LMFDB - Embedded newform 1575.4.a.bo.1.3

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			1.3
Root			$0.329739$ 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)

$f(q)$	$=$	$q+0.428319 q^{2} -7.81654 q^{4} +7.00000 q^{7} -6.77452 q^{8} +O(q^{10})$	$q+0.428319 q^{2} -7.81654 q^{4} +7.00000 q^{7} -6.77452 q^{8} +27.4721 q^{11} -46.5524 q^{13} +2.99823 q^{14} +59.6307 q^{16} -5.20546 q^{17} -91.0007 q^{19} +11.7668 q^{22} +111.563 q^{23} -19.9393 q^{26} -54.7158 q^{28} -0.0763413 q^{29} +201.784 q^{31} +79.7371 q^{32} -2.22960 q^{34} -312.859 q^{37} -38.9773 q^{38} -102.432 q^{41} +257.280 q^{43} -214.737 q^{44} +47.7847 q^{46} +350.994 q^{47} +49.0000 q^{49} +363.879 q^{52} -196.260 q^{53} -47.4217 q^{56} -0.0326984 q^{58} -881.060 q^{59} +737.897 q^{61} +86.4280 q^{62} -442.893 q^{64} +365.021 q^{67} +40.6887 q^{68} -1112.53 q^{71} +261.995 q^{73} -134.004 q^{74} +711.311 q^{76} +192.305 q^{77} +273.829 q^{79} -43.8735 q^{82} +87.1353 q^{83} +110.198 q^{86} -186.110 q^{88} +1090.99 q^{89} -325.867 q^{91} -872.039 q^{92} +150.337 q^{94} +228.830 q^{97} +20.9876 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.428319	0.151434	0.0757168	−	0.997129i	$-0.475876\pi$
$2$	0.428319	0.151434	0.0757168	+	0.997129i	$0.475876\pi$
$3$	0	0
$3$	0	0
$4$	−7.81654	−0.977068
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−6.77452	−0.299394
$9$	0	0
$10$	0	0
$11$	27.4721	0.753013	0.376507	−	0.926414i	$-0.377125\pi$
$11$	27.4721	0.753013	0.376507	+	0.926414i	$0.377125\pi$
$12$	0	0
$13$	−46.5524	−0.993179	−0.496589	−	0.867986i	$-0.665414\pi$
$13$	−46.5524	−0.993179	−0.496589	+	0.867986i	$0.665414\pi$
$14$	2.99823	0.0572365
$15$	0	0
$16$	59.6307	0.931729
$17$	−5.20546	−0.0742653	−0.0371326	−	0.999310i	$-0.511822\pi$
$17$	−5.20546	−0.0742653	−0.0371326	+	0.999310i	$0.511822\pi$
$18$	0	0
$19$	−91.0007	−1.09879	−0.549395	−	0.835563i	$-0.685142\pi$
$19$	−91.0007	−1.09879	−0.549395	+	0.835563i	$0.685142\pi$
$20$	0	0
$21$	0	0
$22$	11.7668	0.114032
$23$	111.563	1.01142	0.505708	−	0.862705i	$-0.331231\pi$
$23$	111.563	1.01142	0.505708	+	0.862705i	$0.331231\pi$
$24$	0	0
$25$	0	0
$26$	−19.9393	−0.150401
$27$	0	0
$28$	−54.7158	−0.369297
$29$	−0.0763413	−0.000488835 0	−0.000244418	−	1.00000i	$-0.500078\pi$
$29$	−0.0763413	−0.000488835 0	−0.000244418		1.00000i	$0.500078\pi$
$30$	0	0
$31$	201.784	1.16908	0.584541	−	0.811364i	$-0.301275\pi$
$31$	201.784	1.16908	0.584541	+	0.811364i	$0.301275\pi$
$32$	79.7371	0.440490
$33$	0	0
$34$	−2.22960	−0.0112463
$35$	0	0
$36$	0	0
$37$	−312.859	−1.39010	−0.695051	−	0.718960i	$-0.744618\pi$
$37$	−312.859	−1.39010	−0.695051	+	0.718960i	$0.744618\pi$
$38$	−38.9773	−0.166394
$39$	0	0
$40$	0	0
$41$	−102.432	−0.390175	−0.195087	−	0.980786i	$-0.562499\pi$
$41$	−102.432	−0.390175	−0.195087	+	0.980786i	$0.562499\pi$
$42$	0	0
$43$	257.280	0.912439	0.456219	−	0.889867i	$-0.349203\pi$
$43$	257.280	0.912439	0.456219	+	0.889867i	$0.349203\pi$
$44$	−214.737	−0.735745
$45$	0	0
$46$	47.7847	0.153162
$47$	350.994	1.08931	0.544657	−	0.838659i	$-0.316660\pi$
$47$	350.994	1.08931	0.544657	+	0.838659i	$0.316660\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	363.879	0.970403
$53$	−196.260	−0.508649	−0.254324	−	0.967119i	$-0.581853\pi$
$53$	−196.260	−0.508649	−0.254324	+	0.967119i	$0.581853\pi$
$54$	0	0
$55$	0	0
$56$	−47.4217	−0.113160
$57$	0	0
$58$	−0.0326984	−7.40261e−5 0
$59$	−881.060	−1.94414	−0.972070	−	0.234692i	$-0.924592\pi$
$59$	−881.060	−1.94414	−0.972070	+	0.234692i	$0.924592\pi$
$60$	0	0
$61$	737.897	1.54882	0.774410	−	0.632684i	$-0.218047\pi$
$61$	737.897	1.54882	0.774410	+	0.632684i	$0.218047\pi$
$62$	86.4280	0.177038
$63$	0	0
$64$	−442.893	−0.865025
$65$	0	0
$66$	0	0
$67$	365.021	0.665589	0.332794	−	0.942999i	$-0.392009\pi$
$67$	365.021	0.665589	0.332794	+	0.942999i	$0.392009\pi$
$68$	40.6887	0.0725622
$69$	0	0
$70$	0	0
$71$	−1112.53	−1.85962	−0.929809	−	0.368042i	$-0.880028\pi$
$71$	−1112.53	−1.85962	−0.929809	+	0.368042i	$0.880028\pi$
$72$	0	0
$73$	261.995	0.420057	0.210029	−	0.977695i	$-0.432644\pi$
$73$	261.995	0.420057	0.210029	+	0.977695i	$0.432644\pi$
$74$	−134.004	−0.210508
$75$	0	0
$76$	711.311	1.07359
$77$	192.305	0.284612
$78$	0	0
$79$	273.829	0.389977	0.194988	−	0.980806i	$-0.437533\pi$
$79$	273.829	0.389977	0.194988	+	0.980806i	$0.437533\pi$
$80$	0	0
$81$	0	0
$82$	−43.8735	−0.0590856
$83$	87.1353	0.115233	0.0576165	−	0.998339i	$-0.481650\pi$
$83$	87.1353	0.115233	0.0576165	+	0.998339i	$0.481650\pi$
$84$	0	0
$85$	0	0
$86$	110.198	0.138174
$87$	0	0
$88$	−186.110	−0.225448
$89$	1090.99	1.29938	0.649690	−	0.760199i	$-0.274899\pi$
