LMFDB - Embedded newform 1575.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.27492 q^{2} -2.82475 q^{4} -7.00000 q^{7} -24.6254 q^{8} +O(q^{10})$	$q+2.27492 q^{2} -2.82475 q^{4} -7.00000 q^{7} -24.6254 q^{8} +40.7492 q^{11} -53.2990 q^{13} -15.9244 q^{14} -33.4228 q^{16} +4.54983 q^{17} +122.598 q^{19} +92.7010 q^{22} +131.347 q^{23} -121.251 q^{26} +19.7733 q^{28} +216.598 q^{29} -251.794 q^{31} +120.969 q^{32} +10.3505 q^{34} -11.8970 q^{37} +278.900 q^{38} +111.752 q^{41} -369.196 q^{43} -115.106 q^{44} +298.804 q^{46} -262.694 q^{47} +49.0000 q^{49} +150.556 q^{52} -567.100 q^{53} +172.378 q^{56} +492.743 q^{58} -839.890 q^{59} -485.794 q^{61} -572.811 q^{62} +542.577 q^{64} +333.691 q^{67} -12.8522 q^{68} -590.248 q^{71} -490.701 q^{73} -27.0647 q^{74} -346.309 q^{76} -285.244 q^{77} +121.691 q^{79} +254.228 q^{82} +609.608 q^{83} -839.890 q^{86} -1003.47 q^{88} -719.038 q^{89} +373.093 q^{91} -371.023 q^{92} -597.608 q^{94} +637.877 q^{97} +111.471 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8	$ 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8 + 6 * q^11 - 16 * q^13 + 21 * q^14 + 137 * q^16 - 6 * q^17 + 64 * q^19 + 276 * q^22 + 6 * q^23 - 318 * q^26 - 119 * q^28 + 252 * q^29 + 40 * q^31 - 279 * q^32 + 66 * q^34 + 248 * q^37 + 588 * q^38 + 450 * q^41 - 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 98 * q^49 + 890 * q^52 - 1104 * q^53 + 609 * q^56 + 306 * q^58 - 804 * q^59 - 428 * q^61 - 2112 * q^62 + 1289 * q^64 - 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 - 1508 * q^76 - 42 * q^77 - 572 * q^79 - 1530 * q^82 + 1944 * q^83 - 804 * q^86 + 1164 * q^88 - 366 * q^89 + 112 * q^91 - 2856 * q^92 - 1920 * q^94 - 808 * 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 $

$2$	2.27492	0.804305	0.402152	−	0.915573i	$-0.368262\pi$
$2$	2.27492	0.804305	0.402152	+	0.915573i	$0.368262\pi$
$3$	0	0
$3$	0	0
$4$	−2.82475	−0.353094
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−24.6254	−1.08830
$9$	0	0
$10$	0	0
$11$	40.7492	1.11694	0.558470	−	0.829525i	$-0.311389\pi$
$11$	40.7492	1.11694	0.558470	+	0.829525i	$0.311389\pi$
$12$	0	0
$13$	−53.2990	−1.13711	−0.568557	−	0.822644i	$-0.692498\pi$
$13$	−53.2990	−1.13711	−0.568557	+	0.822644i	$0.692498\pi$
$14$	−15.9244	−0.303999
$15$	0	0
$16$	−33.4228	−0.522231
$17$	4.54983	0.0649116	0.0324558	−	0.999473i	$-0.489667\pi$
$17$	4.54983	0.0649116	0.0324558	+	0.999473i	$0.489667\pi$
$18$	0	0
$19$	122.598	1.48031	0.740156	−	0.672436i	$-0.234752\pi$
$19$	122.598	1.48031	0.740156	+	0.672436i	$0.234752\pi$
$20$	0	0
$21$	0	0
$22$	92.7010	0.898360
$23$	131.347	1.19077	0.595387	−	0.803439i	$-0.296999\pi$
$23$	131.347	1.19077	0.595387	+	0.803439i	$0.296999\pi$
$24$	0	0
$25$	0	0
$26$	−121.251	−0.914586
$27$	0	0
$28$	19.7733	0.133457
$29$	216.598	1.38694	0.693470	−	0.720486i	$-0.256081\pi$
$29$	216.598	1.38694	0.693470	+	0.720486i	$0.256081\pi$
$30$	0	0
$31$	−251.794	−1.45882	−0.729412	−	0.684075i	$-0.760206\pi$
$31$	−251.794	−1.45882	−0.729412	+	0.684075i	$0.760206\pi$
$32$	120.969	0.668267
$33$	0	0
$34$	10.3505	0.0522087
$35$	0	0
$36$	0	0
$37$	−11.8970	−0.0528610	−0.0264305	−	0.999651i	$-0.508414\pi$
$37$	−11.8970	−0.0528610	−0.0264305	+	0.999651i	$0.508414\pi$
$38$	278.900	1.19062
$39$	0	0
$40$	0	0
$41$	111.752	0.425678	0.212839	−	0.977087i	$-0.431729\pi$
$41$	111.752	0.425678	0.212839	+	0.977087i	$0.431729\pi$
$42$	0	0
$43$	−369.196	−1.30935	−0.654673	−	0.755912i	$-0.727194\pi$
$43$	−369.196	−1.30935	−0.654673	+	0.755912i	$0.727194\pi$
$44$	−115.106	−0.394385
$45$	0	0
$46$	298.804	0.957744
$47$	−262.694	−0.815275	−0.407637	−	0.913144i	$-0.633647\pi$
$47$	−262.694	−0.815275	−0.407637	+	0.913144i	$0.633647\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	150.556	0.401508
$53$	−567.100	−1.46976	−0.734879	−	0.678199i	$-0.762761\pi$
$53$	−567.100	−1.46976	−0.734879	+	0.678199i	$0.762761\pi$
$54$	0	0
$55$	0	0
$56$	172.378	0.411339
$57$	0	0
$58$	492.743	1.11552
$59$	−839.890	−1.85330	−0.926648	−	0.375931i	$-0.877323\pi$
$59$	−839.890	−1.85330	−0.926648	+	0.375931i	$0.877323\pi$
$60$	0	0
$61$	−485.794	−1.01966	−0.509832	−	0.860274i	$-0.670293\pi$
$61$	−485.794	−1.01966	−0.509832	+	0.860274i	$0.670293\pi$
$62$	−572.811	−1.17334
$63$	0	0
$64$	542.577	1.05972
$65$	0	0
$66$	0	0
$67$	333.691	0.608460	0.304230	−	0.952599i	$-0.401601\pi$
$67$	333.691	0.608460	0.304230	+	0.952599i	$0.401601\pi$
$68$	−12.8522	−0.0229199
$69$	0	0
$70$	0	0
$71$	−590.248	−0.986613	−0.493306	−	0.869856i	$-0.664212\pi$
$71$	−590.248	−0.986613	−0.493306	+	0.869856i	$0.664212\pi$
$72$	0	0
$73$	−490.701	−0.786743	−0.393371	−	0.919380i	$-0.628691\pi$
$73$	−490.701	−0.786743	−0.393371	+	0.919380i	$0.628691\pi$
$74$	−27.0647	−0.0425164
$75$	0	0
$76$	−346.309	−0.522689
$77$	−285.244	−0.422164
$78$	0	0
$79$	121.691	0.173308	0.0866539	−	0.996238i	$-0.472383\pi$
