LMFDB - Embedded newform 1875.4.a.k.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,4,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1875.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:	$110.628581261$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.18816 q^{2} -3.00000 q^{3} +18.9170 q^{4} -15.5645 q^{6} -32.8379 q^{7} +56.6394 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.18816 q^{2} -3.00000 q^{3} +18.9170 q^{4} -15.5645 q^{6} -32.8379 q^{7} +56.6394 q^{8} +9.00000 q^{9} +40.3540 q^{11} -56.7511 q^{12} -86.7227 q^{13} -170.369 q^{14} +142.518 q^{16} +58.8915 q^{17} +46.6935 q^{18} -10.2059 q^{19} +98.5138 q^{21} +209.363 q^{22} +92.0018 q^{23} -169.918 q^{24} -449.932 q^{26} -27.0000 q^{27} -621.197 q^{28} +187.828 q^{29} +212.483 q^{31} +286.293 q^{32} -121.062 q^{33} +305.539 q^{34} +170.253 q^{36} -107.979 q^{37} -52.9501 q^{38} +260.168 q^{39} -66.9945 q^{41} +511.106 q^{42} +315.447 q^{43} +763.378 q^{44} +477.320 q^{46} +181.705 q^{47} -427.555 q^{48} +735.330 q^{49} -176.675 q^{51} -1640.54 q^{52} +169.876 q^{53} -140.080 q^{54} -1859.92 q^{56} +30.6178 q^{57} +974.482 q^{58} +861.341 q^{59} -262.518 q^{61} +1102.40 q^{62} -295.541 q^{63} +345.187 q^{64} -628.089 q^{66} +178.935 q^{67} +1114.05 q^{68} -276.005 q^{69} -249.739 q^{71} +509.755 q^{72} -148.657 q^{73} -560.212 q^{74} -193.066 q^{76} -1325.14 q^{77} +1349.79 q^{78} +1135.32 q^{79} +81.0000 q^{81} -347.579 q^{82} -1072.31 q^{83} +1863.59 q^{84} +1636.59 q^{86} -563.484 q^{87} +2285.63 q^{88} +100.655 q^{89} +2847.79 q^{91} +1740.40 q^{92} -637.450 q^{93} +942.715 q^{94} -858.878 q^{96} -302.229 q^{97} +3815.01 q^{98} +363.186 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9}+O(q^{10}) $ 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9	\( 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9} + 96 q^{11} - 399 q^{12} - 156 q^{13} + 92 q^{14} + 845 q^{16} + 46 q^{17} - 9 q^{18} + 182 q^{19} + 186 q^{21} - 158 q^{22} + 286 q^{23} + 81 q^{24} + 478 q^{26} - 648 q^{27} - 701 q^{28} + 1144 q^{29} + 64 q^{31} - 1212 q^{32} - 288 q^{33} + 961 q^{34} + 1197 q^{36} - 762 q^{37} - 474 q^{38} + 468 q^{39} + 1074 q^{41} - 276 q^{42} - 460 q^{43} + 319 q^{44} + 459 q^{46} + 960 q^{47} - 2535 q^{48} + 2680 q^{49} - 138 q^{51} - 2969 q^{52} - 914 q^{53} + 27 q^{54} + 1680 q^{56} - 546 q^{57} - 208 q^{58} + 208 q^{59} + 3520 q^{61} - 334 q^{62} - 558 q^{63} + 5747 q^{64} + 474 q^{66} - 154 q^{67} + 5727 q^{68} - 858 q^{69} - 252 q^{71} - 243 q^{72} - 4414 q^{73} + 5637 q^{74} + 627 q^{76} - 2344 q^{77} - 1434 q^{78} + 1110 q^{79} + 1944 q^{81} - 3714 q^{82} + 1488 q^{83} + 2103 q^{84} + 3036 q^{86} - 3432 q^{87} - 3947 q^{88} + 3402 q^{89} + 3504 q^{91} + 11163 q^{92} - 192 q^{93} + 3408 q^{94} + 3636 q^{96} - 534 q^{97} - 2244 q^{98} + 864 q^{99}+O(q^{100}) \) 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9 + 96 * q^11 - 399 * q^12 - 156 * q^13 + 92 * q^14 + 845 * q^16 + 46 * q^17 - 9 * q^18 + 182 * q^19 + 186 * q^21 - 158 * q^22 + 286 * q^23 + 81 * q^24 + 478 * q^26 - 648 * q^27 - 701 * q^28 + 1144 * q^29 + 64 * q^31 - 1212 * q^32 - 288 * q^33 + 961 * q^34 + 1197 * q^36 - 762 * q^37 - 474 * q^38 + 468 * q^39 + 1074 * q^41 - 276 * q^42 - 460 * q^43 + 319 * q^44 + 459 * q^46 + 960 * q^47 - 2535 * q^48 + 2680 * q^49 - 138 * q^51 - 2969 * q^52 - 914 * q^53 + 27 * q^54 + 1680 * q^56 - 546 * q^57 - 208 * q^58 + 208 * q^59 + 3520 * q^61 - 334 * q^62 - 558 * q^63 + 5747 * q^64 + 474 * q^66 - 154 * q^67 + 5727 * q^68 - 858 * q^69 - 252 * q^71 - 243 * q^72 - 4414 * q^73 + 5637 * q^74 + 627 * q^76 - 2344 * q^77 - 1434 * q^78 + 1110 * q^79 + 1944 * q^81 - 3714 * q^82 + 1488 * q^83 + 2103 * q^84 + 3036 * q^86 - 3432 * q^87 - 3947 * q^88 + 3402 * q^89 + 3504 * q^91 + 11163 * q^92 - 192 * q^93 + 3408 * q^94 + 3636 * q^96 - 534 * q^97 - 2244 * q^98 + 864 * q^99

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 $

$2$	5.18816	1.83429	0.917146	−	0.398550i	$-0.130487\pi$
$2$	5.18816	1.83429	0.917146	+	0.398550i	$0.130487\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	18.9170	2.36463
$5$	0	0
$5$	0	0
$6$	−15.5645	−1.05903
$7$	−32.8379	−1.77308	−0.886541	−	0.462650i	$-0.846899\pi$
$7$	−32.8379	−1.77308	−0.886541	+	0.462650i	$0.846899\pi$
$8$	56.6394	2.50313
$9$	9.00000	0.333333
$10$	0	0
$11$	40.3540	1.10611	0.553054	−	0.833145i	$-0.313462\pi$
$11$	40.3540	1.10611	0.553054	+	0.833145i	$0.313462\pi$
$12$	−56.7511	−1.36522
$13$	−86.7227	−1.85020	−0.925098	−	0.379729i	$-0.876017\pi$
$13$	−86.7227	−1.85020	−0.925098	+	0.379729i	$0.876017\pi$
$14$	−170.369	−3.25235
$15$	0	0
$16$	142.518	2.22685
$17$	58.8915	0.840194	0.420097	−	0.907479i	$-0.361996\pi$
$17$	58.8915	0.840194	0.420097	+	0.907479i	$0.361996\pi$
$18$	46.6935	0.611431
$19$	−10.2059	−0.123232	−0.0616159	−	0.998100i	$-0.519625\pi$
$19$	−10.2059	−0.123232	−0.0616159	+	0.998100i	$0.519625\pi$
$20$	0	0
$21$	98.5138	1.02369
$22$	209.363	2.02893
$23$	92.0018	0.834074	0.417037	−	0.908889i	$-0.363069\pi$
$23$	92.0018	0.834074	0.417037	+	0.908889i	$0.363069\pi$
$24$	−169.918	−1.44518
$25$	0	0
$26$	−449.932	−3.39380
$27$	−27.0000	−0.192450
$28$	−621.197	−4.19268
$29$	187.828	1.20272	0.601358	−	0.798980i	$-0.294627\pi$
$29$	187.828	1.20272	0.601358	+	0.798980i	$0.294627\pi$
$30$	0	0
$31$	212.483	1.23107	0.615534	−	0.788110i	$-0.288940\pi$
$31$	212.483	1.23107	0.615534	+	0.788110i	$0.288940\pi$
$32$	286.293	1.58156
$33$	−121.062	−0.638612
$34$	305.539	1.54116
$35$	0	0
$36$	170.253	0.788210
$37$	−107.979	−0.479773	−0.239887	−	0.970801i	$-0.577110\pi$
$37$	−107.979	−0.479773	−0.239887	+	0.970801i	$0.577110\pi$
