LMFDB - Embedded newform 9075.2.a.cu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.18890$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.456850 q^{2} -1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -4.37780 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.456850 q^{2} -1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -4.37780 q^{7} +1.73205 q^{8} +1.00000 q^{9} +1.79129 q^{12} -0.913701 q^{13} +2.00000 q^{14} +2.79129 q^{16} -1.73205 q^{17} -0.456850 q^{18} -3.46410 q^{19} +4.37780 q^{21} +0.582576 q^{23} -1.73205 q^{24} +0.417424 q^{26} -1.00000 q^{27} +7.84190 q^{28} +9.66930 q^{29} -8.58258 q^{31} -4.73930 q^{32} +0.791288 q^{34} -1.79129 q^{36} -7.58258 q^{37} +1.58258 q^{38} +0.913701 q^{39} +3.46410 q^{41} -2.00000 q^{42} +9.66930 q^{43} -0.266150 q^{46} -3.41742 q^{47} -2.79129 q^{48} +12.1652 q^{49} +1.73205 q^{51} +1.63670 q^{52} -12.1652 q^{53} +0.456850 q^{54} -7.58258 q^{56} +3.46410 q^{57} -4.41742 q^{58} +13.5826 q^{59} -3.55945 q^{61} +3.92095 q^{62} -4.37780 q^{63} -3.41742 q^{64} +4.41742 q^{67} +3.10260 q^{68} -0.582576 q^{69} +8.00000 q^{71} +1.73205 q^{72} -5.10080 q^{73} +3.46410 q^{74} +6.20520 q^{76} -0.417424 q^{78} +0.818350 q^{79} +1.00000 q^{81} -1.58258 q^{82} +15.8745 q^{83} -7.84190 q^{84} -4.41742 q^{86} -9.66930 q^{87} +14.7477 q^{89} +4.00000 q^{91} -1.04356 q^{92} +8.58258 q^{93} +1.56125 q^{94} +4.73930 q^{96} -4.41742 q^{97} -5.55765 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 2 * q^4 + 4 * q^9	$ 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9} - 2 q^{12} + 8 q^{14} + 2 q^{16} - 16 q^{23} + 20 q^{26} - 4 q^{27} - 16 q^{31} - 6 q^{34} + 2 q^{36} - 12 q^{37} - 12 q^{38} - 8 q^{42} - 32 q^{47} - 2 q^{48} + 12 q^{49} - 12 q^{53} - 12 q^{56} - 36 q^{58} + 36 q^{59} - 32 q^{64} + 36 q^{67} + 16 q^{69} + 32 q^{71} - 20 q^{78} + 4 q^{81} + 12 q^{82} - 36 q^{86} + 4 q^{89} + 16 q^{91} - 50 q^{92} + 16 q^{93} - 36 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 + 2 * q^4 + 4 * q^9 - 2 * q^12 + 8 * q^14 + 2 * q^16 - 16 * q^23 + 20 * q^26 - 4 * q^27 - 16 * q^31 - 6 * q^34 + 2 * q^36 - 12 * q^37 - 12 * q^38 - 8 * q^42 - 32 * q^47 - 2 * q^48 + 12 * q^49 - 12 * q^53 - 12 * q^56 - 36 * q^58 + 36 * q^59 - 32 * q^64 + 36 * q^67 + 16 * q^69 + 32 * q^71 - 20 * q^78 + 4 * q^81 + 12 * q^82 - 36 * q^86 + 4 * q^89 + 16 * q^91 - 50 * q^92 + 16 * q^93 - 36 * q^97

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 $

$2$	−0.456850	−0.323042	−0.161521	−	0.986869i	$-0.551640\pi$
$2$	−0.456850	−0.323042	−0.161521	+	0.986869i	$0.551640\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.79129	−0.895644
$5$	0	0
$5$	0	0
$6$	0.456850	0.186508
$7$	−4.37780	−1.65465	−0.827327	−	0.561721i	$-0.810140\pi$
$7$	−4.37780	−1.65465	−0.827327	+	0.561721i	$0.810140\pi$
$8$	1.73205	0.612372
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.79129	0.517100
$13$	−0.913701	−0.253415	−0.126707	−	0.991940i	$-0.540441\pi$
$13$	−0.913701	−0.253415	−0.126707	+	0.991940i	$0.540441\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	2.79129	0.697822
$17$	−1.73205	−0.420084	−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205	−0.420084	−0.210042	+	0.977692i	$0.567360\pi$
$18$	−0.456850	−0.107681
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	4.37780	0.955315
$22$	0	0
$23$	0.582576	0.121475	0.0607377	−	0.998154i	$-0.480655\pi$
$23$	0.582576	0.121475	0.0607377	+	0.998154i	$0.480655\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	0.417424	0.0818636
$27$	−1.00000	−0.192450
$28$	7.84190	1.48198
$29$	9.66930	1.79554	0.897772	−	0.440460i	$-0.145185\pi$
$29$	9.66930	1.79554	0.897772	+	0.440460i	$0.145185\pi$
$30$	0	0
$31$	−8.58258	−1.54148	−0.770738	−	0.637152i	$-0.780112\pi$
$31$	−8.58258	−1.54148	−0.770738	+	0.637152i	$0.780112\pi$
$32$	−4.73930	−0.837798
$33$	0	0
$34$	0.791288	0.135705
$35$	0	0
$36$	−1.79129	−0.298548
$37$	−7.58258	−1.24657	−0.623284	−	0.781996i	$-0.714202\pi$
$37$	−7.58258	−1.24657	−0.623284	+	0.781996i	$0.714202\pi$
$38$	1.58258	0.256728
$39$	0.913701	0.146309
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	−2.00000	−0.308607
$43$	9.66930	1.47456	0.737278	−	0.675590i	$-0.236111\pi$
$43$	9.66930	1.47456	0.737278	+	0.675590i	$0.236111\pi$
$44$	0	0
$45$	0	0
$46$	−0.266150	−0.0392417
$47$	−3.41742	−0.498483	−0.249241	−	0.968441i	$-0.580181\pi$
$47$	−3.41742	−0.498483	−0.249241	+	0.968441i	$0.580181\pi$
$48$	−2.79129	−0.402888
$49$	12.1652	1.73788
$50$	0	0
$51$	1.73205	0.242536
$52$	1.63670	0.226970
$53$	−12.1652	−1.67101	−0.835506	−	0.549481i	$-0.814825\pi$
$53$	−12.1652	−1.67101	−0.835506	+	0.549481i	$0.814825\pi$
$54$	0.456850	0.0621694
$55$	0	0
$56$	−7.58258	−1.01326
$57$	3.46410	0.458831
$58$	−4.41742	−0.580036
$59$	13.5826	1.76830	0.884150	−	0.467202i	$-0.154738\pi$
$59$	13.5826	1.76830	0.884150	+	0.467202i	$0.154738\pi$
$60$	0	0
$61$	−3.55945	−0.455741	−0.227871	−	0.973691i	$-0.573176\pi$
$61$	−3.55945	−0.455741	−0.227871	+	0.973691i	$0.573176\pi$
$62$	3.92095	0.497961
$63$	−4.37780	−0.551551
$64$	−3.41742	−0.427178
$65$	0	0
$66$	0	0