$79$	121.691	0.173308	0.0866539	+	0.996238i	$0.472383\pi$
$80$	0	0
$81$	0	0
$82$	254.228	0.342375
$83$	609.608	0.806183	0.403091	−	0.915160i	$-0.367936\pi$
$83$	609.608	0.806183	0.403091	+	0.915160i	$0.367936\pi$
$84$	0	0
$85$	0	0
$86$	−839.890	−1.05311
$87$	0	0
$88$	−1003.47	−1.21557
$89$	−719.038	−0.856381	−0.428190	−	0.903689i	$-0.640849\pi$
$89$	−719.038	−0.856381	−0.428190	+	0.903689i	$0.640849\pi$
$90$	0	0
$91$	373.093	0.429789
$92$	−371.023	−0.420455
$93$	0	0
$94$	−597.608	−0.655729
$95$	0	0
$96$	0	0
$97$	637.877	0.667697	0.333849	−	0.942627i	$-0.391653\pi$
$97$	637.877	0.667697	0.333849	+	0.942627i	$0.391653\pi$
$98$	111.471	0.114901
$99$	0	0
$100$	0	0
$101$	−671.148	−0.661205	−0.330603	−	0.943770i	$-0.607252\pi$
$101$	−671.148	−0.661205	−0.330603	+	0.943770i	$0.607252\pi$
$102$	0	0
$103$	912.412	0.872841	0.436420	−	0.899743i	$-0.356246\pi$
$103$	912.412	0.872841	0.436420	+	0.899743i	$0.356246\pi$
$104$	1312.51	1.23752
$105$	0	0
$106$	−1290.10	−1.18213
$107$	−116.736	−0.105470	−0.0527350	−	0.998609i	$-0.516794\pi$
$107$	−116.736	−0.105470	−0.0527350	+	0.998609i	$0.516794\pi$
$108$	0	0
$109$	837.176	0.735660	0.367830	−	0.929893i	$-0.380101\pi$
$109$	837.176	0.735660	0.367830	+	0.929893i	$0.380101\pi$
$110$	0	0
$111$	0	0
$112$	233.959	0.197385
$113$	−1086.58	−0.904572	−0.452286	−	0.891873i	$-0.649391\pi$
$113$	−1086.58	−0.904572	−0.452286	+	0.891873i	$0.649391\pi$
$114$	0	0
$115$	0	0
$116$	−611.836	−0.489720
$117$	0	0
$118$	−1910.68	−1.49061
$119$	−31.8488	−0.0245343
$120$	0	0
$121$	329.495	0.247554
$122$	−1105.14	−0.820121
$123$	0	0
$124$	711.256	0.515102
$125$	0	0
$126$	0	0
$127$	537.113	0.375284	0.187642	−	0.982237i	$-0.439916\pi$
$127$	537.113	0.375284	0.187642	+	0.982237i	$0.439916\pi$
$128$	266.564	0.184071
$129$	0	0
$130$	0	0
$131$	−1497.39	−0.998683	−0.499341	−	0.866405i	$-0.666425\pi$
$131$	−1497.39	−0.998683	−0.499341	+	0.866405i	$0.666425\pi$
$132$	0	0
$133$	−858.186	−0.559505
$134$	759.120	0.489388
$135$	0	0
$136$	−112.042	−0.0706433
$137$	−1380.09	−0.860650	−0.430325	−	0.902674i	$-0.641601\pi$
$137$	−1380.09	−0.860650	−0.430325	+	0.902674i	$0.641601\pi$
$138$	0	0
$139$	−141.980	−0.0866374	−0.0433187	−	0.999061i	$-0.513793\pi$
$139$	−141.980	−0.0866374	−0.0433187	+	0.999061i	$0.513793\pi$
$140$	0	0
$141$	0	0
$142$	−1342.76	−0.793537
$143$	−2171.89	−1.27009
$144$	0	0
$145$	0	0
$146$	−1116.30	−0.632781
$147$	0	0
$148$	33.6061	0.0186649
$149$	1943.87	1.06878	0.534390	−	0.845238i	$-0.320542\pi$
$149$	1943.87	1.06878	0.534390	+	0.845238i	$0.320542\pi$
$150$	0	0
$151$	−2654.76	−1.43074	−0.715370	−	0.698746i	$-0.753742\pi$
$151$	−2654.76	−1.43074	−0.715370	+	0.698746i	$0.753742\pi$
$152$	−3019.03	−1.61102
$153$	0	0
$154$	−648.907	−0.339548
$155$	0	0
$156$	0	0
$157$	−1665.22	−0.846489	−0.423244	−	0.906016i	$-0.639109\pi$
$157$	−1665.22	−0.846489	−0.423244	+	0.906016i	$0.639109\pi$
$158$	276.837	0.139392
$159$	0	0
$160$	0	0
$161$	−919.430	−0.450070
$162$	0	0
$163$	33.0732	0.0158926	0.00794629	−	0.999968i	$-0.497471\pi$
$163$	33.0732	0.0158926	0.00794629	+	0.999968i	$0.497471\pi$
$164$	−315.673	−0.150304
$165$	0	0
$166$	1386.81	0.648417
$167$	1654.48	0.766630	0.383315	−	0.923618i	$-0.374782\pi$
$167$	1654.48	0.766630	0.383315	+	0.923618i	$0.374782\pi$
$168$	0	0
$169$	643.784	0.293029
$170$	0	0
$171$	0	0
$172$	1042.89	0.462322
$173$	64.1909	0.0282101	0.0141050	−	0.999901i	$-0.495510\pi$
$173$	64.1909	0.0282101	0.0141050	+	0.999901i	$0.495510\pi$
$174$	0	0
$175$	0	0
$176$	−1361.95	−0.583300
$177$	0	0
$178$	−1635.75	−0.688791
$179$	−3914.68	−1.63462	−0.817309	−	0.576200i	$-0.804535\pi$
$179$	−3914.68	−1.63462	−0.817309	+	0.576200i	$0.804535\pi$
$180$	0	0
$181$	−2058.04	−0.845156	−0.422578	−	0.906327i	$-0.638875\pi$
$181$	−2058.04	−0.845156	−0.422578	+	0.906327i	$0.638875\pi$
$182$	848.756	0.345681
$183$	0	0
$184$	−3234.48	−1.29592
$185$	0	0
$186$	0	0
$187$	185.402	0.0725023
$188$	742.046	0.287869
$189$	0	0
$190$	0	0
$191$	−428.048	−0.162160	−0.0810798	−	0.996708i	$-0.525837\pi$
$191$	−428.048	−0.162160	−0.0810798	+	0.996708i	$0.525837\pi$
$192$	0	0
$193$	−1604.93	−0.598576	−0.299288	−	0.954163i	$-0.596749\pi$
$193$	−1604.93	−0.598576	−0.299288	+	0.954163i	$0.596749\pi$
$194$	1451.12	0.537032
$195$	0	0
$196$	−138.413	−0.0504420
$197$	3738.83	1.35218	0.676092	−	0.736817i	$-0.263672\pi$
$197$	3738.83	1.35218	0.676092	+	0.736817i	$0.263672\pi$
$198$	0	0
$199$	−349.030	−0.124332	−0.0621660	−	0.998066i	$-0.519801\pi$
$199$	−349.030	−0.124332	−0.0621660	+	0.998066i	$0.519801\pi$
$200$	0	0
$201$	0	0
$202$	−1526.81	−0.531810
$203$	−1516.19	−0.524214
$204$	0	0
$205$	0	0
$206$	2075.66	0.702030
$207$	0	0
$208$	1781.40	0.593836
$209$	4995.77	1.65342
$210$	0	0
$211$	2588.58	0.844574	0.422287	−	0.906462i	$-0.361227\pi$
