LMFDB - Embedded newform 1875.2.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1875,2,Mod(1,1875)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1875.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
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)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -3.00000 q^{8} +1.00000 q^{9} +3.23607 q^{11} -1.00000 q^{12} +3.38197 q^{13} +4.47214 q^{14} -1.00000 q^{16} -2.85410 q^{17} +1.00000 q^{18} -3.23607 q^{19} +4.47214 q^{21} +3.23607 q^{22} -4.47214 q^{23} -3.00000 q^{24} +3.38197 q^{26} +1.00000 q^{27} -4.47214 q^{28} +4.38197 q^{29} +7.23607 q^{31} +5.00000 q^{32} +3.23607 q^{33} -2.85410 q^{34} -1.00000 q^{36} +8.09017 q^{37} -3.23607 q^{38} +3.38197 q^{39} -1.38197 q^{41} +4.47214 q^{42} -5.70820 q^{43} -3.23607 q^{44} -4.47214 q^{46} -5.23607 q^{47} -1.00000 q^{48} +13.0000 q^{49} -2.85410 q^{51} -3.38197 q^{52} +1.38197 q^{53} +1.00000 q^{54} -13.4164 q^{56} -3.23607 q^{57} +4.38197 q^{58} -4.00000 q^{59} -0.618034 q^{61} +7.23607 q^{62} +4.47214 q^{63} +7.00000 q^{64} +3.23607 q^{66} +5.23607 q^{67} +2.85410 q^{68} -4.47214 q^{69} -0.763932 q^{71} -3.00000 q^{72} +3.09017 q^{73} +8.09017 q^{74} +3.23607 q^{76} +14.4721 q^{77} +3.38197 q^{78} +1.00000 q^{81} -1.38197 q^{82} +3.52786 q^{83} -4.47214 q^{84} -5.70820 q^{86} +4.38197 q^{87} -9.70820 q^{88} +7.61803 q^{89} +15.1246 q^{91} +4.47214 q^{92} +7.23607 q^{93} -5.23607 q^{94} +5.00000 q^{96} +8.85410 q^{97} +13.0000 q^{98} +3.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 9 q^{13} - 2 q^{16} + q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{22} - 6 q^{24} + 9 q^{26} + 2 q^{27} + 11 q^{29} + 10 q^{31}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 9 * q^13 - 2 * q^16 + q^17 + 2 * q^18 - 2 * q^19 + 2 * q^22 - 6 * q^24 + 9 * q^26 + 2 * q^27 + 11 * q^29 + 10 * q^31 + 10 * q^32 + 2 * q^33 + q^34 - 2 * q^36 + 5 * q^37 - 2 * q^38 + 9 * q^39 - 5 * q^41 + 2 * q^43 - 2 * q^44 - 6 * q^47 - 2 * q^48 + 26 * q^49 + q^51 - 9 * q^52 + 5 * q^53 + 2 * q^54 - 2 * q^57 + 11 * q^58 - 8 * q^59 + q^61 + 10 * q^62 + 14 * q^64 + 2 * q^66 + 6 * q^67 - q^68 - 6 * q^71 - 6 * q^72 - 5 * q^73 + 5 * q^74 + 2 * q^76 + 20 * q^77 + 9 * q^78 + 2 * q^81 - 5 * q^82 + 16 * q^83 + 2 * q^86 + 11 * q^87 - 6 * q^88 + 13 * q^89 - 10 * q^91 + 10 * q^93 - 6 * q^94 + 10 * q^96 + 11 * q^97 + 26 * q^98 + 2 * q^99

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	−1.00000	−0.288675
$13$	3.38197	0.937989	0.468994	−	0.883201i	$-0.344616\pi$
$13$	3.38197	0.937989	0.468994	+	0.883201i	$0.344616\pi$
$14$	4.47214	1.19523
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.85410	−0.692221	−0.346111	−	0.938194i	$-0.612498\pi$
$17$	−2.85410	−0.692221	−0.346111	+	0.938194i	$0.612498\pi$
$18$	1.00000	0.235702
$19$	−3.23607	−0.742405	−0.371202	−	0.928552i	$-0.621054\pi$
$19$	−3.23607	−0.742405	−0.371202	+	0.928552i	$0.621054\pi$
$20$	0	0
$21$	4.47214	0.975900
$22$	3.23607	0.689932
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	3.38197	0.663258
$27$	1.00000	0.192450
$28$	−4.47214	−0.845154
$29$	4.38197	0.813711	0.406855	−	0.913493i	$-0.366625\pi$
$29$	4.38197	0.813711	0.406855	+	0.913493i	$0.366625\pi$
$30$	0	0
$31$	7.23607	1.29964	0.649818	−	0.760090i	$-0.274845\pi$
$31$	7.23607	1.29964	0.649818	+	0.760090i	$0.274845\pi$
$32$	5.00000	0.883883
$33$	3.23607	0.563327
$34$	−2.85410	−0.489474
$35$	0	0
$36$	−1.00000	−0.166667
$37$	8.09017	1.33002	0.665008	−	0.746836i	$-0.268428\pi$
$37$	8.09017	1.33002	0.665008	+	0.746836i	$0.268428\pi$
$38$	−3.23607	−0.524960
$39$	3.38197	0.541548
$40$	0	0
$41$	−1.38197	−0.215827	−0.107913	−	0.994160i	$-0.534417\pi$
$41$	−1.38197	−0.215827	−0.107913	+	0.994160i	$0.534417\pi$
$42$	4.47214	0.690066
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	−3.23607	−0.487856
$45$	0	0
$46$	−4.47214	−0.659380
$47$	−5.23607	−0.763759	−0.381880	−	0.924212i	$-0.624723\pi$
$47$	−5.23607	−0.763759	−0.381880	+	0.924212i	$0.624723\pi$
$48$	−1.00000	−0.144338
$49$	13.0000	1.85714
$50$	0	0
$51$	−2.85410	−0.399654
$52$	−3.38197	−0.468994
$53$	1.38197	0.189828	0.0949138	−	0.995485i	$-0.469742\pi$
$53$	1.38197	0.189828	0.0949138	+	0.995485i	$0.469742\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−13.4164	−1.79284
$57$	−3.23607	−0.428628
$58$	4.38197	0.575380
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−0.618034	−0.0791311	−0.0395656	−	0.999217i	$-0.512597\pi$
$61$	−0.618034	−0.0791311	−0.0395656	+	0.999217i	$0.512597\pi$
$62$	7.23607	0.918982
$63$	4.47214	0.563436
$64$	7.00000	0.875000
$65$	0	0
$66$	3.23607	0.398332
$67$	5.23607	0.639688	0.319844	−	0.947470i	$-0.396370\pi$
$67$	5.23607	0.639688	0.319844	+	0.947470i	$0.396370\pi$
$68$	2.85410	0.346111
$69$	−4.47214	−0.538382
$70$	0	0
$71$	−0.763932	−0.0906621	−0.0453310	−	0.998972i	$-0.514434\pi$
$71$	−0.763932	−0.0906621	−0.0453310	+	0.998972i	$0.514434\pi$
$72$	−3.00000	−0.353553
$73$	3.09017	0.361677	0.180839	−	0.983513i	$-0.442119\pi$
$73$	3.09017	0.361677	0.180839	+	0.983513i	$0.442119\pi$
$74$	8.09017	0.940463
$75$	0	0
$76$	3.23607	0.371202
$77$	14.4721	1.64925
$78$	3.38197	0.382932
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.38197	−0.152613
$83$	3.52786	0.387233	0.193617	−	0.981077i	$-0.437978\pi$