$67$	4.41742	0.539674	0.269837	−	0.962906i	$-0.413030\pi$
$67$	4.41742	0.539674	0.269837	+	0.962906i	$0.413030\pi$
$68$	3.10260	0.376246
$69$	−0.582576	−0.0701339
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.73205	0.204124
$73$	−5.10080	−0.597004	−0.298502	−	0.954409i	$-0.596487\pi$
$73$	−5.10080	−0.597004	−0.298502	+	0.954409i	$0.596487\pi$
$74$	3.46410	0.402694
$75$	0	0
$76$	6.20520	0.711786
$77$	0	0
$78$	−0.417424	−0.0472640
$79$	0.818350	0.0920716	0.0460358	−	0.998940i	$-0.485341\pi$
$79$	0.818350	0.0920716	0.0460358	+	0.998940i	$0.485341\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.58258	−0.174766
$83$	15.8745	1.74245	0.871227	−	0.490881i	$-0.163325\pi$
$83$	15.8745	1.74245	0.871227	+	0.490881i	$0.163325\pi$
$84$	−7.84190	−0.855622
$85$	0	0
$86$	−4.41742	−0.476343
$87$	−9.66930	−1.03666
$88$	0	0
$89$	14.7477	1.56326	0.781628	−	0.623745i	$-0.214390\pi$
$89$	14.7477	1.56326	0.781628	+	0.623745i	$0.214390\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	−1.04356	−0.108799
$93$	8.58258	0.889972
$94$	1.56125	0.161031
$95$	0	0
$96$	4.73930	0.483703
$97$	−4.41742	−0.448521	−0.224261	−	0.974529i	$-0.571997\pi$
$97$	−4.41742	−0.448521	−0.224261	+	0.974529i	$0.571997\pi$
$98$	−5.55765	−0.561408
$99$	0	0
$100$	0	0
$101$	1.82740	0.181833	0.0909166	−	0.995859i	$-0.471020\pi$
$101$	1.82740	0.181833	0.0909166	+	0.995859i	$0.471020\pi$
$102$	−0.791288	−0.0783492
$103$	−7.16515	−0.706003	−0.353002	−	0.935623i	$-0.614839\pi$
$103$	−7.16515	−0.706003	−0.353002	+	0.935623i	$0.614839\pi$
$104$	−1.58258	−0.155184
$105$	0	0
$106$	5.55765	0.539807
$107$	13.0381	1.26044	0.630218	−	0.776418i	$-0.282965\pi$
$107$	13.0381	1.26044	0.630218	+	0.776418i	$0.282965\pi$
$108$	1.79129	0.172367
$109$	−6.92820	−0.663602	−0.331801	−	0.943349i	$-0.607656\pi$
$109$	−6.92820	−0.663602	−0.331801	+	0.943349i	$0.607656\pi$
$110$	0	0
$111$	7.58258	0.719706
$112$	−12.2197	−1.15465
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	−1.58258	−0.148222
$115$	0	0
$116$	−17.3205	−1.60817
$117$	−0.913701	−0.0844716
$118$	−6.20520	−0.571235
$119$	7.58258	0.695094
$120$	0	0
$121$	0	0
$122$	1.62614	0.147223
$123$	−3.46410	−0.312348
$124$	15.3739	1.38061
$125$	0	0
$126$	2.00000	0.178174
$127$	−7.84190	−0.695856	−0.347928	−	0.937521i	$-0.613115\pi$
$127$	−7.84190	−0.695856	−0.347928	+	0.937521i	$0.613115\pi$
$128$	11.0399	0.975795
$129$	−9.66930	−0.851335
$130$	0	0
$131$	20.9753	1.83262	0.916311	−	0.400468i	$-0.131152\pi$
$131$	20.9753	1.83262	0.916311	+	0.400468i	$0.131152\pi$
$132$	0	0
$133$	15.1652	1.31499
$134$	−2.01810	−0.174337
$135$	0	0
$136$	−3.00000	−0.257248
$137$	19.3303	1.65150	0.825750	−	0.564037i	$-0.190752\pi$
$137$	19.3303	1.65150	0.825750	+	0.564037i	$0.190752\pi$
$138$	0.266150	0.0226562
$139$	7.74655	0.657054	0.328527	−	0.944495i	$-0.393448\pi$
$139$	7.74655	0.657054	0.328527	+	0.944495i	$0.393448\pi$
$140$	0	0
$141$	3.41742	0.287799
$142$	−3.65480	−0.306704
$143$	0	0
$144$	2.79129	0.232607
$145$	0	0
$146$	2.33030	0.192857
$147$	−12.1652	−1.00336
$148$	13.5826	1.11648
$149$	−2.74110	−0.224560	−0.112280	−	0.993677i	$-0.535815\pi$
$149$	−2.74110	−0.224560	−0.112280	+	0.993677i	$0.535815\pi$
$150$	0	0
$151$	7.74655	0.630406	0.315203	−	0.949024i	$-0.397927\pi$
$151$	7.74655	0.630406	0.315203	+	0.949024i	$0.397927\pi$
$152$	−6.00000	−0.486664
$153$	−1.73205	−0.140028
$154$	0	0
$155$	0	0
$156$	−1.63670	−0.131041
$157$	−10.7477	−0.857762	−0.428881	−	0.903361i	$-0.641092\pi$
$157$	−10.7477	−0.857762	−0.428881	+	0.903361i	$0.641092\pi$
$158$	−0.373864	−0.0297430
$159$	12.1652	0.964759
$160$	0	0
$161$	−2.55040	−0.201000
$162$	−0.456850	−0.0358935
$163$	20.7477	1.62509	0.812544	−	0.582900i	$-0.198082\pi$
$163$	20.7477	1.62509	0.812544	+	0.582900i	$0.198082\pi$
$164$	−6.20520	−0.484545
$165$	0	0
$166$	−7.25227	−0.562886
$167$	−4.47315	−0.346143	−0.173071	−	0.984909i	$-0.555369\pi$
$167$	−4.47315	−0.346143	−0.173071	+	0.984909i	$0.555369\pi$
$168$	7.58258	0.585008
$169$	−12.1652	−0.935781
$170$	0	0
$171$	−3.46410	−0.264906
$172$	−17.3205	−1.32068
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	4.41742	0.334884
$175$	0	0
$176$	0	0
$177$	−13.5826	−1.02093
$178$	−6.73750	−0.504997
$179$	7.58258	0.566748	0.283374	−	0.959009i	$-0.408546\pi$
$179$	7.58258	0.566748	0.283374	+	0.959009i	$0.408546\pi$
$180$	0	0
$181$	−5.16515	−0.383923	−0.191961	−	0.981402i	$-0.561485\pi$
$181$	−5.16515	−0.383923	−0.191961	+	0.981402i	$0.561485\pi$
$182$	−1.82740	−0.135456
$183$	3.55945	0.263122
$184$	1.00905	0.0743882
$185$	0	0
$186$	−3.92095	−0.287498
$187$	0	0
$188$	6.12159	0.446463
$189$	4.37780	0.318438
$190$	0	0
$191$	−22.7477	−1.64597	−0.822984	−	0.568065i	$-0.807692\pi$
$191$	−22.7477	−1.64597	−0.822984	+	0.568065i	$0.807692\pi$
$192$	3.41742	0.246631
$193$	−16.4068	−1.18099	−0.590494	−	0.807042i	$-0.701067\pi$
$193$	−16.4068	−1.18099	−0.590494	+	0.807042i	$0.701067\pi$
$194$	2.01810	0.144891
$195$	0	0
$196$	−21.7913	−1.55652
$197$	12.4104	0.884205	0.442102	−	0.896965i	$-0.354233\pi$
$197$	12.4104	0.884205	0.442102	+	0.896965i	$0.354233\pi$
$198$	0	0
$199$	−15.7477	−1.11633	−0.558163	−	0.829731i	$-0.688494\pi$
