LMFDB - Embedded newform 1150.4.a.n.1.4

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			1.4
Root			$9.04090$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +5.04090 q^{3} +4.00000 q^{4} -10.0818 q^{6} -5.03071 q^{7} -8.00000 q^{8} -1.58932 q^{9} -5.58219 q^{11} +20.1636 q^{12} -62.7277 q^{13} +10.0614 q^{14} +16.0000 q^{16} +19.7435 q^{17} +3.17864 q^{18} +158.545 q^{19} -25.3593 q^{21} +11.1644 q^{22} +23.0000 q^{23} -40.3272 q^{24} +125.455 q^{26} -144.116 q^{27} -20.1228 q^{28} -35.5033 q^{29} +282.041 q^{31} -32.0000 q^{32} -28.1393 q^{33} -39.4870 q^{34} -6.35728 q^{36} +139.981 q^{37} -317.090 q^{38} -316.204 q^{39} +227.680 q^{41} +50.7186 q^{42} -436.962 q^{43} -22.3288 q^{44} -46.0000 q^{46} -90.2701 q^{47} +80.6544 q^{48} -317.692 q^{49} +99.5250 q^{51} -250.911 q^{52} -330.183 q^{53} +288.232 q^{54} +40.2457 q^{56} +799.209 q^{57} +71.0066 q^{58} -796.203 q^{59} -568.580 q^{61} -564.081 q^{62} +7.99541 q^{63} +64.0000 q^{64} +56.2785 q^{66} -85.1419 q^{67} +78.9740 q^{68} +115.941 q^{69} -369.578 q^{71} +12.7146 q^{72} +310.188 q^{73} -279.963 q^{74} +634.180 q^{76} +28.0824 q^{77} +632.408 q^{78} -1325.46 q^{79} -683.562 q^{81} -455.360 q^{82} +158.806 q^{83} -101.437 q^{84} +873.924 q^{86} -178.969 q^{87} +44.6575 q^{88} -1233.89 q^{89} +315.565 q^{91} +92.0000 q^{92} +1421.74 q^{93} +180.540 q^{94} -161.309 q^{96} +106.389 q^{97} +635.384 q^{98} +8.87189 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	5.04090	0.970122	0.485061	−	0.874480i	$-0.338797\pi$
$3$	5.04090	0.970122	0.485061	+	0.874480i	$0.338797\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−10.0818	−0.685980
$7$	−5.03071	−0.271633	−0.135816	−	0.990734i	$-0.543366\pi$
$7$	−5.03071	−0.271633	−0.135816	+	0.990734i	$0.543366\pi$
$8$	−8.00000	−0.353553
$9$	−1.58932	−0.0588637
$10$	0	0
$11$	−5.58219	−0.153008	−0.0765042	−	0.997069i	$-0.524376\pi$
$11$	−5.58219	−0.153008	−0.0765042	+	0.997069i	$0.524376\pi$
$12$	20.1636	0.485061
$13$	−62.7277	−1.33827	−0.669136	−	0.743140i	$-0.733336\pi$
$13$	−62.7277	−1.33827	−0.669136	+	0.743140i	$0.733336\pi$
$14$	10.0614	0.192073
$15$	0	0
$16$	16.0000	0.250000
$17$	19.7435	0.281677	0.140838	−	0.990033i	$-0.455020\pi$
$17$	19.7435	0.281677	0.140838	+	0.990033i	$0.455020\pi$
$18$	3.17864	0.0416229
$19$	158.545	1.91435	0.957177	−	0.289505i	$-0.0934907\pi$
$19$	158.545	1.91435	0.957177	+	0.289505i	$0.0934907\pi$
$20$	0	0
$21$	−25.3593	−0.263517
$22$	11.1644	0.108193
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−40.3272	−0.342990
$25$	0	0
$26$	125.455	0.946301
$27$	−144.116	−1.02723
$28$	−20.1228	−0.135816
$29$	−35.5033	−0.227338	−0.113669	−	0.993519i	$-0.536260\pi$
$29$	−35.5033	−0.227338	−0.113669	+	0.993519i	$0.536260\pi$
$30$	0	0
$31$	282.041	1.63406	0.817032	−	0.576592i	$-0.195618\pi$
$31$	282.041	1.63406	0.817032	+	0.576592i	$0.195618\pi$
$32$	−32.0000	−0.176777
$33$	−28.1393	−0.148437
$34$	−39.4870	−0.199175
$35$	0	0
$36$	−6.35728	−0.0294319
$37$	139.981	0.621967	0.310984	−	0.950415i	$-0.399342\pi$
$37$	139.981	0.621967	0.310984	+	0.950415i	$0.399342\pi$
$38$	−317.090	−1.35365
$39$	−316.204	−1.29829
$40$	0	0
$41$	227.680	0.867260	0.433630	−	0.901091i	$-0.357232\pi$
$41$	227.680	0.867260	0.433630	+	0.901091i	$0.357232\pi$
$42$	50.7186	0.186335
$43$	−436.962	−1.54968	−0.774838	−	0.632159i	$-0.782169\pi$
$43$	−436.962	−1.54968	−0.774838	+	0.632159i	$0.782169\pi$
$44$	−22.3288	−0.0765042
$45$	0	0
$46$	−46.0000	−0.147442
$47$	−90.2701	−0.280154	−0.140077	−	0.990141i	$-0.544735\pi$
$47$	−90.2701	−0.280154	−0.140077	+	0.990141i	$0.544735\pi$
$48$	80.6544	0.242530
$49$	−317.692	−0.926216
$50$	0	0
$51$	99.5250	0.273261
$52$	−250.911	−0.669136
$53$	−330.183	−0.855737	−0.427869	−	0.903841i	$-0.640735\pi$
$53$	−330.183	−0.855737	−0.427869	+	0.903841i	$0.640735\pi$
$54$	288.232	0.726359
$55$	0	0
$56$	40.2457	0.0960367
$57$	799.209	1.85716
$58$	71.0066	0.160752
$59$	−796.203	−1.75689	−0.878447	−	0.477839i	$-0.841420\pi$
$59$	−796.203	−1.75689	−0.878447	+	0.477839i	$0.841420\pi$
$60$	0	0
$61$	−568.580	−1.19343	−0.596715	−	0.802453i	$-0.703528\pi$
$61$	−568.580	−1.19343	−0.596715	+	0.802453i	$0.703528\pi$
$62$	−564.081	−1.15546
$63$	7.99541	0.0159893
$64$	64.0000	0.125000
$65$	0	0
$66$	56.2785	0.104961
$67$	−85.1419	−0.155250	−0.0776250	−	0.996983i	$-0.524734\pi$
$67$	−85.1419	−0.155250	−0.0776250	+	0.996983i	$0.524734\pi$
$68$	78.9740	0.140838
$69$	115.941	0.202284
$70$	0	0
$71$	−369.578	−0.617758	−0.308879	−	0.951101i	$-0.599954\pi$
$71$	−369.578	−0.617758	−0.308879	+	0.951101i	$0.599954\pi$
$72$	12.7146	0.0208115
$73$	310.188	0.497325	0.248662	−	0.968590i	$-0.420009\pi$
$73$	310.188	0.497325	0.248662	+	0.968590i	$0.420009\pi$
$74$	−279.963	−0.439797
$75$	0	0
$76$	634.180	0.957177
$77$	28.0824	0.0415621
$78$	632.408	0.918027
$79$	−1325.46	−1.88766	−0.943832	−	0.330427i	$-0.892807\pi$
$79$	−1325.46	−1.88766	−0.943832	+	0.330427i	$0.892807\pi$
$80$	0	0
$81$	−683.562	−0.937671
$82$	−455.360	−0.613246
$83$	158.806	0.210014	0.105007	−	0.994471i	$-0.466513\pi$
$83$	158.806	0.210014	0.105007	+	0.994471i	$0.466513\pi$
$84$	−101.437	−0.131758