$89$	1090.99	1.29938	0.649690	+	0.760199i	$0.274899\pi$
$90$	0	0
$91$	−325.867	−0.375386
$92$	−872.039	−0.988221
$93$	0	0
$94$	150.337	0.164959
$95$	0	0
$96$	0	0
$97$	228.830	0.239527	0.119764	−	0.992802i	$-0.461786\pi$
$97$	228.830	0.239527	0.119764	+	0.992802i	$0.461786\pi$
$98$	20.9876	0.0216334
$99$	0	0
$100$	0	0
$101$	590.728	0.581976	0.290988	−	0.956727i	$-0.406016\pi$
$101$	590.728	0.581976	0.290988	+	0.956727i	$0.406016\pi$
$102$	0	0
$103$	−1471.06	−1.40726	−0.703629	−	0.710568i	$-0.748438\pi$
$103$	−1471.06	−1.40726	−0.703629	+	0.710568i	$0.748438\pi$
$104$	315.371	0.297352
$105$	0	0
$106$	−84.0619	−0.0770265
$107$	−1223.29	−1.10523	−0.552617	−	0.833435i	$-0.686371\pi$
$107$	−1223.29	−1.10523	−0.552617	+	0.833435i	$0.686371\pi$
$108$	0	0
$109$	1280.64	1.12535	0.562674	−	0.826679i	$-0.309773\pi$
$109$	1280.64	1.12535	0.562674	+	0.826679i	$0.309773\pi$
$110$	0	0
$111$	0	0
$112$	417.415	0.352161
$113$	−1805.38	−1.50297	−0.751487	−	0.659748i	$-0.770663\pi$
$113$	−1805.38	−1.50297	−0.751487	+	0.659748i	$0.770663\pi$
$114$	0	0
$115$	0	0
$116$	0.596725	0.000477625 0
$117$	0	0
$118$	−377.375	−0.294408
$119$	−36.4382	−0.0280696
$120$	0	0
$121$	−576.284	−0.432971
$122$	316.055	0.234543
$123$	0	0
$124$	−1577.26	−1.14227
$125$	0	0
$126$	0	0
$127$	−1642.42	−1.14757	−0.573786	−	0.819005i	$-0.694526\pi$
$127$	−1642.42	−1.14757	−0.573786	+	0.819005i	$0.694526\pi$
$128$	−827.596	−0.571483
$129$	0	0
$130$	0	0
$131$	−2371.85	−1.58190	−0.790951	−	0.611879i	$-0.790414\pi$
$131$	−2371.85	−1.58190	−0.790951	+	0.611879i	$0.790414\pi$
$132$	0	0
$133$	−637.005	−0.415303
$134$	156.345	0.100792
$135$	0	0
$136$	35.2645	0.0222346
$137$	762.828	0.475714	0.237857	−	0.971300i	$-0.423555\pi$
$137$	762.828	0.475714	0.237857	+	0.971300i	$0.423555\pi$
$138$	0	0
$139$	2025.84	1.23618	0.618092	−	0.786105i	$-0.287906\pi$
$139$	2025.84	1.23618	0.618092	+	0.786105i	$0.287906\pi$
$140$	0	0
$141$	0	0
$142$	−476.517	−0.281609
$143$	−1278.89	−0.747877
$144$	0	0
$145$	0	0
$146$	112.217	0.0636108
$147$	0	0
$148$	2445.48	1.35822
$149$	10.7903	0.00593272	0.00296636	−	0.999996i	$-0.499056\pi$
$149$	10.7903	0.00593272	0.00296636	+	0.999996i	$0.499056\pi$
$150$	0	0
$151$	−2404.48	−1.29585	−0.647925	−	0.761704i	$-0.724363\pi$
$151$	−2404.48	−1.29585	−0.647925	+	0.761704i	$0.724363\pi$
$152$	616.487	0.328972
$153$	0	0
$154$	82.3677	0.0430999
$155$	0	0
$156$	0	0
$157$	−396.624	−0.201618	−0.100809	−	0.994906i	$-0.532143\pi$
$157$	−396.624	−0.201618	−0.100809	+	0.994906i	$0.532143\pi$
$158$	117.286	0.0590556
$159$	0	0
$160$	0	0
$161$	780.943	0.382279
$162$	0	0
$163$	−2751.69	−1.32226	−0.661132	−	0.750270i	$-0.729923\pi$
$163$	−2751.69	−1.32226	−0.661132	+	0.750270i	$0.729923\pi$
$164$	800.663	0.381227
$165$	0	0
$166$	37.3217	0.0174501
$167$	−2079.43	−0.963541	−0.481771	−	0.876297i	$-0.660006\pi$
$167$	−2079.43	−0.963541	−0.481771	+	0.876297i	$0.660006\pi$
$168$	0	0
$169$	−29.8706	−0.0135961
$170$	0	0
$171$	0	0
$172$	−2011.04	−0.891515
$173$	−1929.59	−0.848000	−0.424000	−	0.905662i	$-0.639374\pi$
$173$	−1929.59	−0.848000	−0.424000	+	0.905662i	$0.639374\pi$
$174$	0	0
$175$	0	0
$176$	1638.18	0.701605
$177$	0	0
$178$	467.292	0.196770
$179$	−1638.15	−0.684027	−0.342014	−	0.939695i	$-0.611109\pi$
$179$	−1638.15	−0.684027	−0.342014	+	0.939695i	$0.611109\pi$
$180$	0	0
$181$	−36.6604	−0.0150550	−0.00752749	−	0.999972i	$-0.502396\pi$
$181$	−36.6604	−0.0150550	−0.00752749	+	0.999972i	$0.502396\pi$
$182$	−139.575	−0.0568461
$183$	0	0
$184$	−755.788	−0.302812
$185$	0	0
$186$	0	0
$187$	−143.005	−0.0559227
$188$	−2743.56	−1.06433
$189$	0	0
$190$	0	0
$191$	−1054.82	−0.399603	−0.199801	−	0.979836i	$-0.564030\pi$
$191$	−1054.82	−0.399603	−0.199801	+	0.979836i	$0.564030\pi$
$192$	0	0
$193$	213.661	0.0796873	0.0398436	−	0.999206i	$-0.487314\pi$
$193$	213.661	0.0796873	0.0398436	+	0.999206i	$0.487314\pi$
$194$	98.0121	0.0362725
$195$	0	0
$196$	−383.011	−0.139581
$197$	−3953.62	−1.42987	−0.714933	−	0.699193i	$-0.753543\pi$
$197$	−3953.62	−1.42987	−0.714933	+	0.699193i	$0.753543\pi$
$198$	0	0
$199$	929.168	0.330990	0.165495	−	0.986211i	$-0.447078\pi$
$199$	929.168	0.330990	0.165495	+	0.986211i	$0.447078\pi$
$200$	0	0
$201$	0	0
$202$	253.020	0.0881308
$203$	−0.534389	−0.000184762 0
$204$	0	0
$205$	0	0
$206$	−630.081	−0.213106
$207$	0	0
$208$	−2775.95	−0.925374
$209$	−2499.98	−0.827403
$210$	0	0
$211$	−926.806	−0.302388	−0.151194	−	0.988504i	$-0.548312\pi$
$211$	−926.806	−0.302388	−0.151194	+	0.988504i	$0.548312\pi$
$212$	1534.07	0.496984
$213$	0	0
$214$	−523.959	−0.167370
$215$	0	0
