LMFDB - Embedded newform 1575.4.a.bd.1.1

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:	$3$
Coefficient field:	3.3.2292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.18296$ 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)

$f(q)$	$=$	$q-3.18296 q^{2} +2.13122 q^{4} +7.00000 q^{7} +18.6801 q^{8} +O(q^{10})$	$q-3.18296 q^{2} +2.13122 q^{4} +7.00000 q^{7} +18.6801 q^{8} +68.7204 q^{11} -56.2290 q^{13} -22.2807 q^{14} -76.5076 q^{16} +37.9780 q^{17} -26.0335 q^{19} -218.734 q^{22} +25.5222 q^{23} +178.974 q^{26} +14.9185 q^{28} -148.336 q^{29} +75.7982 q^{31} +94.0799 q^{32} -120.882 q^{34} -120.199 q^{37} +82.8634 q^{38} -345.670 q^{41} -287.989 q^{43} +146.458 q^{44} -81.2360 q^{46} -528.711 q^{47} +49.0000 q^{49} -119.836 q^{52} +361.728 q^{53} +130.761 q^{56} +472.146 q^{58} +705.748 q^{59} -393.171 q^{61} -241.262 q^{62} +312.609 q^{64} -591.202 q^{67} +80.9393 q^{68} +668.829 q^{71} +251.755 q^{73} +382.588 q^{74} -55.4830 q^{76} +481.042 q^{77} +295.651 q^{79} +1100.25 q^{82} -916.511 q^{83} +916.658 q^{86} +1283.70 q^{88} +736.838 q^{89} -393.603 q^{91} +54.3933 q^{92} +1682.86 q^{94} +142.964 q^{97} -155.965 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8	$ 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8} + 66 q^{11} - 102 q^{13} + 7 q^{14} - 69 q^{16} + 152 q^{17} - 138 q^{19} - 186 q^{22} - 180 q^{23} + 98 q^{26} + 21 q^{28} - 170 q^{29} - 366 q^{31} - 151 q^{32} - 36 q^{34} - 252 q^{37} - 234 q^{38} + 206 q^{41} + 108 q^{43} + 306 q^{44} - 672 q^{46} - 24 q^{47} + 147 q^{49} - 78 q^{52} + 354 q^{53} + 147 q^{56} + 858 q^{58} + 880 q^{59} - 870 q^{61} - 1366 q^{62} - 813 q^{64} + 96 q^{67} - 512 q^{68} + 1018 q^{71} - 1554 q^{73} - 980 q^{74} - 450 q^{76} + 462 q^{77} - 1620 q^{79} + 1638 q^{82} - 872 q^{83} + 2932 q^{86} + 1326 q^{88} + 1938 q^{89} - 714 q^{91} - 708 q^{92} + 2112 q^{94} - 1878 q^{97} + 49 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8 + 66 * q^11 - 102 * q^13 + 7 * q^14 - 69 * q^16 + 152 * q^17 - 138 * q^19 - 186 * q^22 - 180 * q^23 + 98 * q^26 + 21 * q^28 - 170 * q^29 - 366 * q^31 - 151 * q^32 - 36 * q^34 - 252 * q^37 - 234 * q^38 + 206 * q^41 + 108 * q^43 + 306 * q^44 - 672 * q^46 - 24 * q^47 + 147 * q^49 - 78 * q^52 + 354 * q^53 + 147 * q^56 + 858 * q^58 + 880 * q^59 - 870 * q^61 - 1366 * q^62 - 813 * q^64 + 96 * q^67 - 512 * q^68 + 1018 * q^71 - 1554 * q^73 - 980 * q^74 - 450 * q^76 + 462 * q^77 - 1620 * q^79 + 1638 * q^82 - 872 * q^83 + 2932 * q^86 + 1326 * q^88 + 1938 * q^89 - 714 * q^91 - 708 * q^92 + 2112 * q^94 - 1878 * q^97 + 49 * 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 $

$2$	−3.18296	−1.12535	−0.562673	−	0.826680i	$-0.690227\pi$
$2$	−3.18296	−1.12535	−0.562673	+	0.826680i	$0.690227\pi$
$3$	0	0
$3$	0	0
$4$	2.13122	0.266402
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	18.6801	0.825551
$9$	0	0
$10$	0	0
$11$	68.7204	1.88363	0.941817	−	0.336127i	$-0.109117\pi$
$11$	68.7204	1.88363	0.941817	+	0.336127i	$0.109117\pi$
$12$	0	0
$13$	−56.2290	−1.19962	−0.599812	−	0.800141i	$-0.704758\pi$
$13$	−56.2290	−1.19962	−0.599812	+	0.800141i	$0.704758\pi$
$14$	−22.2807	−0.425341
$15$	0	0
$16$	−76.5076	−1.19543
$17$	37.9780	0.541825	0.270912	−	0.962604i	$-0.412675\pi$
$17$	37.9780	0.541825	0.270912	+	0.962604i	$0.412675\pi$
$18$	0	0
$19$	−26.0335	−0.314341	−0.157171	−	0.987571i	$-0.550237\pi$
$19$	−26.0335	−0.314341	−0.157171	+	0.987571i	$0.550237\pi$
$20$	0	0
$21$	0	0
$22$	−218.734	−2.11974
$23$	25.5222	0.231380	0.115690	−	0.993285i	$-0.463092\pi$
$23$	25.5222	0.231380	0.115690	+	0.993285i	$0.463092\pi$
$24$	0	0
$25$	0	0
$26$	178.974	1.34999
$27$	0	0
$28$	14.9185	0.100690
$29$	−148.336	−0.949835	−0.474917	−	0.880030i	$-0.657522\pi$
$29$	−148.336	−0.949835	−0.474917	+	0.880030i	$0.657522\pi$
$30$	0	0
$31$	75.7982	0.439153	0.219577	−	0.975595i	$-0.429532\pi$
$31$	75.7982	0.439153	0.219577	+	0.975595i	$0.429532\pi$
$32$	94.0799	0.519723
$33$	0	0
$34$	−120.882	−0.609740
$35$	0	0
$36$	0	0
$37$	−120.199	−0.534070	−0.267035	−	0.963687i	$-0.586044\pi$
$37$	−120.199	−0.534070	−0.267035	+	0.963687i	$0.586044\pi$
$38$	82.8634	0.353743
$39$	0	0
$40$	0	0
$41$	−345.670	−1.31670	−0.658348	−	0.752713i	$-0.728745\pi$
$41$	−345.670	−1.31670	−0.658348	+	0.752713i	$0.728745\pi$
$42$	0	0
$43$	−287.989	−1.02135	−0.510674	−	0.859774i	$-0.670604\pi$
$43$	−287.989	−1.02135	−0.510674	+	0.859774i	$0.670604\pi$
$44$	146.458	0.501804
$45$	0	0
$46$	−81.2360	−0.260383
$47$	−528.711	−1.64086	−0.820430	−	0.571747i	$-0.806266\pi$
$47$	−528.711	−1.64086	−0.820430	+	0.571747i	$0.806266\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−119.836	−0.319582
$53$	361.728	0.937493	0.468747	−	0.883333i	$-0.344706\pi$
$53$	361.728	0.937493	0.468747	+	0.883333i	$0.344706\pi$
$54$	0	0
$55$	0	0
$56$	130.761	0.312029
$57$	0	0
$58$	472.146	1.06889
$59$	705.748	1.55730	0.778649	−	0.627460i	$-0.215905\pi$
