LMFDB - Embedded newform 475.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,4,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	475.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:	$28.0259072527$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-3.00000 q^{2} -7.00000 q^{3} +1.00000 q^{4} +21.0000 q^{6} -11.0000 q^{7} +21.0000 q^{8} +22.0000 q^{9} -36.0000 q^{11} -7.00000 q^{12} -65.0000 q^{13} +33.0000 q^{14} -71.0000 q^{16} +87.0000 q^{17} -66.0000 q^{18} +19.0000 q^{19} +77.0000 q^{21} +108.000 q^{22} +129.000 q^{23} -147.000 q^{24} +195.000 q^{26} +35.0000 q^{27} -11.0000 q^{28} +231.000 q^{29} +110.000 q^{31} +45.0000 q^{32} +252.000 q^{33} -261.000 q^{34} +22.0000 q^{36} +142.000 q^{37} -57.0000 q^{38} +455.000 q^{39} -330.000 q^{41} -231.000 q^{42} -74.0000 q^{43} -36.0000 q^{44} -387.000 q^{46} +336.000 q^{47} +497.000 q^{48} -222.000 q^{49} -609.000 q^{51} -65.0000 q^{52} -501.000 q^{53} -105.000 q^{54} -231.000 q^{56} -133.000 q^{57} -693.000 q^{58} +633.000 q^{59} -88.0000 q^{61} -330.000 q^{62} -242.000 q^{63} +433.000 q^{64} -756.000 q^{66} -119.000 q^{67} +87.0000 q^{68} -903.000 q^{69} -204.000 q^{71} +462.000 q^{72} -407.000 q^{73} -426.000 q^{74} +19.0000 q^{76} +396.000 q^{77} -1365.00 q^{78} +1262.00 q^{79} -839.000 q^{81} +990.000 q^{82} -270.000 q^{83} +77.0000 q^{84} +222.000 q^{86} -1617.00 q^{87} -756.000 q^{88} -30.0000 q^{89} +715.000 q^{91} +129.000 q^{92} -770.000 q^{93} -1008.00 q^{94} -315.000 q^{96} -1406.00 q^{97} +666.000 q^{98} -792.000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	1.00000	0.125000
$5$	0	0
$5$	0	0
$6$	21.0000	1.42887
$7$	−11.0000	−0.593944	−0.296972	−	0.954886i	$-0.595977\pi$
$7$	−11.0000	−0.593944	−0.296972	+	0.954886i	$0.595977\pi$
$8$	21.0000	0.928078
$9$	22.0000	0.814815
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	−7.00000	−0.168394
$13$	−65.0000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−65.0000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	33.0000	0.629973
$15$	0	0
$16$	−71.0000	−1.10938
$17$	87.0000	1.24121	0.620606	−	0.784123i	$-0.286887\pi$
$17$	87.0000	1.24121	0.620606	+	0.784123i	$0.286887\pi$
$18$	−66.0000	−0.864242
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	77.0000	0.800132
$22$	108.000	1.04662
$23$	129.000	1.16949	0.584747	−	0.811216i	$-0.301194\pi$
$23$	129.000	1.16949	0.584747	+	0.811216i	$0.301194\pi$
$24$	−147.000	−1.25026
$25$	0	0
$26$	195.000	1.47087
$27$	35.0000	0.249472
$28$	−11.0000	−0.0742430
$29$	231.000	1.47916	0.739580	−	0.673069i	$-0.235024\pi$
$29$	231.000	1.47916	0.739580	+	0.673069i	$0.235024\pi$
$30$	0	0
$31$	110.000	0.637309	0.318655	−	0.947871i	$-0.396769\pi$
$31$	110.000	0.637309	0.318655	+	0.947871i	$0.396769\pi$
$32$	45.0000	0.248592
$33$	252.000	1.32932
$34$	−261.000	−1.31650
$35$	0	0
$36$	22.0000	0.101852
$37$	142.000	0.630937	0.315468	−	0.948936i	$-0.397838\pi$
$37$	142.000	0.630937	0.315468	+	0.948936i	$0.397838\pi$
$38$	−57.0000	−0.243332
$39$	455.000	1.86816
$40$	0	0
$41$	−330.000	−1.25701	−0.628504	−	0.777806i	$-0.716332\pi$
$41$	−330.000	−1.25701	−0.628504	+	0.777806i	$0.716332\pi$
$42$	−231.000	−0.848668
$43$	−74.0000	−0.262439	−0.131220	−	0.991353i	$-0.541889\pi$
$43$	−74.0000	−0.262439	−0.131220	+	0.991353i	$0.541889\pi$
$44$	−36.0000	−0.123346
$45$	0	0
$46$	−387.000	−1.24044
$47$	336.000	1.04278	0.521390	−	0.853319i	$-0.325414\pi$
$47$	336.000	1.04278	0.521390	+	0.853319i	$0.325414\pi$
$48$	497.000	1.49450
$49$	−222.000	−0.647230
$50$	0	0
$51$	−609.000	−1.67210
$52$	−65.0000	−0.173344
$53$	−501.000	−1.29845	−0.649223	−	0.760598i	$-0.724906\pi$
$53$	−501.000	−1.29845	−0.649223	+	0.760598i	$0.724906\pi$
$54$	−105.000	−0.264605
$55$	0	0
$56$	−231.000	−0.551226
$57$	−133.000	−0.309058
$58$	−693.000	−1.56889
$59$	633.000	1.39677	0.698386	−	0.715721i	$-0.253902\pi$
$59$	633.000	1.39677	0.698386	+	0.715721i	$0.253902\pi$
$60$	0	0
$61$	−88.0000	−0.184709	−0.0923545	−	0.995726i	$-0.529439\pi$
$61$	−88.0000	−0.184709	−0.0923545	+	0.995726i	$0.529439\pi$
$62$	−330.000	−0.675968
$63$	−242.000	−0.483955
$64$	433.000	0.845703
$65$	0	0
$66$	−756.000	−1.40996
$67$	−119.000	−0.216988	−0.108494	−	0.994097i	$-0.534603\pi$
$67$	−119.000	−0.216988	−0.108494	+	0.994097i	$0.534603\pi$
$68$	87.0000	0.155151
$69$	−903.000	−1.57548
$70$	0	0
$71$	−204.000	−0.340991	−0.170495	−	0.985358i	$-0.554537\pi$
$71$	−204.000	−0.340991	−0.170495	+	0.985358i	$0.554537\pi$
$72$	462.000	0.756211
$73$	−407.000	−0.652544	−0.326272	−	0.945276i	$-0.605793\pi$
$73$	−407.000	−0.652544	−0.326272	+	0.945276i	$0.605793\pi$
$74$	−426.000	−0.669209
$75$	0	0
$76$	19.0000	0.0286770
$77$	396.000	0.586083
$78$	−1365.00	−1.98148
$79$	1262.00	1.79729	0.898646	−	0.438674i	$-0.144552\pi$
$79$	1262.00	1.79729	0.898646	+	0.438674i	$0.144552\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	990.000	1.33326
$83$	−270.000	−0.357064	−0.178532	−	0.983934i	$-0.557135\pi$
