LMFDB - Embedded newform 3675.2.a.by.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 2x + 1 $ x^4 - 2x^3 - 3x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.22833$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22833 q^{2} -1.00000 q^{3} +2.96545 q^{4} -2.22833 q^{6} +2.15133 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.22833 q^{2} -1.00000 q^{3} +2.96545 q^{4} -2.22833 q^{6} +2.15133 q^{8} +1.00000 q^{9} +1.77956 q^{11} -2.96545 q^{12} -4.19377 q^{13} -1.13702 q^{16} +0.322905 q^{17} +2.22833 q^{18} -7.66801 q^{19} +3.96545 q^{22} -4.97976 q^{23} -2.15133 q^{24} -9.34511 q^{26} -1.00000 q^{27} -8.08223 q^{29} -4.67067 q^{31} -6.83632 q^{32} -1.77956 q^{33} +0.719538 q^{34} +2.96545 q^{36} +10.2851 q^{37} -17.0868 q^{38} +4.19377 q^{39} -1.69734 q^{41} +8.04621 q^{43} +5.27719 q^{44} -11.0965 q^{46} +10.5813 q^{47} +1.13702 q^{48} -0.322905 q^{51} -12.4364 q^{52} -5.84601 q^{53} -2.22833 q^{54} +7.66801 q^{57} -18.0099 q^{58} -3.58313 q^{59} -10.8284 q^{61} -10.4078 q^{62} -12.9595 q^{64} -3.96545 q^{66} -5.60533 q^{67} +0.957557 q^{68} +4.97976 q^{69} +6.72803 q^{71} +2.15133 q^{72} -5.39021 q^{73} +22.9185 q^{74} -22.7391 q^{76} +9.34511 q^{78} -3.23888 q^{79} +1.00000 q^{81} -3.78222 q^{82} +4.52953 q^{83} +17.9296 q^{86} +8.08223 q^{87} +3.82843 q^{88} +16.2382 q^{89} -14.7672 q^{92} +4.67067 q^{93} +23.5787 q^{94} +6.83632 q^{96} +5.27600 q^{97} +1.77956 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^6 + 4 * q^9	$ 4 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{9} - 4 q^{11} - 2 q^{12} - 6 q^{16} + 4 q^{17} + 2 q^{18} - 8 q^{19} + 6 q^{22} - 12 q^{26} - 4 q^{27} - 4 q^{29} - 8 q^{31} + 2 q^{32} + 4 q^{33} - 8 q^{34} + 2 q^{36} + 16 q^{37} - 4 q^{38} - 24 q^{41} + 20 q^{43} + 14 q^{44} - 6 q^{46} - 8 q^{47} + 6 q^{48} - 4 q^{51} - 16 q^{52} - 20 q^{53} - 2 q^{54} + 8 q^{57} - 6 q^{58} - 8 q^{59} - 32 q^{61} - 28 q^{62} - 12 q^{64} - 6 q^{66} + 12 q^{67} + 12 q^{68} + 4 q^{71} + 34 q^{74} - 40 q^{76} + 12 q^{78} + 4 q^{81} - 16 q^{82} + 20 q^{83} - 14 q^{86} + 4 q^{87} + 4 q^{88} - 8 q^{89} - 10 q^{92} + 8 q^{93} + 32 q^{94} - 2 q^{96} - 24 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^6 + 4 * q^9 - 4 * q^11 - 2 * q^12 - 6 * q^16 + 4 * q^17 + 2 * q^18 - 8 * q^19 + 6 * q^22 - 12 * q^26 - 4 * q^27 - 4 * q^29 - 8 * q^31 + 2 * q^32 + 4 * q^33 - 8 * q^34 + 2 * q^36 + 16 * q^37 - 4 * q^38 - 24 * q^41 + 20 * q^43 + 14 * q^44 - 6 * q^46 - 8 * q^47 + 6 * q^48 - 4 * q^51 - 16 * q^52 - 20 * q^53 - 2 * q^54 + 8 * q^57 - 6 * q^58 - 8 * q^59 - 32 * q^61 - 28 * q^62 - 12 * q^64 - 6 * q^66 + 12 * q^67 + 12 * q^68 + 4 * q^71 + 34 * q^74 - 40 * q^76 + 12 * q^78 + 4 * q^81 - 16 * q^82 + 20 * q^83 - 14 * q^86 + 4 * q^87 + 4 * q^88 - 8 * q^89 - 10 * q^92 + 8 * q^93 + 32 * q^94 - 2 * q^96 - 24 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.22833	1.57567	0.787833	−	0.615889i	$-0.211203\pi$
$2$	2.22833	1.57567	0.787833	+	0.615889i	$0.211203\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.96545	1.48272
$5$	0	0
$5$	0	0
$6$	−2.22833	−0.909711
$7$	0	0
$7$	0	0
$8$	2.15133	0.760611
$9$	1.00000	0.333333
$10$	0	0
$11$	1.77956	0.536558	0.268279	−	0.963341i	$-0.413545\pi$
$11$	1.77956	0.536558	0.268279	+	0.963341i	$0.413545\pi$
$12$	−2.96545	−0.856051
$13$	−4.19377	−1.16314	−0.581572	−	0.813495i	$-0.697562\pi$
$13$	−4.19377	−1.16314	−0.581572	+	0.813495i	$0.697562\pi$
$14$	0	0
$15$	0	0
$16$	−1.13702	−0.284255
$17$	0.322905	0.0783159	0.0391580	−	0.999233i	$-0.487532\pi$
$17$	0.322905	0.0783159	0.0391580	+	0.999233i	$0.487532\pi$
$18$	2.22833	0.525222
$19$	−7.66801	−1.75916	−0.879581	−	0.475749i	$-0.842177\pi$
$19$	−7.66801	−1.75916	−0.879581	+	0.475749i	$0.842177\pi$
$20$	0	0
$21$	0	0
$22$	3.96545	0.845436
$23$	−4.97976	−1.03835	−0.519176	−	0.854667i	$-0.673761\pi$
$23$	−4.97976	−1.03835	−0.519176	+	0.854667i	$0.673761\pi$
$24$	−2.15133	−0.439139
$25$	0	0
$26$	−9.34511	−1.83273
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.08223	−1.50083	−0.750416	−	0.660966i	$-0.770147\pi$
$29$	−8.08223	−1.50083	−0.750416	+	0.660966i	$0.770147\pi$
$30$	0	0
$31$	−4.67067	−0.838877	−0.419439	−	0.907784i	$-0.637773\pi$
$31$	−4.67067	−0.838877	−0.419439	+	0.907784i	$0.637773\pi$
$32$	−6.83632	−1.20850
$33$	−1.77956	−0.309782
$34$	0.719538	0.123400
$35$	0	0
$36$	2.96545	0.494241
$37$	10.2851	1.69086	0.845429	−	0.534088i	$-0.179345\pi$
$37$	10.2851	1.69086	0.845429	+	0.534088i	$0.179345\pi$
$38$	−17.0868	−2.77185
$39$	4.19377	0.671541
$40$	0	0
$41$	−1.69734	−0.265079	−0.132540	−	0.991178i	$-0.542313\pi$
$41$	−1.69734	−0.265079	−0.132540	+	0.991178i	$0.542313\pi$
$42$	0	0
$43$	8.04621	1.22703	0.613517	−	0.789681i	$-0.289754\pi$
$43$	8.04621	1.22703	0.613517	+	0.789681i	$0.289754\pi$
$44$	5.27719	0.795567
$45$	0	0
$46$	−11.0965	−1.63610
$47$	10.5813	1.54344	0.771722	−	0.635960i	$-0.219395\pi$
$47$	10.5813	1.54344	0.771722	+	0.635960i	$0.219395\pi$
$48$	1.13702	0.164115
$49$	0	0
$50$	0	0
$51$	−0.322905	−0.0452157
$52$	−12.4364	−1.72462
$53$	−5.84601	−0.803011	−0.401505	−	0.915857i	$-0.631513\pi$
$53$	−5.84601	−0.803011	−0.401505	+	0.915857i	$0.631513\pi$
$54$	−2.22833	−0.303237
$55$	0	0
$56$	0	0
$57$	7.66801	1.01565
