LMFDB - Embedded newform 1875.4.a.l.1.2

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			1.2
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	$-0.869513\pi$
$2$	−5.18816	−1.83429	−0.917146	+	0.398550i	$0.869513\pi$
$3$	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	$-0.153101\pi$
$7$	32.8379	1.77308	0.886541	+	0.462650i	$0.153101\pi$
$8$	−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	$-0.123983\pi$
$13$	86.7227	1.85020	0.925098	+	0.379729i	$0.123983\pi$
$14$	−170.369	−3.25235
$15$	0	0
$16$	142.518	2.22685
$17$	−58.8915	−0.840194	−0.420097	−	0.907479i	$-0.638004\pi$
$17$	−58.8915	−0.840194	−0.420097	+	0.907479i	$0.638004\pi$
$18$	−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	$-0.636931\pi$
$23$	−92.0018	−0.834074	−0.417037	+	0.908889i	$0.636931\pi$
$24$	−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	$-0.422890\pi$
$37$	107.979	0.479773	0.239887	+	0.970801i	$0.422890\pi$
$38$	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	$-0.688955\pi$
$43$	−315.447	−1.11873	−0.559364	+	0.828922i	$0.688955\pi$
$44$	763.378	2.61554
$45$	0	0
$46$	477.320	1.52994
$47$	−181.705	−0.563923	−0.281962	−	0.959426i	$-0.590985\pi$
$47$	−181.705	−0.563923	−0.281962	+	0.959426i	$0.590985\pi$
$48$	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	$-0.570649\pi$
$53$	−169.876	−0.440268	−0.220134	+	0.975470i	$0.570649\pi$
$54$	−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	$-0.552161\pi$
$67$	−178.935	−0.326275	−0.163137	+	0.986603i	$0.552161\pi$
$68$	−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	$-0.461976\pi$
$73$	148.657	0.238342	0.119171	+	0.992874i	$0.461976\pi$
$74$	−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	$-0.249126\pi$
$83$	1072.31	1.41809	0.709046	+	0.705162i	$0.249126\pi$
$84$	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	$-0.449438\pi$
$97$	302.229	0.316357	0.158179	+	0.987410i	$0.449438\pi$
$98$	−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	$-0.685445\pi$
$103$	−1150.27	−1.10038	−0.550192	+	0.835038i	$0.685445\pi$
$104$	−4911.92	−4.63128
$105$	0	0
$106$	881.342	0.807580
$107$	−851.549	−0.769368	−0.384684	−	0.923048i	$-0.625690\pi$
$107$	−851.549	−0.769368	−0.384684	+	0.923048i	$0.625690\pi$
$108$	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	$-0.508269\pi$
$113$	−62.4052	−0.0519521	−0.0259761	+	0.999663i	$0.508269\pi$
$114$	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	$-0.510781\pi$
$127$	−96.9347	−0.0677288	−0.0338644	+	0.999426i	$0.510781\pi$
$128$	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	$-0.265484\pi$
$137$	2154.80	1.34378	0.671888	+	0.740653i	$0.265484\pi$
$138$	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	$-0.538386\pi$
$157$	−473.314	−0.240602	−0.120301	+	0.992737i	$0.538386\pi$
$158$	−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	$-0.695396\pi$
$163$	−2397.46	−1.15205	−0.576023	+	0.817433i	$0.695396\pi$
$164$	−1267.34	−0.603430
$165$	0	0
$166$	−5563.33	−2.60120
$167$	−1646.65	−0.763004	−0.381502	−	0.924368i	$-0.624593\pi$
$167$	−1646.65	−0.763004	−0.381502	+	0.924368i	$0.624593\pi$
$168$	−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	$-0.656229\pi$
$173$	−2145.03	−0.942678	−0.471339	+	0.881952i	$0.656229\pi$
$174$	−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	$-0.696329\pi$
$193$	−3101.74	−1.15683	−0.578415	+	0.815742i	$0.696329\pi$
$194$	−1568.01	−0.580292
$195$	0	0
$196$	13910.3	5.06934
$197$	−411.645	−0.148875	−0.0744377	−	0.997226i	$-0.523716\pi$
$197$	−411.645	−0.148875	−0.0744377	+	0.997226i	$0.523716\pi$
$198$	−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	$-0.547776\pi$
$223$	−995.899	−0.299060	−0.149530	+	0.988757i	$0.547776\pi$
$224$	−9401.26	−2.80423
$225$	0	0
$226$	323.769	0.0952954
$227$	180.556	0.0527927	0.0263963	−	0.999652i	$-0.491597\pi$
$227$	180.556	0.0527927	0.0263963	+	0.999652i	$0.491597\pi$
$228$	−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	$-0.739823\pi$
$233$	−4866.42	−1.36828	−0.684141	+	0.729350i	$0.739823\pi$
$234$	−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	$-0.317080\pi$
$257$	4478.88	1.08710	0.543551	+	0.839376i	$0.317080\pi$
$258$	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	$-0.513117\pi$
$263$	−351.408	−0.0823908	−0.0411954	+	0.999151i	$0.513117\pi$
