LMFDB - Embedded newform 1875.4.a.l.1.5

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			1.5
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)

$f(q)$	$=$	$q-3.84470 q^{2} +3.00000 q^{3} +6.78175 q^{4} -11.5341 q^{6} -25.3925 q^{7} +4.68382 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.84470 q^{2} +3.00000 q^{3} +6.78175 q^{4} -11.5341 q^{6} -25.3925 q^{7} +4.68382 q^{8} +9.00000 q^{9} -56.8938 q^{11} +20.3452 q^{12} -4.98172 q^{13} +97.6266 q^{14} -72.2619 q^{16} -129.575 q^{17} -34.6023 q^{18} +24.2358 q^{19} -76.1775 q^{21} +218.740 q^{22} -81.3265 q^{23} +14.0514 q^{24} +19.1533 q^{26} +27.0000 q^{27} -172.205 q^{28} -112.900 q^{29} -245.790 q^{31} +240.355 q^{32} -170.681 q^{33} +498.179 q^{34} +61.0357 q^{36} -188.773 q^{37} -93.1793 q^{38} -14.9452 q^{39} +153.477 q^{41} +292.880 q^{42} -353.926 q^{43} -385.840 q^{44} +312.676 q^{46} -411.233 q^{47} -216.786 q^{48} +301.778 q^{49} -388.726 q^{51} -33.7848 q^{52} +346.938 q^{53} -103.807 q^{54} -118.934 q^{56} +72.7073 q^{57} +434.069 q^{58} +270.165 q^{59} +377.746 q^{61} +944.989 q^{62} -228.532 q^{63} -345.999 q^{64} +656.220 q^{66} -769.421 q^{67} -878.747 q^{68} -243.979 q^{69} -656.903 q^{71} +42.1543 q^{72} +370.230 q^{73} +725.778 q^{74} +164.361 q^{76} +1444.68 q^{77} +57.4598 q^{78} -765.237 q^{79} +81.0000 q^{81} -590.072 q^{82} -1319.10 q^{83} -516.616 q^{84} +1360.74 q^{86} -338.701 q^{87} -266.480 q^{88} +271.364 q^{89} +126.498 q^{91} -551.536 q^{92} -737.370 q^{93} +1581.07 q^{94} +721.065 q^{96} +115.231 q^{97} -1160.25 q^{98} -512.044 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.84470	−1.35931	−0.679654	−	0.733533i	$-0.737870\pi$
$2$	−3.84470	−1.35931	−0.679654	+	0.733533i	$0.737870\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	6.78175	0.847719
$5$	0	0
$5$	0	0
$6$	−11.5341	−0.784797
$7$	−25.3925	−1.37107	−0.685533	−	0.728042i	$-0.740431\pi$
$7$	−25.3925	−1.37107	−0.685533	+	0.728042i	$0.740431\pi$
$8$	4.68382	0.206997
$9$	9.00000	0.333333
$10$	0	0
$11$	−56.8938	−1.55947	−0.779733	−	0.626112i	$-0.784645\pi$
$11$	−56.8938	−1.55947	−0.779733	+	0.626112i	$0.784645\pi$
$12$	20.3452	0.489431
$13$	−4.98172	−0.106283	−0.0531416	−	0.998587i	$-0.516923\pi$
$13$	−4.98172	−0.106283	−0.0531416	+	0.998587i	$0.516923\pi$
$14$	97.6266	1.86370
$15$	0	0
$16$	−72.2619	−1.12909
$17$	−129.575	−1.84863	−0.924313	−	0.381636i	$-0.875361\pi$
$17$	−129.575	−1.84863	−0.924313	+	0.381636i	$0.875361\pi$
$18$	−34.6023	−0.453103
$19$	24.2358	0.292635	0.146317	−	0.989238i	$-0.453258\pi$
$19$	24.2358	0.292635	0.146317	+	0.989238i	$0.453258\pi$
$20$	0	0
$21$	−76.1775	−0.791585
$22$	218.740	2.11980
$23$	−81.3265	−0.737293	−0.368647	−	0.929570i	$-0.620179\pi$
$23$	−81.3265	−0.737293	−0.368647	+	0.929570i	$0.620179\pi$
$24$	14.0514	0.119510
$25$	0	0
$26$	19.1533	0.144472
$27$	27.0000	0.192450
$28$	−172.205	−1.16228
$29$	−112.900	−0.722934	−0.361467	−	0.932385i	$-0.617724\pi$
$29$	−112.900	−0.722934	−0.361467	+	0.932385i	$0.617724\pi$
$30$	0	0
$31$	−245.790	−1.42404	−0.712019	−	0.702160i	$-0.752219\pi$
$31$	−245.790	−1.42404	−0.712019	+	0.702160i	$0.752219\pi$
$32$	240.355	1.32779
$33$	−170.681	−0.900359
$34$	498.179	2.51285
$35$	0	0
$36$	61.0357	0.282573
$37$	−188.773	−0.838761	−0.419381	−	0.907811i	$-0.637753\pi$
$37$	−188.773	−0.838761	−0.419381	+	0.907811i	$0.637753\pi$
$38$	−93.1793	−0.397781
$39$	−14.9452	−0.0613626
$40$	0	0
$41$	153.477	0.584610	0.292305	−	0.956325i	$-0.405578\pi$
$41$	153.477	0.584610	0.292305	+	0.956325i	$0.405578\pi$
$42$	292.880	1.07601
$43$	−353.926	−1.25519	−0.627595	−	0.778540i	$-0.715961\pi$
$43$	−353.926	−1.25519	−0.627595	+	0.778540i	$0.715961\pi$
$44$	−385.840	−1.32199
$45$	0	0
$46$	312.676	1.00221
$47$	−411.233	−1.27627	−0.638133	−	0.769926i	$-0.720293\pi$
$47$	−411.233	−1.27627	−0.638133	+	0.769926i	$0.720293\pi$
$48$	−216.786	−0.651881
$49$	301.778	0.879821
$50$	0	0
$51$	−388.726	−1.06730
$52$	−33.7848	−0.0900982
$53$	346.938	0.899162	0.449581	−	0.893239i	$-0.351573\pi$
$53$	346.938	0.899162	0.449581	+	0.893239i	$0.351573\pi$
$54$	−103.807	−0.261599
$55$	0	0
$56$	−118.934	−0.283807
$57$	72.7073	0.168953
$58$	434.069	0.982690
$59$	270.165	0.596143	0.298072	−	0.954544i	$-0.403657\pi$
$59$	270.165	0.596143	0.298072	+	0.954544i	$0.403657\pi$
$60$	0	0
$61$	377.746	0.792876	0.396438	−	0.918062i	$-0.370246\pi$
$61$	377.746	0.792876	0.396438	+	0.918062i	$0.370246\pi$
$62$	944.989	1.93571
$63$	−228.532	−0.457022
$64$	−345.999	−0.675779
$65$	0	0
$66$	656.220	1.22386
$67$	−769.421	−1.40298	−0.701491	−	0.712679i	$-0.747482\pi$
$67$	−769.421	−1.40298	−0.701491	+	0.712679i	$0.747482\pi$
$68$	−878.747	−1.56711
$69$	−243.979	−0.425676
$70$	0	0
$71$	−656.903	−1.09803	−0.549015	−	0.835813i	$-0.684997\pi$
$71$	−656.903	−1.09803	−0.549015	+	0.835813i	$0.684997\pi$
$72$	42.1543	0.0689991
$73$	370.230	0.593591	0.296796	−	0.954941i	$-0.404082\pi$
$73$	370.230	0.593591	0.296796	+	0.954941i	$0.404082\pi$
$74$	725.778	1.14013
$75$	0	0
$76$	164.361	0.248072
$77$	1444.68	2.13813
$78$	57.4598	0.0834107
$79$	−765.237	−1.08982	−0.544911	−	0.838494i	$-0.683437\pi$