$199$	−15.7477	−1.11633	−0.558163	+	0.829731i	$0.688494\pi$
$200$	0	0
$201$	−4.41742	−0.311581
$202$	−0.834849	−0.0587397
$203$	−42.3303	−2.97100
$204$	−3.10260	−0.217226
$205$	0	0
$206$	3.27340	0.228069
$207$	0.582576	0.0404918
$208$	−2.55040	−0.176838
$209$	0	0
$210$	0	0
$211$	−23.4304	−1.61301	−0.806506	−	0.591226i	$-0.798644\pi$
$211$	−23.4304	−1.61301	−0.806506	+	0.591226i	$0.798644\pi$
$212$	21.7913	1.49663
$213$	−8.00000	−0.548151
$214$	−5.95644	−0.407174
$215$	0	0
$216$	−1.73205	−0.117851
$217$	37.5728	2.55061
$218$	3.16515	0.214371
$219$	5.10080	0.344680
$220$	0	0
$221$	1.58258	0.106456
$222$	−3.46410	−0.232495
$223$	16.3303	1.09356	0.546779	−	0.837277i	$-0.315854\pi$
$223$	16.3303	1.09356	0.546779	+	0.837277i	$0.315854\pi$
$224$	20.7477	1.38627
$225$	0	0
$226$	4.11165	0.273503
$227$	20.1570	1.33786	0.668932	−	0.743323i	$-0.266752\pi$
$227$	20.1570	1.33786	0.668932	+	0.743323i	$0.266752\pi$
$228$	−6.20520	−0.410950
$229$	17.0000	1.12339	0.561696	−	0.827344i	$-0.310149\pi$
$229$	17.0000	1.12339	0.561696	+	0.827344i	$0.310149\pi$
$230$	0	0
$231$	0	0
$232$	16.7477	1.09954
$233$	14.1425	0.926503	0.463252	−	0.886227i	$-0.346683\pi$
$233$	14.1425	0.926503	0.463252	+	0.886227i	$0.346683\pi$
$234$	0.417424	0.0272879
$235$	0	0
$236$	−24.3303	−1.58377
$237$	−0.818350	−0.0531576
$238$	−3.46410	−0.224544
$239$	11.1153	0.718989	0.359495	−	0.933147i	$-0.382949\pi$
$239$	11.1153	0.718989	0.359495	+	0.933147i	$0.382949\pi$
$240$	0	0
$241$	−26.1715	−1.68585	−0.842926	−	0.538029i	$-0.819169\pi$
$241$	−26.1715	−1.68585	−0.842926	+	0.538029i	$0.819169\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	6.37600	0.408182
$245$	0	0
$246$	1.58258	0.100901
$247$	3.16515	0.201394
$248$	−14.8655	−0.943957
$249$	−15.8745	−1.00601
$250$	0	0
$251$	−5.58258	−0.352369	−0.176185	−	0.984357i	$-0.556376\pi$
$251$	−5.58258	−0.352369	−0.176185	+	0.984357i	$0.556376\pi$
$252$	7.84190	0.493994
$253$	0	0
$254$	3.58258	0.224791
$255$	0	0
$256$	1.79129	0.111955
$257$	−5.83485	−0.363968	−0.181984	−	0.983302i	$-0.558252\pi$
$257$	−5.83485	−0.363968	−0.181984	+	0.983302i	$0.558252\pi$
$258$	4.41742	0.275017
$259$	33.1950	2.06264
$260$	0	0
$261$	9.66930	0.598515
$262$	−9.58258	−0.592014
$263$	11.4014	0.703038	0.351519	−	0.936181i	$-0.385665\pi$
$263$	11.4014	0.703038	0.351519	+	0.936181i	$0.385665\pi$
$264$	0	0
$265$	0	0
$266$	−6.92820	−0.424795
$267$	−14.7477	−0.902546
$268$	−7.91288	−0.483356
$269$	−8.83485	−0.538670	−0.269335	−	0.963047i	$-0.586804\pi$
$269$	−8.83485	−0.538670	−0.269335	+	0.963047i	$0.586804\pi$
$270$	0	0
$271$	16.6929	1.01402	0.507009	−	0.861940i	$-0.330751\pi$
$271$	16.6929	1.01402	0.507009	+	0.861940i	$0.330751\pi$
$272$	−4.83465	−0.293144
$273$	−4.00000	−0.242091
$274$	−8.83105	−0.533503
$275$	0	0
$276$	1.04356	0.0628150
$277$	1.82740	0.109798	0.0548989	−	0.998492i	$-0.482516\pi$
$277$	1.82740	0.109798	0.0548989	+	0.998492i	$0.482516\pi$
$278$	−3.53901	−0.212256
$279$	−8.58258	−0.513825
$280$	0	0
$281$	−23.7164	−1.41480	−0.707401	−	0.706812i	$-0.750133\pi$
$281$	−23.7164	−1.41480	−0.707401	+	0.706812i	$0.750133\pi$
$282$	−1.56125	−0.0929712
$283$	10.7737	0.640430	0.320215	−	0.947345i	$-0.396245\pi$
$283$	10.7737	0.640430	0.320215	+	0.947345i	$0.396245\pi$
$284$	−14.3303	−0.850347
$285$	0	0
$286$	0	0
$287$	−15.1652	−0.895171
$288$	−4.73930	−0.279266
$289$	−14.0000	−0.823529
$290$	0	0
$291$	4.41742	0.258954
$292$	9.13701	0.534703
$293$	−1.73205	−0.101187	−0.0505937	−	0.998719i	$-0.516111\pi$
$293$	−1.73205	−0.101187	−0.0505937	+	0.998719i	$0.516111\pi$
$294$	5.55765	0.324129
$295$	0	0
$296$	−13.1334	−0.763364
$297$	0	0
$298$	1.25227	0.0725422
$299$	−0.532300	−0.0307837
$300$	0	0
$301$	−42.3303	−2.43988
$302$	−3.53901	−0.203647
$303$	−1.82740	−0.104981
$304$	−9.66930	−0.554573
$305$	0	0
$306$	0.791288	0.0452349
$307$	15.4931	0.884238	0.442119	−	0.896956i	$-0.354227\pi$
$307$	15.4931	0.884238	0.442119	+	0.896956i	$0.354227\pi$
$308$	0	0
$309$	7.16515	0.407611
$310$	0	0
$311$	−11.5826	−0.656788	−0.328394	−	0.944541i	$-0.606507\pi$
$311$	−11.5826	−0.656788	−0.328394	+	0.944541i	$0.606507\pi$
$312$	1.58258	0.0895957
$313$	−5.58258	−0.315546	−0.157773	−	0.987475i	$-0.550431\pi$
$313$	−5.58258	−0.315546	−0.157773	+	0.987475i	$0.550431\pi$
$314$	4.91010	0.277093
$315$	0	0
$316$	−1.46590	−0.0824634
$317$	−30.1652	−1.69424	−0.847122	−	0.531399i	$-0.821667\pi$
$317$	−30.1652	−1.69424	−0.847122	+	0.531399i	$0.821667\pi$
$318$	−5.55765	−0.311658
$319$	0	0
$320$	0	0
$321$	−13.0381	−0.727713
$322$	1.16515	0.0649313
$323$	6.00000	0.333849
$324$	−1.79129	−0.0995160
$325$	0	0
$326$	−9.47860	−0.524971
$327$	6.92820	0.383131
$328$	6.00000	0.331295
$329$	14.9608	0.824816
$330$	0	0
$331$	27.7477	1.52515	0.762577	−	0.646898i	$-0.223934\pi$
$331$	27.7477	1.52515	0.762577	+	0.646898i	$0.223934\pi$
$332$	−28.4358	−1.56062
$333$	−7.58258	−0.415523
$334$	2.04356	0.111819
$335$	0	0
$336$	12.2197	0.666640
$337$	−0.190700	−0.0103881	−0.00519406	−	0.999987i	$-0.501653\pi$
$337$	−0.190700	−0.0103881	−0.00519406	+	0.999987i	$0.501653\pi$
$338$	5.55765	0.302296
$339$	9.00000	0.488813