$85$	0	0
$86$	873.924	1.09579
$87$	−178.969	−0.220545
$88$	44.6575	0.0540967
$89$	−1233.89	−1.46957	−0.734787	−	0.678298i	$-0.762718\pi$
$89$	−1233.89	−1.46957	−0.734787	+	0.678298i	$0.762718\pi$
$90$	0	0
$91$	315.565	0.363519
$92$	92.0000	0.104257
$93$	1421.74	1.58524
$94$	180.540	0.198099
$95$	0	0
$96$	−161.309	−0.171495
$97$	106.389	0.111363	0.0556814	−	0.998449i	$-0.482267\pi$
$97$	106.389	0.111363	0.0556814	+	0.998449i	$0.482267\pi$
$98$	635.384	0.654933
$99$	8.87189	0.00900665
$100$	0	0
$101$	642.676	0.633155	0.316578	−	0.948567i	$-0.397466\pi$
$101$	642.676	0.633155	0.316578	+	0.948567i	$0.397466\pi$
$102$	−199.050	−0.193224
$103$	−1621.99	−1.55165	−0.775823	−	0.630951i	$-0.782665\pi$
$103$	−1621.99	−1.55165	−0.775823	+	0.630951i	$0.782665\pi$
$104$	501.822	0.473150
$105$	0	0
$106$	660.365	0.605098
$107$	−1490.86	−1.34698	−0.673490	−	0.739196i	$-0.735205\pi$
$107$	−1490.86	−1.34698	−0.673490	+	0.739196i	$0.735205\pi$
$108$	−576.464	−0.513613
$109$	1204.00	1.05800	0.529001	−	0.848621i	$-0.322567\pi$
$109$	1204.00	1.05800	0.529001	+	0.848621i	$0.322567\pi$
$110$	0	0
$111$	705.632	0.603384
$112$	−80.4914	−0.0679082
$113$	−276.771	−0.230411	−0.115206	−	0.993342i	$-0.536753\pi$
$113$	−276.771	−0.230411	−0.115206	+	0.993342i	$0.536753\pi$
$114$	−1598.42	−1.31321
$115$	0	0
$116$	−142.013	−0.113669
$117$	99.6944	0.0787757
$118$	1592.41	1.24231
$119$	−99.3239	−0.0765126
$120$	0	0
$121$	−1299.84	−0.976588
$122$	1137.16	0.843883
$123$	1147.71	0.841348
$124$	1128.16	0.817032
$125$	0	0
$126$	−15.9908	−0.0113062
$127$	−1552.79	−1.08494	−0.542471	−	0.840075i	$-0.682511\pi$
$127$	−1552.79	−1.08494	−0.542471	+	0.840075i	$0.682511\pi$
$128$	−128.000	−0.0883883
$129$	−2202.68	−1.50338
$130$	0	0
$131$	−313.873	−0.209337	−0.104669	−	0.994507i	$-0.533378\pi$
$131$	−313.873	−0.209337	−0.104669	+	0.994507i	$0.533378\pi$
$132$	−112.557	−0.0742184
$133$	−797.594	−0.520001
$134$	170.284	0.109778
$135$	0	0
$136$	−157.948	−0.0995877
$137$	1066.09	0.664835	0.332417	−	0.943132i	$-0.392136\pi$
$137$	1066.09	0.664835	0.332417	+	0.943132i	$0.392136\pi$
$138$	−231.881	−0.143037
$139$	−1594.79	−0.973152	−0.486576	−	0.873638i	$-0.661754\pi$
$139$	−1594.79	−0.973152	−0.486576	+	0.873638i	$0.661754\pi$
$140$	0	0
$141$	−455.043	−0.271784
$142$	739.155	0.436821
$143$	350.158	0.204767
$144$	−25.4291	−0.0147159
$145$	0	0
$146$	−620.375	−0.351662
$147$	−1601.45	−0.898542
$148$	559.925	0.310984
$149$	−1096.91	−0.603102	−0.301551	−	0.953450i	$-0.597504\pi$
$149$	−1096.91	−0.603102	−0.301551	+	0.953450i	$0.597504\pi$
$150$	0	0
$151$	2734.41	1.47366	0.736832	−	0.676076i	$-0.236321\pi$
$151$	2734.41	1.47366	0.736832	+	0.676076i	$0.236321\pi$
$152$	−1268.36	−0.676826
$153$	−31.3788	−0.0165805
$154$	−56.1647	−0.0293889
$155$	0	0
$156$	−1264.82	−0.649143
$157$	1439.63	0.731813	0.365907	−	0.930652i	$-0.380759\pi$
$157$	1439.63	0.731813	0.365907	+	0.930652i	$0.380759\pi$
$158$	2650.91	1.33478
$159$	−1664.42	−0.830169
$160$	0	0
$161$	−115.706	−0.0566394
$162$	1367.12	0.663034
$163$	2995.82	1.43957	0.719787	−	0.694195i	$-0.244240\pi$
$163$	2995.82	1.43957	0.719787	+	0.694195i	$0.244240\pi$
$164$	910.721	0.433630
$165$	0	0
$166$	−317.611	−0.148503
$167$	351.532	0.162888	0.0814442	−	0.996678i	$-0.474047\pi$
$167$	351.532	0.162888	0.0814442	+	0.996678i	$0.474047\pi$
$168$	202.874	0.0931673
$169$	1737.76	0.790971
$170$	0	0
$171$	−251.979	−0.112686
$172$	−1747.85	−0.774838
$173$	3701.77	1.62682	0.813411	−	0.581690i	$-0.197608\pi$
$173$	3701.77	1.62682	0.813411	+	0.581690i	$0.197608\pi$
$174$	357.937	0.155949
$175$	0	0
$176$	−89.3150	−0.0382521
$177$	−4013.58	−1.70440
$178$	2467.78	1.03915
$179$	3584.09	1.49658	0.748290	−	0.663372i	$-0.230875\pi$
$179$	3584.09	1.49658	0.748290	+	0.663372i	$0.230875\pi$
$180$	0	0
$181$	783.672	0.321822	0.160911	−	0.986969i	$-0.448557\pi$
$181$	783.672	0.321822	0.160911	+	0.986969i	$0.448557\pi$
$182$	−631.130	−0.257046
$183$	−2866.16	−1.15777
$184$	−184.000	−0.0737210
$185$	0	0
$186$	−2843.48	−1.12093
$187$	−110.212	−0.0430989
$188$	−361.080	−0.140077
$189$	725.005	0.279029
$190$	0	0
$191$	1491.95	0.565201	0.282601	−	0.959238i	$-0.408803\pi$
$191$	1491.95	0.565201	0.282601	+	0.959238i	$0.408803\pi$
$192$	322.618	0.121265
$193$	−3091.09	−1.15286	−0.576430	−	0.817147i	$-0.695554\pi$
$193$	−3091.09	−1.15286	−0.576430	+	0.817147i	$0.695554\pi$
$194$	−212.778	−0.0787454
$195$	0	0
$196$	−1270.77	−0.463108
$197$	−2820.30	−1.01999	−0.509995	−	0.860178i	$-0.670353\pi$
$197$	−2820.30	−1.01999	−0.509995	+	0.860178i	$0.670353\pi$
$198$	−17.7438	−0.00636866
$199$	1316.11	0.468825	0.234413	−	0.972137i	$-0.424683\pi$
$199$	1316.11	0.468825	0.234413	+	0.972137i	$0.424683\pi$
$200$	0	0
$201$	−429.192	−0.150611
$202$	−1285.35	−0.447708
$203$	178.607	0.0617524
$204$	398.100	0.136630
$205$	0	0
$206$	3243.98	1.09718
$207$	−36.5544	−0.0122739
$208$	−1003.64	−0.334568
$209$	−885.028	−0.292912
$210$	0	0
$211$	−3399.19	−1.10905	−0.554525	−	0.832167i	$-0.687100\pi$
$211$	−3399.19	−1.10905	−0.554525	+	0.832167i	$0.687100\pi$
$212$	−1320.73	−0.427869
$213$	−1863.00	−0.599300
$214$	2981.72	0.952458
$215$	0	0