$216$	0	0
$217$	1412.49	0.441871
$218$	548.521	0.170415
$219$	0	0
$220$	0	0
$221$	242.327	0.0737587
$222$	0	0
$223$	−5351.73	−1.60708	−0.803538	−	0.595253i	$-0.797052\pi$
$223$	−5351.73	−1.60708	−0.803538	+	0.595253i	$0.797052\pi$
$224$	558.160	0.166489
$225$	0	0
$226$	−773.279	−0.227601
$227$	−6016.48	−1.75915	−0.879576	−	0.475758i	$-0.842174\pi$
$227$	−6016.48	−1.75915	−0.879576	+	0.475758i	$0.842174\pi$
$228$	0	0
$229$	1210.84	0.349408	0.174704	−	0.984621i	$-0.444103\pi$
$229$	1210.84	0.349408	0.174704	+	0.984621i	$0.444103\pi$
$230$	0	0
$231$	0	0
$232$	0.517176	0.000146355 0
$233$	−3517.31	−0.988957	−0.494478	−	0.869190i	$-0.664641\pi$
$233$	−3517.31	−0.988957	−0.494478	+	0.869190i	$0.664641\pi$
$234$	0	0
$235$	0	0
$236$	6886.84	1.89956
$237$	0	0
$238$	−15.6072	−0.00425068
$239$	6715.89	1.81764	0.908818	−	0.417194i	$-0.136986\pi$
$239$	6715.89	1.81764	0.908818	+	0.417194i	$0.136986\pi$
$240$	0	0
$241$	−1715.70	−0.458582	−0.229291	−	0.973358i	$-0.573641\pi$
$241$	−1715.70	−0.458582	−0.229291	+	0.973358i	$0.573641\pi$
$242$	−246.833	−0.0655663
$243$	0	0
$244$	−5767.80	−1.51330
$245$	0	0
$246$	0	0
$247$	4236.31	1.09129
$248$	−1366.99	−0.350017
$249$	0	0
$250$	0	0
$251$	2464.48	0.619748	0.309874	−	0.950778i	$-0.399713\pi$
$251$	2464.48	0.619748	0.309874	+	0.950778i	$0.399713\pi$
$252$	0	0
$253$	3064.88	0.761609
$254$	−703.482	−0.173781
$255$	0	0
$256$	3188.67	0.778483
$257$	1873.99	0.454850	0.227425	−	0.973796i	$-0.426969\pi$
$257$	1873.99	0.454850	0.227425	+	0.973796i	$0.426969\pi$
$258$	0	0
$259$	−2190.02	−0.525409
$260$	0	0
$261$	0	0
$262$	−1015.91	−0.239553
$263$	−2064.39	−0.484013	−0.242007	−	0.970275i	$-0.577806\pi$
$263$	−2064.39	−0.484013	−0.242007	+	0.970275i	$0.577806\pi$
$264$	0	0
$265$	0	0
$266$	−272.841	−0.0628909
$267$	0	0
$268$	−2853.20	−0.650325
$269$	−5649.86	−1.28059	−0.640294	−	0.768130i	$-0.721188\pi$
$269$	−5649.86	−1.28059	−0.640294	+	0.768130i	$0.721188\pi$
$270$	0	0
$271$	−3094.93	−0.693739	−0.346870	−	0.937913i	$-0.612755\pi$
$271$	−3094.93	−0.693739	−0.346870	+	0.937913i	$0.612755\pi$
$272$	−310.405	−0.0691951
$273$	0	0
$274$	326.734	0.0720391
$275$	0	0
$276$	0	0
$277$	−8962.93	−1.94415	−0.972076	−	0.234665i	$-0.924601\pi$
$277$	−8962.93	−1.94415	−0.972076	+	0.234665i	$0.924601\pi$
$278$	867.706	0.187200
$279$	0	0
$280$	0	0
$281$	5858.94	1.24383	0.621913	−	0.783086i	$-0.286356\pi$
$281$	5858.94	1.24383	0.621913	+	0.783086i	$0.286356\pi$
$282$	0	0
$283$	5819.46	1.22237	0.611186	−	0.791487i	$-0.290693\pi$
$283$	5819.46	1.22237	0.611186	+	0.791487i	$0.290693\pi$
$284$	8696.13	1.81697
$285$	0	0
$286$	−547.774	−0.113254
$287$	−717.023	−0.147472
$288$	0	0
$289$	−4885.90	−0.994485
$290$	0	0
$291$	0	0
$292$	−2047.90	−0.410425
$293$	−5678.78	−1.13228	−0.566139	−	0.824310i	$-0.691564\pi$
$293$	−5678.78	−1.13228	−0.566139	+	0.824310i	$0.691564\pi$
$294$	0	0
$295$	0	0
$296$	2119.47	0.416189
$297$	0	0
$298$	4.62169	0.000898413 0
$299$	−5193.54	−1.00452
$300$	0	0
$301$	1800.96	0.344869
$302$	−1029.88	−0.196235
$303$	0	0
$304$	−5426.44	−1.02377
$305$	0	0
$306$	0	0
$307$	9184.53	1.70745	0.853727	−	0.520720i	$-0.174337\pi$
$307$	9184.53	1.70745	0.853727	+	0.520720i	$0.174337\pi$
$308$	−1503.16	−0.278086
$309$	0	0
$310$	0	0
$311$	−4410.22	−0.804119	−0.402059	−	0.915614i	$-0.631705\pi$
$311$	−4410.22	−0.804119	−0.402059	+	0.915614i	$0.631705\pi$
$312$	0	0
$313$	−4405.28	−0.795530	−0.397765	−	0.917487i	$-0.630214\pi$
$313$	−4405.28	−0.795530	−0.397765	+	0.917487i	$0.630214\pi$
$314$	−169.882	−0.0305318
$315$	0	0
$316$	−2140.40	−0.381034
$317$	7486.86	1.32651	0.663256	−	0.748393i	$-0.269174\pi$
$317$	7486.86	1.32651	0.663256	+	0.748393i	$0.269174\pi$
$318$	0	0
$319$	−2.09726	−0.000368100 0
$320$	0	0
$321$	0	0
$322$	334.493	0.0578899
$323$	473.701	0.0816019
$324$	0	0
$325$	0	0
$326$	−1178.60	−0.200235
$327$	0	0
$328$	693.927	0.116816
$329$	2456.96	0.411722
$330$	0	0
$331$	8860.56	1.47136	0.735680	−	0.677329i	$-0.236863\pi$
$331$	8860.56	1.47136	0.735680	+	0.677329i	$0.236863\pi$
$332$	−681.097	−0.112590
$333$	0	0
$334$	−890.660	−0.145912
$335$	0	0
$336$	0	0
$337$	−8742.01	−1.41308	−0.706539	−	0.707674i	$-0.749745\pi$
$337$	−8742.01	−1.41308	−0.706539	+	0.707674i	$0.749745\pi$
$338$	−12.7942	−0.00205891
$339$	0	0
$340$	0	0
$341$	5543.44	0.880334
$342$	0	0
$343$	343.000	0.0539949
$344$	−1742.95	−0.273179
$345$	0	0
$346$	−826.480	−0.128416
$347$	7285.82	1.12716	0.563579	−	0.826063i	$-0.309424\pi$
$347$	7285.82	1.12716	0.563579	+	0.826063i	$0.309424\pi$
$348$	0	0
$349$	12610.9	1.93423	0.967117	−	0.254330i	$-0.0818550\pi$