$211$	2588.58	0.844574	0.422287	+	0.906462i	$0.361227\pi$
$212$	1601.92	0.518962
$213$	0	0
$214$	−265.565	−0.0848300
$215$	0	0
$216$	0	0
$217$	1762.56	0.551384
$218$	1904.51	0.591695
$219$	0	0
$220$	0	0
$221$	−242.502	−0.0738119
$222$	0	0
$223$	3236.21	0.971804	0.485902	−	0.874013i	$-0.338491\pi$
$223$	3236.21	0.971804	0.485902	+	0.874013i	$0.338491\pi$
$224$	−846.785	−0.252581
$225$	0	0
$226$	−2471.88	−0.727552
$227$	−5631.62	−1.64662	−0.823312	−	0.567589i	$-0.807876\pi$
$227$	−5631.62	−1.64662	−0.823312	+	0.567589i	$0.807876\pi$
$228$	0	0
$229$	3770.25	1.08797	0.543985	−	0.839095i	$-0.316915\pi$
$229$	3770.25	1.08797	0.543985	+	0.839095i	$0.316915\pi$
$230$	0	0
$231$	0	0
$232$	−5333.82	−1.50941
$233$	−6560.90	−1.84472	−0.922358	−	0.386336i	$-0.873741\pi$
$233$	−6560.90	−1.84472	−0.922358	+	0.386336i	$0.873741\pi$
$234$	0	0
$235$	0	0
$236$	2372.48	0.654387
$237$	0	0
$238$	−72.4535	−0.0197330
$239$	771.444	0.208789	0.104394	−	0.994536i	$-0.466710\pi$
$239$	771.444	0.208789	0.104394	+	0.994536i	$0.466710\pi$
$240$	0	0
$241$	1252.10	0.334668	0.167334	−	0.985900i	$-0.446484\pi$
$241$	1252.10	0.334668	0.167334	+	0.985900i	$0.446484\pi$
$242$	749.574	0.199109
$243$	0	0
$244$	1372.25	0.360037
$245$	0	0
$246$	0	0
$247$	−6534.35	−1.68328
$248$	6200.53	1.58764
$249$	0	0
$250$	0	0
$251$	−5166.27	−1.29917	−0.649586	−	0.760288i	$-0.725058\pi$
$251$	−5166.27	−1.29917	−0.649586	+	0.760288i	$0.725058\pi$
$252$	0	0
$253$	5352.29	1.33002
$254$	1221.89	0.301843
$255$	0	0
$256$	−3734.21	−0.911672
$257$	2767.45	0.671707	0.335854	−	0.941914i	$-0.390975\pi$
$257$	2767.45	0.671707	0.335854	+	0.941914i	$0.390975\pi$
$258$	0	0
$259$	83.2791	0.0199796
$260$	0	0
$261$	0	0
$262$	−3406.44	−0.803245
$263$	−4101.78	−0.961699	−0.480849	−	0.876803i	$-0.659672\pi$
$263$	−4101.78	−0.961699	−0.480849	+	0.876803i	$0.659672\pi$
$264$	0	0
$265$	0	0
$266$	−1952.30	−0.450013
$267$	0	0
$268$	−942.594	−0.214844
$269$	6950.84	1.57546	0.787732	−	0.616018i	$-0.211255\pi$
$269$	6950.84	1.57546	0.787732	+	0.616018i	$0.211255\pi$
$270$	0	0
$271$	−7140.29	−1.60052	−0.800262	−	0.599651i	$-0.795306\pi$
$271$	−7140.29	−1.60052	−0.800262	+	0.599651i	$0.795306\pi$
$272$	−152.068	−0.0338988
$273$	0	0
$274$	−3139.59	−0.692225
$275$	0	0
$276$	0	0
$277$	−1320.51	−0.286433	−0.143217	−	0.989691i	$-0.545745\pi$
$277$	−1320.51	−0.286433	−0.143217	+	0.989691i	$0.545745\pi$
$278$	−322.993	−0.0696829
$279$	0	0
$280$	0	0
$281$	204.309	0.0433738	0.0216869	−	0.999765i	$-0.493096\pi$
$281$	204.309	0.0433738	0.0216869	+	0.999765i	$0.493096\pi$
$282$	0	0
$283$	−975.794	−0.204964	−0.102482	−	0.994735i	$-0.532678\pi$
$283$	−975.794	−0.204964	−0.102482	+	0.994735i	$0.532678\pi$
$284$	1667.30	0.348367
$285$	0	0
$286$	−4940.87	−1.02154
$287$	−782.267	−0.160891
$288$	0	0
$289$	−4892.30	−0.995786
$290$	0	0
$291$	0	0
$292$	1386.11	0.277794
$293$	607.919	0.121212	0.0606058	−	0.998162i	$-0.480697\pi$
$293$	607.919	0.121212	0.0606058	+	0.998162i	$0.480697\pi$
$294$	0	0
$295$	0	0
$296$	292.969	0.0575286
$297$	0	0
$298$	4422.14	0.859624
$299$	−7000.67	−1.35405
$300$	0	0
$301$	2584.37	0.494886
$302$	−6039.37	−1.15075
$303$	0	0
$304$	−4097.56	−0.773064
$305$	0	0
$306$	0	0
$307$	−8037.08	−1.49414	−0.747069	−	0.664747i	$-0.768539\pi$
$307$	−8037.08	−1.49414	−0.747069	+	0.664747i	$0.768539\pi$
$308$	805.744	0.149063
$309$	0	0
$310$	0	0
$311$	−5311.60	−0.968468	−0.484234	−	0.874939i	$-0.660902\pi$
$311$	−5311.60	−0.968468	−0.484234	+	0.874939i	$0.660902\pi$
$312$	0	0
$313$	1531.61	0.276587	0.138293	−	0.990391i	$-0.455838\pi$
$313$	1531.61	0.276587	0.138293	+	0.990391i	$0.455838\pi$
$314$	−3788.23	−0.680835
$315$	0	0
$316$	−343.747	−0.0611939
$317$	4219.19	0.747549	0.373775	−	0.927520i	$-0.378063\pi$
$317$	4219.19	0.747549	0.373775	+	0.927520i	$0.378063\pi$
$318$	0	0
$319$	8826.19	1.54913
$320$	0	0
$321$	0	0
$322$	−2091.63	−0.361993
$323$	557.801	0.0960893
$324$	0	0
$325$	0	0
$326$	75.2387	0.0127825
$327$	0	0
$328$	−2751.95	−0.463265
$329$	1838.86	0.308145
$330$	0	0
$331$	8298.19	1.37797	0.688987	−	0.724773i	$-0.258056\pi$
$331$	8298.19	1.37797	0.688987	+	0.724773i	$0.258056\pi$
$332$	−1721.99	−0.284658
$333$	0	0
$334$	3763.79	0.616604
$335$	0	0
$336$	0	0
$337$	4348.44	0.702892	0.351446	−	0.936208i	$-0.385690\pi$
$337$	4348.44	0.702892	0.351446	+	0.936208i	$0.385690\pi$
$338$	1464.56	0.235684
$339$	0	0
$340$	0	0
$341$	−10260.4	−1.62942
$342$	0	0
$343$	−343.000	−0.0539949
$344$	9091.60	1.42496
$345$	0	0
$346$	146.029	0.0226895
$347$	−8345.54	−1.29110	−0.645550	−	0.763718i	$-0.723372\pi$
$347$	−8345.54	−1.29110	−0.645550	+	0.763718i	$0.723372\pi$
$348$	0	0
$349$	−9982.54	−1.53110	−0.765549	−	0.643378i	$-0.777532\pi$
$349$	−9982.54	−1.53110	−0.765549	+	0.643378i	$0.777532\pi$
$350$	0	0
$351$	0	0
$352$	4929.40	0.746414
$353$	8801.59	1.32709	0.663543	−	0.748138i	$-0.269052\pi$