$38$	−52.9501	−0.226043
$39$	260.168	1.06821
$40$	0	0
$41$	−66.9945	−0.255190	−0.127595	−	0.991826i	$-0.540726\pi$
$41$	−66.9945	−0.255190	−0.127595	+	0.991826i	$0.540726\pi$
$42$	511.106	1.87775
$43$	315.447	1.11873	0.559364	−	0.828922i	$-0.311045\pi$
$43$	315.447	1.11873	0.559364	+	0.828922i	$0.311045\pi$
$44$	763.378	2.61554
$45$	0	0
$46$	477.320	1.52994
$47$	181.705	0.563923	0.281962	−	0.959426i	$-0.409015\pi$
$47$	181.705	0.563923	0.281962	+	0.959426i	$0.409015\pi$
$48$	−427.555	−1.28567
$49$	735.330	2.14382
$50$	0	0
$51$	−176.675	−0.485086
$52$	−1640.54	−4.37503
$53$	169.876	0.440268	0.220134	−	0.975470i	$-0.429351\pi$
$53$	169.876	0.440268	0.220134	+	0.975470i	$0.429351\pi$
$54$	−140.080	−0.353010
$55$	0	0
$56$	−1859.92	−4.43826
$57$	30.6178	0.0711479
$58$	974.482	2.20613
$59$	861.341	1.90063	0.950314	−	0.311291i	$-0.100762\pi$
$59$	861.341	1.90063	0.950314	+	0.311291i	$0.100762\pi$
$60$	0	0
$61$	−262.518	−0.551017	−0.275508	−	0.961299i	$-0.588846\pi$
$61$	−262.518	−0.551017	−0.275508	+	0.961299i	$0.588846\pi$
$62$	1102.40	2.25814
$63$	−295.541	−0.591027
$64$	345.187	0.674193
$65$	0	0
$66$	−628.089	−1.17140
$67$	178.935	0.326275	0.163137	−	0.986603i	$-0.447839\pi$
$67$	178.935	0.326275	0.163137	+	0.986603i	$0.447839\pi$
$68$	1114.05	1.98675
$69$	−276.005	−0.481553
$70$	0	0
$71$	−249.739	−0.417444	−0.208722	−	0.977975i	$-0.566930\pi$
$71$	−249.739	−0.417444	−0.208722	+	0.977975i	$0.566930\pi$
$72$	509.755	0.834377
$73$	−148.657	−0.238342	−0.119171	−	0.992874i	$-0.538024\pi$
$73$	−148.657	−0.238342	−0.119171	+	0.992874i	$0.538024\pi$
$74$	−560.212	−0.880045
$75$	0	0
$76$	−193.066	−0.291398
$77$	−1325.14	−1.96122
$78$	1349.79	1.95941
$79$	1135.32	1.61687	0.808437	−	0.588582i	$-0.200314\pi$
$79$	1135.32	1.61687	0.808437	+	0.588582i	$0.200314\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−347.579	−0.468093
$83$	−1072.31	−1.41809	−0.709046	−	0.705162i	$-0.750874\pi$
$83$	−1072.31	−1.41809	−0.709046	+	0.705162i	$0.750874\pi$
$84$	1863.59	2.42065
$85$	0	0
$86$	1636.59	2.05207
$87$	−563.484	−0.694388
$88$	2285.63	2.76873
$89$	100.655	0.119880	0.0599402	−	0.998202i	$-0.480909\pi$
$89$	100.655	0.119880	0.0599402	+	0.998202i	$0.480909\pi$
$90$	0	0
$91$	2847.79	3.28055
$92$	1740.40	1.97228
$93$	−637.450	−0.710757
$94$	942.715	1.03440
$95$	0	0
$96$	−858.878	−0.913113
$97$	−302.229	−0.316357	−0.158179	−	0.987410i	$-0.550562\pi$
$97$	−302.229	−0.316357	−0.158179	+	0.987410i	$0.550562\pi$
$98$	3815.01	3.93239
$99$	363.186	0.368703
$100$	0	0
$101$	1515.34	1.49290	0.746448	−	0.665444i	$-0.231758\pi$
$101$	1515.34	1.49290	0.746448	+	0.665444i	$0.231758\pi$
$102$	−916.617	−0.889790
$103$	1150.27	1.10038	0.550192	−	0.835038i	$-0.314555\pi$
$103$	1150.27	1.10038	0.550192	+	0.835038i	$0.314555\pi$
$104$	−4911.92	−4.63128
$105$	0	0
$106$	881.342	0.807580
$107$	851.549	0.769368	0.384684	−	0.923048i	$-0.374310\pi$
$107$	851.549	0.769368	0.384684	+	0.923048i	$0.374310\pi$
$108$	−510.760	−0.455073
$109$	467.495	0.410806	0.205403	−	0.978677i	$-0.434149\pi$
$109$	467.495	0.410806	0.205403	+	0.978677i	$0.434149\pi$
$110$	0	0
$111$	323.937	0.276997
$112$	−4680.00	−3.94838
$113$	62.4052	0.0519521	0.0259761	−	0.999663i	$-0.491731\pi$
$113$	62.4052	0.0519521	0.0259761	+	0.999663i	$0.491731\pi$
$114$	158.850	0.130506
$115$	0	0
$116$	3553.15	2.84398
$117$	−780.504	−0.616732
$118$	4468.78	3.48631
$119$	−1933.88	−1.48973
$120$	0	0
$121$	297.444	0.223474
$122$	−1361.99	−1.01073
$123$	200.984	0.147334
$124$	4019.55	2.91102
$125$	0	0
$126$	−1533.32	−1.08412
$127$	96.9347	0.0677288	0.0338644	−	0.999426i	$-0.489219\pi$
$127$	96.9347	0.0677288	0.0338644	+	0.999426i	$0.489219\pi$
$128$	−499.454	−0.344890
$129$	−946.342	−0.645898
$130$	0	0
$131$	−2981.14	−1.98827	−0.994136	−	0.108134i	$-0.965512\pi$
$131$	−2981.14	−1.98827	−0.994136	+	0.108134i	$0.965512\pi$
$132$	−2290.13	−1.51008
$133$	335.142	0.218500
$134$	928.346	0.598484
$135$	0	0
$136$	3335.58	2.10312
$137$	−2154.80	−1.34378	−0.671888	−	0.740653i	$-0.734516\pi$
$137$	−2154.80	−1.34378	−0.671888	+	0.740653i	$0.734516\pi$
$138$	−1431.96	−0.883309
$139$	−2405.32	−1.46774	−0.733872	−	0.679288i	$-0.762289\pi$
$139$	−2405.32	−1.46774	−0.733872	+	0.679288i	$0.762289\pi$
$140$	0	0
$141$	−545.115	−0.325581
$142$	−1295.69	−0.765715
$143$	−3499.61	−2.04652
$144$	1282.66	0.742282
$145$	0	0
$146$	−771.255	−0.437189
$147$	−2205.99	−1.23773
$148$	−2042.64	−1.13449
$149$	−282.950	−0.155571	−0.0777857	−	0.996970i	$-0.524785\pi$
$149$	−282.950	−0.155571	−0.0777857	+	0.996970i	$0.524785\pi$
$150$	0	0
$151$	2534.40	1.36587	0.682936	−	0.730479i	$-0.260703\pi$
$151$	2534.40	1.36587	0.682936	+	0.730479i	$0.260703\pi$
$152$	−578.058	−0.308465
$153$	530.024	0.280065
$154$	−6875.05	−3.59745
$155$	0	0
$156$	4921.61	2.52592
$157$	473.314	0.240602	0.120301	−	0.992737i	$-0.461614\pi$
$157$	473.314	0.240602	0.120301	+	0.992737i	$0.461614\pi$
$158$	5890.21	2.96582
$159$	−509.627	−0.254189
$160$	0	0
$161$	−3021.15	−1.47888
$162$	420.241	0.203810
$163$	2397.46	1.15205	0.576023	−	0.817433i	$-0.304604\pi$
$163$	2397.46	1.15205	0.576023	+	0.817433i	$0.304604\pi$
$164$	−1267.34	−0.603430
$165$	0	0
$166$	−5563.33	−2.60120
$167$	1646.65	0.763004	0.381502	−	0.924368i	$-0.375407\pi$
$167$	1646.65	0.763004	0.381502	+	0.924368i	$0.375407\pi$
$168$	5579.76	2.56243
$169$	5323.82	2.42322
$170$	0	0
$171$	−91.8534	−0.0410772
$172$	5967.33	2.64538
$173$	2145.03	0.942678	0.471339	−	0.881952i	$-0.343771\pi$
$173$	2145.03	0.942678	0.471339	+	0.881952i	$0.343771\pi$
$174$	−2923.44	−1.27371
$175$	0	0
$176$	5751.18	2.46313
$177$	−2584.02	−1.09733
$178$	522.212	0.219896