$340$	0	0
$341$	0	0
$342$	1.58258	0.0855759
$343$	−22.6120	−1.22093
$344$	16.7477	0.902977
$345$	0	0
$346$	−3.16515	−0.170160
$347$	−21.9844	−1.18018	−0.590091	−	0.807337i	$-0.700908\pi$
$347$	−21.9844	−1.18018	−0.590091	+	0.807337i	$0.700908\pi$
$348$	17.3205	0.928477
$349$	−25.9808	−1.39072	−0.695359	−	0.718662i	$-0.744755\pi$
$349$	−25.9808	−1.39072	−0.695359	+	0.718662i	$0.744755\pi$
$350$	0	0
$351$	0.913701	0.0487697
$352$	0	0
$353$	−17.8348	−0.949253	−0.474627	−	0.880187i	$-0.657417\pi$
$353$	−17.8348	−0.949253	−0.474627	+	0.880187i	$0.657417\pi$
$354$	6.20520	0.329803
$355$	0	0
$356$	−26.4174	−1.40012
$357$	−7.58258	−0.401312
$358$	−3.46410	−0.183083
$359$	8.75560	0.462103	0.231052	−	0.972942i	$-0.425783\pi$
$359$	8.75560	0.462103	0.231052	+	0.972942i	$0.425783\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	2.35970	0.124023
$363$	0	0
$364$	−7.16515	−0.375556
$365$	0	0
$366$	−1.62614	−0.0849995
$367$	1.25227	0.0653681	0.0326841	−	0.999466i	$-0.489594\pi$
$367$	1.25227	0.0653681	0.0326841	+	0.999466i	$0.489594\pi$
$368$	1.62614	0.0847682
$369$	3.46410	0.180334
$370$	0	0
$371$	53.2566	2.76495
$372$	−15.3739	−0.797098
$373$	13.8564	0.717458	0.358729	−	0.933442i	$-0.383210\pi$
$373$	13.8564	0.717458	0.358729	+	0.933442i	$0.383210\pi$
$374$	0	0
$375$	0	0
$376$	−5.91915	−0.305257
$377$	−8.83485	−0.455018
$378$	−2.00000	−0.102869
$379$	−15.7477	−0.808906	−0.404453	−	0.914559i	$-0.632538\pi$
$379$	−15.7477	−0.808906	−0.404453	+	0.914559i	$0.632538\pi$
$380$	0	0
$381$	7.84190	0.401753
$382$	10.3923	0.531717
$383$	−6.33030	−0.323463	−0.161732	−	0.986835i	$-0.551708\pi$
$383$	−6.33030	−0.323463	−0.161732	+	0.986835i	$0.551708\pi$
$384$	−11.0399	−0.563375
$385$	0	0
$386$	7.49545	0.381509
$387$	9.66930	0.491518
$388$	7.91288	0.401716
$389$	4.74773	0.240719	0.120360	−	0.992730i	$-0.461595\pi$
$389$	4.74773	0.240719	0.120360	+	0.992730i	$0.461595\pi$
$390$	0	0
$391$	−1.00905	−0.0510299
$392$	21.0707	1.06423
$393$	−20.9753	−1.05806
$394$	−5.66970	−0.285635
$395$	0	0
$396$	0	0
$397$	−32.3303	−1.62261	−0.811306	−	0.584622i	$-0.801243\pi$
$397$	−32.3303	−1.62261	−0.811306	+	0.584622i	$0.801243\pi$
$398$	7.19435	0.360620
$399$	−15.1652	−0.759207
$400$	0	0
$401$	14.3303	0.715621	0.357811	−	0.933794i	$-0.383523\pi$
$401$	14.3303	0.715621	0.357811	+	0.933794i	$0.383523\pi$
$402$	2.01810	0.100654
$403$	7.84190	0.390633
$404$	−3.27340	−0.162858
$405$	0	0
$406$	19.3386	0.959759
$407$	0	0
$408$	3.00000	0.148522
$409$	19.4340	0.960947	0.480474	−	0.877009i	$-0.340465\pi$
$409$	19.4340	0.960947	0.480474	+	0.877009i	$0.340465\pi$
$410$	0	0
$411$	−19.3303	−0.953494
$412$	12.8348	0.632328
$413$	−59.4618	−2.92593
$414$	−0.266150	−0.0130806
$415$	0	0
$416$	4.33030	0.212311
$417$	−7.74655	−0.379350
$418$	0	0
$419$	−4.33030	−0.211549	−0.105775	−	0.994390i	$-0.533732\pi$
$419$	−4.33030	−0.211549	−0.105775	+	0.994390i	$0.533732\pi$
$420$	0	0
$421$	−39.3303	−1.91684	−0.958421	−	0.285359i	$-0.907887\pi$
$421$	−39.3303	−1.91684	−0.958421	+	0.285359i	$0.907887\pi$
$422$	10.7042	0.521071
$423$	−3.41742	−0.166161
$424$	−21.0707	−1.02328
$425$	0	0
$426$	3.65480	0.177076
$427$	15.5826	0.754094
$428$	−23.3549	−1.12890
$429$	0	0
$430$	0	0
$431$	−4.18710	−0.201686	−0.100843	−	0.994902i	$-0.532154\pi$
$431$	−4.18710	−0.201686	−0.100843	+	0.994902i	$0.532154\pi$
$432$	−2.79129	−0.134296
$433$	13.5826	0.652737	0.326368	−	0.945243i	$-0.394175\pi$
$433$	13.5826	0.652737	0.326368	+	0.945243i	$0.394175\pi$
$434$	−17.1652	−0.823954
$435$	0	0
$436$	12.4104	0.594351
$437$	−2.01810	−0.0965389
$438$	−2.33030	−0.111346
$439$	−4.28245	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$439$	−4.28245	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$440$	0	0
$441$	12.1652	0.579293
$442$	−0.723000	−0.0343896
$443$	11.1652	0.530472	0.265236	−	0.964183i	$-0.414550\pi$
$443$	11.1652	0.530472	0.265236	+	0.964183i	$0.414550\pi$
$444$	−13.5826	−0.644601
$445$	0	0
$446$	−7.46050	−0.353265
$447$	2.74110	0.129650
$448$	14.9608	0.706832
$449$	21.1652	0.998845	0.499423	−	0.866358i	$-0.333546\pi$
$449$	21.1652	0.998845	0.499423	+	0.866358i	$0.333546\pi$
$450$	0	0
$451$	0	0
$452$	16.1216	0.758296
$453$	−7.74655	−0.363965
$454$	−9.20871	−0.432186
$455$	0	0
$456$	6.00000	0.280976
$457$	−33.0043	−1.54388	−0.771938	−	0.635697i	$-0.780713\pi$
$457$	−33.0043	−1.54388	−0.771938	+	0.635697i	$0.780713\pi$
$458$	−7.76645	−0.362903
$459$	1.73205	0.0808452
$460$	0	0
$461$	18.4249	0.858134	0.429067	−	0.903273i	$-0.358842\pi$
$461$	18.4249	0.858134	0.429067	+	0.903273i	$0.358842\pi$
$462$	0	0
$463$	−19.4955	−0.906031	−0.453015	−	0.891503i	$-0.649652\pi$
$463$	−19.4955	−0.906031	−0.453015	+	0.891503i	$0.649652\pi$
$464$	26.9898	1.25297
$465$	0	0
$466$	−6.46099	−0.299299
$467$	12.5826	0.582252	0.291126	−	0.956685i	$-0.405970\pi$
$467$	12.5826	0.582252	0.291126	+	0.956685i	$0.405970\pi$
$468$	1.63670	0.0756565
$469$	−19.3386	−0.892974
$470$	0	0
$471$	10.7477	0.495229
$472$	23.5257	1.08286
$473$	0	0
$474$	0.373864	0.0171721
$475$	0	0
$476$	−13.5826	−0.622556
$477$	−12.1652	−0.557004
$478$	−5.07803	−0.232264