$216$	1152.93	0.363180
$217$	−1418.86	−0.443865
$218$	−2408.00	−0.748121
$219$	1563.62	0.482466
$220$	0	0
$221$	−1238.46	−0.376960
$222$	−1411.26	−0.426657
$223$	−864.660	−0.259650	−0.129825	−	0.991537i	$-0.541442\pi$
$223$	−864.660	−0.259650	−0.129825	+	0.991537i	$0.541442\pi$
$224$	160.983	0.0480184
$225$	0	0
$226$	553.543	0.162925
$227$	1979.40	0.578754	0.289377	−	0.957215i	$-0.406552\pi$
$227$	1979.40	0.578754	0.289377	+	0.957215i	$0.406552\pi$
$228$	3196.84	0.928578
$229$	−3113.53	−0.898462	−0.449231	−	0.893416i	$-0.648302\pi$
$229$	−3113.53	−0.898462	−0.449231	+	0.893416i	$0.648302\pi$
$230$	0	0
$231$	141.560	0.0403203
$232$	284.026	0.0803761
$233$	−6018.65	−1.69225	−0.846126	−	0.532983i	$-0.821071\pi$
$233$	−6018.65	−1.69225	−0.846126	+	0.532983i	$0.821071\pi$
$234$	−199.389	−0.0557028
$235$	0	0
$236$	−3184.81	−0.878447
$237$	−6681.49	−1.83126
$238$	198.648	0.0541026
$239$	−4224.88	−1.14345	−0.571725	−	0.820445i	$-0.693726\pi$
$239$	−4224.88	−1.14345	−0.571725	+	0.820445i	$0.693726\pi$
$240$	0	0
$241$	1394.24	0.372658	0.186329	−	0.982487i	$-0.440341\pi$
$241$	1394.24	0.372658	0.186329	+	0.982487i	$0.440341\pi$
$242$	2599.68	0.690552
$243$	445.360	0.117571
$244$	−2274.32	−0.596715
$245$	0	0
$246$	−2295.43	−0.594923
$247$	−9945.16	−2.56192
$248$	−2256.32	−0.577729
$249$	800.523	0.203739
$250$	0	0
$251$	−5968.83	−1.50099	−0.750497	−	0.660874i	$-0.770186\pi$
$251$	−5968.83	−1.50099	−0.750497	+	0.660874i	$0.770186\pi$
$252$	31.9816	0.00799466
$253$	−128.390	−0.0319045
$254$	3105.57	0.767169
$255$	0	0
$256$	256.000	0.0625000
$257$	1636.06	0.397100	0.198550	−	0.980091i	$-0.436377\pi$
$257$	1636.06	0.397100	0.198550	+	0.980091i	$0.436377\pi$
$258$	4405.37	1.06305
$259$	−704.206	−0.168947
$260$	0	0
$261$	56.4261	0.0133820
$262$	627.745	0.148024
$263$	−2995.55	−0.702333	−0.351166	−	0.936313i	$-0.614215\pi$
$263$	−2995.55	−0.702333	−0.351166	+	0.936313i	$0.614215\pi$
$264$	225.114	0.0524803
$265$	0	0
$266$	1595.19	0.367696
$267$	−6219.92	−1.42567
$268$	−340.568	−0.0776250
$269$	−5190.17	−1.17640	−0.588198	−	0.808717i	$-0.700162\pi$
$269$	−5190.17	−1.17640	−0.588198	+	0.808717i	$0.700162\pi$
$270$	0	0
$271$	6720.43	1.50641	0.753205	−	0.657786i	$-0.228507\pi$
$271$	6720.43	1.50641	0.753205	+	0.657786i	$0.228507\pi$
$272$	315.896	0.0704192
$273$	1590.73	0.352657
$274$	−2132.18	−0.470109
$275$	0	0
$276$	463.763	0.101142
$277$	6114.74	1.32635	0.663176	−	0.748464i	$-0.269208\pi$
$277$	6114.74	1.32635	0.663176	+	0.748464i	$0.269208\pi$
$278$	3189.58	0.688123
$279$	−448.253	−0.0961871
$280$	0	0
$281$	−3549.07	−0.753450	−0.376725	−	0.926325i	$-0.622950\pi$
$281$	−3549.07	−0.753450	−0.376725	+	0.926325i	$0.622950\pi$
$282$	910.085	0.192180
$283$	−3103.11	−0.651806	−0.325903	−	0.945403i	$-0.605668\pi$
$283$	−3103.11	−0.651806	−0.325903	+	0.945403i	$0.605668\pi$
$284$	−1478.31	−0.308879
$285$	0	0
$286$	−700.316	−0.144792
$287$	−1145.39	−0.235576
$288$	50.8583	0.0104057
$289$	−4523.19	−0.920658
$290$	0	0
$291$	536.297	0.108035
$292$	1240.75	0.248662
$293$	−1218.51	−0.242957	−0.121478	−	0.992594i	$-0.538763\pi$
$293$	−1218.51	−0.242957	−0.121478	+	0.992594i	$0.538763\pi$
$294$	3202.91	0.635365
$295$	0	0
$296$	−1119.85	−0.219899
$297$	804.482	0.157174
$298$	2193.82	0.426458
$299$	−1442.74	−0.279049
$300$	0	0
$301$	2198.23	0.420943
$302$	−5468.82	−1.04204
$303$	3239.67	0.614238
$304$	2536.72	0.478588
$305$	0	0
$306$	62.7575	0.0117242
$307$	−2564.45	−0.476745	−0.238373	−	0.971174i	$-0.576614\pi$
$307$	−2564.45	−0.476745	−0.238373	+	0.971174i	$0.576614\pi$
$308$	112.329	0.0207811
$309$	−8176.30	−1.50529
$310$	0	0
$311$	−92.6253	−0.0168884	−0.00844421	−	0.999964i	$-0.502688\pi$
$311$	−92.6253	−0.0168884	−0.00844421	+	0.999964i	$0.502688\pi$
$312$	2529.63	0.459014
$313$	4276.44	0.772263	0.386132	−	0.922444i	$-0.373811\pi$
$313$	4276.44	0.772263	0.386132	+	0.922444i	$0.373811\pi$
$314$	−2879.25	−0.517470
$315$	0	0
$316$	−5301.82	−0.943832
$317$	2964.24	0.525199	0.262600	−	0.964905i	$-0.415420\pi$
$317$	2964.24	0.525199	0.262600	+	0.964905i	$0.415420\pi$
$318$	3328.84	0.587018
$319$	198.186	0.0347846
$320$	0	0
$321$	−7515.27	−1.30673
$322$	231.413	0.0400501
$323$	3130.23	0.539229
$324$	−2734.25	−0.468836
$325$	0	0
$326$	−5991.63	−1.01793
$327$	6069.24	1.02639
$328$	−1821.44	−0.306623
$329$	454.123	0.0760991
$330$	0	0
$331$	−1348.81	−0.223979	−0.111989	−	0.993709i	$-0.535722\pi$
$331$	−1348.81	−0.223979	−0.111989	+	0.993709i	$0.535722\pi$
$332$	635.223	0.105007
$333$	−222.475	−0.0366113
$334$	−703.064	−0.115180
$335$	0	0
$336$	−405.749	−0.0658792
$337$	5764.30	0.931755	0.465878	−	0.884849i	$-0.345739\pi$
$337$	5764.30	0.931755	0.465878	+	0.884849i	$0.345739\pi$
$338$	−3475.53	−0.559301
$339$	−1395.18	−0.223527
$340$	0	0
$341$	−1574.40	−0.250026
$342$	503.958	0.0796810
$343$	3323.75	0.523223
$344$	3495.70	0.547894
$345$	0	0
$346$	−7403.53	−1.15034
$347$	−4066.16	−0.629057	−0.314528	−	0.949248i	$-0.601846\pi$
$347$	−4066.16	−0.629057	−0.314528	+	0.949248i	$0.601846\pi$
$348$	−715.875	−0.110273
$349$	8407.82	1.28957	0.644786	−	0.764363i	$-0.276947\pi$
$349$	8407.82	1.28957	0.644786	+	0.764363i	$0.276947\pi$
