LMFDB - Embedded newform 945.2.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, 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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.144344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 5x + 9 $ x^4 - x^3 - 8x^2 + 5x + 9
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			$-2.46608$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46608 q^{2} +4.08154 q^{4} -1.00000 q^{5} +1.00000 q^{7} +5.13325 q^{8} +O(q^{10})$	\(q+2.46608 q^{2} +4.08154 q^{4} -1.00000 q^{5} +1.00000 q^{7} +5.13325 q^{8} -2.46608 q^{10} +5.54762 q^{11} +1.38453 q^{13} +2.46608 q^{14} +4.49592 q^{16} -5.74872 q^{17} +2.20109 q^{19} -4.08154 q^{20} +13.6809 q^{22} -5.74872 q^{23} +1.00000 q^{25} +3.41437 q^{26} +4.08154 q^{28} +6.16309 q^{29} +1.18344 q^{31} +0.820777 q^{32} -14.1768 q^{34} -1.00000 q^{35} -2.93216 q^{37} +5.42807 q^{38} -5.13325 q^{40} +2.58563 q^{41} +4.18344 q^{43} +22.6429 q^{44} -14.1768 q^{46} -13.2787 q^{47} +1.00000 q^{49} +2.46608 q^{50} +5.65104 q^{52} +11.2963 q^{53} -5.54762 q^{55} +5.13325 q^{56} +15.1987 q^{58} -4.93216 q^{59} +7.74872 q^{61} +2.91846 q^{62} -6.96773 q^{64} -1.38453 q^{65} -6.91181 q^{67} -23.4636 q^{68} -2.46608 q^{70} -16.1905 q^{71} -3.38453 q^{73} -7.23093 q^{74} +8.98387 q^{76} +5.54762 q^{77} -2.11560 q^{79} -4.49592 q^{80} +6.37636 q^{82} -7.76907 q^{83} +5.74872 q^{85} +10.3167 q^{86} +28.4773 q^{88} -12.2108 q^{89} +1.38453 q^{91} -23.4636 q^{92} -32.7463 q^{94} -2.20109 q^{95} -14.4296 q^{97} +2.46608 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	$ 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{14} + 19 q^{16} + 4 q^{19} - 9 q^{20} + 10 q^{22} + 4 q^{25} + 22 q^{26} + 9 q^{28} + 10 q^{29} + 6 q^{31} - 23 q^{32} - 13 q^{34} - 4 q^{35} + 10 q^{37} + q^{38} + 6 q^{40} + 2 q^{41} + 18 q^{43} + 36 q^{44} - 13 q^{46} - 18 q^{47} + 4 q^{49} - q^{50} - 34 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{56} - 14 q^{58} + 2 q^{59} + 8 q^{61} + 19 q^{62} + 54 q^{64} - 2 q^{65} + 10 q^{67} - 13 q^{68} + q^{70} + 8 q^{71} - 10 q^{73} - 36 q^{74} - 5 q^{76} + 4 q^{77} + 12 q^{79} - 19 q^{80} + 24 q^{82} - 24 q^{83} + 16 q^{86} + 4 q^{88} + 8 q^{89} + 2 q^{91} - 13 q^{92} - 38 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) $ 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + q^10 + 4 * q^11 + 2 * q^13 - q^14 + 19 * q^16 + 4 * q^19 - 9 * q^20 + 10 * q^22 + 4 * q^25 + 22 * q^26 + 9 * q^28 + 10 * q^29 + 6 * q^31 - 23 * q^32 - 13 * q^34 - 4 * q^35 + 10 * q^37 + q^38 + 6 * q^40 + 2 * q^41 + 18 * q^43 + 36 * q^44 - 13 * q^46 - 18 * q^47 + 4 * q^49 - q^50 - 34 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^56 - 14 * q^58 + 2 * q^59 + 8 * q^61 + 19 * q^62 + 54 * q^64 - 2 * q^65 + 10 * q^67 - 13 * q^68 + q^70 + 8 * q^71 - 10 * q^73 - 36 * q^74 - 5 * q^76 + 4 * q^77 + 12 * q^79 - 19 * q^80 + 24 * q^82 - 24 * q^83 + 16 * q^86 + 4 * q^88 + 8 * q^89 + 2 * q^91 - 13 * q^92 - 38 * q^94 - 4 * q^95 + 10 * q^97 - q^98

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 $

$2$	2.46608	1.74378	0.871890	−	0.489701i	$-0.162894\pi$
$2$	2.46608	1.74378	0.871890	+	0.489701i	$0.162894\pi$
$3$	0	0
$3$	0	0