$479$	−30.8353	−1.40890	−0.704451	−	0.709753i	$-0.748807\pi$
$479$	−30.8353	−1.40890	−0.704451	+	0.709753i	$0.748807\pi$
$480$	0	0
$481$	6.92820	0.315899
$482$	11.9564	0.544601
$483$	2.55040	0.116047
$484$	0	0
$485$	0	0
$486$	0.456850	0.0207231
$487$	5.58258	0.252971	0.126485	−	0.991968i	$-0.459630\pi$
$487$	5.58258	0.252971	0.126485	+	0.991968i	$0.459630\pi$
$488$	−6.16515	−0.279083
$489$	−20.7477	−0.938245
$490$	0	0
$491$	−27.7128	−1.25066	−0.625331	−	0.780360i	$-0.715036\pi$
$491$	−27.7128	−1.25066	−0.625331	+	0.780360i	$0.715036\pi$
$492$	6.20520	0.279752
$493$	−16.7477	−0.754280
$494$	−1.44600	−0.0650586
$495$	0	0
$496$	−23.9564	−1.07568
$497$	−35.0224	−1.57097
$498$	7.25227	0.324982
$499$	−10.3303	−0.462448	−0.231224	−	0.972901i	$-0.574273\pi$
$499$	−10.3303	−0.462448	−0.231224	+	0.972901i	$0.574273\pi$
$500$	0	0
$501$	4.47315	0.199846
$502$	2.55040	0.113830
$503$	6.10985	0.272425	0.136212	−	0.990680i	$-0.456507\pi$
$503$	6.10985	0.272425	0.136212	+	0.990680i	$0.456507\pi$
$504$	−7.58258	−0.337755
$505$	0	0
$506$	0	0
$507$	12.1652	0.540273
$508$	14.0471	0.623240
$509$	−26.7477	−1.18557	−0.592786	−	0.805360i	$-0.701972\pi$
$509$	−26.7477	−1.18557	−0.592786	+	0.805360i	$0.701972\pi$
$510$	0	0
$511$	22.3303	0.987834
$512$	−22.8981	−1.01196
$513$	3.46410	0.152944
$514$	2.66565	0.117577
$515$	0	0
$516$	17.3205	0.762493
$517$	0	0
$518$	−15.1652	−0.666318
$519$	−6.92820	−0.304114
$520$	0	0
$521$	36.7477	1.60995	0.804974	−	0.593311i	$-0.202179\pi$
$521$	36.7477	1.60995	0.804974	+	0.593311i	$0.202179\pi$
$522$	−4.41742	−0.193345
$523$	21.8890	0.957140	0.478570	−	0.878050i	$-0.341155\pi$
$523$	21.8890	0.957140	0.478570	+	0.878050i	$0.341155\pi$
$524$	−37.5728	−1.64138
$525$	0	0
$526$	−5.20871	−0.227111
$527$	14.8655	0.647549
$528$	0	0
$529$	−22.6606	−0.985244
$530$	0	0
$531$	13.5826	0.589434
$532$	−27.1652	−1.17776
$533$	−3.16515	−0.137098
$534$	6.73750	0.291560
$535$	0	0
$536$	7.65120	0.330482
$537$	−7.58258	−0.327212
$538$	4.03620	0.174013
$539$	0	0
$540$	0	0
$541$	−22.9934	−0.988564	−0.494282	−	0.869302i	$-0.664569\pi$
$541$	−22.9934	−0.988564	−0.494282	+	0.869302i	$0.664569\pi$
$542$	−7.62614	−0.327571
$543$	5.16515	0.221658
$544$	8.20871	0.351946
$545$	0	0
$546$	1.82740	0.0782055
$547$	−41.0369	−1.75461	−0.877306	−	0.479931i	$-0.840662\pi$
$547$	−41.0369	−1.75461	−0.877306	+	0.479931i	$0.840662\pi$
$548$	−34.6261	−1.47916
$549$	−3.55945	−0.151914
$550$	0	0
$551$	−33.4955	−1.42695
$552$	−1.00905	−0.0429481
$553$	−3.58258	−0.152347
$554$	−0.834849	−0.0354693
$555$	0	0
$556$	−13.8763	−0.588487
$557$	8.46955	0.358867	0.179433	−	0.983770i	$-0.442574\pi$
$557$	8.46955	0.358867	0.179433	+	0.983770i	$0.442574\pi$
$558$	3.92095	0.165987
$559$	−8.83485	−0.373674
$560$	0	0
$561$	0	0
$562$	10.8348	0.457041
$563$	43.5873	1.83699	0.918493	−	0.395437i	$-0.129407\pi$
$563$	43.5873	1.83699	0.918493	+	0.395437i	$0.129407\pi$
$564$	−6.12159	−0.257765
$565$	0	0
$566$	−4.92197	−0.206886
$567$	−4.37780	−0.183850
$568$	13.8564	0.581402
$569$	−37.3821	−1.56714	−0.783570	−	0.621304i	$-0.786603\pi$
$569$	−37.3821	−1.56714	−0.783570	+	0.621304i	$0.786603\pi$
$570$	0	0
$571$	0.627650	0.0262663	0.0131332	−	0.999914i	$-0.495819\pi$
$571$	0.627650	0.0262663	0.0131332	+	0.999914i	$0.495819\pi$
$572$	0	0
$573$	22.7477	0.950300
$574$	6.92820	0.289178
$575$	0	0
$576$	−3.41742	−0.142393
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	6.39590	0.266035
$579$	16.4068	0.681844
$580$	0	0
$581$	−69.4955	−2.88316
$582$	−2.01810	−0.0836530
$583$	0	0
$584$	−8.83485	−0.365589
$585$	0	0
$586$	0.791288	0.0326878
$587$	−7.74773	−0.319783	−0.159891	−	0.987135i	$-0.551114\pi$
$587$	−7.74773	−0.319783	−0.159891	+	0.987135i	$0.551114\pi$
$588$	21.7913	0.898658
$589$	29.7309	1.22504
$590$	0	0
$591$	−12.4104	−0.510496
$592$	−21.1652	−0.869882
$593$	27.7128	1.13803	0.569014	−	0.822328i	$-0.307325\pi$
$593$	27.7128	1.13803	0.569014	+	0.822328i	$0.307325\pi$
$594$	0	0
$595$	0	0
$596$	4.91010	0.201126
$597$	15.7477	0.644512
$598$	0.243181	0.00994442
$599$	6.74773	0.275705	0.137852	−	0.990453i	$-0.455980\pi$
$599$	6.74773	0.275705	0.137852	+	0.990453i	$0.455980\pi$
$600$	0	0
$601$	1.44600	0.0589836	0.0294918	−	0.999565i	$-0.490611\pi$
$601$	1.44600	0.0589836	0.0294918	+	0.999565i	$0.490611\pi$
$602$	19.3386	0.788183
$603$	4.41742	0.179891
$604$	−13.8763	−0.564619
$605$	0	0
$606$	0.834849	0.0339134
$607$	−25.5438	−1.03679	−0.518396	−	0.855141i	$-0.673471\pi$
$607$	−25.5438	−1.03679	−0.518396	+	0.855141i	$0.673471\pi$
$608$	16.4174	0.665814
$609$	42.3303	1.71531
$610$	0	0
$611$	3.12250	0.126323
$612$	3.10260	0.125415
$613$	−1.82740	−0.0738080	−0.0369040	−	0.999319i	$-0.511750\pi$
$613$	−1.82740	−0.0738080	−0.0369040	+	0.999319i	$0.511750\pi$
$614$	−7.07803	−0.285646
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−3.27340	−0.131676
$619$	13.4955	0.542428	0.271214	−	0.962519i	$-0.412575\pi$
$619$	13.4955	0.542428	0.271214	+	0.962519i	$0.412575\pi$
$620$	0	0
$621$	−0.582576	−0.0233780
$622$	5.29150	0.212170
$623$	−64.5626	−2.58665
$624$	2.55040	0.102098
$625$	0	0
$626$	2.55040	0.101935
