LMFDB - Embedded newform 1875.2.a.h.1.3

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 = [4,2,4] 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:	$4$
Coefficient field:	4.4.5125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, a_2]$
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			1.3
Root			$2.12233$ 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)

$f(q)$	$=$	\(q+2.12233 q^{2} +1.00000 q^{3} +2.50430 q^{4} +2.12233 q^{6} +4.35840 q^{7} +1.07029 q^{8} +1.00000 q^{9} -1.57991 q^{11} +2.50430 q^{12} +1.19794 q^{13} +9.24998 q^{14} -2.73708 q^{16} -1.12233 q^{17} +2.12233 q^{18} +7.67867 q^{19} +4.35840 q^{21} -3.35309 q^{22} -2.32027 q^{23} +1.07029 q^{24} +2.54243 q^{26} +1.00000 q^{27} +10.9147 q^{28} -5.50430 q^{29} +4.80206 q^{31} -7.94959 q^{32} -1.57991 q^{33} -2.38197 q^{34} +2.50430 q^{36} -6.37232 q^{37} +16.2967 q^{38} +1.19794 q^{39} -7.47214 q^{41} +9.24998 q^{42} -1.24998 q^{43} -3.95656 q^{44} -4.92439 q^{46} +4.12765 q^{47} -2.73708 q^{48} +11.9957 q^{49} -1.12233 q^{51} +3.00000 q^{52} +3.74568 q^{53} +2.12233 q^{54} +4.66476 q^{56} +7.67867 q^{57} -11.6820 q^{58} +9.15613 q^{59} +1.39588 q^{61} +10.1916 q^{62} +4.35840 q^{63} -11.3975 q^{64} -3.35309 q^{66} -3.86270 q^{67} -2.81066 q^{68} -2.32027 q^{69} -10.6051 q^{71} +1.07029 q^{72} +4.99672 q^{73} -13.5242 q^{74} +19.2297 q^{76} -6.88586 q^{77} +2.54243 q^{78} +14.5531 q^{79} +1.00000 q^{81} -15.8584 q^{82} -8.73603 q^{83} +10.9147 q^{84} -2.65288 q^{86} -5.50430 q^{87} -1.69096 q^{88} -10.0381 q^{89} +5.22110 q^{91} -5.81066 q^{92} +4.80206 q^{93} +8.76025 q^{94} -7.94959 q^{96} -7.49996 q^{97} +25.4588 q^{98} -1.57991 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 15 q^{8} + 4 q^{9} - 7 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{18} + 5 q^{19} + 2 q^{21} - 6 q^{22} + q^{23} + 15 q^{24}+ \cdots - 7 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 2 * q^6 + 2 * q^7 + 15 * q^8 + 4 * q^9 - 7 * q^11 + 8 * q^12 + q^13 + 16 * q^14 + 4 * q^16 + 2 * q^17 + 2 * q^18 + 5 * q^19 + 2 * q^21 - 6 * q^22 + q^23 + 15 * q^24 + 3 * q^26 + 4 * q^27 + 9 * q^28 - 20 * q^29 + 23 * q^31 + 12 * q^32 - 7 * q^33 - 14 * q^34 + 8 * q^36 + 2 * q^37 + 35 * q^38 + q^39 - 12 * q^41 + 16 * q^42 + 16 * q^43 - 29 * q^44 - 17 * q^46 + 2 * q^47 + 4 * q^48 + 8 * q^49 + 2 * q^51 + 12 * q^52 - 4 * q^53 + 2 * q^54 + 5 * q^56 + 5 * q^57 - 25 * q^58 - 15 * q^59 - 2 * q^61 + 9 * q^62 + 2 * q^63 + 23 * q^64 - 6 * q^66 + 2 * q^67 - 11 * q^68 + q^69 - 2 * q^71 + 15 * q^72 + 16 * q^73 - 19 * q^74 + 40 * q^76 + 19 * q^77 + 3 * q^78 + 35 * q^79 + 4 * q^81 - 6 * q^82 + 16 * q^83 + 9 * q^84 + 3 * q^86 - 20 * q^87 - 30 * q^88 - 35 * q^89 - 12 * q^91 - 23 * q^92 + 23 * q^93 - 9 * q^94 + 12 * q^96 + 12 * q^97 - q^98 - 7 * 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 $

$2$	2.12233	1.50072	0.750358	−	0.661031i	$-0.229881\pi$
$2$	2.12233	1.50072	0.750358	+	0.661031i	$0.229881\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.50430	1.25215
$5$	0	0
$5$	0	0
$6$	2.12233	0.866439
$7$	4.35840	1.64732	0.823660	−	0.567083i	$-0.191928\pi$
$7$	4.35840	1.64732	0.823660	+	0.567083i	$0.191928\pi$
$8$	1.07029	0.378405
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.57991	−0.476360	−0.238180	−	0.971221i	$-0.576551\pi$
$11$	−1.57991	−0.476360	−0.238180	+	0.971221i	$0.576551\pi$
$12$	2.50430	0.722929
$13$	1.19794	0.332249	0.166124	−	0.986105i	$-0.446875\pi$
$13$	1.19794	0.332249	0.166124	+	0.986105i	$0.446875\pi$
$14$	9.24998	2.47216
$15$	0	0
$16$	−2.73708	−0.684271
$17$	−1.12233	−0.272206	−0.136103	−	0.990695i	$-0.543458\pi$
$17$	−1.12233	−0.272206	−0.136103	+	0.990695i	$0.543458\pi$
$18$	2.12233	0.500239
$19$	7.67867	1.76161	0.880804	−	0.473480i	$-0.157002\pi$
$19$	7.67867	1.76161	0.880804	+	0.473480i	$0.157002\pi$
$20$	0	0
$21$	4.35840	0.951081
$22$	−3.35309	−0.714881
$23$	−2.32027	−0.483810	−0.241905	−	0.970300i	$-0.577772\pi$
$23$	−2.32027	−0.483810	−0.241905	+	0.970300i	$0.577772\pi$
$24$	1.07029	0.218472
$25$	0	0
$26$	2.54243	0.498611
$27$	1.00000	0.192450
$28$	10.9147	2.06269
$29$	−5.50430	−1.02212	−0.511061	−	0.859544i	$-0.670748\pi$
$29$	−5.50430	−1.02212	−0.511061	+	0.859544i	$0.670748\pi$
$30$	0	0
$31$	4.80206	0.862475	0.431238	−	0.902238i	$-0.358077\pi$
$31$	4.80206	0.862475	0.431238	+	0.902238i	$0.358077\pi$
$32$	−7.94959	−1.40530
$33$	−1.57991	−0.275026
$34$	−2.38197	−0.408504
$35$	0	0
$36$	2.50430	0.417383
$37$	−6.37232	−1.04760	−0.523801	−	0.851841i	$-0.675486\pi$
$37$	−6.37232	−1.04760	−0.523801	+	0.851841i	$0.675486\pi$
$38$	16.2967	2.64368
$39$	1.19794	0.191824
$40$	0	0
$41$	−7.47214	−1.16695	−0.583476	−	0.812131i	$-0.698308\pi$
$41$	−7.47214	−1.16695	−0.583476	+	0.812131i	$0.698308\pi$
$42$	9.24998	1.42730
$43$	−1.24998	−0.190620	−0.0953102	−	0.995448i	$-0.530384\pi$
$43$	−1.24998	−0.190620	−0.0953102	+	0.995448i	$0.530384\pi$
$44$	−3.95656	−0.596473
$45$	0	0
$46$	−4.92439	−0.726062
$47$	4.12765	0.602079	0.301040	−	0.953612i	$-0.402666\pi$
$47$	4.12765	0.602079	0.301040	+	0.953612i	$0.402666\pi$
$48$	−2.73708	−0.395064
$49$	11.9957	1.71367
$50$	0	0
$51$	−1.12233	−0.157158
$52$	3.00000	0.416025
$53$	3.74568	0.514509	0.257254	−	0.966344i	$-0.417182\pi$
$53$	3.74568	0.514509	0.257254	+	0.966344i	$0.417182\pi$
$54$	2.12233	0.288813
$55$	0	0
$56$	4.66476	0.623355
$57$	7.67867	1.01707
$58$	−11.6820	−1.53392
$59$	9.15613	1.19203	0.596013	−	0.802975i	$-0.296751\pi$
$59$	9.15613	1.19203	0.596013	+	0.802975i	$0.296751\pi$
$60$	0	0