$83$	−270.000	−0.357064	−0.178532	+	0.983934i	$0.557135\pi$
$84$	77.0000	0.100017
$85$	0	0
$86$	222.000	0.278359
$87$	−1617.00	−1.99265
$88$	−756.000	−0.915794
$89$	−30.0000	−0.0357303	−0.0178651	−	0.999840i	$-0.505687\pi$
$89$	−30.0000	−0.0357303	−0.0178651	+	0.999840i	$0.505687\pi$
$90$	0	0
$91$	715.000	0.823652
$92$	129.000	0.146187
$93$	−770.000	−0.858551
$94$	−1008.00	−1.10603
$95$	0	0
$96$	−315.000	−0.334891
$97$	−1406.00	−1.47173	−0.735864	−	0.677129i	$-0.763224\pi$
$97$	−1406.00	−1.47173	−0.735864	+	0.677129i	$0.763224\pi$
$98$	666.000	0.686491
$99$	−792.000	−0.804030
$100$	0	0
$101$	1032.00	1.01671	0.508356	−	0.861147i	$-0.330254\pi$
$101$	1032.00	1.01671	0.508356	+	0.861147i	$0.330254\pi$
$102$	1827.00	1.77353
$103$	484.000	0.463009	0.231505	−	0.972834i	$-0.425635\pi$
$103$	484.000	0.463009	0.231505	+	0.972834i	$0.425635\pi$
$104$	−1365.00	−1.28701
$105$	0	0
$106$	1503.00	1.37721
$107$	−1881.00	−1.69947	−0.849734	−	0.527211i	$-0.823238\pi$
$107$	−1881.00	−1.69947	−0.849734	+	0.527211i	$0.823238\pi$
$108$	35.0000	0.0311840
$109$	281.000	0.246926	0.123463	−	0.992349i	$-0.460600\pi$
$109$	281.000	0.246926	0.123463	+	0.992349i	$0.460600\pi$
$110$	0	0
$111$	−994.000	−0.849967
$112$	781.000	0.658907
$113$	−612.000	−0.509488	−0.254744	−	0.967009i	$-0.581991\pi$
$113$	−612.000	−0.509488	−0.254744	+	0.967009i	$0.581991\pi$
$114$	399.000	0.327805
$115$	0	0
$116$	231.000	0.184895
$117$	−1430.00	−1.12994
$118$	−1899.00	−1.48150
$119$	−957.000	−0.737210
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	264.000	0.195913
$123$	2310.00	1.69338
$124$	110.000	0.0796636
$125$	0	0
$126$	726.000	0.513311
$127$	1582.00	1.10535	0.552676	−	0.833396i	$-0.313607\pi$
$127$	1582.00	1.10535	0.552676	+	0.833396i	$0.313607\pi$
$128$	−1659.00	−1.14560
$129$	518.000	0.353545
$130$	0	0
$131$	−1932.00	−1.28855	−0.644273	−	0.764795i	$-0.722840\pi$
$131$	−1932.00	−1.28855	−0.644273	+	0.764795i	$0.722840\pi$
$132$	252.000	0.166165
$133$	−209.000	−0.136260
$134$	357.000	0.230150
$135$	0	0
$136$	1827.00	1.15194
$137$	1515.00	0.944782	0.472391	−	0.881389i	$-0.343391\pi$
$137$	1515.00	0.944782	0.472391	+	0.881389i	$0.343391\pi$
$138$	2709.00	1.67105
$139$	−2284.00	−1.39371	−0.696857	−	0.717210i	$-0.745419\pi$
$139$	−2284.00	−1.39371	−0.696857	+	0.717210i	$0.745419\pi$
$140$	0	0
$141$	−2352.00	−1.40478
$142$	612.000	0.361675
$143$	2340.00	1.36840
$144$	−1562.00	−0.903935
$145$	0	0
$146$	1221.00	0.692128
$147$	1554.00	0.871917
$148$	142.000	0.0788671
$149$	−1356.00	−0.745556	−0.372778	−	0.927921i	$-0.621595\pi$
$149$	−1356.00	−0.745556	−0.372778	+	0.927921i	$0.621595\pi$
$150$	0	0
$151$	2234.00	1.20398	0.601988	−	0.798505i	$-0.294376\pi$
$151$	2234.00	1.20398	0.601988	+	0.798505i	$0.294376\pi$
$152$	399.000	0.212916
$153$	1914.00	1.01136
$154$	−1188.00	−0.621635
$155$	0	0
$156$	455.000	0.233520
$157$	−146.000	−0.0742170	−0.0371085	−	0.999311i	$-0.511815\pi$
$157$	−146.000	−0.0742170	−0.0371085	+	0.999311i	$0.511815\pi$
$158$	−3786.00	−1.90632
$159$	3507.00	1.74920
$160$	0	0
$161$	−1419.00	−0.694614
$162$	2517.00	1.22070
$163$	−1154.00	−0.554529	−0.277265	−	0.960794i	$-0.589428\pi$
$163$	−1154.00	−0.554529	−0.277265	+	0.960794i	$0.589428\pi$
$164$	−330.000	−0.157126
$165$	0	0
$166$	810.000	0.378724
$167$	3558.00	1.64866	0.824330	−	0.566109i	$-0.191552\pi$
$167$	3558.00	1.64866	0.824330	+	0.566109i	$0.191552\pi$
$168$	1617.00	0.742585
$169$	2028.00	0.923077
$170$	0	0
$171$	418.000	0.186931
$172$	−74.0000	−0.0328049
$173$	1482.00	0.651297	0.325648	−	0.945491i	$-0.394417\pi$
$173$	1482.00	0.651297	0.325648	+	0.945491i	$0.394417\pi$
$174$	4851.00	2.11353
$175$	0	0
$176$	2556.00	1.09469
$177$	−4431.00	−1.88166
$178$	90.0000	0.0378977
$179$	−2880.00	−1.20258	−0.601289	−	0.799032i	$-0.705346\pi$
$179$	−2880.00	−1.20258	−0.601289	+	0.799032i	$0.705346\pi$
$180$	0	0
$181$	470.000	0.193010	0.0965050	−	0.995332i	$-0.469234\pi$
$181$	470.000	0.193010	0.0965050	+	0.995332i	$0.469234\pi$
$182$	−2145.00	−0.873615
$183$	616.000	0.248831
$184$	2709.00	1.08538
$185$	0	0
$186$	2310.00	0.910631
$187$	−3132.00	−1.22478
$188$	336.000	0.130347
$189$	−385.000	−0.148173
$190$	0	0
$191$	−2475.00	−0.937616	−0.468808	−	0.883300i	$-0.655316\pi$
$191$	−2475.00	−0.937616	−0.468808	+	0.883300i	$0.655316\pi$
$192$	−3031.00	−1.13929
$193$	−1982.00	−0.739210	−0.369605	−	0.929189i	$-0.620507\pi$
$193$	−1982.00	−0.739210	−0.369605	+	0.929189i	$0.620507\pi$
$194$	4218.00	1.56100
$195$	0	0
$196$	−222.000	−0.0809038
$197$	−4122.00	−1.49076	−0.745382	−	0.666638i	$-0.767733\pi$
$197$	−4122.00	−1.49076	−0.745382	+	0.666638i	$0.767733\pi$
$198$	2376.00	0.852803
$199$	−4075.00	−1.45160	−0.725802	−	0.687904i	$-0.758531\pi$
$199$	−4075.00	−1.45160	−0.725802	+	0.687904i	$0.758531\pi$
$200$	0	0
$201$	833.000	0.292315
$202$	−3096.00	−1.07839
$203$	−2541.00	−0.878538
$204$	−609.000	−0.209012
$205$	0	0