$4$	4.08154	2.04077
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.13325	1.81488
$9$	0	0
$10$	−2.46608	−0.779843
$11$	5.54762	1.67267	0.836336	−	0.548218i	$-0.184693\pi$
$11$	5.54762	1.67267	0.836336	+	0.548218i	$0.184693\pi$
$12$	0	0
$13$	1.38453	0.384001	0.192000	−	0.981395i	$-0.438503\pi$
$13$	1.38453	0.384001	0.192000	+	0.981395i	$0.438503\pi$
$14$	2.46608	0.659087
$15$	0	0
$16$	4.49592	1.12398
$17$	−5.74872	−1.39427	−0.697134	−	0.716940i	$-0.745542\pi$
$17$	−5.74872	−1.39427	−0.697134	+	0.716940i	$0.745542\pi$
$18$	0	0
$19$	2.20109	0.504966	0.252483	−	0.967601i	$-0.418753\pi$
$19$	2.20109	0.504966	0.252483	+	0.967601i	$0.418753\pi$
$20$	−4.08154	−0.912661
$21$	0	0
$22$	13.6809	2.91677
$23$	−5.74872	−1.19869	−0.599345	−	0.800491i	$-0.704572\pi$
$23$	−5.74872	−1.19869	−0.599345	+	0.800491i	$0.704572\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.41437	0.669613
$27$	0	0
$28$	4.08154	0.771339
$29$	6.16309	1.14446	0.572228	−	0.820094i	$-0.306079\pi$
$29$	6.16309	1.14446	0.572228	+	0.820094i	$0.306079\pi$
$30$	0	0
$31$	1.18344	0.212552	0.106276	−	0.994337i	$-0.466107\pi$
$31$	1.18344	0.212552	0.106276	+	0.994337i	$0.466107\pi$
$32$	0.820777	0.145094
$33$	0	0
$34$	−14.1768	−2.43130
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−2.93216	−0.482044	−0.241022	−	0.970520i	$-0.577483\pi$
$37$	−2.93216	−0.482044	−0.241022	+	0.970520i	$0.577483\pi$
$38$	5.42807	0.880550
$39$	0	0
$40$	−5.13325	−0.811638
$41$	2.58563	0.403807	0.201904	−	0.979405i	$-0.435287\pi$
$41$	2.58563	0.403807	0.201904	+	0.979405i	$0.435287\pi$
$42$	0	0
$43$	4.18344	0.637969	0.318984	−	0.947760i	$-0.396658\pi$
$43$	4.18344	0.637969	0.318984	+	0.947760i	$0.396658\pi$
$44$	22.6429	3.41354
$45$	0	0
$46$	−14.1768	−2.09025
$47$	−13.2787	−1.93690	−0.968448	−	0.249217i	$-0.919827\pi$
$47$	−13.2787	−1.93690	−0.968448	+	0.249217i	$0.919827\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.46608	0.348756
$51$	0	0
$52$	5.65104	0.783658
$53$	11.2963	1.55167	0.775836	−	0.630935i	$-0.217328\pi$
$53$	11.2963	1.55167	0.775836	+	0.630935i	$0.217328\pi$
$54$	0	0
$55$	−5.54762	−0.748041
$56$	5.13325	0.685960
$57$	0	0
$58$	15.1987	1.99568
$59$	−4.93216	−0.642112	−0.321056	−	0.947060i	$-0.604038\pi$
$59$	−4.93216	−0.642112	−0.321056	+	0.947060i	$0.604038\pi$
$60$	0	0
$61$	7.74872	0.992122	0.496061	−	0.868288i	$-0.334779\pi$
$61$	7.74872	0.992122	0.496061	+	0.868288i	$0.334779\pi$
$62$	2.91846	0.370644
$63$	0	0
$64$	−6.96773	−0.870966
$65$	−1.38453	−0.171730
$66$	0	0
$67$	−6.91181	−0.844412	−0.422206	−	0.906500i	$-0.638744\pi$
$67$	−6.91181	−0.844412	−0.422206	+	0.906500i	$0.638744\pi$
$68$	−23.4636	−2.84538
$69$	0	0
$70$	−2.46608	−0.294753
$71$	−16.1905	−1.92146	−0.960729	−	0.277489i	$-0.910498\pi$
$71$	−16.1905	−1.92146	−0.960729	+	0.277489i	$0.910498\pi$
$72$	0	0
$73$	−3.38453	−0.396130	−0.198065	−	0.980189i	$-0.563466\pi$
$73$	−3.38453	−0.396130	−0.198065	+	0.980189i	$0.563466\pi$
$74$	−7.23093	−0.840579
$75$	0	0
$76$	8.98387	1.03052
$77$	5.54762	0.632210
$78$	0	0
$79$	−2.11560	−0.238023	−0.119012	−	0.992893i	$-0.537973\pi$
$79$	−2.11560	−0.238023	−0.119012	+	0.992893i	$0.537973\pi$
$80$	−4.49592	−0.502659
$81$	0	0
$82$	6.37636	0.704152