$627$	0	0
$628$	19.2523	0.768249
$629$	13.1334	0.523663
$630$	0	0
$631$	−21.4174	−0.852614	−0.426307	−	0.904578i	$-0.640186\pi$
$631$	−21.4174	−0.852614	−0.426307	+	0.904578i	$0.640186\pi$
$632$	1.41742	0.0563821
$633$	23.4304	0.931273
$634$	13.7810	0.547312
$635$	0	0
$636$	−21.7913	−0.864081
$637$	−11.1153	−0.440404
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−6.74773	−0.266519	−0.133260	−	0.991081i	$-0.542544\pi$
$641$	−6.74773	−0.266519	−0.133260	+	0.991081i	$0.542544\pi$
$642$	5.95644	0.235082
$643$	39.4955	1.55755	0.778774	−	0.627304i	$-0.215842\pi$
$643$	39.4955	1.55755	0.778774	+	0.627304i	$0.215842\pi$
$644$	4.56850	0.180024
$645$	0	0
$646$	−2.74110	−0.107847
$647$	−29.7477	−1.16950	−0.584752	−	0.811212i	$-0.698808\pi$
$647$	−29.7477	−1.16950	−0.584752	+	0.811212i	$0.698808\pi$
$648$	1.73205	0.0680414
$649$	0	0
$650$	0	0
$651$	−37.5728	−1.47259
$652$	−37.1652	−1.45550
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	−3.16515	−0.123767
$655$	0	0
$656$	9.66930	0.377523
$657$	−5.10080	−0.199001
$658$	−6.83485	−0.266450
$659$	18.2342	0.710304	0.355152	−	0.934809i	$-0.384429\pi$
$659$	18.2342	0.710304	0.355152	+	0.934809i	$0.384429\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−12.6766	−0.492688
$663$	−1.58258	−0.0614621
$664$	27.4955	1.06703
$665$	0	0
$666$	3.46410	0.134231
$667$	5.63310	0.218115
$668$	8.01270	0.310021
$669$	−16.3303	−0.631366
$670$	0	0
$671$	0	0
$672$	−20.7477	−0.800361
$673$	−11.3060	−0.435814	−0.217907	−	0.975969i	$-0.569923\pi$
$673$	−11.3060	−0.435814	−0.217907	+	0.975969i	$0.569923\pi$
$674$	0.0871215	0.00335580
$675$	0	0
$676$	21.7913	0.838126
$677$	12.0290	0.462312	0.231156	−	0.972917i	$-0.425749\pi$
$677$	12.0290	0.462312	0.231156	+	0.972917i	$0.425749\pi$
$678$	−4.11165	−0.157907
$679$	19.3386	0.742148
$680$	0	0
$681$	−20.1570	−0.772416
$682$	0	0
$683$	−46.3303	−1.77278	−0.886390	−	0.462939i	$-0.846795\pi$
$683$	−46.3303	−1.77278	−0.886390	+	0.462939i	$0.846795\pi$
$684$	6.20520	0.237262
$685$	0	0
$686$	10.3303	0.394413
$687$	−17.0000	−0.648590
$688$	26.9898	1.02898
$689$	11.1153	0.423459
$690$	0	0
$691$	−0.582576	−0.0221622	−0.0110811	−	0.999939i	$-0.503527\pi$
$691$	−0.582576	−0.0221622	−0.0110811	+	0.999939i	$0.503527\pi$
$692$	−12.4104	−0.471773
$693$	0	0
$694$	10.0436	0.381248
$695$	0	0
$696$	−16.7477	−0.634821
$697$	−6.00000	−0.227266
$698$	11.8693	0.449260
$699$	−14.1425	−0.534917
$700$	0	0
$701$	−21.8890	−0.826737	−0.413368	−	0.910564i	$-0.635648\pi$
$701$	−21.8890	−0.826737	−0.413368	+	0.910564i	$0.635648\pi$
$702$	−0.417424	−0.0157547
$703$	26.2668	0.990672
$704$	0	0
$705$	0	0
$706$	8.14786	0.306649
$707$	−8.00000	−0.300871
$708$	24.3303	0.914389
$709$	−45.0000	−1.69001	−0.845005	−	0.534758i	$-0.820403\pi$
$709$	−45.0000	−1.69001	−0.845005	+	0.534758i	$0.820403\pi$
$710$	0	0
$711$	0.818350	0.0306905
$712$	25.5438	0.957295
$713$	−5.00000	−0.187251
$714$	3.46410	0.129641
$715$	0	0
$716$	−13.5826	−0.507605
$717$	−11.1153	−0.415109
$718$	−4.00000	−0.149279
$719$	−16.3303	−0.609018	−0.304509	−	0.952510i	$-0.598492\pi$
$719$	−16.3303	−0.609018	−0.304509	+	0.952510i	$0.598492\pi$
$720$	0	0
$721$	31.3676	1.16819
$722$	3.19795	0.119015
$723$	26.1715	0.973327
$724$	9.25227	0.343858
$725$	0	0
$726$	0	0
$727$	5.16515	0.191565	0.0957824	−	0.995402i	$-0.469465\pi$
$727$	5.16515	0.191565	0.0957824	+	0.995402i	$0.469465\pi$
$728$	6.92820	0.256776
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.7477	−0.619437
$732$	−6.37600	−0.235664
$733$	13.6657	0.504754	0.252377	−	0.967629i	$-0.418788\pi$
$733$	13.6657	0.504754	0.252377	+	0.967629i	$0.418788\pi$
$734$	−0.572101	−0.0211166
$735$	0	0
$736$	−2.76100	−0.101772
$737$	0	0
$738$	−1.58258	−0.0582554
$739$	−21.9844	−0.808708	−0.404354	−	0.914603i	$-0.632504\pi$
$739$	−21.9844	−0.808708	−0.404354	+	0.914603i	$0.632504\pi$
$740$	0	0
$741$	−3.16515	−0.116275
$742$	−24.3303	−0.893194
$743$	12.6567	0.464328	0.232164	−	0.972677i	$-0.425419\pi$
$743$	12.6567	0.464328	0.232164	+	0.972677i	$0.425419\pi$
$744$	14.8655	0.544994
$745$	0	0
$746$	−6.33030	−0.231769
$747$	15.8745	0.580818
$748$	0	0
$749$	−57.0780	−2.08559
$750$	0	0
$751$	20.0780	0.732658	0.366329	−	0.930485i	$-0.380615\pi$
$751$	20.0780	0.732658	0.366329	+	0.930485i	$0.380615\pi$
$752$	−9.53901	−0.347852
$753$	5.58258	0.203440
$754$	4.03620	0.146990
$755$	0	0
$756$	−7.84190	−0.285207
$757$	−6.83485	−0.248417	−0.124208	−	0.992256i	$-0.539639\pi$
$757$	−6.83485	−0.248417	−0.124208	+	0.992256i	$0.539639\pi$
$758$	7.19435	0.261311
$759$	0	0
$760$	0	0
$761$	14.7701	0.535416	0.267708	−	0.963500i	$-0.413734\pi$
$761$	14.7701	0.535416	0.267708	+	0.963500i	$0.413734\pi$
$762$	−3.58258	−0.129783
$763$	30.3303	1.09803
$764$	40.7477	1.47420
$765$	0	0
$766$	2.89200	0.104492
$767$	−12.4104	−0.448114
$768$	−1.79129	−0.0646375
$769$	45.5101	1.64114	0.820568	−	0.571550i	$-0.193657\pi$
$769$	45.5101	1.64114	0.820568	+	0.571550i	$0.193657\pi$
$770$	0	0
$771$	5.83485	0.210137
$772$	29.3893	1.05774
$773$	−48.4955	−1.74426	−0.872130	−	0.489274i	$-0.837262\pi$
$773$	−48.4955	−1.74426	−0.872130	+	0.489274i	$0.837262\pi$