$83$	3.52786	0.387233	0.193617	+	0.981077i	$0.437978\pi$
$84$	−4.47214	−0.487950
$85$	0	0
$86$	−5.70820	−0.615531
$87$	4.38197	0.469796
$88$	−9.70820	−1.03490
$89$	7.61803	0.807510	0.403755	−	0.914867i	$-0.367705\pi$
$89$	7.61803	0.807510	0.403755	+	0.914867i	$0.367705\pi$
$90$	0	0
$91$	15.1246	1.58549
$92$	4.47214	0.466252
$93$	7.23607	0.750345
$94$	−5.23607	−0.540059
$95$	0	0
$96$	5.00000	0.510310
$97$	8.85410	0.898998	0.449499	−	0.893281i	$-0.351603\pi$
$97$	8.85410	0.898998	0.449499	+	0.893281i	$0.351603\pi$
$98$	13.0000	1.31320
$99$	3.23607	0.325237
$100$	0	0
$101$	−16.5623	−1.64801	−0.824006	−	0.566582i	$-0.808266\pi$
$101$	−16.5623	−1.64801	−0.824006	+	0.566582i	$0.808266\pi$
$102$	−2.85410	−0.282598
$103$	−1.23607	−0.121793	−0.0608967	−	0.998144i	$-0.519396\pi$
$103$	−1.23607	−0.121793	−0.0608967	+	0.998144i	$0.519396\pi$
$104$	−10.1459	−0.994887
$105$	0	0
$106$	1.38197	0.134228
$107$	16.4721	1.59242	0.796211	−	0.605019i	$-0.206835\pi$
$107$	16.4721	1.59242	0.796211	+	0.605019i	$0.206835\pi$
$108$	−1.00000	−0.0962250
$109$	−19.0902	−1.82851	−0.914253	−	0.405143i	$-0.867222\pi$
$109$	−19.0902	−1.82851	−0.914253	+	0.405143i	$0.867222\pi$
$110$	0	0
$111$	8.09017	0.767885
$112$	−4.47214	−0.422577
$113$	2.67376	0.251526	0.125763	−	0.992060i	$-0.459862\pi$
$113$	2.67376	0.251526	0.125763	+	0.992060i	$0.459862\pi$
$114$	−3.23607	−0.303086
$115$	0	0
$116$	−4.38197	−0.406855
$117$	3.38197	0.312663
$118$	−4.00000	−0.368230
$119$	−12.7639	−1.17007
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	−0.618034	−0.0559542
$123$	−1.38197	−0.124608
$124$	−7.23607	−0.649818
$125$	0	0
$126$	4.47214	0.398410
$127$	9.70820	0.861464	0.430732	−	0.902480i	$-0.358255\pi$
$127$	9.70820	0.861464	0.430732	+	0.902480i	$0.358255\pi$
$128$	−3.00000	−0.265165
$129$	−5.70820	−0.502579
$130$	0	0
$131$	14.1803	1.23894	0.619471	−	0.785020i	$-0.287347\pi$
$131$	14.1803	1.23894	0.619471	+	0.785020i	$0.287347\pi$
$132$	−3.23607	−0.281664
$133$	−14.4721	−1.25489
$134$	5.23607	0.452327
$135$	0	0
$136$	8.56231	0.734212
$137$	−5.38197	−0.459812	−0.229906	−	0.973213i	$-0.573842\pi$
$137$	−5.38197	−0.459812	−0.229906	+	0.973213i	$0.573842\pi$
$138$	−4.47214	−0.380693
$139$	−5.05573	−0.428821	−0.214411	−	0.976744i	$-0.568783\pi$
$139$	−5.05573	−0.428821	−0.214411	+	0.976744i	$0.568783\pi$
$140$	0	0
$141$	−5.23607	−0.440956
$142$	−0.763932	−0.0641078
$143$	10.9443	0.915206
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	3.09017	0.255744
$147$	13.0000	1.07222
$148$	−8.09017	−0.665008
$149$	−12.1459	−0.995031	−0.497515	−	0.867455i	$-0.665754\pi$
$149$	−12.1459	−0.995031	−0.497515	+	0.867455i	$0.665754\pi$
$150$	0	0
$151$	16.4721	1.34048	0.670242	−	0.742143i	$-0.266190\pi$
$151$	16.4721	1.34048	0.670242	+	0.742143i	$0.266190\pi$
$152$	9.70820	0.787439
$153$	−2.85410	−0.230740
$154$	14.4721	1.16620
$155$	0	0
$156$	−3.38197	−0.270774
$157$	−13.7984	−1.10123	−0.550615	−	0.834759i	$-0.685607\pi$
$157$	−13.7984	−1.10123	−0.550615	+	0.834759i	$0.685607\pi$
$158$	0	0
$159$	1.38197	0.109597
$160$	0	0
$161$	−20.0000	−1.57622
$162$	1.00000	0.0785674
$163$	2.94427	0.230613	0.115307	−	0.993330i	$-0.463215\pi$
$163$	2.94427	0.230613	0.115307	+	0.993330i	$0.463215\pi$
$164$	1.38197	0.107913
$165$	0	0
$166$	3.52786	0.273815
$167$	−23.4164	−1.81202	−0.906008	−	0.423261i	$-0.860886\pi$
$167$	−23.4164	−1.81202	−0.906008	+	0.423261i	$0.860886\pi$
$168$	−13.4164	−1.03510
$169$	−1.56231	−0.120177
$170$	0	0
$171$	−3.23607	−0.247468
$172$	5.70820	0.435246
$173$	9.90983	0.753430	0.376715	−	0.926329i	$-0.377054\pi$
$173$	9.90983	0.753430	0.376715	+	0.926329i	$0.377054\pi$
$174$	4.38197	0.332196
$175$	0	0
$176$	−3.23607	−0.243928
$177$	−4.00000	−0.300658
$178$	7.61803	0.570996
$179$	6.18034	0.461940	0.230970	−	0.972961i	$-0.425810\pi$
$179$	6.18034	0.461940	0.230970	+	0.972961i	$0.425810\pi$
$180$	0	0
$181$	−9.79837	−0.728307	−0.364154	−	0.931339i	$-0.618642\pi$
$181$	−9.79837	−0.728307	−0.364154	+	0.931339i	$0.618642\pi$
$182$	15.1246	1.12111
$183$	−0.618034	−0.0456864
$184$	13.4164	0.989071
$185$	0	0
$186$	7.23607	0.530574
$187$	−9.23607	−0.675408
$188$	5.23607	0.381880
$189$	4.47214	0.325300
$190$	0	0
$191$	−24.6525	−1.78379	−0.891895	−	0.452242i	$-0.850624\pi$
$191$	−24.6525	−1.78379	−0.891895	+	0.452242i	$0.850624\pi$
$192$	7.00000	0.505181
$193$	7.67376	0.552369	0.276185	−	0.961105i	$-0.410930\pi$
$193$	7.67376	0.552369	0.276185	+	0.961105i	$0.410930\pi$
$194$	8.85410	0.635687
$195$	0	0
$196$	−13.0000	−0.928571
$197$	5.38197	0.383449	0.191725	−	0.981449i	$-0.438592\pi$
$197$	5.38197	0.383449	0.191725	+	0.981449i	$0.438592\pi$
$198$	3.23607	0.229977
$199$	−18.6525	−1.32224	−0.661119	−	0.750281i	$-0.729918\pi$
$199$	−18.6525	−1.32224	−0.661119	+	0.750281i	$0.729918\pi$
$200$	0	0
$201$	5.23607	0.369324
$202$	−16.5623	−1.16532
$203$	19.5967	1.37542
$204$	2.85410	0.199827
$205$	0	0
$206$	−1.23607	−0.0861209
$207$	−4.47214	−0.310835
$208$	−3.38197	−0.234497
$209$	−10.4721	−0.724373
$210$	0	0
$211$	−17.8885	−1.23150	−0.615749	−	0.787942i	$-0.711146\pi$
$211$	−17.8885	−1.23150	−0.615749	+	0.787942i	$0.711146\pi$
$212$	−1.38197	−0.0949138
$213$	−0.763932	−0.0523438
$214$	16.4721	1.12601
$215$	0	0
$216$	−3.00000	−0.204124
$217$	32.3607	2.19679
$218$	−19.0902	−1.29295
$219$	3.09017	0.208814
