	19379.7	0.738596	0.369298	+	0.929311i	$0.379598\pi$
$884$	773.734	0.0294383
$885$	0	0
$886$	7318.53	0.277507
$887$	−22901.7	−0.866925	−0.433463	−	0.901172i	$-0.642708\pi$
$887$	−22901.7	−0.866925	−0.433463	+	0.901172i	$0.642708\pi$
$888$	34257.6	1.29461
$889$	−39788.8	−1.50110
$890$	0	0
$891$	−51055.7	−1.91967
$892$	18055.2	0.677728
$893$	−46010.9	−1.72419
$894$	−56132.9	−2.09996
$895$	0	0
$896$	−3154.31	−0.117610
$897$	933.193	0.0347362
$898$	21059.3	0.782581
$899$	56115.8	2.08183
$900$	0	0
$901$	−12568.3	−0.464719
$902$	−5920.20	−0.218538
$903$	−57122.2	−2.10510
$904$	6659.61	0.245017
$905$	0	0
$906$	62953.9	2.30850
$907$	−42542.2	−1.55743	−0.778716	−	0.627376i	$-0.784129\pi$
$907$	−42542.2	−1.55743	−0.778716	+	0.627376i	$0.784129\pi$
$908$	−11171.4	−0.408300
$909$	46865.9	1.71006
$910$	0	0
$911$	220.864	0.00803243	0.00401622	−	0.999992i	$-0.498722\pi$
$911$	220.864	0.00803243	0.00401622	+	0.999992i	$0.498722\pi$
$912$	−12850.3	−0.466574
$913$	−566.191	−0.0205237
$914$	12886.5	0.466352
$915$	0	0
$916$	5617.99	0.202646
$917$	−48659.1	−1.75231
$918$	−47443.9	−1.70575
$919$	27835.4	0.999135	0.499567	−	0.866275i	$-0.333492\pi$
$919$	27835.4	0.999135	0.499567	+	0.866275i	$0.333492\pi$
$920$	0	0
$921$	−41853.8	−1.49743
$922$	6527.72	0.233166
$923$	−2463.68	−0.0878580
$924$	17353.9	0.617860
$925$	0	0
$926$	19036.2	0.675562
$927$	−111274.	−3.94252
$928$	8136.56	0.287819
$929$	−2172.71	−0.0767323	−0.0383661	−	0.999264i	$-0.512215\pi$
$929$	−2172.71	−0.0767323	−0.0383661	+	0.999264i	$0.512215\pi$
$930$	0	0
$931$	20962.1	0.737921
$932$	−4295.17	−0.150958
$933$	70789.1	2.48396
$934$	38185.3	1.33775
$935$	0	0
$936$	2421.18	0.0845501
$937$	−54906.5	−1.91432	−0.957160	−	0.289560i	$-0.906491\pi$
$937$	−54906.5	−1.91432	−0.957160	+	0.289560i	$0.906491\pi$
$938$	−40415.2	−1.40683
$939$	94984.6	3.30107
$940$	0	0
$941$	5980.06	0.207167	0.103584	−	0.994621i	$-0.466969\pi$
$941$	5980.06	0.207167	0.103584	+	0.994621i	$0.466969\pi$
$942$	−75191.0	−2.60070
$943$	−3915.77	−0.135223
$944$	5650.09	0.194804
$945$	0	0
$946$	7960.33	0.273586
$947$	20268.6	0.695504	0.347752	−	0.937587i	$-0.386945\pi$
$947$	20268.6	0.695504	0.347752	+	0.937587i	$0.386945\pi$
$948$	−6499.24	−0.222664
$949$	2049.92	0.0701193
$950$	0	0
$951$	−80958.6	−2.76053
$952$	9516.97	0.323999
$953$	21797.9	0.740925	0.370463	−	0.928847i	$-0.379199\pi$
$953$	21797.9	0.740925	0.370463	+	0.928847i	$0.379199\pi$
$954$	−39329.1	−1.33472
$955$	0	0
$956$	−10292.7	−0.348212
$957$	−44764.5	−1.51205
$958$	12649.6	0.426608
$959$	−18576.1	−0.625499
$960$	0	0
$961$	18915.6	0.634945
$962$	3389.14	0.113587
$963$	71008.9	2.37615
$964$	−2784.51	−0.0930321
$965$	0	0
$966$	11478.3	0.382308
$967$	−48950.6	−1.62786	−0.813932	−	0.580960i	$-0.802677\pi$
$967$	−48950.6	−1.62786	−0.813932	+	0.580960i	$0.802677\pi$
$968$	8229.62	0.273254
$969$	38771.0	1.28535
$970$	0	0
$971$	−26426.2	−0.873386	−0.436693	−	0.899611i	$-0.643850\pi$
$971$	−26426.2	−0.873386	−0.436693	+	0.899611i	$0.643850\pi$
$972$	−65864.4	−2.17346
$973$	24990.5	0.823389
$974$	−15746.1	−0.518007
$975$	0	0
$976$	−1289.77	−0.0422998
$977$	−5770.09	−0.188947	−0.0944736	−	0.995527i	$-0.530117\pi$
$977$	−5770.09	−0.188947	−0.0944736	+	0.995527i	$0.530117\pi$
$978$	−18546.1	−0.606379
$979$	435.320	0.0142113
$980$	0	0
$981$	−48057.1	−1.56406
$982$	7113.64	0.231166
$983$	484.532	0.0157214	0.00786071	−	0.999969i	$-0.497498\pi$
$983$	484.532	0.0157214	0.00786071	+	0.999969i	$0.497498\pi$
$984$	−13791.3	−0.446799
$985$	0	0
$986$	−24549.0	−0.792901
$987$	144749.	4.66808
$988$	−1271.29	−0.0409365