$350$	0	0
$351$	9040.06	1.37471
$352$	178.630	0.0270483
$353$	−256.068	−0.0386095	−0.0193047	−	0.999814i	$-0.506145\pi$
$353$	−256.068	−0.0386095	−0.0193047	+	0.999814i	$0.506145\pi$
$354$	8027.16	1.20519
$355$	0	0
$356$	−4935.56	−0.734787
$357$	−500.682	−0.0742266
$358$	−7168.19	−1.05824
$359$	−12052.5	−1.77189	−0.885943	−	0.463794i	$-0.846488\pi$
$359$	−12052.5	−1.77189	−0.885943	+	0.463794i	$0.846488\pi$
$360$	0	0
$361$	18277.5	2.66475
$362$	−1567.34	−0.227563
$363$	−6552.36	−0.947410
$364$	1262.26	0.181759
$365$	0	0
$366$	5732.31	0.818669
$367$	9245.40	1.31500	0.657502	−	0.753453i	$-0.271613\pi$
$367$	9245.40	1.31500	0.657502	+	0.753453i	$0.271613\pi$
$368$	368.000	0.0521286
$369$	−361.857	−0.0510502
$370$	0	0
$371$	1661.05	0.232446
$372$	5686.96	0.792621
$373$	869.265	0.120667	0.0603335	−	0.998178i	$-0.480784\pi$
$373$	869.265	0.120667	0.0603335	+	0.998178i	$0.480784\pi$
$374$	220.424	0.0304755
$375$	0	0
$376$	722.161	0.0990495
$377$	2227.04	0.304240
$378$	−1450.01	−0.197303
$379$	1278.76	0.173313	0.0866566	−	0.996238i	$-0.472382\pi$
$379$	1278.76	0.173313	0.0866566	+	0.996238i	$0.472382\pi$
$380$	0	0
$381$	−7827.44	−1.05252
$382$	−2983.89	−0.399658
$383$	5343.75	0.712932	0.356466	−	0.934308i	$-0.383982\pi$
$383$	5343.75	0.712932	0.356466	+	0.934308i	$0.383982\pi$
$384$	−645.235	−0.0857475
$385$	0	0
$386$	6182.19	0.815194
$387$	694.473	0.0912198
$388$	425.557	0.0556814
$389$	−708.346	−0.0923254	−0.0461627	−	0.998934i	$-0.514699\pi$
$389$	−708.346	−0.0923254	−0.0461627	+	0.998934i	$0.514699\pi$
$390$	0	0
$391$	454.101	0.0587336
$392$	2541.54	0.327467
$393$	−1582.20	−0.203083
$394$	5640.60	0.721241
$395$	0	0
$396$	35.4875	0.00450332
$397$	5924.01	0.748911	0.374455	−	0.927245i	$-0.377830\pi$
$397$	5924.01	0.748911	0.374455	+	0.927245i	$0.377830\pi$
$398$	−2632.21	−0.331510
$399$	−4020.59	−0.504464
$400$	0	0
$401$	−11393.6	−1.41888	−0.709441	−	0.704765i	$-0.751052\pi$
$401$	−11393.6	−1.41888	−0.709441	+	0.704765i	$0.751052\pi$
$402$	858.384	0.106498
$403$	−17691.8	−2.18682
$404$	2570.71	0.316578
$405$	0	0
$406$	−357.214	−0.0436656
$407$	−781.402	−0.0951663
$408$	−796.200	−0.0966122
$409$	13804.4	1.66891	0.834457	−	0.551073i	$-0.185782\pi$
$409$	13804.4	1.66891	0.834457	+	0.551073i	$0.185782\pi$
$410$	0	0
$411$	5374.06	0.644971
$412$	−6487.96	−0.775823
$413$	4005.47	0.477230
$414$	73.1087	0.00867898
$415$	0	0
$416$	2007.29	0.236575
$417$	−8039.17	−0.944076
$418$	1770.06	0.207120
$419$	−14795.8	−1.72512	−0.862558	−	0.505958i	$-0.831139\pi$
$419$	−14795.8	−1.72512	−0.862558	+	0.505958i	$0.831139\pi$
$420$	0	0
$421$	5804.66	0.671976	0.335988	−	0.941866i	$-0.390930\pi$
$421$	5804.66	0.671976	0.335988	+	0.941866i	$0.390930\pi$
$422$	6798.37	0.784217
$423$	143.468	0.0164909
$424$	2641.46	0.302549
$425$	0	0
$426$	3726.01	0.423769
$427$	2860.36	0.324175
$428$	−5963.44	−0.673490
$429$	1765.11	0.198649
$430$	0	0
$431$	15268.2	1.70636	0.853181	−	0.521615i	$-0.174670\pi$
$431$	15268.2	1.70636	0.853181	+	0.521615i	$0.174670\pi$
$432$	−2305.85	−0.256807
$433$	−2250.62	−0.249788	−0.124894	−	0.992170i	$-0.539859\pi$
$433$	−2250.62	−0.249788	−0.124894	+	0.992170i	$0.539859\pi$
$434$	2837.73	0.313860
$435$	0	0
$436$	4816.00	0.529001
$437$	3646.53	0.399170
$438$	−3127.25	−0.341155
$439$	10366.2	1.12700	0.563499	−	0.826117i	$-0.309455\pi$
$439$	10366.2	1.12700	0.563499	+	0.826117i	$0.309455\pi$
$440$	0	0
$441$	504.914	0.0545205
$442$	2476.93	0.266551
$443$	3881.11	0.416246	0.208123	−	0.978103i	$-0.433264\pi$
$443$	3881.11	0.416246	0.208123	+	0.978103i	$0.433264\pi$
$444$	2822.53	0.301692
$445$	0	0
$446$	1729.32	0.183600
$447$	−5529.41	−0.585083
$448$	−321.965	−0.0339541
$449$	−10146.5	−1.06647	−0.533233	−	0.845968i	$-0.679023\pi$
$449$	−10146.5	−1.06647	−0.533233	+	0.845968i	$0.679023\pi$
$450$	0	0
$451$	−1270.95	−0.132698
$452$	−1107.09	−0.115206
$453$	13783.9	1.42963
$454$	−3958.79	−0.409241
$455$	0	0
$456$	−6393.68	−0.656604
$457$	4728.87	0.484042	0.242021	−	0.970271i	$-0.422190\pi$
$457$	4728.87	0.484042	0.242021	+	0.970271i	$0.422190\pi$
$458$	6227.06	0.635308
$459$	−2845.35	−0.289346
$460$	0	0
$461$	8380.69	0.846698	0.423349	−	0.905967i	$-0.360854\pi$
$461$	8380.69	0.846698	0.423349	+	0.905967i	$0.360854\pi$
$462$	−283.121	−0.0285108
$463$	−6845.17	−0.687089	−0.343545	−	0.939136i	$-0.611628\pi$
$463$	−6845.17	−0.687089	−0.343545	+	0.939136i	$0.611628\pi$
$464$	−568.053	−0.0568345
$465$	0	0
$466$	12037.3	1.19660
$467$	14884.0	1.47484	0.737418	−	0.675436i	$-0.236045\pi$
$467$	14884.0	1.47484	0.737418	+	0.675436i	$0.236045\pi$
$468$	398.778	0.0393878
$469$	428.324	0.0421710
$470$	0	0
$471$	7257.01	0.709948
$472$	6369.62	0.621156
$473$	2439.20	0.237114
$474$	13363.0	1.29490
$475$	0	0
$476$	−397.295	−0.0382563
$477$	524.766	0.0503719
$478$	8449.76	0.808542
$479$	6048.69	0.576976	0.288488	−	0.957483i	$-0.406847\pi$
$479$	6048.69	0.576976	0.288488	+	0.957483i	$0.406847\pi$
$480$	0	0
$481$	−8780.71	−0.832361
$482$	−2788.47	−0.263509
$483$	−583.264	−0.0549471
$484$	−5199.36	−0.488294
$485$	0	0
$486$	−890.720	−0.0831355
$487$	−3199.35	−0.297693	−0.148846	−	0.988860i	$-0.547556\pi$