$206$	−1452.00	−0.491095
$207$	2838.00	0.952921
$208$	4615.00	1.53843
$209$	−684.000	−0.226379
$210$	0	0
$211$	2765.00	0.902135	0.451067	−	0.892490i	$-0.351043\pi$
$211$	2765.00	0.902135	0.451067	+	0.892490i	$0.351043\pi$
$212$	−501.000	−0.162306
$213$	1428.00	0.459366
$214$	5643.00	1.80256
$215$	0	0
$216$	735.000	0.231530
$217$	−1210.00	−0.378526
$218$	−843.000	−0.261904
$219$	2849.00	0.879076
$220$	0	0
$221$	−5655.00	−1.72125
$222$	2982.00	0.901526
$223$	5326.00	1.59935	0.799676	−	0.600432i	$-0.205005\pi$
$223$	5326.00	1.59935	0.799676	+	0.600432i	$0.205005\pi$
$224$	−495.000	−0.147650
$225$	0	0
$226$	1836.00	0.540393
$227$	1881.00	0.549984	0.274992	−	0.961447i	$-0.411325\pi$
$227$	1881.00	0.549984	0.274992	+	0.961447i	$0.411325\pi$
$228$	−133.000	−0.0386322
$229$	758.000	0.218734	0.109367	−	0.994001i	$-0.465118\pi$
$229$	758.000	0.218734	0.109367	+	0.994001i	$0.465118\pi$
$230$	0	0
$231$	−2772.00	−0.789542
$232$	4851.00	1.37277
$233$	5334.00	1.49975	0.749875	−	0.661579i	$-0.230113\pi$
$233$	5334.00	1.49975	0.749875	+	0.661579i	$0.230113\pi$
$234$	4290.00	1.19849
$235$	0	0
$236$	633.000	0.174597
$237$	−8834.00	−2.42122
$238$	2871.00	0.781930
$239$	−4671.00	−1.26419	−0.632096	−	0.774890i	$-0.717805\pi$
$239$	−4671.00	−1.26419	−0.632096	+	0.774890i	$0.717805\pi$
$240$	0	0
$241$	1748.00	0.467214	0.233607	−	0.972331i	$-0.424947\pi$
$241$	1748.00	0.467214	0.233607	+	0.972331i	$0.424947\pi$
$242$	105.000	0.0278911
$243$	4928.00	1.30095
$244$	−88.0000	−0.0230886
$245$	0	0
$246$	−6930.00	−1.79610
$247$	−1235.00	−0.318142
$248$	2310.00	0.591472
$249$	1890.00	0.481020
$250$	0	0
$251$	3984.00	1.00186	0.500932	−	0.865487i	$-0.332991\pi$
$251$	3984.00	1.00186	0.500932	+	0.865487i	$0.332991\pi$
$252$	−242.000	−0.0604943
$253$	−4644.00	−1.15401
$254$	−4746.00	−1.17240
$255$	0	0
$256$	1513.00	0.369385
$257$	−6126.00	−1.48688	−0.743442	−	0.668800i	$-0.766808\pi$
$257$	−6126.00	−1.48688	−0.743442	+	0.668800i	$0.766808\pi$
$258$	−1554.00	−0.374992
$259$	−1562.00	−0.374741
$260$	0	0
$261$	5082.00	1.20524
$262$	5796.00	1.36671
$263$	6576.00	1.54180	0.770900	−	0.636956i	$-0.219807\pi$
$263$	6576.00	1.54180	0.770900	+	0.636956i	$0.219807\pi$
$264$	5292.00	1.23371
$265$	0	0
$266$	627.000	0.144526
$267$	210.000	0.0481340
$268$	−119.000	−0.0271234
$269$	6078.00	1.37763	0.688814	−	0.724938i	$-0.258131\pi$
$269$	6078.00	1.37763	0.688814	+	0.724938i	$0.258131\pi$
$270$	0	0
$271$	−1015.00	−0.227516	−0.113758	−	0.993508i	$-0.536289\pi$
$271$	−1015.00	−0.227516	−0.113758	+	0.993508i	$0.536289\pi$
$272$	−6177.00	−1.37697
$273$	−5005.00	−1.10958
$274$	−4545.00	−1.00209
$275$	0	0
$276$	−903.000	−0.196936
$277$	1924.00	0.417336	0.208668	−	0.977987i	$-0.433087\pi$
$277$	1924.00	0.417336	0.208668	+	0.977987i	$0.433087\pi$
$278$	6852.00	1.47826
$279$	2420.00	0.519289
$280$	0	0
$281$	−4134.00	−0.877629	−0.438815	−	0.898578i	$-0.644601\pi$
$281$	−4134.00	−0.877629	−0.438815	+	0.898578i	$0.644601\pi$
$282$	7056.00	1.49000
$283$	−5798.00	−1.21786	−0.608932	−	0.793223i	$-0.708402\pi$
$283$	−5798.00	−1.21786	−0.608932	+	0.793223i	$0.708402\pi$
$284$	−204.000	−0.0426238
$285$	0	0
$286$	−7020.00	−1.45140
$287$	3630.00	0.746593
$288$	990.000	0.202557
$289$	2656.00	0.540607
$290$	0	0
$291$	9842.00	1.98264
$292$	−407.000	−0.0815681
$293$	−9315.00	−1.85730	−0.928649	−	0.370960i	$-0.879029\pi$
$293$	−9315.00	−1.85730	−0.928649	+	0.370960i	$0.879029\pi$
$294$	−4662.00	−0.924807
$295$	0	0
$296$	2982.00	0.585558
$297$	−1260.00	−0.246170
$298$	4068.00	0.790782
$299$	−8385.00	−1.62180
$300$	0	0
$301$	814.000	0.155874
$302$	−6702.00	−1.27701
$303$	−7224.00	−1.36966
$304$	−1349.00	−0.254508
$305$	0	0
$306$	−5742.00	−1.07271
$307$	7036.00	1.30803	0.654016	−	0.756481i	$-0.273083\pi$
$307$	7036.00	1.30803	0.654016	+	0.756481i	$0.273083\pi$
$308$	396.000	0.0732604
$309$	−3388.00	−0.623743
$310$	0	0
$311$	−2691.00	−0.490651	−0.245326	−	0.969441i	$-0.578895\pi$
$311$	−2691.00	−0.490651	−0.245326	+	0.969441i	$0.578895\pi$
$312$	9555.00	1.73380
$313$	565.000	0.102031	0.0510155	−	0.998698i	$-0.483754\pi$
$313$	565.000	0.102031	0.0510155	+	0.998698i	$0.483754\pi$
$314$	438.000	0.0787190
$315$	0	0
$316$	1262.00	0.224662
$317$	−2463.00	−0.436391	−0.218195	−	0.975905i	$-0.570017\pi$
$317$	−2463.00	−0.436391	−0.218195	+	0.975905i	$0.570017\pi$
$318$	−10521.0	−1.85531
$319$	−8316.00	−1.45958
$320$	0	0
$321$	13167.0	2.28944
$322$	4257.00	0.736749
$323$	1653.00	0.284753
$324$	−839.000	−0.143861
$325$	0	0
$326$	3462.00	0.588167
$327$	−1967.00	−0.332646
$328$	−6930.00	−1.16660
$329$	−3696.00	−0.619353
$330$	0	0
$331$	7463.00	1.23929	0.619643	−	0.784884i	$-0.287277\pi$
$331$	7463.00	1.23929	0.619643	+	0.784884i	$0.287277\pi$
$332$	−270.000	−0.0446331
$333$	3124.00	0.514097
$334$	−10674.0	−1.74867
$335$	0	0
$336$	−5467.00	−0.887647
$337$	11194.0	1.80942	0.904712	−	0.426023i	$-0.140086\pi$
$337$	11194.0	1.80942	0.904712	+	0.426023i	$0.140086\pi$