$264$	−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	$-0.435768\pi$
$277$	1848.00	0.400850	0.200425	+	0.979709i	$0.435768\pi$
$278$	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	$-0.584018\pi$
$283$	−2484.14	−0.521791	−0.260896	+	0.965367i	$0.584018\pi$
$284$	−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	$-0.551902\pi$
$293$	−1628.31	−0.324666	−0.162333	+	0.986736i	$0.551902\pi$
$294$	−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	$-0.131545\pi$
$307$	9852.49	1.83163	0.915817	+	0.401597i	$0.131545\pi$
$308$	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	$-0.574798\pi$
$313$	−2578.58	−0.465655	−0.232827	+	0.972518i	$0.574798\pi$
$314$	2455.63	0.441335
$315$	0	0
$316$	21476.8	3.82331
$317$	−823.029	−0.145823	−0.0729115	−	0.997338i	$-0.523229\pi$
$317$	−823.029	−0.145823	−0.0729115	+	0.997338i	$0.523229\pi$
$318$	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	$-0.420573\pi$
$337$	3055.46	0.493892	0.246946	+	0.969029i	$0.420573\pi$
$338$	−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	$-0.472232\pi$
$347$	1126.34	0.174252	0.0871259	+	0.996197i	$0.472232\pi$
$348$	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	$-0.603141\pi$
$353$	−4223.26	−0.636775	−0.318388	+	0.947961i	$0.603141\pi$
$354$	−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	$-0.318415\pi$
$367$	7593.50	1.08005	0.540024	+	0.841650i	$0.318415\pi$
$368$	−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	$-0.929819\pi$
$373$	−14058.9	−1.95158	−0.975792	+	0.218699i	$0.929819\pi$
$374$	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	$-0.630079\pi$
$383$	−5957.04	−0.794753	−0.397376	+	0.917656i	$0.630079\pi$
$384$	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	$-0.503229\pi$
$397$	−160.461	−0.0202854	−0.0101427	+	0.999949i	$0.503229\pi$
$398$	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	$-0.437698\pi$
$433$	3504.57	0.388958	0.194479	+	0.980907i	$0.437698\pi$
$434$	−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	$-0.294020\pi$
$443$	11242.6	1.20576	0.602878	+	0.797833i	$0.294020\pi$
$444$	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	$-0.553903\pi$
$457$	−3293.00	−0.337068	−0.168534	+	0.985696i	$0.553903\pi$
$458$	−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	$-0.160745\pi$
$463$	17438.0	1.75035	0.875177	+	0.483802i	$0.160745\pi$
$464$	26768.9	2.67826
$465$	0	0
$466$	25247.8	2.50983
$467$	−9592.87	−0.950547	−0.475273	−	0.879838i	$-0.657651\pi$
$467$	−9592.87	−0.950547	−0.475273	+	0.879838i	$0.657651\pi$
$468$	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	$-0.433303\pi$
$487$	4470.91	0.416008	0.208004	+	0.978128i	$0.433303\pi$
$488$	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	$-0.386032\pi$
$503$	7906.72	0.700881	0.350441	+	0.936585i	$0.386032\pi$
$504$	−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	$-0.546265\pi$
$523$	−3464.63	−0.289670	−0.144835	+	0.989456i	$0.546265\pi$
$524$	−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	$-0.674895\pi$
$547$	−13361.7	−1.04444	−0.522218	+	0.852812i	$0.674895\pi$
$548$	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	$-0.537852\pi$
$557$	−3119.09	−0.237271	−0.118636	+	0.992938i	$0.537852\pi$
$558$	−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	$-0.244682\pi$
$563$	19205.0	1.43764	0.718822	+	0.695194i	$0.244682\pi$
$564$	−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	$-0.496513\pi$
$577$	303.701	0.0219120	0.0109560	+	0.999940i	$0.496513\pi$
$578$	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	$-0.166161\pi$
$587$	24655.6	1.73364	0.866819	+	0.498623i	$0.166161\pi$
$588$	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	$-0.415005\pi$
$593$	7620.47	0.527715	0.263858	+	0.964562i	$0.415005\pi$
$594$	−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	$-0.465245\pi$
$607$	3259.26	0.217940	0.108970	+	0.994045i	$0.465245\pi$
$608$	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	$-0.799133\pi$
$613$	−24508.5	−1.61483	−0.807413	+	0.589986i	$0.799133\pi$
$614$	−51116.3	−3.35975
$615$	0	0
$616$	−75055.2	−4.90919
$617$	25541.5	1.66655	0.833276	−	0.552857i	$-0.186462\pi$
$617$	25541.5	1.66655	0.833276	+	0.552857i	$0.186462\pi$
$618$	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	$-0.494928\pi$
$643$	519.569	0.0318660	0.0159330	+	0.999873i	$0.494928\pi$