$61$	1.39588	0.178724	0.0893620	−	0.995999i	$-0.471517\pi$
$61$	1.39588	0.178724	0.0893620	+	0.995999i	$0.471517\pi$
$62$	10.1916	1.29433
$63$	4.35840	0.549107
$64$	−11.3975	−1.42469
$65$	0	0
$66$	−3.35309	−0.412736
$67$	−3.86270	−0.471904	−0.235952	−	0.971765i	$-0.575821\pi$
$67$	−3.86270	−0.471904	−0.235952	+	0.971765i	$0.575821\pi$
$68$	−2.81066	−0.340843
$69$	−2.32027	−0.279328
$70$	0	0
$71$	−10.6051	−1.25859	−0.629297	−	0.777165i	$-0.716657\pi$
$71$	−10.6051	−1.25859	−0.629297	+	0.777165i	$0.716657\pi$
$72$	1.07029	0.126135
$73$	4.99672	0.584821	0.292411	−	0.956293i	$-0.405543\pi$
$73$	4.99672	0.584821	0.292411	+	0.956293i	$0.405543\pi$
$74$	−13.5242	−1.57215
$75$	0	0
$76$	19.2297	2.20580
$77$	−6.88586	−0.784717
$78$	2.54243	0.287873
$79$	14.5531	1.63735	0.818673	−	0.574259i	$-0.194710\pi$
$79$	14.5531	1.63735	0.818673	+	0.574259i	$0.194710\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−15.8584	−1.75126
$83$	−8.73603	−0.958904	−0.479452	−	0.877568i	$-0.659165\pi$
$83$	−8.73603	−0.958904	−0.479452	+	0.877568i	$0.659165\pi$
$84$	10.9147	1.19090
$85$	0	0
$86$	−2.65288	−0.286067
$87$	−5.50430	−0.590123
$88$	−1.69096	−0.180257
$89$	−10.0381	−1.06404	−0.532020	−	0.846732i	$-0.678567\pi$
$89$	−10.0381	−1.06404	−0.532020	+	0.846732i	$0.678567\pi$
$90$	0	0
$91$	5.22110	0.547320
$92$	−5.81066	−0.605803
$93$	4.80206	0.497950
$94$	8.76025	0.903550
$95$	0	0
$96$	−7.94959	−0.811351
$97$	−7.49996	−0.761506	−0.380753	−	0.924677i	$-0.624335\pi$
$97$	−7.49996	−0.761506	−0.380753	+	0.924677i	$0.624335\pi$
$98$	25.4588	2.57173
$99$	−1.57991	−0.158787
$100$	0	0
$101$	−6.51821	−0.648586	−0.324293	−	0.945957i	$-0.605126\pi$
$101$	−6.51821	−0.648586	−0.324293	+	0.945957i	$0.605126\pi$
$102$	−2.38197	−0.235850
$103$	8.04876	0.793068	0.396534	−	0.918020i	$-0.370213\pi$
$103$	8.04876	0.793068	0.396534	+	0.918020i	$0.370213\pi$
$104$	1.28215	0.125725
$105$	0	0
$106$	7.94959	0.772132
$107$	9.47745	0.916220	0.458110	−	0.888896i	$-0.348527\pi$
$107$	9.47745	0.916220	0.458110	+	0.888896i	$0.348527\pi$
$108$	2.50430	0.240976
$109$	−11.6657	−1.11738	−0.558688	−	0.829378i	$-0.688695\pi$
$109$	−11.6657	−1.11738	−0.558688	+	0.829378i	$0.688695\pi$
$110$	0	0
$111$	−6.37232	−0.604833
$112$	−11.9293	−1.12721
$113$	−14.9254	−1.40406	−0.702030	−	0.712147i	$-0.747723\pi$
$113$	−14.9254	−1.40406	−0.702030	+	0.712147i	$0.747723\pi$
$114$	16.2967	1.52633
$115$	0	0
$116$	−13.7844	−1.27985
$117$	1.19794	0.110750
$118$	19.4324	1.78889
$119$	−4.89158	−0.448410
$120$	0	0
$121$	−8.50390	−0.773082
$122$	2.96252	0.268214
$123$	−7.47214	−0.673740
$124$	12.0258	1.07995
$125$	0	0
$126$	9.24998	0.824054
$127$	−16.9957	−1.50812	−0.754061	−	0.656805i	$-0.771908\pi$
$127$	−16.9957	−1.50812	−0.754061	+	0.656805i	$0.771908\pi$
$128$	−8.29014	−0.732752
$129$	−1.24998	−0.110055
$130$	0	0
$131$	−0.328872	−0.0287337	−0.0143669	−	0.999897i	$-0.504573\pi$
$131$	−0.328872	−0.0287337	−0.0143669	+	0.999897i	$0.504573\pi$
$132$	−3.95656	−0.344374
$133$	33.4667	2.90194
$134$	−8.19794	−0.708194
$135$	0	0
$136$	−1.20122	−0.103004
$137$	4.56271	0.389818	0.194909	−	0.980821i	$-0.437559\pi$
$137$	4.56271	0.389818	0.194909	+	0.980821i	$0.437559\pi$
$138$	−4.92439	−0.419192
$139$	−4.73400	−0.401533	−0.200766	−	0.979639i	$-0.564343\pi$
$139$	−4.73400	−0.401533	−0.200766	+	0.979639i	$0.564343\pi$
$140$	0	0
$141$	4.12765	0.347611
$142$	−22.5076	−1.88879
$143$	−1.89263	−0.158270
$144$	−2.73708	−0.228090
$145$	0	0
$146$	10.6047	0.877651
$147$	11.9957	0.989386
$148$	−15.9582	−1.31175
$149$	4.67644	0.383109	0.191555	−	0.981482i	$-0.438647\pi$
$149$	4.67644	0.383109	0.191555	+	0.981482i	$0.438647\pi$
$150$	0	0
$151$	−6.54178	−0.532362	−0.266181	−	0.963923i	$-0.585762\pi$
$151$	−6.54178	−0.532362	−0.266181	+	0.963923i	$0.585762\pi$
$152$	8.21842	0.666602
$153$	−1.12233	−0.0907353
$154$	−14.6141	−1.17764
$155$	0	0
$156$	3.00000	0.240192
$157$	3.99404	0.318759	0.159379	−	0.987217i	$-0.449051\pi$
$157$	3.99404	0.318759	0.159379	+	0.987217i	$0.449051\pi$
$158$	30.8864	2.45719
$159$	3.74568	0.297052
$160$	0	0
$161$	−10.1127	−0.796991
$162$	2.12233	0.166746
$163$	4.80141	0.376075	0.188038	−	0.982162i	$-0.439787\pi$
$163$	4.80141	0.376075	0.188038	+	0.982162i	$0.439787\pi$
$164$	−18.7125	−1.46120
$165$	0	0
$166$	−18.5408	−1.43904
$167$	23.6600	1.83087	0.915434	−	0.402469i	$-0.131848\pi$
$167$	23.6600	1.83087	0.915434	+	0.402469i	$0.131848\pi$
$168$	4.66476	0.359894
$169$	−11.5649	−0.889611
$170$	0	0
$171$	7.67867	0.587203
$172$	−3.13033	−0.238685
$173$	14.5043	1.10274	0.551371	−	0.834260i	$-0.314105\pi$
$173$	14.5043	1.10274	0.551371	+	0.834260i	$0.314105\pi$
$174$	−11.6820	−0.885607
$175$	0	0
$176$	4.32433	0.325959
$177$	9.15613	0.688217
$178$	−21.3043	−1.59682
$179$	0.796746	0.0595516	0.0297758	−	0.999557i	$-0.490521\pi$
$179$	0.796746	0.0595516	0.0297758	+	0.999557i	$0.490521\pi$
$180$	0	0
$181$	−14.2185	−1.05685	−0.528425	−	0.848980i	$-0.677217\pi$
$181$	−14.2185	−1.05685	−0.528425	+	0.848980i	$0.677217\pi$
$182$	11.0809	0.821372
$183$	1.39588	0.103186
$184$	−2.48337	−0.183076
$185$	0	0
$186$	10.1916	0.747282
$187$	1.77318	0.129668
$188$	10.3369	0.753894
$189$	4.35840	0.317027
$190$	0	0
$191$	−10.3697	−0.750324	−0.375162	−	0.926959i	$-0.622413\pi$
$191$	−10.3697	−0.750324	−0.375162	+	0.926959i	$0.622413\pi$
$192$	−11.3975	−0.822544
$193$	4.34712	0.312913	0.156456	−	0.987685i	$-0.449993\pi$
$193$	4.34712	0.312913	0.156456	+	0.987685i	$0.449993\pi$
$194$	−15.9174	−1.14280
$195$	0	0