$349$	12610.9	1.93423	0.967117	+	0.254330i	$0.0818550\pi$
$350$	0	0
$351$	0	0
$352$	2190.55	0.331695
$353$	1202.59	0.181325	0.0906623	−	0.995882i	$-0.471102\pi$
$353$	1202.59	0.181325	0.0906623	+	0.995882i	$0.471102\pi$
$354$	0	0
$355$	0	0
$356$	−8527.78	−1.26958
$357$	0	0
$358$	−701.649	−0.103585
$359$	−3216.35	−0.472848	−0.236424	−	0.971650i	$-0.575975\pi$
$359$	−3216.35	−0.472848	−0.236424	+	0.971650i	$0.575975\pi$
$360$	0	0
$361$	1422.14	0.207339
$362$	−15.7024	−0.00227983
$363$	0	0
$364$	2547.15	0.366778
$365$	0	0
$366$	0	0
$367$	−1259.66	−0.179165	−0.0895824	−	0.995979i	$-0.528553\pi$
$367$	−1259.66	−0.179165	−0.0895824	+	0.995979i	$0.528553\pi$
$368$	6652.59	0.942365
$369$	0	0
$370$	0	0
$371$	−1373.82	−0.192251
$372$	0	0
$373$	−386.230	−0.0536145	−0.0268073	−	0.999641i	$-0.508534\pi$
$373$	−386.230	−0.0536145	−0.0268073	+	0.999641i	$0.508534\pi$
$374$	−61.2517	−0.00846858
$375$	0	0
$376$	−2377.82	−0.326135
$377$	3.55387	0.000485501 0
$378$	0	0
$379$	14246.0	1.93079	0.965395	−	0.260794i	$-0.0839842\pi$
$379$	14246.0	1.93079	0.965395	+	0.260794i	$0.0839842\pi$
$380$	0	0
$381$	0	0
$382$	−451.800	−0.0605133
$383$	−9298.85	−1.24060	−0.620299	−	0.784365i	$-0.712989\pi$
$383$	−9298.85	−1.24060	−0.620299	+	0.784365i	$0.712989\pi$
$384$	0	0
$385$	0	0
$386$	91.5150	0.0120673
$387$	0	0
$388$	−1788.66	−0.234034
$389$	1043.80	0.136049	0.0680244	−	0.997684i	$-0.478330\pi$
$389$	1043.80	0.136049	0.0680244	+	0.997684i	$0.478330\pi$
$390$	0	0
$391$	−580.738	−0.0751130
$392$	−331.952	−0.0427706
$393$	0	0
$394$	−1693.41	−0.216530
$395$	0	0
$396$	0	0
$397$	4482.04	0.566617	0.283309	−	0.959029i	$-0.408568\pi$
$397$	4482.04	0.566617	0.283309	+	0.959029i	$0.408568\pi$
$398$	397.980	0.0501230
$399$	0	0
$400$	0	0
$401$	−12686.2	−1.57984	−0.789921	−	0.613209i	$-0.789878\pi$
$401$	−12686.2	−1.57984	−0.789921	+	0.613209i	$0.789878\pi$
$402$	0	0
$403$	−9393.55	−1.16111
$404$	−4617.45	−0.568630
$405$	0	0
$406$	−0.228889	−2.79792e−5 0
$407$	−8594.90	−1.04677
$408$	0	0
$409$	−6836.54	−0.826516	−0.413258	−	0.910614i	$-0.635609\pi$
$409$	−6836.54	−0.826516	−0.413258	+	0.910614i	$0.635609\pi$
$410$	0	0
$411$	0	0
$412$	11498.6	1.37499
$413$	−6167.42	−0.734816
$414$	0	0
$415$	0	0
$416$	−3711.96	−0.437485
$417$	0	0
$418$	−1070.79	−0.125297
$419$	5091.64	0.593659	0.296829	−	0.954930i	$-0.404071\pi$
$419$	5091.64	0.593659	0.296829	+	0.954930i	$0.404071\pi$
$420$	0	0
$421$	8373.48	0.969355	0.484677	−	0.874693i	$-0.338937\pi$
$421$	8373.48	0.969355	0.484677	+	0.874693i	$0.338937\pi$
$422$	−396.968	−0.0457918
$423$	0	0
$424$	1329.57	0.152287
$425$	0	0
$426$	0	0
$427$	5165.28	0.585399
$428$	9561.91	1.07989
$429$	0	0
$430$	0	0
$431$	−6027.75	−0.673657	−0.336829	−	0.941566i	$-0.609354\pi$
$431$	−6027.75	−0.673657	−0.336829	+	0.941566i	$0.609354\pi$
$432$	0	0
$433$	11927.5	1.32379	0.661894	−	0.749597i	$-0.269753\pi$
$433$	11927.5	1.32379	0.661894	+	0.749597i	$0.269753\pi$
$434$	604.996	0.0669141
$435$	0	0
$436$	−10010.2	−1.09954
$437$	−10152.3	−1.11133
$438$	0	0
$439$	7914.93	0.860499	0.430250	−	0.902710i	$-0.358426\pi$
$439$	7914.93	0.860499	0.430250	+	0.902710i	$0.358426\pi$
$440$	0	0
$441$	0	0
$442$	103.793	0.0111695
$443$	2522.71	0.270559	0.135279	−	0.990807i	$-0.456807\pi$
$443$	2522.71	0.270559	0.135279	+	0.990807i	$0.456807\pi$
$444$	0	0
$445$	0	0
$446$	−2292.24	−0.243365
$447$	0	0
$448$	−3100.25	−0.326949
$449$	5339.89	0.561258	0.280629	−	0.959816i	$-0.409457\pi$
$449$	5339.89	0.561258	0.280629	+	0.959816i	$0.409457\pi$
$450$	0	0
$451$	−2814.02	−0.293807
$452$	14111.8	1.46851
$453$	0	0
$454$	−2576.97	−0.266395
$455$	0	0
$456$	0	0
$457$	−13765.8	−1.40905	−0.704526	−	0.709679i	$-0.748840\pi$
$457$	−13765.8	−1.40905	−0.704526	+	0.709679i	$0.748840\pi$
$458$	518.625	0.0529122
$459$	0	0
$460$	0	0
$461$	12501.1	1.26298	0.631489	−	0.775384i	$-0.282444\pi$
$461$	12501.1	1.26298	0.631489	+	0.775384i	$0.282444\pi$
$462$	0	0
$463$	3566.32	0.357972	0.178986	−	0.983852i	$-0.442718\pi$
$463$	3566.32	0.357972	0.178986	+	0.983852i	$0.442718\pi$
$464$	−4.55229	−0.000455462 0
$465$	0	0
$466$	−1506.53	−0.149761
$467$	4078.17	0.404101	0.202050	−	0.979375i	$-0.435240\pi$
$467$	4078.17	0.404101	0.202050	+	0.979375i	$0.435240\pi$
$468$	0	0
$469$	2555.15	0.251569
$470$	0	0
$471$	0	0
$472$	5968.76	0.582065
$473$	7068.03	0.687079
$474$	0	0
$475$	0	0
$476$	284.821	0.0274259
$477$	0	0
$478$	2876.54	0.275251
$479$	−19078.0	−1.81982	−0.909911	−	0.414803i	$-0.863850\pi$
$479$	−19078.0	−1.81982	−0.909911	+	0.414803i	$0.863850\pi$
$480$	0	0
$481$	14564.4	1.38062
$482$	−734.868	−0.0694447
$483$	0	0
$484$	4504.55	0.423042
