LMFDB - Embedded newform 1575.4.a.bo.1.5

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:	$5$
Coefficient field:	5.5.78066700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 $ x^5 - x^4 - 18x^3 + 7x^2 + 30*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.37042$ 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)

$f(q)$	$=$	$q+4.88936 q^{2} +15.9059 q^{4} +7.00000 q^{7} +38.6546 q^{8} +O(q^{10})$	$q+4.88936 q^{2} +15.9059 q^{4} +7.00000 q^{7} +38.6546 q^{8} -54.9009 q^{11} -49.7580 q^{13} +34.2255 q^{14} +61.7496 q^{16} -133.661 q^{17} +138.986 q^{19} -268.430 q^{22} -7.32751 q^{23} -243.285 q^{26} +111.341 q^{28} -87.2408 q^{29} -209.479 q^{31} -7.32107 q^{32} -653.519 q^{34} -67.9041 q^{37} +679.555 q^{38} -77.6804 q^{41} -197.692 q^{43} -873.246 q^{44} -35.8269 q^{46} -4.97613 q^{47} +49.0000 q^{49} -791.444 q^{52} +53.0843 q^{53} +270.582 q^{56} -426.552 q^{58} +683.950 q^{59} -26.8658 q^{61} -1024.22 q^{62} -529.792 q^{64} +149.300 q^{67} -2126.00 q^{68} -6.15571 q^{71} +294.545 q^{73} -332.008 q^{74} +2210.70 q^{76} -384.306 q^{77} -938.669 q^{79} -379.807 q^{82} -784.907 q^{83} -966.590 q^{86} -2122.17 q^{88} -275.928 q^{89} -348.306 q^{91} -116.550 q^{92} -24.3301 q^{94} +1165.27 q^{97} +239.579 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) $ 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8	$ 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8} - 66 q^{11} - 2 q^{13} - 7 q^{14} + 155 q^{16} - 108 q^{17} + 174 q^{19} - 506 q^{22} + 116 q^{23} - 446 q^{26} + 189 q^{28} - 370 q^{29} + 342 q^{31} + 55 q^{32} + 112 q^{34} - 408 q^{37} + 34 q^{38} - 802 q^{41} - 584 q^{43} - 290 q^{44} + 640 q^{46} - 716 q^{47} + 245 q^{49} + 338 q^{52} - 98 q^{53} - 231 q^{56} + 482 q^{58} - 704 q^{59} + 650 q^{61} - 2070 q^{62} + 75 q^{64} + 180 q^{67} - 4520 q^{68} - 1470 q^{71} - 534 q^{73} + 1312 q^{74} + 4370 q^{76} - 462 q^{77} - 820 q^{79} - 1338 q^{82} - 1520 q^{83} - 832 q^{86} - 3258 q^{88} - 286 q^{89} - 14 q^{91} - 1288 q^{92} + 2540 q^{94} + 278 q^{97} - 49 q^{98}+O(q^{100}) $ 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8 - 66 * q^11 - 2 * q^13 - 7 * q^14 + 155 * q^16 - 108 * q^17 + 174 * q^19 - 506 * q^22 + 116 * q^23 - 446 * q^26 + 189 * q^28 - 370 * q^29 + 342 * q^31 + 55 * q^32 + 112 * q^34 - 408 * q^37 + 34 * q^38 - 802 * q^41 - 584 * q^43 - 290 * q^44 + 640 * q^46 - 716 * q^47 + 245 * q^49 + 338 * q^52 - 98 * q^53 - 231 * q^56 + 482 * q^58 - 704 * q^59 + 650 * q^61 - 2070 * q^62 + 75 * q^64 + 180 * q^67 - 4520 * q^68 - 1470 * q^71 - 534 * q^73 + 1312 * q^74 + 4370 * q^76 - 462 * q^77 - 820 * q^79 - 1338 * q^82 - 1520 * q^83 - 832 * q^86 - 3258 * q^88 - 286 * q^89 - 14 * q^91 - 1288 * q^92 + 2540 * q^94 + 278 * 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 $

$2$	4.88936	1.72865	0.864325	−	0.502933i	$-0.167746\pi$
$2$	4.88936	1.72865	0.864325	+	0.502933i	$0.167746\pi$
$3$	0	0
$3$	0	0
$4$	15.9059	1.98823
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	38.6546	1.70831
$9$	0	0
$10$	0	0
$11$	−54.9009	−1.50484	−0.752420	−	0.658684i	$-0.771113\pi$
$11$	−54.9009	−1.50484	−0.752420	+	0.658684i	$0.771113\pi$
$12$	0	0
$13$	−49.7580	−1.06157	−0.530784	−	0.847507i	$-0.678103\pi$
$13$	−49.7580	−1.06157	−0.530784	+	0.847507i	$0.678103\pi$
$14$	34.2255	0.653368
$15$	0	0
$16$	61.7496	0.964837
$17$	−133.661	−1.90692	−0.953461	−	0.301517i	$-0.902507\pi$
$17$	−133.661	−1.90692	−0.953461	+	0.301517i	$0.902507\pi$
$18$	0	0
$19$	138.986	1.67819	0.839097	−	0.543983i	$-0.183084\pi$
$19$	138.986	1.67819	0.839097	+	0.543983i	$0.183084\pi$
$20$	0	0
$21$	0	0
$22$	−268.430	−2.60134
$23$	−7.32751	−0.0664301	−0.0332150	−	0.999448i	$-0.510575\pi$
$23$	−7.32751	−0.0664301	−0.0332150	+	0.999448i	$0.510575\pi$
$24$	0	0
$25$	0	0
$26$	−243.285	−1.83508
$27$	0	0
$28$	111.341	0.751481
$29$	−87.2408	−0.558628	−0.279314	−	0.960200i	$-0.590107\pi$
$29$	−87.2408	−0.558628	−0.279314	+	0.960200i	$0.590107\pi$
$30$	0	0
$31$	−209.479	−1.21366	−0.606831	−	0.794831i	$-0.707560\pi$
$31$	−209.479	−1.21366	−0.606831	+	0.794831i	$0.707560\pi$
$32$	−7.32107	−0.0404436
$33$	0	0
$34$	−653.519	−3.29640
$35$	0	0
$36$	0	0
$37$	−67.9041	−0.301713	−0.150856	−	0.988556i	$-0.548203\pi$
$37$	−67.9041	−0.301713	−0.150856	+	0.988556i	$0.548203\pi$
$38$	679.555	2.90101
$39$	0	0
$40$	0	0
$41$	−77.6804	−0.295894	−0.147947	−	0.988995i	$-0.547266\pi$
$41$	−77.6804	−0.295894	−0.147947	+	0.988995i	$0.547266\pi$
$42$	0	0
$43$	−197.692	−0.701112	−0.350556	−	0.936542i	$-0.614007\pi$
$43$	−197.692	−0.701112	−0.350556	+	0.936542i	$0.614007\pi$
$44$	−873.246	−2.99197
$45$	0	0
$46$	−35.8269	−0.114834
$47$	−4.97613	−0.0154435	−0.00772173	−	0.999970i	$-0.502458\pi$
$47$	−4.97613	−0.0154435	−0.00772173	+	0.999970i	$0.502458\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−791.444	−2.11065
$53$	53.0843	0.137579	0.0687895	−	0.997631i	$-0.478086\pi$
$53$	53.0843	0.137579	0.0687895	+	0.997631i	$0.478086\pi$