$338$	−6084.00	−0.979071
$339$	4284.00	0.686357
$340$	0	0
$341$	−3960.00	−0.628874
$342$	−1254.00	−0.198271
$343$	6215.00	0.978363
$344$	−1554.00	−0.243564
$345$	0	0
$346$	−4446.00	−0.690805
$347$	−7806.00	−1.20763	−0.603816	−	0.797124i	$-0.706354\pi$
$347$	−7806.00	−1.20763	−0.603816	+	0.797124i	$0.706354\pi$
$348$	−1617.00	−0.249081
$349$	−8278.00	−1.26966	−0.634830	−	0.772652i	$-0.718930\pi$
$349$	−8278.00	−1.26966	−0.634830	+	0.772652i	$0.718930\pi$
$350$	0	0
$351$	−2275.00	−0.345956
$352$	−1620.00	−0.245302
$353$	−7443.00	−1.12224	−0.561120	−	0.827734i	$-0.689629\pi$
$353$	−7443.00	−1.12224	−0.561120	+	0.827734i	$0.689629\pi$
$354$	13293.0	1.99581
$355$	0	0
$356$	−30.0000	−0.00446628
$357$	6699.00	0.993134
$358$	8640.00	1.27553
$359$	−3705.00	−0.544686	−0.272343	−	0.962200i	$-0.587799\pi$
$359$	−3705.00	−0.544686	−0.272343	+	0.962200i	$0.587799\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−1410.00	−0.204718
$363$	245.000	0.0354247
$364$	715.000	0.102957
$365$	0	0
$366$	−1848.00	−0.263925
$367$	−992.000	−0.141095	−0.0705477	−	0.997508i	$-0.522475\pi$
$367$	−992.000	−0.141095	−0.0705477	+	0.997508i	$0.522475\pi$
$368$	−9159.00	−1.29741
$369$	−7260.00	−1.02423
$370$	0	0
$371$	5511.00	0.771204
$372$	−770.000	−0.107319
$373$	−11135.0	−1.54571	−0.772853	−	0.634585i	$-0.781171\pi$
$373$	−11135.0	−1.54571	−0.772853	+	0.634585i	$0.781171\pi$
$374$	9396.00	1.29908
$375$	0	0
$376$	7056.00	0.967780
$377$	−15015.0	−2.05123
$378$	1155.00	0.157161
$379$	−7135.00	−0.967019	−0.483510	−	0.875339i	$-0.660638\pi$
$379$	−7135.00	−0.967019	−0.483510	+	0.875339i	$0.660638\pi$
$380$	0	0
$381$	−11074.0	−1.48908
$382$	7425.00	0.994492
$383$	−5124.00	−0.683614	−0.341807	−	0.939770i	$-0.611039\pi$
$383$	−5124.00	−0.683614	−0.341807	+	0.939770i	$0.611039\pi$
$384$	11613.0	1.54329
$385$	0	0
$386$	5946.00	0.784050
$387$	−1628.00	−0.213840
$388$	−1406.00	−0.183966
$389$	11004.0	1.43425	0.717127	−	0.696942i	$-0.245457\pi$
$389$	11004.0	1.43425	0.717127	+	0.696942i	$0.245457\pi$
$390$	0	0
$391$	11223.0	1.45159
$392$	−4662.00	−0.600680
$393$	13524.0	1.73587
$394$	12366.0	1.58119
$395$	0	0
$396$	−792.000	−0.100504
$397$	2428.00	0.306947	0.153473	−	0.988153i	$-0.450954\pi$
$397$	2428.00	0.306947	0.153473	+	0.988153i	$0.450954\pi$
$398$	12225.0	1.53966
$399$	1463.00	0.183563
$400$	0	0
$401$	−4146.00	−0.516313	−0.258156	−	0.966103i	$-0.583115\pi$
$401$	−4146.00	−0.516313	−0.258156	+	0.966103i	$0.583115\pi$
$402$	−2499.00	−0.310047
$403$	−7150.00	−0.883789
$404$	1032.00	0.127089
$405$	0	0
$406$	7623.00	0.931830
$407$	−5112.00	−0.622586
$408$	−12789.0	−1.55184
$409$	11630.0	1.40603	0.703015	−	0.711175i	$-0.251837\pi$
$409$	11630.0	1.40603	0.703015	+	0.711175i	$0.251837\pi$
$410$	0	0
$411$	−10605.0	−1.27276
$412$	484.000	0.0578761
$413$	−6963.00	−0.829605
$414$	−8514.00	−1.01073
$415$	0	0
$416$	−2925.00	−0.344735
$417$	15988.0	1.87754
$418$	2052.00	0.240111
$419$	−14670.0	−1.71044	−0.855222	−	0.518261i	$-0.826580\pi$
$419$	−14670.0	−1.71044	−0.855222	+	0.518261i	$0.826580\pi$
$420$	0	0
$421$	−8989.00	−1.04061	−0.520305	−	0.853980i	$-0.674182\pi$
$421$	−8989.00	−1.04061	−0.520305	+	0.853980i	$0.674182\pi$
$422$	−8295.00	−0.956858
$423$	7392.00	0.849672
$424$	−10521.0	−1.20506
$425$	0	0
$426$	−4284.00	−0.487231
$427$	968.000	0.109707
$428$	−1881.00	−0.212434
$429$	−16380.0	−1.84344
$430$	0	0
$431$	−10344.0	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−10344.0	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−2485.00	−0.276758
$433$	4048.00	0.449271	0.224636	−	0.974443i	$-0.427881\pi$
$433$	4048.00	0.449271	0.224636	+	0.974443i	$0.427881\pi$
$434$	3630.00	0.401488
$435$	0	0
$436$	281.000	0.0308657
$437$	2451.00	0.268300
$438$	−8547.00	−0.932401
$439$	7526.00	0.818215	0.409107	−	0.912486i	$-0.365840\pi$
$439$	7526.00	0.818215	0.409107	+	0.912486i	$0.365840\pi$
$440$	0	0
$441$	−4884.00	−0.527373
$442$	16965.0	1.82566
$443$	8796.00	0.943365	0.471682	−	0.881769i	$-0.343647\pi$
$443$	8796.00	0.943365	0.471682	+	0.881769i	$0.343647\pi$
$444$	−994.000	−0.106246
$445$	0	0
$446$	−15978.0	−1.69637
$447$	9492.00	1.00438
$448$	−4763.00	−0.502300
$449$	1638.00	0.172165	0.0860824	−	0.996288i	$-0.472565\pi$
$449$	1638.00	0.172165	0.0860824	+	0.996288i	$0.472565\pi$
$450$	0	0
$451$	11880.0	1.24037
$452$	−612.000	−0.0636860
$453$	−15638.0	−1.62194
$454$	−5643.00	−0.583346
$455$	0	0
$456$	−2793.00	−0.286829
$457$	−10703.0	−1.09555	−0.547774	−	0.836627i	$-0.684525\pi$
$457$	−10703.0	−1.09555	−0.547774	+	0.836627i	$0.684525\pi$
$458$	−2274.00	−0.232002
$459$	3045.00	0.309648
$460$	0	0
$461$	−7350.00	−0.742568	−0.371284	−	0.928519i	$-0.621082\pi$
$461$	−7350.00	−0.742568	−0.371284	+	0.928519i	$0.621082\pi$
$462$	8316.00	0.837436
$463$	−11288.0	−1.13304	−0.566520	−	0.824048i	$-0.691711\pi$
$463$	−11288.0	−1.13304	−0.566520	+	0.824048i	$0.691711\pi$
$464$	−16401.0	−1.64094