$83$	−7.76907	−0.852766	−0.426383	−	0.904543i	$-0.640212\pi$
$83$	−7.76907	−0.852766	−0.426383	+	0.904543i	$0.640212\pi$
$84$	0	0
$85$	5.74872	0.623536
$86$	10.3167	1.11248
$87$	0	0
$88$	28.4773	3.03570
$89$	−12.2108	−1.29435	−0.647173	−	0.762343i	$-0.724049\pi$
$89$	−12.2108	−1.29435	−0.647173	+	0.762343i	$0.724049\pi$
$90$	0	0
$91$	1.38453	0.145139
$92$	−23.4636	−2.44625
$93$	0	0
$94$	−32.7463	−3.37752
$95$	−2.20109	−0.225828
$96$	0	0
$97$	−14.4296	−1.46510	−0.732552	−	0.680712i	$-0.761671\pi$
$97$	−14.4296	−1.46510	−0.732552	+	0.680712i	$0.761671\pi$
$98$	2.46608	0.249112
$99$	0	0
$100$	4.08154	0.408154
$101$	−2.74872	−0.273508	−0.136754	−	0.990605i	$-0.543667\pi$
$101$	−2.74872	−0.273508	−0.136754	+	0.990605i	$0.543667\pi$
$102$	0	0
$103$	17.6252	1.73666	0.868332	−	0.495983i	$-0.165192\pi$
$103$	17.6252	1.73666	0.868332	+	0.495983i	$0.165192\pi$
$104$	7.10716	0.696915
$105$	0	0
$106$	27.8577	2.70578
$107$	−11.0952	−1.07262	−0.536309	−	0.844022i	$-0.680182\pi$
$107$	−11.0952	−1.07262	−0.536309	+	0.844022i	$0.680182\pi$
$108$	0	0
$109$	−15.9596	−1.52865	−0.764324	−	0.644832i	$-0.776927\pi$
$109$	−15.9596	−1.52865	−0.764324	+	0.644832i	$0.776927\pi$
$110$	−13.6809	−1.30442
$111$	0	0
$112$	4.49592	0.424824
$113$	−9.24885	−0.870059	−0.435029	−	0.900416i	$-0.643262\pi$
$113$	−9.24885	−0.870059	−0.435029	+	0.900416i	$0.643262\pi$
$114$	0	0
$115$	5.74872	0.536071
$116$	25.1549	2.33558
$117$	0	0
$118$	−12.1631	−1.11970
$119$	−5.74872	−0.526984
$120$	0	0
$121$	19.7761	1.79783
$122$	19.1089	1.73004
$123$	0	0
$124$	4.83026	0.433770
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.1509	0.989483	0.494741	−	0.869040i	$-0.335263\pi$
$127$	11.1509	0.989483	0.494741	+	0.869040i	$0.335263\pi$
$128$	−18.8245	−1.66387
$129$	0	0
$130$	−3.41437	−0.299460
$131$	−3.29904	−0.288238	−0.144119	−	0.989560i	$-0.546035\pi$
$131$	−3.29904	−0.288238	−0.144119	+	0.989560i	$0.546035\pi$
$132$	0	0
$133$	2.20109	0.190859
$134$	−17.0451	−1.47247
$135$	0	0
$136$	−29.5096	−2.53043
$137$	−11.2963	−0.965112	−0.482556	−	0.875865i	$-0.660291\pi$
$137$	−11.2963	−0.965112	−0.482556	+	0.875865i	$0.660291\pi$
$138$	0	0
$139$	13.4974	1.14484	0.572419	−	0.819961i	$-0.306005\pi$
$139$	13.4974	1.14484	0.572419	+	0.819961i	$0.306005\pi$
$140$	−4.08154	−0.344953
$141$	0	0
$142$	−39.9270	−3.35060
$143$	7.68088	0.642307
$144$	0	0
$145$	−6.16309	−0.511817
$146$	−8.34653	−0.690764
$147$	0	0
$148$	−11.9677	−0.983741
$149$	12.4022	1.01603	0.508013	−	0.861349i	$-0.330380\pi$
$149$	12.4022	1.01603	0.508013	+	0.861349i	$0.330380\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	11.2988	0.916452
$153$	0	0
$154$	13.6809	1.10244
$155$	−1.18344	−0.0950562
$156$	0	0
$157$	7.25833	0.579278	0.289639	−	0.957136i	$-0.406465\pi$
$157$	7.25833	0.579278	0.289639	+	0.957136i	$0.406465\pi$
$158$	−5.21723	−0.415060
$159$	0	0
$160$	−0.820777	−0.0648881
$161$	−5.74872	−0.453062
$162$	0	0
$163$	12.8561	1.00697	0.503486	−	0.864004i	$-0.332051\pi$
$163$	12.8561	1.00697	0.503486	+	0.864004i	$0.332051\pi$
$164$	10.5534	0.824079
$165$	0	0
$166$	−19.1591	−1.48704
$167$	21.0478	1.62872	0.814362	−	0.580357i	$-0.197087\pi$
$167$	21.0478	1.62872	0.814362	+	0.580357i	$0.197087\pi$