$59$	705.748	1.55730	0.778649	+	0.627460i	$0.215905\pi$
$60$	0	0
$61$	−393.171	−0.825253	−0.412627	−	0.910900i	$-0.635389\pi$
$61$	−393.171	−0.825253	−0.412627	+	0.910900i	$0.635389\pi$
$62$	−241.262	−0.494199
$63$	0	0
$64$	312.609	0.610564
$65$	0	0
$66$	0	0
$67$	−591.202	−1.07801	−0.539006	−	0.842302i	$-0.681200\pi$
$67$	−591.202	−1.07801	−0.539006	+	0.842302i	$0.681200\pi$
$68$	80.9393	0.144343
$69$	0	0
$70$	0	0
$71$	668.829	1.11796	0.558981	−	0.829180i	$-0.311192\pi$
$71$	668.829	1.11796	0.558981	+	0.829180i	$0.311192\pi$
$72$	0	0
$73$	251.755	0.403639	0.201819	−	0.979423i	$-0.435315\pi$
$73$	251.755	0.403639	0.201819	+	0.979423i	$0.435315\pi$
$74$	382.588	0.601013
$75$	0	0
$76$	−55.4830	−0.0837412
$77$	481.042	0.711946
$78$	0	0
$79$	295.651	0.421054	0.210527	−	0.977588i	$-0.432482\pi$
$79$	295.651	0.421054	0.210527	+	0.977588i	$0.432482\pi$
$80$	0	0
$81$	0	0
$82$	1100.25	1.48174
$83$	−916.511	−1.21205	−0.606025	−	0.795445i	$-0.707237\pi$
$83$	−916.511	−1.21205	−0.606025	+	0.795445i	$0.707237\pi$
$84$	0	0
$85$	0	0
$86$	916.658	1.14937
$87$	0	0
$88$	1283.70	1.55504
$89$	736.838	0.877580	0.438790	−	0.898590i	$-0.355407\pi$
$89$	736.838	0.877580	0.438790	+	0.898590i	$0.355407\pi$
$90$	0	0
$91$	−393.603	−0.453415
$92$	54.3933	0.0616401
$93$	0	0
$94$	1682.86	1.84653
$95$	0	0
$96$	0	0
$97$	142.964	0.149647	0.0748236	−	0.997197i	$-0.476161\pi$
$97$	142.964	0.149647	0.0748236	+	0.997197i	$0.476161\pi$
$98$	−155.965	−0.160764
$99$	0	0
$100$	0	0
$101$	566.067	0.557681	0.278840	−	0.960337i	$-0.410050\pi$
$101$	566.067	0.557681	0.278840	+	0.960337i	$0.410050\pi$
$102$	0	0
$103$	−1340.79	−1.28264	−0.641318	−	0.767275i	$-0.721612\pi$
$103$	−1340.79	−1.28264	−0.641318	+	0.767275i	$0.721612\pi$
$104$	−1050.36	−0.990351
$105$	0	0
$106$	−1151.36	−1.05500
$107$	−1637.85	−1.47978	−0.739891	−	0.672727i	$-0.765123\pi$
$107$	−1637.85	−1.47978	−0.739891	+	0.672727i	$0.765123\pi$
$108$	0	0
$109$	−586.693	−0.515550	−0.257775	−	0.966205i	$-0.582989\pi$
$109$	−586.693	−0.515550	−0.257775	+	0.966205i	$0.582989\pi$
$110$	0	0
$111$	0	0
$112$	−535.554	−0.451831
$113$	−1617.17	−1.34629	−0.673146	−	0.739509i	$-0.735058\pi$
$113$	−1617.17	−1.34629	−0.673146	+	0.739509i	$0.735058\pi$
$114$	0	0
$115$	0	0
$116$	−316.135	−0.253038
$117$	0	0
$118$	−2246.37	−1.75250
$119$	265.846	0.204790
$120$	0	0
$121$	3391.49	2.54807
$122$	1251.45	0.928695
$123$	0	0
$124$	161.542	0.116991
$125$	0	0
$126$	0	0
$127$	2042.94	1.42741	0.713706	−	0.700446i	$-0.247015\pi$
$127$	2042.94	1.42741	0.713706	+	0.700446i	$0.247015\pi$
$128$	−1747.66	−1.20682
$129$	0	0
$130$	0	0
$131$	2631.04	1.75477	0.877384	−	0.479788i	$-0.159286\pi$
$131$	2631.04	1.75477	0.877384	+	0.479788i	$0.159286\pi$
$132$	0	0
$133$	−182.234	−0.118810
$134$	1881.77	1.21314
$135$	0	0
$136$	709.432	0.447304
$137$	−1588.54	−0.990641	−0.495320	−	0.868710i	$-0.664949\pi$
$137$	−1588.54	−0.990641	−0.495320	+	0.868710i	$0.664949\pi$
$138$	0	0
$139$	1733.49	1.05779	0.528894	−	0.848688i	$-0.322607\pi$
$139$	1733.49	1.05779	0.528894	+	0.848688i	$0.322607\pi$
$140$	0	0
$141$	0	0
$142$	−2128.85	−1.25809
$143$	−3864.07	−2.25965
$144$	0	0
$145$	0	0
$146$	−801.324	−0.454233
$147$	0	0
$148$	−256.170	−0.142277
$149$	1370.68	0.753629	0.376815	−	0.926289i	$-0.377019\pi$
$149$	1370.68	0.753629	0.376815	+	0.926289i	$0.377019\pi$
$150$	0	0
$151$	−3694.08	−1.99086	−0.995429	−	0.0955015i	$-0.969555\pi$
$151$	−3694.08	−1.99086	−0.995429	+	0.0955015i	$0.969555\pi$
$152$	−486.308	−0.259505
$153$	0	0
$154$	−1531.14	−0.801186
$155$	0	0
$156$	0	0
$157$	1054.56	0.536071	0.268036	−	0.963409i	$-0.413626\pi$
$157$	1054.56	0.536071	0.268036	+	0.963409i	$0.413626\pi$
$158$	−941.044	−0.473832
$159$	0	0
$160$	0	0
$161$	178.655	0.0874535
$162$	0	0
$163$	−1380.12	−0.663185	−0.331593	−	0.943423i	$-0.607586\pi$
$163$	−1380.12	−0.663185	−0.331593	+	0.943423i	$0.607586\pi$
$164$	−736.697	−0.350771
$165$	0	0
$166$	2917.22	1.36398
$167$	3143.10	1.45641	0.728205	−	0.685359i	$-0.240355\pi$
$167$	3143.10	1.45641	0.728205	+	0.685359i	$0.240355\pi$
$168$	0	0
$169$	964.696	0.439097
$170$	0	0
$171$	0	0
$172$	−613.767	−0.272089
$173$	3306.12	1.45295	0.726473	−	0.687195i	$-0.241158\pi$
$173$	3306.12	1.45295	0.726473	+	0.687195i	$0.241158\pi$
$174$	0	0
$175$	0	0
$176$	−5257.63	−2.25176
$177$	0	0
$178$	−2345.32	−0.987581
$179$	−2839.60	−1.18571	−0.592854	−	0.805310i	$-0.701999\pi$
$179$	−2839.60	−1.18571	−0.592854	+	0.805310i	$0.701999\pi$
$180$	0	0
$181$	741.522	0.304513	0.152257	−	0.988341i	$-0.451346\pi$
$181$	741.522	0.304513	0.152257	+	0.988341i	$0.451346\pi$
$182$	1252.82	0.510249
$183$	0	0
$184$	476.757	0.191016
$185$	0	0
$186$	0	0
$187$	2609.86	1.02060
$188$	−1126.80	−0.437129
$189$	0	0
$190$	0	0
$191$	−4428.76	−1.67777	−0.838885	−	0.544309i	$-0.816792\pi$
$191$	−4428.76	−1.67777	−0.838885	+	0.544309i	$0.816792\pi$
$192$	0	0
$193$	−815.034	−0.303976	−0.151988	−	0.988382i	$-0.548568\pi$