$465$	0	0
$466$	−16002.0	−1.59073
$467$	−13644.0	−1.35197	−0.675984	−	0.736916i	$-0.736281\pi$
$467$	−13644.0	−1.35197	−0.675984	+	0.736916i	$0.736281\pi$
$468$	−1430.00	−0.141243
$469$	1309.00	0.128878
$470$	0	0
$471$	1022.00	0.0999815
$472$	13293.0	1.29631
$473$	2664.00	0.258966
$474$	26502.0	2.56810
$475$	0	0
$476$	−957.000	−0.0921513
$477$	−11022.0	−1.05799
$478$	14013.0	1.34088
$479$	11472.0	1.09430	0.547149	−	0.837035i	$-0.315713\pi$
$479$	11472.0	1.09430	0.547149	+	0.837035i	$0.315713\pi$
$480$	0	0
$481$	−9230.00	−0.874952
$482$	−5244.00	−0.495555
$483$	9933.00	0.935750
$484$	−35.0000	−0.00328700
$485$	0	0
$486$	−14784.0	−1.37987
$487$	9394.00	0.874092	0.437046	−	0.899439i	$-0.356025\pi$
$487$	9394.00	0.874092	0.437046	+	0.899439i	$0.356025\pi$
$488$	−1848.00	−0.171424
$489$	8078.00	0.747034
$490$	0	0
$491$	−6.00000	−0.000551479 0	−0.000275740	−	1.00000i	$-0.500088\pi$
$491$	−6.00000	−0.000551479 0	−0.000275740		1.00000i	$0.500088\pi$
$492$	2310.00	0.211672
$493$	20097.0	1.83595
$494$	3705.00	0.337441
$495$	0	0
$496$	−7810.00	−0.707015
$497$	2244.00	0.202529
$498$	−5670.00	−0.510198
$499$	8894.00	0.797896	0.398948	−	0.916974i	$-0.369375\pi$
$499$	8894.00	0.797896	0.398948	+	0.916974i	$0.369375\pi$
$500$	0	0
$501$	−24906.0	−2.22099
$502$	−11952.0	−1.06264
$503$	−7107.00	−0.629991	−0.314995	−	0.949093i	$-0.602003\pi$
$503$	−7107.00	−0.629991	−0.314995	+	0.949093i	$0.602003\pi$
$504$	−5082.00	−0.449147
$505$	0	0
$506$	13932.0	1.22402
$507$	−14196.0	−1.24352
$508$	1582.00	0.138169
$509$	−20946.0	−1.82400	−0.911999	−	0.410192i	$-0.865462\pi$
$509$	−20946.0	−1.82400	−0.911999	+	0.410192i	$0.865462\pi$
$510$	0	0
$511$	4477.00	0.387575
$512$	8733.00	0.753804
$513$	665.000	0.0572329
$514$	18378.0	1.57708
$515$	0	0
$516$	518.000	0.0441932
$517$	−12096.0	−1.02898
$518$	4686.00	0.397473
$519$	−10374.0	−0.877395
$520$	0	0
$521$	15696.0	1.31987	0.659937	−	0.751321i	$-0.270583\pi$
$521$	15696.0	1.31987	0.659937	+	0.751321i	$0.270583\pi$
$522$	−15246.0	−1.27835
$523$	6667.00	0.557414	0.278707	−	0.960376i	$-0.410094\pi$
$523$	6667.00	0.557414	0.278707	+	0.960376i	$0.410094\pi$
$524$	−1932.00	−0.161068
$525$	0	0
$526$	−19728.0	−1.63533
$527$	9570.00	0.791036
$528$	−17892.0	−1.47471
$529$	4474.00	0.367716
$530$	0	0
$531$	13926.0	1.13811
$532$	−209.000	−0.0170325
$533$	21450.0	1.74316
$534$	−630.000	−0.0510539
$535$	0	0
$536$	−2499.00	−0.201381
$537$	20160.0	1.62005
$538$	−18234.0	−1.46120
$539$	7992.00	0.638664
$540$	0	0
$541$	−16792.0	−1.33446	−0.667231	−	0.744850i	$-0.732521\pi$
$541$	−16792.0	−1.33446	−0.667231	+	0.744850i	$0.732521\pi$
$542$	3045.00	0.241317
$543$	−3290.00	−0.260014
$544$	3915.00	0.308556
$545$	0	0
$546$	15015.0	1.17689
$547$	1276.00	0.0997401	0.0498700	−	0.998756i	$-0.484119\pi$
$547$	1276.00	0.0997401	0.0498700	+	0.998756i	$0.484119\pi$
$548$	1515.00	0.118098
$549$	−1936.00	−0.150504
$550$	0	0
$551$	4389.00	0.339342
$552$	−18963.0	−1.46217
$553$	−13882.0	−1.06749
$554$	−5772.00	−0.442651
$555$	0	0
$556$	−2284.00	−0.174214
$557$	21144.0	1.60844	0.804219	−	0.594333i	$-0.202584\pi$
$557$	21144.0	1.60844	0.804219	+	0.594333i	$0.202584\pi$
$558$	−7260.00	−0.550789
$559$	4810.00	0.363938
$560$	0	0
$561$	21924.0	1.64997
$562$	12402.0	0.930866
$563$	14652.0	1.09682	0.548409	−	0.836210i	$-0.315234\pi$
$563$	14652.0	1.09682	0.548409	+	0.836210i	$0.315234\pi$
$564$	−2352.00	−0.175598
$565$	0	0
$566$	17394.0	1.29174
$567$	9229.00	0.683565
$568$	−4284.00	−0.316466
$569$	18288.0	1.34740	0.673702	−	0.739003i	$-0.264703\pi$
$569$	18288.0	1.34740	0.673702	+	0.739003i	$0.264703\pi$
$570$	0	0
$571$	19766.0	1.44865	0.724327	−	0.689457i	$-0.242151\pi$
$571$	19766.0	1.44865	0.724327	+	0.689457i	$0.242151\pi$
$572$	2340.00	0.171050
$573$	17325.0	1.26311
$574$	−10890.0	−0.791881
$575$	0	0
$576$	9526.00	0.689091
$577$	−16229.0	−1.17092	−0.585461	−	0.810700i	$-0.699087\pi$
$577$	−16229.0	−1.17092	−0.585461	+	0.810700i	$0.699087\pi$
$578$	−7968.00	−0.573400
$579$	13874.0	0.995827
$580$	0	0
$581$	2970.00	0.212076
$582$	−29526.0	−2.10291
$583$	18036.0	1.28126
$584$	−8547.00	−0.605612
$585$	0	0
$586$	27945.0	1.96996
$587$	3750.00	0.263678	0.131839	−	0.991271i	$-0.457912\pi$
$587$	3750.00	0.263678	0.131839	+	0.991271i	$0.457912\pi$
$588$	1554.00	0.108990
$589$	2090.00	0.146209
$590$	0	0
$591$	28854.0	2.00828
$592$	−10082.0	−0.699945
$593$	22530.0	1.56020	0.780098	−	0.625657i	$-0.215169\pi$
$593$	22530.0	1.56020	0.780098	+	0.625657i	$0.215169\pi$
$594$	3780.00	0.261103
$595$	0	0
$596$	−1356.00	−0.0931945
$597$	28525.0	1.95553
$598$	25155.0	1.72017
$599$	11850.0	0.808310	0.404155	−	0.914690i	$-0.367566\pi$
$599$	11850.0	0.808310	0.404155	+	0.914690i	$0.367566\pi$
$600$	0	0
$601$	−16918.0	−1.14825	−0.574126	−	0.818767i	$-0.694658\pi$
$601$	−16918.0	−1.14825	−0.574126	+	0.818767i	$0.694658\pi$
$602$	−2442.00	−0.165330
$603$	−2618.00	−0.176805
$604$	2234.00	0.150497