$168$	0	0
$169$	−11.0831	−0.852543
$170$	14.1768	1.08731
$171$	0	0
$172$	17.0749	1.30195
$173$	6.97965	0.530653	0.265326	−	0.964159i	$-0.414520\pi$
$173$	6.97965	0.530653	0.265326	+	0.964159i	$0.414520\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	24.9416	1.88005
$177$	0	0
$178$	−30.1129	−2.25706
$179$	26.3441	1.96905	0.984525	−	0.175244i	$-0.0560714\pi$
$179$	26.3441	1.96905	0.984525	+	0.175244i	$0.0560714\pi$
$180$	0	0
$181$	21.0752	1.56650	0.783252	−	0.621704i	$-0.213559\pi$
$181$	21.0752	1.56650	0.783252	+	0.621704i	$0.213559\pi$
$182$	3.41437	0.253090
$183$	0	0
$184$	−29.5096	−2.17548
$185$	2.93216	0.215577
$186$	0	0
$187$	−31.8917	−2.33215
$188$	−54.1975	−3.95276
$189$	0	0
$190$	−5.42807	−0.393794
$191$	−0.854566	−0.0618342	−0.0309171	−	0.999522i	$-0.509843\pi$
$191$	−0.854566	−0.0618342	−0.0309171	+	0.999522i	$0.509843\pi$
$192$	0	0
$193$	16.6334	1.19730	0.598649	−	0.801012i	$-0.295705\pi$
$193$	16.6334	1.19730	0.598649	+	0.801012i	$0.295705\pi$
$194$	−35.5845	−2.55482
$195$	0	0
$196$	4.08154	0.291539
$197$	6.52727	0.465049	0.232524	−	0.972591i	$-0.425301\pi$
$197$	6.52727	0.465049	0.232524	+	0.972591i	$0.425301\pi$
$198$	0	0
$199$	3.84640	0.272664	0.136332	−	0.990663i	$-0.456469\pi$
$199$	3.84640	0.272664	0.136332	+	0.990663i	$0.456469\pi$
$200$	5.13325	0.362976
$201$	0	0
$202$	−6.77855	−0.476937
$203$	6.16309	0.432564
$204$	0	0
$205$	−2.58563	−0.180588
$206$	43.4652	3.02836
$207$	0	0
$208$	6.22475	0.431609
$209$	12.2108	0.844642
$210$	0	0
$211$	−0.210580	−0.0144969	−0.00724846	−	0.999974i	$-0.502307\pi$
$211$	−0.210580	−0.0144969	−0.00724846	+	0.999974i	$0.502307\pi$
$212$	46.1065	3.16661
$213$	0	0
$214$	−27.3618	−1.87041
$215$	−4.18344	−0.285308
$216$	0	0
$217$	1.18344	0.0803371
$218$	−39.3575	−2.66563
$219$	0	0
$220$	−22.6429	−1.52658
$221$	−7.95930	−0.535400
$222$	0	0
$223$	−6.39428	−0.428193	−0.214096	−	0.976813i	$-0.568681\pi$
$223$	−6.39428	−0.428193	−0.214096	+	0.976813i	$0.568681\pi$
$224$	0.820777	0.0548404
$225$	0	0
$226$	−22.8084	−1.51719
$227$	24.7639	1.64364	0.821820	−	0.569747i	$-0.192959\pi$
$227$	24.7639	1.64364	0.821820	+	0.569747i	$0.192959\pi$
$228$	0	0
$229$	−21.3465	−1.41062	−0.705309	−	0.708900i	$-0.749192\pi$
$229$	−21.3465	−1.41062	−0.705309	+	0.708900i	$0.749192\pi$
$230$	14.1768	0.934790
$231$	0	0
$232$	31.6367	2.07705
$233$	−7.01765	−0.459742	−0.229871	−	0.973221i	$-0.573830\pi$
$233$	−7.01765	−0.459742	−0.229871	+	0.973221i	$0.573830\pi$
$234$	0	0
$235$	13.2787	0.866206
$236$	−20.1308	−1.31040
$237$	0	0
$238$	−14.1768	−0.918945
$239$	−3.86431	−0.249962	−0.124981	−	0.992159i	$-0.539887\pi$
$239$	−3.86431	−0.249962	−0.124981	+	0.992159i	$0.539887\pi$
$240$	0	0
$241$	6.44177	0.414951	0.207476	−	0.978240i	$-0.433475\pi$
$241$	6.44177	0.414951	0.207476	+	0.978240i	$0.433475\pi$
$242$	48.7695	3.13502
$243$	0	0
$244$	31.6267	2.02469
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	3.04749	0.193907
$248$	6.07489	0.385756
$249$	0	0
$250$	−2.46608	−0.155969
$251$	−12.5653	−0.793113	−0.396557	−	0.918010i	$-0.629795\pi$
$251$	−12.5653	−0.793113	−0.396557	+	0.918010i	$0.629795\pi$
$252$	0	0
$253$	−31.8917	−2.00502
$254$	27.4990	1.72544
$255$	0	0