$644$	−57151.2	−3.49701
$645$	0	0
$646$	−3118.31	−0.189920
$647$	−4221.51	−0.256514	−0.128257	−	0.991741i	$-0.540938\pi$
$647$	−4221.51	−0.256514	−0.128257	+	0.991741i	$0.540938\pi$
$648$	−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	$-0.737021\pi$
$653$	−22617.0	−1.35539	−0.677696	+	0.735343i	$0.737021\pi$
$654$	−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	$-0.526830\pi$
$673$	−2939.77	−0.168380	−0.0841902	+	0.996450i	$0.526830\pi$
$674$	−15852.2	−0.905943
$675$	0	0
$676$	100711.	5.73003
$677$	1597.59	0.0906947	0.0453473	−	0.998971i	$-0.485561\pi$
$677$	1597.59	0.0906947	0.0453473	+	0.998971i	$0.485561\pi$
$678$	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	$-0.493527\pi$
$683$	725.951	0.0406702	0.0203351	+	0.999793i	$0.493527\pi$
$684$	−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	$-0.354245\pi$
$727$	17330.9	0.884137	0.442068	+	0.896981i	$0.354245\pi$
$728$	−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	$-0.595733\pi$
$733$	−11757.9	−0.592482	−0.296241	+	0.955113i	$0.595733\pi$
$734$	−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	$-0.682750\pi$
$743$	−21998.6	−1.08620	−0.543102	+	0.839667i	$0.682750\pi$
$744$	−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	$-0.622625\pi$
$757$	−15653.3	−0.751559	−0.375779	+	0.926709i	$0.622625\pi$
$758$	−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	$-0.386977\pi$
$773$	14943.5	0.695317	0.347658	+	0.937621i	$0.386977\pi$
$774$	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	$-0.561209\pi$
$787$	−8438.76	−0.382223	−0.191111	+	0.981568i	$0.561209\pi$
$788$	−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	$-0.527070\pi$
$797$	−3822.35	−0.169880	−0.0849402	+	0.996386i	$0.527070\pi$
$798$	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	$-0.334513\pi$
$823$	23458.5	0.993575	0.496788	+	0.867872i	$0.334513\pi$
$824$	65150.7	2.75441
$825$	0	0
$826$	−146746.	−6.18151
$827$	−552.405	−0.0232273	−0.0116137	−	0.999933i	$-0.503697\pi$
$827$	−552.405	−0.0232273	−0.0116137	+	0.999933i	$0.503697\pi$
$828$	−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	$-0.410020\pi$
$853$	13897.9	0.557860	0.278930	+	0.960311i	$0.410020\pi$
$854$	44724.9	1.79210
$855$	0	0
$856$	48231.3	1.92583
$857$	−20750.0	−0.827077	−0.413538	−	0.910487i	$-0.635707\pi$
$857$	−20750.0	−0.827077	−0.413538	+	0.910487i	$0.635707\pi$
$858$	−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	$-0.707176\pi$
$863$	−30720.5	−1.21175	−0.605873	+	0.795561i	$0.707176\pi$
$864$	−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	$-0.575343\pi$
$877$	−12180.4	−0.468987	−0.234493	+	0.972118i	$0.575343\pi$
$878$	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	$-0.457186\pi$
$883$	7037.17	0.268199	0.134099	+	0.990968i	$0.457186\pi$
$884$	−96613.7	−3.67587
$885$	0	0
$886$	−58328.2	−2.21171
$887$	−25367.6	−0.960270	−0.480135	−	0.877195i	$-0.659412\pi$
$887$	−25367.6	−0.960270	−0.480135	+	0.877195i	$0.659412\pi$
$888$	−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	$-0.303983\pi$
$907$	31555.9	1.15523	0.577617	+	0.816308i	$0.303983\pi$
$908$	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	$-0.598860\pi$
$937$	−17531.0	−0.611221	−0.305610	+	0.952157i	$0.598860\pi$
$938$	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	$-0.352524\pi$
$947$	26048.2	0.893824	0.446912	+	0.894578i	$0.352524\pi$
$948$	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	$-0.317768\pi$
$953$	31875.4	1.08347	0.541735	+	0.840549i	$0.317768\pi$
$954$	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	$-0.699672\pi$
$967$	−35299.7	−1.17390	−0.586951	+	0.809623i	$0.699672\pi$
$968$	−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	$-0.371845\pi$
$977$	23931.1	0.783646	0.391823	+	0.920041i	$0.371845\pi$
$978$	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	$-0.174963\pi$
$983$	52560.2	1.70540	0.852700	+	0.522400i	$0.174963\pi$
$984$	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	$-0.599453\pi$
$997$	−19353.1	−0.614763	−0.307381	+	0.951586i	$0.599453\pi$
$998$	−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.l.1.2	yes	24
5.4	even	2		1875.4.a.k.1.23	✓	24

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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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