$58$	−18.0099	−2.36481
$59$	−3.58313	−0.466483	−0.233242	−	0.972419i	$-0.574933\pi$
$59$	−3.58313	−0.466483	−0.233242	+	0.972419i	$0.574933\pi$
$60$	0	0
$61$	−10.8284	−1.38644	−0.693219	−	0.720727i	$-0.743808\pi$
$61$	−10.8284	−1.38644	−0.693219	+	0.720727i	$0.743808\pi$
$62$	−10.4078	−1.32179
$63$	0	0
$64$	−12.9595	−1.61994
$65$	0	0
$66$	−3.96545	−0.488113
$67$	−5.60533	−0.684800	−0.342400	−	0.939554i	$-0.611240\pi$
$67$	−5.60533	−0.684800	−0.342400	+	0.939554i	$0.611240\pi$
$68$	0.957557	0.116121
$69$	4.97976	0.599493
$70$	0	0
$71$	6.72803	0.798471	0.399235	−	0.916848i	$-0.369276\pi$
$71$	6.72803	0.798471	0.399235	+	0.916848i	$0.369276\pi$
$72$	2.15133	0.253537
$73$	−5.39021	−0.630876	−0.315438	−	0.948946i	$-0.602151\pi$
$73$	−5.39021	−0.630876	−0.315438	+	0.948946i	$0.602151\pi$
$74$	22.9185	2.66423
$75$	0	0
$76$	−22.7391	−2.60835
$77$	0	0
$78$	9.34511	1.05812
$79$	−3.23888	−0.364402	−0.182201	−	0.983261i	$-0.558322\pi$
$79$	−3.23888	−0.364402	−0.182201	+	0.983261i	$0.558322\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.78222	−0.417677
$83$	4.52953	0.497180	0.248590	−	0.968609i	$-0.420033\pi$
$83$	4.52953	0.497180	0.248590	+	0.968609i	$0.420033\pi$
$84$	0	0
$85$	0	0
$86$	17.9296	1.93340
$87$	8.08223	0.866506
$88$	3.82843	0.408112
$89$	16.2382	1.72124	0.860622	−	0.509245i	$-0.170075\pi$
$89$	16.2382	1.72124	0.860622	+	0.509245i	$0.170075\pi$
$90$	0	0
$91$	0	0
$92$	−14.7672	−1.53959
$93$	4.67067	0.484326
$94$	23.5787	2.43195
$95$	0	0
$96$	6.83632	0.697729
$97$	5.27600	0.535697	0.267848	−	0.963461i	$-0.413687\pi$
$97$	5.27600	0.535697	0.267848	+	0.963461i	$0.413687\pi$
$98$	0	0
$99$	1.77956	0.178853
$100$	0	0
$101$	−10.5055	−1.04534	−0.522669	−	0.852536i	$-0.675064\pi$
$101$	−10.5055	−1.04534	−0.522669	+	0.852536i	$0.675064\pi$
$102$	−0.719538	−0.0712449
$103$	−6.49644	−0.640113	−0.320057	−	0.947398i	$-0.603702\pi$
$103$	−6.49644	−0.640113	−0.320057	+	0.947398i	$0.603702\pi$
$104$	−9.02220	−0.884700
$105$	0	0
$106$	−13.0268	−1.26528
$107$	−11.9180	−1.15216	−0.576081	−	0.817393i	$-0.695419\pi$
$107$	−11.9180	−1.15216	−0.576081	+	0.817393i	$0.695419\pi$
$108$	−2.96545	−0.285350
$109$	−17.1984	−1.64731	−0.823654	−	0.567093i	$-0.808068\pi$
$109$	−17.1984	−1.64731	−0.823654	+	0.567093i	$0.808068\pi$
$110$	0	0
$111$	−10.2851	−0.976217
$112$	0	0
$113$	14.9393	1.40537	0.702684	−	0.711502i	$-0.251985\pi$
$113$	14.9393	1.40537	0.702684	+	0.711502i	$0.251985\pi$
$114$	17.0868	1.60033
$115$	0	0
$116$	−23.9674	−2.22532
$117$	−4.19377	−0.387715
$118$	−7.98438	−0.735022
$119$	0	0
$120$	0	0
$121$	−7.83316	−0.712106
$122$	−24.1293	−2.18456
$123$	1.69734	0.153044
$124$	−13.8506	−1.24382
$125$	0	0
$126$	0	0
$127$	11.2049	0.994277	0.497138	−	0.867671i	$-0.334384\pi$
$127$	11.2049	0.994277	0.497138	+	0.867671i	$0.334384\pi$
$128$	−15.2054	−1.34398
$129$	−8.04621	−0.708429
$130$	0	0
$131$	−9.11617	−0.796484	−0.398242	−	0.917280i	$-0.630379\pi$
$131$	−9.11617	−0.796484	−0.398242	+	0.917280i	$0.630379\pi$
$132$	−5.27719	−0.459321
$133$	0	0
$134$	−12.4905	−1.07902
$135$	0	0
$136$	0.694676	0.0595679
$137$	−6.28688	−0.537125	−0.268562	−	0.963262i	$-0.586549\pi$
$137$	−6.28688	−0.537125	−0.268562	+	0.963262i	$0.586549\pi$
$138$	11.0965	0.944600
$139$	3.69287	0.313225	0.156613	−	0.987660i	$-0.449943\pi$
$139$	3.69287	0.313225	0.156613	+	0.987660i	$0.449943\pi$
$140$	0	0
$141$	−10.5813	−0.891108
$142$	14.9923	1.25812
$143$	−7.46308	−0.624094
$144$	−1.13702	−0.0947516
$145$	0	0
$146$	−12.0112	−0.994050
$147$	0	0
$148$	30.4999	2.50707
$149$	11.8931	0.974318	0.487159	−	0.873313i	$-0.338033\pi$
$149$	11.8931	0.974318	0.487159	+	0.873313i	$0.338033\pi$
$150$	0	0
$151$	−4.01104	−0.326414	−0.163207	−	0.986592i	$-0.552184\pi$
$151$	−4.01104	−0.326414	−0.163207	+	0.986592i	$0.552184\pi$
$152$	−16.4964	−1.33804
$153$	0.322905	0.0261053
$154$	0	0
$155$	0	0
$156$	12.4364	0.995710
$157$	−0.863588	−0.0689218	−0.0344609	−	0.999406i	$-0.510971\pi$
$157$	−0.863588	−0.0689218	−0.0344609	+	0.999406i	$0.510971\pi$
$158$	−7.21728	−0.574176
$159$	5.84601	0.463619
$160$	0	0
$161$	0	0
$162$	2.22833	0.175074
$163$	−2.43434	−0.190672	−0.0953362	−	0.995445i	$-0.530393\pi$
$163$	−2.43434	−0.190672	−0.0953362	+	0.995445i	$0.530393\pi$
$164$	−5.03336	−0.393039
$165$	0	0
$166$	10.0933	0.783390
$167$	−18.0556	−1.39718	−0.698591	−	0.715522i	$-0.746189\pi$
$167$	−18.0556	−1.39718	−0.698591	+	0.715522i	$0.746189\pi$
$168$	0	0
$169$	4.58775	0.352904
$170$	0	0
$171$	−7.66801	−0.586388
$172$	23.8606	1.81935
$173$	12.8267	0.975199	0.487599	−	0.873067i	$-0.337873\pi$
$173$	12.8267	0.975199	0.487599	+	0.873067i	$0.337873\pi$
$174$	18.0099	1.36532
$175$	0	0
$176$	−2.02340	−0.152519
$177$	3.58313	0.269324
$178$	36.1840	2.71210
$179$	−24.5806	−1.83724	−0.918621	−	0.395140i	$-0.870696\pi$
$179$	−24.5806	−1.83724	−0.918621	+	0.395140i	$0.870696\pi$
$180$	0	0
$181$	−9.74620	−0.724429	−0.362215	−	0.932095i	$-0.617979\pi$
$181$	−9.74620	−0.724429	−0.362215	+	0.932095i	$0.617979\pi$
$182$	0	0
$183$	10.8284	0.800460
$184$	−10.7131	−0.789781
$185$	0	0
$186$	10.4078	0.763136
$187$	0.574629	0.0420210
$188$	31.3784	2.28850
$189$	0	0
$190$	0	0
$191$	5.45192	0.394487	0.197244	−	0.980355i	$-0.436801\pi$