$79$	−765.237	−1.08982	−0.544911	+	0.838494i	$0.683437\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−590.072	−0.794665
$83$	−1319.10	−1.74445	−0.872227	−	0.489101i	$-0.837325\pi$
$83$	−1319.10	−1.74445	−0.872227	+	0.489101i	$0.837325\pi$
$84$	−516.616	−0.671041
$85$	0	0
$86$	1360.74	1.70619
$87$	−338.701	−0.417386
$88$	−266.480	−0.322806
$89$	271.364	0.323197	0.161599	−	0.986857i	$-0.448335\pi$
$89$	271.364	0.323197	0.161599	+	0.986857i	$0.448335\pi$
$90$	0	0
$91$	126.498	0.145721
$92$	−551.536	−0.625017
$93$	−737.370	−0.822168
$94$	1581.07	1.73484
$95$	0	0
$96$	721.065	0.766598
$97$	115.231	0.120618	0.0603089	−	0.998180i	$-0.480791\pi$
$97$	115.231	0.120618	0.0603089	+	0.998180i	$0.480791\pi$
$98$	−1160.25	−1.19595
$99$	−512.044	−0.519822
$100$	0	0
$101$	897.794	0.884494	0.442247	−	0.896893i	$-0.354181\pi$
$101$	897.794	0.884494	0.442247	+	0.896893i	$0.354181\pi$
$102$	1494.54	1.45080
$103$	315.203	0.301533	0.150767	−	0.988569i	$-0.451826\pi$
$103$	315.203	0.301533	0.150767	+	0.988569i	$0.451826\pi$
$104$	−23.3335	−0.0220003
$105$	0	0
$106$	−1333.87	−1.22224
$107$	−1484.60	−1.34132	−0.670661	−	0.741764i	$-0.733989\pi$
$107$	−1484.60	−1.34132	−0.670661	+	0.741764i	$0.733989\pi$
$108$	183.107	0.163144
$109$	1801.37	1.58293	0.791466	−	0.611213i	$-0.209318\pi$
$109$	1801.37	1.58293	0.791466	+	0.611213i	$0.209318\pi$
$110$	0	0
$111$	−566.320	−0.484259
$112$	1834.91	1.54806
$113$	−386.336	−0.321624	−0.160812	−	0.986985i	$-0.551411\pi$
$113$	−386.336	−0.321624	−0.160812	+	0.986985i	$0.551411\pi$
$114$	−279.538	−0.229659
$115$	0	0
$116$	−765.662	−0.612844
$117$	−44.8355	−0.0354277
$118$	−1038.70	−0.810342
$119$	3290.24	2.53459
$120$	0	0
$121$	1905.91	1.43194
$122$	−1452.32	−1.07776
$123$	460.430	0.337525
$124$	−1666.89	−1.20718
$125$	0	0
$126$	878.639	0.621233
$127$	2118.50	1.48021	0.740106	−	0.672491i	$-0.234775\pi$
$127$	2118.50	1.48021	0.740106	+	0.672491i	$0.234775\pi$
$128$	−592.577	−0.409195
$129$	−1061.78	−0.724685
$130$	0	0
$131$	2348.81	1.56654	0.783270	−	0.621682i	$-0.213550\pi$
$131$	2348.81	1.56654	0.783270	+	0.621682i	$0.213550\pi$
$132$	−1157.52	−0.763251
$133$	−615.406	−0.401222
$134$	2958.20	1.90708
$135$	0	0
$136$	−606.907	−0.382661
$137$	−636.581	−0.396984	−0.198492	−	0.980102i	$-0.563604\pi$
$137$	−636.581	−0.396984	−0.198492	+	0.980102i	$0.563604\pi$
$138$	938.029	0.578625
$139$	−484.954	−0.295923	−0.147961	−	0.988993i	$-0.547271\pi$
$139$	−484.954	−0.295923	−0.147961	+	0.988993i	$0.547271\pi$
$140$	0	0
$141$	−1233.70	−0.736852
$142$	2525.60	1.49256
$143$	283.429	0.165745
$144$	−650.357	−0.376364
$145$	0	0
$146$	−1423.42	−0.806873
$147$	905.335	0.507965
$148$	−1280.21	−0.711033
$149$	−1146.60	−0.630425	−0.315212	−	0.949021i	$-0.602076\pi$
$149$	−1146.60	−0.630425	−0.315212	+	0.949021i	$0.602076\pi$
$150$	0	0
$151$	−755.004	−0.406896	−0.203448	−	0.979086i	$-0.565215\pi$
$151$	−755.004	−0.406896	−0.203448	+	0.979086i	$0.565215\pi$
$152$	113.516	0.0605747
$153$	−1166.18	−0.616208
$154$	−5554.35	−2.90638
$155$	0	0
$156$	−101.354	−0.0520182
$157$	−2234.59	−1.13592	−0.567960	−	0.823056i	$-0.692267\pi$
$157$	−2234.59	−1.13592	−0.567960	+	0.823056i	$0.692267\pi$
$158$	2942.11	1.48140
$159$	1040.81	0.519132
$160$	0	0
$161$	2065.08	1.01088
$162$	−311.421	−0.151034
$163$	3234.71	1.55437	0.777185	−	0.629272i	$-0.216647\pi$
$163$	3234.71	1.55437	0.777185	+	0.629272i	$0.216647\pi$
$164$	1040.84	0.495585
$165$	0	0
$166$	5071.54	2.37125
$167$	−1690.49	−0.783317	−0.391659	−	0.920111i	$-0.628099\pi$
$167$	−1690.49	−0.783317	−0.391659	+	0.920111i	$0.628099\pi$
$168$	−356.801	−0.163856
$169$	−2172.18	−0.988704
$170$	0	0
$171$	218.122	0.0975450
$172$	−2400.24	−1.06405
$173$	−4112.74	−1.80743	−0.903716	−	0.428132i	$-0.859172\pi$
$173$	−4112.74	−1.80743	−0.903716	+	0.428132i	$0.859172\pi$
$174$	1302.21	0.567356
$175$	0	0
$176$	4111.25	1.76078
$177$	810.494	0.344183
$178$	−1043.32	−0.439325
$179$	−3816.36	−1.59356	−0.796782	−	0.604267i	$-0.793466\pi$
$179$	−3816.36	−1.59356	−0.796782	+	0.604267i	$0.793466\pi$
$180$	0	0
$181$	1928.05	0.791771	0.395885	−	0.918300i	$-0.370438\pi$
$181$	1928.05	0.791771	0.395885	+	0.918300i	$0.370438\pi$
$182$	−486.349	−0.198080
$183$	1133.24	0.457767
$184$	−380.918	−0.152618
$185$	0	0
$186$	2834.97	1.11758
$187$	7372.03	2.88287
$188$	−2788.88	−1.08191
$189$	−685.597	−0.263862
$190$	0	0
$191$	−1596.40	−0.604774	−0.302387	−	0.953185i	$-0.597783\pi$
$191$	−1596.40	−0.604774	−0.302387	+	0.953185i	$0.597783\pi$
$192$	−1038.00	−0.390161
$193$	−3748.59	−1.39808	−0.699040	−	0.715083i	$-0.746389\pi$
$193$	−3748.59	−1.39808	−0.699040	+	0.715083i	$0.746389\pi$
$194$	−443.028	−0.163957
$195$	0	0
$196$	2046.59	0.745840
$197$	−1584.16	−0.572928	−0.286464	−	0.958091i	$-0.592480\pi$
$197$	−1584.16	−0.572928	−0.286464	+	0.958091i	$0.592480\pi$
$198$	1968.66	0.706599
$199$	3084.05	1.09861	0.549303	−	0.835623i	$-0.314893\pi$
$199$	3084.05	1.09861	0.549303	+	0.835623i	$0.314893\pi$
$200$	0	0
$201$	−2308.26	−0.810012
$202$	−3451.75	−1.20230
$203$	2866.82	0.991190
$204$	−2636.24	−0.904774
$205$	0	0
$206$	−1211.86	−0.409876
$207$	−731.938	−0.245764
$208$	359.989	0.120003