$485$	0	0
$486$	0	0
$487$	−15616.4	−1.45307	−0.726537	−	0.687128i	$-0.758871\pi$
$487$	−15616.4	−1.45307	−0.726537	+	0.687128i	$0.758871\pi$
$488$	−4998.90	−0.463708
$489$	0	0
$490$	0	0
$491$	−7547.46	−0.693711	−0.346856	−	0.937919i	$-0.612751\pi$
$491$	−7547.46	−0.693711	−0.346856	+	0.937919i	$0.612751\pi$
$492$	0	0
$493$	0.397392	3.63035e−5 0
$494$	1814.49	0.165259
$495$	0	0
$496$	12032.5	1.08927
$497$	−7787.70	−0.702870
$498$	0	0
$499$	3288.10	0.294982	0.147491	−	0.989063i	$-0.452880\pi$
$499$	3288.10	0.294982	0.147491	+	0.989063i	$0.452880\pi$
$500$	0	0
$501$	0	0
$502$	1055.58	0.0938506
$503$	−1044.67	−0.0926032	−0.0463016	−	0.998928i	$-0.514744\pi$
$503$	−1044.67	−0.0926032	−0.0463016	+	0.998928i	$0.514744\pi$
$504$	0	0
$505$	0	0
$506$	1312.74	0.115333
$507$	0	0
$508$	12838.1	1.12126
$509$	8783.91	0.764912	0.382456	−	0.923974i	$-0.375078\pi$
$509$	8783.91	0.764912	0.382456	+	0.923974i	$0.375078\pi$
$510$	0	0
$511$	1833.97	0.158767
$512$	7986.54	0.689372
$513$	0	0
$514$	802.666	0.0688796
$515$	0	0
$516$	0	0
$517$	9642.54	0.820268
$518$	−938.025	−0.0795646
$519$	0	0
$520$	0	0
$521$	−9983.79	−0.839535	−0.419767	−	0.907632i	$-0.637888\pi$
$521$	−9983.79	−0.839535	−0.419767	+	0.907632i	$0.637888\pi$
$522$	0	0
$523$	−8177.52	−0.683706	−0.341853	−	0.939754i	$-0.611054\pi$
$523$	−8177.52	−0.683706	−0.341853	+	0.939754i	$0.611054\pi$
$524$	18539.6	1.54563
$525$	0	0
$526$	−884.215	−0.0732958
$527$	−1050.38	−0.0868221
$528$	0	0
$529$	279.364	0.0229608
$530$	0	0
$531$	0	0
$532$	4979.18	0.405780
$533$	4768.45	0.387513
$534$	0	0
$535$	0	0
$536$	−2472.85	−0.199274
$537$	0	0
$538$	−2419.94	−0.193924
$539$	1346.13	0.107573
$540$	0	0
$541$	−3425.16	−0.272198	−0.136099	−	0.990695i	$-0.543457\pi$
$541$	−3425.16	−0.272198	−0.136099	+	0.990695i	$0.543457\pi$
$542$	−1325.61	−0.105055
$543$	0	0
$544$	−415.068	−0.0327131
$545$	0	0
$546$	0	0
$547$	−5955.72	−0.465536	−0.232768	−	0.972532i	$-0.574778\pi$
$547$	−5955.72	−0.465536	−0.232768	+	0.972532i	$0.574778\pi$
$548$	−5962.68	−0.464805
$549$	0	0
$550$	0	0
$551$	6.94712	0.000537127 0
$552$	0	0
$553$	1916.80	0.147397
$554$	−3838.99	−0.294410
$555$	0	0
$556$	−15835.1	−1.20784
$557$	2506.35	0.190660	0.0953298	−	0.995446i	$-0.469609\pi$
$557$	2506.35	0.190660	0.0953298	+	0.995446i	$0.469609\pi$
$558$	0	0
$559$	−11977.0	−0.906215
$560$	0	0
$561$	0	0
$562$	2509.50	0.188357
$563$	−3460.79	−0.259068	−0.129534	−	0.991575i	$-0.541348\pi$
$563$	−3460.79	−0.259068	−0.129534	+	0.991575i	$0.541348\pi$
$564$	0	0
$565$	0	0
$566$	2492.59	0.185108
$567$	0	0
$568$	7536.86	0.556760
$569$	−22561.2	−1.66224	−0.831119	−	0.556095i	$-0.812299\pi$
$569$	−22561.2	−1.66224	−0.831119	+	0.556095i	$0.812299\pi$
$570$	0	0
$571$	−992.585	−0.0727467	−0.0363734	−	0.999338i	$-0.511581\pi$
$571$	−992.585	−0.0727467	−0.0363734	+	0.999338i	$0.511581\pi$
$572$	9996.52	0.730726
$573$	0	0
$574$	−307.114	−0.0223322
$575$	0	0
$576$	0	0
$577$	9202.70	0.663975	0.331987	−	0.943284i	$-0.392281\pi$
$577$	9202.70	0.663975	0.331987	+	0.943284i	$0.392281\pi$
$578$	−2092.72	−0.150598
$579$	0	0
$580$	0	0
$581$	609.947	0.0435540
$582$	0	0
$583$	−5391.67	−0.383019
$584$	−1774.89	−0.125763
$585$	0	0
$586$	−2432.33	−0.171465
$587$	−27046.3	−1.90174	−0.950869	−	0.309593i	$-0.899807\pi$
$587$	−27046.3	−1.90174	−0.950869	+	0.309593i	$0.899807\pi$
$588$	0	0
$589$	−18362.5	−1.28457
$590$	0	0
$591$	0	0
$592$	−18656.0	−1.29520
$593$	26364.0	1.82570	0.912849	−	0.408298i	$-0.133877\pi$
$593$	26364.0	1.82570	0.912849	+	0.408298i	$0.133877\pi$
$594$	0	0
$595$	0	0
$596$	−84.3428	−0.00579667
$597$	0	0
$598$	−2224.49	−0.152117
$599$	−6624.09	−0.451841	−0.225921	−	0.974146i	$-0.572539\pi$
$599$	−6624.09	−0.451841	−0.225921	+	0.974146i	$0.572539\pi$
$600$	0	0
$601$	−3984.03	−0.270403	−0.135201	−	0.990818i	$-0.543168\pi$
$601$	−3984.03	−0.270403	−0.135201	+	0.990818i	$0.543168\pi$
$602$	771.386	0.0522248
$603$	0	0
$604$	18794.7	1.26613
$605$	0	0
$606$	0	0
$607$	−8605.79	−0.575450	−0.287725	−	0.957713i	$-0.592899\pi$
$607$	−8605.79	−0.575450	−0.287725	+	0.957713i	$0.592899\pi$
$608$	−7256.14	−0.484005
$609$	0	0
$610$	0	0
$611$	−16339.6	−1.08188
$612$	0	0
$613$	−22070.4	−1.45418	−0.727091	−	0.686541i	$-0.759128\pi$
$613$	−22070.4	−1.45418	−0.727091	+	0.686541i	$0.759128\pi$
$614$	3933.91	0.258566
$615$	0	0
$616$	−1302.77	−0.0852114
$617$	−17328.8	−1.13068	−0.565340	−	0.824858i	$-0.691255\pi$
$617$	−17328.8	−1.13068	−0.565340	+	0.824858i	$0.691255\pi$
$618$	0	0
$619$	1240.99	0.0805808	0.0402904	−	0.999188i	$-0.487172\pi$
$619$	1240.99	0.0805808	0.0402904	+	0.999188i	$0.487172\pi$