$54$	0	0
$55$	0	0
$56$	270.582	0.645680
$57$	0	0
$58$	−426.552	−0.965673
$59$	683.950	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$59$	683.950	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$60$	0	0
$61$	−26.8658	−0.0563904	−0.0281952	−	0.999602i	$-0.508976\pi$
$61$	−26.8658	−0.0563904	−0.0281952	+	0.999602i	$0.508976\pi$
$62$	−1024.22	−2.09800
$63$	0	0
$64$	−529.792	−1.03475
$65$	0	0
$66$	0	0
$67$	149.300	0.272237	0.136119	−	0.990693i	$-0.456537\pi$
$67$	149.300	0.272237	0.136119	+	0.990693i	$0.456537\pi$
$68$	−2126.00	−3.79140
$69$	0	0
$70$	0	0
$71$	−6.15571	−0.0102894	−0.00514470	−	0.999987i	$-0.501638\pi$
$71$	−6.15571	−0.0102894	−0.00514470	+	0.999987i	$0.501638\pi$
$72$	0	0
$73$	294.545	0.472245	0.236123	−	0.971723i	$-0.424123\pi$
$73$	294.545	0.472245	0.236123	+	0.971723i	$0.424123\pi$
$74$	−332.008	−0.521556
$75$	0	0
$76$	2210.70	3.33664
$77$	−384.306	−0.568776
$78$	0	0
$79$	−938.669	−1.33682	−0.668409	−	0.743794i	$-0.733024\pi$
$79$	−938.669	−1.33682	−0.668409	+	0.743794i	$0.733024\pi$
$80$	0	0
$81$	0	0
$82$	−379.807	−0.511496
$83$	−784.907	−1.03801	−0.519004	−	0.854772i	$-0.673697\pi$
$83$	−784.907	−1.03801	−0.519004	+	0.854772i	$0.673697\pi$
$84$	0	0
$85$	0	0
$86$	−966.590	−1.21198
$87$	0	0
$88$	−2122.17	−2.57073
$89$	−275.928	−0.328633	−0.164316	−	0.986408i	$-0.552542\pi$
$89$	−275.928	−0.328633	−0.164316	+	0.986408i	$0.552542\pi$
$90$	0	0
$91$	−348.306	−0.401235
$92$	−116.550	−0.132078
$93$	0	0
$94$	−24.3301	−0.0266963
$95$	0	0
$96$	0	0
$97$	1165.27	1.21975	0.609873	−	0.792499i	$-0.291220\pi$
$97$	1165.27	1.21975	0.609873	+	0.792499i	$0.291220\pi$
$98$	239.579	0.246950
$99$	0	0
$100$	0	0
$101$	0.162999	0.000160584 0	8.02919e−5	−	1.00000i	$-0.499974\pi$
$101$	0.162999	0.000160584 0	8.02919e−5		1.00000i	$0.499974\pi$
$102$	0	0
$103$	1300.78	1.24437	0.622184	−	0.782871i	$-0.286245\pi$
$103$	1300.78	1.24437	0.622184	+	0.782871i	$0.286245\pi$
$104$	−1923.38	−1.81349
$105$	0	0
$106$	259.548	0.237826
$107$	−1531.63	−1.38382	−0.691908	−	0.721985i	$-0.743230\pi$
$107$	−1531.63	−1.38382	−0.691908	+	0.721985i	$0.743230\pi$
$108$	0	0
$109$	1971.39	1.73234	0.866171	−	0.499748i	$-0.166574\pi$
$109$	1971.39	1.73234	0.866171	+	0.499748i	$0.166574\pi$
$110$	0	0
$111$	0	0
$112$	432.247	0.364674
$113$	262.817	0.218794	0.109397	−	0.993998i	$-0.465108\pi$
$113$	262.817	0.218794	0.109397	+	0.993998i	$0.465108\pi$
$114$	0	0
$115$	0	0
$116$	−1387.64	−1.11068
$117$	0	0
$118$	3344.08	2.60888
$119$	−935.630	−0.720749
$120$	0	0
$121$	1683.11	1.26454
$122$	−131.357	−0.0974793
$123$	0	0
$124$	−3331.94	−2.41304
$125$	0	0
$126$	0	0
$127$	569.649	0.398017	0.199009	−	0.979998i	$-0.436228\pi$
$127$	569.649	0.398017	0.199009	+	0.979998i	$0.436228\pi$
$128$	−2531.78	−1.74828
$129$	0	0
$130$	0	0
$131$	−984.808	−0.656817	−0.328409	−	0.944536i	$-0.606512\pi$
$131$	−984.808	−0.656817	−0.328409	+	0.944536i	$0.606512\pi$
$132$	0	0
$133$	972.905	0.634297
$134$	729.982	0.470603
$135$	0	0
$136$	−5166.63	−3.25761
$137$	1377.35	0.858942	0.429471	−	0.903081i	$-0.358700\pi$
$137$	1377.35	0.858942	0.429471	+	0.903081i	$0.358700\pi$
$138$	0	0
$139$	839.673	0.512375	0.256188	−	0.966627i	$-0.417534\pi$
$139$	839.673	0.512375	0.256188	+	0.966627i	$0.417534\pi$
$140$	0	0
$141$	0	0
$142$	−30.0975	−0.0177868
$143$	2731.76	1.59749
$144$	0	0
$145$	0	0
$146$	1440.14	0.816347
$147$	0	0
$148$	−1080.07	−0.599875
$149$	1470.33	0.808416	0.404208	−	0.914667i	$-0.367547\pi$
$149$	1470.33	0.808416	0.404208	+	0.914667i	$0.367547\pi$
$150$	0	0
$151$	−1695.79	−0.913916	−0.456958	−	0.889488i	$-0.651061\pi$
$151$	−1695.79	−0.913916	−0.456958	+	0.889488i	$0.651061\pi$
$152$	5372.47	2.86687
$153$	0	0
$154$	−1879.01	−0.983215
$155$	0	0
$156$	0	0
$157$	959.433	0.487714	0.243857	−	0.969811i	$-0.421587\pi$
$157$	959.433	0.487714	0.243857	+	0.969811i	$0.421587\pi$
$158$	−4589.49	−2.31089
$159$	0	0
$160$	0	0
$161$	−51.2926	−0.0251082
$162$	0	0
$163$	−3865.27	−1.85737	−0.928685	−	0.370869i	$-0.879060\pi$
$163$	−3865.27	−1.85737	−0.928685	+	0.370869i	$0.879060\pi$
$164$	−1235.57	−0.588305
$165$	0	0
$166$	−3837.69	−1.79435
$167$	465.084	0.215505	0.107752	−	0.994178i	$-0.465635\pi$
$167$	465.084	0.215505	0.107752	+	0.994178i	$0.465635\pi$
$168$	0	0
$169$	278.860	0.126927
$170$	0	0
$171$	0	0
$172$	−3144.47	−1.39397
$173$	−2048.04	−0.900057	−0.450028	−	0.893014i	$-0.648586\pi$
$173$	−2048.04	−0.900057	−0.450028	+	0.893014i	$0.648586\pi$
$174$	0	0
$175$	0	0
$176$	−3390.10	−1.45192
$177$	0	0
$178$	−1349.11	−0.568091
$179$	911.856	0.380756	0.190378	−	0.981711i	$-0.439029\pi$
$179$	911.856	0.380756	0.190378	+	0.981711i	$0.439029\pi$
$180$	0	0
$181$	−2029.81	−0.833562	−0.416781	−	0.909007i	$-0.636842\pi$
$181$	−2029.81	−0.833562	−0.416781	+	0.909007i	$0.636842\pi$
$182$	−1702.99	−0.693595
$183$	0	0
$184$	−283.242	−0.113483
$185$	0	0
$186$	0	0
$187$	7338.13	2.86961
$188$	−79.1496	−0.0307052
$189$	0	0
$190$	0	0