$191$	5.45192	0.394487	0.197244	+	0.980355i	$0.436801\pi$
$192$	12.9595	0.935273
$193$	−24.9911	−1.79890	−0.899449	−	0.437027i	$-0.856032\pi$
$193$	−24.9911	−1.79890	−0.899449	+	0.437027i	$0.856032\pi$
$194$	11.7567	0.844079
$195$	0	0
$196$	0	0
$197$	−20.3497	−1.44986	−0.724929	−	0.688824i	$-0.758127\pi$
$197$	−20.3497	−1.44986	−0.724929	+	0.688824i	$0.758127\pi$
$198$	3.96545	0.281812
$199$	25.5982	1.81461	0.907304	−	0.420475i	$-0.138136\pi$
$199$	25.5982	1.81461	0.907304	+	0.420475i	$0.138136\pi$
$200$	0	0
$201$	5.60533	0.395369
$202$	−23.4098	−1.64710
$203$	0	0
$204$	−0.957557	−0.0670424
$205$	0	0
$206$	−14.4762	−1.00860
$207$	−4.97976	−0.346117
$208$	4.76840	0.330629
$209$	−13.6457	−0.943893
$210$	0	0
$211$	6.16977	0.424745	0.212372	−	0.977189i	$-0.431881\pi$
$211$	6.16977	0.424745	0.212372	+	0.977189i	$0.431881\pi$
$212$	−17.3360	−1.19064
$213$	−6.72803	−0.460997
$214$	−26.5573	−1.81542
$215$	0	0
$216$	−2.15133	−0.146380
$217$	0	0
$218$	−38.3237	−2.59561
$219$	5.39021	0.364237
$220$	0	0
$221$	−1.35419	−0.0910927
$222$	−22.9185	−1.53819
$223$	2.46405	0.165005	0.0825025	−	0.996591i	$-0.473709\pi$
$223$	2.46405	0.165005	0.0825025	+	0.996591i	$0.473709\pi$
$224$	0	0
$225$	0	0
$226$	33.2896	2.21439
$227$	19.9484	1.32402	0.662010	−	0.749495i	$-0.269704\pi$
$227$	19.9484	1.32402	0.662010	+	0.749495i	$0.269704\pi$
$228$	22.7391	1.50593
$229$	−0.816411	−0.0539499	−0.0269750	−	0.999636i	$-0.508587\pi$
$229$	−0.816411	−0.0539499	−0.0269750	+	0.999636i	$0.508587\pi$
$230$	0	0
$231$	0	0
$232$	−17.3875	−1.14155
$233$	9.33868	0.611798	0.305899	−	0.952064i	$-0.401043\pi$
$233$	9.33868	0.611798	0.305899	+	0.952064i	$0.401043\pi$
$234$	−9.34511	−0.610909
$235$	0	0
$236$	−10.6256	−0.691666
$237$	3.23888	0.210388
$238$	0	0
$239$	29.4782	1.90678	0.953392	−	0.301735i	$-0.0975658\pi$
$239$	29.4782	1.90678	0.953392	+	0.301735i	$0.0975658\pi$
$240$	0	0
$241$	23.4893	1.51308	0.756540	−	0.653947i	$-0.226888\pi$
$241$	23.4893	1.51308	0.756540	+	0.653947i	$0.226888\pi$
$242$	−17.4549	−1.12204
$243$	−1.00000	−0.0641500
$244$	−32.1111	−2.05570
$245$	0	0
$246$	3.78222	0.241146
$247$	32.1579	2.04616
$248$	−10.0482	−0.638059
$249$	−4.52953	−0.287047
$250$	0	0
$251$	23.9837	1.51384	0.756918	−	0.653510i	$-0.226704\pi$
$251$	23.9837	1.51384	0.756918	+	0.653510i	$0.226704\pi$
$252$	0	0
$253$	−8.86179	−0.557136
$254$	24.9683	1.56665
$255$	0	0
$256$	−7.96365	−0.497728
$257$	2.20117	0.137305	0.0686526	−	0.997641i	$-0.478130\pi$
$257$	2.20117	0.137305	0.0686526	+	0.997641i	$0.478130\pi$
$258$	−17.9296	−1.11625
$259$	0	0
$260$	0	0
$261$	−8.08223	−0.500277
$262$	−20.3138	−1.25499
$263$	−14.8311	−0.914524	−0.457262	−	0.889332i	$-0.651170\pi$
$263$	−14.8311	−0.914524	−0.457262	+	0.889332i	$0.651170\pi$
$264$	−3.82843	−0.235623
$265$	0	0
$266$	0	0
$267$	−16.2382	−0.993760
$268$	−16.6223	−1.01537
$269$	−7.30070	−0.445132	−0.222566	−	0.974918i	$-0.571443\pi$
$269$	−7.30070	−0.445132	−0.222566	+	0.974918i	$0.571443\pi$
$270$	0	0
$271$	−17.5426	−1.06564	−0.532820	−	0.846229i	$-0.678868\pi$
$271$	−17.5426	−1.06564	−0.532820	+	0.846229i	$0.678868\pi$
$272$	−0.367149	−0.0222617
$273$	0	0
$274$	−14.0092	−0.846329
$275$	0	0
$276$	14.7672	0.888882
$277$	5.62169	0.337775	0.168887	−	0.985635i	$-0.445983\pi$
$277$	5.62169	0.337775	0.168887	+	0.985635i	$0.445983\pi$
$278$	8.22894	0.493539
$279$	−4.67067	−0.279626
$280$	0	0
$281$	−8.67177	−0.517315	−0.258657	−	0.965969i	$-0.583280\pi$
$281$	−8.67177	−0.517315	−0.258657	+	0.965969i	$0.583280\pi$
$282$	−23.5787	−1.40409
$283$	28.8572	1.71538	0.857692	−	0.514164i	$-0.171898\pi$
$283$	28.8572	1.71538	0.857692	+	0.514164i	$0.171898\pi$
$284$	19.9516	1.18391
$285$	0	0
$286$	−16.6302	−0.983364
$287$	0	0
$288$	−6.83632	−0.402834
$289$	−16.8957	−0.993867
$290$	0	0
$291$	−5.27600	−0.309285
$292$	−15.9844	−0.935415
$293$	16.9594	0.990779	0.495390	−	0.868671i	$-0.335025\pi$
$293$	16.9594	0.990779	0.495390	+	0.868671i	$0.335025\pi$
$294$	0	0
$295$	0	0
$296$	22.1266	1.28608
$297$	−1.77956	−0.103261
$298$	26.5017	1.53520
$299$	20.8840	1.20775
$300$	0	0
$301$	0	0
$302$	−8.93792	−0.514320
$303$	10.5055	0.603526
$304$	8.71868	0.500051
$305$	0	0
$306$	0.719538	0.0411333
$307$	25.5898	1.46049	0.730244	−	0.683186i	$-0.239406\pi$
$307$	25.5898	1.46049	0.730244	+	0.683186i	$0.239406\pi$
$308$	0	0
$309$	6.49644	0.369569
$310$	0	0
$311$	−11.2086	−0.635580	−0.317790	−	0.948161i	$-0.602941\pi$
$311$	−11.2086	−0.635580	−0.317790	+	0.948161i	$0.602941\pi$
$312$	9.02220	0.510782
$313$	5.98884	0.338509	0.169255	−	0.985572i	$-0.445864\pi$
$313$	5.98884	0.338509	0.169255	+	0.985572i	$0.445864\pi$
$314$	−1.92436	−0.108598
$315$	0	0
$316$	−9.60472	−0.540308
$317$	19.0060	1.06748	0.533742	−	0.845648i	$-0.320786\pi$
$317$	19.0060	1.06748	0.533742	+	0.845648i	$0.320786\pi$
$318$	13.0268	0.730508
$319$	−14.3828	−0.805283
$320$	0	0
$321$	11.9180	0.665201
$322$	0	0
$323$	−2.47604	−0.137770
$324$	2.96545	0.164747
$325$	0	0
$326$	−5.42451	−0.300436
$327$	17.1984	0.951073
$328$	−3.65153	−0.201622
$329$	0	0
$330$	0	0
$331$	10.2851	0.565319	0.282660	−	0.959220i	$-0.408783\pi$
$331$	10.2851	0.565319	0.282660	+	0.959220i	$0.408783\pi$
$332$	13.4321	0.737180
$333$	10.2851	0.563619
$334$	−40.2337	−2.20149
$335$	0	0