$487$	−3199.35	−0.297693	−0.148846	+	0.988860i	$0.547556\pi$
$488$	4548.64	0.421941
$489$	15101.6	1.39656
$490$	0	0
$491$	−1614.41	−0.148385	−0.0741926	−	0.997244i	$-0.523638\pi$
$491$	−1614.41	−0.148385	−0.0741926	+	0.997244i	$0.523638\pi$
$492$	4590.85	0.420674
$493$	−700.960	−0.0640358
$494$	19890.3	1.81155
$495$	0	0
$496$	4512.65	0.408516
$497$	1859.24	0.167803
$498$	−1601.05	−0.144066
$499$	−5392.04	−0.483729	−0.241864	−	0.970310i	$-0.577759\pi$
$499$	−5392.04	−0.483729	−0.241864	+	0.970310i	$0.577759\pi$
$500$	0	0
$501$	1772.04	0.158022
$502$	11937.7	1.06136
$503$	−12625.4	−1.11917	−0.559583	−	0.828775i	$-0.689039\pi$
$503$	−12625.4	−1.11917	−0.559583	+	0.828775i	$0.689039\pi$
$504$	−63.9633	−0.00565308
$505$	0	0
$506$	256.781	0.0225599
$507$	8759.89	0.767338
$508$	−6211.15	−0.542471
$509$	7065.76	0.615294	0.307647	−	0.951501i	$-0.400459\pi$
$509$	7065.76	0.615294	0.307647	+	0.951501i	$0.400459\pi$
$510$	0	0
$511$	−1560.46	−0.135090
$512$	−512.000	−0.0441942
$513$	−22848.9	−1.96647
$514$	−3272.12	−0.280792
$515$	0	0
$516$	−8810.73	−0.751688
$517$	503.905	0.0428660
$518$	1408.41	0.119463
$519$	18660.2	1.57822
$520$	0	0
$521$	−8399.20	−0.706287	−0.353144	−	0.935569i	$-0.614887\pi$
$521$	−8399.20	−0.706287	−0.353144	+	0.935569i	$0.614887\pi$
$522$	−112.852	−0.00946247
$523$	7002.06	0.585428	0.292714	−	0.956200i	$-0.405442\pi$
$523$	7002.06	0.585428	0.292714	+	0.956200i	$0.405442\pi$
$524$	−1255.49	−0.104669
$525$	0	0
$526$	5991.10	0.496624
$527$	5568.47	0.460278
$528$	−450.228	−0.0371092
$529$	529.000	0.0434783
$530$	0	0
$531$	1265.42	0.103417
$532$	−3190.37	−0.260001
$533$	−14281.9	−1.16063
$534$	12439.8	1.00810
$535$	0	0
$536$	681.136	0.0548891
$537$	18067.1	1.45186
$538$	10380.3	0.831838
$539$	1773.42	0.141719
$540$	0	0
$541$	3338.13	0.265282	0.132641	−	0.991164i	$-0.457654\pi$
$541$	3338.13	0.265282	0.132641	+	0.991164i	$0.457654\pi$
$542$	−13440.9	−1.06519
$543$	3950.41	0.312207
$544$	−631.792	−0.0497939
$545$	0	0
$546$	−3181.46	−0.249366
$547$	−9609.09	−0.751106	−0.375553	−	0.926801i	$-0.622547\pi$
$547$	−9609.09	−0.751106	−0.375553	+	0.926801i	$0.622547\pi$
$548$	4264.37	0.332417
$549$	903.656	0.0702497
$550$	0	0
$551$	−5628.87	−0.435205
$552$	−927.526	−0.0715183
$553$	6667.98	0.512751
$554$	−12229.5	−0.937872
$555$	0	0
$556$	−6379.15	−0.486576
$557$	11931.1	0.907608	0.453804	−	0.891102i	$-0.350067\pi$
$557$	11931.1	0.907608	0.453804	+	0.891102i	$0.350067\pi$
$558$	896.506	0.0680146
$559$	27409.6	2.07389
$560$	0	0
$561$	−555.567	−0.0418112
$562$	7098.13	0.532770
$563$	10082.4	0.754746	0.377373	−	0.926061i	$-0.376827\pi$
$563$	10082.4	0.754746	0.377373	+	0.926061i	$0.376827\pi$
$564$	−1820.17	−0.135892
$565$	0	0
$566$	6206.23	0.460896
$567$	3438.80	0.254702
$568$	2956.62	0.218410
$569$	−18710.5	−1.37853	−0.689267	−	0.724507i	$-0.742067\pi$
$569$	−18710.5	−1.37853	−0.689267	+	0.724507i	$0.742067\pi$
$570$	0	0
$571$	13510.1	0.990159	0.495079	−	0.868848i	$-0.335139\pi$
$571$	13510.1	0.990159	0.495079	+	0.868848i	$0.335139\pi$
$572$	1400.63	0.102383
$573$	7520.75	0.548314
$574$	2290.79	0.166578
$575$	0	0
$576$	−101.717	−0.00735797
$577$	−11582.3	−0.835662	−0.417831	−	0.908525i	$-0.637210\pi$
$577$	−11582.3	−0.835662	−0.417831	+	0.908525i	$0.637210\pi$
$578$	9046.39	0.651004
$579$	−15581.9	−1.11841
$580$	0	0
$581$	−798.905	−0.0570468
$582$	−1072.59	−0.0763926
$583$	1843.14	0.130935
$584$	−2481.50	−0.175831
$585$	0	0
$586$	2437.03	0.171796
$587$	6019.34	0.423245	0.211623	−	0.977351i	$-0.432125\pi$
$587$	6019.34	0.423245	0.211623	+	0.977351i	$0.432125\pi$
$588$	−6405.81	−0.449271
$589$	44716.1	3.12818
$590$	0	0
$591$	−14216.8	−0.989514
$592$	2239.70	0.155492
$593$	−26299.3	−1.82122	−0.910608	−	0.413270i	$-0.864387\pi$
$593$	−26299.3	−1.82122	−0.910608	+	0.413270i	$0.864387\pi$
$594$	−1608.96	−0.111139
$595$	0	0
$596$	−4387.63	−0.301551
$597$	6634.36	0.454818
$598$	2885.47	0.197317
$599$	−27368.0	−1.86682	−0.933409	−	0.358813i	$-0.883181\pi$
$599$	−27368.0	−1.86682	−0.933409	+	0.358813i	$0.883181\pi$
$600$	0	0
$601$	−3472.33	−0.235673	−0.117836	−	0.993033i	$-0.537596\pi$
$601$	−3472.33	−0.235673	−0.117836	+	0.993033i	$0.537596\pi$
$602$	−4396.46	−0.297652
$603$	135.318	0.00913859
$604$	10937.6	0.736832
$605$	0	0
$606$	−6479.33	−0.434332
$607$	−5061.10	−0.338425	−0.169212	−	0.985580i	$-0.554122\pi$
$607$	−5061.10	−0.338425	−0.169212	+	0.985580i	$0.554122\pi$
$608$	−5073.44	−0.338413
$609$	900.339	0.0599074
$610$	0	0
$611$	5662.44	0.374922
$612$	−125.515	−0.00829027
$613$	28272.6	1.86283	0.931417	−	0.363953i	$-0.118573\pi$
$613$	28272.6	1.86283	0.931417	+	0.363953i	$0.118573\pi$
$614$	5128.90	0.337110
$615$	0	0
$616$	−224.659	−0.0146944
$617$	2222.47	0.145013	0.0725066	−	0.997368i	$-0.476900\pi$
$617$	2222.47	0.145013	0.0725066	+	0.997368i	$0.476900\pi$
$618$	16352.6	1.06440
$619$	−10498.2	−0.681679	−0.340840	−	0.940121i	$-0.610711\pi$
$619$	−10498.2	−0.681679	−0.340840	+	0.940121i	$0.610711\pi$
$620$	0	0
$621$	−3314.67	−0.214192
$622$	185.251	0.0119419
$623$	6207.34	0.399185
$624$	−5059.27	−0.324572
$625$	0	0
$626$	−8552.87	−0.546073
$627$	−4461.34	−0.284160