$191$	−4984.11	−1.88815	−0.944077	−	0.329725i	$-0.893044\pi$
$191$	−4984.11	−1.88815	−0.944077	+	0.329725i	$0.893044\pi$
$192$	0	0
$193$	−3393.44	−1.26562	−0.632811	−	0.774306i	$-0.718099\pi$
$193$	−3393.44	−1.26562	−0.632811	+	0.774306i	$0.718099\pi$
$194$	5697.43	2.10852
$195$	0	0
$196$	779.387	0.284033
$197$	−762.475	−0.275757	−0.137878	−	0.990449i	$-0.544028\pi$
$197$	−762.475	−0.275757	−0.137878	+	0.990449i	$0.544028\pi$
$198$	0	0
$199$	−1272.32	−0.453227	−0.226613	−	0.973985i	$-0.572765\pi$
$199$	−1272.32	−0.453227	−0.226613	+	0.973985i	$0.572765\pi$
$200$	0	0
$201$	0	0
$202$	0.796959	0.000277593 0
$203$	−610.686	−0.211142
$204$	0	0
$205$	0	0
$206$	6360.00	2.15108
$207$	0	0
$208$	−3072.53	−1.02424
$209$	−7630.47	−2.52541
$210$	0	0
$211$	2010.72	0.656035	0.328017	−	0.944672i	$-0.393620\pi$
$211$	2010.72	0.656035	0.328017	+	0.944672i	$0.393620\pi$
$212$	844.351	0.273539
$213$	0	0
$214$	−7488.70	−2.39214
$215$	0	0
$216$	0	0
$217$	−1466.35	−0.458721
$218$	9638.85	2.99461
$219$	0	0
$220$	0	0
$221$	6650.73	2.02433
$222$	0	0
$223$	1516.44	0.455374	0.227687	−	0.973734i	$-0.426884\pi$
$223$	1516.44	0.455374	0.227687	+	0.973734i	$0.426884\pi$
$224$	−51.2475	−0.0152862
$225$	0	0
$226$	1285.01	0.378219
$227$	−4101.17	−1.19914	−0.599568	−	0.800323i	$-0.704661\pi$
$227$	−4101.17	−1.19914	−0.599568	+	0.800323i	$0.704661\pi$
$228$	0	0
$229$	−1029.24	−0.297004	−0.148502	−	0.988912i	$-0.547445\pi$
$229$	−1029.24	−0.297004	−0.148502	+	0.988912i	$0.547445\pi$
$230$	0	0
$231$	0	0
$232$	−3372.26	−0.954309
$233$	−5578.41	−1.56847	−0.784236	−	0.620463i	$-0.786945\pi$
$233$	−5578.41	−1.56847	−0.784236	+	0.620463i	$0.786945\pi$
$234$	0	0
$235$	0	0
$236$	10878.8	3.00064
$237$	0	0
$238$	−4574.64	−1.24592
$239$	−5389.67	−1.45870	−0.729348	−	0.684142i	$-0.760177\pi$
$239$	−5389.67	−1.45870	−0.729348	+	0.684142i	$0.760177\pi$
$240$	0	0
$241$	3976.27	1.06280	0.531399	−	0.847122i	$-0.321667\pi$
$241$	3976.27	1.06280	0.531399	+	0.847122i	$0.321667\pi$
$242$	8229.31	2.18595
$243$	0	0
$244$	−427.324	−0.112117
$245$	0	0
$246$	0	0
$247$	−6915.69	−1.78152
$248$	−8097.33	−2.07331
$249$	0	0
$250$	0	0
$251$	−7095.76	−1.78438	−0.892192	−	0.451655i	$-0.850834\pi$
$251$	−7095.76	−1.78438	−0.892192	+	0.451655i	$0.850834\pi$
$252$	0	0
$253$	402.287	0.0999666
$254$	2785.22	0.688033
$255$	0	0
$256$	−8140.43	−1.98741
$257$	2526.56	0.613239	0.306619	−	0.951832i	$-0.400802\pi$
$257$	2526.56	0.613239	0.306619	+	0.951832i	$0.400802\pi$
$258$	0	0
$259$	−475.329	−0.114037
$260$	0	0
$261$	0	0
$262$	−4815.08	−1.13541
$263$	3842.34	0.900871	0.450435	−	0.892809i	$-0.351269\pi$
$263$	3842.34	0.900871	0.450435	+	0.892809i	$0.351269\pi$
$264$	0	0
$265$	0	0
$266$	4756.88	1.09648
$267$	0	0
$268$	2374.75	0.541271
$269$	8411.04	1.90643	0.953216	−	0.302291i	$-0.0977514\pi$
$269$	8411.04	1.90643	0.953216	+	0.302291i	$0.0977514\pi$
$270$	0	0
$271$	5659.73	1.26865	0.634325	−	0.773067i	$-0.281278\pi$
$271$	5659.73	1.26865	0.634325	+	0.773067i	$0.281278\pi$
$272$	−8253.54	−1.83987
$273$	0	0
$274$	6734.37	1.48481
$275$	0	0
$276$	0	0
$277$	−881.409	−0.191187	−0.0955934	−	0.995420i	$-0.530475\pi$
$277$	−881.409	−0.191187	−0.0955934	+	0.995420i	$0.530475\pi$
$278$	4105.47	0.885718
$279$	0	0
$280$	0	0
$281$	3853.71	0.818124	0.409062	−	0.912506i	$-0.365856\pi$
$281$	3853.71	0.818124	0.409062	+	0.912506i	$0.365856\pi$
$282$	0	0
$283$	−1891.60	−0.397328	−0.198664	−	0.980068i	$-0.563660\pi$
$283$	−1891.60	−0.397328	−0.198664	+	0.980068i	$0.563660\pi$
$284$	−97.9118	−0.0204577
$285$	0	0
$286$	13356.6	2.76150
$287$	−543.763	−0.111837
$288$	0	0
$289$	12952.4	2.63635
$290$	0	0
$291$	0	0
$292$	4684.99	0.938933
$293$	4076.18	0.812742	0.406371	−	0.913708i	$-0.366794\pi$
$293$	4076.18	0.812742	0.406371	+	0.913708i	$0.366794\pi$
$294$	0	0
$295$	0	0
$296$	−2624.81	−0.515419
$297$	0	0
$298$	7188.97	1.39747
$299$	364.602	0.0705201
$300$	0	0
$301$	−1383.85	−0.264995
$302$	−8291.33	−1.57984
$303$	0	0
$304$	8582.35	1.61918
$305$	0	0
$306$	0	0
$307$	6201.87	1.15296	0.576481	−	0.817110i	$-0.304425\pi$
$307$	6201.87	1.15296	0.576481	+	0.817110i	$0.304425\pi$
$308$	−6112.72	−1.13086
$309$	0	0
$310$	0	0
$311$	6601.95	1.20374	0.601868	−	0.798595i	$-0.294423\pi$
$311$	6601.95	1.20374	0.601868	+	0.798595i	$0.294423\pi$
$312$	0	0
$313$	2291.01	0.413724	0.206862	−	0.978370i	$-0.433675\pi$
$313$	2291.01	0.413724	0.206862	+	0.978370i	$0.433675\pi$
$314$	4691.02	0.843087
$315$	0	0
$316$	−14930.3	−2.65790
$317$	−3124.07	−0.553517	−0.276759	−	0.960939i	$-0.589260\pi$
$317$	−3124.07	−0.553517	−0.276759	+	0.960939i	$0.589260\pi$
$318$	0	0
$319$	4789.60	0.840646
$320$	0	0
$321$	0	0
$322$	−250.788	−0.0434033
$323$	−18577.1	−3.20018
$324$	0	0
$325$	0	0
$326$	−18898.7	−3.21074
$327$	0	0
$328$	−3002.70	−0.505478
$329$	−34.8329	−0.00583708
$330$	0	0
$331$	9825.49	1.63159	0.815797	−	0.578338i	$-0.196298\pi$
$331$	9825.49	1.63159	0.815797	+	0.578338i	$0.196298\pi$
$332$	−12484.6	−2.06380