$209$	−1378.86	−0.456354
$210$	0	0
$211$	−3534.20	−1.15310	−0.576550	−	0.817062i	$-0.695601\pi$
$211$	−3534.20	−1.15310	−0.576550	+	0.817062i	$0.695601\pi$
$212$	2352.85	0.762237
$213$	−1970.71	−0.633948
$214$	5707.83	1.82327
$215$	0	0
$216$	126.463	0.0398367
$217$	6241.22	1.95245
$218$	−6925.72	−2.15169
$219$	1110.69	0.342710
$220$	0	0
$221$	645.508	0.196478
$222$	2177.33	0.658257
$223$	−5137.40	−1.54272	−0.771359	−	0.636401i	$-0.780422\pi$
$223$	−5137.40	−1.54272	−0.771359	+	0.636401i	$0.780422\pi$
$224$	−6103.21	−1.82048
$225$	0	0
$226$	1485.35	0.437186
$227$	−2973.18	−0.869326	−0.434663	−	0.900593i	$-0.643132\pi$
$227$	−2973.18	−0.869326	−0.434663	+	0.900593i	$0.643132\pi$
$228$	493.082	0.143224
$229$	5055.58	1.45887	0.729436	−	0.684049i	$-0.239782\pi$
$229$	5055.58	1.45887	0.729436	+	0.684049i	$0.239782\pi$
$230$	0	0
$231$	4334.03	1.23445
$232$	−528.805	−0.149645
$233$	−3657.32	−1.02832	−0.514161	−	0.857694i	$-0.671897\pi$
$233$	−3657.32	−1.02832	−0.514161	+	0.857694i	$0.671897\pi$
$234$	172.379	0.0481572
$235$	0	0
$236$	1832.19	0.505362
$237$	−2295.71	−0.629209
$238$	−12650.0	−3.44528
$239$	−3957.00	−1.07095	−0.535475	−	0.844551i	$-0.679867\pi$
$239$	−3957.00	−1.07095	−0.535475	+	0.844551i	$0.679867\pi$
$240$	0	0
$241$	−553.694	−0.147994	−0.0739970	−	0.997258i	$-0.523576\pi$
$241$	−553.694	−0.147994	−0.0739970	+	0.997258i	$0.523576\pi$
$242$	−7327.65	−1.94644
$243$	243.000	0.0641500
$244$	2561.78	0.672135
$245$	0	0
$246$	−1770.22	−0.458800
$247$	−120.736	−0.0311022
$248$	−1151.23	−0.294772
$249$	−3957.29	−1.00716
$250$	0	0
$251$	4933.74	1.24070	0.620349	−	0.784326i	$-0.286991\pi$
$251$	4933.74	1.24070	0.620349	+	0.784326i	$0.286991\pi$
$252$	−1549.85	−0.387426
$253$	4626.97	1.14978
$254$	−8145.02	−2.01206
$255$	0	0
$256$	5046.27	1.23200
$257$	−1662.11	−0.403423	−0.201711	−	0.979445i	$-0.564650\pi$
$257$	−1662.11	−0.403423	−0.201711	+	0.979445i	$0.564650\pi$
$258$	4082.22	0.985070
$259$	4793.43	1.15000
$260$	0	0
$261$	−1016.10	−0.240978
$262$	−9030.48	−2.12941
$263$	−4498.06	−1.05461	−0.527305	−	0.849676i	$-0.676797\pi$
$263$	−4498.06	−1.05461	−0.527305	+	0.849676i	$0.676797\pi$
$264$	−799.441	−0.186372
$265$	0	0
$266$	2366.05	0.545384
$267$	814.093	0.186598
$268$	−5218.02	−1.18933
$269$	−3725.80	−0.844483	−0.422242	−	0.906483i	$-0.638757\pi$
$269$	−3725.80	−0.844483	−0.422242	+	0.906483i	$0.638757\pi$
$270$	0	0
$271$	6350.64	1.42352	0.711760	−	0.702423i	$-0.247898\pi$
$271$	6350.64	1.42352	0.711760	+	0.702423i	$0.247898\pi$
$272$	9363.35	2.08727
$273$	379.495	0.0841322
$274$	2447.47	0.539624
$275$	0	0
$276$	−1654.61	−0.360854
$277$	−1221.75	−0.265010	−0.132505	−	0.991182i	$-0.542302\pi$
$277$	−1221.75	−0.265010	−0.132505	+	0.991182i	$0.542302\pi$
$278$	1864.50	0.402250
$279$	−2212.11	−0.474679
$280$	0	0
$281$	3113.31	0.660941	0.330470	−	0.943816i	$-0.392793\pi$
$281$	3113.31	0.660941	0.330470	+	0.943816i	$0.392793\pi$
$282$	4743.21	1.00161
$283$	7052.28	1.48132	0.740662	−	0.671878i	$-0.234512\pi$
$283$	7052.28	1.48132	0.740662	+	0.671878i	$0.234512\pi$
$284$	−4454.95	−0.930820
$285$	0	0
$286$	−1089.70	−0.225299
$287$	−3897.15	−0.801538
$288$	2163.19	0.442595
$289$	11876.8	2.41741
$290$	0	0
$291$	345.692	0.0696387
$292$	2510.81	0.503198
$293$	2870.81	0.572404	0.286202	−	0.958169i	$-0.407607\pi$
$293$	2870.81	0.572404	0.286202	+	0.958169i	$0.407607\pi$
$294$	−3480.75	−0.690480
$295$	0	0
$296$	−884.180	−0.173621
$297$	−1536.13	−0.300120
$298$	4408.34	0.856941
$299$	405.146	0.0783619
$300$	0	0
$301$	8987.06	1.72095
$302$	2902.77	0.553097
$303$	2693.38	0.510663
$304$	−1751.32	−0.330412
$305$	0	0
$306$	4483.61	0.837617
$307$	2205.87	0.410084	0.205042	−	0.978753i	$-0.434267\pi$
$307$	2205.87	0.410084	0.205042	+	0.978753i	$0.434267\pi$
$308$	9797.43	1.81253
$309$	945.610	0.174090
$310$	0	0
$311$	−1594.53	−0.290731	−0.145366	−	0.989378i	$-0.546436\pi$
$311$	−1594.53	−0.290731	−0.145366	+	0.989378i	$0.546436\pi$
$312$	−70.0004	−0.0127019
$313$	10012.8	1.80816	0.904082	−	0.427360i	$-0.140556\pi$
$313$	10012.8	1.80816	0.904082	+	0.427360i	$0.140556\pi$
$314$	8591.32	1.54406
$315$	0	0
$316$	−5189.65	−0.923862
$317$	−2635.75	−0.466998	−0.233499	−	0.972357i	$-0.575018\pi$
$317$	−2635.75	−0.466998	−0.233499	+	0.972357i	$0.575018\pi$
$318$	−4001.62	−0.705660
$319$	6423.34	1.12739
$320$	0	0
$321$	−4453.79	−0.774412
$322$	−7939.63	−1.37409
$323$	−3140.35	−0.540972
$324$	549.322	0.0941910
$325$	0	0
$326$	−12436.5	−2.11287
$327$	5404.10	0.913906
$328$	718.856	0.121013
$329$	10442.2	1.74984
$330$	0	0
$331$	−9381.49	−1.55786	−0.778932	−	0.627108i	$-0.784238\pi$
$331$	−9381.49	−1.55786	−0.778932	+	0.627108i	$0.784238\pi$
$332$	−8945.78	−1.47881
$333$	−1698.96	−0.279587
$334$	6499.43	1.06477
$335$	0	0
$336$	5504.73	0.893772
$337$	5290.72	0.855204	0.427602	−	0.903967i	$-0.359358\pi$
$337$	5290.72	0.855204	0.427602	+	0.903967i	$0.359358\pi$
$338$	8351.40	1.34395
$339$	−1159.01	−0.185690
$340$	0	0
$341$	13983.9	2.22074
$342$	−838.614	−0.132594
$343$	1046.72	0.164774
$344$	−1657.72	−0.259821
$345$	0	0
$346$	15812.3	2.45686
$347$	2067.82	0.319903	0.159951	−	0.987125i	$-0.448866\pi$
