LMFDB - Embedded newform 1575.4.a.bk.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,16,0,0,28,-9,0,0,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
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} - 24x^{2} - 3x + 46 $ x^4 - 24x^2 - 3x + 46
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.75345$ 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)

$f(q)$	$=$	$q-4.75345 q^{2} +14.5953 q^{4} +7.00000 q^{7} -31.3504 q^{8} -7.31799 q^{11} -4.15422 q^{13} -33.2742 q^{14} +32.2604 q^{16} -53.5216 q^{17} +88.9019 q^{19} +34.7857 q^{22} -156.780 q^{23} +19.7469 q^{26} +102.167 q^{28} -42.2570 q^{29} -14.0248 q^{31} +97.4554 q^{32} +254.412 q^{34} -293.336 q^{37} -422.591 q^{38} +127.214 q^{41} -210.189 q^{43} -106.808 q^{44} +745.244 q^{46} +468.688 q^{47} +49.0000 q^{49} -60.6321 q^{52} -115.973 q^{53} -219.453 q^{56} +200.867 q^{58} +314.090 q^{59} +768.386 q^{61} +66.6662 q^{62} -721.332 q^{64} -717.081 q^{67} -781.164 q^{68} +737.783 q^{71} +477.618 q^{73} +1394.36 q^{74} +1297.55 q^{76} -51.2259 q^{77} -279.262 q^{79} -604.703 q^{82} -776.981 q^{83} +999.123 q^{86} +229.422 q^{88} +29.7626 q^{89} -29.0796 q^{91} -2288.24 q^{92} -2227.88 q^{94} -231.793 q^{97} -232.919 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} + 28 q^{7} - 9 q^{8} - 21 q^{11} + 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 221 q^{22} - 102 q^{23} - 129 q^{26} + 112 q^{28} + 240 q^{29} + 351 q^{31} - 72 q^{32} - 285 q^{34} + 399 q^{37}+ \cdots - 372 q^{97}+O(q^{100}) $ 4 * q + 16 * q^4 + 28 * q^7 - 9 * q^8 - 21 * q^11 + 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 221 * q^22 - 102 * q^23 - 129 * q^26 + 112 * q^28 + 240 * q^29 + 351 * q^31 - 72 * q^32 - 285 * q^34 + 399 * q^37 - 324 * q^38 - 381 * q^41 + 460 * q^43 + 975 * q^44 + 550 * q^46 - 60 * q^47 + 196 * q^49 + 223 * q^52 - 873 * q^53 - 63 * q^56 + 1408 * q^58 + 855 * q^59 + 687 * q^61 + 477 * q^62 - 1285 * q^64 - 503 * q^67 - 1725 * q^68 + 681 * q^71 + 1228 * q^73 + 3369 * q^74 + 1910 * q^76 - 147 * q^77 + 345 * q^79 - 495 * q^82 - 1509 * q^83 + 540 * q^86 + 2266 * q^88 + 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1732 * q^94 - 372 * q^97

Coefficient data

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

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

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

$2$	−4.75345	−1.68060	−0.840299	−	0.542123i	$-0.817621\pi$
$2$	−4.75345	−1.68060	−0.840299	+	0.542123i	$0.817621\pi$
$3$	0	0
$3$	0	0
$4$	14.5953	1.82441
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−31.3504	−1.38551
$9$	0	0
$10$	0	0
$11$	−7.31799	−0.200587	−0.100293	−	0.994958i	$-0.531978\pi$
$11$	−7.31799	−0.200587	−0.100293	+	0.994958i	$0.531978\pi$
$12$	0	0
$13$	−4.15422	−0.0886288	−0.0443144	−	0.999018i	$-0.514110\pi$
$13$	−4.15422	−0.0886288	−0.0443144	+	0.999018i	$0.514110\pi$
$14$	−33.2742	−0.635207
$15$	0	0
$16$	32.2604	0.504068
$17$	−53.5216	−0.763582	−0.381791	−	0.924249i	$-0.624693\pi$
$17$	−53.5216	−0.763582	−0.381791	+	0.924249i	$0.624693\pi$
$18$	0	0
$19$	88.9019	1.07345	0.536724	−	0.843758i	$-0.319662\pi$
$19$	88.9019	1.07345	0.536724	+	0.843758i	$0.319662\pi$
$20$	0	0
$21$	0	0
$22$	34.7857	0.337106
$23$	−156.780	−1.42134	−0.710670	−	0.703526i	$-0.751608\pi$
$23$	−156.780	−1.42134	−0.710670	+	0.703526i	$0.751608\pi$
$24$	0	0
$25$	0	0
$26$	19.7469	0.148949
$27$	0	0
$28$	102.167	0.689563
$29$	−42.2570	−0.270584	−0.135292	−	0.990806i	$-0.543197\pi$
$29$	−42.2570	−0.270584	−0.135292	+	0.990806i	$0.543197\pi$
$30$	0	0
$31$	−14.0248	−0.0812557	−0.0406279	−	0.999174i	$-0.512936\pi$
$31$	−14.0248	−0.0812557	−0.0406279	+	0.999174i	$0.512936\pi$
$32$	97.4554	0.538370
$33$	0	0
$34$	254.412	1.28328
$35$	0	0
$36$	0	0
$37$	−293.336	−1.30335	−0.651677	−	0.758496i	$-0.725934\pi$
$37$	−293.336	−1.30335	−0.651677	+	0.758496i	$0.725934\pi$
$38$	−422.591	−1.80403
$39$	0	0
$40$	0	0
$41$	127.214	0.484571	0.242286	−	0.970205i	$-0.422103\pi$
$41$	127.214	0.484571	0.242286	+	0.970205i	$0.422103\pi$
$42$	0	0
$43$	−210.189	−0.745431	−0.372715	−	0.927946i	$-0.621573\pi$
$43$	−210.189	−0.745431	−0.372715	+	0.927946i	$0.621573\pi$
$44$	−106.808	−0.365953
$45$	0	0
$46$	745.244	2.38870
$47$	468.688	1.45458	0.727289	−	0.686332i	$-0.240780\pi$
$47$	468.688	1.45458	0.727289	+	0.686332i	$0.240780\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−60.6321	−0.161696
$53$	−115.973	−0.300569	−0.150285	−	0.988643i	$-0.548019\pi$
$53$	−115.973	−0.300569	−0.150285	+	0.988643i	$0.548019\pi$
$54$	0	0
$55$	0	0
$56$	−219.453	−0.523672
$57$	0	0
$58$	200.867	0.454743
$59$	314.090	0.693069	0.346534	−	0.938037i	$-0.387358\pi$
$59$	314.090	0.693069	0.346534	+	0.938037i	$0.387358\pi$
$60$	0	0
$61$	768.386	1.61281	0.806407	−	0.591361i	$-0.201409\pi$