$333$	0	0
$334$	2273.96	0.372532
$335$	0	0
$336$	0	0
$337$	−8526.35	−1.37822	−0.689110	−	0.724657i	$-0.741998\pi$
$337$	−8526.35	−1.37822	−0.689110	+	0.724657i	$0.741998\pi$
$338$	1363.45	0.219413
$339$	0	0
$340$	0	0
$341$	11500.6	1.82637
$342$	0	0
$343$	343.000	0.0539949
$344$	−7641.73	−1.19772
$345$	0	0
$346$	−10013.6	−1.55588
$347$	−4164.54	−0.644277	−0.322138	−	0.946693i	$-0.604402\pi$
$347$	−4164.54	−0.644277	−0.322138	+	0.946693i	$0.604402\pi$
$348$	0	0
$349$	2584.60	0.396420	0.198210	−	0.980160i	$-0.436487\pi$
$349$	2584.60	0.396420	0.198210	+	0.980160i	$0.436487\pi$
$350$	0	0
$351$	0	0
$352$	401.933	0.0608611
$353$	−4199.25	−0.633154	−0.316577	−	0.948567i	$-0.602534\pi$
$353$	−4199.25	−0.633154	−0.316577	+	0.948567i	$0.602534\pi$
$354$	0	0
$355$	0	0
$356$	−4388.87	−0.653398
$357$	0	0
$358$	4458.39	0.658194
$359$	990.277	0.145584	0.0727922	−	0.997347i	$-0.476809\pi$
$359$	990.277	0.145584	0.0727922	+	0.997347i	$0.476809\pi$
$360$	0	0
$361$	12458.2	1.81633
$362$	−9924.48	−1.44094
$363$	0	0
$364$	−5540.11	−0.797749
$365$	0	0
$366$	0	0
$367$	4179.24	0.594427	0.297213	−	0.954811i	$-0.403943\pi$
$367$	4179.24	0.594427	0.297213	+	0.954811i	$0.403943\pi$
$368$	−452.471	−0.0640942
$369$	0	0
$370$	0	0
$371$	371.590	0.0519999
$372$	0	0
$373$	−7365.36	−1.02242	−0.511212	−	0.859455i	$-0.670803\pi$
$373$	−7365.36	−1.02242	−0.511212	+	0.859455i	$0.670803\pi$
$374$	35878.8	4.96056
$375$	0	0
$376$	−192.350	−0.0263822
$377$	4340.93	0.593022
$378$	0	0
$379$	−6214.13	−0.842213	−0.421106	−	0.907011i	$-0.638358\pi$
$379$	−6214.13	−0.842213	−0.421106	+	0.907011i	$0.638358\pi$
$380$	0	0
$381$	0	0
$382$	−24369.1	−3.26396
$383$	1755.04	0.234148	0.117074	−	0.993123i	$-0.462649\pi$
$383$	1755.04	0.234148	0.117074	+	0.993123i	$0.462649\pi$
$384$	0	0
$385$	0	0
$386$	−16591.7	−2.18782
$387$	0	0
$388$	18534.6	2.42514
$389$	−8805.69	−1.14773	−0.573864	−	0.818950i	$-0.694556\pi$
$389$	−8805.69	−1.14773	−0.573864	+	0.818950i	$0.694556\pi$
$390$	0	0
$391$	979.406	0.126677
$392$	1894.08	0.244044
$393$	0	0
$394$	−3728.01	−0.476687
$395$	0	0
$396$	0	0
$397$	−13717.2	−1.73413	−0.867063	−	0.498199i	$-0.833995\pi$
$397$	−13717.2	−1.73413	−0.867063	+	0.498199i	$0.833995\pi$
$398$	−6220.82	−0.783471
$399$	0	0
$400$	0	0
$401$	−307.220	−0.0382590	−0.0191295	−	0.999817i	$-0.506089\pi$
$401$	−307.220	−0.0382590	−0.0191295	+	0.999817i	$0.506089\pi$
$402$	0	0
$403$	10423.3	1.28839
$404$	2.59263	0.000319278 0
$405$	0	0
$406$	−2985.86	−0.364990
$407$	3728.00	0.454029
$408$	0	0
$409$	12390.7	1.49799	0.748997	−	0.662573i	$-0.230536\pi$
$409$	12390.7	1.49799	0.748997	+	0.662573i	$0.230536\pi$
$410$	0	0
$411$	0	0
$412$	20690.1	2.47409
$413$	4787.65	0.570423
$414$	0	0
$415$	0	0
$416$	364.282	0.0429336
$417$	0	0
$418$	−37308.2	−4.36555
$419$	−2664.10	−0.310620	−0.155310	−	0.987866i	$-0.549638\pi$
$419$	−2664.10	−0.310620	−0.155310	+	0.987866i	$0.549638\pi$
$420$	0	0
$421$	−6851.11	−0.793118	−0.396559	−	0.918009i	$-0.629796\pi$
$421$	−6851.11	−0.793118	−0.396559	+	0.918009i	$0.629796\pi$
$422$	9831.12	1.13406
$423$	0	0
$424$	2051.95	0.235027
$425$	0	0
$426$	0	0
$427$	−188.061	−0.0213136
$428$	−24361.9	−2.75135
$429$	0	0
$430$	0	0
$431$	7651.38	0.855114	0.427557	−	0.903988i	$-0.359374\pi$
$431$	7651.38	0.855114	0.427557	+	0.903988i	$0.359374\pi$
$432$	0	0
$433$	691.572	0.0767548	0.0383774	−	0.999263i	$-0.487781\pi$
$433$	691.572	0.0767548	0.0383774	+	0.999263i	$0.487781\pi$
$434$	−7169.53	−0.792969
$435$	0	0
$436$	31356.7	3.44430
$437$	−1018.42	−0.111483
$438$	0	0
$439$	10621.7	1.15478	0.577389	−	0.816469i	$-0.304072\pi$
$439$	10621.7	1.15478	0.577389	+	0.816469i	$0.304072\pi$
$440$	0	0
$441$	0	0
$442$	32517.8	3.49936
$443$	3133.67	0.336084	0.168042	−	0.985780i	$-0.446256\pi$
$443$	3133.67	0.336084	0.168042	+	0.985780i	$0.446256\pi$
$444$	0	0
$445$	0	0
$446$	7414.43	0.787182
$447$	0	0
$448$	−3708.54	−0.391099
$449$	9144.92	0.961192	0.480596	−	0.876942i	$-0.340420\pi$
$449$	9144.92	0.961192	0.480596	+	0.876942i	$0.340420\pi$
$450$	0	0
$451$	4264.72	0.445272
$452$	4180.33	0.435014
$453$	0	0
$454$	−20052.1	−2.07289
$455$	0	0
$456$	0	0
$457$	−8142.60	−0.833468	−0.416734	−	0.909029i	$-0.636825\pi$
$457$	−8142.60	−0.833468	−0.416734	+	0.909029i	$0.636825\pi$
$458$	−5032.32	−0.513417
$459$	0	0
$460$	0	0
$461$	9287.29	0.938291	0.469145	−	0.883121i	$-0.344562\pi$
$461$	9287.29	0.938291	0.469145	+	0.883121i	$0.344562\pi$
$462$	0	0
$463$	−2440.53	−0.244970	−0.122485	−	0.992470i	$-0.539086\pi$
$463$	−2440.53	−0.244970	−0.122485	+	0.992470i	$0.539086\pi$
$464$	−5387.08	−0.538985
$465$	0	0
$466$	−27274.9	−2.71134
$467$	−12066.3	−1.19564	−0.597818	−	0.801632i	$-0.703966\pi$
$467$	−12066.3	−1.19564	−0.597818	+	0.801632i	$0.703966\pi$
$468$	0	0
$469$	1045.10	0.102896
$470$	0	0
$471$	0	0
$472$	26437.8	2.57818
$473$	10853.5	1.05506
$474$	0	0
$475$	0	0
$476$	−14882.0	−1.43302
$477$	0	0
$478$	−26352.0	−2.52158
$479$	−395.211	−0.0376987	−0.0188493	−	0.999822i	$-0.506000\pi$