$347$	2067.82	0.319903	0.159951	+	0.987125i	$0.448866\pi$
$348$	−2296.99	−0.353826
$349$	−5629.03	−0.863367	−0.431684	−	0.902025i	$-0.642080\pi$
$349$	−5629.03	−0.863367	−0.431684	+	0.902025i	$0.642080\pi$
$350$	0	0
$351$	−134.507	−0.0204542
$352$	−13674.7	−2.07064
$353$	11516.6	1.73644	0.868222	−	0.496177i	$-0.165263\pi$
$353$	11516.6	1.73644	0.868222	+	0.496177i	$0.165263\pi$
$354$	−3116.11	−0.467851
$355$	0	0
$356$	1840.32	0.273980
$357$	9870.72	1.46334
$358$	14672.8	2.16614
$359$	−4504.68	−0.662251	−0.331126	−	0.943587i	$-0.607428\pi$
$359$	−4504.68	−0.662251	−0.331126	+	0.943587i	$0.607428\pi$
$360$	0	0
$361$	−6271.63	−0.914365
$362$	−7412.77	−1.07626
$363$	5717.72	0.826729
$364$	857.880	0.123531
$365$	0	0
$366$	−4356.96	−0.622246
$367$	1278.30	0.181816	0.0909080	−	0.995859i	$-0.471023\pi$
$367$	1278.30	0.181816	0.0909080	+	0.995859i	$0.471023\pi$
$368$	5876.80	0.832472
$369$	1381.29	0.194870
$370$	0	0
$371$	−8809.62	−1.23281
$372$	−5000.66	−0.696967
$373$	7998.62	1.11033	0.555164	−	0.831741i	$-0.312655\pi$
$373$	7998.62	1.11033	0.555164	+	0.831741i	$0.312655\pi$
$374$	−28343.3	−3.91871
$375$	0	0
$376$	−1926.14	−0.264184
$377$	562.439	0.0768357
$378$	2635.92	0.358669
$379$	1191.93	0.161544	0.0807721	−	0.996733i	$-0.474261\pi$
$379$	1191.93	0.161544	0.0807721	+	0.996733i	$0.474261\pi$
$380$	0	0
$381$	6355.51	0.854600
$382$	6137.70	0.822074
$383$	7251.83	0.967497	0.483749	−	0.875207i	$-0.339275\pi$
$383$	7251.83	0.967497	0.483749	+	0.875207i	$0.339275\pi$
$384$	−1777.73	−0.236249
$385$	0	0
$386$	14412.2	1.90042
$387$	−3185.33	−0.418397
$388$	781.466	0.102250
$389$	9137.12	1.19093	0.595463	−	0.803383i	$-0.296969\pi$
$389$	9137.12	1.19093	0.595463	+	0.803383i	$0.296969\pi$
$390$	0	0
$391$	10537.9	1.36298
$392$	1413.47	0.182121
$393$	7046.43	0.904442
$394$	6090.63	0.778786
$395$	0	0
$396$	−3472.56	−0.440663
$397$	−8211.56	−1.03810	−0.519051	−	0.854743i	$-0.673714\pi$
$397$	−8211.56	−1.03810	−0.519051	+	0.854743i	$0.673714\pi$
$398$	−11857.3	−1.49334
$399$	−1846.22	−0.231645
$400$	0	0
$401$	−5175.68	−0.644542	−0.322271	−	0.946647i	$-0.604446\pi$
$401$	−5175.68	−0.644542	−0.322271	+	0.946647i	$0.604446\pi$
$402$	8874.59	1.10106
$403$	1224.46	0.151351
$404$	6088.62	0.749802
$405$	0	0
$406$	−11022.1	−1.34733
$407$	10740.0	1.30802
$408$	−1820.72	−0.220929
$409$	−7139.46	−0.863139	−0.431569	−	0.902080i	$-0.642040\pi$
$409$	−7139.46	−0.863139	−0.431569	+	0.902080i	$0.642040\pi$
$410$	0	0
$411$	−1909.74	−0.229199
$412$	2137.63	0.255615
$413$	−6860.15	−0.817351
$414$	2814.09	0.334069
$415$	0	0
$416$	−1197.38	−0.141121
$417$	−1454.86	−0.170851
$418$	5301.33	0.620326
$419$	11634.1	1.35647	0.678235	−	0.734845i	$-0.262745\pi$
$419$	11634.1	1.35647	0.678235	+	0.734845i	$0.262745\pi$
$420$	0	0
$421$	−3651.13	−0.422673	−0.211336	−	0.977413i	$-0.567782\pi$
$421$	−3651.13	−0.422673	−0.211336	+	0.977413i	$0.567782\pi$
$422$	13587.9	1.56742
$423$	−3701.10	−0.425422
$424$	1624.99	0.186124
$425$	0	0
$426$	7576.80	0.861730
$427$	−9591.91	−1.08708
$428$	−10068.2	−1.13706
$429$	850.288	0.0956930
$430$	0	0
$431$	6408.88	0.716253	0.358126	−	0.933673i	$-0.383416\pi$
$431$	6408.88	0.716253	0.358126	+	0.933673i	$0.383416\pi$
$432$	−1951.07	−0.217294
$433$	−3172.69	−0.352124	−0.176062	−	0.984379i	$-0.556336\pi$
$433$	−3172.69	−0.352124	−0.176062	+	0.984379i	$0.556336\pi$
$434$	−23995.6	−2.65398
$435$	0	0
$436$	12216.4	1.34188
$437$	−1971.01	−0.215758
$438$	−4270.27	−0.465848
$439$	8476.19	0.921519	0.460759	−	0.887525i	$-0.347577\pi$
$439$	8476.19	0.921519	0.460759	+	0.887525i	$0.347577\pi$
$440$	0	0
$441$	2716.01	0.293274
$442$	−2481.79	−0.267074
$443$	−15713.9	−1.68531	−0.842654	−	0.538455i	$-0.819008\pi$
$443$	−15713.9	−1.68531	−0.842654	+	0.538455i	$0.819008\pi$
$444$	−3840.64	−0.410515
$445$	0	0
$446$	19751.8	2.09703
$447$	−3439.80	−0.363976
$448$	8785.77	0.926537
$449$	−6523.11	−0.685622	−0.342811	−	0.939404i	$-0.611379\pi$
$449$	−6523.11	−0.685622	−0.342811	+	0.939404i	$0.611379\pi$
$450$	0	0
$451$	−8731.87	−0.911680
$452$	−2620.04	−0.272646
$453$	−2265.01	−0.234922
$454$	11431.0	1.18168
$455$	0	0
$456$	340.547	0.0349728
$457$	15753.4	1.61250	0.806248	−	0.591577i	$-0.201495\pi$
$457$	15753.4	1.61250	0.806248	+	0.591577i	$0.201495\pi$
$458$	−19437.2	−1.98306
$459$	−3498.53	−0.355768
$460$	0	0
$461$	−2107.27	−0.212896	−0.106448	−	0.994318i	$-0.533948\pi$
$461$	−2107.27	−0.212896	−0.106448	+	0.994318i	$0.533948\pi$
$462$	−16663.1	−1.67800
$463$	9316.07	0.935107	0.467554	−	0.883965i	$-0.345135\pi$
$463$	9316.07	0.935107	0.467554	+	0.883965i	$0.345135\pi$
$464$	8158.40	0.816259
$465$	0	0
$466$	14061.3	1.39781
$467$	−705.989	−0.0699556	−0.0349778	−	0.999388i	$-0.511136\pi$
$467$	−705.989	−0.0699556	−0.0349778	+	0.999388i	$0.511136\pi$
$468$	−304.063	−0.0300327
$469$	19537.5	1.92358
$470$	0	0
$471$	−6703.76	−0.655823
$472$	1265.40	0.123400
$473$	20136.2	1.95743
$474$	8826.33	0.855289
$475$	0	0
$476$	22313.6	2.14862
$477$	3122.44	0.299721
$478$	15213.5	1.45575
$479$	3947.03	0.376502	0.188251	−	0.982121i	$-0.439718\pi$
$479$	3947.03	0.376502	0.188251	+	0.982121i	$0.439718\pi$