$179$	−860.453	−0.359292	−0.179646	−	0.983731i	$-0.557495\pi$
$179$	−860.453	−0.359292	−0.179646	+	0.983731i	$0.557495\pi$
$180$	0	0
$181$	2719.12	1.11663	0.558316	−	0.829628i	$-0.311448\pi$
$181$	2719.12	1.11663	0.558316	+	0.829628i	$0.311448\pi$
$182$	14774.8	6.01749
$183$	787.555	0.318130
$184$	5210.93	2.08780
$185$	0	0
$186$	−3307.19	−1.30374
$187$	2376.51	0.929345
$188$	3437.32	1.33347
$189$	886.624	0.341230
$190$	0	0
$191$	−3911.54	−1.48183	−0.740914	−	0.671600i	$-0.765607\pi$
$191$	−3911.54	−1.48183	−0.740914	+	0.671600i	$0.765607\pi$
$192$	−1035.56	−0.389246
$193$	3101.74	1.15683	0.578415	−	0.815742i	$-0.303671\pi$
$193$	3101.74	1.15683	0.578415	+	0.815742i	$0.303671\pi$
$194$	−1568.01	−0.580292
$195$	0	0
$196$	13910.3	5.06934
$197$	411.645	0.148875	0.0744377	−	0.997226i	$-0.476284\pi$
$197$	411.645	0.148875	0.0744377	+	0.997226i	$0.476284\pi$
$198$	1884.27	0.676309
$199$	−1809.24	−0.644490	−0.322245	−	0.946656i	$-0.604438\pi$
$199$	−1809.24	−0.644490	−0.322245	+	0.946656i	$0.604438\pi$
$200$	0	0
$201$	−536.806	−0.188375
$202$	7861.86	2.73841
$203$	−6167.88	−2.13251
$204$	−3342.16	−1.14705
$205$	0	0
$206$	5967.79	2.01843
$207$	828.016	0.278025
$208$	−12359.6	−4.12010
$209$	−411.850	−0.136308
$210$	0	0
$211$	−4532.22	−1.47872	−0.739362	−	0.673308i	$-0.764873\pi$
$211$	−4532.22	−1.47872	−0.739362	+	0.673308i	$0.764873\pi$
$212$	3213.54	1.04107
$213$	749.217	0.241012
$214$	4417.98	1.41125
$215$	0	0
$216$	−1529.26	−0.481728
$217$	−6977.51	−2.18278
$218$	2425.44	0.753539
$219$	445.970	0.137607
$220$	0	0
$221$	−5107.23	−1.55452
$222$	1680.64	0.508094
$223$	995.899	0.299060	0.149530	−	0.988757i	$-0.452224\pi$
$223$	995.899	0.299060	0.149530	+	0.988757i	$0.452224\pi$
$224$	−9401.26	−2.80423
$225$	0	0
$226$	323.769	0.0952954
$227$	−180.556	−0.0527927	−0.0263963	−	0.999652i	$-0.508403\pi$
$227$	−180.556	−0.0527927	−0.0263963	+	0.999652i	$0.508403\pi$
$228$	579.199	0.168238
$229$	583.351	0.168336	0.0841680	−	0.996452i	$-0.473177\pi$
$229$	583.351	0.168336	0.0841680	+	0.996452i	$0.473177\pi$
$230$	0	0
$231$	3975.42	1.13231
$232$	10638.5	3.01056
$233$	4866.42	1.36828	0.684141	−	0.729350i	$-0.260177\pi$
$233$	4866.42	1.36828	0.684141	+	0.729350i	$0.260177\pi$
$234$	−4049.38	−1.13127
$235$	0	0
$236$	16294.0	4.49429
$237$	−3405.95	−0.933503
$238$	−10033.3	−2.73260
$239$	3177.95	0.860102	0.430051	−	0.902805i	$-0.358496\pi$
$239$	3177.95	0.860102	0.430051	+	0.902805i	$0.358496\pi$
$240$	0	0
$241$	−3072.28	−0.821174	−0.410587	−	0.911821i	$-0.634676\pi$
$241$	−3072.28	−0.821174	−0.410587	+	0.911821i	$0.634676\pi$
$242$	1543.19	0.409918
$243$	−243.000	−0.0641500
$244$	−4966.07	−1.30295
$245$	0	0
$246$	1042.74	0.270254
$247$	885.086	0.228003
$248$	12034.9	3.08153
$249$	3216.94	0.818736
$250$	0	0
$251$	4578.87	1.15146	0.575729	−	0.817641i	$-0.304718\pi$
$251$	4578.87	1.15146	0.575729	+	0.817641i	$0.304718\pi$
$252$	−5590.77	−1.39756
$253$	3712.64	0.922576
$254$	502.913	0.124235
$255$	0	0
$256$	−5352.75	−1.30682
$257$	−4478.88	−1.08710	−0.543551	−	0.839376i	$-0.682920\pi$
$257$	−4478.88	−1.08710	−0.543551	+	0.839376i	$0.682920\pi$
$258$	−4909.78	−1.18477
$259$	3545.80	0.850677
$260$	0	0
$261$	1690.45	0.400905
$262$	−15466.7	−3.64707
$263$	351.408	0.0823908	0.0411954	−	0.999151i	$-0.486883\pi$
$263$	351.408	0.0823908	0.0411954	+	0.999151i	$0.486883\pi$
$264$	−6856.88	−1.59853
$265$	0	0
$266$	1738.77	0.400793
$267$	−301.964	−0.0692130
$268$	3384.93	0.771520
$269$	1251.16	0.283586	0.141793	−	0.989896i	$-0.454713\pi$
$269$	1251.16	0.283586	0.141793	+	0.989896i	$0.454713\pi$
$270$	0	0
$271$	−5370.40	−1.20380	−0.601898	−	0.798573i	$-0.705589\pi$
$271$	−5370.40	−1.20380	−0.601898	+	0.798573i	$0.705589\pi$
$272$	8393.11	1.87098
$273$	−8543.38	−1.89403
$274$	−11179.5	−2.46488
$275$	0	0
$276$	−5221.21	−1.13869
$277$	−1848.00	−0.400850	−0.200425	−	0.979709i	$-0.564232\pi$
$277$	−1848.00	−0.400850	−0.200425	+	0.979709i	$0.564232\pi$
$278$	−12479.2	−2.69227
$279$	1912.35	0.410356
$280$	0	0
$281$	3838.35	0.814864	0.407432	−	0.913236i	$-0.366424\pi$
$281$	3838.35	0.814864	0.407432	+	0.913236i	$0.366424\pi$
$282$	−2828.15	−0.597211
$283$	2484.14	0.521791	0.260896	−	0.965367i	$-0.415982\pi$
$283$	2484.14	0.521791	0.260896	+	0.965367i	$0.415982\pi$
$284$	−4724.32	−0.987102
$285$	0	0
$286$	−18156.5	−3.75391
$287$	2199.96	0.452473
$288$	2576.63	0.527186
$289$	−1444.79	−0.294075
$290$	0	0
$291$	906.686	0.182649
$292$	−2812.14	−0.563590
$293$	1628.31	0.324666	0.162333	−	0.986736i	$-0.448098\pi$
$293$	1628.31	0.324666	0.162333	+	0.986736i	$0.448098\pi$
$294$	−11445.0	−2.27037
$295$	0	0
$296$	−6115.86	−1.20094
$297$	−1089.56	−0.212871
$298$	−1467.99	−0.285363
$299$	−7978.64	−1.54320
$300$	0	0
$301$	−10358.6	−1.98360
$302$	13148.9	2.50541
$303$	−4546.03	−0.861923
$304$	−1454.53	−0.274418
$305$	0	0
$306$	2749.85	0.513720
$307$	−9852.49	−1.83163	−0.915817	−	0.401597i	$-0.868455\pi$
$307$	−9852.49	−1.83163	−0.915817	+	0.401597i	$0.868455\pi$
$308$	−25067.8	−4.63756
$309$	−3450.81	−0.635307
$310$	0	0
$311$	−4336.40	−0.790657	−0.395329	−	0.918540i	$-0.629369\pi$
$311$	−4336.40	−0.790657	−0.395329	+	0.918540i	$0.629369\pi$
$312$	14735.8	2.67387
$313$	2578.58	0.465655	0.232827	−	0.972518i	$-0.425202\pi$
$313$	2578.58	0.465655	0.232827	+	0.972518i	$0.425202\pi$
$314$	2455.63	0.441335
$315$	0	0
$316$	21476.8	3.82331
$317$	823.029	0.145823	0.0729115	−	0.997338i	$-0.476771\pi$
$317$	823.029	0.145823	0.0729115	+	0.997338i	$0.476771\pi$
$318$	−2644.03	−0.466257
$319$	7579.60	1.33033
$320$	0	0
$321$	−2554.65	−0.444195
$322$	−15674.2	−2.71270
$323$	−601.043	−0.103539
$324$	1532.28	0.262737
$325$	0	0