$620$	0	0
$621$	0	0
$622$	−1888.98	−0.121771
$623$	7636.94	0.491119
$624$	0	0
$625$	0	0
$626$	−1886.86	−0.120470
$627$	0	0
$628$	3100.23	0.196995
$629$	1628.58	0.103236
$630$	0	0
$631$	10004.0	0.631143	0.315572	−	0.948902i	$-0.397804\pi$
$631$	10004.0	0.631143	0.315572	+	0.948902i	$0.397804\pi$
$632$	−1855.06	−0.116757
$633$	0	0
$634$	3206.76	0.200878
$635$	0	0
$636$	0	0
$637$	−2281.07	−0.141883
$638$	−0.898294	−5.57426e−5 0
$639$	0	0
$640$	0	0
$641$	23107.3	1.42384	0.711921	−	0.702260i	$-0.247826\pi$
$641$	23107.3	1.42384	0.711921	+	0.702260i	$0.247826\pi$
$642$	0	0
$643$	11629.9	0.713281	0.356641	−	0.934242i	$-0.383922\pi$
$643$	11629.9	0.713281	0.356641	+	0.934242i	$0.383922\pi$
$644$	−6104.27	−0.373513
$645$	0	0
$646$	202.895	0.0123573
$647$	−6371.50	−0.387156	−0.193578	−	0.981085i	$-0.562009\pi$
$647$	−6371.50	−0.387156	−0.193578	+	0.981085i	$0.562009\pi$
$648$	0	0
$649$	−24204.6	−1.46396
$650$	0	0
$651$	0	0
$652$	21508.7	1.29194
$653$	20264.5	1.21441	0.607207	−	0.794544i	$-0.292290\pi$
$653$	20264.5	1.21441	0.607207	+	0.794544i	$0.292290\pi$
$654$	0	0
$655$	0	0
$656$	−6108.08	−0.363537
$657$	0	0
$658$	1052.36	0.0623485
$659$	7132.74	0.421627	0.210813	−	0.977526i	$-0.432389\pi$
$659$	7132.74	0.421627	0.210813	+	0.977526i	$0.432389\pi$
$660$	0	0
$661$	10555.8	0.621140	0.310570	−	0.950550i	$-0.399480\pi$
$661$	10555.8	0.621140	0.310570	+	0.950550i	$0.399480\pi$
$662$	3795.14	0.222813
$663$	0	0
$664$	−590.300	−0.0345001
$665$	0	0
$666$	0	0
$667$	−8.51689	−0.000494416 0
$668$	16254.0	0.941445
$669$	0	0
$670$	0	0
$671$	20271.6	1.16628
$672$	0	0
$673$	3828.08	0.219259	0.109630	−	0.993973i	$-0.465034\pi$
$673$	3828.08	0.219259	0.109630	+	0.993973i	$0.465034\pi$
$674$	−3744.37	−0.213988
$675$	0	0
$676$	233.485	0.0132843
$677$	24660.6	1.39998	0.699989	−	0.714154i	$-0.253188\pi$
$677$	24660.6	1.39998	0.699989	+	0.714154i	$0.253188\pi$
$678$	0	0
$679$	1601.81	0.0905328
$680$	0	0
$681$	0	0
$682$	2374.36	0.133312
$683$	18562.2	1.03992	0.519958	−	0.854192i	$-0.325948\pi$
$683$	18562.2	1.03992	0.519958	+	0.854192i	$0.325948\pi$
$684$	0	0
$685$	0	0
$686$	146.913	0.00817665
$687$	0	0
$688$	15341.8	0.850146
$689$	9136.38	0.505179
$690$	0	0
$691$	27335.9	1.50493	0.752465	−	0.658632i	$-0.228865\pi$
$691$	27335.9	1.50493	0.752465	+	0.658632i	$0.228865\pi$
$692$	15082.7	0.828554
$693$	0	0
$694$	3120.66	0.170689
$695$	0	0
$696$	0	0
$697$	533.205	0.0289764
$698$	5401.50	0.292908
$699$	0	0
$700$	0	0
$701$	30924.1	1.66617	0.833087	−	0.553142i	$-0.186571\pi$
$701$	30924.1	1.66617	0.833087	+	0.553142i	$0.186571\pi$
$702$	0	0
$703$	28470.4	1.52743
$704$	−12167.2	−0.651375
$705$	0	0
$706$	515.093	0.0274586
$707$	4135.09	0.219966
$708$	0	0
$709$	−28329.0	−1.50059	−0.750295	−	0.661103i	$-0.770088\pi$
$709$	−28329.0	−1.50059	−0.750295	+	0.661103i	$0.770088\pi$
$710$	0	0
$711$	0	0
$712$	−7390.95	−0.389027
$713$	22511.7	1.18243
$714$	0	0
$715$	0	0
$716$	12804.6	0.668341
$717$	0	0
$718$	−1377.62	−0.0716051
$719$	12563.4	0.651651	0.325825	−	0.945430i	$-0.394358\pi$
$719$	12563.4	0.651651	0.325825	+	0.945430i	$0.394358\pi$
$720$	0	0
$721$	−10297.4	−0.531893
$722$	609.127	0.0313980
$723$	0	0
$724$	286.558	0.0147097
$725$	0	0
$726$	0	0
$727$	−13523.2	−0.689888	−0.344944	−	0.938623i	$-0.612102\pi$
$727$	−13523.2	−0.689888	−0.344944	+	0.938623i	$0.612102\pi$
$728$	2207.59	0.112389
$729$	0	0
$730$	0	0
$731$	−1339.26	−0.0677625
$732$	0	0
$733$	21325.0	1.07457	0.537284	−	0.843402i	$-0.319450\pi$
$733$	21325.0	1.07457	0.537284	+	0.843402i	$0.319450\pi$
$734$	−539.534	−0.0271316
$735$	0	0
$736$	8895.74	0.445518
$737$	10027.9	0.501197
$738$	0	0
$739$	−7403.75	−0.368540	−0.184270	−	0.982876i	$-0.558992\pi$
$739$	−7403.75	−0.368540	−0.184270	+	0.982876i	$0.558992\pi$
$740$	0	0
$741$	0	0
$742$	−588.433	−0.0291133
$743$	−27131.6	−1.33965	−0.669825	−	0.742519i	$-0.733631\pi$
$743$	−27131.6	−1.33965	−0.669825	+	0.742519i	$0.733631\pi$
$744$	0	0
$745$	0	0
$746$	−165.430	−0.00811904
$747$	0	0
$748$	1117.80	0.0546403
$749$	−8563.04	−0.417739
$750$	0	0
$751$	−29393.3	−1.42820	−0.714098	−	0.700046i	$-0.753163\pi$
$751$	−29393.3	−1.42820	−0.714098	+	0.700046i	$0.753163\pi$
$752$	20930.0	1.01495
$753$	0	0
$754$	1.52219	7.35211e−5 0
$755$	0	0
$756$	0	0
$757$	37648.5	1.80761	0.903804	−	0.427946i	$-0.140763\pi$
$757$	37648.5	1.80761	0.903804	+	0.427946i	$0.140763\pi$
$758$	6101.84	0.292386
$759$	0	0
$760$	0	0
$761$	35633.2	1.69738	0.848688	−	0.528894i	$-0.177393\pi$
$761$	35633.2	1.69738	0.848688	+	0.528894i	$0.177393\pi$
$762$	0	0
$763$	8964.46	0.425341
$764$	8245.05	0.390439
$765$	0	0