$628$	5758.51	0.365907
$629$	2763.72	0.175194
$630$	0	0
$631$	−11402.2	−0.719355	−0.359678	−	0.933077i	$-0.617113\pi$
$631$	−11402.2	−0.719355	−0.359678	+	0.933077i	$0.617113\pi$
$632$	10603.6	0.667390
$633$	−17135.0	−1.07591
$634$	−5928.47	−0.371372
$635$	0	0
$636$	−6657.67	−0.415085
$637$	19928.1	1.23953
$638$	−396.372	−0.0245964
$639$	587.377	0.0363635
$640$	0	0
$641$	26128.3	1.60999	0.804996	−	0.593280i	$-0.202167\pi$
$641$	26128.3	1.60999	0.804996	+	0.593280i	$0.202167\pi$
$642$	15030.5	0.924001
$643$	−863.502	−0.0529599	−0.0264799	−	0.999649i	$-0.508430\pi$
$643$	−863.502	−0.0529599	−0.0264799	+	0.999649i	$0.508430\pi$
$644$	−462.825	−0.0283197
$645$	0	0
$646$	−6260.47	−0.381292
$647$	5477.43	0.332828	0.166414	−	0.986056i	$-0.446781\pi$
$647$	5477.43	0.332828	0.166414	+	0.986056i	$0.446781\pi$
$648$	5468.50	0.331517
$649$	4444.55	0.268820
$650$	0	0
$651$	−7152.36	−0.430604
$652$	11983.3	0.719787
$653$	−19233.2	−1.15261	−0.576305	−	0.817235i	$-0.695506\pi$
$653$	−19233.2	−1.15261	−0.576305	+	0.817235i	$0.695506\pi$
$654$	−12138.5	−0.725768
$655$	0	0
$656$	3642.88	0.216815
$657$	−492.987	−0.0292744
$658$	−908.245	−0.0538102
$659$	9952.51	0.588307	0.294154	−	0.955758i	$-0.404962\pi$
$659$	9952.51	0.588307	0.294154	+	0.955758i	$0.404962\pi$
$660$	0	0
$661$	3047.96	0.179352	0.0896762	−	0.995971i	$-0.471417\pi$
$661$	3047.96	0.179352	0.0896762	+	0.995971i	$0.471417\pi$
$662$	2697.61	0.158377
$663$	−6242.98	−0.365697
$664$	−1270.45	−0.0742513
$665$	0	0
$666$	444.950	0.0258881
$667$	−816.576	−0.0474032
$668$	1406.13	0.0814442
$669$	−4358.67	−0.251892
$670$	0	0
$671$	3173.92	0.182605
$672$	811.498	0.0465836
$673$	−16245.9	−0.930508	−0.465254	−	0.885177i	$-0.654037\pi$
$673$	−16245.9	−0.930508	−0.465254	+	0.885177i	$0.654037\pi$
$674$	−11528.6	−0.658850
$675$	0	0
$676$	6951.05	0.395486
$677$	30056.6	1.70631	0.853154	−	0.521659i	$-0.174687\pi$
$677$	30056.6	1.70631	0.853154	+	0.521659i	$0.174687\pi$
$678$	2790.35	0.158057
$679$	−535.213	−0.0302498
$680$	0	0
$681$	9977.94	0.561462
$682$	3148.81	0.176795
$683$	−10064.8	−0.563866	−0.281933	−	0.959434i	$-0.590976\pi$
$683$	−10064.8	−0.563866	−0.281933	+	0.959434i	$0.590976\pi$
$684$	−1007.92	−0.0563430
$685$	0	0
$686$	−6647.50	−0.369975
$687$	−15695.0	−0.871617
$688$	−6991.39	−0.387419
$689$	20711.6	1.14521
$690$	0	0
$691$	10260.7	0.564886	0.282443	−	0.959284i	$-0.408855\pi$
$691$	10260.7	0.564886	0.282443	+	0.959284i	$0.408855\pi$
$692$	14807.1	0.813411
$693$	−44.6319	−0.00244650
$694$	8132.31	0.444810
$695$	0	0
$696$	1431.75	0.0779746
$697$	4495.21	0.244287
$698$	−16815.6	−0.911865
$699$	−30339.4	−1.64169
$700$	0	0
$701$	−19815.6	−1.06766	−0.533828	−	0.845593i	$-0.679247\pi$
$701$	−19815.6	−1.06766	−0.533828	+	0.845593i	$0.679247\pi$
$702$	−18080.1	−0.972066
$703$	22193.3	1.19067
$704$	−357.260	−0.0191261
$705$	0	0
$706$	512.136	0.0273010
$707$	−3233.12	−0.171986
$708$	−16054.3	−0.852201
$709$	−30533.5	−1.61736	−0.808681	−	0.588247i	$-0.799818\pi$
$709$	−30533.5	−1.61736	−0.808681	+	0.588247i	$0.799818\pi$
$710$	0	0
$711$	2106.57	0.111115
$712$	9871.12	0.519573
$713$	6486.93	0.340726
$714$	1001.36	0.0524861
$715$	0	0
$716$	14336.4	0.748290
$717$	−21297.2	−1.10929
$718$	24105.0	1.25291
$719$	9125.45	0.473327	0.236663	−	0.971592i	$-0.423946\pi$
$719$	9125.45	0.473327	0.236663	+	0.971592i	$0.423946\pi$
$720$	0	0
$721$	8159.77	0.421478
$722$	−36555.0	−1.88426
$723$	7028.21	0.361524
$724$	3134.69	0.160911
$725$	0	0
$726$	13104.7	0.669920
$727$	−2635.53	−0.134452	−0.0672259	−	0.997738i	$-0.521415\pi$
$727$	−2635.53	−0.134452	−0.0672259	+	0.997738i	$0.521415\pi$
$728$	−2524.52	−0.128523
$729$	20701.2	1.05173
$730$	0	0
$731$	−8627.17	−0.436508
$732$	−11464.6	−0.578886
$733$	−10778.0	−0.543102	−0.271551	−	0.962424i	$-0.587537\pi$
$733$	−10778.0	−0.543102	−0.271551	+	0.962424i	$0.587537\pi$
$734$	−18490.8	−0.929848
$735$	0	0
$736$	−736.000	−0.0368605
$737$	475.278	0.0237545
$738$	723.714	0.0360979
$739$	−1758.72	−0.0875449	−0.0437725	−	0.999042i	$-0.513938\pi$
$739$	−1758.72	−0.0875449	−0.0437725	+	0.999042i	$0.513938\pi$
$740$	0	0
$741$	−50132.6	−2.48538
$742$	−3322.11	−0.164364
$743$	−9737.97	−0.480823	−0.240412	−	0.970671i	$-0.577282\pi$
$743$	−9737.97	−0.480823	−0.240412	+	0.970671i	$0.577282\pi$
$744$	−11373.9	−0.560467
$745$	0	0
$746$	−1738.53	−0.0853245
$747$	−252.393	−0.0123622
$748$	−440.848	−0.0215495
$749$	7500.08	0.365884
$750$	0	0
$751$	−23795.7	−1.15621	−0.578107	−	0.815961i	$-0.696208\pi$
$751$	−23795.7	−1.15621	−0.578107	+	0.815961i	$0.696208\pi$
$752$	−1444.32	−0.0700385
$753$	−30088.3	−1.45615
$754$	−4454.08	−0.215130
$755$	0	0
$756$	2900.02	0.139514
$757$	−23647.6	−1.13538	−0.567692	−	0.823241i	$-0.692163\pi$
$757$	−23647.6	−1.13538	−0.567692	+	0.823241i	$0.692163\pi$
$758$	−2557.53	−0.122551
$759$	−647.203	−0.0309512
$760$	0	0
$761$	−4606.09	−0.219410	−0.109705	−	0.993964i	$-0.534991\pi$
$761$	−4606.09	−0.219410	−0.109705	+	0.993964i	$0.534991\pi$
$762$	15654.9	0.744248
$763$	−6056.98	−0.287388
$764$	5967.79	0.282601
$765$	0	0
$766$	−10687.5	−0.504119
$767$	49944.0	2.35120
$768$	1290.47	0.0606326