$193$	−815.034	−0.303976	−0.151988	+	0.988382i	$0.548568\pi$
$194$	−455.048	−0.168405
$195$	0	0
$196$	104.430	0.0380574
$197$	−1609.61	−0.582133	−0.291067	−	0.956703i	$-0.594010\pi$
$197$	−1609.61	−0.582133	−0.291067	+	0.956703i	$0.594010\pi$
$198$	0	0
$199$	−273.622	−0.0974700	−0.0487350	−	0.998812i	$-0.515519\pi$
$199$	−273.622	−0.0974700	−0.0487350	+	0.998812i	$0.515519\pi$
$200$	0	0
$201$	0	0
$202$	−1801.77	−0.627584
$203$	−1038.35	−0.359004
$204$	0	0
$205$	0	0
$206$	4267.66	1.44341
$207$	0	0
$208$	4301.95	1.43407
$209$	−1789.03	−0.592104
$210$	0	0
$211$	−5157.70	−1.68280	−0.841399	−	0.540414i	$-0.818267\pi$
$211$	−5157.70	−1.68280	−0.841399	+	0.540414i	$0.818267\pi$
$212$	770.920	0.249750
$213$	0	0
$214$	5213.20	1.66527
$215$	0	0
$216$	0	0
$217$	530.587	0.165984
$218$	1867.42	0.580172
$219$	0	0
$220$	0	0
$221$	−2135.46	−0.649986
$222$	0	0
$223$	−1444.38	−0.433733	−0.216867	−	0.976201i	$-0.569584\pi$
$223$	−1444.38	−0.433733	−0.216867	+	0.976201i	$0.569584\pi$
$224$	658.559	0.196437
$225$	0	0
$226$	5147.40	1.51504
$227$	1635.20	0.478114	0.239057	−	0.971006i	$-0.423162\pi$
$227$	1635.20	0.478114	0.239057	+	0.971006i	$0.423162\pi$
$228$	0	0
$229$	−2807.34	−0.810105	−0.405052	−	0.914293i	$-0.632747\pi$
$229$	−2807.34	−0.810105	−0.405052	+	0.914293i	$0.632747\pi$
$230$	0	0
$231$	0	0
$232$	−2770.92	−0.784137
$233$	579.805	0.163023	0.0815113	−	0.996672i	$-0.474025\pi$
$233$	579.805	0.163023	0.0815113	+	0.996672i	$0.474025\pi$
$234$	0	0
$235$	0	0
$236$	1504.10	0.414867
$237$	0	0
$238$	−846.177	−0.230460
$239$	1695.51	0.458886	0.229443	−	0.973322i	$-0.426310\pi$
$239$	1695.51	0.458886	0.229443	+	0.973322i	$0.426310\pi$
$240$	0	0
$241$	−1182.39	−0.316035	−0.158018	−	0.987436i	$-0.550510\pi$
$241$	−1182.39	−0.316035	−0.158018	+	0.987436i	$0.550510\pi$
$242$	−10795.0	−2.86746
$243$	0	0
$244$	−837.933	−0.219849
$245$	0	0
$246$	0	0
$247$	1463.84	0.377091
$248$	1415.92	0.362544
$249$	0	0
$250$	0	0
$251$	−2411.68	−0.606469	−0.303234	−	0.952916i	$-0.598066\pi$
$251$	−2411.68	−0.606469	−0.303234	+	0.952916i	$0.598066\pi$
$252$	0	0
$253$	1753.89	0.435835
$254$	−6502.58	−1.60633
$255$	0	0
$256$	3061.85	0.747523
$257$	1054.74	0.256003	0.128002	−	0.991774i	$-0.459144\pi$
$257$	1054.74	0.256003	0.128002	+	0.991774i	$0.459144\pi$
$258$	0	0
$259$	−841.392	−0.201859
$260$	0	0
$261$	0	0
$262$	−8374.48	−1.97472
$263$	−3390.76	−0.794993	−0.397496	−	0.917604i	$-0.630121\pi$
$263$	−3390.76	−0.794993	−0.397496	+	0.917604i	$0.630121\pi$
$264$	0	0
$265$	0	0
$266$	580.044	0.133702
$267$	0	0
$268$	−1259.98	−0.287184
$269$	8792.06	1.99279	0.996397	−	0.0848151i	$-0.0270300\pi$
$269$	8792.06	1.99279	0.996397	+	0.0848151i	$0.0270300\pi$
$270$	0	0
$271$	1593.15	0.357111	0.178556	−	0.983930i	$-0.442858\pi$
$271$	1593.15	0.357111	0.178556	+	0.983930i	$0.442858\pi$
$272$	−2905.61	−0.647715
$273$	0	0
$274$	5056.24	1.11481
$275$	0	0
$276$	0	0
$277$	208.765	0.0452833	0.0226417	−	0.999744i	$-0.492792\pi$
$277$	208.765	0.0452833	0.0226417	+	0.999744i	$0.492792\pi$
$278$	−5517.62	−1.19038
$279$	0	0
$280$	0	0
$281$	−2445.06	−0.519075	−0.259538	−	0.965733i	$-0.583570\pi$
$281$	−2445.06	−0.519075	−0.259538	+	0.965733i	$0.583570\pi$
$282$	0	0
$283$	−3894.53	−0.818042	−0.409021	−	0.912525i	$-0.634130\pi$
$283$	−3894.53	−0.818042	−0.409021	+	0.912525i	$0.634130\pi$
$284$	1425.42	0.297828
$285$	0	0
$286$	12299.2	2.54289
$287$	−2419.69	−0.497665
$288$	0	0
$289$	−3470.67	−0.706426
$290$	0	0
$291$	0	0
$292$	536.543	0.107530
$293$	−5746.21	−1.14572	−0.572862	−	0.819652i	$-0.694167\pi$
$293$	−5746.21	−1.14572	−0.572862	+	0.819652i	$0.694167\pi$
$294$	0	0
$295$	0	0
$296$	−2245.33	−0.440902
$297$	0	0
$298$	−4362.83	−0.848093
$299$	−1435.09	−0.277569
$300$	0	0
$301$	−2015.93	−0.386033
$302$	11758.1	2.24040
$303$	0	0
$304$	1991.76	0.375774
$305$	0	0
$306$	0	0
$307$	−1979.55	−0.368008	−0.184004	−	0.982925i	$-0.558906\pi$
$307$	−1979.55	−0.368008	−0.184004	+	0.982925i	$0.558906\pi$
$308$	1025.21	0.189664
$309$	0	0
$310$	0	0
$311$	−3495.57	−0.637348	−0.318674	−	0.947864i	$-0.603238\pi$
$311$	−3495.57	−0.637348	−0.318674	+	0.947864i	$0.603238\pi$
$312$	0	0
$313$	1448.68	0.261611	0.130805	−	0.991408i	$-0.458244\pi$
$313$	1448.68	0.261611	0.130805	+	0.991408i	$0.458244\pi$
$314$	−3356.62	−0.603265
$315$	0	0
$316$	630.096	0.112170
$317$	−9453.34	−1.67493	−0.837465	−	0.546492i	$-0.815963\pi$
$317$	−9453.34	−1.67493	−0.837465	+	0.546492i	$0.815963\pi$
$318$	0	0
$319$	−10193.7	−1.78914
$320$	0	0
$321$	0	0
$322$	−568.652	−0.0984154
$323$	−988.699	−0.170318
$324$	0	0
$325$	0	0
$326$	4392.86	0.746313
$327$	0	0
$328$	−6457.14	−1.08700
$329$	−3700.98	−0.620187
$330$	0	0
$331$	−791.305	−0.131402	−0.0657010	−	0.997839i	$-0.520928\pi$
$331$	−791.305	−0.131402	−0.0657010	+	0.997839i	$0.520928\pi$
$332$	−1953.28	−0.322893
$333$	0	0
$334$	−10004.4	−1.63896
$335$	0	0
$336$	0	0