$326$	12438.4	2.11319
$327$	−1402.49	−0.237179
$328$	−3794.53	−0.638774
$329$	−5966.82	−0.999882
$330$	0	0
$331$	2324.54	0.386007	0.193004	−	0.981198i	$-0.438177\pi$
$331$	2324.54	0.386007	0.193004	+	0.981198i	$0.438177\pi$
$332$	−20285.0	−3.35326
$333$	−971.810	−0.159924
$334$	8543.10	1.39957
$335$	0	0
$336$	14040.0	2.27960
$337$	−3055.46	−0.493892	−0.246946	−	0.969029i	$-0.579427\pi$
$337$	−3055.46	−0.493892	−0.246946	+	0.969029i	$0.579427\pi$
$338$	27620.9	4.44490
$339$	−187.216	−0.0299946
$340$	0	0
$341$	8574.54	1.36169
$342$	−476.551	−0.0753477
$343$	−12883.3	−2.02808
$344$	17866.8	2.80032
$345$	0	0
$346$	11128.7	1.72915
$347$	−1126.34	−0.174252	−0.0871259	−	0.996197i	$-0.527768\pi$
$347$	−1126.34	−0.174252	−0.0871259	+	0.996197i	$0.527768\pi$
$348$	−10659.4	−1.64197
$349$	3423.60	0.525104	0.262552	−	0.964918i	$-0.415436\pi$
$349$	3423.60	0.525104	0.262552	+	0.964918i	$0.415436\pi$
$350$	0	0
$351$	2341.51	0.356070
$352$	11553.0	1.74937
$353$	4223.26	0.636775	0.318388	−	0.947961i	$-0.396859\pi$
$353$	4223.26	0.636775	0.318388	+	0.947961i	$0.396859\pi$
$354$	−13406.3	−2.01282
$355$	0	0
$356$	1904.09	0.283473
$357$	5801.63	0.860097
$358$	−4464.17	−0.659047
$359$	−2070.39	−0.304376	−0.152188	−	0.988352i	$-0.548632\pi$
$359$	−2070.39	−0.304376	−0.152188	+	0.988352i	$0.548632\pi$
$360$	0	0
$361$	−6754.84	−0.984814
$362$	14107.2	2.04823
$363$	−892.333	−0.129023
$364$	53871.8	7.75729
$365$	0	0
$366$	4085.96	0.583543
$367$	−7593.50	−1.08005	−0.540024	−	0.841650i	$-0.681585\pi$
$367$	−7593.50	−1.08005	−0.540024	+	0.841650i	$0.681585\pi$
$368$	13111.9	1.85736
$369$	−602.951	−0.0850633
$370$	0	0
$371$	−5578.36	−0.780631
$372$	−12058.7	−1.68068
$373$	14058.9	1.95158	0.975792	−	0.218699i	$-0.0701814\pi$
$373$	14058.9	1.95158	0.975792	+	0.218699i	$0.0701814\pi$
$374$	12329.7	1.70469
$375$	0	0
$376$	10291.7	1.41157
$377$	−16288.9	−2.22526
$378$	4599.95	0.625915
$379$	12121.7	1.64287	0.821437	−	0.570300i	$-0.193173\pi$
$379$	12121.7	1.64287	0.821437	+	0.570300i	$0.193173\pi$
$380$	0	0
$381$	−290.804	−0.0391033
$382$	−20293.7	−2.71811
$383$	5957.04	0.794753	0.397376	−	0.917656i	$-0.369921\pi$
$383$	5957.04	0.794753	0.397376	+	0.917656i	$0.369921\pi$
$384$	1498.36	0.199122
$385$	0	0
$386$	16092.3	2.12197
$387$	2839.03	0.372909
$388$	−5717.27	−0.748068
$389$	−5797.52	−0.755645	−0.377823	−	0.925878i	$-0.623327\pi$
$389$	−5797.52	−0.755645	−0.377823	+	0.925878i	$0.623327\pi$
$390$	0	0
$391$	5418.13	0.700784
$392$	41648.7	5.36626
$393$	8943.43	1.14793
$394$	2135.68	0.273081
$395$	0	0
$396$	6870.40	0.871845
$397$	160.461	0.0202854	0.0101427	−	0.999949i	$-0.496771\pi$
$397$	160.461	0.0202854	0.0101427	+	0.999949i	$0.496771\pi$
$398$	−9386.63	−1.18218
$399$	−1005.43	−0.126151
$400$	0	0
$401$	−7261.25	−0.904263	−0.452131	−	0.891951i	$-0.649336\pi$
$401$	−7261.25	−0.904263	−0.452131	+	0.891951i	$0.649336\pi$
$402$	−2785.04	−0.345535
$403$	−18427.1	−2.27772
$404$	28665.8	3.53015
$405$	0	0
$406$	−32000.0	−3.91165
$407$	−4357.38	−0.530681
$408$	−10006.7	−1.21423
$409$	−38.3236	−0.00463321	−0.00231660	−	0.999997i	$-0.500737\pi$
$409$	−38.3236	−0.00463321	−0.00231660	+	0.999997i	$0.500737\pi$
$410$	0	0
$411$	6464.41	0.775830
$412$	21759.7	2.60200
$413$	−28284.7	−3.36997
$414$	4295.88	0.509979
$415$	0	0
$416$	−24828.1	−2.92619
$417$	7215.95	0.847402
$418$	−2136.75	−0.250028
$419$	−9808.93	−1.14367	−0.571835	−	0.820369i	$-0.693768\pi$
$419$	−9808.93	−1.14367	−0.571835	+	0.820369i	$0.693768\pi$
$420$	0	0
$421$	13901.5	1.60930	0.804652	−	0.593746i	$-0.202352\pi$
$421$	13901.5	1.60930	0.804652	+	0.593746i	$0.202352\pi$
$422$	−23513.9	−2.71241
$423$	1635.34	0.187974
$424$	9621.65	1.10205
$425$	0	0
$426$	3887.06	0.442086
$427$	8620.56	0.976998
$428$	16108.8	1.81927
$429$	10498.8	1.18156
$430$	0	0
$431$	919.823	0.102799	0.0513994	−	0.998678i	$-0.483632\pi$
$431$	919.823	0.102799	0.0513994	+	0.998678i	$0.483632\pi$
$432$	−3847.99	−0.428557
$433$	−3504.57	−0.388958	−0.194479	−	0.980907i	$-0.562302\pi$
$433$	−3504.57	−0.388958	−0.194479	+	0.980907i	$0.562302\pi$
$434$	−36200.5	−4.00387
$435$	0	0
$436$	8843.62	0.971405
$437$	−938.965	−0.102784
$438$	2313.77	0.252411
$439$	−608.406	−0.0661449	−0.0330725	−	0.999453i	$-0.510529\pi$
$439$	−608.406	−0.0661449	−0.0330725	+	0.999453i	$0.510529\pi$
$440$	0	0
$441$	6617.97	0.714606
$442$	−26497.1	−2.85145
$443$	−11242.6	−1.20576	−0.602878	−	0.797833i	$-0.705980\pi$
$443$	−11242.6	−1.20576	−0.602878	+	0.797833i	$0.705980\pi$
$444$	6127.92	0.654996
$445$	0	0
$446$	5166.89	0.548563
$447$	848.849	0.0898191
$448$	−11335.2	−1.19540
$449$	7559.04	0.794506	0.397253	−	0.917709i	$-0.369964\pi$
$449$	7559.04	0.794506	0.397253	+	0.917709i	$0.369964\pi$
$450$	0	0
$451$	−2703.50	−0.282268
$452$	1180.52	0.122848
$453$	−7603.20	−0.788586
$454$	−936.755	−0.0968372
$455$	0	0
$456$	1734.18	0.178093
$457$	3293.00	0.337068	0.168534	−	0.985696i	$-0.446097\pi$
$457$	3293.00	0.337068	0.168534	+	0.985696i	$0.446097\pi$
$458$	3026.52	0.308777
$459$	−1590.07	−0.161695
$460$	0	0
$461$	−4329.40	−0.437398	−0.218699	−	0.975792i	$-0.570181\pi$
$461$	−4329.40	−0.437398	−0.218699	+	0.975792i	$0.570181\pi$
$462$	20625.2	2.07699
$463$	−17438.0	−1.75035	−0.875177	−	0.483802i	$-0.839255\pi$
$463$	−17438.0	−1.75035	−0.875177	+	0.483802i	$0.839255\pi$
$464$	26768.9	2.67826
$465$	0	0
$466$	25247.8	2.50983
$467$	9592.87	0.950547	0.475273	−	0.879838i	$-0.342349\pi$
$467$	9592.87	0.950547	0.475273	+	0.879838i	$0.342349\pi$
$468$	−14764.8	−1.45834
$469$	−5875.86	−0.578512
$470$	0	0
$471$	−1419.94	−0.138912
$472$	48785.9	4.75753
$473$	12729.6	1.23743
$474$	−17670.6	−1.71232
$475$	0	0
$476$	−36583.2	−3.52267