$220$	0	0
$221$	−9.65248	−0.649296
$222$	8.09017	0.542977
$223$	−14.1803	−0.949586	−0.474793	−	0.880098i	$-0.657477\pi$
$223$	−14.1803	−0.949586	−0.474793	+	0.880098i	$0.657477\pi$
$224$	22.3607	1.49404
$225$	0	0
$226$	2.67376	0.177856
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	3.23607	0.214314
$229$	−24.9787	−1.65064	−0.825320	−	0.564665i	$-0.809005\pi$
$229$	−24.9787	−1.65064	−0.825320	+	0.564665i	$0.809005\pi$
$230$	0	0
$231$	14.4721	0.952197
$232$	−13.1459	−0.863070
$233$	14.6180	0.957659	0.478830	−	0.877908i	$-0.341061\pi$
$233$	14.6180	0.957659	0.478830	+	0.877908i	$0.341061\pi$
$234$	3.38197	0.221086
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	−12.7639	−0.827363
$239$	−7.05573	−0.456397	−0.228199	−	0.973615i	$-0.573284\pi$
$239$	−7.05573	−0.456397	−0.228199	+	0.973615i	$0.573284\pi$
$240$	0	0
$241$	−1.03444	−0.0666343	−0.0333171	−	0.999445i	$-0.510607\pi$
$241$	−1.03444	−0.0666343	−0.0333171	+	0.999445i	$0.510607\pi$
$242$	−0.527864	−0.0339324
$243$	1.00000	0.0641500
$244$	0.618034	0.0395656
$245$	0	0
$246$	−1.38197	−0.0881109
$247$	−10.9443	−0.696367
$248$	−21.7082	−1.37847
$249$	3.52786	0.223569
$250$	0	0
$251$	−26.9443	−1.70071	−0.850354	−	0.526212i	$-0.823612\pi$
$251$	−26.9443	−1.70071	−0.850354	+	0.526212i	$0.823612\pi$
$252$	−4.47214	−0.281718
$253$	−14.4721	−0.909855
$254$	9.70820	0.609147
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−12.7984	−0.798341	−0.399170	−	0.916877i	$-0.630702\pi$
$257$	−12.7984	−0.798341	−0.399170	+	0.916877i	$0.630702\pi$
$258$	−5.70820	−0.355377
$259$	36.1803	2.24814
$260$	0	0
$261$	4.38197	0.271237
$262$	14.1803	0.876064
$263$	11.8885	0.733079	0.366540	−	0.930402i	$-0.380542\pi$
$263$	11.8885	0.733079	0.366540	+	0.930402i	$0.380542\pi$
$264$	−9.70820	−0.597499
$265$	0	0
$266$	−14.4721	−0.887344
$267$	7.61803	0.466216
$268$	−5.23607	−0.319844
$269$	−0.145898	−0.00889556	−0.00444778	−	0.999990i	$-0.501416\pi$
$269$	−0.145898	−0.00889556	−0.00444778	+	0.999990i	$0.501416\pi$
$270$	0	0
$271$	22.1803	1.34736	0.673680	−	0.739023i	$-0.264713\pi$
$271$	22.1803	1.34736	0.673680	+	0.739023i	$0.264713\pi$
$272$	2.85410	0.173055
$273$	15.1246	0.915383
$274$	−5.38197	−0.325136
$275$	0	0
$276$	4.47214	0.269191
$277$	−5.79837	−0.348391	−0.174195	−	0.984711i	$-0.555732\pi$
$277$	−5.79837	−0.348391	−0.174195	+	0.984711i	$0.555732\pi$
$278$	−5.05573	−0.303222
$279$	7.23607	0.433212
$280$	0	0
$281$	−17.3820	−1.03692	−0.518461	−	0.855102i	$-0.673495\pi$
$281$	−17.3820	−1.03692	−0.518461	+	0.855102i	$0.673495\pi$
$282$	−5.23607	−0.311803
$283$	0.291796	0.0173455	0.00867274	−	0.999962i	$-0.497239\pi$
$283$	0.291796	0.0173455	0.00867274	+	0.999962i	$0.497239\pi$
$284$	0.763932	0.0453310
$285$	0	0
$286$	10.9443	0.647148
$287$	−6.18034	−0.364814
$288$	5.00000	0.294628
$289$	−8.85410	−0.520830
$290$	0	0
$291$	8.85410	0.519037
$292$	−3.09017	−0.180839
$293$	3.79837	0.221903	0.110952	−	0.993826i	$-0.464610\pi$
$293$	3.79837	0.221903	0.110952	+	0.993826i	$0.464610\pi$
$294$	13.0000	0.758175
$295$	0	0
$296$	−24.2705	−1.41069
$297$	3.23607	0.187776
$298$	−12.1459	−0.703593
$299$	−15.1246	−0.874679
$300$	0	0
$301$	−25.5279	−1.47140
$302$	16.4721	0.947865
$303$	−16.5623	−0.951480
$304$	3.23607	0.185601
$305$	0	0
$306$	−2.85410	−0.163158
$307$	1.34752	0.0769073	0.0384536	−	0.999260i	$-0.487757\pi$
$307$	1.34752	0.0769073	0.0384536	+	0.999260i	$0.487757\pi$
$308$	−14.4721	−0.824626
$309$	−1.23607	−0.0703175
$310$	0	0
$311$	4.29180	0.243365	0.121683	−	0.992569i	$-0.461171\pi$
$311$	4.29180	0.243365	0.121683	+	0.992569i	$0.461171\pi$
$312$	−10.1459	−0.574398
$313$	8.47214	0.478873	0.239437	−	0.970912i	$-0.423037\pi$
$313$	8.47214	0.478873	0.239437	+	0.970912i	$0.423037\pi$
$314$	−13.7984	−0.778687
$315$	0	0
$316$	0	0
$317$	−24.8328	−1.39475	−0.697375	−	0.716706i	$-0.745649\pi$
$317$	−24.8328	−1.39475	−0.697375	+	0.716706i	$0.745649\pi$
$318$	1.38197	0.0774968
$319$	14.1803	0.793947
$320$	0	0
$321$	16.4721	0.919385
$322$	−20.0000	−1.11456
$323$	9.23607	0.513909
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	2.94427	0.163068
$327$	−19.0902	−1.05569
$328$	4.14590	0.228919
$329$	−23.4164	−1.29099
$330$	0	0
$331$	10.9443	0.601552	0.300776	−	0.953695i	$-0.402754\pi$
$331$	10.9443	0.601552	0.300776	+	0.953695i	$0.402754\pi$
$332$	−3.52786	−0.193617
$333$	8.09017	0.443339
$334$	−23.4164	−1.28129
$335$	0	0
$336$	−4.47214	−0.243975
$337$	−14.9443	−0.814066	−0.407033	−	0.913413i	$-0.633437\pi$
$337$	−14.9443	−0.814066	−0.407033	+	0.913413i	$0.633437\pi$
$338$	−1.56231	−0.0849782
$339$	2.67376	0.145219
$340$	0	0
$341$	23.4164	1.26807
$342$	−3.23607	−0.174987
$343$	26.8328	1.44884
$344$	17.1246	0.923297
$345$	0	0
$346$	9.90983	0.532756
$347$	−33.7082	−1.80955	−0.904776	−	0.425889i	$-0.859962\pi$
$347$	−33.7082	−1.80955	−0.904776	+	0.425889i	$0.859962\pi$
$348$	−4.38197	−0.234898
$349$	25.0344	1.34006	0.670031	−	0.742333i	$-0.266281\pi$
$349$	25.0344	1.34006	0.670031	+	0.742333i	$0.266281\pi$
$350$	0	0
$351$	3.38197	0.180516
$352$	16.1803	0.862415
$353$	−30.3607	−1.61594	−0.807968	−	0.589226i	$-0.799433\pi$
$353$	−30.3607	−1.61594	−0.807968	+	0.589226i	$0.799433\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−7.61803	−0.403755
$357$	−12.7639	−0.675539
$358$	6.18034	0.326641