$337$	−5959.06	−0.963237	−0.481618	−	0.876381i	$-0.659951\pi$
$337$	−5959.06	−0.963237	−0.481618	+	0.876381i	$0.659951\pi$
$338$	−3070.59	−0.494136
$339$	0	0
$340$	0	0
$341$	5208.88	0.827204
$342$	0	0
$343$	343.000	0.0539949
$344$	−5379.67	−0.843175
$345$	0	0
$346$	−10523.2	−1.63507
$347$	−4560.34	−0.705509	−0.352755	−	0.935716i	$-0.614755\pi$
$347$	−4560.34	−0.705509	−0.352755	+	0.935716i	$0.614755\pi$
$348$	0	0
$349$	−9404.92	−1.44250	−0.721252	−	0.692673i	$-0.756433\pi$
$349$	−9404.92	−1.44250	−0.721252	+	0.692673i	$0.756433\pi$
$350$	0	0
$351$	0	0
$352$	6465.20	0.978967
$353$	1391.37	0.209788	0.104894	−	0.994483i	$-0.466550\pi$
$353$	1391.37	0.209788	0.104894	+	0.994483i	$0.466550\pi$
$354$	0	0
$355$	0	0
$356$	1570.36	0.233789
$357$	0	0
$358$	9038.32	1.33433
$359$	−4306.80	−0.633159	−0.316579	−	0.948566i	$-0.602534\pi$
$359$	−4306.80	−0.633159	−0.316579	+	0.948566i	$0.602534\pi$
$360$	0	0
$361$	−6181.26	−0.901189
$362$	−2360.23	−0.342682
$363$	0	0
$364$	−838.852	−0.120791
$365$	0	0
$366$	0	0
$367$	−6405.46	−0.911069	−0.455534	−	0.890218i	$-0.650552\pi$
$367$	−6405.46	−0.911069	−0.455534	+	0.890218i	$0.650552\pi$
$368$	−1952.64	−0.276599
$369$	0	0
$370$	0	0
$371$	2532.09	0.354339
$372$	0	0
$373$	10293.2	1.42885	0.714425	−	0.699712i	$-0.246688\pi$
$373$	10293.2	1.42885	0.714425	+	0.699712i	$0.246688\pi$
$374$	−8307.08	−1.14853
$375$	0	0
$376$	−9876.37	−1.35461
$377$	8340.75	1.13944
$378$	0	0
$379$	3832.35	0.519405	0.259702	−	0.965689i	$-0.416375\pi$
$379$	3832.35	0.519405	0.259702	+	0.965689i	$0.416375\pi$
$380$	0	0
$381$	0	0
$382$	14096.6	1.88807
$383$	−13514.8	−1.80307	−0.901536	−	0.432705i	$-0.857559\pi$
$383$	−13514.8	−1.80307	−0.901536	+	0.432705i	$0.857559\pi$
$384$	0	0
$385$	0	0
$386$	2594.22	0.342078
$387$	0	0
$388$	304.687	0.0398663
$389$	6444.44	0.839965	0.419982	−	0.907532i	$-0.362036\pi$
$389$	6444.44	0.839965	0.419982	+	0.907532i	$0.362036\pi$
$390$	0	0
$391$	969.282	0.125367
$392$	915.324	0.117936
$393$	0	0
$394$	5123.33	0.655101
$395$	0	0
$396$	0	0
$397$	−2811.16	−0.355386	−0.177693	−	0.984086i	$-0.556863\pi$
$397$	−2811.16	−0.355386	−0.177693	+	0.984086i	$0.556863\pi$
$398$	870.926	0.109687
$399$	0	0
$400$	0	0
$401$	−9918.99	−1.23524	−0.617619	−	0.786477i	$-0.711903\pi$
$401$	−9918.99	−1.23524	−0.617619	+	0.786477i	$0.711903\pi$
$402$	0	0
$403$	−4262.05	−0.526819
$404$	1206.41	0.148567
$405$	0	0
$406$	3305.02	0.404003
$407$	−8260.11	−1.00599
$408$	0	0
$409$	−562.052	−0.0679504	−0.0339752	−	0.999423i	$-0.510817\pi$
$409$	−562.052	−0.0679504	−0.0339752	+	0.999423i	$0.510817\pi$
$410$	0	0
$411$	0	0
$412$	−2857.50	−0.341697
$413$	4940.24	0.588603
$414$	0	0
$415$	0	0
$416$	−5290.01	−0.623472
$417$	0	0
$418$	5694.40	0.666322
$419$	−354.260	−0.0413049	−0.0206525	−	0.999787i	$-0.506574\pi$
$419$	−354.260	−0.0413049	−0.0206525	+	0.999787i	$0.506574\pi$
$420$	0	0
$421$	−2969.20	−0.343729	−0.171864	−	0.985121i	$-0.554979\pi$
$421$	−2969.20	−0.343729	−0.171864	+	0.985121i	$0.554979\pi$
$422$	16416.7	1.89373
$423$	0	0
$424$	6757.11	0.773948
$425$	0	0
$426$	0	0
$427$	−2752.20	−0.311916
$428$	−3490.61	−0.394217
$429$	0	0
$430$	0	0
$431$	4583.08	0.512202	0.256101	−	0.966650i	$-0.417562\pi$
$431$	4583.08	0.512202	0.256101	+	0.966650i	$0.417562\pi$
$432$	0	0
$433$	247.808	0.0275033	0.0137516	−	0.999905i	$-0.495623\pi$
$433$	247.808	0.0275033	0.0137516	+	0.999905i	$0.495623\pi$
$434$	−1688.84	−0.186790
$435$	0	0
$436$	−1250.37	−0.137344
$437$	−664.431	−0.0727324
$438$	0	0
$439$	−3793.19	−0.412390	−0.206195	−	0.978511i	$-0.566108\pi$
$439$	−3793.19	−0.412390	−0.206195	+	0.978511i	$0.566108\pi$
$440$	0	0
$441$	0	0
$442$	6797.09	0.731458
$443$	13157.7	1.41115	0.705575	−	0.708635i	$-0.250689\pi$
$443$	13157.7	1.41115	0.705575	+	0.708635i	$0.250689\pi$
$444$	0	0
$445$	0	0
$446$	4597.38	0.488100
$447$	0	0
$448$	2188.26	0.230772
$449$	8705.75	0.915033	0.457517	−	0.889201i	$-0.348739\pi$
$449$	8705.75	0.915033	0.457517	+	0.889201i	$0.348739\pi$
$450$	0	0
$451$	−23754.6	−2.48017
$452$	−3446.55	−0.358655
$453$	0	0
$454$	−5204.76	−0.538043
$455$	0	0
$456$	0	0
$457$	11239.4	1.15045	0.575227	−	0.817994i	$-0.304914\pi$
$457$	11239.4	1.15045	0.575227	+	0.817994i	$0.304914\pi$
$458$	8935.63	0.911648
$459$	0	0
$460$	0	0
$461$	12534.3	1.26633	0.633167	−	0.774015i	$-0.281754\pi$
$461$	12534.3	1.26633	0.633167	+	0.774015i	$0.281754\pi$
$462$	0	0
$463$	−12563.8	−1.26110	−0.630549	−	0.776150i	$-0.717170\pi$
$463$	−12563.8	−1.26110	−0.630549	+	0.776150i	$0.717170\pi$
$464$	11348.8	1.13546
$465$	0	0
$466$	−1845.49	−0.183457
$467$	−4817.97	−0.477407	−0.238703	−	0.971093i	$-0.576722\pi$
$467$	−4817.97	−0.477407	−0.238703	+	0.971093i	$0.576722\pi$
$468$	0	0
$469$	−4138.41	−0.407450
$470$	0	0
$471$	0	0
$472$	13183.4	1.28563
$473$	−19790.7	−1.92384
$474$	0	0
$475$	0	0
$476$	566.575	0.0545566
$477$	0	0
$478$	−5396.75	−0.516405
$479$	7088.81	0.676192	0.338096	−	0.941112i	$-0.390217\pi$