$769$	17760.9	0.832867	0.416433	−	0.909166i	$-0.363280\pi$
$769$	17760.9	0.832867	0.416433	+	0.909166i	$0.363280\pi$
$770$	0	0
$771$	8247.22	0.385235
$772$	−12364.4	−0.576430
$773$	−7147.47	−0.332570	−0.166285	−	0.986078i	$-0.553177\pi$
$773$	−7147.47	−0.332570	−0.166285	+	0.986078i	$0.553177\pi$
$774$	−1388.95	−0.0645021
$775$	0	0
$776$	−851.114	−0.0393727
$777$	−3549.83	−0.163899
$778$	1416.69	0.0652839
$779$	36097.6	1.66024
$780$	0	0
$781$	2063.05	0.0945222
$782$	−908.201	−0.0415310
$783$	5116.59	0.233528
$784$	−5083.07	−0.231554
$785$	0	0
$786$	3164.40	0.143601
$787$	31719.1	1.43668	0.718338	−	0.695694i	$-0.244903\pi$
$787$	31719.1	1.43668	0.718338	+	0.695694i	$0.244903\pi$
$788$	−11281.2	−0.509995
$789$	−15100.3	−0.681348
$790$	0	0
$791$	1392.36	0.0625872
$792$	−70.9751	−0.00318433
$793$	35665.7	1.59713
$794$	−11848.0	−0.529560
$795$	0	0
$796$	5264.42	0.234413
$797$	−30828.3	−1.37013	−0.685066	−	0.728481i	$-0.740227\pi$
$797$	−30828.3	−1.37013	−0.685066	+	0.728481i	$0.740227\pi$
$798$	8041.18	0.356710
$799$	−1782.25	−0.0789129
$800$	0	0
$801$	1961.05	0.0865046
$802$	22787.3	1.00330
$803$	−1731.53	−0.0760949
$804$	−1716.77	−0.0753057
$805$	0	0
$806$	35383.5	1.54632
$807$	−26163.2	−1.14125
$808$	−5141.41	−0.223854
$809$	36679.2	1.59403	0.797015	−	0.603959i	$-0.206411\pi$
$809$	36679.2	1.59403	0.797015	+	0.603959i	$0.206411\pi$
$810$	0	0
$811$	−12210.0	−0.528672	−0.264336	−	0.964431i	$-0.585153\pi$
$811$	−12210.0	−0.528672	−0.264336	+	0.964431i	$0.585153\pi$
$812$	714.427	0.0308762
$813$	33877.0	1.46140
$814$	1562.80	0.0672927
$815$	0	0
$816$	1592.40	0.0683152
$817$	−69278.2	−2.96663
$818$	−27608.9	−1.18010
$819$	−501.534	−0.0213981
$820$	0	0
$821$	−9996.47	−0.424944	−0.212472	−	0.977167i	$-0.568151\pi$
$821$	−9996.47	−0.424944	−0.212472	+	0.977167i	$0.568151\pi$
$822$	−10748.1	−0.456063
$823$	42868.8	1.81569	0.907845	−	0.419306i	$-0.137727\pi$
$823$	42868.8	1.81569	0.907845	+	0.419306i	$0.137727\pi$
$824$	12975.9	0.548590
$825$	0	0
$826$	−8010.93	−0.337453
$827$	15833.4	0.665759	0.332880	−	0.942969i	$-0.391980\pi$
$827$	15833.4	0.665759	0.332880	+	0.942969i	$0.391980\pi$
$828$	−146.217	−0.00613697
$829$	−17722.3	−0.742484	−0.371242	−	0.928536i	$-0.621068\pi$
$829$	−17722.3	−0.742484	−0.371242	+	0.928536i	$0.621068\pi$
$830$	0	0
$831$	30823.8	1.28672
$832$	−4014.57	−0.167284
$833$	−6272.35	−0.260893
$834$	16078.3	0.667563
$835$	0	0
$836$	−3540.11	−0.146456
$837$	−40646.5	−1.67855
$838$	29591.7	1.21984
$839$	−40528.4	−1.66769	−0.833847	−	0.551996i	$-0.813866\pi$
$839$	−40528.4	−1.66769	−0.833847	+	0.551996i	$0.813866\pi$
$840$	0	0
$841$	−23128.5	−0.948317
$842$	−11609.3	−0.475159
$843$	−17890.5	−0.730938
$844$	−13596.7	−0.554525
$845$	0	0
$846$	−286.936	−0.0116608
$847$	6539.11	0.265273
$848$	−5282.92	−0.213934
$849$	−15642.5	−0.632331
$850$	0	0
$851$	3219.57	0.129689
$852$	−7452.02	−0.299650
$853$	−8116.97	−0.325814	−0.162907	−	0.986641i	$-0.552087\pi$
$853$	−8116.97	−0.325814	−0.162907	+	0.986641i	$0.552087\pi$
$854$	−5720.73	−0.229226
$855$	0	0
$856$	11926.9	0.476229
$857$	−16285.5	−0.649126	−0.324563	−	0.945864i	$-0.605217\pi$
$857$	−16285.5	−0.649126	−0.324563	+	0.945864i	$0.605217\pi$
$858$	−3530.22	−0.140466
$859$	−26487.5	−1.05209	−0.526043	−	0.850458i	$-0.676325\pi$
$859$	−26487.5	−1.05209	−0.526043	+	0.850458i	$0.676325\pi$
$860$	0	0
$861$	−5773.81	−0.228538
$862$	−30536.3	−1.20658
$863$	−40396.2	−1.59340	−0.796699	−	0.604376i	$-0.793422\pi$
$863$	−40396.2	−1.59340	−0.796699	+	0.604376i	$0.793422\pi$
$864$	4611.71	0.181590
$865$	0	0
$866$	4501.25	0.176627
$867$	−22801.0	−0.893151
$868$	−5675.46	−0.221933
$869$	7398.94	0.288828
$870$	0	0
$871$	5340.76	0.207767
$872$	−9632.00	−0.374060
$873$	−169.087	−0.00655523
$874$	−7293.07	−0.282256
$875$	0	0
$876$	6254.50	0.241233
$877$	−10962.9	−0.422109	−0.211054	−	0.977474i	$-0.567690\pi$
$877$	−10962.9	−0.422109	−0.211054	+	0.977474i	$0.567690\pi$
$878$	−20732.4	−0.796908
$879$	−6142.40	−0.235698
$880$	0	0
$881$	23906.5	0.914222	0.457111	−	0.889410i	$-0.348884\pi$
$881$	23906.5	0.914222	0.457111	+	0.889410i	$0.348884\pi$
$882$	−1009.83	−0.0385518
$883$	30509.1	1.16275	0.581377	−	0.813634i	$-0.302514\pi$
$883$	30509.1	1.16275	0.581377	+	0.813634i	$0.302514\pi$
$884$	−4953.86	−0.188480
$885$	0	0
$886$	−7762.22	−0.294331
$887$	40034.3	1.51547	0.757733	−	0.652565i	$-0.226307\pi$
$887$	40034.3	1.51547	0.757733	+	0.652565i	$0.226307\pi$
$888$	−5645.06	−0.213328
$889$	7811.62	0.294706
$890$	0	0
$891$	3815.77	0.143472
$892$	−3458.64	−0.129825
$893$	−14311.9	−0.536314
$894$	11058.8	0.413716
$895$	0	0
$896$	643.931	0.0240092
$897$	−7272.69	−0.270711
$898$	20293.0	0.754106
$899$	−10013.4	−0.371485
$900$	0	0
$901$	−6518.96	−0.241041
$902$	2541.91	0.0938318
$903$	11081.1	0.408366
$904$	2214.17	0.0814626
$905$	0	0
$906$	−27567.8	−1.01090
$907$	2843.48	0.104097	0.0520486	−	0.998645i	$-0.483425\pi$
$907$	2843.48	0.104097	0.0520486	+	0.998645i	$0.483425\pi$
$908$	7917.58	0.289377
$909$	−1021.42	−0.0372699
$910$	0	0
$911$	42268.6	1.53723	0.768617	−	0.639710i	$-0.220945\pi$
$911$	42268.6	1.53723	0.768617	+	0.639710i	$0.220945\pi$
$912$	12787.4	0.464289