$766$	−3982.87	−0.187868
$767$	41015.5	1.93088
$768$	0	0
$769$	−3571.28	−0.167469	−0.0837345	−	0.996488i	$-0.526685\pi$
$769$	−3571.28	−0.167469	−0.0837345	+	0.996488i	$0.526685\pi$
$770$	0	0
$771$	0	0
$772$	−1670.09	−0.0778599
$773$	−16250.3	−0.756122	−0.378061	−	0.925781i	$-0.623409\pi$
$773$	−16250.3	−0.756122	−0.378061	+	0.925781i	$0.623409\pi$
$774$	0	0
$775$	0	0
$776$	−1550.21	−0.0717131
$777$	0	0
$778$	447.081	0.0206024
$779$	9321.37	0.428720
$780$	0	0
$781$	−30563.5	−1.40032
$782$	−248.741	−0.0113746
$783$	0	0
$784$	2921.90	0.133104
$785$	0	0
$786$	0	0
$787$	13156.5	0.595906	0.297953	−	0.954581i	$-0.403696\pi$
$787$	13156.5	0.595906	0.297953	+	0.954581i	$0.403696\pi$
$788$	30903.6	1.39708
$789$	0	0
$790$	0	0
$791$	−12637.7	−0.568071
$792$	0	0
$793$	−34350.9	−1.53826
$794$	1919.74	0.0858049
$795$	0	0
$796$	−7262.88	−0.323399
$797$	−13868.8	−0.616385	−0.308192	−	0.951324i	$-0.599724\pi$
$797$	−13868.8	−0.616385	−0.308192	+	0.951324i	$0.599724\pi$
$798$	0	0
$799$	−1827.09	−0.0808982
$800$	0	0
$801$	0	0
$802$	−5433.72	−0.239241
$803$	7197.55	0.316309
$804$	0	0
$805$	0	0
$806$	−4023.43	−0.175831
$807$	0	0
$808$	−4001.90	−0.174240
$809$	7042.67	0.306066	0.153033	−	0.988221i	$-0.451096\pi$
$809$	7042.67	0.306066	0.153033	+	0.988221i	$0.451096\pi$
$810$	0	0
$811$	−8985.64	−0.389061	−0.194531	−	0.980896i	$-0.562318\pi$
$811$	−8985.64	−0.389061	−0.194531	+	0.980896i	$0.562318\pi$
$812$	4.17708	0.000180525 0
$813$	0	0
$814$	−3681.36	−0.158515
$815$	0	0
$816$	0	0
$817$	−23412.7	−1.00258
$818$	−2928.22	−0.125162
$819$	0	0
$820$	0	0
$821$	1293.90	0.0550031	0.0275016	−	0.999622i	$-0.491245\pi$
$821$	1293.90	0.0550031	0.0275016	+	0.999622i	$0.491245\pi$
$822$	0	0
$823$	28480.7	1.20629	0.603144	−	0.797632i	$-0.293914\pi$
$823$	28480.7	1.20629	0.603144	+	0.797632i	$0.293914\pi$
$824$	9965.71	0.421325
$825$	0	0
$826$	−2641.62	−0.111276
$827$	8825.09	0.371074	0.185537	−	0.982637i	$-0.440597\pi$
$827$	8825.09	0.371074	0.185537	+	0.982637i	$0.440597\pi$
$828$	0	0
$829$	−6673.08	−0.279572	−0.139786	−	0.990182i	$-0.544642\pi$
$829$	−6673.08	−0.279572	−0.139786	+	0.990182i	$0.544642\pi$
$830$	0	0
$831$	0	0
$832$	20617.7	0.859124
$833$	−255.068	−0.0106093
$834$	0	0
$835$	0	0
$836$	19541.2	0.808429
$837$	0	0
$838$	2180.85	0.0898999
$839$	7026.24	0.289121	0.144561	−	0.989496i	$-0.453823\pi$
$839$	7026.24	0.289121	0.144561	+	0.989496i	$0.453823\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	3586.52	0.146793
$843$	0	0
$844$	7244.42	0.295454
$845$	0	0
$846$	0	0
$847$	−4033.99	−0.163648
$848$	−11703.1	−0.473923
$849$	0	0
$850$	0	0
$851$	−34903.6	−1.40597
$852$	0	0
$853$	7181.30	0.288257	0.144128	−	0.989559i	$-0.453962\pi$
$853$	7181.30	0.288257	0.144128	+	0.989559i	$0.453962\pi$
$854$	2212.39	0.0886491
$855$	0	0
$856$	8287.21	0.330901
$857$	−23061.9	−0.919228	−0.459614	−	0.888119i	$-0.652012\pi$
$857$	−23061.9	−0.919228	−0.459614	+	0.888119i	$0.652012\pi$
$858$	0	0
$859$	32264.0	1.28153	0.640764	−	0.767738i	$-0.278618\pi$
$859$	32264.0	1.28153	0.640764	+	0.767738i	$0.278618\pi$
$860$	0	0
$861$	0	0
$862$	−2581.80	−0.102014
$863$	−2831.11	−0.111671	−0.0558354	−	0.998440i	$-0.517782\pi$
$863$	−2831.11	−0.111671	−0.0558354	+	0.998440i	$0.517782\pi$
$864$	0	0
$865$	0	0
$866$	5108.79	0.200466
$867$	0	0
$868$	−11040.8	−0.431738
$869$	7522.66	0.293658
$870$	0	0
$871$	−16992.6	−0.661048
$872$	−8675.71	−0.336923
$873$	0	0
$874$	−4348.44	−0.168293
$875$	0	0
$876$	0	0
$877$	−17952.1	−0.691218	−0.345609	−	0.938379i	$-0.612328\pi$
$877$	−17952.1	−0.691218	−0.345609	+	0.938379i	$0.612328\pi$
$878$	3390.12	0.130309
$879$	0	0
$880$	0	0
$881$	−25983.2	−0.993641	−0.496821	−	0.867853i	$-0.665499\pi$
$881$	−25983.2	−0.993641	−0.496821	+	0.867853i	$0.665499\pi$
$882$	0	0
$883$	−10296.7	−0.392424	−0.196212	−	0.980562i	$-0.562864\pi$
$883$	−10296.7	−0.392424	−0.196212	+	0.980562i	$0.562864\pi$
$884$	−1894.16	−0.0720672
$885$	0	0
$886$	1080.52	0.0409717
$887$	−14925.6	−0.564996	−0.282498	−	0.959268i	$-0.591163\pi$
$887$	−14925.6	−0.564996	−0.282498	+	0.959268i	$0.591163\pi$
$888$	0	0
$889$	−11497.0	−0.433741
$890$	0	0
$891$	0	0
$892$	41832.0	1.57022
$893$	−31940.7	−1.19693
$894$	0	0
$895$	0	0
$896$	−5793.17	−0.216000
$897$	0	0
$898$	2287.17	0.0849933
$899$	−15.4045	−0.000571488 0
$900$	0	0
$901$	1021.62	0.0377749
$902$	−1205.30	−0.0444922
$903$	0	0
$904$	12230.6	0.449982
$905$	0	0
$906$	0	0
$907$	24744.9	0.905889	0.452944	−	0.891539i	$-0.350374\pi$
$907$	24744.9	0.905889	0.452944	+	0.891539i	$0.350374\pi$
$908$	47028.0	1.71881
$909$	0	0
$910$	0	0
$911$	−4378.72	−0.159246	−0.0796232	−	0.996825i	$-0.525372\pi$