$774$	−4.41742	−0.158781
$775$	0	0
$776$	−7.65120	−0.274662
$777$	−33.1950	−1.19086
$778$	−2.16900	−0.0777624
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	0.460985	0.0164848
$783$	−9.66930	−0.345553
$784$	33.9564	1.21273
$785$	0	0
$786$	9.58258	0.341799
$787$	−14.2378	−0.507523	−0.253762	−	0.967267i	$-0.581668\pi$
$787$	−14.2378	−0.507523	−0.253762	+	0.967267i	$0.581668\pi$
$788$	−22.2306	−0.791933
$789$	−11.4014	−0.405899
$790$	0	0
$791$	39.4002	1.40091
$792$	0	0
$793$	3.25227	0.115492
$794$	14.7701	0.524171
$795$	0	0
$796$	28.2087	0.999831
$797$	4.33030	0.153387	0.0766936	−	0.997055i	$-0.475564\pi$
$797$	4.33030	0.153387	0.0766936	+	0.997055i	$0.475564\pi$
$798$	6.92820	0.245256
$799$	5.91915	0.209405
$800$	0	0
$801$	14.7477	0.521085
$802$	−6.54680	−0.231176
$803$	0	0
$804$	7.91288	0.279066
$805$	0	0
$806$	−3.58258	−0.126191
$807$	8.83485	0.311001
$808$	3.16515	0.111350
$809$	−8.75560	−0.307831	−0.153915	−	0.988084i	$-0.549188\pi$
$809$	−8.75560	−0.307831	−0.153915	+	0.988084i	$0.549188\pi$
$810$	0	0
$811$	54.7980	1.92422	0.962109	−	0.272667i	$-0.0879056\pi$
$811$	54.7980	1.92422	0.962109	+	0.272667i	$0.0879056\pi$
$812$	75.8258	2.66096
$813$	−16.6929	−0.585444
$814$	0	0
$815$	0	0
$816$	4.83465	0.169247
$817$	−33.4955	−1.17186
$818$	−8.87841	−0.310426
$819$	4.00000	0.139771
$820$	0	0
$821$	18.7665	0.654956	0.327478	−	0.944859i	$-0.393801\pi$
$821$	18.7665	0.654956	0.327478	+	0.944859i	$0.393801\pi$
$822$	8.83105	0.308018
$823$	19.1652	0.668055	0.334028	−	0.942563i	$-0.391592\pi$
$823$	19.1652	0.668055	0.334028	+	0.942563i	$0.391592\pi$
$824$	−12.4104	−0.432337
$825$	0	0
$826$	27.1652	0.945197
$827$	5.67290	0.197266	0.0986331	−	0.995124i	$-0.468553\pi$
$827$	5.67290	0.197266	0.0986331	+	0.995124i	$0.468553\pi$
$828$	−1.04356	−0.0362662
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	0	0
$831$	−1.82740	−0.0633918
$832$	3.12250	0.108253
$833$	−21.0707	−0.730055
$834$	3.53901	0.122546
$835$	0	0
$836$	0	0
$837$	8.58258	0.296657
$838$	1.97830	0.0683392
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	64.4955	2.22398
$842$	17.9681	0.619220
$843$	23.7164	0.816837
$844$	41.9705	1.44468
$845$	0	0
$846$	1.56125	0.0536769
$847$	0	0
$848$	−33.9564	−1.16607
$849$	−10.7737	−0.369753
$850$	0	0
$851$	−4.41742	−0.151427
$852$	14.3303	0.490948
$853$	21.3567	0.731240	0.365620	−	0.930764i	$-0.380857\pi$
$853$	21.3567	0.731240	0.365620	+	0.930764i	$0.380857\pi$
$854$	−7.11890	−0.243604
$855$	0	0
$856$	22.5826	0.771857
$857$	6.83285	0.233406	0.116703	−	0.993167i	$-0.462767\pi$
$857$	6.83285	0.233406	0.116703	+	0.993167i	$0.462767\pi$
$858$	0	0
$859$	48.6606	1.66028	0.830139	−	0.557556i	$-0.188261\pi$
$859$	48.6606	1.66028	0.830139	+	0.557556i	$0.188261\pi$
$860$	0	0
$861$	15.1652	0.516827
$862$	1.91288	0.0651529
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	4.73930	0.161234
$865$	0	0
$866$	−6.20520	−0.210861
$867$	14.0000	0.475465
$868$	−67.3037	−2.28444
$869$	0	0
$870$	0	0
$871$	−4.03620	−0.136762
$872$	−12.0000	−0.406371
$873$	−4.41742	−0.149507
$874$	0.921970	0.0311861
$875$	0	0
$876$	−9.13701	−0.308711
$877$	20.5939	0.695407	0.347703	−	0.937605i	$-0.386962\pi$
$877$	20.5939	0.695407	0.347703	+	0.937605i	$0.386962\pi$
$878$	1.95644	0.0660266
$879$	1.73205	0.0584206
$880$	0	0
$881$	−29.4955	−0.993727	−0.496864	−	0.867829i	$-0.665515\pi$
$881$	−29.4955	−0.993727	−0.496864	+	0.867829i	$0.665515\pi$
$882$	−5.55765	−0.187136
$883$	−34.2432	−1.15237	−0.576187	−	0.817318i	$-0.695460\pi$
$883$	−34.2432	−1.15237	−0.576187	+	0.817318i	$0.695460\pi$
$884$	−2.83485	−0.0953463
$885$	0	0
$886$	−5.10080	−0.171365
$887$	−29.7309	−0.998266	−0.499133	−	0.866525i	$-0.666348\pi$
$887$	−29.7309	−0.998266	−0.499133	+	0.866525i	$0.666348\pi$
$888$	13.1334	0.440728
$889$	34.3303	1.15140
$890$	0	0
$891$	0	0
$892$	−29.2523	−0.979439
$893$	11.8383	0.396154
$894$	−1.25227	−0.0418823
$895$	0	0
$896$	−48.3303	−1.61460
$897$	0.532300	0.0177730
$898$	−9.66930	−0.322669
$899$	−82.9875	−2.76779
$900$	0	0
$901$	21.0707	0.701965
$902$	0	0
$903$	42.3303	1.40866
$904$	−15.5885	−0.518464
$905$	0	0
$906$	3.53901	0.117576
$907$	−10.8348	−0.359765	−0.179883	−	0.983688i	$-0.557572\pi$
$907$	−10.8348	−0.359765	−0.179883	+	0.983688i	$0.557572\pi$
$908$	−36.1069	−1.19825
$909$	1.82740	0.0606111
$910$	0	0
$911$	−17.2523	−0.571593	−0.285797	−	0.958290i	$-0.592258\pi$
$911$	−17.2523	−0.571593	−0.285797	+	0.958290i	$0.592258\pi$
$912$	9.66930	0.320183
$913$	0	0
$914$	15.0780	0.498737
$915$	0	0
$916$	−30.4519	−1.00616
$917$	−91.8258	−3.03235
$918$	−0.791288	−0.0261164
$919$	−6.73750	−0.222250	−0.111125	−	0.993806i	$-0.535445\pi$
$919$	−6.73750	−0.222250	−0.111125	+	0.993806i	$0.535445\pi$
$920$	0	0
$921$	−15.4931	−0.510515
$922$	−8.41742	−0.277213
$923$	−7.30960	−0.240599
$924$	0	0
$925$	0	0
$926$	8.90650	0.292686
$927$	−7.16515	−0.235334
$928$	−45.8258	−1.50430
$929$	20.3303	0.667016	0.333508	−	0.942747i	$-0.391768\pi$
$929$	20.3303	0.667016	0.333508	+	0.942747i	$0.391768\pi$
$930$	0	0
$931$	−42.1413	−1.38113
$932$	−25.3332	−0.829817
$933$	11.5826	0.379197
$934$	−5.74835	−0.188092