$479$	7088.81	0.676192	0.338096	+	0.941112i	$0.390217\pi$
$480$	0	0
$481$	6758.66	0.640683
$482$	3763.50	0.355649
$483$	0	0
$484$	7227.99	0.678812
$485$	0	0
$486$	0	0
$487$	16131.5	1.50101	0.750503	−	0.660867i	$-0.229811\pi$
$487$	16131.5	1.50101	0.750503	+	0.660867i	$0.229811\pi$
$488$	−7344.48	−0.681289
$489$	0	0
$490$	0	0
$491$	3961.20	0.364087	0.182043	−	0.983290i	$-0.441729\pi$
$491$	3961.20	0.364087	0.182043	+	0.983290i	$0.441729\pi$
$492$	0	0
$493$	−5633.49	−0.514644
$494$	−4659.32	−0.424358
$495$	0	0
$496$	−5799.14	−0.524978
$497$	4681.80	0.422550
$498$	0	0
$499$	−12981.0	−1.16455	−0.582274	−	0.812992i	$-0.697837\pi$
$499$	−12981.0	−1.16455	−0.582274	+	0.812992i	$0.697837\pi$
$500$	0	0
$501$	0	0
$502$	7676.26	0.682486
$503$	9611.24	0.851976	0.425988	−	0.904729i	$-0.359927\pi$
$503$	9611.24	0.851976	0.425988	+	0.904729i	$0.359927\pi$
$504$	0	0
$505$	0	0
$506$	−5582.57	−0.490465
$507$	0	0
$508$	4353.94	0.380265
$509$	2605.16	0.226860	0.113430	−	0.993546i	$-0.463816\pi$
$509$	2605.16	0.226860	0.113430	+	0.993546i	$0.463816\pi$
$510$	0	0
$511$	1762.28	0.152561
$512$	4235.53	0.365597
$513$	0	0
$514$	−3357.19	−0.288092
$515$	0	0
$516$	0	0
$517$	−36333.2	−3.09078
$518$	2678.12	0.227162
$519$	0	0
$520$	0	0
$521$	342.674	0.0288154	0.0144077	−	0.999896i	$-0.495414\pi$
$521$	342.674	0.0288154	0.0144077	+	0.999896i	$0.495414\pi$
$522$	0	0
$523$	−10494.7	−0.877441	−0.438720	−	0.898624i	$-0.644568\pi$
$523$	−10494.7	−0.877441	−0.438720	+	0.898624i	$0.644568\pi$
$524$	5607.31	0.467474
$525$	0	0
$526$	10792.6	0.894642
$527$	2878.66	0.237944
$528$	0	0
$529$	−11515.6	−0.946463
$530$	0	0
$531$	0	0
$532$	−388.381	−0.0316512
$533$	19436.7	1.57954
$534$	0	0
$535$	0	0
$536$	−11043.7	−0.889953
$537$	0	0
$538$	−27984.8	−2.24258
$539$	3367.30	0.269090
$540$	0	0
$541$	−4026.52	−0.319988	−0.159994	−	0.987118i	$-0.551148\pi$
$541$	−4026.52	−0.319988	−0.159994	+	0.987118i	$0.551148\pi$
$542$	−5070.93	−0.401873
$543$	0	0
$544$	3572.97	0.281599
$545$	0	0
$546$	0	0
$547$	−16182.8	−1.26495	−0.632475	−	0.774581i	$-0.717961\pi$
$547$	−16182.8	−1.26495	−0.632475	+	0.774581i	$0.717961\pi$
$548$	−3385.51	−0.263909
$549$	0	0
$550$	0	0
$551$	3861.69	0.298573
$552$	0	0
$553$	2069.55	0.159144
$554$	−664.491	−0.0509594
$555$	0	0
$556$	3694.44	0.281797
$557$	10934.3	0.831779	0.415890	−	0.909415i	$-0.363470\pi$
$557$	10934.3	0.831779	0.415890	+	0.909415i	$0.363470\pi$
$558$	0	0
$559$	16193.3	1.22523
$560$	0	0
$561$	0	0
$562$	7782.52	0.584139
$563$	−164.542	−0.0123173	−0.00615863	−	0.999981i	$-0.501960\pi$
$563$	−164.542	−0.0123173	−0.00615863	+	0.999981i	$0.501960\pi$
$564$	0	0
$565$	0	0
$566$	12396.1	0.920579
$567$	0	0
$568$	12493.8	0.922935
$569$	−21924.4	−1.61532	−0.807660	−	0.589648i	$-0.799266\pi$
$569$	−21924.4	−1.61532	−0.807660	+	0.589648i	$0.799266\pi$
$570$	0	0
$571$	−6765.55	−0.495848	−0.247924	−	0.968779i	$-0.579748\pi$
$571$	−6765.55	−0.495848	−0.247924	+	0.968779i	$0.579748\pi$
$572$	−8235.18	−0.601976
$573$	0	0
$574$	7701.77	0.560044
$575$	0	0
$576$	0	0
$577$	−26134.4	−1.88560	−0.942798	−	0.333365i	$-0.891816\pi$
$577$	−26134.4	−1.88560	−0.942798	+	0.333365i	$0.891816\pi$
$578$	11047.0	0.794973
$579$	0	0
$580$	0	0
$581$	−6415.58	−0.458112
$582$	0	0
$583$	24858.1	1.76589
$584$	4702.80	0.333225
$585$	0	0
$586$	18289.9	1.28933
$587$	−6342.06	−0.445937	−0.222968	−	0.974826i	$-0.571575\pi$
$587$	−6342.06	−0.445937	−0.222968	+	0.974826i	$0.571575\pi$
$588$	0	0
$589$	−1973.29	−0.138044
$590$	0	0
$591$	0	0
$592$	9196.14	0.638444
$593$	24801.2	1.71748	0.858739	−	0.512413i	$-0.171248\pi$
$593$	24801.2	1.71748	0.858739	+	0.512413i	$0.171248\pi$
$594$	0	0
$595$	0	0
$596$	2921.22	0.200768
$597$	0	0
$598$	4567.82	0.312361
$599$	−12339.2	−0.841681	−0.420841	−	0.907135i	$-0.638265\pi$
$599$	−12339.2	−0.841681	−0.420841	+	0.907135i	$0.638265\pi$
$600$	0	0
$601$	3221.02	0.218616	0.109308	−	0.994008i	$-0.465137\pi$
$601$	3221.02	0.218616	0.109308	+	0.994008i	$0.465137\pi$
$602$	6416.60	0.434421
$603$	0	0
$604$	−7872.87	−0.530369
$605$	0	0
$606$	0	0
$607$	20076.6	1.34248	0.671239	−	0.741241i	$-0.265762\pi$
$607$	20076.6	1.34248	0.671239	+	0.741241i	$0.265762\pi$
$608$	−2449.23	−0.163370
$609$	0	0
$610$	0	0
$611$	29728.9	1.96842
$612$	0	0
$613$	7428.07	0.489424	0.244712	−	0.969596i	$-0.421307\pi$
$613$	7428.07	0.489424	0.244712	+	0.969596i	$0.421307\pi$
$614$	6300.81	0.414137
$615$	0	0
$616$	8985.92	0.587748
$617$	−2200.68	−0.143592	−0.0717959	−	0.997419i	$-0.522873\pi$
$617$	−2200.68	−0.143592	−0.0717959	+	0.997419i	$0.522873\pi$
$618$	0	0
$619$	−5674.34	−0.368451	−0.184225	−	0.982884i	$-0.558978\pi$
$619$	−5674.34	−0.368451	−0.184225	+	0.982884i	$0.558978\pi$
$620$	0	0
$621$	0	0
$622$	11126.2	0.717237
$623$	5157.86	0.331694
$624$	0	0
$625$	0	0
$626$	−4611.08	−0.294403
$627$	0	0
$628$	2247.50	0.142810
$629$	−4564.92	−0.289372
$630$	0	0