$196$	30.0407	2.14577
$197$	1.87990	0.133937	0.0669686	−	0.997755i	$-0.478667\pi$
$197$	1.87990	0.133937	0.0669686	+	0.997755i	$0.478667\pi$
$198$	−3.35309	−0.238294
$199$	4.26028	0.302003	0.151002	−	0.988534i	$-0.451750\pi$
$199$	4.26028	0.302003	0.151002	+	0.988534i	$0.451750\pi$
$200$	0	0
$201$	−3.86270	−0.272454
$202$	−13.8338	−0.973344
$203$	−23.9899	−1.68376
$204$	−2.81066	−0.196786
$205$	0	0
$206$	17.0821	1.19017
$207$	−2.32027	−0.161270
$208$	−3.27886	−0.227348
$209$	−12.1316	−0.839159
$210$	0	0
$211$	−9.29671	−0.640012	−0.320006	−	0.947416i	$-0.603685\pi$
$211$	−9.29671	−0.640012	−0.320006	+	0.947416i	$0.603685\pi$
$212$	9.38031	0.644242
$213$	−10.6051	−0.726649
$214$	20.1143	1.37499
$215$	0	0
$216$	1.07029	0.0728241
$217$	20.9293	1.42077
$218$	−24.7586	−1.67686
$219$	4.99672	0.337647
$220$	0	0
$221$	−1.34449	−0.0904400
$222$	−13.5242	−0.907683
$223$	−3.39758	−0.227519	−0.113759	−	0.993508i	$-0.536289\pi$
$223$	−3.39758	−0.227519	−0.113759	+	0.993508i	$0.536289\pi$
$224$	−34.6475	−2.31498
$225$	0	0
$226$	−31.6766	−2.10710
$227$	26.7973	1.77860	0.889300	−	0.457324i	$-0.151192\pi$
$227$	26.7973	1.77860	0.889300	+	0.457324i	$0.151192\pi$
$228$	19.2297	1.27352
$229$	−1.18200	−0.0781085	−0.0390543	−	0.999237i	$-0.512435\pi$
$229$	−1.18200	−0.0781085	−0.0390543	+	0.999237i	$0.512435\pi$
$230$	0	0
$231$	−6.88586	−0.453057
$232$	−5.89121	−0.386777
$233$	7.09483	0.464798	0.232399	−	0.972621i	$-0.425342\pi$
$233$	7.09483	0.464798	0.232399	+	0.972621i	$0.425342\pi$
$234$	2.54243	0.166204
$235$	0	0
$236$	22.9297	1.49260
$237$	14.5531	0.945323
$238$	−10.3816	−0.672937
$239$	0.0427926	0.00276802	0.00138401	−	0.999999i	$-0.499559\pi$
$239$	0.0427926	0.00276802	0.00138401	+	0.999999i	$0.499559\pi$
$240$	0	0
$241$	−11.7711	−0.758242	−0.379121	−	0.925347i	$-0.623774\pi$
$241$	−11.7711	−0.758242	−0.379121	+	0.925347i	$0.623774\pi$
$242$	−18.0481	−1.16018
$243$	1.00000	0.0641500
$244$	3.49570	0.223789
$245$	0	0
$246$	−15.8584	−1.01109
$247$	9.19859	0.585292
$248$	5.13961	0.326365
$249$	−8.73603	−0.553623
$250$	0	0
$251$	17.4764	1.10310	0.551550	−	0.834142i	$-0.314036\pi$
$251$	17.4764	1.10310	0.551550	+	0.834142i	$0.314036\pi$
$252$	10.9147	0.687564
$253$	3.66581	0.230468
$254$	−36.0705	−2.26326
$255$	0	0
$256$	5.20057	0.325036
$257$	15.4671	0.964814	0.482407	−	0.875947i	$-0.339763\pi$
$257$	15.4671	0.964814	0.482407	+	0.875947i	$0.339763\pi$
$258$	−2.65288	−0.165161
$259$	−27.7731	−1.72574
$260$	0	0
$261$	−5.50430	−0.340708
$262$	−0.697977	−0.0431211
$263$	1.44759	0.0892624	0.0446312	−	0.999004i	$-0.485789\pi$
$263$	1.44759	0.0892624	0.0446312	+	0.999004i	$0.485789\pi$
$264$	−1.69096	−0.104071
$265$	0	0
$266$	71.0276	4.35498
$267$	−10.0381	−0.614323
$268$	−9.67336	−0.590895
$269$	−8.70617	−0.530825	−0.265412	−	0.964135i	$-0.585508\pi$
$269$	−8.70617	−0.530825	−0.265412	+	0.964135i	$0.585508\pi$
$270$	0	0
$271$	−28.6338	−1.73938	−0.869689	−	0.493600i	$-0.835681\pi$
$271$	−28.6338	−1.73938	−0.869689	+	0.493600i	$0.835681\pi$
$272$	3.07192	0.186263
$273$	5.22110	0.315995
$274$	9.68359	0.585007
$275$	0	0
$276$	−5.81066	−0.349761
$277$	5.30210	0.318572	0.159286	−	0.987232i	$-0.449081\pi$
$277$	5.30210	0.318572	0.159286	+	0.987232i	$0.449081\pi$
$278$	−10.0471	−0.602587
$279$	4.80206	0.287492
$280$	0	0
$281$	−24.0832	−1.43668	−0.718342	−	0.695691i	$-0.755098\pi$
$281$	−24.0832	−1.43668	−0.718342	+	0.695691i	$0.755098\pi$
$282$	8.76025	0.521665
$283$	−27.1143	−1.61178	−0.805889	−	0.592066i	$-0.798312\pi$
$283$	−27.1143	−1.61178	−0.805889	+	0.592066i	$0.798312\pi$
$284$	−26.5583	−1.57595
$285$	0	0
$286$	−4.01680	−0.237518
$287$	−32.5666	−1.92234
$288$	−7.94959	−0.468434
$289$	−15.7404	−0.925904
$290$	0	0
$291$	−7.49996	−0.439656
$292$	12.5133	0.732284
$293$	−20.3016	−1.18603	−0.593017	−	0.805190i	$-0.702063\pi$
$293$	−20.3016	−1.18603	−0.593017	+	0.805190i	$0.702063\pi$
$294$	25.4588	1.48479
$295$	0	0
$296$	−6.82024	−0.396418
$297$	−1.57991	−0.0916754
$298$	9.92497	0.574938
$299$	−2.77955	−0.160745
$300$	0	0
$301$	−5.44792	−0.314013
$302$	−13.8838	−0.798925
$303$	−6.51821	−0.374462
$304$	−21.0172	−1.20542
$305$	0	0
$306$	−2.38197	−0.136168
$307$	−2.51330	−0.143442	−0.0717208	−	0.997425i	$-0.522849\pi$
$307$	−2.51330	−0.143442	−0.0717208	+	0.997425i	$0.522849\pi$
$308$	−17.2443	−0.982583
$309$	8.04876	0.457878
$310$	0	0
$311$	4.82417	0.273554	0.136777	−	0.990602i	$-0.456326\pi$
$311$	4.82417	0.273554	0.136777	+	0.990602i	$0.456326\pi$
$312$	1.28215	0.0725872
$313$	−5.98812	−0.338468	−0.169234	−	0.985576i	$-0.554129\pi$
$313$	−5.98812	−0.338468	−0.169234	+	0.985576i	$0.554129\pi$
$314$	8.47668	0.478366
$315$	0	0
$316$	36.4452	2.05020
$317$	17.3531	0.974646	0.487323	−	0.873222i	$-0.337973\pi$
$317$	17.3531	0.974646	0.487323	+	0.873222i	$0.337973\pi$
$318$	7.94959	0.445791
$319$	8.69627	0.486898
$320$	0	0
$321$	9.47745	0.528980
$322$	−21.4625	−1.19606
$323$	−8.61803	−0.479520
$324$	2.50430	0.139128
$325$	0	0
$326$	10.1902	0.564383
$327$	−11.6657	−0.645117
$328$	−7.99737	−0.441581
$329$	17.9899	0.991818
$330$	0	0
$331$	20.3355	1.11774	0.558870	−	0.829255i	$-0.311235\pi$
$331$	20.3355	1.11774	0.558870	+	0.829255i	$0.311235\pi$
$332$	−21.8776	−1.20069
$333$	−6.37232	−0.349201
$334$	50.2145	2.74761
$335$	0	0
$336$	−11.9293	−0.650797
$337$	34.2571	1.86610	0.933052	−	0.359741i	$-0.117135\pi$
$337$	34.2571	1.86610	0.933052	+	0.359741i	$0.117135\pi$
$338$	−24.5447	−1.33505
$339$	−14.9254	−0.810635
$340$	0	0
$341$	−7.58680	−0.410848
$342$	16.2967	0.881225