$935$	0	0
$936$	−1.58258	−0.0517281
$937$	−9.86001	−0.322112	−0.161056	−	0.986945i	$-0.551490\pi$
$937$	−9.86001	−0.322112	−0.161056	+	0.986945i	$0.551490\pi$
$938$	8.83485	0.288468
$939$	5.58258	0.182180
$940$	0	0
$941$	−41.0369	−1.33777	−0.668883	−	0.743368i	$-0.733227\pi$
$941$	−41.0369	−1.33777	−0.668883	+	0.743368i	$0.733227\pi$
$942$	−4.91010	−0.159980
$943$	2.01810	0.0657184
$944$	37.9129	1.23396
$945$	0	0
$946$	0	0
$947$	24.9129	0.809560	0.404780	−	0.914414i	$-0.367348\pi$
$947$	24.9129	0.809560	0.404780	+	0.914414i	$0.367348\pi$
$948$	1.46590	0.0476102
$949$	4.66061	0.151290
$950$	0	0
$951$	30.1652	0.978172
$952$	13.1334	0.425656
$953$	−20.7846	−0.673280	−0.336640	−	0.941634i	$-0.609290\pi$
$953$	−20.7846	−0.673280	−0.336640	+	0.941634i	$0.609290\pi$
$954$	5.55765	0.179936
$955$	0	0
$956$	−19.9107	−0.643958
$957$	0	0
$958$	14.0871	0.455134
$959$	−84.6242	−2.73266
$960$	0	0
$961$	42.6606	1.37615
$962$	−3.16515	−0.102049
$963$	13.0381	0.420145
$964$	46.8806	1.50992
$965$	0	0
$966$	−1.16515	−0.0374881
$967$	4.71940	0.151766	0.0758829	−	0.997117i	$-0.475822\pi$
$967$	4.71940	0.151766	0.0758829	+	0.997117i	$0.475822\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	25.5826	0.820984	0.410492	−	0.911864i	$-0.365357\pi$
$971$	25.5826	0.820984	0.410492	+	0.911864i	$0.365357\pi$
$972$	1.79129	0.0574556
$973$	−33.9129	−1.08720
$974$	−2.55040	−0.0817201
$975$	0	0
$976$	−9.93545	−0.318026
$977$	−12.1652	−0.389198	−0.194599	−	0.980883i	$-0.562340\pi$
$977$	−12.1652	−0.389198	−0.194599	+	0.980883i	$0.562340\pi$
$978$	9.47860	0.303092
$979$	0	0
$980$	0	0
$981$	−6.92820	−0.221201
$982$	12.6606	0.404016
$983$	−21.4174	−0.683110	−0.341555	−	0.939862i	$-0.610954\pi$
$983$	−21.4174	−0.683110	−0.341555	+	0.939862i	$0.610954\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	7.65120	0.243664
$987$	−14.9608	−0.476208
$988$	−5.66970	−0.180377
$989$	5.63310	0.179122
$990$	0	0
$991$	−37.2432	−1.18307	−0.591534	−	0.806280i	$-0.701478\pi$
$991$	−37.2432	−1.18307	−0.591534	+	0.806280i	$0.701478\pi$
$992$	40.6754	1.29145
$993$	−27.7477	−0.880548
$994$	16.0000	0.507489
$995$	0	0
$996$	28.4358	0.901023
$997$	41.4183	1.31173	0.655866	−	0.754878i	$-0.272304\pi$
$997$	41.4183	1.31173	0.655866	+	0.754878i	$0.272304\pi$
$998$	4.71940	0.149390
$999$	7.58258	0.239902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cu.1.2		4
5.4	even	2		1815.2.a.t.1.3	yes	4
11.10	odd	2	inner	9075.2.a.cu.1.3		4
15.14	odd	2		5445.2.a.bl.1.2		4
55.54	odd	2		1815.2.a.t.1.2	✓	4
165.164	even	2		5445.2.a.bl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.t.1.2	✓	4	55.54	odd	2
1815.2.a.t.1.3	yes	4	5.4	even	2
5445.2.a.bl.1.2		4	15.14	odd	2
5445.2.a.bl.1.3		4	165.164	even	2
9075.2.a.cu.1.2		4	1.1	even	1	trivial
9075.2.a.cu.1.3		4	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.456850 q^{2} -1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -4.37780 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.456850 q^{2} -1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -4.37780 q^{7} +1.73205 q^{8} +1.00000 q^{9} +1.79129 q^{12} -0.913701 q^{13} +2.00000 q^{14} +2.79129 q^{16} -1.73205 q^{17} -0.456850 q^{18} -3.46410 q^{19} +4.37780 q^{21} +0.582576 q^{23} -1.73205 q^{24} +0.417424 q^{26} -1.00000 q^{27} +7.84190 q^{28} +9.66930 q^{29} -8.58258 q^{31} -4.73930 q^{32} +0.791288 q^{34} -1.79129 q^{36} -7.58258 q^{37} +1.58258 q^{38} +0.913701 q^{39} +3.46410 q^{41} -2.00000 q^{42} +9.66930 q^{43} -0.266150 q^{46} -3.41742 q^{47} -2.79129 q^{48} +12.1652 q^{49} +1.73205 q^{51} +1.63670 q^{52} -12.1652 q^{53} +0.456850 q^{54} -7.58258 q^{56} +3.46410 q^{57} -4.41742 q^{58} +13.5826 q^{59} -3.55945 q^{61} +3.92095 q^{62} -4.37780 q^{63} -3.41742 q^{64} +4.41742 q^{67} +3.10260 q^{68} -0.582576 q^{69} +8.00000 q^{71} +1.73205 q^{72} -5.10080 q^{73} +3.46410 q^{74} +6.20520 q^{76} -0.417424 q^{78} +0.818350 q^{79} +1.00000 q^{81} -1.58258 q^{82} +15.8745 q^{83} -7.84190 q^{84} -4.41742 q^{86} -9.66930 q^{87} +14.7477 q^{89} +4.00000 q^{91} -1.04356 q^{92} +8.58258 q^{93} +1.56125 q^{94} +4.73930 q^{96} -4.41742 q^{97} -5.55765 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 2 * q^4 + 4 * q^9	\( 4 q - 4 q^{3} + 2 q^{4} + 4 q^{9} - 2 q^{12} + 8 q^{14} + 2 q^{16} - 16 q^{23} + 20 q^{26} - 4 q^{27} - 16 q^{31} - 6 q^{34} + 2 q^{36} - 12 q^{37} - 12 q^{38} - 8 q^{42} - 32 q^{47} - 2 q^{48} + 12 q^{49} - 12 q^{53} - 12 q^{56} - 36 q^{58} + 36 q^{59} - 32 q^{64} + 36 q^{67} + 16 q^{69} + 32 q^{71} - 20 q^{78} + 4 q^{81} + 12 q^{82} - 36 q^{86} + 4 q^{89} + 16 q^{91} - 50 q^{92} + 16 q^{93} - 36 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 + 2 * q^4 + 4 * q^9 - 2 * q^12 + 8 * q^14 + 2 * q^16 - 16 * q^23 + 20 * q^26 - 4 * q^27 - 16 * q^31 - 6 * q^34 + 2 * q^36 - 12 * q^37 - 12 * q^38 - 8 * q^42 - 32 * q^47 - 2 * q^48 + 12 * q^49 - 12 * q^53 - 12 * q^56 - 36 * q^58 + 36 * q^59 - 32 * q^64 + 36 * q^67 + 16 * q^69 + 32 * q^71 - 20 * q^78 + 4 * q^81 + 12 * q^82 - 36 * q^86 + 4 * q^89 + 16 * q^91 - 50 * q^92 + 16 * q^93 - 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more