$480$	0	0
$481$	940.417	0.0891462
$482$	2128.79	0.201169
$483$	6195.24	0.583630
$484$	12925.4	1.21388
$485$	0	0
$486$	−934.263	−0.0871997
$487$	18485.7	1.72005	0.860026	−	0.510250i	$-0.170447\pi$
$487$	18485.7	1.72005	0.860026	+	0.510250i	$0.170447\pi$
$488$	1769.29	0.164123
$489$	9704.14	0.897416
$490$	0	0
$491$	−1069.93	−0.0983410	−0.0491705	−	0.998790i	$-0.515658\pi$
$491$	−1069.93	−0.0983410	−0.0491705	+	0.998790i	$0.515658\pi$
$492$	3122.52	0.286126
$493$	14629.1	1.33643
$494$	464.194	0.0422774
$495$	0	0
$496$	17761.2	1.60787
$497$	16680.4	1.50547
$498$	15214.6	1.36904
$499$	18165.3	1.62964	0.814819	−	0.579716i	$-0.196837\pi$
$499$	18165.3	1.62964	0.814819	+	0.579716i	$0.196837\pi$
$500$	0	0
$501$	−5071.47	−0.452248
$502$	−18968.8	−1.68649
$503$	2735.45	0.242480	0.121240	−	0.992623i	$-0.461313\pi$
$503$	2735.45	0.242480	0.121240	+	0.992623i	$0.461313\pi$
$504$	−1070.40	−0.0946023
$505$	0	0
$506$	−17789.3	−1.56291
$507$	−6516.55	−0.570828
$508$	14367.2	1.25480
$509$	−9399.52	−0.818520	−0.409260	−	0.912418i	$-0.634213\pi$
$509$	−9399.52	−0.818520	−0.409260	+	0.912418i	$0.634213\pi$
$510$	0	0
$511$	−9401.06	−0.813852
$512$	−14660.8	−1.26547
$513$	654.365	0.0563176
$514$	6390.33	0.548376
$515$	0	0
$516$	−7200.71	−0.614329
$517$	23396.6	1.99029
$518$	−18429.3	−1.56320
$519$	−12338.2	−1.04352
$520$	0	0
$521$	−5473.70	−0.460283	−0.230141	−	0.973157i	$-0.573919\pi$
$521$	−5473.70	−0.460283	−0.230141	+	0.973157i	$0.573919\pi$
$522$	3906.62	0.327563
$523$	−11063.4	−0.924989	−0.462494	−	0.886622i	$-0.653045\pi$
$523$	−11063.4	−0.924989	−0.462494	+	0.886622i	$0.653045\pi$
$524$	15929.0	1.32798
$525$	0	0
$526$	17293.7	1.43354
$527$	31848.3	2.63251
$528$	12333.8	1.01659
$529$	−5553.01	−0.456399
$530$	0	0
$531$	2431.48	0.198714
$532$	−4173.53	−0.340123
$533$	−764.578	−0.0621342
$534$	−3129.95	−0.253644
$535$	0	0
$536$	−3603.83	−0.290414
$537$	−11449.1	−0.920044
$538$	14324.6	1.14791
$539$	−17169.3	−1.37205
$540$	0	0
$541$	−2887.19	−0.229445	−0.114723	−	0.993398i	$-0.536598\pi$
$541$	−2887.19	−0.229445	−0.114723	+	0.993398i	$0.536598\pi$
$542$	−24416.3	−1.93500
$543$	5784.14	0.457129
$544$	−31144.1	−2.45458
$545$	0	0
$546$	−1459.05	−0.114362
$547$	3908.06	0.305478	0.152739	−	0.988267i	$-0.451191\pi$
$547$	3908.06	0.305478	0.152739	+	0.988267i	$0.451191\pi$
$548$	−4317.14	−0.336531
$549$	3399.71	0.264292
$550$	0	0
$551$	−2736.23	−0.211556
$552$	−1142.75	−0.0881139
$553$	19431.3	1.49422
$554$	4697.27	0.360231
$555$	0	0
$556$	−3288.84	−0.250859
$557$	19340.0	1.47121	0.735605	−	0.677411i	$-0.236898\pi$
$557$	19340.0	1.47121	0.735605	+	0.677411i	$0.236898\pi$
$558$	8504.90	0.645235
$559$	1763.16	0.133406
$560$	0	0
$561$	22116.1	1.66443
$562$	−11969.7	−0.898422
$563$	−659.518	−0.0493701	−0.0246851	−	0.999695i	$-0.507858\pi$
$563$	−659.518	−0.0493701	−0.0246851	+	0.999695i	$0.507858\pi$
$564$	−8366.64	−0.624643
$565$	0	0
$566$	−27113.9	−2.01357
$567$	−2056.79	−0.152341
$568$	−3076.81	−0.227289
$569$	−13523.8	−0.996393	−0.498196	−	0.867064i	$-0.666004\pi$
$569$	−13523.8	−0.996393	−0.498196	+	0.867064i	$0.666004\pi$
$570$	0	0
$571$	−21561.8	−1.58026	−0.790132	−	0.612936i	$-0.789988\pi$
$571$	−21561.8	−1.58026	−0.790132	+	0.612936i	$0.789988\pi$
$572$	1922.15	0.140505
$573$	−4789.21	−0.349166
$574$	14983.4	1.08954
$575$	0	0
$576$	−3113.99	−0.225260
$577$	−15184.0	−1.09553	−0.547764	−	0.836633i	$-0.684521\pi$
$577$	−15184.0	−1.09553	−0.547764	+	0.836633i	$0.684521\pi$
$578$	−45662.6	−3.28601
$579$	−11245.8	−0.807181
$580$	0	0
$581$	33495.1	2.39176
$582$	−1329.09	−0.0946604
$583$	−19738.6	−1.40221
$584$	1734.09	0.122872
$585$	0	0
$586$	−11037.4	−0.778074
$587$	26690.1	1.87669	0.938347	−	0.345696i	$-0.112357\pi$
$587$	26690.1	1.87669	0.938347	+	0.345696i	$0.112357\pi$
$588$	6139.76	0.430611
$589$	−5956.90	−0.416723
$590$	0	0
$591$	−4752.49	−0.330780
$592$	13641.1	0.947038
$593$	9280.64	0.642682	0.321341	−	0.946964i	$-0.395867\pi$
$593$	9280.64	0.642682	0.321341	+	0.946964i	$0.395867\pi$
$594$	5905.98	0.407955
$595$	0	0
$596$	−7775.96	−0.534423
$597$	9252.15	0.634280
$598$	−1557.67	−0.106518
$599$	−25222.4	−1.72047	−0.860234	−	0.509899i	$-0.829683\pi$
$599$	−25222.4	−1.72047	−0.860234	+	0.509899i	$0.829683\pi$
$600$	0	0
$601$	9246.22	0.627556	0.313778	−	0.949496i	$-0.398405\pi$
$601$	9246.22	0.627556	0.313778	+	0.949496i	$0.398405\pi$
$602$	−34552.6	−2.33930
$603$	−6924.79	−0.467661
$604$	−5120.24	−0.344933
$605$	0	0
$606$	−10355.3	−0.694148
$607$	19028.3	1.27238	0.636192	−	0.771531i	$-0.280509\pi$
$607$	19028.3	1.27238	0.636192	+	0.771531i	$0.280509\pi$
$608$	5825.18	0.388557
$609$	8600.47	0.572264
$610$	0	0
$611$	2048.65	0.135646
$612$	−7908.72	−0.522371
$613$	−10873.1	−0.716411	−0.358206	−	0.933643i	$-0.616611\pi$
$613$	−10873.1	−0.716411	−0.358206	+	0.933643i	$0.616611\pi$
$614$	−8480.92	−0.557430
$615$	0	0
$616$	6766.60	0.442588
$617$	−6527.77	−0.425929	−0.212965	−	0.977060i	$-0.568312\pi$
$617$	−6527.77	−0.425929	−0.212965	+	0.977060i	$0.568312\pi$
$618$	−3635.59	−0.236642
$619$	−2684.74	−0.174328	−0.0871639	−	0.996194i	$-0.527780\pi$
$619$	−2684.74	−0.174328	−0.0871639	+	0.996194i	$0.527780\pi$