$343$	21.7731	1.17564
$344$	−1.33784	−0.0721318
$345$	0	0
$346$	30.7830	1.65490
$347$	−0.0493616	−0.00264987	−0.00132494	−	0.999999i	$-0.500422\pi$
$347$	−0.0493616	−0.00264987	−0.00132494	+	0.999999i	$0.500422\pi$
$348$	−13.7844	−0.738922
$349$	−7.47437	−0.400094	−0.200047	−	0.979786i	$-0.564109\pi$
$349$	−7.47437	−0.400094	−0.200047	+	0.979786i	$0.564109\pi$
$350$	0	0
$351$	1.19794	0.0639413
$352$	12.5596	0.669429
$353$	−18.5650	−0.988115	−0.494057	−	0.869429i	$-0.664487\pi$
$353$	−18.5650	−0.988115	−0.494057	+	0.869429i	$0.664487\pi$
$354$	19.4324	1.03282
$355$	0	0
$356$	−25.1385	−1.33234
$357$	−4.89158	−0.258890
$358$	1.69096	0.0893700
$359$	10.6044	0.559681	0.279841	−	0.960046i	$-0.409718\pi$
$359$	10.6044	0.559681	0.279841	+	0.960046i	$0.409718\pi$
$360$	0	0
$361$	39.9620	2.10327
$362$	−30.1763	−1.58603
$363$	−8.50390	−0.446339
$364$	13.0752	0.685327
$365$	0	0
$366$	2.96252	0.154854
$367$	−11.6176	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$367$	−11.6176	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$368$	6.35078	0.331057
$369$	−7.47214	−0.388984
$370$	0	0
$371$	16.3252	0.847561
$372$	12.0258	0.623509
$373$	−23.1272	−1.19748	−0.598742	−	0.800942i	$-0.704332\pi$
$373$	−23.1272	−1.19748	−0.598742	+	0.800942i	$0.704332\pi$
$374$	3.76328	0.194595
$375$	0	0
$376$	4.41779	0.227830
$377$	−6.59382	−0.339599
$378$	9.24998	0.475768
$379$	24.8912	1.27857	0.639287	−	0.768968i	$-0.279230\pi$
$379$	24.8912	1.27857	0.639287	+	0.768968i	$0.279230\pi$
$380$	0	0
$381$	−16.9957	−0.870714
$382$	−22.0079	−1.12602
$383$	−15.7221	−0.803363	−0.401681	−	0.915780i	$-0.631574\pi$
$383$	−15.7221	−0.803363	−0.401681	+	0.915780i	$0.631574\pi$
$384$	−8.29014	−0.423054
$385$	0	0
$386$	9.22604	0.469593
$387$	−1.24998	−0.0635401
$388$	−18.7822	−0.953519
$389$	−8.50633	−0.431288	−0.215644	−	0.976472i	$-0.569185\pi$
$389$	−8.50633	−0.431288	−0.215644	+	0.976472i	$0.569185\pi$
$390$	0	0
$391$	2.60412	0.131696
$392$	12.8389	0.648460
$393$	−0.328872	−0.0165894
$394$	3.98977	0.201002
$395$	0	0
$396$	−3.95656	−0.198824
$397$	26.2549	1.31770	0.658848	−	0.752276i	$-0.271044\pi$
$397$	26.2549	1.31770	0.658848	+	0.752276i	$0.271044\pi$
$398$	9.04174	0.453222
$399$	33.4667	1.67543
$400$	0	0
$401$	−25.2815	−1.26250	−0.631250	−	0.775579i	$-0.717458\pi$
$401$	−25.2815	−1.26250	−0.631250	+	0.775579i	$0.717458\pi$
$402$	−8.19794	−0.408876
$403$	5.75258	0.286556
$404$	−16.3236	−0.812127
$405$	0	0
$406$	−50.9147	−2.52685
$407$	10.0677	0.499035
$408$	−1.20122	−0.0594695
$409$	33.7932	1.67097	0.835483	−	0.549517i	$-0.185188\pi$
$409$	33.7932	1.67097	0.835483	+	0.549517i	$0.185188\pi$
$410$	0	0
$411$	4.56271	0.225062
$412$	20.1565	0.993039
$413$	39.9061	1.96365
$414$	−4.92439	−0.242021
$415$	0	0
$416$	−9.52313	−0.466910
$417$	−4.73400	−0.231825
$418$	−25.7473	−1.25934
$419$	7.93332	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$419$	7.93332	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$420$	0	0
$421$	7.48795	0.364941	0.182470	−	0.983211i	$-0.441591\pi$
$421$	7.48795	0.364941	0.182470	+	0.983211i	$0.441591\pi$
$422$	−19.7307	−0.960476
$423$	4.12765	0.200693
$424$	4.00897	0.194693
$425$	0	0
$426$	−22.5076	−1.09049
$427$	6.08380	0.294416
$428$	23.7344	1.14724
$429$	−1.89263	−0.0913771
$430$	0	0
$431$	23.3470	1.12459	0.562294	−	0.826937i	$-0.309919\pi$
$431$	23.3470	1.12459	0.562294	+	0.826937i	$0.309919\pi$
$432$	−2.73708	−0.131688
$433$	−3.35677	−0.161316	−0.0806581	−	0.996742i	$-0.525702\pi$
$433$	−3.35677	−0.161316	−0.0806581	+	0.996742i	$0.525702\pi$
$434$	44.4190	2.13218
$435$	0	0
$436$	−29.2145	−1.39912
$437$	−17.8166	−0.852285
$438$	10.6047	0.506712
$439$	−9.48402	−0.452648	−0.226324	−	0.974052i	$-0.572671\pi$
$439$	−9.48402	−0.452648	−0.226324	+	0.974052i	$0.572671\pi$
$440$	0	0
$441$	11.9957	0.571222
$442$	−2.85345	−0.135725
$443$	9.65446	0.458697	0.229349	−	0.973344i	$-0.426340\pi$
$443$	9.65446	0.458697	0.229349	+	0.973344i	$0.426340\pi$
$444$	−15.9582	−0.757342
$445$	0	0
$446$	−7.21080	−0.341441
$447$	4.67644	0.221188
$448$	−49.6749	−2.34692
$449$	31.5260	1.48780	0.743902	−	0.668289i	$-0.232973\pi$
$449$	31.5260	1.48780	0.743902	+	0.668289i	$0.232973\pi$
$450$	0	0
$451$	11.8053	0.555888
$452$	−37.3776	−1.75809
$453$	−6.54178	−0.307360
$454$	56.8729	2.66918
$455$	0	0
$456$	8.21842	0.384863
$457$	−9.94467	−0.465192	−0.232596	−	0.972573i	$-0.574722\pi$
$457$	−9.94467	−0.465192	−0.232596	+	0.972573i	$0.574722\pi$
$458$	−2.50859	−0.117219
$459$	−1.12233	−0.0523860
$460$	0	0
$461$	−23.6836	−1.10305	−0.551527	−	0.834157i	$-0.685955\pi$
$461$	−23.6836	−1.10305	−0.551527	+	0.834157i	$0.685955\pi$
$462$	−14.6141	−0.679909
$463$	27.6504	1.28502	0.642511	−	0.766277i	$-0.277893\pi$
$463$	27.6504	1.28502	0.642511	+	0.766277i	$0.277893\pi$
$464$	15.0657	0.699409
$465$	0	0
$466$	15.0576	0.697530
$467$	4.35840	0.201683	0.100841	−	0.994903i	$-0.467847\pi$
$467$	4.35840	0.201683	0.100841	+	0.994903i	$0.467847\pi$
$468$	3.00000	0.138675
$469$	−16.8352	−0.777377
$470$	0	0
$471$	3.99404	0.184035
$472$	9.79973	0.451069
$473$	1.97485	0.0908038
$474$	30.8864	1.41866
$475$	0	0
$476$	−12.2500	−0.561477
$477$	3.74568	0.171503
$478$	0.0908201	0.00415401
$479$	12.8143	0.585499	0.292750	−	0.956189i	$-0.405430\pi$
$479$	12.8143	0.585499	0.292750	+	0.956189i	$0.405430\pi$
$480$	0	0
$481$	−7.63365	−0.348064
$482$	−24.9822	−1.13791
$483$	−10.1127	−0.460143
$484$	−21.2963	−0.968014
$485$	0	0
$486$	2.12233	0.0962710
$487$	−6.92209	−0.313670	−0.156835	−	0.987625i	$-0.550129\pi$