$631$	−4650.60	−0.293403	−0.146701	−	0.989181i	$-0.546866\pi$
$631$	−4650.60	−0.293403	−0.146701	+	0.989181i	$0.546866\pi$
$632$	5522.78	0.347602
$633$	0	0
$634$	30089.6	1.88487
$635$	0	0
$636$	0	0
$637$	−2755.22	−0.171375
$638$	32446.0	2.01340
$639$	0	0
$640$	0	0
$641$	−15808.8	−0.974119	−0.487060	−	0.873369i	$-0.661931\pi$
$641$	−15808.8	−0.974119	−0.487060	+	0.873369i	$0.661931\pi$
$642$	0	0
$643$	−18829.6	−1.15485	−0.577424	−	0.816444i	$-0.695942\pi$
$643$	−18829.6	−1.15485	−0.577424	+	0.816444i	$0.695942\pi$
$644$	380.753	0.0232978
$645$	0	0
$646$	3146.99	0.191667
$647$	21519.7	1.30761	0.653807	−	0.756661i	$-0.273171\pi$
$647$	21519.7	1.30761	0.653807	+	0.756661i	$0.273171\pi$
$648$	0	0
$649$	48499.2	2.93338
$650$	0	0
$651$	0	0
$652$	−2941.33	−0.176674
$653$	4298.98	0.257629	0.128815	−	0.991669i	$-0.458883\pi$
$653$	4298.98	0.257629	0.128815	+	0.991669i	$0.458883\pi$
$654$	0	0
$655$	0	0
$656$	26446.4	1.57402
$657$	0	0
$658$	11780.1	0.697925
$659$	1199.37	0.0708964	0.0354482	−	0.999372i	$-0.488714\pi$
$659$	1199.37	0.0708964	0.0354482	+	0.999372i	$0.488714\pi$
$660$	0	0
$661$	14967.5	0.880739	0.440369	−	0.897817i	$-0.354847\pi$
$661$	14967.5	0.880739	0.440369	+	0.897817i	$0.354847\pi$
$662$	2518.69	0.147873
$663$	0	0
$664$	−17120.5	−1.00061
$665$	0	0
$666$	0	0
$667$	−3785.85	−0.219773
$668$	6698.63	0.387991
$669$	0	0
$670$	0	0
$671$	−27018.9	−1.55447
$672$	0	0
$673$	23664.3	1.35541	0.677706	−	0.735333i	$-0.262974\pi$
$673$	23664.3	1.35541	0.677706	+	0.735333i	$0.262974\pi$
$674$	18967.4	1.08397
$675$	0	0
$676$	2055.98	0.116976
$677$	−4413.97	−0.250580	−0.125290	−	0.992120i	$-0.539986\pi$
$677$	−4413.97	−0.250580	−0.125290	+	0.992120i	$0.539986\pi$
$678$	0	0
$679$	1000.75	0.0565613
$680$	0	0
$681$	0	0
$682$	−16579.6	−0.930890
$683$	−26357.7	−1.47664	−0.738322	−	0.674449i	$-0.764381\pi$
$683$	−26357.7	−1.47664	−0.738322	+	0.674449i	$0.764381\pi$
$684$	0	0
$685$	0	0
$686$	−1091.75	−0.0607629
$687$	0	0
$688$	22033.4	1.22095
$689$	−20339.6	−1.12464
$690$	0	0
$691$	19098.2	1.05142	0.525708	−	0.850665i	$-0.323800\pi$
$691$	19098.2	1.05142	0.525708	+	0.850665i	$0.323800\pi$
$692$	7046.05	0.387068
$693$	0	0
$694$	14515.4	0.793942
$695$	0	0
$696$	0	0
$697$	−13127.9	−0.713419
$698$	29935.4	1.62331
$699$	0	0
$700$	0	0
$701$	7008.54	0.377616	0.188808	−	0.982014i	$-0.439538\pi$
$701$	7008.54	0.377616	0.188808	+	0.982014i	$0.439538\pi$
$702$	0	0
$703$	3129.20	0.167880
$704$	21482.6	1.15008
$705$	0	0
$706$	−4428.67	−0.236084
$707$	3962.47	0.210784
$708$	0	0
$709$	31728.6	1.68066	0.840332	−	0.542071i	$-0.182360\pi$
$709$	31728.6	1.68066	0.840332	+	0.542071i	$0.182360\pi$
$710$	0	0
$711$	0	0
$712$	13764.2	0.724487
$713$	1934.54	0.101611
$714$	0	0
$715$	0	0
$716$	−6051.80	−0.315875
$717$	0	0
$718$	13708.4	0.712522
$719$	8033.78	0.416703	0.208352	−	0.978054i	$-0.433190\pi$
$719$	8033.78	0.416703	0.208352	+	0.978054i	$0.433190\pi$
$720$	0	0
$721$	−9385.50	−0.484791
$722$	19674.7	1.01415
$723$	0	0
$724$	1580.34	0.0811229
$725$	0	0
$726$	0	0
$727$	−20514.4	−1.04654	−0.523272	−	0.852166i	$-0.675289\pi$
$727$	−20514.4	−1.04654	−0.523272	+	0.852166i	$0.675289\pi$
$728$	−7352.53	−0.374317
$729$	0	0
$730$	0	0
$731$	−10937.3	−0.553391
$732$	0	0
$733$	39200.4	1.97531	0.987654	−	0.156653i	$-0.0500705\pi$
$733$	39200.4	1.97531	0.987654	+	0.156653i	$0.0500705\pi$
$734$	20388.3	1.02527
$735$	0	0
$736$	2401.12	0.120254
$737$	−40627.6	−2.03058
$738$	0	0
$739$	−18386.2	−0.915221	−0.457611	−	0.889153i	$-0.651295\pi$
$739$	−18386.2	−0.915221	−0.457611	+	0.889153i	$0.651295\pi$
$740$	0	0
$741$	0	0
$742$	−8059.55	−0.398754
$743$	10118.8	0.499628	0.249814	−	0.968294i	$-0.419631\pi$
$743$	10118.8	0.499628	0.249814	+	0.968294i	$0.419631\pi$
$744$	0	0
$745$	0	0
$746$	−32762.8	−1.60795
$747$	0	0
$748$	5562.18	0.271890
$749$	−11464.9	−0.559305
$750$	0	0
$751$	−16116.3	−0.783077	−0.391538	−	0.920162i	$-0.628057\pi$
$751$	−16116.3	−0.783077	−0.391538	+	0.920162i	$0.628057\pi$
$752$	40450.4	1.96154
$753$	0	0
$754$	−26548.3	−1.28227
$755$	0	0
$756$	0	0
$757$	16856.5	0.809324	0.404662	−	0.914466i	$-0.367389\pi$
$757$	16856.5	0.809324	0.404662	+	0.914466i	$0.367389\pi$
$758$	−12198.2	−0.584510
$759$	0	0
$760$	0	0
$761$	−19917.5	−0.948765	−0.474383	−	0.880319i	$-0.657329\pi$
$761$	−19917.5	−0.948765	−0.474383	+	0.880319i	$0.657329\pi$
$762$	0	0
$763$	−4106.85	−0.194860
$764$	−9438.65	−0.446961
$765$	0	0
$766$	43017.2	2.02908
$767$	−39683.5	−1.86817
$768$	0	0
$769$	2378.22	0.111523	0.0557613	−	0.998444i	$-0.482241\pi$
$769$	2378.22	0.111523	0.0557613	+	0.998444i	$0.482241\pi$
$770$	0	0
$771$	0	0
$772$	−1737.01	−0.0809799
$773$	−7153.02	−0.332828	−0.166414	−	0.986056i	$-0.553219\pi$
$773$	−7153.02	−0.332828	−0.166414	+	0.986056i	$0.553219\pi$
$774$	0	0
$775$	0	0
$776$	2670.58	0.123541
$777$	0	0
$778$	−20512.4	−0.945250
$779$	8998.99	0.413892
$780$	0	0
$781$	45962.1	2.10583
$782$	−3085.18	−0.141082
$783$	0	0