$913$	−886.483	−0.0321340
$914$	−9457.74	−0.342269
$915$	0	0
$916$	−12454.1	−0.449231
$917$	1579.00	0.0568629
$918$	5690.71	0.204598
$919$	−12699.6	−0.455844	−0.227922	−	0.973679i	$-0.573193\pi$
$919$	−12699.6	−0.455844	−0.227922	+	0.973679i	$0.573193\pi$
$920$	0	0
$921$	−12927.1	−0.462501
$922$	−16761.4	−0.598706
$923$	23182.8	0.826728
$924$	566.242	0.0201602
$925$	0	0
$926$	13690.3	0.485845
$927$	2577.86	0.0913357
$928$	1136.11	0.0401880
$929$	−24889.6	−0.879011	−0.439505	−	0.898240i	$-0.644846\pi$
$929$	−24889.6	−0.879011	−0.439505	+	0.898240i	$0.644846\pi$
$930$	0	0
$931$	−50368.5	−1.77310
$932$	−24074.6	−0.846126
$933$	−466.915	−0.0163838
$934$	−29768.0	−1.04287
$935$	0	0
$936$	−797.555	−0.0278514
$937$	−9585.48	−0.334199	−0.167099	−	0.985940i	$-0.553440\pi$
$937$	−9585.48	−0.334199	−0.167099	+	0.985940i	$0.553440\pi$
$938$	−856.649	−0.0298194
$939$	21557.1	0.749189
$940$	0	0
$941$	−21146.9	−0.732592	−0.366296	−	0.930498i	$-0.619374\pi$
$941$	−21146.9	−0.732592	−0.366296	+	0.930498i	$0.619374\pi$
$942$	−14514.0	−0.502009
$943$	5236.65	0.180836
$944$	−12739.2	−0.439224
$945$	0	0
$946$	−4878.41	−0.167665
$947$	−48747.2	−1.67272	−0.836362	−	0.548177i	$-0.815322\pi$
$947$	−48747.2	−1.67272	−0.836362	+	0.548177i	$0.815322\pi$
$948$	−26726.0	−0.915632
$949$	−19457.3	−0.665556
$950$	0	0
$951$	14942.4	0.509507
$952$	794.591	0.0270513
$953$	−24925.9	−0.847249	−0.423625	−	0.905838i	$-0.639242\pi$
$953$	−24925.9	−0.847249	−0.423625	+	0.905838i	$0.639242\pi$
$954$	−1049.53	−0.0356183
$955$	0	0
$956$	−16899.5	−0.571725
$957$	999.036	0.0337453
$958$	−12097.4	−0.407984
$959$	−5363.20	−0.180591
$960$	0	0
$961$	49755.9	1.67017
$962$	17561.4	0.588568
$963$	2369.45	0.0792882
$964$	5576.95	0.186329
$965$	0	0
$966$	1166.53	0.0388534
$967$	1881.92	0.0625838	0.0312919	−	0.999510i	$-0.490038\pi$
$967$	1881.92	0.0625838	0.0312919	+	0.999510i	$0.490038\pi$
$968$	10398.7	0.345276
$969$	15779.2	0.523117
$970$	0	0
$971$	−14637.3	−0.483763	−0.241881	−	0.970306i	$-0.577764\pi$
$971$	−14637.3	−0.483763	−0.241881	+	0.970306i	$0.577764\pi$
$972$	1781.44	0.0587857
$973$	8022.92	0.264340
$974$	6398.70	0.210501
$975$	0	0
$976$	−9097.28	−0.298358
$977$	−42591.4	−1.39470	−0.697348	−	0.716733i	$-0.745637\pi$
$977$	−42591.4	−1.39470	−0.697348	+	0.716733i	$0.745637\pi$
$978$	−30203.2	−0.987518
$979$	6887.81	0.224857
$980$	0	0
$981$	−1913.54	−0.0622780
$982$	3228.81	0.104924
$983$	48858.3	1.58529	0.792644	−	0.609684i	$-0.208704\pi$
$983$	48858.3	1.58529	0.792644	+	0.609684i	$0.208704\pi$
$984$	−9181.71	−0.297462
$985$	0	0
$986$	1401.92	0.0452801
$987$	2289.19	0.0738254
$988$	−39780.6	−1.28096
$989$	−10050.1	−0.323130
$990$	0	0
$991$	−56522.1	−1.81179	−0.905895	−	0.423502i	$-0.860801\pi$
$991$	−56522.1	−1.81179	−0.905895	+	0.423502i	$0.860801\pi$
$992$	−9025.30	−0.288864
$993$	−6799.19	−0.217287
$994$	−3718.48	−0.118655
$995$	0	0
$996$	3202.09	0.101870
$997$	20767.9	0.659706	0.329853	−	0.944032i	$-0.393001\pi$
$997$	20767.9	0.659706	0.329853	+	0.944032i	$0.393001\pi$
$998$	10784.1	0.342048
$999$	−20173.5	−0.638902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.n.1.4		4
5.2	odd	4		1150.4.b.o.599.1		8
5.3	odd	4		1150.4.b.o.599.8		8
5.4	even	2		230.4.a.j.1.1	✓	4
15.14	odd	2		2070.4.a.bg.1.3		4
20.19	odd	2		1840.4.a.k.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.1	✓	4	5.4	even	2
1150.4.a.n.1.4		4	1.1	even	1	trivial
1150.4.b.o.599.1		8	5.2	odd	4
1150.4.b.o.599.8		8	5.3	odd	4
1840.4.a.k.1.4		4	20.19	odd	2
2070.4.a.bg.1.3		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +5.04090 q^{3} +4.00000 q^{4} -10.0818 q^{6} -5.03071 q^{7} -8.00000 q^{8} -1.58932 q^{9} -5.58219 q^{11} +20.1636 q^{12} -62.7277 q^{13} +10.0614 q^{14} +16.0000 q^{16} +19.7435 q^{17} +3.17864 q^{18} +158.545 q^{19} -25.3593 q^{21} +11.1644 q^{22} +23.0000 q^{23} -40.3272 q^{24} +125.455 q^{26} -144.116 q^{27} -20.1228 q^{28} -35.5033 q^{29} +282.041 q^{31} -32.0000 q^{32} -28.1393 q^{33} -39.4870 q^{34} -6.35728 q^{36} +139.981 q^{37} -317.090 q^{38} -316.204 q^{39} +227.680 q^{41} +50.7186 q^{42} -436.962 q^{43} -22.3288 q^{44} -46.0000 q^{46} -90.2701 q^{47} +80.6544 q^{48} -317.692 q^{49} +99.5250 q^{51} -250.911 q^{52} -330.183 q^{53} +288.232 q^{54} +40.2457 q^{56} +799.209 q^{57} +71.0066 q^{58} -796.203 q^{59} -568.580 q^{61} -564.081 q^{62} +7.99541 q^{63} +64.0000 q^{64} +56.2785 q^{66} -85.1419 q^{67} +78.9740 q^{68} +115.941 q^{69} -369.578 q^{71} +12.7146 q^{72} +310.188 q^{73} -279.963 q^{74} +634.180 q^{76} +28.0824 q^{77} +632.408 q^{78} -1325.46 q^{79} -683.562 q^{81} -455.360 q^{82} +158.806 q^{83} -101.437 q^{84} +873.924 q^{86} -178.969 q^{87} +44.6575 q^{88} -1233.89 q^{89} +315.565 q^{91} +92.0000 q^{92} +1421.74 q^{93} +180.540 q^{94} -161.309 q^{96} +106.389 q^{97} +635.384 q^{98} +8.87189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 14 q^{3} + 16 q^{4} + 28 q^{6} - 8 q^{7} - 32 q^{8} + 64 q^{9} + 21 q^{11} - 56 q^{12} - 70 q^{13} + 16 q^{14} + 64 q^{16} - 56 q^{17} - 128 q^{18} + 173 q^{19} - 120 q^{21} - 42 q^{22}+ \cdots - 2745 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more