$336$	0	0
$337$	12.2389	0.666694	0.333347	−	0.942804i	$-0.391822\pi$
$337$	12.2389	0.666694	0.333347	+	0.942804i	$0.391822\pi$
$338$	10.2230	0.556058
$339$	−14.9393	−0.811390
$340$	0	0
$341$	−8.31175	−0.450106
$342$	−17.0868	−0.923951
$343$	0	0
$344$	17.3101	0.933296
$345$	0	0
$346$	28.5822	1.53659
$347$	−30.6524	−1.64551	−0.822753	−	0.568399i	$-0.807563\pi$
$347$	−30.6524	−1.64551	−0.822753	+	0.568399i	$0.807563\pi$
$348$	23.9674	1.28479
$349$	−14.5369	−0.778144	−0.389072	−	0.921207i	$-0.627204\pi$
$349$	−14.5369	−0.778144	−0.389072	+	0.921207i	$0.627204\pi$
$350$	0	0
$351$	4.19377	0.223847
$352$	−12.1656	−0.648431
$353$	9.43531	0.502191	0.251096	−	0.967962i	$-0.419209\pi$
$353$	9.43531	0.502191	0.251096	+	0.967962i	$0.419209\pi$
$354$	7.98438	0.424365
$355$	0	0
$356$	48.1535	2.55213
$357$	0	0
$358$	−54.7737	−2.89488
$359$	−16.8415	−0.888863	−0.444431	−	0.895813i	$-0.646594\pi$
$359$	−16.8415	−0.888863	−0.444431	+	0.895813i	$0.646594\pi$
$360$	0	0
$361$	39.7984	2.09465
$362$	−21.7177	−1.14146
$363$	7.83316	0.411134
$364$	0	0
$365$	0	0
$366$	24.1293	1.26126
$367$	24.6955	1.28910	0.644548	−	0.764564i	$-0.277045\pi$
$367$	24.6955	1.28910	0.644548	+	0.764564i	$0.277045\pi$
$368$	5.66208	0.295156
$369$	−1.69734	−0.0883598
$370$	0	0
$371$	0	0
$372$	13.8506	0.718122
$373$	−29.1322	−1.50841	−0.754205	−	0.656639i	$-0.771977\pi$
$373$	−29.1322	−1.50841	−0.754205	+	0.656639i	$0.771977\pi$
$374$	1.28046	0.0662111
$375$	0	0
$376$	22.7639	1.17396
$377$	33.8950	1.74568
$378$	0	0
$379$	23.2908	1.19637	0.598184	−	0.801359i	$-0.295889\pi$
$379$	23.2908	1.19637	0.598184	+	0.801359i	$0.295889\pi$
$380$	0	0
$381$	−11.2049	−0.574046
$382$	12.1487	0.621580
$383$	−19.3249	−0.987455	−0.493727	−	0.869617i	$-0.664366\pi$
$383$	−19.3249	−0.987455	−0.493727	+	0.869617i	$0.664366\pi$
$384$	15.2054	0.775949
$385$	0	0
$386$	−55.6883	−2.83446
$387$	8.04621	0.409012
$388$	15.6457	0.794290
$389$	1.95465	0.0991049	0.0495524	−	0.998772i	$-0.484221\pi$
$389$	1.95465	0.0991049	0.0495524	+	0.998772i	$0.484221\pi$
$390$	0	0
$391$	−1.60799	−0.0813195
$392$	0	0
$393$	9.11617	0.459850
$394$	−45.3459	−2.28449
$395$	0	0
$396$	5.27719	0.265189
$397$	32.0159	1.60683	0.803417	−	0.595417i	$-0.203013\pi$
$397$	32.0159	1.60683	0.803417	+	0.595417i	$0.203013\pi$
$398$	57.0412	2.85922
$399$	0	0
$400$	0	0
$401$	0.768517	0.0383779	0.0191889	−	0.999816i	$-0.493892\pi$
$401$	0.768517	0.0383779	0.0191889	+	0.999816i	$0.493892\pi$
$402$	12.4905	0.622970
$403$	19.5877	0.975735
$404$	−31.1536	−1.54995
$405$	0	0
$406$	0	0
$407$	18.3029	0.907243
$408$	−0.694676	−0.0343916
$409$	−28.6310	−1.41571	−0.707857	−	0.706356i	$-0.750338\pi$
$409$	−28.6310	−1.41571	−0.707857	+	0.706356i	$0.750338\pi$
$410$	0	0
$411$	6.28688	0.310109
$412$	−19.2648	−0.949111
$413$	0	0
$414$	−11.0965	−0.545365
$415$	0	0
$416$	28.6700	1.40566
$417$	−3.69287	−0.180841
$418$	−30.4071	−1.48726
$419$	−10.0663	−0.491773	−0.245886	−	0.969299i	$-0.579079\pi$
$419$	−10.0663	−0.491773	−0.245886	+	0.969299i	$0.579079\pi$
$420$	0	0
$421$	16.9648	0.826816	0.413408	−	0.910546i	$-0.364338\pi$
$421$	16.9648	0.826816	0.413408	+	0.910546i	$0.364338\pi$
$422$	13.7483	0.669256
$423$	10.5813	0.514482
$424$	−12.5767	−0.610779
$425$	0	0
$426$	−14.9923	−0.726378
$427$	0	0
$428$	−35.3423	−1.70834
$429$	7.46308	0.360321
$430$	0	0
$431$	11.6047	0.558981	0.279490	−	0.960148i	$-0.409834\pi$
$431$	11.6047	0.558981	0.279490	+	0.960148i	$0.409834\pi$
$432$	1.13702	0.0547049
$433$	−4.43556	−0.213159	−0.106580	−	0.994304i	$-0.533990\pi$
$433$	−4.43556	−0.213159	−0.106580	+	0.994304i	$0.533990\pi$
$434$	0	0
$435$	0	0
$436$	−51.0009	−2.44250
$437$	38.1849	1.82663
$438$	12.0112	0.573915
$439$	7.30787	0.348786	0.174393	−	0.984676i	$-0.444204\pi$
$439$	7.30787	0.348786	0.174393	+	0.984676i	$0.444204\pi$
$440$	0	0
$441$	0	0
$442$	−3.01758	−0.143532
$443$	13.4209	0.637647	0.318823	−	0.947814i	$-0.396712\pi$
$443$	13.4209	0.637647	0.318823	+	0.947814i	$0.396712\pi$
$444$	−30.4999	−1.44746
$445$	0	0
$446$	5.49072	0.259993
$447$	−11.8931	−0.562523
$448$	0	0
$449$	13.6670	0.644987	0.322494	−	0.946572i	$-0.395479\pi$
$449$	13.6670	0.644987	0.322494	+	0.946572i	$0.395479\pi$
$450$	0	0
$451$	−3.02051	−0.142230
$452$	44.3016	2.08377
$453$	4.01104	0.188455
$454$	44.4515	2.08621
$455$	0	0
$456$	16.4964	0.772517
$457$	−0.559123	−0.0261547	−0.0130773	−	0.999914i	$-0.504163\pi$
$457$	−0.559123	−0.0261547	−0.0130773	+	0.999914i	$0.504163\pi$
$458$	−1.81923	−0.0850071
$459$	−0.322905	−0.0150719
$460$	0	0
$461$	−38.1170	−1.77528	−0.887642	−	0.460534i	$-0.847658\pi$
$461$	−38.1170	−1.77528	−0.887642	+	0.460534i	$0.847658\pi$
$462$	0	0
$463$	−10.7412	−0.499184	−0.249592	−	0.968351i	$-0.580296\pi$
$463$	−10.7412	−0.499184	−0.249592	+	0.968351i	$0.580296\pi$
$464$	9.18965	0.426619
$465$	0	0
$466$	20.8097	0.963989
$467$	−0.681130	−0.0315189	−0.0157595	−	0.999876i	$-0.505017\pi$
$467$	−0.681130	−0.0315189	−0.0157595	+	0.999876i	$0.505017\pi$
$468$	−12.4364	−0.574874
$469$	0	0
$470$	0	0
$471$	0.863588	0.0397920
$472$	−7.70849	−0.354812
$473$	14.3187	0.658375
$474$	7.21728	0.331501
$475$	0	0
$476$	0	0
$477$	−5.84601	−0.267670
$478$	65.6870	3.00445
$479$	−11.8157	−0.539873	−0.269936	−	0.962878i	$-0.587003\pi$