$359$	−10.5836	−0.558581	−0.279290	−	0.960207i	$-0.590099\pi$
$359$	−10.5836	−0.558581	−0.279290	+	0.960207i	$0.590099\pi$
$360$	0	0
$361$	−8.52786	−0.448835
$362$	−9.79837	−0.514991
$363$	−0.527864	−0.0277057
$364$	−15.1246	−0.792745
$365$	0	0
$366$	−0.618034	−0.0323052
$367$	−6.00000	−0.313197	−0.156599	−	0.987662i	$-0.550053\pi$
$367$	−6.00000	−0.313197	−0.156599	+	0.987662i	$0.550053\pi$
$368$	4.47214	0.233126
$369$	−1.38197	−0.0719423
$370$	0	0
$371$	6.18034	0.320867
$372$	−7.23607	−0.375173
$373$	29.4164	1.52312	0.761562	−	0.648092i	$-0.224433\pi$
$373$	29.4164	1.52312	0.761562	+	0.648092i	$0.224433\pi$
$374$	−9.23607	−0.477586
$375$	0	0
$376$	15.7082	0.810089
$377$	14.8197	0.763251
$378$	4.47214	0.230022
$379$	23.4164	1.20282	0.601410	−	0.798941i	$-0.294606\pi$
$379$	23.4164	1.20282	0.601410	+	0.798941i	$0.294606\pi$
$380$	0	0
$381$	9.70820	0.497366
$382$	−24.6525	−1.26133
$383$	20.8328	1.06451	0.532254	−	0.846585i	$-0.321345\pi$
$383$	20.8328	1.06451	0.532254	+	0.846585i	$0.321345\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	7.67376	0.390584
$387$	−5.70820	−0.290164
$388$	−8.85410	−0.449499
$389$	−37.0902	−1.88055	−0.940273	−	0.340421i	$-0.889430\pi$
$389$	−37.0902	−1.88055	−0.940273	+	0.340421i	$0.889430\pi$
$390$	0	0
$391$	12.7639	0.645500
$392$	−39.0000	−1.96980
$393$	14.1803	0.715304
$394$	5.38197	0.271140
$395$	0	0
$396$	−3.23607	−0.162619
$397$	10.9443	0.549277	0.274639	−	0.961548i	$-0.411442\pi$
$397$	10.9443	0.549277	0.274639	+	0.961548i	$0.411442\pi$
$398$	−18.6525	−0.934964
$399$	−14.4721	−0.724513
$400$	0	0
$401$	33.4508	1.67046	0.835228	−	0.549904i	$-0.185336\pi$
$401$	33.4508	1.67046	0.835228	+	0.549904i	$0.185336\pi$
$402$	5.23607	0.261151
$403$	24.4721	1.21904
$404$	16.5623	0.824006
$405$	0	0
$406$	19.5967	0.972570
$407$	26.1803	1.29771
$408$	8.56231	0.423897
$409$	25.7984	1.27565	0.637824	−	0.770182i	$-0.279835\pi$
$409$	25.7984	1.27565	0.637824	+	0.770182i	$0.279835\pi$
$410$	0	0
$411$	−5.38197	−0.265473
$412$	1.23607	0.0608967
$413$	−17.8885	−0.880238
$414$	−4.47214	−0.219793
$415$	0	0
$416$	16.9098	0.829073
$417$	−5.05573	−0.247580
$418$	−10.4721	−0.512209
$419$	32.9443	1.60943	0.804717	−	0.593659i	$-0.202317\pi$
$419$	32.9443	1.60943	0.804717	+	0.593659i	$0.202317\pi$
$420$	0	0
$421$	23.1459	1.12806	0.564031	−	0.825754i	$-0.309250\pi$
$421$	23.1459	1.12806	0.564031	+	0.825754i	$0.309250\pi$
$422$	−17.8885	−0.870801
$423$	−5.23607	−0.254586
$424$	−4.14590	−0.201343
$425$	0	0
$426$	−0.763932	−0.0370126
$427$	−2.76393	−0.133756
$428$	−16.4721	−0.796211
$429$	10.9443	0.528394
$430$	0	0
$431$	−10.6525	−0.513112	−0.256556	−	0.966529i	$-0.582588\pi$
$431$	−10.6525	−0.513112	−0.256556	+	0.966529i	$0.582588\pi$
$432$	−1.00000	−0.0481125
$433$	24.9787	1.20040	0.600200	−	0.799850i	$-0.295088\pi$
$433$	24.9787	1.20040	0.600200	+	0.799850i	$0.295088\pi$
$434$	32.3607	1.55336
$435$	0	0
$436$	19.0902	0.914253
$437$	14.4721	0.692296
$438$	3.09017	0.147654
$439$	−6.18034	−0.294972	−0.147486	−	0.989064i	$-0.547118\pi$
$439$	−6.18034	−0.294972	−0.147486	+	0.989064i	$0.547118\pi$
$440$	0	0
$441$	13.0000	0.619048
$442$	−9.65248	−0.459121
$443$	−30.7639	−1.46164	−0.730819	−	0.682571i	$-0.760862\pi$
$443$	−30.7639	−1.46164	−0.730819	+	0.682571i	$0.760862\pi$
$444$	−8.09017	−0.383942
$445$	0	0
$446$	−14.1803	−0.671459
$447$	−12.1459	−0.574481
$448$	31.3050	1.47902
$449$	−7.79837	−0.368028	−0.184014	−	0.982924i	$-0.558909\pi$
$449$	−7.79837	−0.368028	−0.184014	+	0.982924i	$0.558909\pi$
$450$	0	0
$451$	−4.47214	−0.210585
$452$	−2.67376	−0.125763
$453$	16.4721	0.773928
$454$	−20.0000	−0.938647
$455$	0	0
$456$	9.70820	0.454628
$457$	22.3607	1.04599	0.522994	−	0.852336i	$-0.324815\pi$
$457$	22.3607	1.04599	0.522994	+	0.852336i	$0.324815\pi$
$458$	−24.9787	−1.16718
$459$	−2.85410	−0.133218
$460$	0	0
$461$	−8.20163	−0.381988	−0.190994	−	0.981591i	$-0.561171\pi$
$461$	−8.20163	−0.381988	−0.190994	+	0.981591i	$0.561171\pi$
$462$	14.4721	0.673305
$463$	23.2361	1.07987	0.539936	−	0.841706i	$-0.318448\pi$
$463$	23.2361	1.07987	0.539936	+	0.841706i	$0.318448\pi$
$464$	−4.38197	−0.203428
$465$	0	0
$466$	14.6180	0.677167
$467$	5.81966	0.269302	0.134651	−	0.990893i	$-0.457009\pi$
$467$	5.81966	0.269302	0.134651	+	0.990893i	$0.457009\pi$
$468$	−3.38197	−0.156331
$469$	23.4164	1.08127
$470$	0	0
$471$	−13.7984	−0.635796
$472$	12.0000	0.552345
$473$	−18.4721	−0.849350
$474$	0	0
$475$	0	0
$476$	12.7639	0.585034
$477$	1.38197	0.0632759
$478$	−7.05573	−0.322721
$479$	1.05573	0.0482374	0.0241187	−	0.999709i	$-0.492322\pi$
$479$	1.05573	0.0482374	0.0241187	+	0.999709i	$0.492322\pi$
$480$	0	0
$481$	27.3607	1.24754
$482$	−1.03444	−0.0471175
$483$	−20.0000	−0.910032
$484$	0.527864	0.0239938
$485$	0	0
$486$	1.00000	0.0453609
$487$	−3.70820	−0.168035	−0.0840174	−	0.996464i	$-0.526775\pi$
$487$	−3.70820	−0.168035	−0.0840174	+	0.996464i	$0.526775\pi$
$488$	1.85410	0.0839313
$489$	2.94427	0.133145
$490$	0	0
$491$	−29.8885	−1.34885	−0.674426	−	0.738343i	$-0.735609\pi$
$491$	−29.8885	−1.34885	−0.674426	+	0.738343i	$0.735609\pi$
$492$	1.38197	0.0623038
$493$	−12.5066	−0.563268
$494$	−10.9443	−0.492406
$495$	0	0
$496$	−7.23607	−0.324909
$497$	−3.41641	−0.153247
$498$	3.52786	0.158087