$479$	−395.211	−0.0376987	−0.0188493	+	0.999822i	$0.506000\pi$
$480$	0	0
$481$	3378.77	0.320289
$482$	19441.4	1.83721
$483$	0	0
$484$	26771.2	2.51420
$485$	0	0
$486$	0	0
$487$	9609.06	0.894102	0.447051	−	0.894508i	$-0.352474\pi$
$487$	9609.06	0.894102	0.447051	+	0.894508i	$0.352474\pi$
$488$	−1038.49	−0.0963323
$489$	0	0
$490$	0	0
$491$	−10941.0	−1.00562	−0.502810	−	0.864397i	$-0.667701\pi$
$491$	−10941.0	−1.00562	−0.502810	+	0.864397i	$0.667701\pi$
$492$	0	0
$493$	11660.7	1.06526
$494$	−33813.3	−3.07962
$495$	0	0
$496$	−12935.2	−1.17099
$497$	−43.0899	−0.00388903
$498$	0	0
$499$	9269.90	0.831618	0.415809	−	0.909452i	$-0.363498\pi$
$499$	9269.90	0.831618	0.415809	+	0.909452i	$0.363498\pi$
$500$	0	0
$501$	0	0
$502$	−34693.8	−3.08458
$503$	15085.4	1.33723	0.668615	−	0.743609i	$-0.266888\pi$
$503$	15085.4	1.33723	0.668615	+	0.743609i	$0.266888\pi$
$504$	0	0
$505$	0	0
$506$	1966.93	0.172807
$507$	0	0
$508$	9060.76	0.791351
$509$	−8650.44	−0.753289	−0.376645	−	0.926358i	$-0.622922\pi$
$509$	−8650.44	−0.753289	−0.376645	+	0.926358i	$0.622922\pi$
$510$	0	0
$511$	2061.82	0.178492
$512$	−19547.3	−1.68726
$513$	0	0
$514$	12353.3	1.06008
$515$	0	0
$516$	0	0
$517$	273.194	0.0232399
$518$	−2324.06	−0.197130
$519$	0	0
$520$	0	0
$521$	−10661.6	−0.896535	−0.448268	−	0.893899i	$-0.647959\pi$
$521$	−10661.6	−0.896535	−0.448268	+	0.893899i	$0.647959\pi$
$522$	0	0
$523$	−3449.22	−0.288382	−0.144191	−	0.989550i	$-0.546058\pi$
$523$	−3449.22	−0.288382	−0.144191	+	0.989550i	$0.546058\pi$
$524$	−15664.2	−1.30591
$525$	0	0
$526$	18786.6	1.55729
$527$	27999.3	2.31436
$528$	0	0
$529$	−12113.3	−0.995587
$530$	0	0
$531$	0	0
$532$	15474.9	1.26113
$533$	3865.22	0.314111
$534$	0	0
$535$	0	0
$536$	5771.14	0.465066
$537$	0	0
$538$	41124.6	3.29555
$539$	−2690.14	−0.214977
$540$	0	0
$541$	13403.2	1.06516	0.532578	−	0.846381i	$-0.321223\pi$
$541$	13403.2	1.06516	0.532578	+	0.846381i	$0.321223\pi$
$542$	27672.5	2.19305
$543$	0	0
$544$	978.544	0.0771227
$545$	0	0
$546$	0	0
$547$	−3670.29	−0.286893	−0.143446	−	0.989658i	$-0.545818\pi$
$547$	−3670.29	−0.286893	−0.143446	+	0.989658i	$0.545818\pi$
$548$	21907.9	1.70778
$549$	0	0
$550$	0	0
$551$	−12125.3	−0.937486
$552$	0	0
$553$	−6570.69	−0.505269
$554$	−4309.53	−0.330495
$555$	0	0
$556$	13355.7	1.01872
$557$	−521.169	−0.0396456	−0.0198228	−	0.999804i	$-0.506310\pi$
$557$	−521.169	−0.0396456	−0.0198228	+	0.999804i	$0.506310\pi$
$558$	0	0
$559$	9836.78	0.744278
$560$	0	0
$561$	0	0
$562$	18842.2	1.41425
$563$	17970.2	1.34521	0.672606	−	0.740000i	$-0.265175\pi$
$563$	17970.2	1.34521	0.672606	+	0.740000i	$0.265175\pi$
$564$	0	0
$565$	0	0
$566$	−9248.71	−0.686842
$567$	0	0
$568$	−237.947	−0.0175775
$569$	21808.6	1.60679	0.803396	−	0.595445i	$-0.203024\pi$
$569$	21808.6	1.60679	0.803396	+	0.595445i	$0.203024\pi$
$570$	0	0
$571$	6604.75	0.484064	0.242032	−	0.970268i	$-0.422186\pi$
$571$	6604.75	0.484064	0.242032	+	0.970268i	$0.422186\pi$
$572$	43451.0	3.17618
$573$	0	0
$574$	−2658.65	−0.193327
$575$	0	0
$576$	0	0
$577$	−25886.9	−1.86774	−0.933868	−	0.357618i	$-0.883589\pi$
$577$	−25886.9	−1.86774	−0.933868	+	0.357618i	$0.883589\pi$
$578$	63328.9	4.55733
$579$	0	0
$580$	0	0
$581$	−5494.35	−0.392330
$582$	0	0
$583$	−2914.37	−0.207034
$584$	11385.5	0.806741
$585$	0	0
$586$	19929.9	1.40495
$587$	4949.88	0.348046	0.174023	−	0.984742i	$-0.444323\pi$
$587$	4949.88	0.348046	0.174023	+	0.984742i	$0.444323\pi$
$588$	0	0
$589$	−29114.7	−2.03676
$590$	0	0
$591$	0	0
$592$	−4193.05	−0.291104
$593$	−31.3988	−0.00217436	−0.00108718	−	0.999999i	$-0.500346\pi$
$593$	−31.3988	−0.00217436	−0.00108718	+	0.999999i	$0.500346\pi$
$594$	0	0
$595$	0	0
$596$	23386.8	1.60732
$597$	0	0
$598$	1782.67	0.121905
$599$	11870.4	0.809704	0.404852	−	0.914382i	$-0.367323\pi$
$599$	11870.4	0.809704	0.404852	+	0.914382i	$0.367323\pi$
$600$	0	0
$601$	−967.320	−0.0656536	−0.0328268	−	0.999461i	$-0.510451\pi$
$601$	−967.320	−0.0656536	−0.0328268	+	0.999461i	$0.510451\pi$
$602$	−6766.13	−0.458084
$603$	0	0
$604$	−26973.0	−1.81708
$605$	0	0
$606$	0	0
$607$	−9518.28	−0.636466	−0.318233	−	0.948013i	$-0.603089\pi$
$607$	−9518.28	−0.636466	−0.318233	+	0.948013i	$0.603089\pi$
$608$	−1017.53	−0.0678721
$609$	0	0
$610$	0	0
$611$	247.602	0.0163943
$612$	0	0
$613$	−3607.19	−0.237672	−0.118836	−	0.992914i	$-0.537916\pi$
$613$	−3607.19	−0.237672	−0.118836	+	0.992914i	$0.537916\pi$
$614$	30323.2	1.99307
$615$	0	0
$616$	−14855.2	−0.971645
$617$	−22473.2	−1.46635	−0.733173	−	0.680042i	$-0.761962\pi$
$617$	−22473.2	−1.46635	−0.733173	+	0.680042i	$0.761962\pi$
$618$	0	0
$619$	19200.1	1.24672	0.623358	−	0.781936i	$-0.285768\pi$
$619$	19200.1	1.24672	0.623358	+	0.781936i	$0.285768\pi$
$620$	0	0
$621$	0	0
$622$	32279.3	2.08084
$623$	−1931.50	−0.124211
$624$	0	0
$625$	0	0
$626$	11201.6	0.715184
$627$	0	0
$628$	15260.6	0.969689
$629$	9076.17	0.575343
$630$	0	0
$631$	−6980.64	−0.440404	−0.220202	−	0.975454i	$-0.570672\pi$