$256$	−32.4873	−2.03046
$257$	−6.15091	−0.383683	−0.191842	−	0.981426i	$-0.561446\pi$
$257$	−6.15091	−0.383683	−0.191842	+	0.981426i	$0.561446\pi$
$258$	0	0
$259$	−2.93216	−0.182195
$260$	−5.65104	−0.350463
$261$	0	0
$262$	−8.13569	−0.502624
$263$	4.69306	0.289386	0.144693	−	0.989477i	$-0.453781\pi$
$263$	4.69306	0.289386	0.144693	+	0.989477i	$0.453781\pi$
$264$	0	0
$265$	−11.2963	−0.693929
$266$	5.42807	0.332816
$267$	0	0
$268$	−28.2108	−1.72325
$269$	−25.2936	−1.54218	−0.771090	−	0.636726i	$-0.780288\pi$
$269$	−25.2936	−1.54218	−0.771090	+	0.636726i	$0.780288\pi$
$270$	0	0
$271$	−9.37235	−0.569330	−0.284665	−	0.958627i	$-0.591882\pi$
$271$	−9.37235	−0.569330	−0.284665	+	0.958627i	$0.591882\pi$
$272$	−25.8457	−1.56713
$273$	0	0
$274$	−27.8577	−1.68294
$275$	5.54762	0.334534
$276$	0	0
$277$	15.1306	0.909107	0.454553	−	0.890719i	$-0.349799\pi$
$277$	15.1306	0.909107	0.454553	+	0.890719i	$0.349799\pi$
$278$	33.2857	1.99635
$279$	0	0
$280$	−5.13325	−0.306770
$281$	20.8725	1.24515	0.622574	−	0.782561i	$-0.286087\pi$
$281$	20.8725	1.24515	0.622574	+	0.782561i	$0.286087\pi$
$282$	0	0
$283$	4.03531	0.239874	0.119937	−	0.992781i	$-0.461731\pi$
$283$	4.03531	0.239874	0.119937	+	0.992781i	$0.461731\pi$
$284$	−66.0822	−3.92126
$285$	0	0
$286$	18.9416	1.12004
$287$	2.58563	0.152625
$288$	0	0
$289$	16.0478	0.943986
$290$	−15.1987	−0.892496
$291$	0	0
$292$	−13.8141	−0.808411
$293$	15.6130	0.912123	0.456062	−	0.889948i	$-0.349260\pi$
$293$	15.6130	0.912123	0.456062	+	0.889948i	$0.349260\pi$
$294$	0	0
$295$	4.93216	0.287161
$296$	−15.0515	−0.874851
$297$	0	0
$298$	30.5848	1.77173
$299$	−7.95930	−0.460298
$300$	0	0
$301$	4.18344	0.241129
$302$	19.7286	1.13526
$303$	0	0
$304$	9.89593	0.567571
$305$	−7.74872	−0.443690
$306$	0	0
$307$	13.7609	0.785376	0.392688	−	0.919672i	$-0.371545\pi$
$307$	13.7609	0.785376	0.392688	+	0.919672i	$0.371545\pi$
$308$	22.6429	1.29020
$309$	0	0
$310$	−2.91846	−0.165757
$311$	34.1905	1.93876	0.969382	−	0.245556i	$-0.0789704\pi$
$311$	34.1905	1.93876	0.969382	+	0.245556i	$0.0789704\pi$
$312$	0	0
$313$	−8.15360	−0.460869	−0.230434	−	0.973088i	$-0.574015\pi$
$313$	−8.15360	−0.460869	−0.230434	+	0.973088i	$0.574015\pi$
$314$	17.8996	1.01013
$315$	0	0
$316$	−8.63490	−0.485751
$317$	−20.4323	−1.14759	−0.573796	−	0.818998i	$-0.694530\pi$
$317$	−20.4323	−1.14759	−0.573796	+	0.818998i	$0.694530\pi$
$318$	0	0
$319$	34.1905	1.91430
$320$	6.96773	0.389508
$321$	0	0
$322$	−14.1768	−0.790042
$323$	−12.6535	−0.704058
$324$	0	0
$325$	1.38453	0.0768002
$326$	31.7043	1.75594
$327$	0	0
$328$	13.2727	0.732861
$329$	−13.2787	−0.732078
$330$	0	0
$331$	−8.69306	−0.477814	−0.238907	−	0.971043i	$-0.576789\pi$
$331$	−8.69306	−0.477814	−0.238907	+	0.971043i	$0.576789\pi$
$332$	−31.7098	−1.74030
$333$	0	0
$334$	51.9054	2.84014
$335$	6.91181	0.377632
$336$	0	0
$337$	3.63312	0.197909	0.0989543	−	0.995092i	$-0.468450\pi$
$337$	3.63312	0.197909	0.0989543	+	0.995092i	$0.468450\pi$
$338$	−27.3317	−1.48665
$339$	0	0
$340$	23.4636	1.27249
$341$	6.56528	0.355530
$342$	0	0
$343$	1.00000	0.0539949
$344$	21.4747	1.15784
$345$	0	0
$346$	17.2124	0.925342
$347$	8.95956	0.480974	0.240487	−	0.970652i	$-0.422693\pi$