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	−23.4164	−1.04617
$502$	−26.9443	−1.20258
$503$	35.4164	1.57914	0.789570	−	0.613661i	$-0.210304\pi$
$503$	35.4164	1.57914	0.789570	+	0.613661i	$0.210304\pi$
$504$	−13.4164	−0.597614
$505$	0	0
$506$	−14.4721	−0.643365
$507$	−1.56231	−0.0693844
$508$	−9.70820	−0.430732
$509$	26.8541	1.19029	0.595144	−	0.803619i	$-0.297095\pi$
$509$	26.8541	1.19029	0.595144	+	0.803619i	$0.297095\pi$
$510$	0	0
$511$	13.8197	0.611346
$512$	−11.0000	−0.486136
$513$	−3.23607	−0.142876
$514$	−12.7984	−0.564512
$515$	0	0
$516$	5.70820	0.251290
$517$	−16.9443	−0.745208
$518$	36.1803	1.58967
$519$	9.90983	0.434993
$520$	0	0
$521$	2.03444	0.0891305	0.0445653	−	0.999006i	$-0.485810\pi$
$521$	2.03444	0.0891305	0.0445653	+	0.999006i	$0.485810\pi$
$522$	4.38197	0.191793
$523$	−12.2918	−0.537483	−0.268741	−	0.963212i	$-0.586608\pi$
$523$	−12.2918	−0.537483	−0.268741	+	0.963212i	$0.586608\pi$
$524$	−14.1803	−0.619471
$525$	0	0
$526$	11.8885	0.518365
$527$	−20.6525	−0.899636
$528$	−3.23607	−0.140832
$529$	−3.00000	−0.130435
$530$	0	0
$531$	−4.00000	−0.173585
$532$	14.4721	0.627447
$533$	−4.67376	−0.202443
$534$	7.61803	0.329665
$535$	0	0
$536$	−15.7082	−0.678491
$537$	6.18034	0.266701
$538$	−0.145898	−0.00629011
$539$	42.0689	1.81204
$540$	0	0
$541$	−35.3262	−1.51879	−0.759397	−	0.650628i	$-0.774506\pi$
$541$	−35.3262	−1.51879	−0.759397	+	0.650628i	$0.774506\pi$
$542$	22.1803	0.952727
$543$	−9.79837	−0.420488
$544$	−14.2705	−0.611843
$545$	0	0
$546$	15.1246	0.647274
$547$	−2.36068	−0.100935	−0.0504677	−	0.998726i	$-0.516071\pi$
$547$	−2.36068	−0.100935	−0.0504677	+	0.998726i	$0.516071\pi$
$548$	5.38197	0.229906
$549$	−0.618034	−0.0263770
$550$	0	0
$551$	−14.1803	−0.604103
$552$	13.4164	0.571040
$553$	0	0
$554$	−5.79837	−0.246349
$555$	0	0
$556$	5.05573	0.214411
$557$	22.2705	0.943632	0.471816	−	0.881697i	$-0.343599\pi$
$557$	22.2705	0.943632	0.471816	+	0.881697i	$0.343599\pi$
$558$	7.23607	0.306327
$559$	−19.3050	−0.816512
$560$	0	0
$561$	−9.23607	−0.389947
$562$	−17.3820	−0.733214
$563$	−17.5279	−0.738711	−0.369356	−	0.929288i	$-0.620422\pi$
$563$	−17.5279	−0.738711	−0.369356	+	0.929288i	$0.620422\pi$
$564$	5.23607	0.220478
$565$	0	0
$566$	0.291796	0.0122651
$567$	4.47214	0.187812
$568$	2.29180	0.0961616
$569$	−30.2705	−1.26901	−0.634503	−	0.772920i	$-0.718795\pi$
$569$	−30.2705	−1.26901	−0.634503	+	0.772920i	$0.718795\pi$
$570$	0	0
$571$	−21.2361	−0.888702	−0.444351	−	0.895853i	$-0.646566\pi$
$571$	−21.2361	−0.888702	−0.444351	+	0.895853i	$0.646566\pi$
$572$	−10.9443	−0.457603
$573$	−24.6525	−1.02987
$574$	−6.18034	−0.257962
$575$	0	0
$576$	7.00000	0.291667
$577$	−30.9443	−1.28823	−0.644113	−	0.764930i	$-0.722774\pi$
$577$	−30.9443	−1.28823	−0.644113	+	0.764930i	$0.722774\pi$
$578$	−8.85410	−0.368282
$579$	7.67376	0.318911
$580$	0	0
$581$	15.7771	0.654544
$582$	8.85410	0.367014
$583$	4.47214	0.185217
$584$	−9.27051	−0.383616
$585$	0	0
$586$	3.79837	0.156909
$587$	−14.5836	−0.601929	−0.300965	−	0.953635i	$-0.597309\pi$
$587$	−14.5836	−0.601929	−0.300965	+	0.953635i	$0.597309\pi$
$588$	−13.0000	−0.536111
$589$	−23.4164	−0.964856
$590$	0	0
$591$	5.38197	0.221384
$592$	−8.09017	−0.332504
$593$	−32.7426	−1.34458	−0.672290	−	0.740288i	$-0.734689\pi$
$593$	−32.7426	−1.34458	−0.672290	+	0.740288i	$0.734689\pi$
$594$	3.23607	0.132777
$595$	0	0
$596$	12.1459	0.497515
$597$	−18.6525	−0.763395
$598$	−15.1246	−0.618491
$599$	−12.4721	−0.509598	−0.254799	−	0.966994i	$-0.582009\pi$
$599$	−12.4721	−0.509598	−0.254799	+	0.966994i	$0.582009\pi$
$600$	0	0
$601$	−24.3262	−0.992288	−0.496144	−	0.868240i	$-0.665251\pi$
$601$	−24.3262	−0.992288	−0.496144	+	0.868240i	$0.665251\pi$
$602$	−25.5279	−1.04044
$603$	5.23607	0.213229
$604$	−16.4721	−0.670242
$605$	0	0
$606$	−16.5623	−0.672798
$607$	−39.2361	−1.59254	−0.796271	−	0.604940i	$-0.793197\pi$
$607$	−39.2361	−1.59254	−0.796271	+	0.604940i	$0.793197\pi$
$608$	−16.1803	−0.656199
$609$	19.5967	0.794100
$610$	0	0
$611$	−17.7082	−0.716397
$612$	2.85410	0.115370
$613$	39.5066	1.59566	0.797828	−	0.602885i	$-0.205982\pi$
$613$	39.5066	1.59566	0.797828	+	0.602885i	$0.205982\pi$
$614$	1.34752	0.0543816
$615$	0	0
$616$	−43.4164	−1.74930
$617$	26.9787	1.08612	0.543061	−	0.839693i	$-0.317265\pi$
$617$	26.9787	1.08612	0.543061	+	0.839693i	$0.317265\pi$
$618$	−1.23607	−0.0497219
$619$	28.3607	1.13991	0.569956	−	0.821675i	$-0.306960\pi$
$619$	28.3607	1.13991	0.569956	+	0.821675i	$0.306960\pi$
$620$	0	0
$621$	−4.47214	−0.179461
$622$	4.29180	0.172085
$623$	34.0689	1.36494
$624$	−3.38197	−0.135387
$625$	0	0
$626$	8.47214	0.338615
$627$	−10.4721	−0.418217
$628$	13.7984	0.550615
$629$	−23.0902	−0.920665
$630$	0	0
$631$	10.2918	0.409710	0.204855	−	0.978792i	$-0.434328\pi$
$631$	10.2918	0.409710	0.204855	+	0.978792i	$0.434328\pi$
$632$	0	0
$633$	−17.8885	−0.711006
$634$	−24.8328	−0.986237
$635$	0	0
$636$	−1.38197	−0.0547985
$637$	43.9656	1.74198
$638$	14.1803	0.561405
$639$	−0.763932	−0.0302207
$640$	0	0
$641$	38.9443	1.53821	0.769103	−	0.639125i	$-0.220703\pi$
$641$	38.9443	1.53821	0.769103	+	0.639125i	$0.220703\pi$
$642$	16.4721	0.650103
$643$	13.8885	0.547711	0.273855	−	0.961771i	$-0.411701\pi$
$643$	13.8885	0.547711	0.273855	+	0.961771i	$0.411701\pi$
$644$	20.0000	0.788110
$645$	0	0
$646$	9.23607	0.363388