$61$	768.386	1.61281	0.806407	+	0.591361i	$0.201409\pi$
$62$	66.6662	0.136558
$63$	0	0
$64$	−721.332	−1.40885
$65$	0	0
$66$	0	0
$67$	−717.081	−1.30754	−0.653772	−	0.756692i	$-0.726814\pi$
$67$	−717.081	−1.30754	−0.653772	+	0.756692i	$0.726814\pi$
$68$	−781.164	−1.39309
$69$	0	0
$70$	0	0
$71$	737.783	1.23322	0.616611	−	0.787268i	$-0.288505\pi$
$71$	737.783	1.23322	0.616611	+	0.787268i	$0.288505\pi$
$72$	0	0
$73$	477.618	0.765767	0.382884	−	0.923797i	$-0.374931\pi$
$73$	477.618	0.765767	0.382884	+	0.923797i	$0.374931\pi$
$74$	1394.36	2.19042
$75$	0	0
$76$	1297.55	1.95841
$77$	−51.2259	−0.0758147
$78$	0	0
$79$	−279.262	−0.397714	−0.198857	−	0.980029i	$-0.563723\pi$
$79$	−279.262	−0.397714	−0.198857	+	0.980029i	$0.563723\pi$
$80$	0	0
$81$	0	0
$82$	−604.703	−0.814370
$83$	−776.981	−1.02753	−0.513764	−	0.857932i	$-0.671749\pi$
$83$	−776.981	−1.02753	−0.513764	+	0.857932i	$0.671749\pi$
$84$	0	0
$85$	0	0
$86$	999.123	1.25277
$87$	0	0
$88$	229.422	0.277914
$89$	29.7626	0.0354476	0.0177238	−	0.999843i	$-0.494358\pi$
$89$	29.7626	0.0354476	0.0177238	+	0.999843i	$0.494358\pi$
$90$	0	0
$91$	−29.0796	−0.0334985
$92$	−2288.24	−2.59311
$93$	0	0
$94$	−2227.88	−2.44456
$95$	0	0
$96$	0	0
$97$	−231.793	−0.242629	−0.121314	−	0.992614i	$-0.538711\pi$
$97$	−231.793	−0.242629	−0.121314	+	0.992614i	$0.538711\pi$
$98$	−232.919	−0.240086
$99$	0	0
$100$	0	0
$101$	1898.26	1.87014	0.935069	−	0.354467i	$-0.115338\pi$
$101$	1898.26	1.87014	0.935069	+	0.354467i	$0.115338\pi$
$102$	0	0
$103$	1375.67	1.31601	0.658003	−	0.753015i	$-0.271401\pi$
$103$	1375.67	1.31601	0.658003	+	0.753015i	$0.271401\pi$
$104$	130.237	0.122796
$105$	0	0
$106$	551.274	0.505136
$107$	−166.359	−0.150304	−0.0751521	−	0.997172i	$-0.523944\pi$
$107$	−166.359	−0.150304	−0.0751521	+	0.997172i	$0.523944\pi$
$108$	0	0
$109$	−1346.00	−1.18279	−0.591393	−	0.806383i	$-0.701422\pi$
$109$	−1346.00	−1.18279	−0.591393	+	0.806383i	$0.701422\pi$
$110$	0	0
$111$	0	0
$112$	225.822	0.190520
$113$	1322.25	1.10077	0.550386	−	0.834911i	$-0.314481\pi$
$113$	1322.25	1.10077	0.550386	+	0.834911i	$0.314481\pi$
$114$	0	0
$115$	0	0
$116$	−616.754	−0.493657
$117$	0	0
$118$	−1493.01	−1.16477
$119$	−374.651	−0.288607
$120$	0	0
$121$	−1277.45	−0.959765
$122$	−3652.48	−2.71049
$123$	0	0
$124$	−204.696	−0.148244
$125$	0	0
$126$	0	0
$127$	2111.85	1.47556	0.737781	−	0.675040i	$-0.235874\pi$
$127$	2111.85	1.47556	0.737781	+	0.675040i	$0.235874\pi$
$128$	2649.18	1.82935
$129$	0	0
$130$	0	0
$131$	209.773	0.139908	0.0699541	−	0.997550i	$-0.477715\pi$
$131$	209.773	0.139908	0.0699541	+	0.997550i	$0.477715\pi$
$132$	0	0
$133$	622.313	0.405725
$134$	3408.61	2.19746
$135$	0	0
$136$	1677.93	1.05795
$137$	−2386.27	−1.48812	−0.744062	−	0.668111i	$-0.767103\pi$
$137$	−2386.27	−1.48812	−0.744062	+	0.668111i	$0.767103\pi$
$138$	0	0
$139$	1215.45	0.741675	0.370838	−	0.928698i	$-0.379071\pi$
$139$	1215.45	0.741675	0.370838	+	0.928698i	$0.379071\pi$
$140$	0	0
$141$	0	0
$142$	−3507.01	−2.07255
$143$	30.4006	0.0177778
$144$	0	0
$145$	0	0
$146$	−2270.34	−1.28695
$147$	0	0
$148$	−4281.33	−2.37786
$149$	−2849.51	−1.56672	−0.783359	−	0.621570i	$-0.786495\pi$
$149$	−2849.51	−1.56672	−0.783359	+	0.621570i	$0.786495\pi$
$150$	0	0
$151$	−2643.84	−1.42485	−0.712426	−	0.701747i	$-0.752403\pi$
$151$	−2643.84	−1.42485	−0.712426	+	0.701747i	$0.752403\pi$
$152$	−2787.11	−1.48727
$153$	0	0
$154$	243.500	0.127414
$155$	0	0
$156$	0	0
$157$	−2563.09	−1.30291	−0.651454	−	0.758688i	$-0.725841\pi$
$157$	−2563.09	−1.30291	−0.651454	+	0.758688i	$0.725841\pi$
$158$	1327.46	0.668398
$159$	0	0
$160$	0	0
$161$	−1097.46	−0.537216
$162$	0	0
$163$	−1403.46	−0.674400	−0.337200	−	0.941433i	$-0.609480\pi$
$163$	−1403.46	−0.674400	−0.337200	+	0.941433i	$0.609480\pi$
$164$	1856.72	0.884057
$165$	0	0
$166$	3693.34	1.72686
$167$	2658.97	1.23208	0.616040	−	0.787715i	$-0.288736\pi$
$167$	2658.97	1.23208	0.616040	+	0.787715i	$0.288736\pi$
$168$	0	0
$169$	−2179.74	−0.992145
$170$	0	0
$171$	0	0
$172$	−3067.77	−1.35997
$173$	−3763.19	−1.65382	−0.826909	−	0.562336i	$-0.809903\pi$
$173$	−3763.19	−1.65382	−0.826909	+	0.562336i	$0.809903\pi$
$174$	0	0
$175$	0	0
$176$	−236.081	−0.101109
$177$	0	0
$178$	−141.475	−0.0595731
$179$	1056.94	0.441336	0.220668	−	0.975349i	$-0.429176\pi$
$179$	1056.94	0.441336	0.220668	+	0.975349i	$0.429176\pi$
$180$	0	0
$181$	537.439	0.220705	0.110352	−	0.993893i	$-0.464802\pi$
$181$	537.439	0.220705	0.110352	+	0.993893i	$0.464802\pi$
$182$	138.228	0.0562976
$183$	0	0
$184$	4915.11	1.96927
$185$	0	0
$186$	0	0
$187$	391.670	0.153165
$188$	6840.64	2.65375
$189$	0	0
$190$	0	0
$191$	3236.34	1.22604	0.613020	−	0.790068i	$-0.289955\pi$
$191$	3236.34	1.22604	0.613020	+	0.790068i	$0.289955\pi$
$192$	0	0
$193$	4620.71	1.72335	0.861673	−	0.507464i	$-0.169417\pi$