$479$	−11.8157	−0.539873	−0.269936	+	0.962878i	$0.587003\pi$
$480$	0	0
$481$	−43.1333	−1.96671
$482$	52.3419	2.38411
$483$	0	0
$484$	−23.2288	−1.05586
$485$	0	0
$486$	−2.22833	−0.101079
$487$	−0.890411	−0.0403484	−0.0201742	−	0.999796i	$-0.506422\pi$
$487$	−0.890411	−0.0403484	−0.0201742	+	0.999796i	$0.506422\pi$
$488$	−23.2955	−1.05454
$489$	2.43434	0.110085
$490$	0	0
$491$	−32.3434	−1.45964	−0.729819	−	0.683640i	$-0.760396\pi$
$491$	−32.3434	−1.45964	−0.729819	+	0.683640i	$0.760396\pi$
$492$	5.03336	0.226921
$493$	−2.60979	−0.117539
$494$	71.6584	3.22406
$495$	0	0
$496$	5.31065	0.238455
$497$	0	0
$498$	−10.0933	−0.452290
$499$	0.686292	0.0307226	0.0153613	−	0.999882i	$-0.495110\pi$
$499$	0.686292	0.0307226	0.0153613	+	0.999882i	$0.495110\pi$
$500$	0	0
$501$	18.0556	0.806663
$502$	53.4435	2.38530
$503$	−27.1443	−1.21031	−0.605153	−	0.796109i	$-0.706888\pi$
$503$	−27.1443	−1.21031	−0.605153	+	0.796109i	$0.706888\pi$
$504$	0	0
$505$	0	0
$506$	−19.7470	−0.877860
$507$	−4.58775	−0.203749
$508$	33.2276	1.47424
$509$	−40.6000	−1.79956	−0.899782	−	0.436339i	$-0.856275\pi$
$509$	−40.6000	−1.79956	−0.899782	+	0.436339i	$0.856275\pi$
$510$	0	0
$511$	0	0
$512$	12.6652	0.559730
$513$	7.66801	0.338551
$514$	4.90493	0.216347
$515$	0	0
$516$	−23.8606	−1.05040
$517$	18.8301	0.828148
$518$	0	0
$519$	−12.8267	−0.563031
$520$	0	0
$521$	−29.6424	−1.29866	−0.649330	−	0.760507i	$-0.724950\pi$
$521$	−29.6424	−1.29866	−0.649330	+	0.760507i	$0.724950\pi$
$522$	−18.0099	−0.788270
$523$	42.5431	1.86028	0.930139	−	0.367207i	$-0.119686\pi$
$523$	42.5431	1.86028	0.930139	+	0.367207i	$0.119686\pi$
$524$	−27.0335	−1.18096
$525$	0	0
$526$	−33.0485	−1.44098
$527$	−1.50818	−0.0656975
$528$	2.02340	0.0880570
$529$	1.79800	0.0781739
$530$	0	0
$531$	−3.58313	−0.155494
$532$	0	0
$533$	7.11825	0.308325
$534$	−36.1840	−1.56583
$535$	0	0
$536$	−12.0589	−0.520866
$537$	24.5806	1.06073
$538$	−16.2684	−0.701379
$539$	0	0
$540$	0	0
$541$	3.07858	0.132358	0.0661792	−	0.997808i	$-0.478919\pi$
$541$	3.07858	0.132358	0.0661792	+	0.997808i	$0.478919\pi$
$542$	−39.0908	−1.67909
$543$	9.74620	0.418250
$544$	−2.20748	−0.0946449
$545$	0	0
$546$	0	0
$547$	−11.1358	−0.476134	−0.238067	−	0.971249i	$-0.576514\pi$
$547$	−11.1358	−0.476134	−0.238067	+	0.971249i	$0.576514\pi$
$548$	−18.6434	−0.796408
$549$	−10.8284	−0.462146
$550$	0	0
$551$	61.9746	2.64021
$552$	10.7131	0.455980
$553$	0	0
$554$	12.5270	0.532220
$555$	0	0
$556$	10.9510	0.464427
$557$	−20.4816	−0.867835	−0.433917	−	0.900953i	$-0.642869\pi$
$557$	−20.4816	−0.867835	−0.433917	+	0.900953i	$0.642869\pi$
$558$	−10.4078	−0.440597
$559$	−33.7440	−1.42722
$560$	0	0
$561$	−0.574629	−0.0242609
$562$	−19.3236	−0.815115
$563$	−5.83931	−0.246098	−0.123049	−	0.992401i	$-0.539267\pi$
$563$	−5.83931	−0.246098	−0.123049	+	0.992401i	$0.539267\pi$
$564$	−31.3784	−1.32127
$565$	0	0
$566$	64.3033	2.70287
$567$	0	0
$568$	14.4742	0.607325
$569$	−23.3045	−0.976976	−0.488488	−	0.872571i	$-0.662451\pi$
$569$	−23.3045	−0.976976	−0.488488	+	0.872571i	$0.662451\pi$
$570$	0	0
$571$	−32.1415	−1.34508	−0.672541	−	0.740060i	$-0.734797\pi$
$571$	−32.1415	−1.34508	−0.672541	+	0.740060i	$0.734797\pi$
$572$	−22.1314	−0.925359
$573$	−5.45192	−0.227757
$574$	0	0
$575$	0	0
$576$	−12.9595	−0.539980
$577$	−5.69010	−0.236882	−0.118441	−	0.992961i	$-0.537790\pi$
$577$	−5.69010	−0.236882	−0.118441	+	0.992961i	$0.537790\pi$
$578$	−37.6492	−1.56600
$579$	24.9911	1.03859
$580$	0	0
$581$	0	0
$582$	−11.7567	−0.487329
$583$	−10.4033	−0.430862
$584$	−11.5961	−0.479851
$585$	0	0
$586$	37.7911	1.56114
$587$	−8.71798	−0.359830	−0.179915	−	0.983682i	$-0.557582\pi$
$587$	−8.71798	−0.359830	−0.179915	+	0.983682i	$0.557582\pi$
$588$	0	0
$589$	35.8148	1.47572
$590$	0	0
$591$	20.3497	0.837076
$592$	−11.6943	−0.480634
$593$	−43.6426	−1.79219	−0.896094	−	0.443865i	$-0.853607\pi$
$593$	−43.6426	−1.79219	−0.896094	+	0.443865i	$0.853607\pi$
$594$	−3.96545	−0.162704
$595$	0	0
$596$	35.2683	1.44464
$597$	−25.5982	−1.04766
$598$	46.5364	1.90301
$599$	10.7685	0.439990	0.219995	−	0.975501i	$-0.429396\pi$
$599$	10.7685	0.439990	0.219995	+	0.975501i	$0.429396\pi$
$600$	0	0
$601$	22.1072	0.901772	0.450886	−	0.892582i	$-0.351108\pi$
$601$	22.1072	0.901772	0.450886	+	0.892582i	$0.351108\pi$
$602$	0	0
$603$	−5.60533	−0.228267
$604$	−11.8945	−0.483982
$605$	0	0
$606$	23.4098	0.950956
$607$	5.79338	0.235146	0.117573	−	0.993064i	$-0.462489\pi$
$607$	5.79338	0.235146	0.117573	+	0.993064i	$0.462489\pi$
$608$	52.4210	2.12595
$609$	0	0
$610$	0	0
$611$	−44.3757	−1.79525
$612$	0.957557	0.0387070
$613$	−7.69652	−0.310860	−0.155430	−	0.987847i	$-0.549676\pi$
$613$	−7.69652	−0.310860	−0.155430	+	0.987847i	$0.549676\pi$
$614$	57.0225	2.30124
$615$	0	0
$616$	0	0
$617$	−31.1716	−1.25492	−0.627461	−	0.778648i	$-0.715906\pi$
$617$	−31.1716	−1.25492	−0.627461	+	0.778648i	$0.715906\pi$
$618$	14.4762	0.582318
$619$	−31.6052	−1.27032	−0.635160	−	0.772380i	$-0.719066\pi$
$619$	−31.6052	−1.27032	−0.635160	+	0.772380i	$0.719066\pi$
$620$	0	0
$621$	4.97976	0.199831
$622$	−24.9764	−1.00146
$623$	0	0
$624$	−4.76840	−0.190889
$625$	0	0
$626$	13.3451	0.533378
$627$	13.6457	0.544957
$628$	−2.56092	−0.102192
$629$	3.32110	0.132421