$647$	10.3607	0.407320	0.203660	−	0.979042i	$-0.434716\pi$
$647$	10.3607	0.407320	0.203660	+	0.979042i	$0.434716\pi$
$648$	−3.00000	−0.117851
$649$	−12.9443	−0.508107
$650$	0	0
$651$	32.3607	1.26832
$652$	−2.94427	−0.115307
$653$	45.1033	1.76503	0.882515	−	0.470285i	$-0.155849\pi$
$653$	45.1033	1.76503	0.882515	+	0.470285i	$0.155849\pi$
$654$	−19.0902	−0.746485
$655$	0	0
$656$	1.38197	0.0539567
$657$	3.09017	0.120559
$658$	−23.4164	−0.912867
$659$	−6.58359	−0.256460	−0.128230	−	0.991744i	$-0.540930\pi$
$659$	−6.58359	−0.256460	−0.128230	+	0.991744i	$0.540930\pi$
$660$	0	0
$661$	25.4164	0.988584	0.494292	−	0.869296i	$-0.335427\pi$
$661$	25.4164	0.988584	0.494292	+	0.869296i	$0.335427\pi$
$662$	10.9443	0.425361
$663$	−9.65248	−0.374871
$664$	−10.5836	−0.410723
$665$	0	0
$666$	8.09017	0.313488
$667$	−19.5967	−0.758789
$668$	23.4164	0.906008
$669$	−14.1803	−0.548244
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	22.3607	0.862582
$673$	29.7984	1.14864	0.574321	−	0.818630i	$-0.305266\pi$
$673$	29.7984	1.14864	0.574321	+	0.818630i	$0.305266\pi$
$674$	−14.9443	−0.575632
$675$	0	0
$676$	1.56231	0.0600887
$677$	37.4164	1.43803	0.719015	−	0.694995i	$-0.244593\pi$
$677$	37.4164	1.43803	0.719015	+	0.694995i	$0.244593\pi$
$678$	2.67376	0.102685
$679$	39.5967	1.51958
$680$	0	0
$681$	−20.0000	−0.766402
$682$	23.4164	0.896661
$683$	7.41641	0.283781	0.141890	−	0.989882i	$-0.454682\pi$
$683$	7.41641	0.283781	0.141890	+	0.989882i	$0.454682\pi$
$684$	3.23607	0.123734
$685$	0	0
$686$	26.8328	1.02448
$687$	−24.9787	−0.952997
$688$	5.70820	0.217623
$689$	4.67376	0.178056
$690$	0	0
$691$	−21.2361	−0.807858	−0.403929	−	0.914790i	$-0.632356\pi$
$691$	−21.2361	−0.807858	−0.403929	+	0.914790i	$0.632356\pi$
$692$	−9.90983	−0.376715
$693$	14.4721	0.549751
$694$	−33.7082	−1.27955
$695$	0	0
$696$	−13.1459	−0.498294
$697$	3.94427	0.149400
$698$	25.0344	0.947568
$699$	14.6180	0.552905
$700$	0	0
$701$	0.437694	0.0165315	0.00826574	−	0.999966i	$-0.497369\pi$
$701$	0.437694	0.0165315	0.00826574	+	0.999966i	$0.497369\pi$
$702$	3.38197	0.127644
$703$	−26.1803	−0.987410
$704$	22.6525	0.853747
$705$	0	0
$706$	−30.3607	−1.14264
$707$	−74.0689	−2.78565
$708$	4.00000	0.150329
$709$	−6.72949	−0.252731	−0.126366	−	0.991984i	$-0.540331\pi$
$709$	−6.72949	−0.252731	−0.126366	+	0.991984i	$0.540331\pi$
$710$	0	0
$711$	0	0
$712$	−22.8541	−0.856494
$713$	−32.3607	−1.21192
$714$	−12.7639	−0.477678
$715$	0	0
$716$	−6.18034	−0.230970
$717$	−7.05573	−0.263501
$718$	−10.5836	−0.394976
$719$	−35.8885	−1.33842	−0.669208	−	0.743075i	$-0.733367\pi$
$719$	−35.8885	−1.33842	−0.669208	+	0.743075i	$0.733367\pi$
$720$	0	0
$721$	−5.52786	−0.205868
$722$	−8.52786	−0.317374
$723$	−1.03444	−0.0384713
$724$	9.79837	0.364154
$725$	0	0
$726$	−0.527864	−0.0195909
$727$	15.3475	0.569208	0.284604	−	0.958645i	$-0.408138\pi$
$727$	15.3475	0.569208	0.284604	+	0.958645i	$0.408138\pi$
$728$	−45.3738	−1.68167
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.2918	0.602574
$732$	0.618034	0.0228432
$733$	2.58359	0.0954272	0.0477136	−	0.998861i	$-0.484807\pi$
$733$	2.58359	0.0954272	0.0477136	+	0.998861i	$0.484807\pi$
$734$	−6.00000	−0.221464
$735$	0	0
$736$	−22.3607	−0.824226
$737$	16.9443	0.624150
$738$	−1.38197	−0.0508709
$739$	17.7082	0.651407	0.325703	−	0.945472i	$-0.394399\pi$
$739$	17.7082	0.651407	0.325703	+	0.945472i	$0.394399\pi$
$740$	0	0
$741$	−10.9443	−0.402048
$742$	6.18034	0.226887
$743$	0.875388	0.0321149	0.0160574	−	0.999871i	$-0.494889\pi$
$743$	0.875388	0.0321149	0.0160574	+	0.999871i	$0.494889\pi$
$744$	−21.7082	−0.795861
$745$	0	0
$746$	29.4164	1.07701
$747$	3.52786	0.129078
$748$	9.23607	0.337704
$749$	73.6656	2.69168
$750$	0	0
$751$	5.34752	0.195134	0.0975670	−	0.995229i	$-0.468894\pi$
$751$	5.34752	0.195134	0.0975670	+	0.995229i	$0.468894\pi$
$752$	5.23607	0.190940
$753$	−26.9443	−0.981904
$754$	14.8197	0.539700
$755$	0	0
$756$	−4.47214	−0.162650
$757$	27.3820	0.995214	0.497607	−	0.867402i	$-0.334212\pi$
$757$	27.3820	0.995214	0.497607	+	0.867402i	$0.334212\pi$
$758$	23.4164	0.850522
$759$	−14.4721	−0.525305
$760$	0	0
$761$	−9.74265	−0.353171	−0.176585	−	0.984285i	$-0.556505\pi$
$761$	−9.74265	−0.353171	−0.176585	+	0.984285i	$0.556505\pi$
$762$	9.70820	0.351691
$763$	−85.3738	−3.09074
$764$	24.6525	0.891895
$765$	0	0
$766$	20.8328	0.752720
$767$	−13.5279	−0.488463
$768$	−17.0000	−0.613435
$769$	−5.63932	−0.203359	−0.101680	−	0.994817i	$-0.532422\pi$
$769$	−5.63932	−0.203359	−0.101680	+	0.994817i	$0.532422\pi$
$770$	0	0
$771$	−12.7984	−0.460922
$772$	−7.67376	−0.276185
$773$	−35.9230	−1.29206	−0.646030	−	0.763312i	$-0.723572\pi$
$773$	−35.9230	−1.29206	−0.646030	+	0.763312i	$0.723572\pi$
$774$	−5.70820	−0.205177
$775$	0	0
$776$	−26.5623	−0.953531
$777$	36.1803	1.29796
$778$	−37.0902	−1.32975
$779$	4.47214	0.160231
$780$	0	0
$781$	−2.47214	−0.0884600
$782$	12.7639	0.456437
$783$	4.38197	0.156599
$784$	−13.0000	−0.464286
$785$	0	0
$786$	14.1803	0.505796
$787$	8.18034	0.291598	0.145799	−	0.989314i	$-0.453425\pi$
$787$	8.18034	0.291598	0.145799	+	0.989314i	$0.453425\pi$
$788$	−5.38197	−0.191725
$789$	11.8885	0.423243
$790$	0	0
$791$	11.9574	0.425157
$792$	−9.70820	−0.344966
$793$	−2.09017	−0.0742241
$794$	10.9443	0.388398
$795$	0	0
$796$	18.6525	0.661119