$605$	0	0
$606$	21672.0	1.45275
$607$	−2702.00	−0.180677	−0.0903384	−	0.995911i	$-0.528795\pi$
$607$	−2702.00	−0.180677	−0.0903384	+	0.995911i	$0.528795\pi$
$608$	855.000	0.0570310
$609$	17787.0	1.18352
$610$	0	0
$611$	−21840.0	−1.44608
$612$	1914.00	0.126420
$613$	−18884.0	−1.24424	−0.622119	−	0.782923i	$-0.713728\pi$
$613$	−18884.0	−1.24424	−0.622119	+	0.782923i	$0.713728\pi$
$614$	−21108.0	−1.38738
$615$	0	0
$616$	8316.00	0.543931
$617$	−9798.00	−0.639307	−0.319654	−	0.947534i	$-0.603567\pi$
$617$	−9798.00	−0.639307	−0.319654	+	0.947534i	$0.603567\pi$
$618$	10164.0	0.661579
$619$	1172.00	0.0761012	0.0380506	−	0.999276i	$-0.487885\pi$
$619$	1172.00	0.0761012	0.0380506	+	0.999276i	$0.487885\pi$
$620$	0	0
$621$	4515.00	0.291756
$622$	8073.00	0.520414
$623$	330.000	0.0212218
$624$	−32305.0	−2.07249
$625$	0	0
$626$	−1695.00	−0.108220
$627$	4788.00	0.304967
$628$	−146.000	−0.00927712
$629$	12354.0	0.783126
$630$	0	0
$631$	920.000	0.0580422	0.0290211	−	0.999579i	$-0.490761\pi$
$631$	920.000	0.0580422	0.0290211	+	0.999579i	$0.490761\pi$
$632$	26502.0	1.66803
$633$	−19355.0	−1.21531
$634$	7389.00	0.462862
$635$	0	0
$636$	3507.00	0.218650
$637$	14430.0	0.897547
$638$	24948.0	1.54812
$639$	−4488.00	−0.277844
$640$	0	0
$641$	−9240.00	−0.569357	−0.284679	−	0.958623i	$-0.591887\pi$
$641$	−9240.00	−0.569357	−0.284679	+	0.958623i	$0.591887\pi$
$642$	−39501.0	−2.42832
$643$	−8246.00	−0.505739	−0.252870	−	0.967500i	$-0.581374\pi$
$643$	−8246.00	−0.505739	−0.252870	+	0.967500i	$0.581374\pi$
$644$	−1419.00	−0.0868268
$645$	0	0
$646$	−4959.00	−0.302027
$647$	−11475.0	−0.697262	−0.348631	−	0.937260i	$-0.613353\pi$
$647$	−11475.0	−0.697262	−0.348631	+	0.937260i	$0.613353\pi$
$648$	−17619.0	−1.06812
$649$	−22788.0	−1.37829
$650$	0	0
$651$	8470.00	0.509932
$652$	−1154.00	−0.0693161
$653$	−2196.00	−0.131602	−0.0658010	−	0.997833i	$-0.520960\pi$
$653$	−2196.00	−0.131602	−0.0658010	+	0.997833i	$0.520960\pi$
$654$	5901.00	0.352825
$655$	0	0
$656$	23430.0	1.39449
$657$	−8954.00	−0.531703
$658$	11088.0	0.656923
$659$	−21513.0	−1.27167	−0.635833	−	0.771827i	$-0.719343\pi$
$659$	−21513.0	−1.27167	−0.635833	+	0.771827i	$0.719343\pi$
$660$	0	0
$661$	31853.0	1.87434	0.937170	−	0.348874i	$-0.113436\pi$
$661$	31853.0	1.87434	0.937170	+	0.348874i	$0.113436\pi$
$662$	−22389.0	−1.31446
$663$	39585.0	2.31878
$664$	−5670.00	−0.331384
$665$	0	0
$666$	−9372.00	−0.545282
$667$	29799.0	1.72987
$668$	3558.00	0.206083
$669$	−37282.0	−2.15457
$670$	0	0
$671$	3168.00	0.182264
$672$	3465.00	0.198907
$673$	−27308.0	−1.56411	−0.782055	−	0.623209i	$-0.785828\pi$
$673$	−27308.0	−1.56411	−0.782055	+	0.623209i	$0.785828\pi$
$674$	−33582.0	−1.91918
$675$	0	0
$676$	2028.00	0.115385
$677$	12609.0	0.715810	0.357905	−	0.933758i	$-0.383491\pi$
$677$	12609.0	0.715810	0.357905	+	0.933758i	$0.383491\pi$
$678$	−12852.0	−0.727991
$679$	15466.0	0.874125
$680$	0	0
$681$	−13167.0	−0.740911
$682$	11880.0	0.667022
$683$	−32292.0	−1.80911	−0.904553	−	0.426362i	$-0.859795\pi$
$683$	−32292.0	−1.80911	−0.904553	+	0.426362i	$0.859795\pi$
$684$	418.000	0.0233664
$685$	0	0
$686$	−18645.0	−1.03771
$687$	−5306.00	−0.294667
$688$	5254.00	0.291144
$689$	32565.0	1.80062
$690$	0	0
$691$	−16558.0	−0.911572	−0.455786	−	0.890089i	$-0.650642\pi$
$691$	−16558.0	−0.911572	−0.455786	+	0.890089i	$0.650642\pi$
$692$	1482.00	0.0814121
$693$	8712.00	0.477549
$694$	23418.0	1.28089
$695$	0	0
$696$	−33957.0	−1.84933
$697$	−28710.0	−1.56021
$698$	24834.0	1.34668
$699$	−37338.0	−2.02039
$700$	0	0
$701$	−23712.0	−1.27759	−0.638794	−	0.769377i	$-0.720567\pi$
$701$	−23712.0	−1.27759	−0.638794	+	0.769377i	$0.720567\pi$
$702$	6825.00	0.366942
$703$	2698.00	0.144747
$704$	−15588.0	−0.834510
$705$	0	0
$706$	22329.0	1.19032
$707$	−11352.0	−0.603870
$708$	−4431.00	−0.235208
$709$	10334.0	0.547393	0.273696	−	0.961816i	$-0.411754\pi$
$709$	10334.0	0.547393	0.273696	+	0.961816i	$0.411754\pi$
$710$	0	0
$711$	27764.0	1.46446
$712$	−630.000	−0.0331605
$713$	14190.0	0.745329
$714$	−20097.0	−1.05338
$715$	0	0
$716$	−2880.00	−0.150322
$717$	32697.0	1.70306
$718$	11115.0	0.577727
$719$	1353.00	0.0701786	0.0350893	−	0.999384i	$-0.488828\pi$
$719$	1353.00	0.0701786	0.0350893	+	0.999384i	$0.488828\pi$
$720$	0	0
$721$	−5324.00	−0.275002
$722$	−1083.00	−0.0558242
$723$	−12236.0	−0.629408
$724$	470.000	0.0241263
$725$	0	0
$726$	−735.000	−0.0375736
$727$	−8687.00	−0.443168	−0.221584	−	0.975141i	$-0.571123\pi$
$727$	−8687.00	−0.443168	−0.221584	+	0.975141i	$0.571123\pi$
$728$	15015.0	0.764413
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−6438.00	−0.325743
$732$	616.000	0.0311038
$733$	−29144.0	−1.46857	−0.734283	−	0.678844i	$-0.762481\pi$
$733$	−29144.0	−1.46857	−0.734283	+	0.678844i	$0.762481\pi$
$734$	2976.00	0.149654
$735$	0	0
$736$	5805.00	0.290727
$737$	4284.00	0.214116
$738$	21780.0	1.08636
$739$	−2050.00	−0.102044	−0.0510220	−	0.998698i	$-0.516248\pi$