$477$	1528.88	0.146756
$478$	16487.7	1.57768
$479$	11983.7	1.14311	0.571555	−	0.820563i	$-0.306340\pi$
$479$	11983.7	1.14311	0.571555	+	0.820563i	$0.306340\pi$
$480$	0	0
$481$	9364.22	0.887675
$482$	−15939.5	−1.50627
$483$	9063.45	0.853833
$484$	5626.77	0.528434
$485$	0	0
$486$	−1260.72	−0.117670
$487$	−4470.91	−0.416008	−0.208004	−	0.978128i	$-0.566697\pi$
$487$	−4470.91	−0.416008	−0.208004	+	0.978128i	$0.566697\pi$
$488$	−14868.9	−1.37927
$489$	−7192.38	−0.665135
$490$	0	0
$491$	10104.3	0.928723	0.464361	−	0.885646i	$-0.346284\pi$
$491$	10104.3	0.928723	0.464361	+	0.885646i	$0.346284\pi$
$492$	3802.02	0.348390
$493$	11061.5	1.01051
$494$	4591.97	0.418224
$495$	0	0
$496$	30282.7	2.74140
$497$	8200.91	0.740163
$498$	16690.0	1.50180
$499$	4996.46	0.448241	0.224121	−	0.974561i	$-0.428049\pi$
$499$	4996.46	0.448241	0.224121	+	0.974561i	$0.428049\pi$
$500$	0	0
$501$	−4939.95	−0.440521
$502$	23755.9	2.11211
$503$	−7906.72	−0.700881	−0.350441	−	0.936585i	$-0.613968\pi$
$503$	−7906.72	−0.700881	−0.350441	+	0.936585i	$0.613968\pi$
$504$	−16739.3	−1.47942
$505$	0	0
$506$	19261.8	1.69227
$507$	−15971.5	−1.39905
$508$	1833.72	0.160154
$509$	9971.90	0.868363	0.434181	−	0.900825i	$-0.357038\pi$
$509$	9971.90	0.868363	0.434181	+	0.900825i	$0.357038\pi$
$510$	0	0
$511$	4881.58	0.422599
$512$	−23775.3	−2.05221
$513$	275.560	0.0237160
$514$	−23237.2	−1.99406
$515$	0	0
$516$	−17902.0	−1.52731
$517$	7332.52	0.623760
$518$	18396.2	1.56039
$519$	−6435.08	−0.544255
$520$	0	0
$521$	3086.11	0.259510	0.129755	−	0.991546i	$-0.458581\pi$
$521$	3086.11	0.259510	0.129755	+	0.991546i	$0.458581\pi$
$522$	8770.33	0.735378
$523$	3464.63	0.289670	0.144835	−	0.989456i	$-0.453735\pi$
$523$	3464.63	0.289670	0.144835	+	0.989456i	$0.453735\pi$
$524$	−56394.4	−4.70153
$525$	0	0
$526$	1823.16	0.151129
$527$	12513.5	1.03434
$528$	−17253.5	−1.42209
$529$	−3702.67	−0.304320
$530$	0	0
$531$	7752.07	0.633543
$532$	6339.89	0.516672
$533$	5809.94	0.472151
$534$	−1566.64	−0.126957
$535$	0	0
$536$	10134.8	0.816709
$537$	2581.36	0.207437
$538$	6491.22	0.520179
$539$	29673.5	2.37129
$540$	0	0
$541$	−4277.19	−0.339909	−0.169954	−	0.985452i	$-0.554362\pi$
$541$	−4277.19	−0.339909	−0.169954	+	0.985452i	$0.554362\pi$
$542$	−27862.5	−2.20811
$543$	−8157.35	−0.644688
$544$	16860.2	1.32881
$545$	0	0
$546$	−44324.5	−3.47420
$547$	13361.7	1.04444	0.522218	−	0.852812i	$-0.325105\pi$
$547$	13361.7	1.04444	0.522218	+	0.852812i	$0.325105\pi$
$548$	−40762.5	−3.17753
$549$	−2362.67	−0.183672
$550$	0	0
$551$	−1916.96	−0.148213
$552$	−15632.8	−1.20539
$553$	−37281.4	−2.86685
$554$	−9587.71	−0.735276
$555$	0	0
$556$	−45501.5	−3.47067
$557$	3119.09	0.237271	0.118636	−	0.992938i	$-0.462148\pi$
$557$	3119.09	0.237271	0.118636	+	0.992938i	$0.462148\pi$
$558$	9921.58	0.752713
$559$	−27356.5	−2.06987
$560$	0	0
$561$	−7129.52	−0.536557
$562$	19914.0	1.49470
$563$	−19205.0	−1.43764	−0.718822	−	0.695194i	$-0.755318\pi$
$563$	−19205.0	−1.43764	−0.718822	+	0.695194i	$0.755318\pi$
$564$	−10312.0	−0.769879
$565$	0	0
$566$	12888.1	0.957118
$567$	−2659.87	−0.197009
$568$	−14145.1	−1.04492
$569$	15248.8	1.12348	0.561742	−	0.827313i	$-0.310131\pi$
$569$	15248.8	1.12348	0.561742	+	0.827313i	$0.310131\pi$
$570$	0	0
$571$	−8357.86	−0.612549	−0.306274	−	0.951943i	$-0.599083\pi$
$571$	−8357.86	−0.612549	−0.306274	+	0.951943i	$0.599083\pi$
$572$	−66202.2	−4.83925
$573$	11734.6	0.855534
$574$	11413.8	0.829967
$575$	0	0
$576$	3106.68	0.224731
$577$	−303.701	−0.0219120	−0.0109560	−	0.999940i	$-0.503487\pi$
$577$	−303.701	−0.0219120	−0.0109560	+	0.999940i	$0.503487\pi$
$578$	−7495.80	−0.539419
$579$	−9305.23	−0.667896
$580$	0	0
$581$	35212.5	2.51439
$582$	4704.04	0.335032
$583$	6855.16	0.486984
$584$	−8419.83	−0.596601
$585$	0	0
$586$	8447.95	0.595532
$587$	−24655.6	−1.73364	−0.866819	−	0.498623i	$-0.833839\pi$
$587$	−24655.6	−1.73364	−0.866819	+	0.498623i	$0.833839\pi$
$588$	−41730.8	−2.92678
$589$	−2168.59	−0.151707
$590$	0	0
$591$	−1234.93	−0.0859533
$592$	−15389.0	−1.06838
$593$	−7620.47	−0.527715	−0.263858	−	0.964562i	$-0.584995\pi$
$593$	−7620.47	−0.527715	−0.263858	+	0.964562i	$0.584995\pi$
$594$	−5652.80	−0.390467
$595$	0	0
$596$	−5352.57	−0.367869
$597$	5427.72	0.372097
$598$	−41394.5	−2.83068
$599$	−20541.8	−1.40119	−0.700596	−	0.713558i	$-0.747083\pi$
$599$	−20541.8	−1.40119	−0.700596	+	0.713558i	$0.747083\pi$
$600$	0	0
$601$	12396.1	0.841346	0.420673	−	0.907212i	$-0.361794\pi$
$601$	12396.1	0.841346	0.420673	+	0.907212i	$0.361794\pi$
$602$	−53742.3	−3.63850
$603$	1610.42	0.108758
$604$	47943.4	3.22978
$605$	0	0
$606$	−23585.6	−1.58102
$607$	−3259.26	−0.217940	−0.108970	−	0.994045i	$-0.534755\pi$
$607$	−3259.26	−0.217940	−0.108970	+	0.994045i	$0.534755\pi$
$608$	−2921.88	−0.194898
$609$	18503.6	1.23121
$610$	0	0
$611$	−15757.9	−1.04337
$612$	10026.5	0.662249
$613$	24508.5	1.61483	0.807413	−	0.589986i	$-0.200867\pi$
$613$	24508.5	1.61483	0.807413	+	0.589986i	$0.200867\pi$
$614$	−51116.3	−3.35975
$615$	0	0
$616$	−75055.2	−4.90919
$617$	−25541.5	−1.66655	−0.833276	−	0.552857i	$-0.813538\pi$
$617$	−25541.5	−1.66655	−0.833276	+	0.552857i	$0.813538\pi$
$618$	−17903.4	−1.16534
$619$	10548.4	0.684938	0.342469	−	0.939529i	$-0.388737\pi$
$619$	10548.4	0.684938	0.342469	+	0.939529i	$0.388737\pi$
$620$	0	0
$621$	−2484.05	−0.160518
$622$	−22497.9	−1.45030
$623$	−3305.29	−0.212558
$624$	37078.7	2.37874
$625$	0	0
$626$	13378.1	0.854147
$627$	1235.55	0.0786972
$628$	8953.70	0.568936
$629$	−6359.04	−0.403102
$630$	0	0
$631$	4923.75	0.310636	0.155318	−	0.987865i	$-0.450360\pi$
$631$	4923.75	0.310636	0.155318	+	0.987865i	$0.450360\pi$