$631$	−6980.64	−0.440404	−0.220202	+	0.975454i	$0.570672\pi$
$632$	−36283.9	−2.28370
$633$	0	0
$634$	−15274.7	−0.956838
$635$	0	0
$636$	0	0
$637$	−2438.14	−0.151653
$638$	23418.1	1.45318
$639$	0	0
$640$	0	0
$641$	1083.82	0.0667837	0.0333919	−	0.999442i	$-0.489369\pi$
$641$	1083.82	0.0667837	0.0333919	+	0.999442i	$0.489369\pi$
$642$	0	0
$643$	4151.42	0.254613	0.127307	−	0.991863i	$-0.459367\pi$
$643$	4151.42	0.254613	0.127307	+	0.991863i	$0.459367\pi$
$644$	−815.853	−0.0499210
$645$	0	0
$646$	−90830.3	−5.53200
$647$	−3014.99	−0.183202	−0.0916008	−	0.995796i	$-0.529198\pi$
$647$	−3014.99	−0.183202	−0.0916008	+	0.995796i	$0.529198\pi$
$648$	0	0
$649$	−37549.4	−2.27110
$650$	0	0
$651$	0	0
$652$	−61480.5	−3.69288
$653$	−21130.7	−1.26632	−0.633162	−	0.774019i	$-0.718243\pi$
$653$	−21130.7	−1.26632	−0.633162	+	0.774019i	$0.718243\pi$
$654$	0	0
$655$	0	0
$656$	−4796.73	−0.285489
$657$	0	0
$658$	−170.311	−0.0100903
$659$	9961.24	0.588824	0.294412	−	0.955679i	$-0.404876\pi$
$659$	9961.24	0.588824	0.294412	+	0.955679i	$0.404876\pi$
$660$	0	0
$661$	1581.43	0.0930569	0.0465285	−	0.998917i	$-0.485184\pi$
$661$	1581.43	0.0930569	0.0465285	+	0.998917i	$0.485184\pi$
$662$	48040.4	2.82046
$663$	0	0
$664$	−30340.3	−1.77324
$665$	0	0
$666$	0	0
$667$	639.258	0.0371097
$668$	7397.56	0.428473
$669$	0	0
$670$	0	0
$671$	1474.96	0.0848586
$672$	0	0
$673$	−11101.6	−0.635865	−0.317932	−	0.948113i	$-0.602988\pi$
$673$	−11101.6	−0.635865	−0.317932	+	0.948113i	$0.602988\pi$
$674$	−41688.4	−2.38246
$675$	0	0
$676$	4435.50	0.252361
$677$	−18249.8	−1.03603	−0.518017	−	0.855370i	$-0.673330\pi$
$677$	−18249.8	−1.03603	−0.518017	+	0.855370i	$0.673330\pi$
$678$	0	0
$679$	8156.90	0.461021
$680$	0	0
$681$	0	0
$682$	56230.5	3.15715
$683$	−14144.2	−0.792403	−0.396202	−	0.918164i	$-0.629672\pi$
$683$	−14144.2	−0.792403	−0.396202	+	0.918164i	$0.629672\pi$
$684$	0	0
$685$	0	0
$686$	1677.05	0.0933384
$687$	0	0
$688$	−12207.4	−0.676458
$689$	−2641.37	−0.146049
$690$	0	0
$691$	19621.7	1.08024	0.540120	−	0.841588i	$-0.318379\pi$
$691$	19621.7	1.08024	0.540120	+	0.841588i	$0.318379\pi$
$692$	−32575.9	−1.78952
$693$	0	0
$694$	−20361.9	−1.11373
$695$	0	0
$696$	0	0
$697$	10382.9	0.564246
$698$	12637.1	0.685272
$699$	0	0
$700$	0	0
$701$	−21609.1	−1.16428	−0.582142	−	0.813087i	$-0.697785\pi$
$701$	−21609.1	−1.16428	−0.582142	+	0.813087i	$0.697785\pi$
$702$	0	0
$703$	−9437.75	−0.506332
$704$	29086.0	1.55713
$705$	0	0
$706$	−20531.6	−1.09450
$707$	1.14099	6.06950e−5 0
$708$	0	0
$709$	−12557.2	−0.665158	−0.332579	−	0.943075i	$-0.607919\pi$
$709$	−12557.2	−0.665158	−0.332579	+	0.943075i	$0.607919\pi$
$710$	0	0
$711$	0	0
$712$	−10665.9	−0.561406
$713$	1534.96	0.0806237
$714$	0	0
$715$	0	0
$716$	14503.9	0.757031
$717$	0	0
$718$	4841.82	0.251665
$719$	−36151.9	−1.87516	−0.937580	−	0.347771i	$-0.886939\pi$
$719$	−36151.9	−1.87516	−0.937580	+	0.347771i	$0.886939\pi$
$720$	0	0
$721$	9105.48	0.470327
$722$	60912.8	3.13980
$723$	0	0
$724$	−32285.9	−1.65731
$725$	0	0
$726$	0	0
$727$	−12007.2	−0.612550	−0.306275	−	0.951943i	$-0.599083\pi$
$727$	−12007.2	−0.612550	−0.306275	+	0.951943i	$0.599083\pi$
$728$	−13463.6	−0.685434
$729$	0	0
$730$	0	0
$731$	26423.9	1.33697
$732$	0	0
$733$	15920.2	0.802220	0.401110	−	0.916030i	$-0.368625\pi$
$733$	15920.2	0.802220	0.401110	+	0.916030i	$0.368625\pi$
$734$	20433.8	1.02756
$735$	0	0
$736$	53.6452	0.00268667
$737$	−8196.70	−0.409674
$738$	0	0
$739$	−26581.1	−1.32314	−0.661571	−	0.749882i	$-0.730110\pi$
$739$	−26581.1	−1.32314	−0.661571	+	0.749882i	$0.730110\pi$
$740$	0	0
$741$	0	0
$742$	1816.84	0.0898897
$743$	−6601.38	−0.325950	−0.162975	−	0.986630i	$-0.552109\pi$
$743$	−6601.38	−0.325950	−0.162975	+	0.986630i	$0.552109\pi$
$744$	0	0
$745$	0	0
$746$	−36011.9	−1.76741
$747$	0	0
$748$	116719.	5.70546
$749$	−10721.4	−0.523034
$750$	0	0
$751$	−29809.6	−1.44842	−0.724212	−	0.689578i	$-0.757796\pi$
$751$	−29809.6	−1.44842	−0.724212	+	0.689578i	$0.757796\pi$
$752$	−307.274	−0.0149004
$753$	0	0
$754$	21224.4	1.02513
$755$	0	0
$756$	0	0
$757$	8179.09	0.392700	0.196350	−	0.980534i	$-0.437091\pi$
$757$	8179.09	0.392700	0.196350	+	0.980534i	$0.437091\pi$
$758$	−30383.2	−1.45589
$759$	0	0
$760$	0	0
$761$	31616.8	1.50605	0.753027	−	0.657990i	$-0.228593\pi$
$761$	31616.8	1.50605	0.753027	+	0.657990i	$0.228593\pi$
$762$	0	0
$763$	13799.8	0.654763
$764$	−79276.5	−3.75409
$765$	0	0
$766$	8581.05	0.404760
$767$	−34032.0	−1.60212
$768$	0	0
$769$	24651.3	1.15598	0.577991	−	0.816043i	$-0.303837\pi$
$769$	24651.3	1.15598	0.577991	+	0.816043i	$0.303837\pi$
$770$	0	0
$771$	0	0
$772$	−53975.6	−2.51635
$773$	−7888.63	−0.367056	−0.183528	−	0.983015i	$-0.558752\pi$
$773$	−7888.63	−0.367056	−0.183528	+	0.983015i	$0.558752\pi$
$774$	0	0
$775$	0	0
$776$	45043.1	2.08370
$777$	0	0
$778$	−43054.2	−1.98402
$779$	−10796.5	−0.496566
$780$	0	0
$781$	337.954	0.0154839
$782$	4788.67	0.218980
$783$	0	0
$784$	3025.73	0.137834