$630$	0	0
$631$	28.0045	1.11484	0.557421	−	0.830230i	$-0.311791\pi$
$631$	28.0045	1.11484	0.557421	+	0.830230i	$0.311791\pi$
$632$	−6.96790	−0.277168
$633$	−6.16977	−0.245226
$634$	42.3516	1.68200
$635$	0	0
$636$	17.3360	0.687418
$637$	0	0
$638$	−32.0496	−1.26886
$639$	6.72803	0.266157
$640$	0	0
$641$	26.0692	1.02967	0.514835	−	0.857289i	$-0.327853\pi$
$641$	26.0692	1.02967	0.514835	+	0.857289i	$0.327853\pi$
$642$	26.5573	1.04813
$643$	−44.1293	−1.74029	−0.870145	−	0.492796i	$-0.835975\pi$
$643$	−44.1293	−1.74029	−0.870145	+	0.492796i	$0.835975\pi$
$644$	0	0
$645$	0	0
$646$	−5.51743	−0.217080
$647$	6.34863	0.249590	0.124795	−	0.992183i	$-0.460173\pi$
$647$	6.34863	0.249590	0.124795	+	0.992183i	$0.460173\pi$
$648$	2.15133	0.0845123
$649$	−6.37639	−0.250295
$650$	0	0
$651$	0	0
$652$	−7.21891	−0.282714
$653$	24.6432	0.964363	0.482181	−	0.876071i	$-0.339845\pi$
$653$	24.6432	0.964363	0.482181	+	0.876071i	$0.339845\pi$
$654$	38.3237	1.49857
$655$	0	0
$656$	1.92990	0.0753501
$657$	−5.39021	−0.210292
$658$	0	0
$659$	23.4325	0.912802	0.456401	−	0.889774i	$-0.349138\pi$
$659$	23.4325	0.912802	0.456401	+	0.889774i	$0.349138\pi$
$660$	0	0
$661$	3.74758	0.145764	0.0728819	−	0.997341i	$-0.476780\pi$
$661$	3.74758	0.145764	0.0728819	+	0.997341i	$0.476780\pi$
$662$	22.9185	0.890754
$663$	1.35419	0.0525924
$664$	9.74451	0.378160
$665$	0	0
$666$	22.9185	0.888075
$667$	40.2475	1.55839
$668$	−53.5428	−2.07163
$669$	−2.46405	−0.0952657
$670$	0	0
$671$	−19.2698	−0.743904
$672$	0	0
$673$	21.7808	0.839589	0.419795	−	0.907619i	$-0.362102\pi$
$673$	21.7808	0.839589	0.419795	+	0.907619i	$0.362102\pi$
$674$	27.2722	1.05049
$675$	0	0
$676$	13.6047	0.523258
$677$	−19.4392	−0.747111	−0.373555	−	0.927608i	$-0.621861\pi$
$677$	−19.4392	−0.747111	−0.373555	+	0.927608i	$0.621861\pi$
$678$	−33.2896	−1.27848
$679$	0	0
$680$	0	0
$681$	−19.9484	−0.764423
$682$	−18.5213	−0.709217
$683$	−2.25560	−0.0863081	−0.0431541	−	0.999068i	$-0.513741\pi$
$683$	−2.25560	−0.0863081	−0.0431541	+	0.999068i	$0.513741\pi$
$684$	−22.7391	−0.869450
$685$	0	0
$686$	0	0
$687$	0.816411	0.0311480
$688$	−9.14869	−0.348791
$689$	24.5168	0.934017
$690$	0	0
$691$	−15.6608	−0.595764	−0.297882	−	0.954603i	$-0.596280\pi$
$691$	−15.6608	−0.595764	−0.297882	+	0.954603i	$0.596280\pi$
$692$	38.0370	1.44595
$693$	0	0
$694$	−68.3036	−2.59277
$695$	0	0
$696$	17.3875	0.659073
$697$	−0.548078	−0.0207599
$698$	−32.3930	−1.22609
$699$	−9.33868	−0.353221
$700$	0	0
$701$	−13.2776	−0.501486	−0.250743	−	0.968054i	$-0.580675\pi$
$701$	−13.2776	−0.501486	−0.250743	+	0.968054i	$0.580675\pi$
$702$	9.34511	0.352708
$703$	−78.8661	−2.97449
$704$	−23.0623	−0.869192
$705$	0	0
$706$	21.0250	0.791285
$707$	0	0
$708$	10.6256	0.399333
$709$	24.5445	0.921787	0.460894	−	0.887455i	$-0.347529\pi$
$709$	24.5445	0.921787	0.460894	+	0.887455i	$0.347529\pi$
$710$	0	0
$711$	−3.23888	−0.121467
$712$	34.9337	1.30920
$713$	23.2588	0.871050
$714$	0	0
$715$	0	0
$716$	−72.8925	−2.72412
$717$	−29.4782	−1.10088
$718$	−37.5285	−1.40055
$719$	−7.17550	−0.267601	−0.133800	−	0.991008i	$-0.542718\pi$
$719$	−7.17550	−0.267601	−0.133800	+	0.991008i	$0.542718\pi$
$720$	0	0
$721$	0	0
$722$	88.6839	3.30047
$723$	−23.4893	−0.873577
$724$	−28.9018	−1.07413
$725$	0	0
$726$	17.4549	0.647810
$727$	2.86052	0.106091	0.0530455	−	0.998592i	$-0.483107\pi$
$727$	2.86052	0.106091	0.0530455	+	0.998592i	$0.483107\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.59816	0.0960964
$732$	32.1111	1.18686
$733$	23.8693	0.881633	0.440817	−	0.897597i	$-0.354689\pi$
$733$	23.8693	0.881633	0.440817	+	0.897597i	$0.354689\pi$
$734$	55.0298	2.03119
$735$	0	0
$736$	34.0432	1.25485
$737$	−9.97502	−0.367435
$738$	−3.78222	−0.139226
$739$	−39.8651	−1.46646	−0.733231	−	0.679980i	$-0.761989\pi$
$739$	−39.8651	−1.46646	−0.733231	+	0.679980i	$0.761989\pi$
$740$	0	0
$741$	−32.1579	−1.18135
$742$	0	0
$743$	18.2217	0.668489	0.334245	−	0.942486i	$-0.391519\pi$
$743$	18.2217	0.668489	0.334245	+	0.942486i	$0.391519\pi$
$744$	10.0482	0.368384
$745$	0	0
$746$	−64.9162	−2.37675
$747$	4.52953	0.165727
$748$	1.70403	0.0623056
$749$	0	0
$750$	0	0
$751$	−3.38642	−0.123572	−0.0617861	−	0.998089i	$-0.519680\pi$
$751$	−3.38642	−0.123572	−0.0617861	+	0.998089i	$0.519680\pi$
$752$	−12.0312	−0.438732
$753$	−23.9837	−0.874014
$754$	75.5293	2.75061
$755$	0	0
$756$	0	0
$757$	−8.35419	−0.303638	−0.151819	−	0.988408i	$-0.548513\pi$
$757$	−8.35419	−0.303638	−0.151819	+	0.988408i	$0.548513\pi$
$758$	51.8996	1.88508
$759$	8.86179	0.321662
$760$	0	0
$761$	−24.1698	−0.876154	−0.438077	−	0.898938i	$-0.644340\pi$
$761$	−24.1698	−0.876154	−0.438077	+	0.898938i	$0.644340\pi$
$762$	−24.9683	−0.904505
$763$	0	0
$764$	16.1674	0.584915
$765$	0	0
$766$	−43.0621	−1.55590
$767$	15.0268	0.542587
$768$	7.96365	0.287363
$769$	41.1976	1.48562	0.742811	−	0.669501i	$-0.233492\pi$
$769$	41.1976	1.48562	0.742811	+	0.669501i	$0.233492\pi$
$770$	0	0
$771$	−2.20117	−0.0792732
$772$	−74.1097	−2.66727
$773$	20.0652	0.721694	0.360847	−	0.932625i	$-0.382488\pi$
$773$	20.0652	0.721694	0.360847	+	0.932625i	$0.382488\pi$
$774$	17.9296	0.644466
$775$	0	0
$776$	11.3504	0.407457
$777$	0	0
$778$	4.35561	0.156156
$779$	13.0152	0.466318
$780$	0	0