$193$	4620.71	1.72335	0.861673	+	0.507464i	$0.169417\pi$
$194$	1101.82	0.407762
$195$	0	0
$196$	715.170	0.260630
$197$	2519.57	0.911230	0.455615	−	0.890177i	$-0.349419\pi$
$197$	2519.57	0.911230	0.455615	+	0.890177i	$0.349419\pi$
$198$	0	0
$199$	−2121.77	−0.755819	−0.377910	−	0.925842i	$-0.623357\pi$
$199$	−2121.77	−0.755819	−0.377910	+	0.925842i	$0.623357\pi$
$200$	0	0
$201$	0	0
$202$	−9023.28	−3.14295
$203$	−295.799	−0.102271
$204$	0	0
$205$	0	0
$206$	−6539.17	−2.21168
$207$	0	0
$208$	−134.017	−0.0446750
$209$	−650.583	−0.215319
$210$	0	0
$211$	−1557.91	−0.508297	−0.254149	−	0.967165i	$-0.581795\pi$
$211$	−1557.91	−0.508297	−0.254149	+	0.967165i	$0.581795\pi$
$212$	−1692.67	−0.548362
$213$	0	0
$214$	790.780	0.252601
$215$	0	0
$216$	0	0
$217$	−98.1736	−0.0307118
$218$	6398.16	1.98779
$219$	0	0
$220$	0	0
$221$	222.341	0.0676754
$222$	0	0
$223$	1319.83	0.396333	0.198167	−	0.980168i	$-0.436501\pi$
$223$	1319.83	0.396333	0.198167	+	0.980168i	$0.436501\pi$
$224$	682.188	0.203485
$225$	0	0
$226$	−6285.27	−1.84995
$227$	−6442.42	−1.88369	−0.941847	−	0.336043i	$-0.890911\pi$
$227$	−6442.42	−1.88369	−0.941847	+	0.336043i	$0.890911\pi$
$228$	0	0
$229$	4654.11	1.34302	0.671511	−	0.740995i	$-0.265646\pi$
$229$	4654.11	1.34302	0.671511	+	0.740995i	$0.265646\pi$
$230$	0	0
$231$	0	0
$232$	1324.78	0.374896
$233$	2628.61	0.739080	0.369540	−	0.929215i	$-0.379515\pi$
$233$	2628.61	0.739080	0.369540	+	0.929215i	$0.379515\pi$
$234$	0	0
$235$	0	0
$236$	4584.24	1.26444
$237$	0	0
$238$	1780.89	0.485033
$239$	−586.369	−0.158699	−0.0793495	−	0.996847i	$-0.525284\pi$
$239$	−586.369	−0.158699	−0.0793495	+	0.996847i	$0.525284\pi$
$240$	0	0
$241$	3141.71	0.839731	0.419865	−	0.907586i	$-0.362077\pi$
$241$	3141.71	0.839731	0.419865	+	0.907586i	$0.362077\pi$
$242$	6072.28	1.61298
$243$	0	0
$244$	11214.8	2.94244
$245$	0	0
$246$	0	0
$247$	−369.318	−0.0951383
$248$	439.683	0.112580
$249$	0	0
$250$	0	0
$251$	2929.38	0.736656	0.368328	−	0.929696i	$-0.379930\pi$
$251$	2929.38	0.736656	0.368328	+	0.929696i	$0.379930\pi$
$252$	0	0
$253$	1147.31	0.285102
$254$	−10038.6	−2.47983
$255$	0	0
$256$	−6822.07	−1.66554
$257$	388.552	0.0943081	0.0471541	−	0.998888i	$-0.484985\pi$
$257$	388.552	0.0943081	0.0471541	+	0.998888i	$0.484985\pi$
$258$	0	0
$259$	−2053.35	−0.492622
$260$	0	0
$261$	0	0
$262$	−997.147	−0.235130
$263$	−766.349	−0.179677	−0.0898387	−	0.995956i	$-0.528635\pi$
$263$	−766.349	−0.179677	−0.0898387	+	0.995956i	$0.528635\pi$
$264$	0	0
$265$	0	0
$266$	−2958.14	−0.681861
$267$	0	0
$268$	−10466.0	−2.38550
$269$	−2842.03	−0.644170	−0.322085	−	0.946711i	$-0.604384\pi$
$269$	−2842.03	−0.644170	−0.322085	+	0.946711i	$0.604384\pi$
$270$	0	0
$271$	3512.72	0.787390	0.393695	−	0.919241i	$-0.371197\pi$
$271$	3512.72	0.787390	0.393695	+	0.919241i	$0.371197\pi$
$272$	−1726.63	−0.384897
$273$	0	0
$274$	11343.0	2.50094
$275$	0	0
$276$	0	0
$277$	6388.71	1.38578	0.692889	−	0.721044i	$-0.256337\pi$
$277$	6388.71	1.38578	0.692889	+	0.721044i	$0.256337\pi$
$278$	−5777.57	−1.24646
$279$	0	0
$280$	0	0
$281$	2126.77	0.451503	0.225752	−	0.974185i	$-0.427516\pi$
$281$	2126.77	0.451503	0.225752	+	0.974185i	$0.427516\pi$
$282$	0	0
$283$	8332.13	1.75015	0.875077	−	0.483983i	$-0.160810\pi$
$283$	8332.13	1.75015	0.875077	+	0.483983i	$0.160810\pi$
$284$	10768.2	2.24990
$285$	0	0
$286$	−144.508	−0.0298773
$287$	890.495	0.183151
$288$	0	0
$289$	−2048.44	−0.416942
$290$	0	0
$291$	0	0
$292$	6970.98	1.39707
$293$	8654.11	1.72552	0.862762	−	0.505610i	$-0.168733\pi$
$293$	8654.11	1.72552	0.862762	+	0.505610i	$0.168733\pi$
$294$	0	0
$295$	0	0
$296$	9196.21	1.80581
$297$	0	0
$298$	13545.0	2.63302
$299$	651.297	0.125972
$300$	0	0
$301$	−1471.32	−0.281746
$302$	12567.4	2.39460
$303$	0	0
$304$	2868.01	0.541090
$305$	0	0
$306$	0	0
$307$	5318.56	0.988749	0.494375	−	0.869249i	$-0.335397\pi$
$307$	5318.56	0.988749	0.494375	+	0.869249i	$0.335397\pi$
$308$	−747.657	−0.138317
$309$	0	0
$310$	0	0
$311$	7097.87	1.29416	0.647079	−	0.762423i	$-0.275990\pi$
$311$	7097.87	1.29416	0.647079	+	0.762423i	$0.275990\pi$
$312$	0	0
$313$	−2080.78	−0.375759	−0.187879	−	0.982192i	$-0.560161\pi$
$313$	−2080.78	−0.375759	−0.187879	+	0.982192i	$0.560161\pi$
$314$	12183.5	2.18967
$315$	0	0
$316$	−4075.91	−0.725595
$317$	−2644.61	−0.468568	−0.234284	−	0.972168i	$-0.575275\pi$
$317$	−2644.61	−0.468568	−0.234284	+	0.972168i	$0.575275\pi$
$318$	0	0
$319$	309.236	0.0542756
$320$	0	0
$321$	0	0
$322$	5216.71	0.902844
$323$	−4758.17	−0.819665
$324$	0	0
$325$	0	0
$326$	6671.26	1.13340
$327$	0	0
$328$	−3988.20	−0.671376
$329$	3280.81	0.549779
$330$	0	0
$331$	−1831.16	−0.304077	−0.152039	−	0.988375i	$-0.548584\pi$
$331$	−1831.16	−0.304077	−0.152039	+	0.988375i	$0.548584\pi$
$332$	−11340.3	−1.87463
$333$	0	0