$620$	0	0
$621$	−2195.81	−0.141892
$622$	6130.49	0.395193
$623$	−6890.61	−0.443125
$624$	1079.97	0.0692841
$625$	0	0
$626$	−38496.1	−2.45785
$627$	−4136.59	−0.263476
$628$	−15154.4	−0.962940
$629$	24460.4	1.55055
$630$	0	0
$631$	−18279.6	−1.15325	−0.576623	−	0.817011i	$-0.695630\pi$
$631$	−18279.6	−1.15325	−0.576623	+	0.817011i	$0.695630\pi$
$632$	−3584.23	−0.225590
$633$	−10602.6	−0.665742
$634$	10133.7	0.634795
$635$	0	0
$636$	7058.54	0.440078
$637$	−1503.38	−0.0935101
$638$	−24695.8	−1.53247
$639$	−5912.13	−0.366010
$640$	0	0
$641$	−5539.92	−0.341363	−0.170682	−	0.985326i	$-0.554597\pi$
$641$	−5539.92	−0.341363	−0.170682	+	0.985326i	$0.554597\pi$
$642$	17123.5	1.05266
$643$	−1756.75	−0.107744	−0.0538720	−	0.998548i	$-0.517156\pi$
$643$	−1756.75	−0.107744	−0.0538720	+	0.998548i	$0.517156\pi$
$644$	14004.9	0.856939
$645$	0	0
$646$	12073.7	0.735348
$647$	−9954.72	−0.604885	−0.302442	−	0.953168i	$-0.597802\pi$
$647$	−9954.72	−0.604885	−0.302442	+	0.953168i	$0.597802\pi$
$648$	379.389	0.0229997
$649$	−15370.7	−0.929665
$650$	0	0
$651$	18723.6	1.12725
$652$	21937.0	1.31767
$653$	3322.45	0.199108	0.0995540	−	0.995032i	$-0.468258\pi$
$653$	3322.45	0.199108	0.0995540	+	0.995032i	$0.468258\pi$
$654$	−20777.2	−1.24228
$655$	0	0
$656$	−11090.5	−0.660078
$657$	3332.07	0.197864
$658$	−40147.3	−2.37858
$659$	−403.836	−0.0238714	−0.0119357	−	0.999929i	$-0.503799\pi$
$659$	−403.836	−0.0238714	−0.0119357	+	0.999929i	$0.503799\pi$
$660$	0	0
$661$	−19243.6	−1.13236	−0.566180	−	0.824282i	$-0.691579\pi$
$661$	−19243.6	−1.13236	−0.566180	+	0.824282i	$0.691579\pi$
$662$	36069.0	2.11762
$663$	1936.53	0.113437
$664$	−6178.41	−0.361097
$665$	0	0
$666$	6532.00	0.380045
$667$	9181.79	0.533014
$668$	−11464.5	−0.664032
$669$	−15412.2	−0.890688
$670$	0	0
$671$	−21491.4	−1.23646
$672$	−18309.6	−1.05106
$673$	−23572.8	−1.35017	−0.675084	−	0.737741i	$-0.735893\pi$
$673$	−23572.8	−1.35017	−0.675084	+	0.737741i	$0.735893\pi$
$674$	−20341.2	−1.16249
$675$	0	0
$676$	−14731.2	−0.838143
$677$	−18366.2	−1.04265	−0.521323	−	0.853360i	$-0.674561\pi$
$677$	−18366.2	−1.04265	−0.521323	+	0.853360i	$0.674561\pi$
$678$	4456.05	0.252409
$679$	−2926.00	−0.165375
$680$	0	0
$681$	−8919.54	−0.501905
$682$	−53764.1	−3.01867
$683$	−22763.0	−1.27526	−0.637630	−	0.770342i	$-0.720085\pi$
$683$	−22763.0	−1.27526	−0.637630	+	0.770342i	$0.720085\pi$
$684$	1479.25	0.0826907
$685$	0	0
$686$	−4024.32	−0.223978
$687$	15166.7	0.842281
$688$	25575.4	1.41723
$689$	−1728.35	−0.0955659
$690$	0	0
$691$	−17479.5	−0.962302	−0.481151	−	0.876638i	$-0.659781\pi$
$691$	−17479.5	−0.962302	−0.481151	+	0.876638i	$0.659781\pi$
$692$	−27891.6	−1.53219
$693$	13002.1	0.712710
$694$	−7950.14	−0.434846
$695$	0	0
$696$	−1586.41	−0.0863978
$697$	−19886.8	−1.08072
$698$	21642.0	1.17358
$699$	−10972.0	−0.593702
$700$	0	0
$701$	3168.85	0.170736	0.0853680	−	0.996349i	$-0.472793\pi$
$701$	3168.85	0.170736	0.0853680	+	0.996349i	$0.472793\pi$
$702$	517.138	0.0278036
$703$	−4575.07	−0.245451
$704$	19685.2	1.05385
$705$	0	0
$706$	−44277.8	−2.36036
$707$	−22797.2	−1.21270
$708$	5496.57	0.291771
$709$	9259.39	0.490471	0.245235	−	0.969464i	$-0.421135\pi$
$709$	9259.39	0.490471	0.245235	+	0.969464i	$0.421135\pi$
$710$	0	0
$711$	−6887.14	−0.363274
$712$	1271.02	0.0669010
$713$	19989.2	1.04993
$714$	−37950.0	−1.98914
$715$	0	0
$716$	−25881.6	−1.35089
$717$	−11871.0	−0.618313
$718$	17319.2	0.900203
$719$	−12203.8	−0.632995	−0.316497	−	0.948593i	$-0.602507\pi$
$719$	−12203.8	−0.632995	−0.316497	+	0.948593i	$0.602507\pi$
$720$	0	0
$721$	−8003.80	−0.413422
$722$	24112.6	1.24290
$723$	−1661.08	−0.0854444
$724$	13075.5	0.671199
$725$	0	0
$726$	−21983.0	−1.12378
$727$	36065.2	1.83987	0.919933	−	0.392076i	$-0.128243\pi$
$727$	36065.2	1.83987	0.919933	+	0.392076i	$0.128243\pi$
$728$	592.495	0.0301639
$729$	729.000	0.0370370
$730$	0	0
$731$	45860.1	2.32038
$732$	7685.33	0.388058
$733$	17563.8	0.885038	0.442519	−	0.896759i	$-0.354085\pi$
$733$	17563.8	0.885038	0.442519	+	0.896759i	$0.354085\pi$
$734$	−4914.67	−0.247144
$735$	0	0
$736$	−19547.2	−0.978968
$737$	43775.3	2.18790
$738$	−5310.65	−0.264888
$739$	−5775.83	−0.287507	−0.143753	−	0.989614i	$-0.545917\pi$
$739$	−5775.83	−0.287507	−0.143753	+	0.989614i	$0.545917\pi$
$740$	0	0
$741$	−362.207	−0.0179568
$742$	33870.4	1.67577
$743$	−9877.37	−0.487706	−0.243853	−	0.969812i	$-0.578411\pi$
$743$	−9877.37	−0.487706	−0.243853	+	0.969812i	$0.578411\pi$
$744$	−3453.70	−0.170187
$745$	0	0
$746$	−30752.3	−1.50928
$747$	−11871.9	−0.581485
$748$	49995.3	2.44386
$749$	37697.6	1.83904
$750$	0	0
$751$	−2277.02	−0.110639	−0.0553193	−	0.998469i	$-0.517618\pi$
$751$	−2277.02	−0.110639	−0.0553193	+	0.998469i	$0.517618\pi$
$752$	29716.5	1.44102
$753$	14801.2	0.716317
$754$	−2162.41	−0.104443
$755$	0	0
$756$	−4649.55	−0.223680
$757$	−24708.1	−1.18631	−0.593153	−	0.805090i	$-0.702117\pi$
$757$	−24708.1	−1.18631	−0.593153	+	0.805090i	$0.702117\pi$
$758$	−4582.61	−0.219588
$759$	13880.9	0.663828
$760$	0	0
$761$	4733.97	0.225501	0.112751	−	0.993623i	$-0.464034\pi$
$761$	4733.97	0.225501	0.112751	+	0.993623i	$0.464034\pi$