$784$	−3748.87	−0.170776
$785$	0	0
$786$	0	0
$787$	28137.0	1.27443	0.637214	−	0.770687i	$-0.280087\pi$
$787$	28137.0	1.27443	0.637214	+	0.770687i	$0.280087\pi$
$788$	−3430.44	−0.155082
$789$	0	0
$790$	0	0
$791$	−11320.2	−0.508851
$792$	0	0
$793$	22107.6	0.989993
$794$	8947.81	0.399932
$795$	0	0
$796$	−583.147	−0.0259662
$797$	−40807.1	−1.81363	−0.906815	−	0.421530i	$-0.861493\pi$
$797$	−40807.1	−1.81363	−0.906815	+	0.421530i	$0.861493\pi$
$798$	0	0
$799$	−20079.4	−0.889059
$800$	0	0
$801$	0	0
$802$	31571.7	1.39007
$803$	17300.7	0.760308
$804$	0	0
$805$	0	0
$806$	13565.9	0.592853
$807$	0	0
$808$	10574.2	0.460394
$809$	−15233.3	−0.662020	−0.331010	−	0.943627i	$-0.607389\pi$
$809$	−15233.3	−0.662020	−0.331010	+	0.943627i	$0.607389\pi$
$810$	0	0
$811$	−26847.5	−1.16245	−0.581223	−	0.813744i	$-0.697426\pi$
$811$	−26847.5	−1.16245	−0.581223	+	0.813744i	$0.697426\pi$
$812$	−2212.95	−0.0956393
$813$	0	0
$814$	26291.6	1.13209
$815$	0	0
$816$	0	0
$817$	7497.36	0.321052
$818$	1788.99	0.0764676
$819$	0	0
$820$	0	0
$821$	−23640.8	−1.00496	−0.502479	−	0.864590i	$-0.667578\pi$
$821$	−23640.8	−1.00496	−0.502479	+	0.864590i	$0.667578\pi$
$822$	0	0
$823$	−21191.6	−0.897562	−0.448781	−	0.893642i	$-0.648142\pi$
$823$	−21191.6	−0.897562	−0.448781	+	0.893642i	$0.648142\pi$
$824$	−25046.0	−1.05888
$825$	0	0
$826$	−15724.6	−0.662382
$827$	8177.77	0.343856	0.171928	−	0.985110i	$-0.445000\pi$
$827$	8177.77	0.343856	0.171928	+	0.985110i	$0.445000\pi$
$828$	0	0
$829$	28931.0	1.21208	0.606040	−	0.795434i	$-0.292757\pi$
$829$	28931.0	1.21208	0.606040	+	0.795434i	$0.292757\pi$
$830$	0	0
$831$	0	0
$832$	−17577.7	−0.732448
$833$	1860.92	0.0774035
$834$	0	0
$835$	0	0
$836$	−3812.81	−0.157738
$837$	0	0
$838$	1127.60	0.0464823
$839$	41631.2	1.71307	0.856537	−	0.516085i	$-0.172611\pi$
$839$	41631.2	1.71307	0.856537	+	0.516085i	$0.172611\pi$
$840$	0	0
$841$	−2385.58	−0.0978137
$842$	9450.83	0.386814
$843$	0	0
$844$	−10992.2	−0.448301
$845$	0	0
$846$	0	0
$847$	23740.4	0.963081
$848$	−27674.9	−1.12071
$849$	0	0
$850$	0	0
$851$	−3067.74	−0.123573
$852$	0	0
$853$	−38242.1	−1.53503	−0.767517	−	0.641029i	$-0.778508\pi$
$853$	−38242.1	−1.53503	−0.767517	+	0.641029i	$0.778508\pi$
$854$	8760.13	0.351014
$855$	0	0
$856$	−30595.1	−1.22164
$857$	−1118.45	−0.0445805	−0.0222903	−	0.999752i	$-0.507096\pi$
$857$	−1118.45	−0.0445805	−0.0222903	+	0.999752i	$0.507096\pi$
$858$	0	0
$859$	34103.0	1.35458	0.677288	−	0.735718i	$-0.263155\pi$
$859$	34103.0	1.35458	0.677288	+	0.735718i	$0.263155\pi$
$860$	0	0
$861$	0	0
$862$	−14587.7	−0.576405
$863$	12332.6	0.486450	0.243225	−	0.969970i	$-0.421795\pi$
$863$	12332.6	0.486450	0.243225	+	0.969970i	$0.421795\pi$
$864$	0	0
$865$	0	0
$866$	−788.764	−0.0309507
$867$	0	0
$868$	1130.80	0.0442186
$869$	20317.2	0.793112
$870$	0	0
$871$	33242.6	1.29321
$872$	−10959.5	−0.425613
$873$	0	0
$874$	2114.86	0.0818491
$875$	0	0
$876$	0	0
$877$	17287.9	0.665644	0.332822	−	0.942990i	$-0.391999\pi$
$877$	17287.9	0.665644	0.332822	+	0.942990i	$0.391999\pi$
$878$	12073.6	0.464081
$879$	0	0
$880$	0	0
$881$	15797.0	0.604103	0.302052	−	0.953292i	$-0.402329\pi$
$881$	15797.0	0.604103	0.302052	+	0.953292i	$0.402329\pi$
$882$	0	0
$883$	40601.9	1.54741	0.773706	−	0.633545i	$-0.218401\pi$
$883$	40601.9	1.54741	0.773706	+	0.633545i	$0.218401\pi$
$884$	−4551.13	−0.173157
$885$	0	0
$886$	−41880.3	−1.58803
$887$	3654.03	0.138320	0.0691602	−	0.997606i	$-0.477968\pi$
$887$	3654.03	0.138320	0.0691602	+	0.997606i	$0.477968\pi$
$888$	0	0
$889$	14300.6	0.539511
$890$	0	0
$891$	0	0
$892$	−3078.28	−0.115547
$893$	13764.2	0.515791
$894$	0	0
$895$	0	0
$896$	−12233.6	−0.456135
$897$	0	0
$898$	−27710.0	−1.02973
$899$	−11243.6	−0.417123
$900$	0	0
$901$	13737.7	0.507957
$902$	75609.7	2.79105
$903$	0	0
$904$	−30209.0	−1.11143
$905$	0	0
$906$	0	0
$907$	−8732.15	−0.319676	−0.159838	−	0.987143i	$-0.551097\pi$
$907$	−8732.15	−0.319676	−0.159838	+	0.987143i	$0.551097\pi$
$908$	3484.96	0.127370
$909$	0	0
$910$	0	0
$911$	29118.5	1.05899	0.529495	−	0.848313i	$-0.322381\pi$
$911$	29118.5	1.05899	0.529495	+	0.848313i	$0.322381\pi$
$912$	0	0
$913$	−62983.0	−2.28306
$914$	−35774.6	−1.29466
$915$	0	0
$916$	−5983.04	−0.215814
$917$	18417.3	0.663240
$918$	0	0
$919$	−36147.4	−1.29749	−0.648745	−	0.761006i	$-0.724706\pi$
$919$	−36147.4	−1.29749	−0.648745	+	0.761006i	$0.724706\pi$
$920$	0	0
$921$	0	0
$922$	−39896.1	−1.42506
$923$	−37607.5	−1.34113
$924$	0	0
$925$	0	0
$926$	39990.0	1.41917
$927$	0	0
$928$	−13955.4	−0.493651
$929$	380.568	0.0134403	0.00672015	−	0.999977i	$-0.497861\pi$
$929$	380.568	0.0134403	0.00672015	+	0.999977i	$0.497861\pi$
$930$	0	0
$931$	−1275.64	−0.0449059
$932$	1235.69	0.0434295
$933$	0	0
$934$	15335.4	0.537247
$935$	0	0
$936$	0	0
$937$	11577.5	0.403650	0.201825	−	0.979422i	$-0.435313\pi$
$937$	11577.5	0.403650	0.201825	+	0.979422i	$0.435313\pi$
$938$	13172.4	0.458522
$939$	0	0
$940$	0	0