$334$	−12639.3	−2.07063
$335$	0	0
$336$	0	0
$337$	9307.67	1.50451	0.752257	−	0.658870i	$-0.228965\pi$
$337$	9307.67	1.50451	0.752257	+	0.658870i	$0.228965\pi$
$338$	10361.3	1.66740
$339$	0	0
$340$	0	0
$341$	102.633	0.0162988
$342$	0	0
$343$	343.000	0.0539949
$344$	6589.52	1.03280
$345$	0	0
$346$	17888.2	2.77940
$347$	10802.7	1.67124	0.835619	−	0.549310i	$-0.185109\pi$
$347$	10802.7	1.67124	0.835619	+	0.549310i	$0.185109\pi$
$348$	0	0
$349$	2242.94	0.344017	0.172009	−	0.985095i	$-0.444974\pi$
$349$	2242.94	0.344017	0.172009	+	0.985095i	$0.444974\pi$
$350$	0	0
$351$	0	0
$352$	−713.177	−0.107990
$353$	−3295.68	−0.496916	−0.248458	−	0.968643i	$-0.579924\pi$
$353$	−3295.68	−0.496916	−0.248458	+	0.968643i	$0.579924\pi$
$354$	0	0
$355$	0	0
$356$	434.395	0.0646710
$357$	0	0
$358$	−5024.09	−0.741709
$359$	4634.55	0.681344	0.340672	−	0.940182i	$-0.389345\pi$
$359$	4634.55	0.681344	0.340672	+	0.940182i	$0.389345\pi$
$360$	0	0
$361$	1044.55	0.152289
$362$	−2554.69	−0.370916
$363$	0	0
$364$	−424.425	−0.0611152
$365$	0	0
$366$	0	0
$367$	6316.17	0.898369	0.449185	−	0.893439i	$-0.351715\pi$
$367$	6316.17	0.898369	0.449185	+	0.893439i	$0.351715\pi$
$368$	−5057.76	−0.716452
$369$	0	0
$370$	0	0
$371$	−811.814	−0.113604
$372$	0	0
$373$	−7880.83	−1.09398	−0.546989	−	0.837140i	$-0.684226\pi$
$373$	−7880.83	−1.09398	−0.546989	+	0.837140i	$0.684226\pi$
$374$	−1861.79	−0.257408
$375$	0	0
$376$	−14693.6	−2.01533
$377$	175.545	0.0239815
$378$	0	0
$379$	9853.13	1.33541	0.667706	−	0.744425i	$-0.267276\pi$
$379$	9853.13	1.33541	0.667706	+	0.744425i	$0.267276\pi$
$380$	0	0
$381$	0	0
$382$	−15383.8	−2.06048
$383$	−1193.49	−0.159228	−0.0796141	−	0.996826i	$-0.525369\pi$
$383$	−1193.49	−0.159228	−0.0796141	+	0.996826i	$0.525369\pi$
$384$	0	0
$385$	0	0
$386$	−21964.3	−2.89625
$387$	0	0
$388$	−3383.09	−0.442655
$389$	10425.7	1.35888	0.679442	−	0.733729i	$-0.262222\pi$
$389$	10425.7	1.35888	0.679442	+	0.733729i	$0.262222\pi$
$390$	0	0
$391$	8391.09	1.08531
$392$	−1536.17	−0.197929
$393$	0	0
$394$	−11976.7	−1.53141
$395$	0	0
$396$	0	0
$397$	3650.05	0.461437	0.230719	−	0.973021i	$-0.425892\pi$
$397$	3650.05	0.461437	0.230719	+	0.973021i	$0.425892\pi$
$398$	10085.7	1.27023
$399$	0	0
$400$	0	0
$401$	8202.17	1.02144	0.510719	−	0.859747i	$-0.329379\pi$
$401$	8202.17	1.02144	0.510719	+	0.859747i	$0.329379\pi$
$402$	0	0
$403$	58.2622	0.00720160
$404$	27705.7	3.41190
$405$	0	0
$406$	1406.07	0.171877
$407$	2146.63	0.261436
$408$	0	0
$409$	9097.72	1.09989	0.549943	−	0.835202i	$-0.314649\pi$
$409$	9097.72	1.09989	0.549943	+	0.835202i	$0.314649\pi$
$410$	0	0
$411$	0	0
$412$	20078.3	2.40094
$413$	2198.63	0.261955
$414$	0	0
$415$	0	0
$416$	−404.852	−0.0477151
$417$	0	0
$418$	3092.51	0.361866
$419$	4704.01	0.548462	0.274231	−	0.961664i	$-0.411577\pi$
$419$	4704.01	0.548462	0.274231	+	0.961664i	$0.411577\pi$
$420$	0	0
$421$	1596.95	0.184871	0.0924355	−	0.995719i	$-0.470535\pi$
$421$	1596.95	0.184871	0.0924355	+	0.995719i	$0.470535\pi$
$422$	7405.44	0.854244
$423$	0	0
$424$	3635.82	0.416441
$425$	0	0
$426$	0	0
$427$	5378.70	0.609587
$428$	−2428.06	−0.274217
$429$	0	0
$430$	0	0
$431$	−6235.23	−0.696845	−0.348423	−	0.937338i	$-0.613283\pi$
$431$	−6235.23	−0.696845	−0.348423	+	0.937338i	$0.613283\pi$
$432$	0	0
$433$	2363.94	0.262364	0.131182	−	0.991358i	$-0.458123\pi$
$433$	2363.94	0.262364	0.131182	+	0.991358i	$0.458123\pi$
$434$	466.663	0.0516142
$435$	0	0
$436$	−19645.3	−2.15789
$437$	−13938.0	−1.52573
$438$	0	0
$439$	−17537.7	−1.90667	−0.953335	−	0.301916i	$-0.902374\pi$
$439$	−17537.7	−1.90667	−0.953335	+	0.301916i	$0.902374\pi$
$440$	0	0
$441$	0	0
$442$	−1056.89	−0.113735
$443$	4488.83	0.481424	0.240712	−	0.970597i	$-0.422619\pi$
$443$	4488.83	0.481424	0.240712	+	0.970597i	$0.422619\pi$
$444$	0	0
$445$	0	0
$446$	−6273.74	−0.666077
$447$	0	0
$448$	−5049.33	−0.532496
$449$	−13188.3	−1.38618	−0.693090	−	0.720851i	$-0.743751\pi$
$449$	−13188.3	−1.38618	−0.693090	+	0.720851i	$0.743751\pi$
$450$	0	0
$451$	−930.947	−0.0971986
$452$	19298.7	2.00826
$453$	0	0
$454$	30623.7	3.16573
$455$	0	0
$456$	0	0
$457$	8085.04	0.827576	0.413788	−	0.910373i	$-0.364206\pi$
$457$	8085.04	0.827576	0.413788	+	0.910373i	$0.364206\pi$
$458$	−22123.1	−2.25708
$459$	0	0
$460$	0	0
$461$	825.258	0.0833754	0.0416877	−	0.999131i	$-0.486727\pi$
$461$	825.258	0.0833754	0.0416877	+	0.999131i	$0.486727\pi$
$462$	0	0
$463$	16607.8	1.66702	0.833509	−	0.552505i	$-0.186328\pi$
$463$	16607.8	1.66702	0.833509	+	0.552505i	$0.186328\pi$
$464$	−1363.23	−0.136393
$465$	0	0
$466$	−12495.0	−1.24210
$467$	−6003.84	−0.594914	−0.297457	−	0.954735i	$-0.596138\pi$
$467$	−6003.84	−0.594914	−0.297457	+	0.954735i	$0.596138\pi$
$468$	0	0
$469$	−5019.57	−0.494205
$470$	0	0
$471$	0	0
$472$	−9846.86	−0.960251
$473$	1538.16	0.149524