$785$	0	0
$786$	0	0
$787$	8592.94	0.389206	0.194603	−	0.980882i	$-0.437658\pi$
$787$	8592.94	0.389206	0.194603	+	0.980882i	$0.437658\pi$
$788$	−12127.8	−0.548268
$789$	0	0
$790$	0	0
$791$	1839.72	0.0826964
$792$	0	0
$793$	1336.79	0.0598623
$794$	−67068.5	−2.99770
$795$	0	0
$796$	−20237.3	−0.901120
$797$	−37498.1	−1.66656	−0.833282	−	0.552848i	$-0.813541\pi$
$797$	−37498.1	−1.66656	−0.833282	+	0.552848i	$0.813541\pi$
$798$	0	0
$799$	665.116	0.0294495
$800$	0	0
$801$	0	0
$802$	−1502.11	−0.0661364
$803$	−16170.8	−0.710653
$804$	0	0
$805$	0	0
$806$	50963.1	2.22717
$807$	0	0
$808$	6.30065	0.000274327 0
$809$	−33834.0	−1.47039	−0.735193	−	0.677858i	$-0.762908\pi$
$809$	−33834.0	−1.47039	−0.735193	+	0.677858i	$0.762908\pi$
$810$	0	0
$811$	16234.0	0.702899	0.351450	−	0.936207i	$-0.385689\pi$
$811$	16234.0	0.702899	0.351450	+	0.936207i	$0.385689\pi$
$812$	−9713.48	−0.419799
$813$	0	0
$814$	18227.5	0.784858
$815$	0	0
$816$	0	0
$817$	−27476.6	−1.17660
$818$	60582.5	2.58951
$819$	0	0
$820$	0	0
$821$	26551.4	1.12868	0.564342	−	0.825541i	$-0.309130\pi$
$821$	26551.4	1.12868	0.564342	+	0.825541i	$0.309130\pi$
$822$	0	0
$823$	−651.954	−0.0276132	−0.0138066	−	0.999905i	$-0.504395\pi$
$823$	−651.954	−0.0276132	−0.0138066	+	0.999905i	$0.504395\pi$
$824$	50281.3	2.12577
$825$	0	0
$826$	23408.6	0.986063
$827$	13248.0	0.557046	0.278523	−	0.960430i	$-0.410155\pi$
$827$	13248.0	0.557046	0.278523	+	0.960430i	$0.410155\pi$
$828$	0	0
$829$	−36828.9	−1.54297	−0.771484	−	0.636248i	$-0.780485\pi$
$829$	−36828.9	−1.54297	−0.771484	+	0.636248i	$0.780485\pi$
$830$	0	0
$831$	0	0
$832$	26361.4	1.09846
$833$	−6549.41	−0.272417
$834$	0	0
$835$	0	0
$836$	−121369.	−5.02111
$837$	0	0
$838$	−13025.7	−0.536953
$839$	10641.5	0.437884	0.218942	−	0.975738i	$-0.429739\pi$
$839$	10641.5	0.437884	0.218942	+	0.975738i	$0.429739\pi$
$840$	0	0
$841$	−16778.0	−0.687935
$842$	−33497.6	−1.37102
$843$	0	0
$844$	31982.2	1.30435
$845$	0	0
$846$	0	0
$847$	11781.7	0.477952
$848$	3277.93	0.132741
$849$	0	0
$850$	0	0
$851$	497.568	0.0200428
$852$	0	0
$853$	−2819.13	−0.113160	−0.0565799	−	0.998398i	$-0.518020\pi$
$853$	−2819.13	−0.113160	−0.0565799	+	0.998398i	$0.518020\pi$
$854$	−919.497	−0.0368437
$855$	0	0
$856$	−59204.6	−2.36399
$857$	−38853.9	−1.54869	−0.774343	−	0.632766i	$-0.781919\pi$
$857$	−38853.9	−1.54869	−0.774343	+	0.632766i	$0.781919\pi$
$858$	0	0
$859$	8959.42	0.355869	0.177935	−	0.984042i	$-0.443058\pi$
$859$	8959.42	0.355869	0.177935	+	0.984042i	$0.443058\pi$
$860$	0	0
$861$	0	0
$862$	37410.4	1.47819
$863$	−32649.8	−1.28785	−0.643924	−	0.765090i	$-0.722695\pi$
$863$	−32649.8	−1.28785	−0.643924	+	0.765090i	$0.722695\pi$
$864$	0	0
$865$	0	0
$866$	3381.35	0.132682
$867$	0	0
$868$	−23323.6	−0.912045
$869$	51533.8	2.01170
$870$	0	0
$871$	−7428.87	−0.288999
$872$	76203.5	2.95937
$873$	0	0
$874$	−4979.45	−0.192714
$875$	0	0
$876$	0	0
$877$	20229.3	0.778899	0.389450	−	0.921048i	$-0.372665\pi$
$877$	20229.3	0.778899	0.389450	+	0.921048i	$0.372665\pi$
$878$	51933.5	1.99621
$879$	0	0
$880$	0	0
$881$	−4356.08	−0.166584	−0.0832918	−	0.996525i	$-0.526543\pi$
$881$	−4356.08	−0.166584	−0.0832918	+	0.996525i	$0.526543\pi$
$882$	0	0
$883$	−43349.9	−1.65214	−0.826070	−	0.563567i	$-0.809429\pi$
$883$	−43349.9	−1.65214	−0.826070	+	0.563567i	$0.809429\pi$
$884$	105786.	4.02484
$885$	0	0
$886$	15321.6	0.580971
$887$	−9754.67	−0.369256	−0.184628	−	0.982809i	$-0.559108\pi$
$887$	−9754.67	−0.369256	−0.184628	+	0.982809i	$0.559108\pi$
$888$	0	0
$889$	3987.54	0.150436
$890$	0	0
$891$	0	0
$892$	24120.3	0.905389
$893$	−691.614	−0.0259171
$894$	0	0
$895$	0	0
$896$	−17722.4	−0.660786
$897$	0	0
$898$	44712.8	1.66157
$899$	18275.1	0.677986
$900$	0	0
$901$	−7095.32	−0.262352
$902$	20851.8	0.769720
$903$	0	0
$904$	10159.1	0.373768
$905$	0	0
$906$	0	0
$907$	7155.16	0.261944	0.130972	−	0.991386i	$-0.458190\pi$
$907$	7155.16	0.261944	0.130972	+	0.991386i	$0.458190\pi$
$908$	−65232.6	−2.38416
$909$	0	0
$910$	0	0
$911$	47255.4	1.71860	0.859298	−	0.511475i	$-0.170901\pi$
$911$	47255.4	1.71860	0.859298	+	0.511475i	$0.170901\pi$
$912$	0	0
$913$	43092.1	1.56204
$914$	−39812.1	−1.44077
$915$	0	0
$916$	−16370.9	−0.590514
$917$	−6893.66	−0.248254
$918$	0	0
$919$	30942.1	1.11065	0.555324	−	0.831634i	$-0.312594\pi$
$919$	30942.1	1.11065	0.555324	+	0.831634i	$0.312594\pi$
$920$	0	0
$921$	0	0
$922$	45408.9	1.62198
$923$	306.296	0.0109229
$924$	0	0
$925$	0	0
$926$	−11932.6	−0.423468
$927$	0	0
$928$	638.696	0.0225929
$929$	−12147.4	−0.429004	−0.214502	−	0.976724i	$-0.568813\pi$
$929$	−12147.4	−0.429004	−0.214502	+	0.976724i	$0.568813\pi$
$930$	0	0
$931$	6810.33	0.239742
$932$	−88729.4	−3.11849
$933$	0	0
$934$	−58996.6	−2.06684
$935$	0	0
$936$	0	0
$937$	13388.6	0.466794	0.233397	−	0.972381i	$-0.425016\pi$
$937$	13388.6	0.466794	0.233397	+	0.972381i	$0.425016\pi$
$938$	5109.87	0.177871
$939$	0	0
$940$	0	0
$941$	8035.36	0.278369	0.139184	−	0.990266i	$-0.455552\pi$