$762$	−24435.1	−1.16167
$763$	−45741.2	−2.17030
$764$	−10826.4	−0.512678
$765$	0	0
$766$	−27881.2	−1.31513
$767$	−1345.89	−0.0633600
$768$	15138.8	0.711296
$769$	5565.64	0.260991	0.130495	−	0.991449i	$-0.458343\pi$
$769$	5565.64	0.260991	0.130495	+	0.991449i	$0.458343\pi$
$770$	0	0
$771$	−4986.33	−0.232916
$772$	−25422.0	−1.18518
$773$	−6981.69	−0.324856	−0.162428	−	0.986720i	$-0.551933\pi$
$773$	−6981.69	−0.324856	−0.162428	+	0.986720i	$0.551933\pi$
$774$	12246.7	0.568730
$775$	0	0
$776$	539.720	0.0249676
$777$	14380.3	0.663951
$778$	−35129.5	−1.61884
$779$	3719.62	0.171077
$780$	0	0
$781$	37373.8	1.71234
$782$	−40515.1	−1.85271
$783$	−3048.31	−0.139129
$784$	−21807.1	−0.993398
$785$	0	0
$786$	−27091.5	−1.22942
$787$	−31972.3	−1.44815	−0.724073	−	0.689723i	$-0.757732\pi$
$787$	−31972.3	−1.44815	−0.724073	+	0.689723i	$0.757732\pi$
$788$	−10743.4	−0.485682
$789$	−13494.2	−0.608879
$790$	0	0
$791$	9810.04	0.440967
$792$	−2398.32	−0.107602
$793$	−1881.83	−0.0842694
$794$	31571.0	1.41110
$795$	0	0
$796$	20915.3	0.931308
$797$	−167.785	−0.00745702	−0.00372851	−	0.999993i	$-0.501187\pi$
$797$	−167.785	−0.00745702	−0.00372851	+	0.999993i	$0.501187\pi$
$798$	7098.16	0.314877
$799$	53285.6	2.35934
$800$	0	0
$801$	2442.28	0.107732
$802$	19899.0	0.876131
$803$	−21063.8	−0.925686
$804$	−15654.1	−0.686662
$805$	0	0
$806$	−4707.68	−0.205733
$807$	−11177.4	−0.487562
$808$	4205.10	0.183088
$809$	9943.62	0.432137	0.216069	−	0.976378i	$-0.430677\pi$
$809$	9943.62	0.432137	0.216069	+	0.976378i	$0.430677\pi$
$810$	0	0
$811$	−13276.9	−0.574866	−0.287433	−	0.957801i	$-0.592802\pi$
$811$	−13276.9	−0.574866	−0.287433	+	0.957801i	$0.592802\pi$
$812$	19442.1	0.840250
$813$	19051.9	0.821870
$814$	−41292.3	−1.77800
$815$	0	0
$816$	28090.1	1.20508
$817$	−8577.66	−0.367313
$818$	27449.1	1.17327
$819$	1138.49	0.0485737
$820$	0	0
$821$	28582.4	1.21502	0.607510	−	0.794312i	$-0.292168\pi$
$821$	28582.4	1.21502	0.607510	+	0.794312i	$0.292168\pi$
$822$	7342.40	0.311552
$823$	−29219.7	−1.23759	−0.618793	−	0.785554i	$-0.712378\pi$
$823$	−29219.7	−1.23759	−0.618793	+	0.785554i	$0.712378\pi$
$824$	1476.35	0.0624166
$825$	0	0
$826$	26375.3	1.11103
$827$	21299.3	0.895586	0.447793	−	0.894137i	$-0.352210\pi$
$827$	21299.3	0.895586	0.447793	+	0.894137i	$0.352210\pi$
$828$	−4963.82	−0.208339
$829$	−2367.09	−0.0991704	−0.0495852	−	0.998770i	$-0.515790\pi$
$829$	−2367.09	−0.0991704	−0.0495852	+	0.998770i	$0.515790\pi$
$830$	0	0
$831$	−3665.25	−0.153004
$832$	1723.67	0.0718239
$833$	−39103.0	−1.62646
$834$	5593.51	0.232239
$835$	0	0
$836$	−9351.11	−0.386860
$837$	−6636.33	−0.274056
$838$	−44729.5	−1.84386
$839$	−9255.11	−0.380837	−0.190418	−	0.981703i	$-0.560984\pi$
$839$	−9255.11	−0.380837	−0.190418	+	0.981703i	$0.560984\pi$
$840$	0	0
$841$	−11642.5	−0.477367
$842$	14037.5	0.574543
$843$	9339.92	0.381594
$844$	−23968.0	−0.977504
$845$	0	0
$846$	14229.6	0.578279
$847$	−48395.7	−1.96328
$848$	−25070.4	−1.01524
$849$	21156.8	0.855242
$850$	0	0
$851$	15352.3	0.618413
$852$	−13364.9	−0.537409
$853$	−2998.79	−0.120371	−0.0601856	−	0.998187i	$-0.519169\pi$
$853$	−2998.79	−0.120371	−0.0601856	+	0.998187i	$0.519169\pi$
$854$	36878.1	1.47768
$855$	0	0
$856$	−6953.58	−0.277650
$857$	11304.4	0.450583	0.225292	−	0.974291i	$-0.427667\pi$
$857$	11304.4	0.450583	0.225292	+	0.974291i	$0.427667\pi$
$858$	−3269.11	−0.130076
$859$	37199.5	1.47757	0.738783	−	0.673943i	$-0.235401\pi$
$859$	37199.5	1.47757	0.738783	+	0.673943i	$0.235401\pi$
$860$	0	0
$861$	−11691.5	−0.462768
$862$	−24640.2	−0.973608
$863$	6719.15	0.265032	0.132516	−	0.991181i	$-0.457694\pi$
$863$	6719.15	0.265032	0.132516	+	0.991181i	$0.457694\pi$
$864$	6489.58	0.255533
$865$	0	0
$866$	12198.0	0.478645
$867$	35630.3	1.39569
$868$	42326.4	1.65513
$869$	43537.3	1.69954
$870$	0	0
$871$	3833.04	0.149113
$872$	8437.27	0.327663
$873$	1037.08	0.0402059
$874$	7577.94	0.293281
$875$	0	0
$876$	7532.42	0.290522
$877$	−23364.1	−0.899602	−0.449801	−	0.893129i	$-0.648505\pi$
$877$	−23364.1	−0.899602	−0.449801	+	0.893129i	$0.648505\pi$
$878$	−32588.5	−1.25263
$879$	8612.43	0.330478
$880$	0	0
$881$	−26991.7	−1.03221	−0.516103	−	0.856526i	$-0.672618\pi$
$881$	−26991.7	−1.03221	−0.516103	+	0.856526i	$0.672618\pi$
$882$	−10442.2	−0.398649
$883$	−7492.93	−0.285569	−0.142784	−	0.989754i	$-0.545606\pi$
$883$	−7492.93	−0.285569	−0.142784	+	0.989754i	$0.545606\pi$
$884$	4377.68	0.166558
$885$	0	0
$886$	60415.4	2.29085
$887$	35983.4	1.36212	0.681062	−	0.732226i	$-0.261518\pi$
$887$	35983.4	1.36212	0.681062	+	0.732226i	$0.261518\pi$
$888$	−2652.54	−0.100240
$889$	−53794.1	−2.02947
$890$	0	0
$891$	−4608.40	−0.173274
$892$	−34840.6	−1.30779
$893$	−9966.54	−0.373480
$894$	13225.0	0.494755
$895$	0	0
$896$	15047.0	0.561033
$897$	1215.44	0.0452422
$898$	25079.4	0.931972
$899$	27749.8	1.02949
$900$	0	0
$901$	−44954.6	−1.66221
$902$	33571.4	1.23925
$903$	26961.2	0.993590
$904$	−1809.53	−0.0665753
$905$	0	0
$906$	8708.30	0.319331
$907$	11901.5	0.435704	0.217852	−	0.975982i	$-0.430095\pi$
$907$	11901.5	0.435704	0.217852	+	0.975982i	$0.430095\pi$
$908$	−20163.4	−0.736944
$909$	8080.15	0.294831