$353$	8801.59	1.32709	0.663543	+	0.748138i	$0.269052\pi$
$354$	0	0
$355$	0	0
$356$	2031.10	0.302383
$357$	0	0
$358$	−8905.56	−1.31473
$359$	524.039	0.0770409	0.0385205	−	0.999258i	$-0.487736\pi$
$359$	524.039	0.0770409	0.0385205	+	0.999258i	$0.487736\pi$
$360$	0	0
$361$	8171.27	1.19132
$362$	−4681.88	−0.679763
$363$	0	0
$364$	−1053.90	−0.151756
$365$	0	0
$366$	0	0
$367$	6362.72	0.904991	0.452495	−	0.891767i	$-0.350534\pi$
$367$	6362.72	0.904991	0.452495	+	0.891767i	$0.350534\pi$
$368$	−4389.99	−0.621858
$369$	0	0
$370$	0	0
$371$	3969.70	0.555516
$372$	0	0
$373$	11265.8	1.56387	0.781935	−	0.623361i	$-0.214233\pi$
$373$	11265.8	1.56387	0.781935	+	0.623361i	$0.214233\pi$
$374$	421.774	0.0583140
$375$	0	0
$376$	6468.96	0.887263
$377$	−11544.5	−1.57711
$378$	0	0
$379$	−1151.71	−0.156094	−0.0780470	−	0.996950i	$-0.524868\pi$
$379$	−1151.71	−0.156094	−0.0780470	+	0.996950i	$0.524868\pi$
$380$	0	0
$381$	0	0
$382$	−973.774	−0.130426
$383$	−151.554	−0.0202195	−0.0101097	−	0.999949i	$-0.503218\pi$
$383$	−151.554	−0.0202195	−0.0101097	+	0.999949i	$0.503218\pi$
$384$	0	0
$385$	0	0
$386$	−3651.08	−0.481437
$387$	0	0
$388$	−1801.84	−0.235760
$389$	−4794.18	−0.624870	−0.312435	−	0.949939i	$-0.601145\pi$
$389$	−4794.18	−0.624870	−0.312435	+	0.949939i	$0.601145\pi$
$390$	0	0
$391$	597.608	0.0772950
$392$	−1206.65	−0.155471
$393$	0	0
$394$	8505.52	1.08757
$395$	0	0
$396$	0	0
$397$	4623.94	0.584556	0.292278	−	0.956333i	$-0.405587\pi$
$397$	4623.94	0.584556	0.292278	+	0.956333i	$0.405587\pi$
$398$	−794.014	−0.100001
$399$	0	0
$400$	0	0
$401$	3610.63	0.449642	0.224821	−	0.974400i	$-0.427820\pi$
$401$	3610.63	0.449642	0.224821	+	0.974400i	$0.427820\pi$
$402$	0	0
$403$	13420.4	1.65885
$404$	1895.83	0.233467
$405$	0	0
$406$	−3449.20	−0.421628
$407$	−484.794	−0.0590426
$408$	0	0
$409$	8959.57	1.08318	0.541592	−	0.840641i	$-0.317822\pi$
$409$	8959.57	1.08318	0.541592	+	0.840641i	$0.317822\pi$
$410$	0	0
$411$	0	0
$412$	−2577.34	−0.308195
$413$	5879.23	0.700480
$414$	0	0
$415$	0	0
$416$	−6447.54	−0.759896
$417$	0	0
$418$	11365.0	1.32985
$419$	7078.28	0.825290	0.412645	−	0.910892i	$-0.364605\pi$
$419$	7078.28	0.825290	0.412645	+	0.910892i	$0.364605\pi$
$420$	0	0
$421$	11551.5	1.33725	0.668626	−	0.743599i	$-0.266883\pi$
$421$	11551.5	1.33725	0.668626	+	0.743599i	$0.266883\pi$
$422$	5888.80	0.679295
$423$	0	0
$424$	13965.1	1.59954
$425$	0	0
$426$	0	0
$427$	3400.56	0.385397
$428$	329.750	0.0372408
$429$	0	0
$430$	0	0
$431$	4064.38	0.454232	0.227116	−	0.973868i	$-0.427070\pi$
$431$	4064.38	0.454232	0.227116	+	0.973868i	$0.427070\pi$
$432$	0	0
$433$	−17456.3	−1.93740	−0.968701	−	0.248229i	$-0.920151\pi$
$433$	−17456.3	−1.93740	−0.968701	+	0.248229i	$0.920151\pi$
$434$	4009.67	0.443480
$435$	0	0
$436$	−2364.81	−0.259757
$437$	16102.9	1.76271
$438$	0	0
$439$	4595.39	0.499604	0.249802	−	0.968297i	$-0.419635\pi$
$439$	4595.39	0.499604	0.249802	+	0.968297i	$0.419635\pi$
$440$	0	0
$441$	0	0
$442$	−551.671	−0.0593672
$443$	−306.214	−0.0328412	−0.0164206	−	0.999865i	$-0.505227\pi$
$443$	−306.214	−0.0328412	−0.0164206	+	0.999865i	$0.505227\pi$
$444$	0	0
$445$	0	0
$446$	7362.10	0.781627
$447$	0	0
$448$	−3798.04	−0.400537
$449$	−9229.22	−0.970053	−0.485026	−	0.874500i	$-0.661190\pi$
$449$	−9229.22	−0.970053	−0.485026	+	0.874500i	$0.661190\pi$
$450$	0	0
$451$	4553.82	0.475457
$452$	3069.31	0.319399
$453$	0	0
$454$	−12811.5	−1.32439
$455$	0	0
$456$	0	0
$457$	10992.2	1.12515	0.562577	−	0.826745i	$-0.309810\pi$
$457$	10992.2	1.12515	0.562577	+	0.826745i	$0.309810\pi$
$458$	8577.01	0.875059
$459$	0	0
$460$	0	0
$461$	−7387.88	−0.746394	−0.373197	−	0.927752i	$-0.621739\pi$
$461$	−7387.88	−0.746394	−0.373197	+	0.927752i	$0.621739\pi$
$462$	0	0
$463$	−10163.8	−1.02020	−0.510101	−	0.860114i	$-0.670392\pi$
$463$	−10163.8	−1.02020	−0.510101	+	0.860114i	$0.670392\pi$
$464$	−7239.30	−0.724302
$465$	0	0
$466$	−14925.5	−1.48371
$467$	−15814.6	−1.56705	−0.783524	−	0.621362i	$-0.786580\pi$
$467$	−15814.6	−1.56705	−0.783524	+	0.621362i	$0.786580\pi$
$468$	0	0
$469$	−2335.84	−0.229976
$470$	0	0
$471$	0	0
$472$	20682.6	2.01694
$473$	−15044.4	−1.46246
$474$	0	0
$475$	0	0
$476$	89.9651	0.00866290
$477$	0	0
$478$	1754.97	0.167930
$479$	1444.85	0.137823	0.0689113	−	0.997623i	$-0.478047\pi$
$479$	1444.85	0.137823	0.0689113	+	0.997623i	$0.478047\pi$
$480$	0	0
$481$	634.099	0.0601090
$482$	2848.43	0.269175
$483$	0	0
$484$	−930.742	−0.0874100
$485$	0	0
$486$	0	0
$487$	489.402	0.0455378	0.0227689	−	0.999741i	$-0.492752\pi$
$487$	489.402	0.0455378	0.0227689	+	0.999741i	$0.492752\pi$
$488$	11962.9	1.10970
$489$	0	0
$490$	0	0
$491$	3941.30	0.362257	0.181129	−	0.983459i	$-0.442025\pi$
$491$	3941.30	0.362257	0.181129	+	0.983459i	$0.442025\pi$
$492$	0	0
$493$	985.485	0.0900284
$494$	−14865.1	−1.35387
$495$	0	0
$496$	8415.65	0.761843