$739$	−2050.00	−0.102044	−0.0510220	+	0.998698i	$0.516248\pi$
$740$	0	0
$741$	8645.00	0.428586
$742$	−16533.0	−0.817986
$743$	−5142.00	−0.253892	−0.126946	−	0.991910i	$-0.540517\pi$
$743$	−5142.00	−0.253892	−0.126946	+	0.991910i	$0.540517\pi$
$744$	−16170.0	−0.796802
$745$	0	0
$746$	33405.0	1.63947
$747$	−5940.00	−0.290941
$748$	−3132.00	−0.153098
$749$	20691.0	1.00939
$750$	0	0
$751$	−27070.0	−1.31531	−0.657655	−	0.753319i	$-0.728452\pi$
$751$	−27070.0	−1.31531	−0.657655	+	0.753319i	$0.728452\pi$
$752$	−23856.0	−1.15683
$753$	−27888.0	−1.34966
$754$	45045.0	2.17565
$755$	0	0
$756$	−385.000	−0.0185216
$757$	−10694.0	−0.513448	−0.256724	−	0.966485i	$-0.582643\pi$
$757$	−10694.0	−0.513448	−0.256724	+	0.966485i	$0.582643\pi$
$758$	21405.0	1.02568
$759$	32508.0	1.55463
$760$	0	0
$761$	−21285.0	−1.01390	−0.506952	−	0.861974i	$-0.669228\pi$
$761$	−21285.0	−1.01390	−0.506952	+	0.861974i	$0.669228\pi$
$762$	33222.0	1.57940
$763$	−3091.00	−0.146660
$764$	−2475.00	−0.117202
$765$	0	0
$766$	15372.0	0.725082
$767$	−41145.0	−1.93698
$768$	−10591.0	−0.497617
$769$	12809.0	0.600656	0.300328	−	0.953836i	$-0.402904\pi$
$769$	12809.0	0.600656	0.300328	+	0.953836i	$0.402904\pi$
$770$	0	0
$771$	42882.0	2.00306
$772$	−1982.00	−0.0924012
$773$	1893.00	0.0880808	0.0440404	−	0.999030i	$-0.485977\pi$
$773$	1893.00	0.0880808	0.0440404	+	0.999030i	$0.485977\pi$
$774$	4884.00	0.226811
$775$	0	0
$776$	−29526.0	−1.36588
$777$	10934.0	0.504833
$778$	−33012.0	−1.52126
$779$	−6270.00	−0.288377
$780$	0	0
$781$	7344.00	0.336478
$782$	−33669.0	−1.53964
$783$	8085.00	0.369009
$784$	15762.0	0.718021
$785$	0	0
$786$	−40572.0	−1.84116
$787$	32101.0	1.45397	0.726987	−	0.686652i	$-0.240920\pi$
$787$	32101.0	1.45397	0.726987	+	0.686652i	$0.240920\pi$
$788$	−4122.00	−0.186345
$789$	−46032.0	−2.07704
$790$	0	0
$791$	6732.00	0.302607
$792$	−16632.0	−0.746203
$793$	5720.00	0.256145
$794$	−7284.00	−0.325566
$795$	0	0
$796$	−4075.00	−0.181450
$797$	−21753.0	−0.966789	−0.483394	−	0.875403i	$-0.660596\pi$
$797$	−21753.0	−0.966789	−0.483394	+	0.875403i	$0.660596\pi$
$798$	−4389.00	−0.194698
$799$	29232.0	1.29431
$800$	0	0
$801$	−660.000	−0.0291135
$802$	12438.0	0.547632
$803$	14652.0	0.643908
$804$	833.000	0.0365394
$805$	0	0
$806$	21450.0	0.937400
$807$	−42546.0	−1.85587
$808$	21672.0	0.943587
$809$	−14247.0	−0.619157	−0.309578	−	0.950874i	$-0.600188\pi$
$809$	−14247.0	−0.619157	−0.309578	+	0.950874i	$0.600188\pi$
$810$	0	0
$811$	−25045.0	−1.08440	−0.542200	−	0.840249i	$-0.682409\pi$
$811$	−25045.0	−1.08440	−0.542200	+	0.840249i	$0.682409\pi$
$812$	−2541.00	−0.109817
$813$	7105.00	0.306498
$814$	15336.0	0.660352
$815$	0	0
$816$	43239.0	1.85499
$817$	−1406.00	−0.0602077
$818$	−34890.0	−1.49132
$819$	15730.0	0.671124
$820$	0	0
$821$	−22224.0	−0.944730	−0.472365	−	0.881403i	$-0.656599\pi$
$821$	−22224.0	−0.944730	−0.472365	+	0.881403i	$0.656599\pi$
$822$	31815.0	1.34997
$823$	31867.0	1.34971	0.674856	−	0.737949i	$-0.264206\pi$
$823$	31867.0	1.34971	0.674856	+	0.737949i	$0.264206\pi$
$824$	10164.0	0.429708
$825$	0	0
$826$	20889.0	0.879929
$827$	4233.00	0.177988	0.0889939	−	0.996032i	$-0.471635\pi$
$827$	4233.00	0.177988	0.0889939	+	0.996032i	$0.471635\pi$
$828$	2838.00	0.119115
$829$	26561.0	1.11279	0.556394	−	0.830918i	$-0.312184\pi$
$829$	26561.0	1.11279	0.556394	+	0.830918i	$0.312184\pi$
$830$	0	0
$831$	−13468.0	−0.562214
$832$	−28145.0	−1.17278
$833$	−19314.0	−0.803350
$834$	−47964.0	−1.99144
$835$	0	0
$836$	−684.000	−0.0282974
$837$	3850.00	0.158991
$838$	44010.0	1.81420
$839$	3792.00	0.156036	0.0780181	−	0.996952i	$-0.475141\pi$
$839$	3792.00	0.156036	0.0780181	+	0.996952i	$0.475141\pi$
$840$	0	0
$841$	28972.0	1.18791
$842$	26967.0	1.10373
$843$	28938.0	1.18230
$844$	2765.00	0.112767
$845$	0	0
$846$	−22176.0	−0.901213
$847$	385.000	0.0156184
$848$	35571.0	1.44046
$849$	40586.0	1.64065
$850$	0	0
$851$	18318.0	0.737877
$852$	1428.00	0.0574207
$853$	−26858.0	−1.07808	−0.539039	−	0.842281i	$-0.681212\pi$
$853$	−26858.0	−1.07808	−0.539039	+	0.842281i	$0.681212\pi$
$854$	−2904.00	−0.116362
$855$	0	0
$856$	−39501.0	−1.57724
$857$	−6672.00	−0.265941	−0.132970	−	0.991120i	$-0.542452\pi$
$857$	−6672.00	−0.265941	−0.132970	+	0.991120i	$0.542452\pi$
$858$	49140.0	1.95526
$859$	−8764.00	−0.348107	−0.174053	−	0.984736i	$-0.555687\pi$
$859$	−8764.00	−0.348107	−0.174053	+	0.984736i	$0.555687\pi$
$860$	0	0
$861$	−25410.0	−1.00577
$862$	31032.0	1.22616
$863$	6012.00	0.237139	0.118569	−	0.992946i	$-0.462169\pi$
$863$	6012.00	0.237139	0.118569	+	0.992946i	$0.462169\pi$
$864$	1575.00	0.0620169
$865$	0	0
$866$	−12144.0	−0.476524
$867$	−18592.0	−0.728278
$868$	−1210.00	−0.0473158
$869$	−45432.0	−1.77350
$870$	0	0
$871$	7735.00	0.300908
$872$	5901.00	0.229166
$873$	−30932.0	−1.19919
$874$	−7353.00	−0.284575
$875$	0	0
$876$	2849.00	0.109884
$877$	−47207.0	−1.81764	−0.908818	−	0.417192i	$-0.863014\pi$