$911$	−4378.72	−0.159246	−0.0796232	+	0.996825i	$0.525372\pi$
$912$	0	0
$913$	2393.79	0.0867720
$914$	−5896.15	−0.213378
$915$	0	0
$916$	−9464.58	−0.341396
$917$	−16602.9	−0.597903
$918$	0	0
$919$	26026.1	0.934192	0.467096	−	0.884207i	$-0.345300\pi$
$919$	26026.1	0.934192	0.467096	+	0.884207i	$0.345300\pi$
$920$	0	0
$921$	0	0
$922$	5354.45	0.191257
$923$	51790.9	1.84693
$924$	0	0
$925$	0	0
$926$	1527.52	0.0542090
$927$	0	0
$928$	−6.08724	−0.000215327 0
$929$	−4240.78	−0.149769	−0.0748846	−	0.997192i	$-0.523859\pi$
$929$	−4240.78	−0.149769	−0.0748846	+	0.997192i	$0.523859\pi$
$930$	0	0
$931$	−4459.04	−0.156970
$932$	27493.2	0.966278
$933$	0	0
$934$	1746.76	0.0611944
$935$	0	0
$936$	0	0
$937$	−664.833	−0.0231794	−0.0115897	−	0.999933i	$-0.503689\pi$
$937$	−664.833	−0.0231794	−0.0115897	+	0.999933i	$0.503689\pi$
$938$	1094.42	0.0380960
$939$	0	0
$940$	0	0
$941$	−9520.60	−0.329822	−0.164911	−	0.986308i	$-0.552734\pi$
$941$	−9520.60	−0.329822	−0.164911	+	0.986308i	$0.552734\pi$
$942$	0	0
$943$	−11427.6	−0.394629
$944$	−52538.2	−1.81141
$945$	0	0
$946$	3027.37	0.104047
$947$	−17066.3	−0.585618	−0.292809	−	0.956171i	$-0.594590\pi$
$947$	−17066.3	−0.585618	−0.292809	+	0.956171i	$0.594590\pi$
$948$	0	0
$949$	−12196.5	−0.417192
$950$	0	0
$951$	0	0
$952$	246.852	0.00840389
$953$	−6991.85	−0.237658	−0.118829	−	0.992915i	$-0.537914\pi$
$953$	−6991.85	−0.237658	−0.118829	+	0.992915i	$0.537914\pi$
$954$	0	0
$955$	0	0
$956$	−52495.0	−1.77595
$957$	0	0
$958$	−8171.46	−0.275582
$959$	5339.80	0.179803
$960$	0	0
$961$	10925.9	0.366751
$962$	6238.20	0.209072
$963$	0	0
$964$	13410.9	0.448065
$965$	0	0
$966$	0	0
$967$	18731.2	0.622911	0.311455	−	0.950261i	$-0.399184\pi$
$967$	18731.2	0.622911	0.311455	+	0.950261i	$0.399184\pi$
$968$	3904.05	0.129629
$969$	0	0
$970$	0	0
$971$	45780.2	1.51303	0.756517	−	0.653974i	$-0.226900\pi$
$971$	45780.2	1.51303	0.756517	+	0.653974i	$0.226900\pi$
$972$	0	0
$973$	14180.9	0.467234
$974$	−6688.80	−0.220044
$975$	0	0
$976$	44001.3	1.44308
$977$	20137.9	0.659436	0.329718	−	0.944080i	$-0.393046\pi$
$977$	20137.9	0.659436	0.329718	+	0.944080i	$0.393046\pi$
$978$	0	0
$979$	29971.8	0.978451
$980$	0	0
$981$	0	0
$982$	−3232.72	−0.105051
$983$	6220.73	0.201842	0.100921	−	0.994894i	$-0.467821\pi$
$983$	6220.73	0.201842	0.100921	+	0.994894i	$0.467821\pi$
$984$	0	0
$985$	0	0
$986$	0.170210	5.49757e−6 0
$987$	0	0
$988$	−33113.3	−1.06627
$989$	28703.0	0.922855
$990$	0	0
$991$	−10296.5	−0.330049	−0.165025	−	0.986289i	$-0.552770\pi$
$991$	−10296.5	−0.330049	−0.165025	+	0.986289i	$0.552770\pi$
$992$	16089.7	0.514968
$993$	0	0
$994$	−3335.62	−0.106438
$995$	0	0
$996$	0	0
$997$	−27392.6	−0.870144	−0.435072	−	0.900396i	$-0.643277\pi$
$997$	−27392.6	−0.870144	−0.435072	+	0.900396i	$0.643277\pi$
$998$	1408.36	0.0446701
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bo.1.3		5
3.2	odd	2		525.4.a.x.1.3		5
5.2	odd	4		315.4.d.b.64.6		10
5.3	odd	4		315.4.d.b.64.5		10
5.4	even	2		1575.4.a.bp.1.3		5
15.2	even	4		105.4.d.b.64.5	✓	10
15.8	even	4		105.4.d.b.64.6	yes	10
15.14	odd	2		525.4.a.w.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.b.64.5	✓	10	15.2	even	4
105.4.d.b.64.6	yes	10	15.8	even	4
315.4.d.b.64.5		10	5.3	odd	4
315.4.d.b.64.6		10	5.2	odd	4
525.4.a.w.1.3		5	15.14	odd	2
525.4.a.x.1.3		5	3.2	odd	2
1575.4.a.bo.1.3		5	1.1	even	1	trivial
1575.4.a.bp.1.3		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)

\(f(q)\)	\(=\)	\(q+0.428319 q^{2} -7.81654 q^{4} +7.00000 q^{7} -6.77452 q^{8} +O(q^{10})\)	\(q+0.428319 q^{2} -7.81654 q^{4} +7.00000 q^{7} -6.77452 q^{8} +27.4721 q^{11} -46.5524 q^{13} +2.99823 q^{14} +59.6307 q^{16} -5.20546 q^{17} -91.0007 q^{19} +11.7668 q^{22} +111.563 q^{23} -19.9393 q^{26} -54.7158 q^{28} -0.0763413 q^{29} +201.784 q^{31} +79.7371 q^{32} -2.22960 q^{34} -312.859 q^{37} -38.9773 q^{38} -102.432 q^{41} +257.280 q^{43} -214.737 q^{44} +47.7847 q^{46} +350.994 q^{47} +49.0000 q^{49} +363.879 q^{52} -196.260 q^{53} -47.4217 q^{56} -0.0326984 q^{58} -881.060 q^{59} +737.897 q^{61} +86.4280 q^{62} -442.893 q^{64} +365.021 q^{67} +40.6887 q^{68} -1112.53 q^{71} +261.995 q^{73} -134.004 q^{74} +711.311 q^{76} +192.305 q^{77} +273.829 q^{79} -43.8735 q^{82} +87.1353 q^{83} +110.198 q^{86} -186.110 q^{88} +1090.99 q^{89} -325.867 q^{91} -872.039 q^{92} +150.337 q^{94} +228.830 q^{97} +20.9876 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 27 q^{4} + 35 q^{7} - 33 q^{8}+O(q^{10}) \) 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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