$347$	8.95956	0.480974	0.240487	+	0.970652i	$0.422693\pi$
$348$	0	0
$349$	−8.94434	−0.478779	−0.239390	−	0.970924i	$-0.576947\pi$
$349$	−8.94434	−0.478779	−0.239390	+	0.970924i	$0.576947\pi$
$350$	2.46608	0.131817
$351$	0	0
$352$	4.55336	0.242695
$353$	6.90475	0.367503	0.183751	−	0.982973i	$-0.441176\pi$
$353$	6.90475	0.367503	0.183751	+	0.982973i	$0.441176\pi$
$354$	0	0
$355$	16.1905	0.859302
$356$	−49.8391	−2.64147
$357$	0	0
$358$	64.9666	3.43359
$359$	24.0369	1.26862	0.634309	−	0.773079i	$-0.281285\pi$
$359$	24.0369	1.26862	0.634309	+	0.773079i	$0.281285\pi$
$360$	0	0
$361$	−14.1552	−0.745010
$362$	51.9730	2.73164
$363$	0	0
$364$	5.65104	0.296195
$365$	3.38453	0.177155
$366$	0	0
$367$	24.6931	1.28897	0.644484	−	0.764618i	$-0.277072\pi$
$367$	24.6931	1.28897	0.644484	+	0.764618i	$0.277072\pi$
$368$	−25.8457	−1.34730
$369$	0	0
$370$	7.23093	0.375918
$371$	11.2963	0.586477
$372$	0	0
$373$	−34.1498	−1.76821	−0.884105	−	0.467289i	$-0.845231\pi$
$373$	−34.1498	−1.76821	−0.884105	+	0.467289i	$0.845231\pi$
$374$	−78.6475	−4.06676
$375$	0	0
$376$	−68.1628	−3.51523
$377$	8.53301	0.439472
$378$	0	0
$379$	6.19161	0.318042	0.159021	−	0.987275i	$-0.449166\pi$
$379$	6.19161	0.318042	0.159021	+	0.987275i	$0.449166\pi$
$380$	−8.98387	−0.460863
$381$	0	0
$382$	−2.10743	−0.107825
$383$	−9.87650	−0.504665	−0.252333	−	0.967641i	$-0.581198\pi$
$383$	−9.87650	−0.504665	−0.252333	+	0.967641i	$0.581198\pi$
$384$	0	0
$385$	−5.54762	−0.282733
$386$	41.0192	2.08782
$387$	0	0
$388$	−58.8950	−2.98994
$389$	21.7012	1.10030	0.550148	−	0.835067i	$-0.314571\pi$
$389$	21.7012	1.10030	0.550148	+	0.835067i	$0.314571\pi$
$390$	0	0
$391$	33.0478	1.67130
$392$	5.13325	0.259268
$393$	0	0
$394$	16.0968	0.810943
$395$	2.11560	0.106447
$396$	0	0
$397$	5.62521	0.282321	0.141161	−	0.989987i	$-0.454917\pi$
$397$	5.62521	0.282321	0.141161	+	0.989987i	$0.454917\pi$
$398$	9.48552	0.475466
$399$	0	0
$400$	4.49592	0.224796
$401$	−32.4570	−1.62083	−0.810413	−	0.585860i	$-0.800757\pi$
$401$	−32.4570	−1.62083	−0.810413	+	0.585860i	$0.800757\pi$
$402$	0	0
$403$	1.63851	0.0816202
$404$	−11.2190	−0.558167
$405$	0	0
$406$	15.1987	0.754297
$407$	−16.2665	−0.806301
$408$	0	0
$409$	−8.19161	−0.405049	−0.202525	−	0.979277i	$-0.564915\pi$
$409$	−8.19161	−0.405049	−0.202525	+	0.979277i	$0.564915\pi$
$410$	−6.37636	−0.314906
$411$	0	0
$412$	71.9381	3.54414
$413$	−4.93216	−0.242696
$414$	0	0
$415$	7.76907	0.381369
$416$	1.13639	0.0557163
$417$	0	0
$418$	30.1129	1.47287
$419$	15.7772	0.770769	0.385384	−	0.922756i	$-0.374069\pi$
$419$	15.7772	0.770769	0.385384	+	0.922756i	$0.374069\pi$
$420$	0	0
$421$	11.0955	0.540762	0.270381	−	0.962753i	$-0.412850\pi$
$421$	11.0955	0.540762	0.270381	+	0.962753i	$0.412850\pi$
$422$	−0.519307	−0.0252795
$423$	0	0
$424$	57.9870	2.81610
$425$	−5.74872	−0.278854
$426$	0	0
$427$	7.74872	0.374987
$428$	−45.2857	−2.18897
$429$	0	0
$430$	−10.3167	−0.497515
$431$	−0.213276	−0.0102732	−0.00513658	−	0.999987i	$-0.501635\pi$
$431$	−0.213276	−0.0102732	−0.00513658	+	0.999987i	$0.501635\pi$
$432$	0	0
$433$	18.5425	0.891095	0.445548	−	0.895258i	$-0.353009\pi$
$433$	18.5425	0.891095	0.445548	+	0.895258i	$0.353009\pi$
$434$	2.91846	0.140090
$435$	0	0
$436$	−65.1397	−3.11962