$877$	−47207.0	−1.81764	−0.908818	+	0.417192i	$0.863014\pi$
$878$	−22578.0	−0.867848
$879$	65205.0	2.50206
$880$	0	0
$881$	21618.0	0.826707	0.413354	−	0.910571i	$-0.364357\pi$
$881$	21618.0	0.826707	0.413354	+	0.910571i	$0.364357\pi$
$882$	14652.0	0.559363
$883$	−25706.0	−0.979701	−0.489850	−	0.871807i	$-0.662949\pi$
$883$	−25706.0	−0.979701	−0.489850	+	0.871807i	$0.662949\pi$
$884$	−5655.00	−0.215156
$885$	0	0
$886$	−26388.0	−1.00059
$887$	−3336.00	−0.126282	−0.0631409	−	0.998005i	$-0.520112\pi$
$887$	−3336.00	−0.126282	−0.0631409	+	0.998005i	$0.520112\pi$
$888$	−20874.0	−0.788835
$889$	−17402.0	−0.656518
$890$	0	0
$891$	30204.0	1.13566
$892$	5326.00	0.199919
$893$	6384.00	0.239230
$894$	−28476.0	−1.06530
$895$	0	0
$896$	18249.0	0.680420
$897$	58695.0	2.18480
$898$	−4914.00	−0.182608
$899$	25410.0	0.942682
$900$	0	0
$901$	−43587.0	−1.61165
$902$	−35640.0	−1.31561
$903$	−5698.00	−0.209986
$904$	−12852.0	−0.472844
$905$	0	0
$906$	46914.0	1.72032
$907$	18979.0	0.694804	0.347402	−	0.937716i	$-0.387064\pi$
$907$	18979.0	0.694804	0.347402	+	0.937716i	$0.387064\pi$
$908$	1881.00	0.0687480
$909$	22704.0	0.828431
$910$	0	0
$911$	3600.00	0.130926	0.0654629	−	0.997855i	$-0.479148\pi$
$911$	3600.00	0.130926	0.0654629	+	0.997855i	$0.479148\pi$
$912$	9443.00	0.342861
$913$	9720.00	0.352338
$914$	32109.0	1.16200
$915$	0	0
$916$	758.000	0.0273417
$917$	21252.0	0.765325
$918$	−9135.00	−0.328431
$919$	13655.0	0.490138	0.245069	−	0.969506i	$-0.421189\pi$
$919$	13655.0	0.490138	0.245069	+	0.969506i	$0.421189\pi$
$920$	0	0
$921$	−49252.0	−1.76212
$922$	22050.0	0.787612
$923$	13260.0	0.472869
$924$	−2772.00	−0.0986928
$925$	0	0
$926$	33864.0	1.20177
$927$	10648.0	0.377267
$928$	10395.0	0.367708
$929$	−35043.0	−1.23759	−0.618796	−	0.785551i	$-0.712379\pi$
$929$	−35043.0	−1.23759	−0.618796	+	0.785551i	$0.712379\pi$
$930$	0	0
$931$	−4218.00	−0.148485
$932$	5334.00	0.187469
$933$	18837.0	0.660981
$934$	40932.0	1.43398
$935$	0	0
$936$	−30030.0	−1.04868
$937$	7045.00	0.245624	0.122812	−	0.992430i	$-0.460809\pi$
$937$	7045.00	0.245624	0.122812	+	0.992430i	$0.460809\pi$
$938$	−3927.00	−0.136696
$939$	−3955.00	−0.137451
$940$	0	0
$941$	3399.00	0.117752	0.0588758	−	0.998265i	$-0.481248\pi$
$941$	3399.00	0.117752	0.0588758	+	0.998265i	$0.481248\pi$
$942$	−3066.00	−0.106046
$943$	−42570.0	−1.47006
$944$	−44943.0	−1.54954
$945$	0	0
$946$	−7992.00	−0.274675
$947$	22884.0	0.785248	0.392624	−	0.919699i	$-0.371567\pi$
$947$	22884.0	0.785248	0.392624	+	0.919699i	$0.371567\pi$
$948$	−8834.00	−0.302653
$949$	26455.0	0.904916
$950$	0	0
$951$	17241.0	0.587884
$952$	−20097.0	−0.684189
$953$	−50520.0	−1.71721	−0.858606	−	0.512636i	$-0.828669\pi$
$953$	−50520.0	−1.71721	−0.858606	+	0.512636i	$0.828669\pi$
$954$	33066.0	1.12217
$955$	0	0
$956$	−4671.00	−0.158024
$957$	58212.0	1.96628
$958$	−34416.0	−1.16068
$959$	−16665.0	−0.561148
$960$	0	0
$961$	−17691.0	−0.593837
$962$	27690.0	0.928026
$963$	−41382.0	−1.38475
$964$	1748.00	0.0584018
$965$	0	0
$966$	−29799.0	−0.992513
$967$	7468.00	0.248350	0.124175	−	0.992260i	$-0.460372\pi$
$967$	7468.00	0.248350	0.124175	+	0.992260i	$0.460372\pi$
$968$	−735.000	−0.0244047
$969$	−11571.0	−0.383606
$970$	0	0
$971$	11172.0	0.369234	0.184617	−	0.982811i	$-0.440895\pi$
$971$	11172.0	0.369234	0.184617	+	0.982811i	$0.440895\pi$
$972$	4928.00	0.162619
$973$	25124.0	0.827789
$974$	−28182.0	−0.927115
$975$	0	0
$976$	6248.00	0.204911
$977$	−35586.0	−1.16530	−0.582649	−	0.812724i	$-0.697984\pi$
$977$	−35586.0	−1.16530	−0.582649	+	0.812724i	$0.697984\pi$
$978$	−24234.0	−0.792350
$979$	1080.00	0.0352574
$980$	0	0
$981$	6182.00	0.201199
$982$	18.0000	0.000584932 0
$983$	−29694.0	−0.963471	−0.481735	−	0.876317i	$-0.659993\pi$
$983$	−29694.0	−0.963471	−0.481735	+	0.876317i	$0.659993\pi$
$984$	48510.0	1.57159
$985$	0	0
$986$	−60291.0	−1.94732
$987$	25872.0	0.834362
$988$	−1235.00	−0.0397678
$989$	−9546.00	−0.306921
$990$	0	0
$991$	−46546.0	−1.49201	−0.746005	−	0.665940i	$-0.768031\pi$
$991$	−46546.0	−1.49201	−0.746005	+	0.665940i	$0.768031\pi$
$992$	4950.00	0.158430
$993$	−52241.0	−1.66950
$994$	−6732.00	−0.214815
$995$	0	0
$996$	1890.00	0.0601275
$997$	−6338.00	−0.201330	−0.100665	−	0.994920i	$-0.532097\pi$
$997$	−6338.00	−0.201330	−0.100665	+	0.994920i	$0.532097\pi$
$998$	−26682.0	−0.846297
$999$	4970.00	0.157401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.a.b.1.1		1
5.2	odd	4		475.4.b.d.324.1		2
5.3	odd	4		475.4.b.d.324.2		2
5.4	even	2		95.4.a.c.1.1	✓	1
15.14	odd	2		855.4.a.c.1.1		1
20.19	odd	2		1520.4.a.a.1.1		1
95.94	odd	2		1805.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.4.a.c.1.1	✓	1	5.4	even	2
475.4.a.b.1.1		1	1.1	even	1	trivial
475.4.b.d.324.1		2	5.2	odd	4
475.4.b.d.324.2		2	5.3	odd	4
855.4.a.c.1.1		1	15.14	odd	2
1520.4.a.a.1.1		1	20.19	odd	2
1805.4.a.c.1.1		1	95.94	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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