$632$	64303.7	4.04725
$633$	13596.7	0.853742
$634$	4270.01	0.267482
$635$	0	0
$636$	−9640.63	−0.601063
$637$	−63769.8	−3.96648
$638$	39324.2	2.44022
$639$	−2247.65	−0.139148
$640$	0	0
$641$	−21564.0	−1.32874	−0.664372	−	0.747402i	$-0.731301\pi$
$641$	−21564.0	−1.32874	−0.664372	+	0.747402i	$0.731301\pi$
$642$	−13253.9	−0.814784
$643$	−519.569	−0.0318660	−0.0159330	−	0.999873i	$-0.505072\pi$
$643$	−519.569	−0.0318660	−0.0159330	+	0.999873i	$0.505072\pi$
$644$	−57151.2	−3.49701
$645$	0	0
$646$	−3118.31	−0.189920
$647$	4221.51	0.256514	0.128257	−	0.991741i	$-0.459062\pi$
$647$	4221.51	0.256514	0.128257	+	0.991741i	$0.459062\pi$
$648$	4587.79	0.278126
$649$	34758.6	2.10230
$650$	0	0
$651$	20932.5	1.26023
$652$	45352.9	2.72417
$653$	22617.0	1.35539	0.677696	−	0.735343i	$-0.262979\pi$
$653$	22617.0	1.35539	0.677696	+	0.735343i	$0.262979\pi$
$654$	−7276.32	−0.435056
$655$	0	0
$656$	−9547.94	−0.568269
$657$	−1337.91	−0.0794472
$658$	−30956.8	−1.83408
$659$	20202.2	1.19418	0.597091	−	0.802173i	$-0.296323\pi$
$659$	20202.2	1.19418	0.597091	+	0.802173i	$0.296323\pi$
$660$	0	0
$661$	−24225.5	−1.42551	−0.712757	−	0.701411i	$-0.752554\pi$
$661$	−24225.5	−1.42551	−0.712757	+	0.701411i	$0.752554\pi$
$662$	12060.1	0.708050
$663$	15321.7	0.897504
$664$	−60735.2	−3.54967
$665$	0	0
$666$	−5041.91	−0.293348
$667$	17280.5	1.00315
$668$	31149.8	1.80422
$669$	−2987.70	−0.172662
$670$	0	0
$671$	−10593.7	−0.609484
$672$	28203.8	1.61902
$673$	2939.77	0.168380	0.0841902	−	0.996450i	$-0.473170\pi$
$673$	2939.77	0.168380	0.0841902	+	0.996450i	$0.473170\pi$
$674$	−15852.2	−0.905943
$675$	0	0
$676$	100711.	5.73003
$677$	−1597.59	−0.0906947	−0.0453473	−	0.998971i	$-0.514439\pi$
$677$	−1597.59	−0.0906947	−0.0453473	+	0.998971i	$0.514439\pi$
$678$	−971.306	−0.0550188
$679$	9924.57	0.560928
$680$	0	0
$681$	541.669	0.0304799
$682$	44486.1	2.49775
$683$	−725.951	−0.0406702	−0.0203351	−	0.999793i	$-0.506473\pi$
$683$	−725.951	−0.0406702	−0.0203351	+	0.999793i	$0.506473\pi$
$684$	−1737.60	−0.0971325
$685$	0	0
$686$	−66840.7	−3.72010
$687$	−1750.05	−0.0971888
$688$	44957.0	2.49124
$689$	−14732.1	−0.814582
$690$	0	0
$691$	15079.3	0.830163	0.415082	−	0.909784i	$-0.363753\pi$
$691$	15079.3	0.830163	0.415082	+	0.909784i	$0.363753\pi$
$692$	40577.5	2.22908
$693$	−11926.3	−0.653740
$694$	−5843.66	−0.319629
$695$	0	0
$696$	−31915.4	−1.73815
$697$	−3945.41	−0.214409
$698$	17762.2	0.963194
$699$	−14599.3	−0.789978
$700$	0	0
$701$	−36102.8	−1.94520	−0.972600	−	0.232484i	$-0.925315\pi$
$701$	−36102.8	−1.94520	−0.972600	+	0.232484i	$0.925315\pi$
$702$	12148.2	0.653137
$703$	1102.03	0.0591233
$704$	13929.7	0.745730
$705$	0	0
$706$	21911.0	1.16803
$707$	−49760.8	−2.64703
$708$	−48882.1	−2.59478
$709$	556.398	0.0294725	0.0147362	−	0.999891i	$-0.495309\pi$
$709$	556.398	0.0294725	0.0147362	+	0.999891i	$0.495309\pi$
$710$	0	0
$711$	10217.8	0.538958
$712$	5701.02	0.300077
$713$	19548.8	1.02680
$714$	30099.8	1.57767
$715$	0	0
$716$	−16277.2	−0.849593
$717$	−9533.84	−0.496580
$718$	−10741.5	−0.558314
$719$	−8401.29	−0.435765	−0.217883	−	0.975975i	$-0.569915\pi$
$719$	−8401.29	−0.435765	−0.217883	+	0.975975i	$0.569915\pi$
$720$	0	0
$721$	−37772.5	−1.95107
$722$	−35045.2	−1.80644
$723$	9216.84	0.474105
$724$	51437.6	2.64042
$725$	0	0
$726$	−4629.57	−0.236666
$727$	−17330.9	−0.884137	−0.442068	−	0.896981i	$-0.645755\pi$
$727$	−17330.9	−0.884137	−0.442068	+	0.896981i	$0.645755\pi$
$728$	161297.	8.21165
$729$	729.000	0.0370370
$730$	0	0
$731$	18577.2	0.939948
$732$	14898.2	0.752259
$733$	11757.9	0.592482	0.296241	−	0.955113i	$-0.404267\pi$
$733$	11757.9	0.592482	0.296241	+	0.955113i	$0.404267\pi$
$734$	−39396.3	−1.98112
$735$	0	0
$736$	26339.4	1.31914
$737$	7220.75	0.360895
$738$	−3128.21	−0.156031
$739$	23600.6	1.17478	0.587390	−	0.809304i	$-0.300156\pi$
$739$	23600.6	1.17478	0.587390	+	0.809304i	$0.300156\pi$
$740$	0	0
$741$	−2655.26	−0.131637
$742$	−28941.5	−1.43191
$743$	21998.6	1.08620	0.543102	−	0.839667i	$-0.317250\pi$
$743$	21998.6	1.08620	0.543102	+	0.839667i	$0.317250\pi$
$744$	−36104.8	−1.77912
$745$	0	0
$746$	72939.8	3.57978
$747$	−9650.82	−0.472697
$748$	44956.5	2.19756
$749$	−27963.1	−1.36415
$750$	0	0
$751$	5286.80	0.256881	0.128441	−	0.991717i	$-0.459003\pi$
$751$	5286.80	0.256881	0.128441	+	0.991717i	$0.459003\pi$
$752$	25896.3	1.25577
$753$	−13736.6	−0.664794
$754$	−84509.7	−4.08178
$755$	0	0
$756$	16772.3	0.806882
$757$	15653.3	0.751559	0.375779	−	0.926709i	$-0.377375\pi$
$757$	15653.3	0.751559	0.375779	+	0.926709i	$0.377375\pi$
$758$	62889.3	3.01351
$759$	−11137.9	−0.532649
$760$	0	0
$761$	25290.3	1.20469	0.602347	−	0.798234i	$-0.294232\pi$
$761$	25290.3	1.20469	0.602347	+	0.798234i	$0.294232\pi$
$762$	−1508.74	−0.0717268
$763$	−15351.6	−0.728393
$764$	−73994.8	−3.50398
$765$	0	0
$766$	30906.1	1.45781
$767$	−74697.8	−3.51654
$768$	16058.2	0.754494
$769$	−1858.49	−0.0871509	−0.0435754	−	0.999050i	$-0.513875\pi$
$769$	−1858.49	−0.0871509	−0.0435754	+	0.999050i	$0.513875\pi$
$770$	0	0
$771$	13436.7	0.627638
$772$	58675.8	2.73548
$773$	−14943.5	−0.695317	−0.347658	−	0.937621i	$-0.613023\pi$
$773$	−14943.5	−0.695317	−0.347658	+	0.937621i	$0.613023\pi$
$774$	14729.3	0.684025
$775$	0	0
$776$	−17118.1	−0.791885
$777$	−10637.4	−0.491139
$778$	−30078.5	−1.38607
$779$	683.742	0.0314475
$780$	0	0
$781$	−10078.0	−0.461738
$782$	28110.1	1.28544
$783$	−5071.35	−0.231463
$784$	104798.	4.77396
$785$	0	0
$786$	46400.0	2.10564
$787$	8438.76	0.382223	0.191111	−	0.981568i	$-0.438791\pi$
$787$	8438.76	0.382223	0.191111	+	0.981568i	$0.438791\pi$
$788$	7787.10	0.352035
$789$	−1054.23	−0.0475683
$790$	0	0
$791$	−2049.26	−0.0921154