$497$	4131.73	0.372905
$498$	0	0
$499$	11.0894	0.000994850 0	0.000497425	−	1.00000i	$-0.499842\pi$
$499$	11.0894	0.000994850 0	0.000497425		1.00000i	$0.499842\pi$
$500$	0	0
$501$	0	0
$502$	−11752.8	−1.04493
$503$	7088.41	0.628343	0.314172	−	0.949366i	$-0.398273\pi$
$503$	7088.41	0.628343	0.314172	+	0.949366i	$0.398273\pi$
$504$	0	0
$505$	0	0
$506$	12176.0	1.06974
$507$	0	0
$508$	−1517.21	−0.132511
$509$	−17588.4	−1.53162	−0.765810	−	0.643067i	$-0.777662\pi$
$509$	−17588.4	−1.53162	−0.765810	+	0.643067i	$0.777662\pi$
$510$	0	0
$511$	3434.91	0.297361
$512$	−10627.5	−0.917333
$513$	0	0
$514$	6295.72	0.540257
$515$	0	0
$516$	0	0
$517$	−10704.6	−0.910613
$518$	189.453	0.0160697
$519$	0	0
$520$	0	0
$521$	11646.6	0.979360	0.489680	−	0.871902i	$-0.337114\pi$
$521$	11646.6	0.979360	0.489680	+	0.871902i	$0.337114\pi$
$522$	0	0
$523$	−8965.82	−0.749614	−0.374807	−	0.927103i	$-0.622291\pi$
$523$	−8965.82	−0.749614	−0.374807	+	0.927103i	$0.622291\pi$
$524$	4229.75	0.352629
$525$	0	0
$526$	−9331.22	−0.773499
$527$	−1145.62	−0.0946946
$528$	0	0
$529$	5085.08	0.417941
$530$	0	0
$531$	0	0
$532$	2424.16	0.197558
$533$	−5956.30	−0.484045
$534$	0	0
$535$	0	0
$536$	−8217.28	−0.662187
$537$	0	0
$538$	15812.6	1.26715
$539$	1996.71	0.159563
$540$	0	0
$541$	−195.272	−0.0155183	−0.00775914	−	0.999970i	$-0.502470\pi$
$541$	−195.272	−0.0155183	−0.00775914	+	0.999970i	$0.502470\pi$
$542$	−16243.6	−1.28731
$543$	0	0
$544$	550.390	0.0433783
$545$	0	0
$546$	0	0
$547$	1399.26	0.109375	0.0546874	−	0.998504i	$-0.482584\pi$
$547$	1399.26	0.109375	0.0546874	+	0.998504i	$0.482584\pi$
$548$	3898.41	0.303890
$549$	0	0
$550$	0	0
$551$	26554.5	2.05310
$552$	0	0
$553$	−851.837	−0.0655042
$554$	−3004.06	−0.230380
$555$	0	0
$556$	401.059	0.0305911
$557$	43.0467	0.00327459	0.00163730	−	0.999999i	$-0.499479\pi$
$557$	43.0467	0.00327459	0.00163730	+	0.999999i	$0.499479\pi$
$558$	0	0
$559$	19677.8	1.48888
$560$	0	0
$561$	0	0
$562$	464.786	0.0348858
$563$	−19232.9	−1.43973	−0.719865	−	0.694114i	$-0.755797\pi$
$563$	−19232.9	−1.43973	−0.719865	+	0.694114i	$0.755797\pi$
$564$	0	0
$565$	0	0
$566$	−2219.85	−0.164854
$567$	0	0
$568$	14535.1	1.07373
$569$	−5163.98	−0.380466	−0.190233	−	0.981739i	$-0.560924\pi$
$569$	−5163.98	−0.380466	−0.190233	+	0.981739i	$0.560924\pi$
$570$	0	0
$571$	−10231.9	−0.749899	−0.374950	−	0.927045i	$-0.622340\pi$
$571$	−10231.9	−0.749899	−0.374950	+	0.927045i	$0.622340\pi$
$572$	6135.05	0.448460
$573$	0	0
$574$	−1779.59	−0.129406
$575$	0	0
$576$	0	0
$577$	−16563.7	−1.19507	−0.597537	−	0.801842i	$-0.703854\pi$
$577$	−16563.7	−1.19507	−0.597537	+	0.801842i	$0.703854\pi$
$578$	−11129.6	−0.800916
$579$	0	0
$580$	0	0
$581$	−4267.26	−0.304708
$582$	0	0
$583$	−23108.8	−1.64163
$584$	12083.7	0.856212
$585$	0	0
$586$	1382.96	0.0974910
$587$	16020.6	1.12648	0.563239	−	0.826294i	$-0.309555\pi$
$587$	16020.6	1.12648	0.563239	+	0.826294i	$0.309555\pi$
$588$	0	0
$589$	−30869.4	−2.15951
$590$	0	0
$591$	0	0
$592$	397.631	0.0276057
$593$	−6771.14	−0.468900	−0.234450	−	0.972128i	$-0.575329\pi$
$593$	−6771.14	−0.468900	−0.234450	+	0.972128i	$0.575329\pi$
$594$	0	0
$595$	0	0
$596$	−5490.95	−0.377379
$597$	0	0
$598$	−15926.0	−1.08906
$599$	−11070.2	−0.755120	−0.377560	−	0.925985i	$-0.623237\pi$
$599$	−11070.2	−0.755120	−0.377560	+	0.925985i	$0.623237\pi$
$600$	0	0
$601$	−24187.7	−1.64166	−0.820830	−	0.571173i	$-0.806489\pi$
$601$	−24187.7	−1.64166	−0.820830	+	0.571173i	$0.806489\pi$
$602$	5879.23	0.398039
$603$	0	0
$604$	7499.05	0.505185
$605$	0	0
$606$	0	0
$607$	10074.1	0.673631	0.336816	−	0.941571i	$-0.390650\pi$
$607$	10074.1	0.673631	0.336816	+	0.941571i	$0.390650\pi$
$608$	14830.6	0.989243
$609$	0	0
$610$	0	0
$611$	14001.3	0.927060
$612$	0	0
$613$	11114.6	0.732323	0.366161	−	0.930551i	$-0.380672\pi$
$613$	11114.6	0.732323	0.366161	+	0.930551i	$0.380672\pi$
$614$	−18283.7	−1.20174
$615$	0	0
$616$	7024.26	0.459441
$617$	20496.4	1.33737	0.668683	−	0.743548i	$-0.266858\pi$
$617$	20496.4	1.33737	0.668683	+	0.743548i	$0.266858\pi$
$618$	0	0
$619$	−16714.4	−1.08532	−0.542658	−	0.839954i	$-0.682582\pi$
$619$	−16714.4	−1.08532	−0.542658	+	0.839954i	$0.682582\pi$
$620$	0	0
$621$	0	0
$622$	−12083.5	−0.778943
$623$	5033.27	0.323682
$624$	0	0
$625$	0	0
$626$	3484.28	0.222460
$627$	0	0
$628$	4703.82	0.298890
$629$	−54.1295	−0.00343129
$630$	0	0
$631$	9168.53	0.578437	0.289218	−	0.957263i	$-0.406605\pi$
$631$	9168.53	0.578437	0.289218	+	0.957263i	$0.406605\pi$
$632$	−2996.69	−0.188611
$633$	0	0
$634$	9598.30	0.601257
$635$	0	0
$636$	0	0
$637$	−2611.65	−0.162445
$638$	20078.9	1.24597
$639$	0	0
$640$	0	0
$641$	4273.37	0.263319	0.131660	−	0.991295i	$-0.457969\pi$
$641$	4273.37	0.263319	0.131660	+	0.991295i	$0.457969\pi$
$642$	0	0
$643$	−2955.75	−0.181281	−0.0906404	−	0.995884i	$-0.528891\pi$
$643$	−2955.75	−0.181281	−0.0906404	+	0.995884i	$0.528891\pi$