$781$	11.9730	0.428426
$782$	−3.58313	−0.128132
$783$	8.08223	0.288835
$784$	0	0
$785$	0	0
$786$	20.3138	0.724570
$787$	16.5640	0.590442	0.295221	−	0.955429i	$-0.404607\pi$
$787$	16.5640	0.590442	0.295221	+	0.955429i	$0.404607\pi$
$788$	−60.3460	−2.14974
$789$	14.8311	0.528001
$790$	0	0
$791$	0	0
$792$	3.82843	0.136037
$793$	45.4120	1.61263
$794$	71.3420	2.53183
$795$	0	0
$796$	75.9101	2.69056
$797$	−12.6624	−0.448527	−0.224263	−	0.974529i	$-0.571998\pi$
$797$	−12.6624	−0.448527	−0.224263	+	0.974529i	$0.571998\pi$
$798$	0	0
$799$	3.41676	0.120876
$800$	0	0
$801$	16.2382	0.573748
$802$	1.71251	0.0604707
$803$	−9.59221	−0.338502
$804$	16.6223	0.586223
$805$	0	0
$806$	43.6479	1.53743
$807$	7.30070	0.256997
$808$	−22.6009	−0.795096
$809$	−32.0020	−1.12513	−0.562565	−	0.826753i	$-0.690185\pi$
$809$	−32.0020	−1.12513	−0.562565	+	0.826753i	$0.690185\pi$
$810$	0	0
$811$	46.9147	1.64740	0.823699	−	0.567028i	$-0.191907\pi$
$811$	46.9147	1.64740	0.823699	+	0.567028i	$0.191907\pi$
$812$	0	0
$813$	17.5426	0.615247
$814$	40.7850	1.42951
$815$	0	0
$816$	0.367149	0.0128528
$817$	−61.6984	−2.15855
$818$	−63.7994	−2.23069
$819$	0	0
$820$	0	0
$821$	10.7935	0.376695	0.188348	−	0.982102i	$-0.439687\pi$
$821$	10.7935	0.376695	0.188348	+	0.982102i	$0.439687\pi$
$822$	14.0092	0.488628
$823$	−5.64049	−0.196615	−0.0983075	−	0.995156i	$-0.531343\pi$
$823$	−5.64049	−0.196615	−0.0983075	+	0.995156i	$0.531343\pi$
$824$	−13.9760	−0.486877
$825$	0	0
$826$	0	0
$827$	17.7900	0.618620	0.309310	−	0.950961i	$-0.399902\pi$
$827$	17.7900	0.618620	0.309310	+	0.950961i	$0.399902\pi$
$828$	−14.7672	−0.513196
$829$	27.6764	0.961242	0.480621	−	0.876928i	$-0.340411\pi$
$829$	27.6764	0.961242	0.480621	+	0.876928i	$0.340411\pi$
$830$	0	0
$831$	−5.62169	−0.195014
$832$	54.3493	1.88422
$833$	0	0
$834$	−8.22894	−0.284945
$835$	0	0
$836$	−40.4656	−1.39953
$837$	4.67067	0.161442
$838$	−22.4311	−0.774869
$839$	−30.6423	−1.05789	−0.528945	−	0.848656i	$-0.677412\pi$
$839$	−30.6423	−1.05789	−0.528945	+	0.848656i	$0.677412\pi$
$840$	0	0
$841$	36.3224	1.25250
$842$	37.8032	1.30279
$843$	8.67177	0.298672
$844$	18.2961	0.629779
$845$	0	0
$846$	23.5787	0.810651
$847$	0	0
$848$	6.64702	0.228260
$849$	−28.8572	−0.990377
$850$	0	0
$851$	−51.2172	−1.75570
$852$	−19.9516	−0.683532
$853$	−47.8305	−1.63769	−0.818843	−	0.574017i	$-0.805384\pi$
$853$	−47.8305	−1.63769	−0.818843	+	0.574017i	$0.805384\pi$
$854$	0	0
$855$	0	0
$856$	−25.6397	−0.876346
$857$	20.0572	0.685143	0.342571	−	0.939492i	$-0.388702\pi$
$857$	20.0572	0.685143	0.342571	+	0.939492i	$0.388702\pi$
$858$	16.6302	0.567745
$859$	11.6010	0.395820	0.197910	−	0.980220i	$-0.436585\pi$
$859$	11.6010	0.395820	0.197910	+	0.980220i	$0.436585\pi$
$860$	0	0
$861$	0	0
$862$	25.8592	0.880767
$863$	−44.2115	−1.50498	−0.752489	−	0.658605i	$-0.771147\pi$
$863$	−44.2115	−1.50498	−0.752489	+	0.658605i	$0.771147\pi$
$864$	6.83632	0.232576
$865$	0	0
$866$	−9.88388	−0.335868
$867$	16.8957	0.573809
$868$	0	0
$869$	−5.76378	−0.195523
$870$	0	0
$871$	23.5075	0.796521
$872$	−36.9995	−1.25296
$873$	5.27600	0.178566
$874$	85.0884	2.87816
$875$	0	0
$876$	15.9844	0.540062
$877$	−43.4692	−1.46785	−0.733926	−	0.679230i	$-0.762314\pi$
$877$	−43.4692	−1.46785	−0.733926	+	0.679230i	$0.762314\pi$
$878$	16.2843	0.549570
$879$	−16.9594	−0.572027
$880$	0	0
$881$	−0.238049	−0.00802007	−0.00401003	−	0.999992i	$-0.501276\pi$
$881$	−0.238049	−0.00802007	−0.00401003	+	0.999992i	$0.501276\pi$
$882$	0	0
$883$	20.2839	0.682608	0.341304	−	0.939953i	$-0.389131\pi$
$883$	20.2839	0.682608	0.341304	+	0.939953i	$0.389131\pi$
$884$	−4.01578	−0.135065
$885$	0	0
$886$	29.9062	1.00472
$887$	27.8618	0.935507	0.467754	−	0.883859i	$-0.345063\pi$
$887$	27.8618	0.935507	0.467754	+	0.883859i	$0.345063\pi$
$888$	−22.1266	−0.742521
$889$	0	0
$890$	0	0
$891$	1.77956	0.0596175
$892$	7.30701	0.244657
$893$	−81.1377	−2.71517
$894$	−26.5017	−0.886348
$895$	0	0
$896$	0	0
$897$	−20.8840	−0.697296
$898$	30.4547	1.01628
$899$	37.7494	1.25901
$900$	0	0
$901$	−1.88770	−0.0628885
$902$	−6.73070	−0.224108
$903$	0	0
$904$	32.1393	1.06894
$905$	0	0
$906$	8.93792	0.296943
$907$	38.7727	1.28743	0.643713	−	0.765267i	$-0.277393\pi$
$907$	38.7727	1.28743	0.643713	+	0.765267i	$0.277393\pi$
$908$	59.1558	1.96315
$909$	−10.5055	−0.348446
$910$	0	0
$911$	30.5205	1.01119	0.505594	−	0.862771i	$-0.331273\pi$
$911$	30.5205	1.01119	0.505594	+	0.862771i	$0.331273\pi$
$912$	−8.71868	−0.288704
$913$	8.06057	0.266766
$914$	−1.24591	−0.0412110
$915$	0	0
$916$	−2.42102	−0.0799928
$917$	0	0
$918$	−0.719538	−0.0237483
$919$	17.4712	0.576322	0.288161	−	0.957582i	$-0.406956\pi$
$919$	17.4712	0.576322	0.288161	+	0.957582i	$0.406956\pi$
$920$	0	0
$921$	−25.5898	−0.843213
$922$	−84.9371	−2.79725
$923$	−28.2159	−0.928736
$924$	0	0
$925$	0	0
$926$	−23.9348	−0.786547
$927$	−6.49644	−0.213371
$928$	55.2527	1.81376
$929$	6.35054	0.208354	0.104177	−	0.994559i	$-0.466779\pi$
$929$	6.35054	0.208354	0.104177	+	0.994559i	$0.466779\pi$
$930$	0	0
$931$	0	0
$932$	27.6934	0.907126
$933$	11.2086	0.366952
$934$	−1.51778	−0.0496633
$935$	0	0
$936$	−9.02220	−0.294900
$937$	−3.90227	−0.127482	−0.0637408	−	0.997966i	$-0.520303\pi$
$937$	−3.90227	−0.127482	−0.0637408	+	0.997966i	$0.520303\pi$