$474$	0	0
$475$	0	0
$476$	−5468.15	−0.526538
$477$	0	0
$478$	2787.28	0.266709
$479$	−20386.0	−1.94459	−0.972295	−	0.233759i	$-0.924897\pi$
$479$	−20386.0	−1.94459	−0.972295	+	0.233759i	$0.924897\pi$
$480$	0	0
$481$	1218.58	0.115515
$482$	−14933.9	−1.41125
$483$	0	0
$484$	−18644.7	−1.75101
$485$	0	0
$486$	0	0
$487$	9422.61	0.876754	0.438377	−	0.898791i	$-0.355553\pi$
$487$	9422.61	0.876754	0.438377	+	0.898791i	$0.355553\pi$
$488$	−24089.2	−2.23456
$489$	0	0
$490$	0	0
$491$	−5908.30	−0.543051	−0.271525	−	0.962431i	$-0.587528\pi$
$491$	−5908.30	−0.543051	−0.271525	+	0.962431i	$0.587528\pi$
$492$	0	0
$493$	2261.66	0.206613
$494$	1755.54	0.159889
$495$	0	0
$496$	−452.445	−0.0409584
$497$	5164.48	0.466114
$498$	0	0
$499$	−3237.65	−0.290455	−0.145227	−	0.989398i	$-0.546391\pi$
$499$	−3237.65	−0.290455	−0.145227	+	0.989398i	$0.546391\pi$
$500$	0	0
$501$	0	0
$502$	−13924.7	−1.23802
$503$	−1112.05	−0.0985760	−0.0492880	−	0.998785i	$-0.515695\pi$
$503$	−1112.05	−0.0985760	−0.0492880	+	0.998785i	$0.515695\pi$
$504$	0	0
$505$	0	0
$506$	−5453.68	−0.479142
$507$	0	0
$508$	30823.1	2.69203
$509$	−15737.5	−1.37044	−0.685220	−	0.728336i	$-0.740294\pi$
$509$	−15737.5	−1.37044	−0.685220	+	0.728336i	$0.740294\pi$
$510$	0	0
$511$	3343.33	0.289433
$512$	11235.0	0.969765
$513$	0	0
$514$	−1846.96	−0.158494
$515$	0	0
$516$	0	0
$517$	−3429.85	−0.291769
$518$	9760.51	0.827900
$519$	0	0
$520$	0	0
$521$	5009.73	0.421267	0.210633	−	0.977565i	$-0.432447\pi$
$521$	5009.73	0.421267	0.210633	+	0.977565i	$0.432447\pi$
$522$	0	0
$523$	14162.2	1.18407	0.592035	−	0.805912i	$-0.298325\pi$
$523$	14162.2	1.18407	0.592035	+	0.805912i	$0.298325\pi$
$524$	3061.70	0.255250
$525$	0	0
$526$	3642.80	0.301965
$527$	750.630	0.0620454
$528$	0	0
$529$	12412.8	1.02020
$530$	0	0
$531$	0	0
$532$	9082.85	0.740209
$533$	−528.474	−0.0429470
$534$	0	0
$535$	0	0
$536$	22480.8	1.81161
$537$	0	0
$538$	13509.5	1.08259
$539$	−358.581	−0.0286553
$540$	0	0
$541$	14815.3	1.17738	0.588689	−	0.808360i	$-0.299644\pi$
$541$	14815.3	1.17738	0.588689	+	0.808360i	$0.299644\pi$
$542$	−16697.5	−1.32329
$543$	0	0
$544$	−5215.97	−0.411090
$545$	0	0
$546$	0	0
$547$	14248.3	1.11373	0.556867	−	0.830601i	$-0.312003\pi$
$547$	14248.3	1.11373	0.556867	+	0.830601i	$0.312003\pi$
$548$	−34828.3	−2.71495
$549$	0	0
$550$	0	0
$551$	−3756.73	−0.290458
$552$	0	0
$553$	−1954.83	−0.150322
$554$	−30368.4	−2.32894
$555$	0	0
$556$	17739.8	1.35312
$557$	−9394.37	−0.714636	−0.357318	−	0.933983i	$-0.616309\pi$
$557$	−9394.37	−0.714636	−0.357318	+	0.933983i	$0.616309\pi$
$558$	0	0
$559$	873.173	0.0660667
$560$	0	0
$561$	0	0
$562$	−10109.5	−0.758796
$563$	8572.89	0.641748	0.320874	−	0.947122i	$-0.396023\pi$
$563$	8572.89	0.641748	0.320874	+	0.947122i	$0.396023\pi$
$564$	0	0
$565$	0	0
$566$	−39606.4	−2.94131
$567$	0	0
$568$	−23129.8	−1.70864
$569$	−8954.65	−0.659751	−0.329876	−	0.944024i	$-0.607007\pi$
$569$	−8954.65	−0.659751	−0.329876	+	0.944024i	$0.607007\pi$
$570$	0	0
$571$	21592.5	1.58252	0.791260	−	0.611479i	$-0.209425\pi$
$571$	21592.5	1.58252	0.791260	+	0.611479i	$0.209425\pi$
$572$	443.705	0.0324340
$573$	0	0
$574$	−4232.92	−0.307803
$575$	0	0
$576$	0	0
$577$	−17587.5	−1.26894	−0.634470	−	0.772947i	$-0.718782\pi$
$577$	−17587.5	−1.26894	−0.634470	+	0.772947i	$0.718782\pi$
$578$	9737.14	0.700712
$579$	0	0
$580$	0	0
$581$	−5438.87	−0.388369
$582$	0	0
$583$	848.692	0.0602903
$584$	−14973.5	−1.06098
$585$	0	0
$586$	−41136.9	−2.89991
$587$	4613.88	0.324421	0.162210	−	0.986756i	$-0.448138\pi$
$587$	4613.88	0.324421	0.162210	+	0.986756i	$0.448138\pi$
$588$	0	0
$589$	−1246.83	−0.0872237
$590$	0	0
$591$	0	0
$592$	−9463.12	−0.656980
$593$	8688.07	0.601647	0.300823	−	0.953680i	$-0.402739\pi$
$593$	8688.07	0.601647	0.300823	+	0.953680i	$0.402739\pi$
$594$	0	0
$595$	0	0
$596$	−41589.4	−2.85834
$597$	0	0
$598$	−3095.91	−0.211708
$599$	−7361.43	−0.502137	−0.251068	−	0.967969i	$-0.580782\pi$
$599$	−7361.43	−0.502137	−0.251068	+	0.967969i	$0.580782\pi$
$600$	0	0
$601$	16441.9	1.11594	0.557971	−	0.829861i	$-0.311580\pi$
$601$	16441.9	1.11594	0.557971	+	0.829861i	$0.311580\pi$
$602$	6993.86	0.473503
$603$	0	0
$604$	−38587.6	−2.59952
$605$	0	0
$606$	0	0
$607$	−21024.5	−1.40586	−0.702931	−	0.711258i	$-0.748126\pi$
$607$	−21024.5	−1.40586	−0.702931	+	0.711258i	$0.748126\pi$
$608$	8663.97	0.577912
$609$	0	0
$610$	0	0
$611$	−1947.03	−0.128917
$612$	0	0
$613$	3278.79	0.216034	0.108017	−	0.994149i	$-0.465550\pi$
$613$	3278.79	0.216034	0.108017	+	0.994149i	$0.465550\pi$
$614$	−25281.5	−1.66169
$615$	0	0
$616$	1605.95	0.105042
$617$	−136.641	−0.00891567	−0.00445783	−	0.999990i	$-0.501419\pi$
$617$	−136.641	−0.00891567	−0.00445783	+	0.999990i	$0.501419\pi$
$618$	0	0
$619$	−13109.0	−0.851205	−0.425603	−	0.904910i	$-0.639938\pi$