$910$	0	0
$911$	−23273.7	−0.846422	−0.423211	−	0.906031i	$-0.639097\pi$
$911$	−23273.7	−0.846422	−0.423211	+	0.906031i	$0.639097\pi$
$912$	−5253.96	−0.190763
$913$	75048.5	2.72042
$914$	−60567.0	−2.19188
$915$	0	0
$916$	34285.6	1.23671
$917$	−59642.2	−2.14783
$918$	13450.8	0.483598
$919$	22949.6	0.823763	0.411882	−	0.911237i	$-0.364872\pi$
$919$	22949.6	0.823763	0.411882	+	0.911237i	$0.364872\pi$
$920$	0	0
$921$	6617.61	0.236762
$922$	8101.82	0.289392
$923$	3272.51	0.116702
$924$	29392.3	1.04647
$925$	0	0
$926$	−35817.5	−1.27110
$927$	2836.83	0.100511
$928$	−27136.2	−0.959902
$929$	−3026.51	−0.106885	−0.0534427	−	0.998571i	$-0.517019\pi$
$929$	−3026.51	−0.106885	−0.0534427	+	0.998571i	$0.517019\pi$
$930$	0	0
$931$	7313.83	0.257466
$932$	−24803.0	−0.871727
$933$	−4783.59	−0.167854
$934$	2714.32	0.0950912
$935$	0	0
$936$	−210.001	−0.00733345
$937$	16973.3	0.591774	0.295887	−	0.955223i	$-0.404385\pi$
$937$	16973.3	0.591774	0.295887	+	0.955223i	$0.404385\pi$
$938$	−75116.0	−2.61474
$939$	30038.3	1.04394
$940$	0	0
$941$	7850.76	0.271974	0.135987	−	0.990711i	$-0.456579\pi$
$941$	7850.76	0.271974	0.135987	+	0.990711i	$0.456579\pi$
$942$	25774.0	0.891466
$943$	−12481.7	−0.431029
$944$	−19522.6	−0.673100
$945$	0	0
$946$	−77417.8	−2.66075
$947$	−12900.5	−0.442671	−0.221336	−	0.975198i	$-0.571042\pi$
$947$	−12900.5	−0.442671	−0.221336	+	0.975198i	$0.571042\pi$
$948$	−15568.9	−0.533392
$949$	−1844.38	−0.0630888
$950$	0	0
$951$	−7907.25	−0.269622
$952$	15410.9	0.524653
$953$	−38633.5	−1.31318	−0.656591	−	0.754247i	$-0.728002\pi$
$953$	−38633.5	−1.31318	−0.656591	+	0.754247i	$0.728002\pi$
$954$	−12004.9	−0.407413
$955$	0	0
$956$	−26835.4	−0.907864
$957$	19270.0	0.650900
$958$	−15175.2	−0.511782
$959$	16164.4	0.544291
$960$	0	0
$961$	30621.7	1.02788
$962$	−3615.63	−0.121177
$963$	−13361.4	−0.447107
$964$	−3755.01	−0.125457
$965$	0	0
$966$	−23818.9	−0.793333
$967$	−49404.1	−1.64295	−0.821473	−	0.570247i	$-0.806847\pi$
$967$	−49404.1	−1.64295	−0.821473	+	0.570247i	$0.806847\pi$
$968$	8926.92	0.296407
$969$	−9421.06	−0.312330
$970$	0	0
$971$	−18298.5	−0.604766	−0.302383	−	0.953186i	$-0.597782\pi$
$971$	−18298.5	−0.604766	−0.302383	+	0.953186i	$0.597782\pi$
$972$	1647.96	0.0543812
$973$	12314.2	0.405730
$974$	−71071.9	−2.33808
$975$	0	0
$976$	−27296.6	−0.895229
$977$	−49315.9	−1.61490	−0.807449	−	0.589937i	$-0.799153\pi$
$977$	−49315.9	−1.61490	−0.807449	+	0.589937i	$0.799153\pi$
$978$	−37309.6	−1.21987
$979$	−15439.0	−0.504015
$980$	0	0
$981$	16212.3	0.527644
$982$	4113.58	0.133676
$983$	−61321.2	−1.98967	−0.994833	−	0.101524i	$-0.967628\pi$
$983$	−61321.2	−1.98967	−0.994833	+	0.101524i	$0.967628\pi$
$984$	2156.57	0.0698667
$985$	0	0
$986$	−56244.6	−1.81663
$987$	31326.7	1.01027
$988$	−818.800	−0.0263659
$989$	28783.6	0.925444
$990$	0	0
$991$	15367.1	0.492586	0.246293	−	0.969195i	$-0.420787\pi$
$991$	15367.1	0.492586	0.246293	+	0.969195i	$0.420787\pi$
$992$	−59076.8	−1.89082
$993$	−28144.5	−0.899434
$994$	−64131.2	−2.04640
$995$	0	0
$996$	−26837.3	−0.853789
$997$	16668.2	0.529475	0.264737	−	0.964321i	$-0.414715\pi$
$997$	16668.2	0.529475	0.264737	+	0.964321i	$0.414715\pi$
$998$	−69840.1	−2.21518
$999$	−5096.88	−0.161420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.4.a.l.1.5	yes	24
5.4	even	2		1875.4.a.k.1.20	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.4.a.k.1.20	✓	24	5.4	even	2
1875.4.a.l.1.5	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)

\(f(q)\)	\(=\)	\(q-3.84470 q^{2} +3.00000 q^{3} +6.78175 q^{4} -11.5341 q^{6} -25.3925 q^{7} +4.68382 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.84470 q^{2} +3.00000 q^{3} +6.78175 q^{4} -11.5341 q^{6} -25.3925 q^{7} +4.68382 q^{8} +9.00000 q^{9} -56.8938 q^{11} +20.3452 q^{12} -4.98172 q^{13} +97.6266 q^{14} -72.2619 q^{16} -129.575 q^{17} -34.6023 q^{18} +24.2358 q^{19} -76.1775 q^{21} +218.740 q^{22} -81.3265 q^{23} +14.0514 q^{24} +19.1533 q^{26} +27.0000 q^{27} -172.205 q^{28} -112.900 q^{29} -245.790 q^{31} +240.355 q^{32} -170.681 q^{33} +498.179 q^{34} +61.0357 q^{36} -188.773 q^{37} -93.1793 q^{38} -14.9452 q^{39} +153.477 q^{41} +292.880 q^{42} -353.926 q^{43} -385.840 q^{44} +312.676 q^{46} -411.233 q^{47} -216.786 q^{48} +301.778 q^{49} -388.726 q^{51} -33.7848 q^{52} +346.938 q^{53} -103.807 q^{54} -118.934 q^{56} +72.7073 q^{57} +434.069 q^{58} +270.165 q^{59} +377.746 q^{61} +944.989 q^{62} -228.532 q^{63} -345.999 q^{64} +656.220 q^{66} -769.421 q^{67} -878.747 q^{68} -243.979 q^{69} -656.903 q^{71} +42.1543 q^{72} +370.230 q^{73} +725.778 q^{74} +164.361 q^{76} +1444.68 q^{77} +57.4598 q^{78} -765.237 q^{79} +81.0000 q^{81} -590.072 q^{82} -1319.10 q^{83} -516.616 q^{84} +1360.74 q^{86} -338.701 q^{87} -266.480 q^{88} +271.364 q^{89} +126.498 q^{91} -551.536 q^{92} -737.370 q^{93} +1581.07 q^{94} +721.065 q^{96} +115.231 q^{97} -1160.25 q^{98} -512.044 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 72 q^{3} + 133 q^{4} + 3 q^{6} + 62 q^{7} + 27 q^{8} + 216 q^{9}+O(q^{10}) \) 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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