$941$	−37835.3	−1.31073	−0.655364	−	0.755313i	$-0.727485\pi$
$941$	−37835.3	−1.31073	−0.655364	+	0.755313i	$0.727485\pi$
$942$	0	0
$943$	−8822.25	−0.304657
$944$	−53995.1	−1.86164
$945$	0	0
$946$	62993.0	2.16499
$947$	32719.8	1.12276	0.561379	−	0.827559i	$-0.310271\pi$
$947$	32719.8	1.12276	0.561379	+	0.827559i	$0.310271\pi$
$948$	0	0
$949$	−14155.9	−0.484215
$950$	0	0
$951$	0	0
$952$	4966.03	0.169065
$953$	−42719.5	−1.45207	−0.726033	−	0.687659i	$-0.758638\pi$
$953$	−42719.5	−1.45207	−0.726033	+	0.687659i	$0.758638\pi$
$954$	0	0
$955$	0	0
$956$	3613.51	0.122248
$957$	0	0
$958$	−22563.4	−0.760949
$959$	−11119.8	−0.374427
$960$	0	0
$961$	−24045.6	−0.807144
$962$	−21512.5	−0.720989
$963$	0	0
$964$	−2519.93	−0.0841924
$965$	0	0
$966$	0	0
$967$	56616.7	1.88280	0.941402	−	0.337286i	$-0.109509\pi$
$967$	56616.7	1.88280	0.941402	+	0.337286i	$0.109509\pi$
$968$	63353.3	2.10357
$969$	0	0
$970$	0	0
$971$	26230.8	0.866927	0.433463	−	0.901171i	$-0.357291\pi$
$971$	26230.8	0.866927	0.433463	+	0.901171i	$0.357291\pi$
$972$	0	0
$973$	12134.4	0.399806
$974$	−51346.0	−1.68915
$975$	0	0
$976$	30080.6	0.986534
$977$	−35339.9	−1.15724	−0.578620	−	0.815597i	$-0.696409\pi$
$977$	−35339.9	−1.15724	−0.578620	+	0.815597i	$0.696409\pi$
$978$	0	0
$979$	50635.8	1.65304
$980$	0	0
$981$	0	0
$982$	−12608.3	−0.409723
$983$	−25377.6	−0.823419	−0.411710	−	0.911315i	$-0.635068\pi$
$983$	−25377.6	−0.823419	−0.411710	+	0.911315i	$0.635068\pi$
$984$	0	0
$985$	0	0
$986$	17931.1	0.579152
$987$	0	0
$988$	3119.75	0.100458
$989$	−7350.12	−0.236320
$990$	0	0
$991$	−15086.3	−0.483585	−0.241792	−	0.970328i	$-0.577735\pi$
$991$	−15086.3	−0.483585	−0.241792	+	0.970328i	$0.577735\pi$
$992$	7131.08	0.228238
$993$	0	0
$994$	−14902.0	−0.475515
$995$	0	0
$996$	0	0
$997$	1393.06	0.0442513	0.0221256	−	0.999755i	$-0.492957\pi$
$997$	1393.06	0.0442513	0.0221256	+	0.999755i	$0.492957\pi$
$998$	41318.0	1.31052
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bd.1.1		3
3.2	odd	2		525.4.a.q.1.3		3
5.2	odd	4		315.4.d.a.64.2		6
5.3	odd	4		315.4.d.a.64.5		6
5.4	even	2		1575.4.a.bc.1.3		3
15.2	even	4		105.4.d.a.64.5	yes	6
15.8	even	4		105.4.d.a.64.2	✓	6
15.14	odd	2		525.4.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.a.64.2	✓	6	15.8	even	4
105.4.d.a.64.5	yes	6	15.2	even	4
315.4.d.a.64.2		6	5.2	odd	4
315.4.d.a.64.5		6	5.3	odd	4
525.4.a.q.1.3		3	3.2	odd	2
525.4.a.r.1.1		3	15.14	odd	2
1575.4.a.bc.1.3		3	5.4	even	2
1575.4.a.bd.1.1		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-3.18296 q^{2} +2.13122 q^{4} +7.00000 q^{7} +18.6801 q^{8} +O(q^{10})\)	\(q-3.18296 q^{2} +2.13122 q^{4} +7.00000 q^{7} +18.6801 q^{8} +68.7204 q^{11} -56.2290 q^{13} -22.2807 q^{14} -76.5076 q^{16} +37.9780 q^{17} -26.0335 q^{19} -218.734 q^{22} +25.5222 q^{23} +178.974 q^{26} +14.9185 q^{28} -148.336 q^{29} +75.7982 q^{31} +94.0799 q^{32} -120.882 q^{34} -120.199 q^{37} +82.8634 q^{38} -345.670 q^{41} -287.989 q^{43} +146.458 q^{44} -81.2360 q^{46} -528.711 q^{47} +49.0000 q^{49} -119.836 q^{52} +361.728 q^{53} +130.761 q^{56} +472.146 q^{58} +705.748 q^{59} -393.171 q^{61} -241.262 q^{62} +312.609 q^{64} -591.202 q^{67} +80.9393 q^{68} +668.829 q^{71} +251.755 q^{73} +382.588 q^{74} -55.4830 q^{76} +481.042 q^{77} +295.651 q^{79} +1100.25 q^{82} -916.511 q^{83} +916.658 q^{86} +1283.70 q^{88} +736.838 q^{89} -393.603 q^{91} +54.3933 q^{92} +1682.86 q^{94} +142.964 q^{97} -155.965 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8	\( 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8} + 66 q^{11} - 102 q^{13} + 7 q^{14} - 69 q^{16} + 152 q^{17} - 138 q^{19} - 186 q^{22} - 180 q^{23} + 98 q^{26} + 21 q^{28} - 170 q^{29} - 366 q^{31} - 151 q^{32} - 36 q^{34} - 252 q^{37} - 234 q^{38} + 206 q^{41} + 108 q^{43} + 306 q^{44} - 672 q^{46} - 24 q^{47} + 147 q^{49} - 78 q^{52} + 354 q^{53} + 147 q^{56} + 858 q^{58} + 880 q^{59} - 870 q^{61} - 1366 q^{62} - 813 q^{64} + 96 q^{67} - 512 q^{68} + 1018 q^{71} - 1554 q^{73} - 980 q^{74} - 450 q^{76} + 462 q^{77} - 1620 q^{79} + 1638 q^{82} - 872 q^{83} + 2932 q^{86} + 1326 q^{88} + 1938 q^{89} - 714 q^{91} - 708 q^{92} + 2112 q^{94} - 1878 q^{97} + 49 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8 + 66 * q^11 - 102 * q^13 + 7 * q^14 - 69 * q^16 + 152 * q^17 - 138 * q^19 - 186 * q^22 - 180 * q^23 + 98 * q^26 + 21 * q^28 - 170 * q^29 - 366 * q^31 - 151 * q^32 - 36 * q^34 - 252 * q^37 - 234 * q^38 + 206 * q^41 + 108 * q^43 + 306 * q^44 - 672 * q^46 - 24 * q^47 + 147 * q^49 - 78 * q^52 + 354 * q^53 + 147 * q^56 + 858 * q^58 + 880 * q^59 - 870 * q^61 - 1366 * q^62 - 813 * q^64 + 96 * q^67 - 512 * q^68 + 1018 * q^71 - 1554 * q^73 - 980 * q^74 - 450 * q^76 + 462 * q^77 - 1620 * q^79 + 1638 * q^82 - 872 * q^83 + 2932 * q^86 + 1326 * q^88 + 1938 * q^89 - 714 * q^91 - 708 * q^92 + 2112 * q^94 - 1878 * q^97 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more