$792$	20570.6	0.922911
$793$	22766.3	1.01949
$794$	832.497	0.0372093
$795$	0	0
$796$	−34225.5	−1.52398
$797$	3822.35	0.169880	0.0849402	−	0.996386i	$-0.472930\pi$
$797$	3822.35	0.169880	0.0849402	+	0.996386i	$0.472930\pi$
$798$	−5216.31	−0.231398
$799$	10700.9	0.473805
$800$	0	0
$801$	905.891	0.0399602
$802$	−37672.5	−1.65868
$803$	−5998.89	−0.263632
$804$	−10154.8	−0.445437
$805$	0	0
$806$	−95602.9	−4.17800
$807$	−3753.48	−0.163728
$808$	85828.2	3.73691
$809$	26327.5	1.14416	0.572080	−	0.820198i	$-0.306137\pi$
$809$	26327.5	1.14416	0.572080	+	0.820198i	$0.306137\pi$
$810$	0	0
$811$	−32273.9	−1.39740	−0.698699	−	0.715416i	$-0.746237\pi$
$811$	−32273.9	−1.39740	−0.698699	+	0.715416i	$0.746237\pi$
$812$	−116678.	−5.04261
$813$	16111.2	0.695012
$814$	−22606.8	−0.973424
$815$	0	0
$816$	−25179.3	−1.08021
$817$	−3219.44	−0.137863
$818$	−198.829	−0.00849866
$819$	25630.1	1.09352
$820$	0	0
$821$	−41177.7	−1.75044	−0.875221	−	0.483724i	$-0.839284\pi$
$821$	−41177.7	−1.75044	−0.875221	+	0.483724i	$0.839284\pi$
$822$	33538.4	1.42310
$823$	−23458.5	−0.993575	−0.496788	−	0.867872i	$-0.665487\pi$
$823$	−23458.5	−0.993575	−0.496788	+	0.867872i	$0.665487\pi$
$824$	65150.7	2.75441
$825$	0	0
$826$	−146746.	−6.18151
$827$	552.405	0.0232273	0.0116137	−	0.999933i	$-0.496303\pi$
$827$	552.405	0.0232273	0.0116137	+	0.999933i	$0.496303\pi$
$828$	15663.6	0.657426
$829$	−43677.5	−1.82989	−0.914947	−	0.403575i	$-0.867767\pi$
$829$	−43677.5	−1.82989	−0.914947	+	0.403575i	$0.867767\pi$
$830$	0	0
$831$	5543.99	0.231431
$832$	−29935.5	−1.24739
$833$	43304.7	1.80122
$834$	37437.6	1.55438
$835$	0	0
$836$	−7790.99	−0.322317
$837$	−5737.05	−0.236919
$838$	−50890.3	−2.09782
$839$	327.507	0.0134765	0.00673827	−	0.999977i	$-0.497855\pi$
$839$	327.507	0.0134765	0.00673827	+	0.999977i	$0.497855\pi$
$840$	0	0
$841$	10890.3	0.446525
$842$	72123.2	2.95194
$843$	−11515.0	−0.470462
$844$	−85736.2	−3.49664
$845$	0	0
$846$	8484.44	0.344800
$847$	−9767.46	−0.396238
$848$	24210.4	0.980409
$849$	−7452.42	−0.301256
$850$	0	0
$851$	−9934.25	−0.400167
$852$	14173.0	0.569903
$853$	−13897.9	−0.557860	−0.278930	−	0.960311i	$-0.589980\pi$
$853$	−13897.9	−0.557860	−0.278930	+	0.960311i	$0.589980\pi$
$854$	44724.9	1.79210
$855$	0	0
$856$	48231.3	1.92583
$857$	20750.0	0.827077	0.413538	−	0.910487i	$-0.364293\pi$
$857$	20750.0	0.827077	0.413538	+	0.910487i	$0.364293\pi$
$858$	54469.6	2.16732
$859$	−42544.8	−1.68988	−0.844942	−	0.534858i	$-0.820365\pi$
$859$	−42544.8	−1.68988	−0.844942	+	0.534858i	$0.820365\pi$
$860$	0	0
$861$	−6599.89	−0.261235
$862$	4772.19	0.188563
$863$	30720.5	1.21175	0.605873	−	0.795561i	$-0.292824\pi$
$863$	30720.5	1.21175	0.605873	+	0.795561i	$0.292824\pi$
$864$	−7729.90	−0.304371
$865$	0	0
$866$	−18182.3	−0.713464
$867$	4334.37	0.169784
$868$	−131994.	−5.16148
$869$	45814.6	1.78844
$870$	0	0
$871$	−15517.7	−0.603673
$872$	26478.6	1.02830
$873$	−2720.06	−0.105452
$874$	−4871.50	−0.188537
$875$	0	0
$876$	8436.43	0.325389
$877$	12180.4	0.468987	0.234493	−	0.972118i	$-0.424657\pi$
$877$	12180.4	0.468987	0.234493	+	0.972118i	$0.424657\pi$
$878$	−3156.51	−0.121329
$879$	−4884.94	−0.187446
$880$	0	0
$881$	9457.65	0.361676	0.180838	−	0.983513i	$-0.442119\pi$
$881$	9457.65	0.361676	0.180838	+	0.983513i	$0.442119\pi$
$882$	34335.1	1.31080
$883$	−7037.17	−0.268199	−0.134099	−	0.990968i	$-0.542814\pi$
$883$	−7037.17	−0.268199	−0.134099	+	0.990968i	$0.542814\pi$
$884$	−96613.7	−3.67587
$885$	0	0
$886$	−58328.2	−2.21171
$887$	25367.6	0.960270	0.480135	−	0.877195i	$-0.340588\pi$
$887$	25367.6	0.960270	0.480135	+	0.877195i	$0.340588\pi$
$888$	18347.6	0.693361
$889$	−3183.13	−0.120089
$890$	0	0
$891$	3268.67	0.122901
$892$	18839.5	0.707166
$893$	−1854.47	−0.0694932
$894$	4403.97	0.164755
$895$	0	0
$896$	16401.0	0.611518
$897$	23935.9	0.890967
$898$	39217.5	1.45736
$899$	39910.3	1.48062
$900$	0	0
$901$	10004.2	0.369910
$902$	−14026.2	−0.517761
$903$	31075.9	1.14523
$904$	3534.60	0.130043
$905$	0	0
$906$	−39446.7	−1.44650
$907$	−31555.9	−1.15523	−0.577617	−	0.816308i	$-0.696017\pi$
$907$	−31555.9	−1.15523	−0.577617	+	0.816308i	$0.696017\pi$
$908$	−3415.59	−0.124835
$909$	13638.1	0.497632
$910$	0	0
$911$	−29498.4	−1.07281	−0.536403	−	0.843962i	$-0.680217\pi$
$911$	−29498.4	−1.07281	−0.536403	+	0.843962i	$0.680217\pi$
$912$	4363.60	0.158435
$913$	−43272.1	−1.56856
$914$	17084.6	0.618281
$915$	0	0
$916$	11035.3	0.398052
$917$	97894.6	3.52537
$918$	−8249.55	−0.296597
$919$	28553.9	1.02492	0.512462	−	0.858710i	$-0.328733\pi$
$919$	28553.9	1.02492	0.512462	+	0.858710i	$0.328733\pi$
$920$	0	0
$921$	29557.5	1.05749
$922$	−22461.7	−0.802316
$923$	21658.0	0.772354
$924$	75203.3	2.67750
$925$	0	0
$926$	−90471.4	−3.21066
$927$	10352.4	0.366795
$928$	53773.7	1.90216
$929$	−33215.5	−1.17305	−0.586526	−	0.809930i	$-0.699505\pi$
$929$	−33215.5	−1.17305	−0.586526	+	0.809930i	$0.699505\pi$
$930$	0	0
$931$	−7504.73	−0.264187
$932$	92058.3	3.23548
$933$	13009.2	0.456486
$934$	49769.4	1.74358
$935$	0	0
$936$	−44207.3	−1.54376
$937$	17531.0	0.611221	0.305610	−	0.952157i	$-0.401140\pi$
$937$	17531.0	0.611221	0.305610	+	0.952157i	$0.401140\pi$
$938$	−30485.0	−1.06116
$939$	−7735.74	−0.268846
$940$	0	0
$941$	42607.1	1.47604	0.738019	−	0.674780i	$-0.235762\pi$
$941$	42607.1	1.47604	0.738019	+	0.674780i	$0.235762\pi$
$942$	−7366.89	−0.254805
$943$	−6163.62	−0.212847
$944$	122757.	4.23241
$945$	0	0
$946$	66043.1	2.26982
$947$	−26048.2	−0.893824	−0.446912	−	0.894578i	$-0.647476\pi$
$947$	−26048.2	−0.893824	−0.446912	+	0.894578i	$0.647476\pi$
$948$	−64430.5	−2.20739
$949$	12891.9	0.440979
$950$	0	0
$951$	−2469.09	−0.0841910