$644$	2597.16	0.158917
$645$	0	0
$646$	1268.95	0.0772851
$647$	22701.2	1.37941	0.689704	−	0.724091i	$-0.257741\pi$
$647$	22701.2	1.37941	0.689704	+	0.724091i	$0.257741\pi$
$648$	0	0
$649$	−34224.8	−2.07002
$650$	0	0
$651$	0	0
$652$	−93.4235	−0.00561158
$653$	1537.81	0.0921582	0.0460791	−	0.998938i	$-0.485327\pi$
$653$	1537.81	0.0921582	0.0460791	+	0.998938i	$0.485327\pi$
$654$	0	0
$655$	0	0
$656$	−3735.08	−0.222302
$657$	0	0
$658$	4183.26	0.247842
$659$	−12338.1	−0.729323	−0.364661	−	0.931140i	$-0.618815\pi$
$659$	−12338.1	−0.729323	−0.364661	+	0.931140i	$0.618815\pi$
$660$	0	0
$661$	1845.10	0.108572	0.0542859	−	0.998525i	$-0.482712\pi$
$661$	1845.10	0.108572	0.0542859	+	0.998525i	$0.482712\pi$
$662$	18877.7	1.10831
$663$	0	0
$664$	−15011.8	−0.877369
$665$	0	0
$666$	0	0
$667$	28449.5	1.65153
$668$	−4673.48	−0.270692
$669$	0	0
$670$	0	0
$671$	−19795.7	−1.13890
$672$	0	0
$673$	−23955.4	−1.37208	−0.686041	−	0.727563i	$-0.740653\pi$
$673$	−23955.4	−1.37208	−0.686041	+	0.727563i	$0.740653\pi$
$674$	9892.34	0.565339
$675$	0	0
$676$	−1818.53	−0.103467
$677$	−3678.26	−0.208814	−0.104407	−	0.994535i	$-0.533294\pi$
$677$	−3678.26	−0.208814	−0.104407	+	0.994535i	$0.533294\pi$
$678$	0	0
$679$	−4465.14	−0.252366
$680$	0	0
$681$	0	0
$682$	−23341.6	−1.31055
$683$	4390.87	0.245991	0.122996	−	0.992407i	$-0.460750\pi$
$683$	4390.87	0.245991	0.122996	+	0.992407i	$0.460750\pi$
$684$	0	0
$685$	0	0
$686$	−780.297	−0.0434284
$687$	0	0
$688$	12339.6	0.683781
$689$	30225.8	1.67128
$690$	0	0
$691$	10371.7	0.570994	0.285497	−	0.958380i	$-0.407841\pi$
$691$	10371.7	0.570994	0.285497	+	0.958380i	$0.407841\pi$
$692$	−181.323	−0.00996081
$693$	0	0
$694$	−18985.4	−1.03844
$695$	0	0
$696$	0	0
$697$	508.455	0.0276314
$698$	−22709.4	−1.23147
$699$	0	0
$700$	0	0
$701$	−109.675	−0.00590922	−0.00295461	−	0.999996i	$-0.500940\pi$
$701$	−109.675	−0.00590922	−0.00295461	+	0.999996i	$0.500940\pi$
$702$	0	0
$703$	−1458.55	−0.0782508
$704$	22109.6	1.18364
$705$	0	0
$706$	20022.9	1.06738
$707$	4698.03	0.249912
$708$	0	0
$709$	26918.8	1.42589	0.712944	−	0.701221i	$-0.247361\pi$
$709$	26918.8	1.42589	0.712944	+	0.701221i	$0.247361\pi$
$710$	0	0
$711$	0	0
$712$	17706.6	0.931999
$713$	−33072.4	−1.73713
$714$	0	0
$715$	0	0
$716$	11058.0	0.577174
$717$	0	0
$718$	1192.14	0.0619644
$719$	−15170.8	−0.786889	−0.393445	−	0.919348i	$-0.628717\pi$
$719$	−15170.8	−0.786889	−0.393445	+	0.919348i	$0.628717\pi$
$720$	0	0
$721$	−6386.88	−0.329903
$722$	18589.0	0.958185
$723$	0	0
$724$	5813.46	0.298419
$725$	0	0
$726$	0	0
$727$	33286.9	1.69813	0.849066	−	0.528288i	$-0.177166\pi$
$727$	33286.9	1.69813	0.849066	+	0.528288i	$0.177166\pi$
$728$	−9187.57	−0.467739
$729$	0	0
$730$	0	0
$731$	−1679.78	−0.0849917
$732$	0	0
$733$	−20544.0	−1.03521	−0.517607	−	0.855619i	$-0.673177\pi$
$733$	−20544.0	−1.03521	−0.517607	+	0.855619i	$0.673177\pi$
$734$	14474.7	0.727888
$735$	0	0
$736$	15889.0	0.795755
$737$	13597.6	0.679614
$738$	0	0
$739$	34357.2	1.71022	0.855109	−	0.518449i	$-0.173490\pi$
$739$	34357.2	1.71022	0.855109	+	0.518449i	$0.173490\pi$
$740$	0	0
$741$	0	0
$742$	9030.73	0.446804
$743$	8166.99	0.403254	0.201627	−	0.979462i	$-0.435377\pi$
$743$	8166.99	0.403254	0.201627	+	0.979462i	$0.435377\pi$
$744$	0	0
$745$	0	0
$746$	25628.9	1.25783
$747$	0	0
$748$	−523.715	−0.0256001
$749$	817.151	0.0398639
$750$	0	0
$751$	17080.1	0.829909	0.414954	−	0.909842i	$-0.363798\pi$
$751$	17080.1	0.829909	0.414954	+	0.909842i	$0.363798\pi$
$752$	8779.97	0.425761
$753$	0	0
$754$	−26262.7	−1.26848
$755$	0	0
$756$	0	0
$757$	16324.0	0.783758	0.391879	−	0.920017i	$-0.371825\pi$
$757$	16324.0	0.783758	0.391879	+	0.920017i	$0.371825\pi$
$758$	−2620.06	−0.125547
$759$	0	0
$760$	0	0
$761$	32366.2	1.54175	0.770875	−	0.636986i	$-0.219819\pi$
$761$	32366.2	1.54175	0.770875	+	0.636986i	$0.219819\pi$
$762$	0	0
$763$	−5860.23	−0.278053
$764$	1209.13	0.0572576
$765$	0	0
$766$	−344.774	−0.0162626
$767$	44765.3	2.10741
$768$	0	0
$769$	−7948.44	−0.372728	−0.186364	−	0.982481i	$-0.559670\pi$
$769$	−7948.44	−0.372728	−0.186364	+	0.982481i	$0.559670\pi$
$770$	0	0
$771$	0	0
$772$	4533.52	0.211354
$773$	17819.3	0.829127	0.414564	−	0.910020i	$-0.363934\pi$
$773$	17819.3	0.829127	0.414564	+	0.910020i	$0.363934\pi$
$774$	0	0
$775$	0	0
$776$	−15708.0	−0.726655
$777$	0	0
$778$	−10906.4	−0.502586
$779$	13700.6	0.630136
$780$	0	0
$781$	−24052.1	−1.10199
$782$	1359.51	0.0621687
$783$	0	0
$784$	−1637.72	−0.0746044
$785$	0	0
$786$	0	0
$787$	−2912.38	−0.131912	−0.0659562	−	0.997823i	$-0.521010\pi$
$787$	−2912.38	−0.131912	−0.0659562	+	0.997823i	$0.521010\pi$
$788$	−10561.3	−0.477448
$789$	0	0
$790$	0	0
$791$	7606.05	0.341896
$792$	0	0
$793$	25892.3	1.15948
$794$	10519.1	0.470162
$795$	0	0
$796$	985.923	0.0439009
$797$	−33789.1	−1.50172	−0.750861	−	0.660460i	$-0.770361\pi$