$619$	−13109.0	−0.851205	−0.425603	+	0.904910i	$0.639938\pi$
$620$	0	0
$621$	0	0
$622$	−33739.4	−2.17496
$623$	208.338	0.0133979
$624$	0	0
$625$	0	0
$626$	9890.87	0.631500
$627$	0	0
$628$	−37409.0	−2.37704
$629$	15699.8	0.995219
$630$	0	0
$631$	−351.608	−0.0221827	−0.0110913	−	0.999938i	$-0.503531\pi$
$631$	−351.608	−0.0221827	−0.0110913	+	0.999938i	$0.503531\pi$
$632$	8754.98	0.551035
$633$	0	0
$634$	12571.0	0.787474
$635$	0	0
$636$	0	0
$637$	−203.557	−0.0126613
$638$	−1469.94	−0.0912155
$639$	0	0
$640$	0	0
$641$	17551.0	1.08147	0.540736	−	0.841192i	$-0.318146\pi$
$641$	17551.0	1.08147	0.540736	+	0.841192i	$0.318146\pi$
$642$	0	0
$643$	20683.7	1.26856	0.634281	−	0.773103i	$-0.281296\pi$
$643$	20683.7	1.26856	0.634281	+	0.773103i	$0.281296\pi$
$644$	−16017.7	−0.980103
$645$	0	0
$646$	22617.7	1.37753
$647$	−1782.40	−0.108305	−0.0541526	−	0.998533i	$-0.517246\pi$
$647$	−1782.40	−0.108305	−0.0541526	+	0.998533i	$0.517246\pi$
$648$	0	0
$649$	−2298.51	−0.139020
$650$	0	0
$651$	0	0
$652$	−20483.9	−1.23038
$653$	3642.28	0.218275	0.109137	−	0.994027i	$-0.465191\pi$
$653$	3642.28	0.218275	0.109137	+	0.994027i	$0.465191\pi$
$654$	0	0
$655$	0	0
$656$	4103.95	0.244257
$657$	0	0
$658$	−15595.2	−0.923957
$659$	14883.2	0.879768	0.439884	−	0.898055i	$-0.355020\pi$
$659$	14883.2	0.879768	0.439884	+	0.898055i	$0.355020\pi$
$660$	0	0
$661$	−1871.82	−0.110145	−0.0550723	−	0.998482i	$-0.517539\pi$
$661$	−1871.82	−0.110145	−0.0550723	+	0.998482i	$0.517539\pi$
$662$	8704.33	0.511032
$663$	0	0
$664$	24358.7	1.42365
$665$	0	0
$666$	0	0
$667$	6625.04	0.384592
$668$	38808.5	2.24782
$669$	0	0
$670$	0	0
$671$	−5623.03	−0.323509
$672$	0	0
$673$	−22208.8	−1.27205	−0.636023	−	0.771670i	$-0.719422\pi$
$673$	−22208.8	−1.27205	−0.636023	+	0.771670i	$0.719422\pi$
$674$	−44243.6	−2.52848
$675$	0	0
$676$	−31814.0	−1.81008
$677$	31340.2	1.77918	0.889589	−	0.456762i	$-0.150991\pi$
$677$	31340.2	1.77918	0.889589	+	0.456762i	$0.150991\pi$
$678$	0	0
$679$	−1622.55	−0.0917051
$680$	0	0
$681$	0	0
$682$	−487.862	−0.0273918
$683$	−27666.5	−1.54997	−0.774985	−	0.631979i	$-0.782243\pi$
$683$	−27666.5	−1.54997	−0.774985	+	0.631979i	$0.782243\pi$
$684$	0	0
$685$	0	0
$686$	−1630.43	−0.0907438
$687$	0	0
$688$	−6780.77	−0.375748
$689$	481.780	0.0266391
$690$	0	0
$691$	−7978.64	−0.439250	−0.219625	−	0.975584i	$-0.570483\pi$
$691$	−7978.64	−0.439250	−0.219625	+	0.975584i	$0.570483\pi$
$692$	−54925.0	−3.01724
$693$	0	0
$694$	−51350.1	−2.80868
$695$	0	0
$696$	0	0
$697$	−6808.67	−0.370010
$698$	−10661.7	−0.578155
$699$	0	0
$700$	0	0
$701$	16299.3	0.878197	0.439099	−	0.898439i	$-0.355298\pi$
$701$	16299.3	0.878197	0.439099	+	0.898439i	$0.355298\pi$
$702$	0	0
$703$	−26078.1	−1.39908
$704$	5278.70	0.282597
$705$	0	0
$706$	15665.8	0.835116
$707$	13287.8	0.706845
$708$	0	0
$709$	−36223.4	−1.91876	−0.959379	−	0.282122i	$-0.908962\pi$
$709$	−36223.4	−1.91876	−0.959379	+	0.282122i	$0.908962\pi$
$710$	0	0
$711$	0	0
$712$	−933.071	−0.0491128
$713$	2198.80	0.115492
$714$	0	0
$715$	0	0
$716$	15426.3	0.805179
$717$	0	0
$718$	−22030.1	−1.14507
$719$	636.264	0.0330023	0.0165011	−	0.999864i	$-0.494747\pi$
$719$	636.264	0.0330023	0.0165011	+	0.999864i	$0.494747\pi$
$720$	0	0
$721$	9629.67	0.497403
$722$	−4965.21	−0.255936
$723$	0	0
$724$	7844.08	0.402656
$725$	0	0
$726$	0	0
$727$	−33098.8	−1.68854	−0.844270	−	0.535918i	$-0.819965\pi$
$727$	−33098.8	−1.68854	−0.844270	+	0.535918i	$0.819965\pi$
$728$	911.657	0.0464124
$729$	0	0
$730$	0	0
$731$	11249.7	0.569198
$732$	0	0
$733$	−12019.8	−0.605678	−0.302839	−	0.953042i	$-0.597934\pi$
$733$	−12019.8	−0.605678	−0.302839	+	0.953042i	$0.597934\pi$
$734$	−30023.6	−1.50980
$735$	0	0
$736$	−15279.0	−0.765207
$737$	5247.59	0.262276
$738$	0	0
$739$	5332.38	0.265433	0.132716	−	0.991154i	$-0.457630\pi$
$739$	5332.38	0.265433	0.132716	+	0.991154i	$0.457630\pi$
$740$	0	0
$741$	0	0
$742$	3858.92	0.190924
$743$	−6547.51	−0.323290	−0.161645	−	0.986849i	$-0.551680\pi$
$743$	−6547.51	−0.323290	−0.161645	+	0.986849i	$0.551680\pi$
$744$	0	0
$745$	0	0
$746$	37461.1	1.83854
$747$	0	0
$748$	5716.55	0.279435
$749$	−1164.51	−0.0568097
$750$	0	0
$751$	19316.3	0.938565	0.469282	−	0.883048i	$-0.344513\pi$
$751$	19316.3	0.938565	0.469282	+	0.883048i	$0.344513\pi$
$752$	15120.0	0.733206
$753$	0	0
$754$	−834.446	−0.0403033
$755$	0	0
$756$	0	0
$757$	40483.3	1.94371	0.971857	−	0.235573i	$-0.0756967\pi$
$757$	40483.3	1.94371	0.971857	+	0.235573i	$0.0756967\pi$
$758$	−46836.3	−2.24429
$759$	0	0
$760$	0	0
$761$	5470.00	0.260561	0.130281	−	0.991477i	$-0.458412\pi$
$761$	5470.00	0.260561	0.130281	+	0.991477i	$0.458412\pi$
$762$	0	0
$763$	−9422.02	−0.447051
$764$	47235.4	2.23680
$765$	0	0
$766$	5673.19	0.267599
$767$	−1304.80	−0.0614258
$768$	0	0
$769$	17692.4	0.829653	0.414826	−	0.909901i	$-0.363842\pi$