$437$	−12.6535	−0.605298
$438$	0	0
$439$	−36.2559	−1.73040	−0.865200	−	0.501427i	$-0.832809\pi$
$439$	−36.2559	−1.73040	−0.865200	+	0.501427i	$0.832809\pi$
$440$	−28.4773	−1.35760
$441$	0	0
$442$	−19.6283	−0.933621
$443$	−27.2108	−1.29283	−0.646413	−	0.762988i	$-0.723732\pi$
$443$	−27.2108	−1.29283	−0.646413	+	0.762988i	$0.723732\pi$
$444$	0	0
$445$	12.2108	0.578849
$446$	−15.7688	−0.746675
$447$	0	0
$448$	−6.96773	−0.329194
$449$	−9.96469	−0.470263	−0.235131	−	0.971964i	$-0.575552\pi$
$449$	−9.96469	−0.470263	−0.235131	+	0.971964i	$0.575552\pi$
$450$	0	0
$451$	14.3441	0.675437
$452$	−37.7496	−1.77559
$453$	0	0
$454$	61.0698	2.86615
$455$	−1.38453	−0.0649080
$456$	0	0
$457$	−23.4895	−1.09879	−0.549397	−	0.835562i	$-0.685142\pi$
$457$	−23.4895	−1.09879	−0.549397	+	0.835562i	$0.685142\pi$
$458$	−52.6422	−2.45981
$459$	0	0
$460$	23.4636	1.09400
$461$	33.7357	1.57123	0.785614	−	0.618717i	$-0.212347\pi$
$461$	33.7357	1.57123	0.785614	+	0.618717i	$0.212347\pi$
$462$	0	0
$463$	−32.5644	−1.51340	−0.756699	−	0.653764i	$-0.773189\pi$
$463$	−32.5644	−1.51340	−0.756699	+	0.653764i	$0.773189\pi$
$464$	27.7087	1.28635
$465$	0	0
$466$	−17.3061	−0.801689
$467$	−34.1783	−1.58158	−0.790792	−	0.612085i	$-0.790331\pi$
$467$	−34.1783	−1.58158	−0.790792	+	0.612085i	$0.790331\pi$
$468$	0	0
$469$	−6.91181	−0.319158
$470$	32.7463	1.51047
$471$	0	0
$472$	−25.3180	−1.16536
$473$	23.2081	1.06711
$474$	0	0
$475$	2.20109	0.100993
$476$	−23.4636	−1.07545
$477$	0	0
$478$	−9.52970	−0.435879
$479$	10.9322	0.499503	0.249752	−	0.968310i	$-0.419651\pi$
$479$	10.9322	0.499503	0.249752	+	0.968310i	$0.419651\pi$
$480$	0	0
$481$	−4.05967	−0.185105
$482$	15.8859	0.723584
$483$	0	0
$484$	80.7171	3.66896
$485$	14.4296	0.655214
$486$	0	0
$487$	−41.7631	−1.89247	−0.946233	−	0.323486i	$-0.895145\pi$
$487$	−41.7631	−1.89247	−0.946233	+	0.323486i	$0.895145\pi$
$488$	39.7761	1.80058
$489$	0	0
$490$	−2.46608	−0.111406
$491$	7.80951	0.352438	0.176219	−	0.984351i	$-0.443613\pi$
$491$	7.80951	0.352438	0.176219	+	0.984351i	$0.443613\pi$
$492$	0	0
$493$	−35.4299	−1.59568
$494$	7.51535	0.338132
$495$	0	0
$496$	5.32064	0.238904
$497$	−16.1905	−0.726243
$498$	0	0
$499$	−19.0150	−0.851226	−0.425613	−	0.904905i	$-0.639942\pi$
$499$	−19.0150	−0.851226	−0.425613	+	0.904905i	$0.639942\pi$
$500$	−4.08154	−0.182532
$501$	0	0
$502$	−30.9870	−1.38302
$503$	21.0955	0.940602	0.470301	−	0.882506i	$-0.344145\pi$
$503$	21.0955	0.940602	0.470301	+	0.882506i	$0.344145\pi$
$504$	0	0
$505$	2.74872	0.122316
$506$	−78.6475	−3.49631
$507$	0	0
$508$	45.5129	2.01931
$509$	26.8835	1.19159	0.595796	−	0.803136i	$-0.296837\pi$
$509$	26.8835	1.19159	0.595796	+	0.803136i	$0.296837\pi$
$510$	0	0
$511$	−3.38453	−0.149723
$512$	−42.4672	−1.87680
$513$	0	0
$514$	−15.1686	−0.669059
$515$	−17.6252	−0.776660
$516$	0	0
$517$	−73.6651	−3.23979
$518$	−7.23093	−0.317709
$519$	0	0
$520$	−7.10716	−0.311670
$521$	9.87927	0.432819	0.216409	−	0.976303i	$-0.430565\pi$
$521$	9.87927	0.432819	0.216409	+	0.976303i	$0.430565\pi$
$522$	0	0
$523$	−3.43472	−0.150190	−0.0750950	−	0.997176i	$-0.523926\pi$
$523$	−3.43472	−0.150190	−0.0750950	+	0.997176i	$0.523926\pi$
$524$	−13.4652	−0.588229
$525$	0	0
$526$	11.5734	0.504626