$797$	−33789.1	−1.50172	−0.750861	+	0.660460i	$0.770361\pi$
$798$	0	0
$799$	−1195.22	−0.0529208
$800$	0	0
$801$	0	0
$802$	8213.90	0.361649
$803$	−19995.7	−0.878744
$804$	0	0
$805$	0	0
$806$	30530.2	1.33422
$807$	0	0
$808$	16527.3	0.719589
$809$	−1252.13	−0.0544159	−0.0272079	−	0.999630i	$-0.508662\pi$
$809$	−1252.13	−0.0544159	−0.0272079	+	0.999630i	$0.508662\pi$
$810$	0	0
$811$	−31913.1	−1.38178	−0.690889	−	0.722961i	$-0.742781\pi$
$811$	−31913.1	−1.38178	−0.690889	+	0.722961i	$0.742781\pi$
$812$	4282.85	0.185097
$813$	0	0
$814$	−1102.87	−0.0474882
$815$	0	0
$816$	0	0
$817$	−45262.7	−1.93824
$818$	20382.3	0.871210
$819$	0	0
$820$	0	0
$821$	−30742.4	−1.30684	−0.653421	−	0.756995i	$-0.726667\pi$
$821$	−30742.4	−1.30684	−0.653421	+	0.756995i	$0.726667\pi$
$822$	0	0
$823$	13822.6	0.585449	0.292724	−	0.956197i	$-0.405438\pi$
$823$	13822.6	0.585449	0.292724	+	0.956197i	$0.405438\pi$
$824$	−22468.5	−0.949913
$825$	0	0
$826$	13374.8	0.563399
$827$	−42107.1	−1.77051	−0.885253	−	0.465110i	$-0.846015\pi$
$827$	−42107.1	−1.77051	−0.885253	+	0.465110i	$0.846015\pi$
$828$	0	0
$829$	−38763.8	−1.62403	−0.812015	−	0.583636i	$-0.801629\pi$
$829$	−38763.8	−1.62403	−0.812015	+	0.583636i	$0.801629\pi$
$830$	0	0
$831$	0	0
$832$	−28918.8	−1.20502
$833$	222.942	0.00927308
$834$	0	0
$835$	0	0
$836$	−14111.8	−0.583812
$837$	0	0
$838$	16102.5	0.663784
$839$	−16896.3	−0.695262	−0.347631	−	0.937631i	$-0.613014\pi$
$839$	−16896.3	−0.695262	−0.347631	+	0.937631i	$0.613014\pi$
$840$	0	0
$841$	22525.7	0.923601
$842$	26278.6	1.07556
$843$	0	0
$844$	−7312.09	−0.298214
$845$	0	0
$846$	0	0
$847$	−2306.47	−0.0935668
$848$	18954.0	0.767552
$849$	0	0
$850$	0	0
$851$	−1562.64	−0.0629455
$852$	0	0
$853$	−46429.3	−1.86367	−0.931833	−	0.362887i	$-0.881791\pi$
$853$	−46429.3	−1.86367	−0.931833	+	0.362887i	$0.881791\pi$
$854$	7735.99	0.309977
$855$	0	0
$856$	2874.67	0.114783
$857$	21206.4	0.845272	0.422636	−	0.906300i	$-0.361105\pi$
$857$	21206.4	0.845272	0.422636	+	0.906300i	$0.361105\pi$
$858$	0	0
$859$	13876.2	0.551163	0.275581	−	0.961278i	$-0.411130\pi$
$859$	13876.2	0.551163	0.275581	+	0.961278i	$0.411130\pi$
$860$	0	0
$861$	0	0
$862$	9246.12	0.365341
$863$	14337.1	0.565515	0.282757	−	0.959191i	$-0.408751\pi$
$863$	14337.1	0.565515	0.282757	+	0.959191i	$0.408751\pi$
$864$	0	0
$865$	0	0
$866$	−39711.6	−1.55826
$867$	0	0
$868$	−4978.79	−0.194690
$869$	4958.81	0.193574
$870$	0	0
$871$	−17785.4	−0.691889
$872$	−20615.8	−0.800619
$873$	0	0
$874$	36632.8	1.41776
$875$	0	0
$876$	0	0
$877$	24369.3	0.938304	0.469152	−	0.883118i	$-0.344560\pi$
$877$	24369.3	0.938304	0.469152	+	0.883118i	$0.344560\pi$
$878$	10454.1	0.401834
$879$	0	0
$880$	0	0
$881$	26127.0	0.999140	0.499570	−	0.866273i	$-0.333491\pi$
$881$	26127.0	0.999140	0.499570	+	0.866273i	$0.333491\pi$
$882$	0	0
$883$	15713.1	0.598855	0.299428	−	0.954119i	$-0.403204\pi$
$883$	15713.1	0.598855	0.299428	+	0.954119i	$0.403204\pi$
$884$	685.007	0.0260625
$885$	0	0
$886$	−696.611	−0.0264143
$887$	13139.5	0.497385	0.248692	−	0.968583i	$-0.419999\pi$
$887$	13139.5	0.497385	0.248692	+	0.968583i	$0.419999\pi$
$888$	0	0
$889$	−3759.79	−0.141844
$890$	0	0
$891$	0	0
$892$	−9141.48	−0.343138
$893$	−32205.8	−1.20686
$894$	0	0
$895$	0	0
$896$	−1865.95	−0.0695725
$897$	0	0
$898$	−20995.7	−0.780218
$899$	−54538.1	−2.02330
$900$	0	0
$901$	−2580.21	−0.0954043
$902$	10359.6	0.382412
$903$	0	0
$904$	26757.4	0.984446
$905$	0	0
$906$	0	0
$907$	3799.71	0.139104	0.0695519	−	0.997578i	$-0.477843\pi$
$907$	3799.71	0.139104	0.0695519	+	0.997578i	$0.477843\pi$
$908$	15907.9	0.581413
$909$	0	0
$910$	0	0
$911$	51528.4	1.87400	0.936998	−	0.349334i	$-0.113592\pi$
$911$	51528.4	1.87400	0.936998	+	0.349334i	$0.113592\pi$
$912$	0	0
$913$	24841.0	0.900458
$914$	25006.4	0.904966
$915$	0	0
$916$	−10650.0	−0.384156
$917$	10481.7	0.377467
$918$	0	0
$919$	−16984.7	−0.609657	−0.304828	−	0.952407i	$-0.598599\pi$
$919$	−16984.7	−0.609657	−0.304828	+	0.952407i	$0.598599\pi$
$920$	0	0
$921$	0	0
$922$	−16806.8	−0.600329
$923$	31459.6	1.12189
$924$	0	0
$925$	0	0
$926$	−23121.9	−0.820553
$927$	0	0
$928$	26201.7	0.926846
$929$	5451.85	0.192540	0.0962699	−	0.995355i	$-0.469309\pi$
$929$	5451.85	0.192540	0.0962699	+	0.995355i	$0.469309\pi$
$930$	0	0
$931$	6007.30	0.211473
$932$	18532.9	0.651358
$933$	0	0
$934$	−35976.8	−1.26038
$935$	0	0
$936$	0	0
$937$	−42429.4	−1.47930	−0.739652	−	0.672989i	$-0.765010\pi$
$937$	−42429.4	−1.47930	−0.739652	+	0.672989i	$0.765010\pi$
$938$	−5313.84	−0.184971
$939$	0	0
$940$	0	0
$941$	32977.9	1.14245	0.571226	−	0.820793i	$-0.306468\pi$
$941$	32977.9	1.14245	0.571226	+	0.820793i	$0.306468\pi$
$942$	0	0
$943$	14678.4	0.506886
$944$	28071.5	0.967848
$945$	0	0
$946$	−34224.8	−1.17626
$947$	−23753.4	−0.815082	−0.407541	−	0.913187i	$-0.633614\pi$