$769$	17692.4	0.829653	0.414826	+	0.909901i	$0.363842\pi$
$770$	0	0
$771$	0	0
$772$	67440.6	3.14409
$773$	3338.50	0.155340	0.0776699	−	0.996979i	$-0.475252\pi$
$773$	3338.50	0.155340	0.0776699	+	0.996979i	$0.475252\pi$
$774$	0	0
$775$	0	0
$776$	7266.81	0.336164
$777$	0	0
$778$	−49558.2	−2.28374
$779$	11309.5	0.520161
$780$	0	0
$781$	−5399.08	−0.247368
$782$	−39886.7	−1.82397
$783$	0	0
$784$	1580.76	0.0720097
$785$	0	0
$786$	0	0
$787$	11738.0	0.531657	0.265829	−	0.964020i	$-0.414354\pi$
$787$	11738.0	0.531657	0.265829	+	0.964020i	$0.414354\pi$
$788$	36773.9	1.66246
$789$	0	0
$790$	0	0
$791$	9255.77	0.416052
$792$	0	0
$793$	−3192.05	−0.142942
$794$	−17350.3	−0.775491
$795$	0	0
$796$	−30967.8	−1.37893
$797$	20150.5	0.895568	0.447784	−	0.894142i	$-0.352213\pi$
$797$	20150.5	0.895568	0.447784	+	0.894142i	$0.352213\pi$
$798$	0	0
$799$	−25084.9	−1.11069
$800$	0	0
$801$	0	0
$802$	−38988.6	−1.71663
$803$	−3495.20	−0.153603
$804$	0	0
$805$	0	0
$806$	−276.946	−0.0121030
$807$	0	0
$808$	−59511.3	−2.59109
$809$	3381.53	0.146957	0.0734785	−	0.997297i	$-0.476590\pi$
$809$	3381.53	0.146957	0.0734785	+	0.997297i	$0.476590\pi$
$810$	0	0
$811$	11375.3	0.492528	0.246264	−	0.969203i	$-0.420797\pi$
$811$	11375.3	0.492528	0.246264	+	0.969203i	$0.420797\pi$
$812$	−4317.28	−0.186585
$813$	0	0
$814$	−10203.9	−0.439369
$815$	0	0
$816$	0	0
$817$	−18686.2	−0.800181
$818$	−43245.6	−1.84847
$819$	0	0
$820$	0	0
$821$	9610.94	0.408556	0.204278	−	0.978913i	$-0.434515\pi$
$821$	9610.94	0.408556	0.204278	+	0.978913i	$0.434515\pi$
$822$	0	0
$823$	13767.3	0.583109	0.291555	−	0.956554i	$-0.405827\pi$
$823$	13767.3	0.583109	0.291555	+	0.956554i	$0.405827\pi$
$824$	−43127.8	−1.82333
$825$	0	0
$826$	−10451.1	−0.440242
$827$	−31978.6	−1.34462	−0.672312	−	0.740268i	$-0.734699\pi$
$827$	−31978.6	−1.34462	−0.672312	+	0.740268i	$0.734699\pi$
$828$	0	0
$829$	18477.2	0.774111	0.387055	−	0.922056i	$-0.373492\pi$
$829$	18477.2	0.774111	0.387055	+	0.922056i	$0.373492\pi$
$830$	0	0
$831$	0	0
$832$	2996.58	0.124865
$833$	−2622.56	−0.109083
$834$	0	0
$835$	0	0
$836$	−9495.45	−0.392831
$837$	0	0
$838$	−22360.3	−0.921745
$839$	38552.6	1.58639	0.793196	−	0.608967i	$-0.208416\pi$
$839$	38552.6	1.58639	0.793196	+	0.608967i	$0.208416\pi$
$840$	0	0
$841$	−22603.3	−0.926784
$842$	−7591.04	−0.310694
$843$	0	0
$844$	−22738.1	−0.927344
$845$	0	0
$846$	0	0
$847$	−8942.13	−0.362757
$848$	−3741.34	−0.151507
$849$	0	0
$850$	0	0
$851$	45989.1	1.85251
$852$	0	0
$853$	3080.15	0.123637	0.0618185	−	0.998087i	$-0.480310\pi$
$853$	3080.15	0.123637	0.0618185	+	0.998087i	$0.480310\pi$
$854$	−25567.4	−1.02447
$855$	0	0
$856$	5215.43	0.208247
$857$	28362.3	1.13050	0.565249	−	0.824920i	$-0.308780\pi$
$857$	28362.3	1.13050	0.565249	+	0.824920i	$0.308780\pi$
$858$	0	0
$859$	−4928.50	−0.195761	−0.0978803	−	0.995198i	$-0.531206\pi$
$859$	−4928.50	−0.195761	−0.0978803	+	0.995198i	$0.531206\pi$
$860$	0	0
$861$	0	0
$862$	29638.9	1.17112
$863$	−39910.8	−1.57425	−0.787126	−	0.616792i	$-0.788432\pi$
$863$	−39910.8	−1.57425	−0.787126	+	0.616792i	$0.788432\pi$
$864$	0	0
$865$	0	0
$866$	−11236.9	−0.440928
$867$	0	0
$868$	−1432.87	−0.0560309
$869$	2043.63	0.0797762
$870$	0	0
$871$	2978.92	0.115886
$872$	42197.8	1.63876
$873$	0	0
$874$	66253.6	2.56414
$875$	0	0
$876$	0	0
$877$	−29231.5	−1.12552	−0.562759	−	0.826621i	$-0.690260\pi$
$877$	−29231.5	−1.12552	−0.562759	+	0.826621i	$0.690260\pi$
$878$	83364.5	3.20435
$879$	0	0
$880$	0	0
$881$	23473.3	0.897657	0.448829	−	0.893618i	$-0.351841\pi$
$881$	23473.3	0.897657	0.448829	+	0.893618i	$0.351841\pi$
$882$	0	0
$883$	21250.7	0.809902	0.404951	−	0.914338i	$-0.367289\pi$
$883$	21250.7	0.809902	0.404951	+	0.914338i	$0.367289\pi$
$884$	3245.13	0.123468
$885$	0	0
$886$	−21337.4	−0.809080
$887$	48748.6	1.84534	0.922671	−	0.385588i	$-0.126001\pi$
$887$	48748.6	1.84534	0.922671	+	0.385588i	$0.126001\pi$
$888$	0	0
$889$	14783.0	0.557710
$890$	0	0
$891$	0	0
$892$	19263.3	0.723075
$893$	41667.2	1.56141
$894$	0	0
$895$	0	0
$896$	18544.2	0.691428
$897$	0	0
$898$	62690.0	2.32961
$899$	592.646	0.0219865
$900$	0	0
$901$	6207.08	0.229509
$902$	4425.21	0.163352
$903$	0	0
$904$	−41453.2	−1.52513
$905$	0	0
$906$	0	0
$907$	−13144.5	−0.481209	−0.240604	−	0.970623i	$-0.577346\pi$
$907$	−13144.5	−0.481209	−0.240604	+	0.970623i	$0.577346\pi$
$908$	−94029.0	−3.43663
$909$	0	0
$910$	0	0
$911$	29731.7	1.08129	0.540644	−	0.841251i	$-0.318181\pi$
$911$	29731.7	1.08129	0.540644	+	0.841251i	$0.318181\pi$
$912$	0	0
$913$	5685.94	0.206108
$914$	−38431.8	−1.39082
$915$	0	0
$916$	67928.1	2.45022
$917$	1468.41	0.0528803
$918$	0	0
$919$	−47715.4	−1.71272	−0.856358	−	0.516383i	$-0.827278\pi$
$919$	−47715.4	−1.71272	−0.856358	+	0.516383i	$0.827278\pi$
$920$	0	0
$921$	0	0
$922$	−3922.82	−0.140121