$938$	0	0
$939$	−5.98884	−0.195438
$940$	0	0
$941$	−53.7237	−1.75134	−0.875672	−	0.482907i	$-0.839581\pi$
$941$	−53.7237	−1.75134	−0.875672	+	0.482907i	$0.839581\pi$
$942$	1.92436	0.0626990
$943$	8.45232	0.275246
$944$	4.07408	0.132600
$945$	0	0
$946$	31.9068	1.03738
$947$	−21.6762	−0.704383	−0.352192	−	0.935928i	$-0.614563\pi$
$947$	−21.6762	−0.704383	−0.352192	+	0.935928i	$0.614563\pi$
$948$	9.60472	0.311947
$949$	22.6053	0.733800
$950$	0	0
$951$	−19.0060	−0.616312
$952$	0	0
$953$	29.6009	0.958866	0.479433	−	0.877578i	$-0.340842\pi$
$953$	29.6009	0.958866	0.479433	+	0.877578i	$0.340842\pi$
$954$	−13.0268	−0.421759
$955$	0	0
$956$	87.4159	2.82723
$957$	14.3828	0.464930
$958$	−26.3292	−0.850659
$959$	0	0
$960$	0	0
$961$	−9.18482	−0.296285
$962$	−96.1152	−3.09888
$963$	−11.9180	−0.384054
$964$	69.6563	2.24348
$965$	0	0
$966$	0	0
$967$	24.6511	0.792727	0.396363	−	0.918094i	$-0.370272\pi$
$967$	24.6511	0.792727	0.396363	+	0.918094i	$0.370272\pi$
$968$	−16.8517	−0.541635
$969$	2.47604	0.0795418
$970$	0	0
$971$	39.5625	1.26962	0.634811	−	0.772667i	$-0.281078\pi$
$971$	39.5625	1.26962	0.634811	+	0.772667i	$0.281078\pi$
$972$	−2.96545	−0.0951167
$973$	0	0
$974$	−1.98413	−0.0635756
$975$	0	0
$976$	12.3121	0.394102
$977$	25.9256	0.829434	0.414717	−	0.909950i	$-0.363881\pi$
$977$	25.9256	0.829434	0.414717	+	0.909950i	$0.363881\pi$
$978$	5.42451	0.173457
$979$	28.8968	0.923547
$980$	0	0
$981$	−17.1984	−0.549103
$982$	−72.0718	−2.29990
$983$	−28.6108	−0.912543	−0.456272	−	0.889841i	$-0.650815\pi$
$983$	−28.6108	−0.912543	−0.456272	+	0.889841i	$0.650815\pi$
$984$	3.65153	0.116407
$985$	0	0
$986$	−5.81547	−0.185202
$987$	0	0
$988$	95.3626	3.03389
$989$	−40.0682	−1.27409
$990$	0	0
$991$	−15.8985	−0.505033	−0.252517	−	0.967593i	$-0.581258\pi$
$991$	−15.8985	−0.505033	−0.252517	+	0.967593i	$0.581258\pi$
$992$	31.9302	1.01378
$993$	−10.2851	−0.326387
$994$	0	0
$995$	0	0
$996$	−13.4321	−0.425611
$997$	−29.1750	−0.923981	−0.461990	−	0.886885i	$-0.652865\pi$
$997$	−29.1750	−0.923981	−0.461990	+	0.886885i	$0.652865\pi$
$998$	1.52928	0.0484086
$999$	−10.2851	−0.325406

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.by.1.4	yes	4
5.4	even	2		3675.2.a.bo.1.1	yes	4
7.6	odd	2		3675.2.a.ca.1.4	yes	4
35.34	odd	2		3675.2.a.bm.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3675.2.a.bm.1.1	✓	4	35.34	odd	2
3675.2.a.bo.1.1	yes	4	5.4	even	2
3675.2.a.by.1.4	yes	4	1.1	even	1	trivial
3675.2.a.ca.1.4	yes	4	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22833 q^{2} -1.00000 q^{3} +2.96545 q^{4} -2.22833 q^{6} +2.15133 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.22833 q^{2} -1.00000 q^{3} +2.96545 q^{4} -2.22833 q^{6} +2.15133 q^{8} +1.00000 q^{9} +1.77956 q^{11} -2.96545 q^{12} -4.19377 q^{13} -1.13702 q^{16} +0.322905 q^{17} +2.22833 q^{18} -7.66801 q^{19} +3.96545 q^{22} -4.97976 q^{23} -2.15133 q^{24} -9.34511 q^{26} -1.00000 q^{27} -8.08223 q^{29} -4.67067 q^{31} -6.83632 q^{32} -1.77956 q^{33} +0.719538 q^{34} +2.96545 q^{36} +10.2851 q^{37} -17.0868 q^{38} +4.19377 q^{39} -1.69734 q^{41} +8.04621 q^{43} +5.27719 q^{44} -11.0965 q^{46} +10.5813 q^{47} +1.13702 q^{48} -0.322905 q^{51} -12.4364 q^{52} -5.84601 q^{53} -2.22833 q^{54} +7.66801 q^{57} -18.0099 q^{58} -3.58313 q^{59} -10.8284 q^{61} -10.4078 q^{62} -12.9595 q^{64} -3.96545 q^{66} -5.60533 q^{67} +0.957557 q^{68} +4.97976 q^{69} +6.72803 q^{71} +2.15133 q^{72} -5.39021 q^{73} +22.9185 q^{74} -22.7391 q^{76} +9.34511 q^{78} -3.23888 q^{79} +1.00000 q^{81} -3.78222 q^{82} +4.52953 q^{83} +17.9296 q^{86} +8.08223 q^{87} +3.82843 q^{88} +16.2382 q^{89} -14.7672 q^{92} +4.67067 q^{93} +23.5787 q^{94} +6.83632 q^{96} +5.27600 q^{97} +1.77956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^6 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{9} - 4 q^{11} - 2 q^{12} - 6 q^{16} + 4 q^{17} + 2 q^{18} - 8 q^{19} + 6 q^{22} - 12 q^{26} - 4 q^{27} - 4 q^{29} - 8 q^{31} + 2 q^{32} + 4 q^{33} - 8 q^{34} + 2 q^{36} + 16 q^{37} - 4 q^{38} - 24 q^{41} + 20 q^{43} + 14 q^{44} - 6 q^{46} - 8 q^{47} + 6 q^{48} - 4 q^{51} - 16 q^{52} - 20 q^{53} - 2 q^{54} + 8 q^{57} - 6 q^{58} - 8 q^{59} - 32 q^{61} - 28 q^{62} - 12 q^{64} - 6 q^{66} + 12 q^{67} + 12 q^{68} + 4 q^{71} + 34 q^{74} - 40 q^{76} + 12 q^{78} + 4 q^{81} - 16 q^{82} + 20 q^{83} - 14 q^{86} + 4 q^{87} + 4 q^{88} - 8 q^{89} - 10 q^{92} + 8 q^{93} + 32 q^{94} - 2 q^{96} - 24 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^6 + 4 * q^9 - 4 * q^11 - 2 * q^12 - 6 * q^16 + 4 * q^17 + 2 * q^18 - 8 * q^19 + 6 * q^22 - 12 * q^26 - 4 * q^27 - 4 * q^29 - 8 * q^31 + 2 * q^32 + 4 * q^33 - 8 * q^34 + 2 * q^36 + 16 * q^37 - 4 * q^38 - 24 * q^41 + 20 * q^43 + 14 * q^44 - 6 * q^46 - 8 * q^47 + 6 * q^48 - 4 * q^51 - 16 * q^52 - 20 * q^53 - 2 * q^54 + 8 * q^57 - 6 * q^58 - 8 * q^59 - 32 * q^61 - 28 * q^62 - 12 * q^64 - 6 * q^66 + 12 * q^67 + 12 * q^68 + 4 * q^71 + 34 * q^74 - 40 * q^76 + 12 * q^78 + 4 * q^81 - 16 * q^82 + 20 * q^83 - 14 * q^86 + 4 * q^87 + 4 * q^88 - 8 * q^89 - 10 * q^92 + 8 * q^93 + 32 * q^94 - 2 * q^96 - 24 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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