$797$	38.1033	1.34969	0.674845	−	0.737960i	$-0.264211\pi$
$797$	38.1033	1.34969	0.674845	+	0.737960i	$0.264211\pi$
$798$	−14.4721	−0.512308
$799$	14.9443	0.528690
$800$	0	0
$801$	7.61803	0.269170
$802$	33.4508	1.18119
$803$	10.0000	0.352892
$804$	−5.23607	−0.184662
$805$	0	0
$806$	24.4721	0.861994
$807$	−0.145898	−0.00513585
$808$	49.6869	1.74798
$809$	−23.2148	−0.816188	−0.408094	−	0.912940i	$-0.633806\pi$
$809$	−23.2148	−0.816188	−0.408094	+	0.912940i	$0.633806\pi$
$810$	0	0
$811$	9.23607	0.324322	0.162161	−	0.986764i	$-0.448154\pi$
$811$	9.23607	0.324322	0.162161	+	0.986764i	$0.448154\pi$
$812$	−19.5967	−0.687711
$813$	22.1803	0.777898
$814$	26.1803	0.917620
$815$	0	0
$816$	2.85410	0.0999136
$817$	18.4721	0.646258
$818$	25.7984	0.902019
$819$	15.1246	0.528497
$820$	0	0
$821$	5.05573	0.176446	0.0882231	−	0.996101i	$-0.471881\pi$
$821$	5.05573	0.176446	0.0882231	+	0.996101i	$0.471881\pi$
$822$	−5.38197	−0.187718
$823$	−14.7639	−0.514638	−0.257319	−	0.966326i	$-0.582839\pi$
$823$	−14.7639	−0.514638	−0.257319	+	0.966326i	$0.582839\pi$
$824$	3.70820	0.129181
$825$	0	0
$826$	−17.8885	−0.622422
$827$	22.0689	0.767410	0.383705	−	0.923456i	$-0.374648\pi$
$827$	22.0689	0.767410	0.383705	+	0.923456i	$0.374648\pi$
$828$	4.47214	0.155417
$829$	−26.5066	−0.920611	−0.460306	−	0.887760i	$-0.652260\pi$
$829$	−26.5066	−0.920611	−0.460306	+	0.887760i	$0.652260\pi$
$830$	0	0
$831$	−5.79837	−0.201143
$832$	23.6738	0.820740
$833$	−37.1033	−1.28555
$834$	−5.05573	−0.175066
$835$	0	0
$836$	10.4721	0.362186
$837$	7.23607	0.250115
$838$	32.9443	1.13804
$839$	−44.0689	−1.52143	−0.760713	−	0.649088i	$-0.775151\pi$
$839$	−44.0689	−1.52143	−0.760713	+	0.649088i	$0.775151\pi$
$840$	0	0
$841$	−9.79837	−0.337875
$842$	23.1459	0.797660
$843$	−17.3820	−0.598667
$844$	17.8885	0.615749
$845$	0	0
$846$	−5.23607	−0.180020
$847$	−2.36068	−0.0811139
$848$	−1.38197	−0.0474569
$849$	0.291796	0.0100144
$850$	0	0
$851$	−36.1803	−1.24025
$852$	0.763932	0.0261719
$853$	29.8541	1.02218	0.511092	−	0.859526i	$-0.329241\pi$
$853$	29.8541	1.02218	0.511092	+	0.859526i	$0.329241\pi$
$854$	−2.76393	−0.0945798
$855$	0	0
$856$	−49.4164	−1.68902
$857$	−3.52786	−0.120510	−0.0602548	−	0.998183i	$-0.519191\pi$
$857$	−3.52786	−0.120510	−0.0602548	+	0.998183i	$0.519191\pi$
$858$	10.9443	0.373631
$859$	3.41641	0.116566	0.0582832	−	0.998300i	$-0.481437\pi$
$859$	3.41641	0.116566	0.0582832	+	0.998300i	$0.481437\pi$
$860$	0	0
$861$	−6.18034	−0.210625
$862$	−10.6525	−0.362825
$863$	51.1246	1.74030	0.870151	−	0.492785i	$-0.164021\pi$
$863$	51.1246	1.74030	0.870151	+	0.492785i	$0.164021\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	24.9787	0.848811
$867$	−8.85410	−0.300701
$868$	−32.3607	−1.09839
$869$	0	0
$870$	0	0
$871$	17.7082	0.600020
$872$	57.2705	1.93942
$873$	8.85410	0.299666
$874$	14.4721	0.489527
$875$	0	0
$876$	−3.09017	−0.104407
$877$	18.2148	0.615069	0.307535	−	0.951537i	$-0.400496\pi$
$877$	18.2148	0.615069	0.307535	+	0.951537i	$0.400496\pi$
$878$	−6.18034	−0.208576
$879$	3.79837	0.128116
$880$	0	0
$881$	25.0557	0.844149	0.422074	−	0.906561i	$-0.361302\pi$
$881$	25.0557	0.844149	0.422074	+	0.906561i	$0.361302\pi$
$882$	13.0000	0.437733
$883$	−0.180340	−0.00606892	−0.00303446	−	0.999995i	$-0.500966\pi$
$883$	−0.180340	−0.00606892	−0.00303446	+	0.999995i	$0.500966\pi$
$884$	9.65248	0.324648
$885$	0	0
$886$	−30.7639	−1.03353
$887$	17.1246	0.574988	0.287494	−	0.957782i	$-0.407178\pi$
$887$	17.1246	0.574988	0.287494	+	0.957782i	$0.407178\pi$
$888$	−24.2705	−0.814465
$889$	43.4164	1.45614
$890$	0	0
$891$	3.23607	0.108412
$892$	14.1803	0.474793
$893$	16.9443	0.567018
$894$	−12.1459	−0.406220
$895$	0	0
$896$	−13.4164	−0.448211
$897$	−15.1246	−0.504996
$898$	−7.79837	−0.260235
$899$	31.7082	1.05753
$900$	0	0
$901$	−3.94427	−0.131403
$902$	−4.47214	−0.148906
$903$	−25.5279	−0.849514
$904$	−8.02129	−0.266784
$905$	0	0
$906$	16.4721	0.547250
$907$	−33.1246	−1.09988	−0.549942	−	0.835203i	$-0.685350\pi$
$907$	−33.1246	−1.09988	−0.549942	+	0.835203i	$0.685350\pi$
$908$	20.0000	0.663723
$909$	−16.5623	−0.549337
$910$	0	0
$911$	−4.18034	−0.138501	−0.0692504	−	0.997599i	$-0.522061\pi$
$911$	−4.18034	−0.138501	−0.0692504	+	0.997599i	$0.522061\pi$
$912$	3.23607	0.107157
$913$	11.4164	0.377828
$914$	22.3607	0.739626
$915$	0	0
$916$	24.9787	0.825320
$917$	63.4164	2.09419
$918$	−2.85410	−0.0941994
$919$	−49.1935	−1.62274	−0.811372	−	0.584530i	$-0.801279\pi$
$919$	−49.1935	−1.62274	−0.811372	+	0.584530i	$0.801279\pi$
$920$	0	0
$921$	1.34752	0.0444024
$922$	−8.20163	−0.270106
$923$	−2.58359	−0.0850400
$924$	−14.4721	−0.476098
$925$	0	0
$926$	23.2361	0.763585
$927$	−1.23607	−0.0405978
$928$	21.9098	0.719225
$929$	2.09017	0.0685763	0.0342881	−	0.999412i	$-0.489084\pi$
$929$	2.09017	0.0685763	0.0342881	+	0.999412i	$0.489084\pi$
$930$	0	0
$931$	−42.0689	−1.37875
$932$	−14.6180	−0.478830
$933$	4.29180	0.140507
$934$	5.81966	0.190425
$935$	0	0
$936$	−10.1459	−0.331629
$937$	11.6869	0.381795	0.190897	−	0.981610i	$-0.438860\pi$
$937$	11.6869	0.381795	0.190897	+	0.981610i	$0.438860\pi$
$938$	23.4164	0.764573
$939$	8.47214	0.276478
$940$	0	0
$941$	5.32624	0.173630	0.0868152	−	0.996224i	$-0.472331\pi$
$941$	5.32624	0.173630	0.0868152	+	0.996224i	$0.472331\pi$
$942$	−13.7984	−0.449575