$487$	−6.92209	−0.313670	−0.156835	+	0.987625i	$0.550129\pi$
$488$	1.49400	0.0676301
$489$	4.80141	0.217127
$490$	0	0
$491$	26.8161	1.21020	0.605098	−	0.796151i	$-0.293134\pi$
$491$	26.8161	1.21020	0.605098	+	0.796151i	$0.293134\pi$
$492$	−18.7125	−0.843623
$493$	6.17766	0.278228
$494$	19.5225	0.878358
$495$	0	0
$496$	−13.1436	−0.590167
$497$	−46.2213	−2.07331
$498$	−18.5408	−0.830832
$499$	2.75460	0.123313	0.0616565	−	0.998097i	$-0.480362\pi$
$499$	2.75460	0.123313	0.0616565	+	0.998097i	$0.480362\pi$
$500$	0	0
$501$	23.6600	1.05705
$502$	37.0907	1.65544
$503$	15.7102	0.700483	0.350241	−	0.936659i	$-0.386100\pi$
$503$	15.7102	0.700483	0.350241	+	0.936659i	$0.386100\pi$
$504$	4.66476	0.207785
$505$	0	0
$506$	7.78008	0.345867
$507$	−11.5649	−0.513617
$508$	−42.5622	−1.88839
$509$	−41.0884	−1.82121	−0.910606	−	0.413277i	$-0.864384\pi$
$509$	−41.0884	−1.82121	−0.910606	+	0.413277i	$0.864384\pi$
$510$	0	0
$511$	21.7777	0.963388
$512$	27.6176	1.22054
$513$	7.67867	0.339022
$514$	32.8264	1.44791
$515$	0	0
$516$	−3.13033	−0.137805
$517$	−6.52129	−0.286806
$518$	−58.9438	−2.58984
$519$	14.5043	0.636668
$520$	0	0
$521$	36.1710	1.58468	0.792341	−	0.610079i	$-0.208862\pi$
$521$	36.1710	1.58468	0.792341	+	0.610079i	$0.208862\pi$
$522$	−11.6820	−0.511305
$523$	28.0490	1.22650	0.613248	−	0.789891i	$-0.289863\pi$
$523$	28.0490	1.22650	0.613248	+	0.789891i	$0.289863\pi$
$524$	−0.823595	−0.0359789
$525$	0	0
$526$	3.07228	0.133958
$527$	−5.38951	−0.234771
$528$	4.32433	0.188192
$529$	−17.6163	−0.765927
$530$	0	0
$531$	9.15613	0.397342
$532$	83.8108	3.63366
$533$	−8.95117	−0.387718
$534$	−21.3043	−0.921925
$535$	0	0
$536$	−4.13422	−0.178571
$537$	0.796746	0.0343821
$538$	−18.4774	−0.796617
$539$	−18.9520	−0.816321
$540$	0	0
$541$	3.45822	0.148681	0.0743403	−	0.997233i	$-0.476315\pi$
$541$	3.45822	0.148681	0.0743403	+	0.997233i	$0.476315\pi$
$542$	−60.7704	−2.61031
$543$	−14.2185	−0.610173
$544$	8.92209	0.382531
$545$	0	0
$546$	11.0809	0.474220
$547$	13.9635	0.597034	0.298517	−	0.954404i	$-0.403508\pi$
$547$	13.9635	0.597034	0.298517	+	0.954404i	$0.403508\pi$
$548$	11.4264	0.488111
$549$	1.39588	0.0595747
$550$	0	0
$551$	−42.2657	−1.80058
$552$	−2.48337	−0.105699
$553$	63.4281	2.69724
$554$	11.2528	0.478086
$555$	0	0
$556$	−11.8554	−0.502779
$557$	−6.59585	−0.279475	−0.139738	−	0.990189i	$-0.544626\pi$
$557$	−6.59585	−0.279475	−0.139738	+	0.990189i	$0.544626\pi$
$558$	10.1916	0.431444
$559$	−1.49740	−0.0633334
$560$	0	0
$561$	1.77318	0.0748638
$562$	−51.1126	−2.15605
$563$	−16.0218	−0.675238	−0.337619	−	0.941283i	$-0.609621\pi$
$563$	−16.0218	−0.675238	−0.337619	+	0.941283i	$0.609621\pi$
$564$	10.3369	0.435261
$565$	0	0
$566$	−57.5456	−2.41882
$567$	4.35840	0.183036
$568$	−11.3505	−0.476258
$569$	23.0914	0.968042	0.484021	−	0.875056i	$-0.339176\pi$
$569$	23.0914	0.968042	0.484021	+	0.875056i	$0.339176\pi$
$570$	0	0
$571$	−34.0102	−1.42328	−0.711640	−	0.702544i	$-0.752047\pi$
$571$	−34.0102	−1.42328	−0.711640	+	0.702544i	$0.752047\pi$
$572$	−4.73972	−0.198178
$573$	−10.3697	−0.433200
$574$	−69.1171	−2.88489
$575$	0	0
$576$	−11.3975	−0.474896
$577$	−6.46727	−0.269236	−0.134618	−	0.990898i	$-0.542981\pi$
$577$	−6.46727	−0.269236	−0.134618	+	0.990898i	$0.542981\pi$
$578$	−33.4063	−1.38952
$579$	4.34712	0.180660
$580$	0	0
$581$	−38.0751	−1.57962
$582$	−15.9174	−0.659798
$583$	−5.91782	−0.245091
$584$	5.34794	0.221299
$585$	0	0
$586$	−43.0868	−1.77990
$587$	38.5012	1.58911	0.794557	−	0.607189i	$-0.207703\pi$
$587$	38.5012	1.58911	0.794557	+	0.607189i	$0.207703\pi$
$588$	30.0407	1.23886
$589$	36.8735	1.51934
$590$	0	0
$591$	1.87990	0.0773287
$592$	17.4416	0.716843
$593$	4.93069	0.202479	0.101240	−	0.994862i	$-0.467719\pi$
$593$	4.93069	0.202479	0.101240	+	0.994862i	$0.467719\pi$
$594$	−3.35309	−0.137579
$595$	0	0
$596$	11.7112	0.479710
$597$	4.26028	0.174362
$598$	−5.89913	−0.241233
$599$	−35.0268	−1.43116	−0.715578	−	0.698533i	$-0.753836\pi$
$599$	−35.0268	−1.43116	−0.715578	+	0.698533i	$0.753836\pi$
$600$	0	0
$601$	−4.90570	−0.200108	−0.100054	−	0.994982i	$-0.531901\pi$
$601$	−4.90570	−0.200108	−0.100054	+	0.994982i	$0.531901\pi$
$602$	−11.5623	−0.471244
$603$	−3.86270	−0.157301
$604$	−16.3826	−0.666597
$605$	0	0
$606$	−13.8338	−0.561961
$607$	48.6955	1.97649	0.988244	−	0.152884i	$-0.0488562\pi$
$607$	48.6955	1.97649	0.988244	+	0.152884i	$0.0488562\pi$
$608$	−61.0423	−2.47559
$609$	−23.9899	−0.972122
$610$	0	0
$611$	4.94467	0.200040
$612$	−2.81066	−0.113614
$613$	17.2139	0.695262	0.347631	−	0.937631i	$-0.386986\pi$
$613$	17.2139	0.695262	0.347631	+	0.937631i	$0.386986\pi$
$614$	−5.33406	−0.215265
$615$	0	0
$616$	−7.36988	−0.296941
$617$	−31.6092	−1.27254	−0.636268	−	0.771468i	$-0.719523\pi$
$617$	−31.6092	−1.27254	−0.636268	+	0.771468i	$0.719523\pi$
$618$	17.0821	0.687145
$619$	−22.8196	−0.917198	−0.458599	−	0.888643i	$-0.651649\pi$
$619$	−22.8196	−0.917198	−0.458599	+	0.888643i	$0.651649\pi$
$620$	0	0
$621$	−2.32027	−0.0931094
$622$	10.2385	0.410526
$623$	−43.7502	−1.75281
$624$	−3.27886	−0.131259
$625$	0	0
$626$	−12.7088	−0.507945
$627$	−12.1316	−0.484489
$628$	10.0023	0.399134
$629$	7.15186	0.285163
$630$	0	0
$631$	38.0499	1.51474	0.757372	−	0.652984i	$-0.226483\pi$
$631$	38.0499	1.51474	0.757372	+	0.652984i	$0.226483\pi$
$632$	15.5760	0.619581
$633$	−9.29671	−0.369511
$634$	36.8290	1.46267
$635$	0	0
$636$	9.38031	0.371953
$637$	14.3701	0.569363
$638$	18.4564	0.730696
$639$	−10.6051	−0.419531
$640$	0	0