$952$	−109534.	−3.72900
$953$	−31875.4	−1.08347	−0.541735	−	0.840549i	$-0.682232\pi$
$953$	−31875.4	−1.08347	−0.541735	+	0.840549i	$0.682232\pi$
$954$	7932.08	0.269193
$955$	0	0
$956$	60117.4	2.03382
$957$	−22738.8	−0.768068
$958$	62173.5	2.09680
$959$	70759.3	2.38263
$960$	0	0
$961$	15358.1	0.515529
$962$	48583.1	1.62826
$963$	7663.94	0.256456
$964$	−58118.4	−1.94177
$965$	0	0
$966$	47022.7	1.56618
$967$	35299.7	1.17390	0.586951	−	0.809623i	$-0.300328\pi$
$967$	35299.7	1.17390	0.586951	+	0.809623i	$0.300328\pi$
$968$	16847.1	0.559386
$969$	1803.13	0.0597780
$970$	0	0
$971$	−11520.1	−0.380738	−0.190369	−	0.981713i	$-0.560969\pi$
$971$	−11520.1	−0.380738	−0.190369	+	0.981713i	$0.560969\pi$
$972$	−4596.84	−0.151691
$973$	78985.7	2.60243
$974$	−23195.8	−0.763081
$975$	0	0
$976$	−37413.6	−1.22703
$977$	−23931.1	−0.783646	−0.391823	−	0.920041i	$-0.628155\pi$
$977$	−23931.1	−0.783646	−0.391823	+	0.920041i	$0.628155\pi$
$978$	−37315.3	−1.22005
$979$	4061.81	0.132601
$980$	0	0
$981$	4207.46	0.136935
$982$	52423.0	1.70355
$983$	−52560.2	−1.70540	−0.852700	−	0.522400i	$-0.825037\pi$
$983$	−52560.2	−1.70540	−0.852700	+	0.522400i	$0.825037\pi$
$984$	11383.6	0.368796
$985$	0	0
$986$	57388.7	1.85358
$987$	17900.4	0.577282
$988$	16743.2	0.539142
$989$	29021.7	0.933102
$990$	0	0
$991$	−51650.9	−1.65565	−0.827823	−	0.560989i	$-0.810421\pi$
$991$	−51650.9	−1.65565	−0.827823	+	0.560989i	$0.810421\pi$
$992$	60832.4	1.94701
$993$	−6973.63	−0.222861
$994$	42547.7	1.35768
$995$	0	0
$996$	60855.0	1.93601
$997$	19353.1	0.614763	0.307381	−	0.951586i	$-0.400547\pi$
$997$	19353.1	0.614763	0.307381	+	0.951586i	$0.400547\pi$
$998$	25922.5	0.822206
$999$	2915.43	0.0923324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.4.a.k.1.23	✓	24
5.4	even	2		1875.4.a.l.1.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.4.a.k.1.23	✓	24	1.1	even	1	trivial
1875.4.a.l.1.2	yes	24	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+5.18816 q^{2} -3.00000 q^{3} +18.9170 q^{4} -15.5645 q^{6} -32.8379 q^{7} +56.6394 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.18816 q^{2} -3.00000 q^{3} +18.9170 q^{4} -15.5645 q^{6} -32.8379 q^{7} +56.6394 q^{8} +9.00000 q^{9} +40.3540 q^{11} -56.7511 q^{12} -86.7227 q^{13} -170.369 q^{14} +142.518 q^{16} +58.8915 q^{17} +46.6935 q^{18} -10.2059 q^{19} +98.5138 q^{21} +209.363 q^{22} +92.0018 q^{23} -169.918 q^{24} -449.932 q^{26} -27.0000 q^{27} -621.197 q^{28} +187.828 q^{29} +212.483 q^{31} +286.293 q^{32} -121.062 q^{33} +305.539 q^{34} +170.253 q^{36} -107.979 q^{37} -52.9501 q^{38} +260.168 q^{39} -66.9945 q^{41} +511.106 q^{42} +315.447 q^{43} +763.378 q^{44} +477.320 q^{46} +181.705 q^{47} -427.555 q^{48} +735.330 q^{49} -176.675 q^{51} -1640.54 q^{52} +169.876 q^{53} -140.080 q^{54} -1859.92 q^{56} +30.6178 q^{57} +974.482 q^{58} +861.341 q^{59} -262.518 q^{61} +1102.40 q^{62} -295.541 q^{63} +345.187 q^{64} -628.089 q^{66} +178.935 q^{67} +1114.05 q^{68} -276.005 q^{69} -249.739 q^{71} +509.755 q^{72} -148.657 q^{73} -560.212 q^{74} -193.066 q^{76} -1325.14 q^{77} +1349.79 q^{78} +1135.32 q^{79} +81.0000 q^{81} -347.579 q^{82} -1072.31 q^{83} +1863.59 q^{84} +1636.59 q^{86} -563.484 q^{87} +2285.63 q^{88} +100.655 q^{89} +2847.79 q^{91} +1740.40 q^{92} -637.450 q^{93} +942.715 q^{94} -858.878 q^{96} -302.229 q^{97} +3815.01 q^{98} +363.186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9}+O(q^{10}) \) 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9	\( 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9} + 96 q^{11} - 399 q^{12} - 156 q^{13} + 92 q^{14} + 845 q^{16} + 46 q^{17} - 9 q^{18} + 182 q^{19} + 186 q^{21} - 158 q^{22} + 286 q^{23} + 81 q^{24} + 478 q^{26} - 648 q^{27} - 701 q^{28} + 1144 q^{29} + 64 q^{31} - 1212 q^{32} - 288 q^{33} + 961 q^{34} + 1197 q^{36} - 762 q^{37} - 474 q^{38} + 468 q^{39} + 1074 q^{41} - 276 q^{42} - 460 q^{43} + 319 q^{44} + 459 q^{46} + 960 q^{47} - 2535 q^{48} + 2680 q^{49} - 138 q^{51} - 2969 q^{52} - 914 q^{53} + 27 q^{54} + 1680 q^{56} - 546 q^{57} - 208 q^{58} + 208 q^{59} + 3520 q^{61} - 334 q^{62} - 558 q^{63} + 5747 q^{64} + 474 q^{66} - 154 q^{67} + 5727 q^{68} - 858 q^{69} - 252 q^{71} - 243 q^{72} - 4414 q^{73} + 5637 q^{74} + 627 q^{76} - 2344 q^{77} - 1434 q^{78} + 1110 q^{79} + 1944 q^{81} - 3714 q^{82} + 1488 q^{83} + 2103 q^{84} + 3036 q^{86} - 3432 q^{87} - 3947 q^{88} + 3402 q^{89} + 3504 q^{91} + 11163 q^{92} - 192 q^{93} + 3408 q^{94} + 3636 q^{96} - 534 q^{97} - 2244 q^{98} + 864 q^{99}+O(q^{100}) \) 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9 + 96 * q^11 - 399 * q^12 - 156 * q^13 + 92 * q^14 + 845 * q^16 + 46 * q^17 - 9 * q^18 + 182 * q^19 + 186 * q^21 - 158 * q^22 + 286 * q^23 + 81 * q^24 + 478 * q^26 - 648 * q^27 - 701 * q^28 + 1144 * q^29 + 64 * q^31 - 1212 * q^32 - 288 * q^33 + 961 * q^34 + 1197 * q^36 - 762 * q^37 - 474 * q^38 + 468 * q^39 + 1074 * q^41 - 276 * q^42 - 460 * q^43 + 319 * q^44 + 459 * q^46 + 960 * q^47 - 2535 * q^48 + 2680 * q^49 - 138 * q^51 - 2969 * q^52 - 914 * q^53 + 27 * q^54 + 1680 * q^56 - 546 * q^57 - 208 * q^58 + 208 * q^59 + 3520 * q^61 - 334 * q^62 - 558 * q^63 + 5747 * q^64 + 474 * q^66 - 154 * q^67 + 5727 * q^68 - 858 * q^69 - 252 * q^71 - 243 * q^72 - 4414 * q^73 + 5637 * q^74 + 627 * q^76 - 2344 * q^77 - 1434 * q^78 + 1110 * q^79 + 1944 * q^81 - 3714 * q^82 + 1488 * q^83 + 2103 * q^84 + 3036 * q^86 - 3432 * q^87 - 3947 * q^88 + 3402 * q^89 + 3504 * q^91 + 11163 * q^92 - 192 * q^93 + 3408 * q^94 + 3636 * q^96 - 534 * q^97 - 2244 * q^98 + 864 * q^99

Introduction

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