$923$	−3064.92	−0.109299
$924$	0	0
$925$	0	0
$926$	−78944.4	−2.80159
$927$	0	0
$928$	−4118.18	−0.145674
$929$	19795.2	0.699094	0.349547	−	0.936919i	$-0.386335\pi$
$929$	19795.2	0.699094	0.349547	+	0.936919i	$0.386335\pi$
$930$	0	0
$931$	4356.19	0.153350
$932$	38365.3	1.34839
$933$	0	0
$934$	28539.0	0.999811
$935$	0	0
$936$	0	0
$937$	−17993.9	−0.627359	−0.313680	−	0.949529i	$-0.601562\pi$
$937$	−17993.9	−0.627359	−0.313680	+	0.949529i	$0.601562\pi$
$938$	23860.3	0.830560
$939$	0	0
$940$	0	0
$941$	34667.7	1.20099	0.600497	−	0.799627i	$-0.294970\pi$
$941$	34667.7	1.20099	0.600497	+	0.799627i	$0.294970\pi$
$942$	0	0
$943$	−19944.5	−0.688740
$944$	10132.7	0.349354
$945$	0	0
$946$	−7311.57	−0.251289
$947$	56727.3	1.94656	0.973279	−	0.229627i	$-0.0737505\pi$
$947$	56727.3	1.94656	0.973279	+	0.229627i	$0.0737505\pi$
$948$	0	0
$949$	−1984.13	−0.0678690
$950$	0	0
$951$	0	0
$952$	11745.5	0.399867
$953$	−46267.5	−1.57267	−0.786333	−	0.617802i	$-0.788023\pi$
$953$	−46267.5	−1.57267	−0.786333	+	0.617802i	$0.788023\pi$
$954$	0	0
$955$	0	0
$956$	−8558.23	−0.289532
$957$	0	0
$958$	96903.7	3.26807
$959$	−16703.9	−0.562458
$960$	0	0
$961$	−29594.3	−0.993398
$962$	−5792.48	−0.194134
$963$	0	0
$964$	45854.1	1.53201
$965$	0	0
$966$	0	0
$967$	26514.2	0.881738	0.440869	−	0.897571i	$-0.354670\pi$
$967$	26514.2	0.881738	0.440869	+	0.897571i	$0.354670\pi$
$968$	40048.5	1.32976
$969$	0	0
$970$	0	0
$971$	−38864.6	−1.28447	−0.642237	−	0.766506i	$-0.721994\pi$
$971$	−38864.6	−1.28447	−0.642237	+	0.766506i	$0.721994\pi$
$972$	0	0
$973$	8508.13	0.280327
$974$	−44789.9	−1.47347
$975$	0	0
$976$	24788.4	0.812968
$977$	33774.5	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$977$	33774.5	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$978$	0	0
$979$	−217.803	−0.00711032
$980$	0	0
$981$	0	0
$982$	28084.8	0.912651
$983$	−23536.8	−0.763691	−0.381846	−	0.924226i	$-0.624711\pi$
$983$	−23536.8	−0.763691	−0.381846	+	0.924226i	$0.624711\pi$
$984$	0	0
$985$	0	0
$986$	−10750.7	−0.347234
$987$	0	0
$988$	−5390.31	−0.173572
$989$	32953.3	1.05951
$990$	0	0
$991$	29012.6	0.929984	0.464992	−	0.885315i	$-0.346057\pi$
$991$	29012.6	0.929984	0.464992	+	0.885315i	$0.346057\pi$
$992$	−1366.79	−0.0437457
$993$	0	0
$994$	−24549.1	−0.783350
$995$	0	0
$996$	0	0
$997$	14027.1	0.445579	0.222790	−	0.974867i	$-0.428484\pi$
$997$	14027.1	0.445579	0.222790	+	0.974867i	$0.428484\pi$
$998$	15390.0	0.488138
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bk.1.1		4
3.2	odd	2		525.4.a.u.1.4	yes	4
5.4	even	2		1575.4.a.bj.1.4		4
15.2	even	4		525.4.d.n.274.8		8
15.8	even	4		525.4.d.n.274.1		8
15.14	odd	2		525.4.a.t.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.t.1.1	✓	4	15.14	odd	2
525.4.a.u.1.4	yes	4	3.2	odd	2
525.4.d.n.274.1		8	15.8	even	4
525.4.d.n.274.8		8	15.2	even	4
1575.4.a.bj.1.4		4	5.4	even	2
1575.4.a.bk.1.1		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-4.75345 q^{2} +14.5953 q^{4} +7.00000 q^{7} -31.3504 q^{8} -7.31799 q^{11} -4.15422 q^{13} -33.2742 q^{14} +32.2604 q^{16} -53.5216 q^{17} +88.9019 q^{19} +34.7857 q^{22} -156.780 q^{23} +19.7469 q^{26} +102.167 q^{28} -42.2570 q^{29} -14.0248 q^{31} +97.4554 q^{32} +254.412 q^{34} -293.336 q^{37} -422.591 q^{38} +127.214 q^{41} -210.189 q^{43} -106.808 q^{44} +745.244 q^{46} +468.688 q^{47} +49.0000 q^{49} -60.6321 q^{52} -115.973 q^{53} -219.453 q^{56} +200.867 q^{58} +314.090 q^{59} +768.386 q^{61} +66.6662 q^{62} -721.332 q^{64} -717.081 q^{67} -781.164 q^{68} +737.783 q^{71} +477.618 q^{73} +1394.36 q^{74} +1297.55 q^{76} -51.2259 q^{77} -279.262 q^{79} -604.703 q^{82} -776.981 q^{83} +999.123 q^{86} +229.422 q^{88} +29.7626 q^{89} -29.0796 q^{91} -2288.24 q^{92} -2227.88 q^{94} -231.793 q^{97} -232.919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 28 q^{7} - 9 q^{8} - 21 q^{11} + 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 221 q^{22} - 102 q^{23} - 129 q^{26} + 112 q^{28} + 240 q^{29} + 351 q^{31} - 72 q^{32} - 285 q^{34} + 399 q^{37}+ \cdots - 372 q^{97}+O(q^{100}) \) 4 * q + 16 * q^4 + 28 * q^7 - 9 * q^8 - 21 * q^11 + 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 221 * q^22 - 102 * q^23 - 129 * q^26 + 112 * q^28 + 240 * q^29 + 351 * q^31 - 72 * q^32 - 285 * q^34 + 399 * q^37 - 324 * q^38 - 381 * q^41 + 460 * q^43 + 975 * q^44 + 550 * q^46 - 60 * q^47 + 196 * q^49 + 223 * q^52 - 873 * q^53 - 63 * q^56 + 1408 * q^58 + 855 * q^59 + 687 * q^61 + 477 * q^62 - 1285 * q^64 - 503 * q^67 - 1725 * q^68 + 681 * q^71 + 1228 * q^73 + 3369 * q^74 + 1910 * q^76 - 147 * q^77 + 345 * q^79 - 495 * q^82 - 1509 * q^83 + 540 * q^86 + 2266 * q^88 + 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1732 * q^94 - 372 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more