$641$	26.5108	1.04711	0.523557	−	0.851991i	$-0.324605\pi$
$641$	26.5108	1.04711	0.523557	+	0.851991i	$0.324605\pi$
$642$	20.1143	0.793849
$643$	−36.0014	−1.41976	−0.709879	−	0.704324i	$-0.751250\pi$
$643$	−36.0014	−1.41976	−0.709879	+	0.704324i	$0.751250\pi$
$644$	−25.3252	−0.997952
$645$	0	0
$646$	−18.2903	−0.719624
$647$	41.7840	1.64270	0.821349	−	0.570426i	$-0.193222\pi$
$647$	41.7840	1.64270	0.821349	+	0.570426i	$0.193222\pi$
$648$	1.07029	0.0420450
$649$	−14.4658	−0.567833
$650$	0	0
$651$	20.9293	0.820284
$652$	12.0242	0.470903
$653$	21.4404	0.839027	0.419513	−	0.907749i	$-0.362201\pi$
$653$	21.4404	0.839027	0.419513	+	0.907749i	$0.362201\pi$
$654$	−24.7586	−0.968137
$655$	0	0
$656$	20.4519	0.798511
$657$	4.99672	0.194940
$658$	38.1807	1.48844
$659$	−16.8319	−0.655678	−0.327839	−	0.944734i	$-0.606320\pi$
$659$	−16.8319	−0.655678	−0.327839	+	0.944734i	$0.606320\pi$
$660$	0	0
$661$	2.03846	0.0792868	0.0396434	−	0.999214i	$-0.487378\pi$
$661$	2.03846	0.0792868	0.0396434	+	0.999214i	$0.487378\pi$
$662$	43.1587	1.67741
$663$	−1.34449	−0.0522156
$664$	−9.35010	−0.362854
$665$	0	0
$666$	−13.5242	−0.524051
$667$	12.7715	0.494514
$668$	59.2518	2.29252
$669$	−3.39758	−0.131358
$670$	0	0
$671$	−2.20536	−0.0851369
$672$	−34.6475	−1.33656
$673$	35.3776	1.36371	0.681854	−	0.731489i	$-0.261174\pi$
$673$	35.3776	1.36371	0.681854	+	0.731489i	$0.261174\pi$
$674$	72.7050	2.80049
$675$	0	0
$676$	−28.9621	−1.11393
$677$	44.4173	1.70710	0.853548	−	0.521014i	$-0.174446\pi$
$677$	44.4173	1.70710	0.853548	+	0.521014i	$0.174446\pi$
$678$	−31.6766	−1.21653
$679$	−32.6879	−1.25444
$680$	0	0
$681$	26.7973	1.02688
$682$	−16.1017	−0.616567
$683$	42.4952	1.62603	0.813016	−	0.582241i	$-0.197824\pi$
$683$	42.4952	1.62603	0.813016	+	0.582241i	$0.197824\pi$
$684$	19.2297	0.735266
$685$	0	0
$686$	46.2098	1.76430
$687$	−1.18200	−0.0450960
$688$	3.42130	0.130436
$689$	4.48710	0.170945
$690$	0	0
$691$	28.0808	1.06825	0.534123	−	0.845407i	$-0.320642\pi$
$691$	28.0808	1.06825	0.534123	+	0.845407i	$0.320642\pi$
$692$	36.3231	1.38080
$693$	−6.88586	−0.261572
$694$	−0.104762	−0.00397671
$695$	0	0
$696$	−5.89121	−0.223306
$697$	8.38623	0.317651
$698$	−15.8631	−0.600427
$699$	7.09483	0.268351
$700$	0	0
$701$	46.4314	1.75369	0.876845	−	0.480772i	$-0.159644\pi$
$701$	46.4314	1.75369	0.876845	+	0.480772i	$0.159644\pi$
$702$	2.54243	0.0959578
$703$	−48.9309	−1.84547
$704$	18.0070	0.678664
$705$	0	0
$706$	−39.4011	−1.48288
$707$	−28.4090	−1.06843
$708$	22.9297	0.861750
$709$	48.6331	1.82646	0.913228	−	0.407450i	$-0.133582\pi$
$709$	48.6331	1.82646	0.913228	+	0.407450i	$0.133582\pi$
$710$	0	0
$711$	14.5531	0.545782
$712$	−10.7437	−0.402638
$713$	−11.1421	−0.417275
$714$	−10.3816	−0.388520
$715$	0	0
$716$	1.99529	0.0745675
$717$	0.0427926	0.00159812
$718$	22.5062	0.839923
$719$	−21.1954	−0.790456	−0.395228	−	0.918583i	$-0.629334\pi$
$719$	−21.1954	−0.790456	−0.395228	+	0.918583i	$0.629334\pi$
$720$	0	0
$721$	35.0797	1.30644
$722$	84.8128	3.15641
$723$	−11.7711	−0.437771
$724$	−35.6073	−1.32333
$725$	0	0
$726$	−18.0481	−0.669828
$727$	13.5911	0.504066	0.252033	−	0.967719i	$-0.418901\pi$
$727$	13.5911	0.504066	0.252033	+	0.967719i	$0.418901\pi$
$728$	5.58810	0.207109
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.40290	0.0518880
$732$	3.49570	0.129205
$733$	18.6531	0.688967	0.344484	−	0.938792i	$-0.388054\pi$
$733$	18.6531	0.688967	0.344484	+	0.938792i	$0.388054\pi$
$734$	−24.6565	−0.910088
$735$	0	0
$736$	18.4452	0.679900
$737$	6.10270	0.224796
$738$	−15.8584	−0.583754
$739$	26.4452	0.972803	0.486401	−	0.873735i	$-0.338309\pi$
$739$	26.4452	0.972803	0.486401	+	0.873735i	$0.338309\pi$
$740$	0	0
$741$	9.19859	0.337919
$742$	34.6475	1.27195
$743$	−29.0191	−1.06461	−0.532304	−	0.846553i	$-0.678674\pi$
$743$	−29.0191	−1.06461	−0.532304	+	0.846553i	$0.678674\pi$
$744$	5.13961	0.188427
$745$	0	0
$746$	−49.0837	−1.79708
$747$	−8.73603	−0.319635
$748$	4.44058	0.162364
$749$	41.3065	1.50931
$750$	0	0
$751$	15.9489	0.581985	0.290992	−	0.956725i	$-0.406015\pi$
$751$	15.9489	0.581985	0.290992	+	0.956725i	$0.406015\pi$
$752$	−11.2977	−0.411985
$753$	17.4764	0.636875
$754$	−13.9943	−0.509642
$755$	0	0
$756$	10.9147	0.396965
$757$	−16.4183	−0.596734	−0.298367	−	0.954451i	$-0.596442\pi$
$757$	−16.4183	−0.596734	−0.298367	+	0.954451i	$0.596442\pi$
$758$	52.8274	1.91878
$759$	3.66581	0.133061
$760$	0	0
$761$	−49.2715	−1.78609	−0.893045	−	0.449967i	$-0.851436\pi$
$761$	−49.2715	−1.78609	−0.893045	+	0.449967i	$0.851436\pi$
$762$	−36.0705	−1.30670
$763$	−50.8440	−1.84068
$764$	−25.9688	−0.939518
$765$	0	0
$766$	−33.3676	−1.20562
$767$	10.9685	0.396049
$768$	5.20057	0.187660
$769$	49.4236	1.78226	0.891129	−	0.453749i	$-0.149914\pi$
$769$	49.4236	1.78226	0.891129	+	0.453749i	$0.149914\pi$
$770$	0	0
$771$	15.4671	0.557036
$772$	10.8865	0.391814
$773$	25.9084	0.931860	0.465930	−	0.884822i	$-0.345720\pi$
$773$	25.9084	0.931860	0.465930	+	0.884822i	$0.345720\pi$
$774$	−2.65288	−0.0953557
$775$	0	0
$776$	−8.02715	−0.288158
$777$	−27.7731	−0.996355
$778$	−18.0533	−0.647241
$779$	−57.3761	−2.05571
$780$	0	0
$781$	16.7551	0.599543
$782$	5.52681	0.197638
$783$	−5.50430	−0.196708
$784$	−32.8331	−1.17261
$785$	0	0
$786$	−0.697977	−0.0248960
$787$	30.7155	1.09489	0.547444	−	0.836842i	$-0.315601\pi$
$787$	30.7155	1.09489	0.547444	+	0.836842i	$0.315601\pi$
$788$	4.70783	0.167710
$789$	1.44759	0.0515357
$790$	0	0
$791$	−65.0508	−2.31294
$792$	−1.69096	−0.0600857
$793$	1.67218	0.0593808
$794$	55.7216	1.97749