$947$	−23753.4	−0.815082	−0.407541	+	0.913187i	$0.633614\pi$
$948$	0	0
$949$	26153.9	0.894616
$950$	0	0
$951$	0	0
$952$	784.291	0.0267006
$953$	−28074.3	−0.954267	−0.477134	−	0.878831i	$-0.658324\pi$
$953$	−28074.3	−0.954267	−0.477134	+	0.878831i	$0.658324\pi$
$954$	0	0
$955$	0	0
$956$	−2179.14	−0.0737221
$957$	0	0
$958$	3286.92	0.110851
$959$	9660.63	0.325295
$960$	0	0
$961$	33609.2	1.12817
$962$	1442.52	0.0483460
$963$	0	0
$964$	−3536.88	−0.118169
$965$	0	0
$966$	0	0
$967$	11150.3	0.370806	0.185403	−	0.982663i	$-0.440641\pi$
$967$	11150.3	0.370806	0.185403	+	0.982663i	$0.440641\pi$
$968$	−8113.95	−0.269414
$969$	0	0
$970$	0	0
$971$	−6059.04	−0.200251	−0.100126	−	0.994975i	$-0.531924\pi$
$971$	−6059.04	−0.200251	−0.100126	+	0.994975i	$0.531924\pi$
$972$	0	0
$973$	993.861	0.0327459
$974$	1113.35	0.0366263
$975$	0	0
$976$	16236.6	0.532500
$977$	5700.49	0.186668	0.0933341	−	0.995635i	$-0.470248\pi$
$977$	5700.49	0.186668	0.0933341	+	0.995635i	$0.470248\pi$
$978$	0	0
$979$	−29300.2	−0.956526
$980$	0	0
$981$	0	0
$982$	8966.12	0.291365
$983$	197.480	0.00640757	0.00320378	−	0.999995i	$-0.498980\pi$
$983$	197.480	0.00640757	0.00320378	+	0.999995i	$0.498980\pi$
$984$	0	0
$985$	0	0
$986$	2241.90	0.0724103
$987$	0	0
$988$	18457.9	0.594357
$989$	−48492.9	−1.55913
$990$	0	0
$991$	20620.8	0.660990	0.330495	−	0.943808i	$-0.392784\pi$
$991$	20620.8	0.660990	0.330495	+	0.943808i	$0.392784\pi$
$992$	−30459.3	−0.974884
$993$	0	0
$994$	9399.35	0.299929
$995$	0	0
$996$	0	0
$997$	−19326.8	−0.613928	−0.306964	−	0.951721i	$-0.599313\pi$
$997$	−19326.8	−0.613928	−0.306964	+	0.951721i	$0.599313\pi$
$998$	25.2275	0.000800162 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.p.1.2		2
3.2	odd	2		525.4.a.n.1.1		2
5.4	even	2		63.4.a.e.1.1		2
15.2	even	4		525.4.d.g.274.2		4
15.8	even	4		525.4.d.g.274.3		4
15.14	odd	2		21.4.a.c.1.2	✓	2
20.19	odd	2		1008.4.a.ba.1.2		2
35.4	even	6		441.4.e.q.226.2		4
35.9	even	6		441.4.e.q.361.2		4
35.19	odd	6		441.4.e.p.361.2		4
35.24	odd	6		441.4.e.p.226.2		4
35.34	odd	2		441.4.a.r.1.1		2
60.59	even	2		336.4.a.m.1.1		2
105.44	odd	6		147.4.e.l.67.1		4
105.59	even	6		147.4.e.m.79.1		4
105.74	odd	6		147.4.e.l.79.1		4
105.89	even	6		147.4.e.m.67.1		4
105.104	even	2		147.4.a.i.1.2		2
120.29	odd	2		1344.4.a.bg.1.2		2
120.59	even	2		1344.4.a.bo.1.2		2
420.419	odd	2		2352.4.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.2	✓	2	15.14	odd	2
63.4.a.e.1.1		2	5.4	even	2
147.4.a.i.1.2		2	105.104	even	2
147.4.e.l.67.1		4	105.44	odd	6
147.4.e.l.79.1		4	105.74	odd	6
147.4.e.m.67.1		4	105.89	even	6
147.4.e.m.79.1		4	105.59	even	6
336.4.a.m.1.1		2	60.59	even	2
441.4.a.r.1.1		2	35.34	odd	2
441.4.e.p.226.2		4	35.24	odd	6
441.4.e.p.361.2		4	35.19	odd	6
441.4.e.q.226.2		4	35.4	even	6
441.4.e.q.361.2		4	35.9	even	6
525.4.a.n.1.1		2	3.2	odd	2
525.4.d.g.274.2		4	15.2	even	4
525.4.d.g.274.3		4	15.8	even	4
1008.4.a.ba.1.2		2	20.19	odd	2
1344.4.a.bg.1.2		2	120.29	odd	2
1344.4.a.bo.1.2		2	120.59	even	2
1575.4.a.p.1.2		2	1.1	even	1	trivial
2352.4.a.bz.1.2		2	420.419	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+2.27492 q^{2} -2.82475 q^{4} -7.00000 q^{7} -24.6254 q^{8} +O(q^{10})\)	\(q+2.27492 q^{2} -2.82475 q^{4} -7.00000 q^{7} -24.6254 q^{8} +40.7492 q^{11} -53.2990 q^{13} -15.9244 q^{14} -33.4228 q^{16} +4.54983 q^{17} +122.598 q^{19} +92.7010 q^{22} +131.347 q^{23} -121.251 q^{26} +19.7733 q^{28} +216.598 q^{29} -251.794 q^{31} +120.969 q^{32} +10.3505 q^{34} -11.8970 q^{37} +278.900 q^{38} +111.752 q^{41} -369.196 q^{43} -115.106 q^{44} +298.804 q^{46} -262.694 q^{47} +49.0000 q^{49} +150.556 q^{52} -567.100 q^{53} +172.378 q^{56} +492.743 q^{58} -839.890 q^{59} -485.794 q^{61} -572.811 q^{62} +542.577 q^{64} +333.691 q^{67} -12.8522 q^{68} -590.248 q^{71} -490.701 q^{73} -27.0647 q^{74} -346.309 q^{76} -285.244 q^{77} +121.691 q^{79} +254.228 q^{82} +609.608 q^{83} -839.890 q^{86} -1003.47 q^{88} -719.038 q^{89} +373.093 q^{91} -371.023 q^{92} -597.608 q^{94} +637.877 q^{97} +111.471 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8	\( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 17 * q^4 - 14 * q^7 - 87 * q^8 + 6 * q^11 - 16 * q^13 + 21 * q^14 + 137 * q^16 - 6 * q^17 + 64 * q^19 + 276 * q^22 + 6 * q^23 - 318 * q^26 - 119 * q^28 + 252 * q^29 + 40 * q^31 - 279 * q^32 + 66 * q^34 + 248 * q^37 + 588 * q^38 + 450 * q^41 - 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 98 * q^49 + 890 * q^52 - 1104 * q^53 + 609 * q^56 + 306 * q^58 - 804 * q^59 - 428 * q^61 - 2112 * q^62 + 1289 * q^64 - 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 - 1508 * q^76 - 42 * q^77 - 572 * q^79 - 1530 * q^82 + 1944 * q^83 - 804 * q^86 + 1164 * q^88 - 366 * q^89 + 112 * q^91 - 2856 * q^92 - 1920 * q^94 - 808 * q^97 - 147 * q^98

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