$989$	5265.16	0.169285
$990$	0	0
$991$	−26511.5	−0.849813	−0.424907	−	0.905237i	$-0.639693\pi$
$991$	−26511.5	−0.849813	−0.424907	+	0.905237i	$0.639693\pi$
$992$	7062.27	0.226035
$993$	48591.9	1.55289
$994$	−30303.4	−0.966966
$995$	0	0
$996$	−1318.96	−0.0419606
$997$	4628.20	0.147018	0.0735088	−	0.997295i	$-0.476580\pi$
$997$	4628.20	0.147018	0.0735088	+	0.997295i	$0.476580\pi$
$998$	3866.74	0.122645
$999$	−207816.	−6.58158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.n.1.1		4
5.2	odd	4		1150.4.b.o.599.4		8
5.3	odd	4		1150.4.b.o.599.5		8
5.4	even	2		230.4.a.j.1.4	✓	4
15.14	odd	2		2070.4.a.bg.1.1		4
20.19	odd	2		1840.4.a.k.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.4	✓	4	5.4	even	2
1150.4.a.n.1.1		4	1.1	even	1	trivial
1150.4.b.o.599.4		8	5.2	odd	4
1150.4.b.o.599.5		8	5.3	odd	4
1840.4.a.k.1.1		4	20.19	odd	2
2070.4.a.bg.1.1		4	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -10.1257 q^{3} +4.00000 q^{4} +20.2514 q^{6} +24.6431 q^{7} -8.00000 q^{8} +75.5301 q^{9} -17.3867 q^{11} -40.5029 q^{12} -4.00699 q^{13} -49.2862 q^{14} +16.0000 q^{16} -48.2740 q^{17} -151.060 q^{18} +79.3172 q^{19} -249.529 q^{21} +34.7734 q^{22} +23.0000 q^{23} +81.0057 q^{24} +8.01398 q^{26} -491.402 q^{27} +98.5723 q^{28} -254.267 q^{29} -220.696 q^{31} -32.0000 q^{32} +176.053 q^{33} +96.5480 q^{34} +302.120 q^{36} +422.904 q^{37} -158.634 q^{38} +40.5736 q^{39} -170.251 q^{41} +499.058 q^{42} +228.920 q^{43} -69.5468 q^{44} -46.0000 q^{46} -580.087 q^{47} -162.011 q^{48} +264.282 q^{49} +488.809 q^{51} -16.0280 q^{52} +260.354 q^{53} +982.803 q^{54} -197.145 q^{56} -803.144 q^{57} +508.535 q^{58} +353.130 q^{59} -80.6108 q^{61} +441.392 q^{62} +1861.29 q^{63} +64.0000 q^{64} -352.106 q^{66} +820.011 q^{67} -193.096 q^{68} -232.891 q^{69} +614.845 q^{71} -604.241 q^{72} -511.586 q^{73} -845.808 q^{74} +317.269 q^{76} -428.462 q^{77} -81.1472 q^{78} +160.464 q^{79} +2936.48 q^{81} +340.502 q^{82} +32.5646 q^{83} -998.115 q^{84} -457.840 q^{86} +2574.64 q^{87} +139.094 q^{88} -25.0375 q^{89} -98.7445 q^{91} +92.0000 q^{92} +2234.70 q^{93} +1160.17 q^{94} +324.023 q^{96} +249.798 q^{97} -528.563 q^{98} -1313.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 14 q^{3} + 16 q^{4} + 28 q^{6} - 8 q^{7} - 32 q^{8} + 64 q^{9} + 21 q^{11} - 56 q^{12} - 70 q^{13} + 16 q^{14} + 64 q^{16} - 56 q^{17} - 128 q^{18} + 173 q^{19} - 120 q^{21} - 42 q^{22}+ \cdots - 2745 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 14 * q^3 + 16 * q^4 + 28 * q^6 - 8 * q^7 - 32 * q^8 + 64 * q^9 + 21 * q^11 - 56 * q^12 - 70 * q^13 + 16 * q^14 + 64 * q^16 - 56 * q^17 - 128 * q^18 + 173 * q^19 - 120 * q^21 - 42 * q^22 + 92 * q^23 + 112 * q^24 + 140 * q^26 - 389 * q^27 - 32 * q^28 - 118 * q^29 + 17 * q^31 - 128 * q^32 + 89 * q^33 + 112 * q^34 + 256 * q^36 + 343 * q^37 - 346 * q^38 - 221 * q^39 + 139 * q^41 + 240 * q^42 + 50 * q^43 + 84 * q^44 - 184 * q^46 - 367 * q^47 - 224 * q^48 - 124 * q^49 + 439 * q^51 - 280 * q^52 + 353 * q^53 + 778 * q^54 + 64 * q^56 + 238 * q^57 + 236 * q^58 - 453 * q^59 - 327 * q^61 - 34 * q^62 + 1723 * q^63 + 256 * q^64 - 178 * q^66 + 455 * q^67 - 224 * q^68 - 322 * q^69 + 195 * q^71 - 512 * q^72 + 633 * q^73 - 686 * q^74 + 692 * q^76 + 2 * q^77 + 442 * q^78 - 1140 * q^79 + 1456 * q^81 - 278 * q^82 + 1199 * q^83 - 480 * q^84 - 100 * q^86 + 1775 * q^87 - 168 * q^88 - 2170 * q^89 + 557 * q^91 + 368 * q^92 + 3241 * q^93 + 734 * q^94 + 448 * q^96 + 703 * q^97 + 248 * q^98 - 2745 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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