$795$	0	0
$796$	10.6690	0.378154
$797$	−33.7141	−1.19421	−0.597107	−	0.802161i	$-0.703683\pi$
$797$	−33.7141	−1.19421	−0.597107	+	0.802161i	$0.703683\pi$
$798$	71.0276	2.51435
$799$	−4.63260	−0.163890
$800$	0	0
$801$	−10.0381	−0.354680
$802$	−53.6559	−1.89465
$803$	−7.89434	−0.278585
$804$	−9.67336	−0.341153
$805$	0	0
$806$	12.2089	0.430040
$807$	−8.70617	−0.306472
$808$	−6.97639	−0.245429
$809$	−0.923014	−0.0324514	−0.0162257	−	0.999868i	$-0.505165\pi$
$809$	−0.923014	−0.0324514	−0.0162257	+	0.999868i	$0.505165\pi$
$810$	0	0
$811$	26.7815	0.940424	0.470212	−	0.882553i	$-0.344177\pi$
$811$	26.7815	0.940424	0.470212	+	0.882553i	$0.344177\pi$
$812$	−60.0780	−2.10832
$813$	−28.6338	−1.00423
$814$	21.3669	0.748910
$815$	0	0
$816$	3.07192	0.107539
$817$	−9.59820	−0.335799
$818$	71.7204	2.50765
$819$	5.22110	0.182440
$820$	0	0
$821$	−36.8489	−1.28604	−0.643018	−	0.765851i	$-0.722318\pi$
$821$	−36.8489	−1.28604	−0.643018	+	0.765851i	$0.722318\pi$
$822$	9.68359	0.337754
$823$	20.4262	0.712012	0.356006	−	0.934484i	$-0.384138\pi$
$823$	20.4262	0.712012	0.356006	+	0.934484i	$0.384138\pi$
$824$	8.61452	0.300101
$825$	0	0
$826$	84.6940	2.94688
$827$	4.33017	0.150575	0.0752874	−	0.997162i	$-0.476013\pi$
$827$	4.33017	0.150575	0.0752874	+	0.997162i	$0.476013\pi$
$828$	−5.81066	−0.201934
$829$	−0.614348	−0.0213372	−0.0106686	−	0.999943i	$-0.503396\pi$
$829$	−0.614348	−0.0213372	−0.0106686	+	0.999943i	$0.503396\pi$
$830$	0	0
$831$	5.30210	0.183928
$832$	−13.6535	−0.473351
$833$	−13.4631	−0.466470
$834$	−10.0471	−0.347904
$835$	0	0
$836$	−30.3811	−1.05075
$837$	4.80206	0.165983
$838$	16.8372	0.581630
$839$	5.39981	0.186422	0.0932111	−	0.995646i	$-0.470287\pi$
$839$	5.39981	0.186422	0.0932111	+	0.995646i	$0.470287\pi$
$840$	0	0
$841$	1.29731	0.0447349
$842$	15.8919	0.547672
$843$	−24.0832	−0.829469
$844$	−23.2817	−0.801391
$845$	0	0
$846$	8.76025	0.301183
$847$	−37.0634	−1.27351
$848$	−10.2522	−0.352063
$849$	−27.1143	−0.930561
$850$	0	0
$851$	14.7855	0.506841
$852$	−26.5583	−0.909874
$853$	−39.7935	−1.36250	−0.681252	−	0.732049i	$-0.738564\pi$
$853$	−39.7935	−1.36250	−0.681252	+	0.732049i	$0.738564\pi$
$854$	12.9119	0.441835
$855$	0	0
$856$	10.1436	0.346702
$857$	−42.8643	−1.46422	−0.732109	−	0.681188i	$-0.761464\pi$
$857$	−42.8643	−1.46422	−0.732109	+	0.681188i	$0.761464\pi$
$858$	−4.01680	−0.137131
$859$	−4.56494	−0.155754	−0.0778769	−	0.996963i	$-0.524814\pi$
$859$	−4.56494	−0.155754	−0.0778769	+	0.996963i	$0.524814\pi$
$860$	0	0
$861$	−32.5666	−1.10987
$862$	49.5502	1.68769
$863$	−37.6273	−1.28085	−0.640424	−	0.768022i	$-0.721241\pi$
$863$	−37.6273	−1.28085	−0.640424	+	0.768022i	$0.721241\pi$
$864$	−7.94959	−0.270450
$865$	0	0
$866$	−7.12419	−0.242090
$867$	−15.7404	−0.534571
$868$	52.4133	1.77902
$869$	−22.9925	−0.779966
$870$	0	0
$871$	−4.62728	−0.156790
$872$	−12.4857	−0.422821
$873$	−7.49996	−0.253835
$874$	−37.8128	−1.27904
$875$	0	0
$876$	12.5133	0.422784
$877$	52.4041	1.76956	0.884781	−	0.466008i	$-0.154308\pi$
$877$	52.4041	1.76956	0.884781	+	0.466008i	$0.154308\pi$
$878$	−20.1283	−0.679296
$879$	−20.3016	−0.684757
$880$	0	0
$881$	−7.93778	−0.267430	−0.133715	−	0.991020i	$-0.542691\pi$
$881$	−7.93778	−0.267430	−0.133715	+	0.991020i	$0.542691\pi$
$882$	25.4588	0.857242
$883$	31.6919	1.06652	0.533259	−	0.845952i	$-0.320967\pi$
$883$	31.6919	1.06652	0.533259	+	0.845952i	$0.320967\pi$
$884$	−3.36700	−0.113244
$885$	0	0
$886$	20.4900	0.688374
$887$	2.82104	0.0947213	0.0473606	−	0.998878i	$-0.484919\pi$
$887$	2.82104	0.0947213	0.0473606	+	0.998878i	$0.484919\pi$
$888$	−6.82024	−0.228872
$889$	−74.0739	−2.48436
$890$	0	0
$891$	−1.57991	−0.0529288
$892$	−8.50856	−0.284888
$893$	31.6949	1.06063
$894$	9.92497	0.331941
$895$	0	0
$896$	−36.1318	−1.20708
$897$	−2.77955	−0.0928064
$898$	66.9087	2.23277
$899$	−26.4320	−0.881556
$900$	0	0
$901$	−4.20390	−0.140052
$902$	25.0547	0.834231
$903$	−5.44792	−0.181295
$904$	−15.9745	−0.531304
$905$	0	0
$906$	−13.8838	−0.461259
$907$	−44.1799	−1.46697	−0.733485	−	0.679705i	$-0.762108\pi$
$907$	−44.1799	−1.46697	−0.733485	+	0.679705i	$0.762108\pi$
$908$	67.1085	2.22707
$909$	−6.51821	−0.216195
$910$	0	0
$911$	−34.5260	−1.14390	−0.571949	−	0.820289i	$-0.693812\pi$
$911$	−34.5260	−1.14390	−0.571949	+	0.820289i	$0.693812\pi$
$912$	−21.0172	−0.695948
$913$	13.8021	0.456783
$914$	−21.1059	−0.698122
$915$	0	0
$916$	−2.96007	−0.0978036
$917$	−1.43336	−0.0473336
$918$	−2.38197	−0.0786166
$919$	−0.531075	−0.0175185	−0.00875927	−	0.999962i	$-0.502788\pi$
$919$	−0.531075	−0.0175185	−0.00875927	+	0.999962i	$0.502788\pi$
$920$	0	0
$921$	−2.51330	−0.0828161
$922$	−50.2645	−1.65537
$923$	−12.7043	−0.418166
$924$	−17.2443	−0.567295
$925$	0	0
$926$	58.6833	1.92845
$927$	8.04876	0.264356
$928$	43.7569	1.43639
$929$	46.1961	1.51565	0.757823	−	0.652460i	$-0.226263\pi$
$929$	46.1961	1.51565	0.757823	+	0.652460i	$0.226263\pi$
$930$	0	0
$931$	92.1108	3.01881
$932$	17.7676	0.581997
$933$	4.82417	0.157936
$934$	9.24998	0.302669
$935$	0	0
$936$	1.28215	0.0419082
$937$	−30.1220	−0.984042	−0.492021	−	0.870583i	$-0.663742\pi$
$937$	−30.1220	−0.984042	−0.492021	+	0.870583i	$0.663742\pi$
$938$	−35.7299	−1.16662
$939$	−5.98812	−0.195415
$940$	0	0
$941$	57.0783	1.86070	0.930350	−	0.366672i	$-0.119503\pi$
$941$	57.0783	1.86070	0.930350	+	0.366672i	$0.119503\pi$
$942$	8.47668	0.276185
$943$	17.3374	0.564583
$944$	−25.0611	−0.815668
$945$	0	0
$946$	4.19130	0.136271
$947$	−41.1250	−1.33638	−0.668192	−	0.743989i	$-0.732931\pi$