$941$	8035.36	0.278369	0.139184	+	0.990266i	$0.455552\pi$
$942$	0	0
$943$	569.204	0.0196562
$944$	42233.6	1.45613
$945$	0	0
$946$	53066.6	1.82383
$947$	17647.7	0.605567	0.302784	−	0.953059i	$-0.402084\pi$
$947$	17647.7	0.605567	0.302784	+	0.953059i	$0.402084\pi$
$948$	0	0
$949$	−14656.0	−0.501321
$950$	0	0
$951$	0	0
$952$	−36166.4	−1.23126
$953$	35629.1	1.21106	0.605529	−	0.795823i	$-0.292961\pi$
$953$	35629.1	1.21106	0.605529	+	0.795823i	$0.292961\pi$
$954$	0	0
$955$	0	0
$956$	−85727.3	−2.90023
$957$	0	0
$958$	−1932.33	−0.0651679
$959$	9641.45	0.324649
$960$	0	0
$961$	14090.4	0.472977
$962$	16520.1	0.553667
$963$	0	0
$964$	63246.0	2.11309
$965$	0	0
$966$	0	0
$967$	−35954.1	−1.19566	−0.597831	−	0.801622i	$-0.703971\pi$
$967$	−35954.1	−1.19566	−0.597831	+	0.801622i	$0.703971\pi$
$968$	65059.8	2.16023
$969$	0	0
$970$	0	0
$971$	−59266.0	−1.95874	−0.979370	−	0.202074i	$-0.935232\pi$
$971$	−59266.0	−1.95874	−0.979370	+	0.202074i	$0.935232\pi$
$972$	0	0
$973$	5877.71	0.193660
$974$	46982.2	1.54559
$975$	0	0
$976$	−1658.95	−0.0544076
$977$	35518.9	1.16310	0.581551	−	0.813510i	$-0.302446\pi$
$977$	35518.9	1.16310	0.581551	+	0.813510i	$0.302446\pi$
$978$	0	0
$979$	15148.7	0.494540
$980$	0	0
$981$	0	0
$982$	−53494.4	−1.73837
$983$	−37683.6	−1.22271	−0.611353	−	0.791358i	$-0.709374\pi$
$983$	−37683.6	−1.22271	−0.611353	+	0.791358i	$0.709374\pi$
$984$	0	0
$985$	0	0
$986$	57013.6	1.84146
$987$	0	0
$988$	−110000.	−3.54207
$989$	1448.59	0.0465749
$990$	0	0
$991$	21371.1	0.685041	0.342521	−	0.939510i	$-0.388719\pi$
$991$	21371.1	0.685041	0.342521	+	0.939510i	$0.388719\pi$
$992$	1533.61	0.0490848
$993$	0	0
$994$	−210.682	−0.00672278
$995$	0	0
$996$	0	0
$997$	−42934.0	−1.36383	−0.681913	−	0.731434i	$-0.738852\pi$
$997$	−42934.0	−1.36383	−0.681913	+	0.731434i	$0.738852\pi$
$998$	45323.9	1.43758
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bo.1.5		5
3.2	odd	2		525.4.a.x.1.1		5
5.2	odd	4		315.4.d.b.64.9		10
5.3	odd	4		315.4.d.b.64.2		10
5.4	even	2		1575.4.a.bp.1.1		5
15.2	even	4		105.4.d.b.64.2	✓	10
15.8	even	4		105.4.d.b.64.9	yes	10
15.14	odd	2		525.4.a.w.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.b.64.2	✓	10	15.2	even	4
105.4.d.b.64.9	yes	10	15.8	even	4
315.4.d.b.64.2		10	5.3	odd	4
315.4.d.b.64.9		10	5.2	odd	4
525.4.a.w.1.5		5	15.14	odd	2
525.4.a.x.1.1		5	3.2	odd	2
1575.4.a.bo.1.5		5	1.1	even	1	trivial
1575.4.a.bp.1.1		5	5.4	even	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)

\(f(q)\)	\(=\)	\(q+4.88936 q^{2} +15.9059 q^{4} +7.00000 q^{7} +38.6546 q^{8} +O(q^{10})\)	\(q+4.88936 q^{2} +15.9059 q^{4} +7.00000 q^{7} +38.6546 q^{8} -54.9009 q^{11} -49.7580 q^{13} +34.2255 q^{14} +61.7496 q^{16} -133.661 q^{17} +138.986 q^{19} -268.430 q^{22} -7.32751 q^{23} -243.285 q^{26} +111.341 q^{28} -87.2408 q^{29} -209.479 q^{31} -7.32107 q^{32} -653.519 q^{34} -67.9041 q^{37} +679.555 q^{38} -77.6804 q^{41} -197.692 q^{43} -873.246 q^{44} -35.8269 q^{46} -4.97613 q^{47} +49.0000 q^{49} -791.444 q^{52} +53.0843 q^{53} +270.582 q^{56} -426.552 q^{58} +683.950 q^{59} -26.8658 q^{61} -1024.22 q^{62} -529.792 q^{64} +149.300 q^{67} -2126.00 q^{68} -6.15571 q^{71} +294.545 q^{73} -332.008 q^{74} +2210.70 q^{76} -384.306 q^{77} -938.669 q^{79} -379.807 q^{82} -784.907 q^{83} -966.590 q^{86} -2122.17 q^{88} -275.928 q^{89} -348.306 q^{91} -116.550 q^{92} -24.3301 q^{94} +1165.27 q^{97} +239.579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) \) 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8	\( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8} - 66 q^{11} - 2 q^{13} - 7 q^{14} + 155 q^{16} - 108 q^{17} + 174 q^{19} - 506 q^{22} + 116 q^{23} - 446 q^{26} + 189 q^{28} - 370 q^{29} + 342 q^{31} + 55 q^{32} + 112 q^{34} - 408 q^{37} + 34 q^{38} - 802 q^{41} - 584 q^{43} - 290 q^{44} + 640 q^{46} - 716 q^{47} + 245 q^{49} + 338 q^{52} - 98 q^{53} - 231 q^{56} + 482 q^{58} - 704 q^{59} + 650 q^{61} - 2070 q^{62} + 75 q^{64} + 180 q^{67} - 4520 q^{68} - 1470 q^{71} - 534 q^{73} + 1312 q^{74} + 4370 q^{76} - 462 q^{77} - 820 q^{79} - 1338 q^{82} - 1520 q^{83} - 832 q^{86} - 3258 q^{88} - 286 q^{89} - 14 q^{91} - 1288 q^{92} + 2540 q^{94} + 278 q^{97} - 49 q^{98}+O(q^{100}) \) 5 * q - q^2 + 27 * q^4 + 35 * q^7 - 33 * q^8 - 66 * q^11 - 2 * q^13 - 7 * q^14 + 155 * q^16 - 108 * q^17 + 174 * q^19 - 506 * q^22 + 116 * q^23 - 446 * q^26 + 189 * q^28 - 370 * q^29 + 342 * q^31 + 55 * q^32 + 112 * q^34 - 408 * q^37 + 34 * q^38 - 802 * q^41 - 584 * q^43 - 290 * q^44 + 640 * q^46 - 716 * q^47 + 245 * q^49 + 338 * q^52 - 98 * q^53 - 231 * q^56 + 482 * q^58 - 704 * q^59 + 650 * q^61 - 2070 * q^62 + 75 * q^64 + 180 * q^67 - 4520 * q^68 - 1470 * q^71 - 534 * q^73 + 1312 * q^74 + 4370 * q^76 - 462 * q^77 - 820 * q^79 - 1338 * q^82 - 1520 * q^83 - 832 * q^86 - 3258 * q^88 - 286 * q^89 - 14 * q^91 - 1288 * q^92 + 2540 * q^94 + 278 * q^97 - 49 * q^98

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