$943$	6.18034	0.201260
$944$	4.00000	0.130189
$945$	0	0
$946$	−18.4721	−0.600581
$947$	29.4164	0.955905	0.477952	−	0.878386i	$-0.341379\pi$
$947$	29.4164	0.955905	0.477952	+	0.878386i	$0.341379\pi$
$948$	0	0
$949$	10.4508	0.339249
$950$	0	0
$951$	−24.8328	−0.805259
$952$	38.2918	1.24104
$953$	10.0902	0.326853	0.163426	−	0.986556i	$-0.447745\pi$
$953$	10.0902	0.326853	0.163426	+	0.986556i	$0.447745\pi$
$954$	1.38197	0.0447428
$955$	0	0
$956$	7.05573	0.228199
$957$	14.1803	0.458385
$958$	1.05573	0.0341090
$959$	−24.0689	−0.777225
$960$	0	0
$961$	21.3607	0.689054
$962$	27.3607	0.882144
$963$	16.4721	0.530807
$964$	1.03444	0.0333171
$965$	0	0
$966$	−20.0000	−0.643489
$967$	34.8328	1.12015	0.560074	−	0.828443i	$-0.310773\pi$
$967$	34.8328	1.12015	0.560074	+	0.828443i	$0.310773\pi$
$968$	1.58359	0.0508986
$969$	9.23607	0.296705
$970$	0	0
$971$	−6.58359	−0.211278	−0.105639	−	0.994405i	$-0.533689\pi$
$971$	−6.58359	−0.211278	−0.105639	+	0.994405i	$0.533689\pi$
$972$	−1.00000	−0.0320750
$973$	−22.6099	−0.724840
$974$	−3.70820	−0.118819
$975$	0	0
$976$	0.618034	0.0197828
$977$	−45.7426	−1.46344	−0.731718	−	0.681607i	$-0.761281\pi$
$977$	−45.7426	−1.46344	−0.731718	+	0.681607i	$0.761281\pi$
$978$	2.94427	0.0941474
$979$	24.6525	0.787897
$980$	0	0
$981$	−19.0902	−0.609502
$982$	−29.8885	−0.953782
$983$	21.7082	0.692384	0.346192	−	0.938164i	$-0.387475\pi$
$983$	21.7082	0.692384	0.346192	+	0.938164i	$0.387475\pi$
$984$	4.14590	0.132166
$985$	0	0
$986$	−12.5066	−0.398291
$987$	−23.4164	−0.745352
$988$	10.9443	0.348184
$989$	25.5279	0.811739
$990$	0	0
$991$	−3.12461	−0.0992566	−0.0496283	−	0.998768i	$-0.515804\pi$
$991$	−3.12461	−0.0992566	−0.0496283	+	0.998768i	$0.515804\pi$
$992$	36.1803	1.14873
$993$	10.9443	0.347306
$994$	−3.41641	−0.108362
$995$	0	0
$996$	−3.52786	−0.111785
$997$	9.41641	0.298221	0.149110	−	0.988821i	$-0.452359\pi$
$997$	9.41641	0.298221	0.149110	+	0.988821i	$0.452359\pi$
$998$	−6.00000	−0.189927
$999$	8.09017	0.255962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.d.1.2		2
3.2	odd	2		5625.2.a.a.1.2		2
5.2	odd	4		1875.2.b.b.1249.4		4
5.3	odd	4		1875.2.b.b.1249.1		4
5.4	even	2		1875.2.a.a.1.1		2
15.14	odd	2		5625.2.a.h.1.1		2
25.3	odd	20		375.2.i.a.49.1		8
25.4	even	10		375.2.g.a.76.1		4
25.6	even	5		75.2.g.a.61.1	yes	4
25.8	odd	20		375.2.i.a.199.2		8
25.17	odd	20		375.2.i.a.199.1		8
25.19	even	10		375.2.g.a.301.1		4
25.21	even	5		75.2.g.a.16.1	✓	4
25.22	odd	20		375.2.i.a.49.2		8
75.56	odd	10		225.2.h.a.136.1		4
75.71	odd	10		225.2.h.a.91.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.a.16.1	✓	4	25.21	even	5
75.2.g.a.61.1	yes	4	25.6	even	5
225.2.h.a.91.1		4	75.71	odd	10
225.2.h.a.136.1		4	75.56	odd	10
375.2.g.a.76.1		4	25.4	even	10
375.2.g.a.301.1		4	25.19	even	10
375.2.i.a.49.1		8	25.3	odd	20
375.2.i.a.49.2		8	25.22	odd	20
375.2.i.a.199.1		8	25.17	odd	20
375.2.i.a.199.2		8	25.8	odd	20
1875.2.a.a.1.1		2	5.4	even	2
1875.2.a.d.1.2		2	1.1	even	1	trivial
1875.2.b.b.1249.1		4	5.3	odd	4
1875.2.b.b.1249.4		4	5.2	odd	4
5625.2.a.a.1.2		2	3.2	odd	2
5625.2.a.h.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -3.00000 q^{8} +1.00000 q^{9} +3.23607 q^{11} -1.00000 q^{12} +3.38197 q^{13} +4.47214 q^{14} -1.00000 q^{16} -2.85410 q^{17} +1.00000 q^{18} -3.23607 q^{19} +4.47214 q^{21} +3.23607 q^{22} -4.47214 q^{23} -3.00000 q^{24} +3.38197 q^{26} +1.00000 q^{27} -4.47214 q^{28} +4.38197 q^{29} +7.23607 q^{31} +5.00000 q^{32} +3.23607 q^{33} -2.85410 q^{34} -1.00000 q^{36} +8.09017 q^{37} -3.23607 q^{38} +3.38197 q^{39} -1.38197 q^{41} +4.47214 q^{42} -5.70820 q^{43} -3.23607 q^{44} -4.47214 q^{46} -5.23607 q^{47} -1.00000 q^{48} +13.0000 q^{49} -2.85410 q^{51} -3.38197 q^{52} +1.38197 q^{53} +1.00000 q^{54} -13.4164 q^{56} -3.23607 q^{57} +4.38197 q^{58} -4.00000 q^{59} -0.618034 q^{61} +7.23607 q^{62} +4.47214 q^{63} +7.00000 q^{64} +3.23607 q^{66} +5.23607 q^{67} +2.85410 q^{68} -4.47214 q^{69} -0.763932 q^{71} -3.00000 q^{72} +3.09017 q^{73} +8.09017 q^{74} +3.23607 q^{76} +14.4721 q^{77} +3.38197 q^{78} +1.00000 q^{81} -1.38197 q^{82} +3.52786 q^{83} -4.47214 q^{84} -5.70820 q^{86} +4.38197 q^{87} -9.70820 q^{88} +7.61803 q^{89} +15.1246 q^{91} +4.47214 q^{92} +7.23607 q^{93} -5.23607 q^{94} +5.00000 q^{96} +8.85410 q^{97} +13.0000 q^{98} +3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} + 9 q^{13} - 2 q^{16} + q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{22} - 6 q^{24} + 9 q^{26} + 2 q^{27} + 11 q^{29} + 10 q^{31}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^11 - 2 * q^12 + 9 * q^13 - 2 * q^16 + q^17 + 2 * q^18 - 2 * q^19 + 2 * q^22 - 6 * q^24 + 9 * q^26 + 2 * q^27 + 11 * q^29 + 10 * q^31 + 10 * q^32 + 2 * q^33 + q^34 - 2 * q^36 + 5 * q^37 - 2 * q^38 + 9 * q^39 - 5 * q^41 + 2 * q^43 - 2 * q^44 - 6 * q^47 - 2 * q^48 + 26 * q^49 + q^51 - 9 * q^52 + 5 * q^53 + 2 * q^54 - 2 * q^57 + 11 * q^58 - 8 * q^59 + q^61 + 10 * q^62 + 14 * q^64 + 2 * q^66 + 6 * q^67 - q^68 - 6 * q^71 - 6 * q^72 - 5 * q^73 + 5 * q^74 + 2 * q^76 + 20 * q^77 + 9 * q^78 + 2 * q^81 - 5 * q^82 + 16 * q^83 + 2 * q^86 + 11 * q^87 - 6 * q^88 + 13 * q^89 - 10 * q^91 + 10 * q^93 - 6 * q^94 + 10 * q^96 + 11 * q^97 + 26 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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