$947$	−41.1250	−1.33638	−0.668192	+	0.743989i	$0.732931\pi$
$948$	36.4452	1.18369
$949$	5.98576	0.194306
$950$	0	0
$951$	17.3531	0.562712
$952$	−5.23542	−0.169681
$953$	34.8143	1.12775	0.563873	−	0.825861i	$-0.309311\pi$
$953$	34.8143	1.12775	0.563873	+	0.825861i	$0.309311\pi$
$954$	7.94959	0.257377
$955$	0	0
$956$	0.107165	0.00346598
$957$	8.69627	0.281111
$958$	27.1962	0.878668
$959$	19.8861	0.642156
$960$	0	0
$961$	−7.94022	−0.256136
$962$	−16.2012	−0.522346
$963$	9.47745	0.305407
$964$	−29.4783	−0.949432
$965$	0	0
$966$	−21.4625	−0.690544
$967$	−37.2697	−1.19851	−0.599257	−	0.800557i	$-0.704537\pi$
$967$	−37.2697	−1.19851	−0.599257	+	0.800557i	$0.704537\pi$
$968$	−9.10165	−0.292538
$969$	−8.61803	−0.276851
$970$	0	0
$971$	−26.0092	−0.834677	−0.417338	−	0.908751i	$-0.637037\pi$
$971$	−26.0092	−0.834677	−0.417338	+	0.908751i	$0.637037\pi$
$972$	2.50430	0.0803254
$973$	−20.6327	−0.661453
$974$	−14.6910	−0.470729
$975$	0	0
$976$	−3.82064	−0.122296
$977$	0.458579	0.0146712	0.00733561	−	0.999973i	$-0.497665\pi$
$977$	0.458579	0.0146712	0.00733561	+	0.999973i	$0.497665\pi$
$978$	10.1902	0.325846
$979$	15.8593	0.506865
$980$	0	0
$981$	−11.6657	−0.372458
$982$	56.9128	1.81616
$983$	8.94565	0.285322	0.142661	−	0.989772i	$-0.454434\pi$
$983$	8.94565	0.285322	0.142661	+	0.989772i	$0.454434\pi$
$984$	−7.99737	−0.254947
$985$	0	0
$986$	13.1111	0.417541
$987$	17.9899	0.572626
$988$	23.0360	0.732874
$989$	2.90030	0.0922241
$990$	0	0
$991$	−56.5337	−1.79585	−0.897926	−	0.440147i	$-0.854926\pi$
$991$	−56.5337	−1.79585	−0.897926	+	0.440147i	$0.854926\pi$
$992$	−38.1744	−1.21204
$993$	20.3355	0.645327
$994$	−98.0970	−3.11145
$995$	0	0
$996$	−21.8776	−0.693219
$997$	39.6114	1.25451	0.627253	−	0.778816i	$-0.284179\pi$
$997$	39.6114	1.25451	0.627253	+	0.778816i	$0.284179\pi$
$998$	5.84619	0.185058
$999$	−6.37232	−0.201611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.h.1.3		4
3.2	odd	2		5625.2.a.i.1.2		4
5.2	odd	4		1875.2.b.c.1249.7		8
5.3	odd	4		1875.2.b.c.1249.2		8
5.4	even	2		1875.2.a.e.1.2		4
15.14	odd	2		5625.2.a.n.1.3		4
25.2	odd	20		375.2.i.b.274.4		16
25.9	even	10		375.2.g.b.151.1		8
25.11	even	5		75.2.g.b.46.2	yes	8
25.12	odd	20		375.2.i.b.349.1		16
25.13	odd	20		375.2.i.b.349.4		16
25.14	even	10		375.2.g.b.226.1		8
25.16	even	5		75.2.g.b.31.2	✓	8
25.23	odd	20		375.2.i.b.274.1		16
75.11	odd	10		225.2.h.c.46.1		8
75.41	odd	10		225.2.h.c.181.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.31.2	✓	8	25.16	even	5
75.2.g.b.46.2	yes	8	25.11	even	5
225.2.h.c.46.1		8	75.11	odd	10
225.2.h.c.181.1		8	75.41	odd	10
375.2.g.b.151.1		8	25.9	even	10
375.2.g.b.226.1		8	25.14	even	10
375.2.i.b.274.1		16	25.23	odd	20
375.2.i.b.274.4		16	25.2	odd	20
375.2.i.b.349.1		16	25.12	odd	20
375.2.i.b.349.4		16	25.13	odd	20
1875.2.a.e.1.2		4	5.4	even	2
1875.2.a.h.1.3		4	1.1	even	1	trivial
1875.2.b.c.1249.2		8	5.3	odd	4
1875.2.b.c.1249.7		8	5.2	odd	4
5625.2.a.i.1.2		4	3.2	odd	2
5625.2.a.n.1.3		4	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)

\(f(q)\)	\(=\)	\(q+2.12233 q^{2} +1.00000 q^{3} +2.50430 q^{4} +2.12233 q^{6} +4.35840 q^{7} +1.07029 q^{8} +1.00000 q^{9} -1.57991 q^{11} +2.50430 q^{12} +1.19794 q^{13} +9.24998 q^{14} -2.73708 q^{16} -1.12233 q^{17} +2.12233 q^{18} +7.67867 q^{19} +4.35840 q^{21} -3.35309 q^{22} -2.32027 q^{23} +1.07029 q^{24} +2.54243 q^{26} +1.00000 q^{27} +10.9147 q^{28} -5.50430 q^{29} +4.80206 q^{31} -7.94959 q^{32} -1.57991 q^{33} -2.38197 q^{34} +2.50430 q^{36} -6.37232 q^{37} +16.2967 q^{38} +1.19794 q^{39} -7.47214 q^{41} +9.24998 q^{42} -1.24998 q^{43} -3.95656 q^{44} -4.92439 q^{46} +4.12765 q^{47} -2.73708 q^{48} +11.9957 q^{49} -1.12233 q^{51} +3.00000 q^{52} +3.74568 q^{53} +2.12233 q^{54} +4.66476 q^{56} +7.67867 q^{57} -11.6820 q^{58} +9.15613 q^{59} +1.39588 q^{61} +10.1916 q^{62} +4.35840 q^{63} -11.3975 q^{64} -3.35309 q^{66} -3.86270 q^{67} -2.81066 q^{68} -2.32027 q^{69} -10.6051 q^{71} +1.07029 q^{72} +4.99672 q^{73} -13.5242 q^{74} +19.2297 q^{76} -6.88586 q^{77} +2.54243 q^{78} +14.5531 q^{79} +1.00000 q^{81} -15.8584 q^{82} -8.73603 q^{83} +10.9147 q^{84} -2.65288 q^{86} -5.50430 q^{87} -1.69096 q^{88} -10.0381 q^{89} +5.22110 q^{91} -5.81066 q^{92} +4.80206 q^{93} +8.76025 q^{94} -7.94959 q^{96} -7.49996 q^{97} +25.4588 q^{98} -1.57991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 15 q^{8} + 4 q^{9} - 7 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{18} + 5 q^{19} + 2 q^{21} - 6 q^{22} + q^{23} + 15 q^{24}+ \cdots - 7 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 2 * q^6 + 2 * q^7 + 15 * q^8 + 4 * q^9 - 7 * q^11 + 8 * q^12 + q^13 + 16 * q^14 + 4 * q^16 + 2 * q^17 + 2 * q^18 + 5 * q^19 + 2 * q^21 - 6 * q^22 + q^23 + 15 * q^24 + 3 * q^26 + 4 * q^27 + 9 * q^28 - 20 * q^29 + 23 * q^31 + 12 * q^32 - 7 * q^33 - 14 * q^34 + 8 * q^36 + 2 * q^37 + 35 * q^38 + q^39 - 12 * q^41 + 16 * q^42 + 16 * q^43 - 29 * q^44 - 17 * q^46 + 2 * q^47 + 4 * q^48 + 8 * q^49 + 2 * q^51 + 12 * q^52 - 4 * q^53 + 2 * q^54 + 5 * q^56 + 5 * q^57 - 25 * q^58 - 15 * q^59 - 2 * q^61 + 9 * q^62 + 2 * q^63 + 23 * q^64 - 6 * q^66 + 2 * q^67 - 11 * q^68 + q^69 - 2 * q^71 + 15 * q^72 + 16 * q^73 - 19 * q^74 + 40 * q^76 + 19 * q^77 + 3 * q^78 + 35 * q^79 + 4 * q^81 - 6 * q^82 + 16 * q^83 + 9 * q^84 + 3 * q^86 - 20 * q^87 - 30 * q^88 - 35 * q^89 - 12 * q^91 - 23 * q^92 + 23 * q^93 - 9 * q^94 + 12 * q^96 + 12 * q^97 - q^98 - 7 * q^99

Introduction

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