$527$	−6.80326	−0.296355
$528$	0	0
$529$	10.0478	0.436859
$530$	−27.8577	−1.21006
$531$	0	0
$532$	8.98387	0.389500
$533$	3.57989	0.155062
$534$	0	0
$535$	11.0952	0.479689
$536$	−35.4800	−1.53250
$537$	0	0
$538$	−62.3761	−2.68923
$539$	5.54762	0.238953
$540$	0	0
$541$	22.0071	0.946157	0.473079	−	0.881020i	$-0.343143\pi$
$541$	22.0071	0.946157	0.473079	+	0.881020i	$0.343143\pi$
$542$	−23.1130	−0.992787
$543$	0	0
$544$	−4.71841	−0.202300
$545$	15.9596	0.683632
$546$	0	0
$547$	−22.4649	−0.960530	−0.480265	−	0.877123i	$-0.659459\pi$
$547$	−22.4649	−0.960530	−0.480265	+	0.877123i	$0.659459\pi$
$548$	−46.1065	−1.96957
$549$	0	0
$550$	13.6809	0.583354
$551$	13.5655	0.577911
$552$	0	0
$553$	−2.11560	−0.0899643
$554$	37.3131	1.58528
$555$	0	0
$556$	55.0904	2.33635
$557$	−16.7691	−0.710528	−0.355264	−	0.934766i	$-0.615609\pi$
$557$	−16.7691	−0.710528	−0.355264	+	0.934766i	$0.615609\pi$
$558$	0	0
$559$	5.79212	0.244980
$560$	−4.49592	−0.189987
$561$	0	0
$562$	51.4732	2.17127
$563$	46.5644	1.96246	0.981228	−	0.192850i	$-0.0617731\pi$
$563$	46.5644	1.96246	0.981228	+	0.192850i	$0.0617731\pi$
$564$	0	0
$565$	9.24885	0.389102
$566$	9.95139	0.418288
$567$	0	0
$568$	−83.1099	−3.48721
$569$	25.1306	1.05353	0.526764	−	0.850012i	$-0.323405\pi$
$569$	25.1306	1.05353	0.526764	+	0.850012i	$0.323405\pi$
$570$	0	0
$571$	41.9148	1.75408	0.877041	−	0.480416i	$-0.159514\pi$
$571$	41.9148	1.75408	0.877041	+	0.480416i	$0.159514\pi$
$572$	31.3498	1.31080
$573$	0	0
$574$	6.37636	0.266144
$575$	−5.74872	−0.239738
$576$	0	0
$577$	−25.0097	−1.04117	−0.520585	−	0.853810i	$-0.674286\pi$
$577$	−25.0097	−1.04117	−0.520585	+	0.853810i	$0.674286\pi$
$578$	39.5750	1.64610
$579$	0	0
$580$	−25.1549	−1.04450
$581$	−7.76907	−0.322315
$582$	0	0
$583$	62.6678	2.59544
$584$	−17.3737	−0.718928
$585$	0	0
$586$	38.5030	1.59054
$587$	−15.2784	−0.630608	−0.315304	−	0.948991i	$-0.602106\pi$
$587$	−15.2784	−0.630608	−0.315304	+	0.948991i	$0.602106\pi$
$588$	0	0
$589$	2.60486	0.107332
$590$	12.1631	0.500746
$591$	0	0
$592$	−13.1827	−0.541807
$593$	41.3660	1.69870	0.849350	−	0.527831i	$-0.176994\pi$
$593$	41.3660	1.69870	0.849350	+	0.527831i	$0.176994\pi$
$594$	0	0
$595$	5.74872	0.235674
$596$	50.6201	2.07348
$597$	0	0
$598$	−19.6283	−0.802659
$599$	3.15334	0.128842	0.0644210	−	0.997923i	$-0.479480\pi$
$599$	3.15334	0.128842	0.0644210	+	0.997923i	$0.479480\pi$
$600$	0	0
$601$	−17.4822	−0.713114	−0.356557	−	0.934274i	$-0.616050\pi$
$601$	−17.4822	−0.713114	−0.356557	+	0.934274i	$0.616050\pi$
$602$	10.3167	0.420477
$603$	0	0
$604$	32.6524	1.32861
$605$	−19.7761	−0.804014
$606$	0	0
$607$	33.6283	1.36493	0.682464	−	0.730919i	$-0.260908\pi$
$607$	33.6283	1.36493	0.682464	+	0.730919i	$0.260908\pi$
$608$	1.80661	0.0732676
$609$	0	0
$610$	−19.1089	−0.773699
$611$	−18.3848	−0.743769
$612$	0	0
$613$	1.68489	0.0680520	0.0340260	−	0.999421i	$-0.489167\pi$
$613$	1.68489	0.0680520	0.0340260	+	0.999421i	$0.489167\pi$
$614$	33.9355	1.36952
$615$	0	0
$616$	28.4773	1.14739
$617$	−8.22388	−0.331081	−0.165540	−	0.986203i	$-0.552937\pi$
$617$	−8.22388	−0.331081	−0.165540	+	0.986203i	$0.552937\pi$
$618$	0	0
$619$	19.3085	0.776075	0.388038	−	0.921644i	$-0.373153\pi$
$619$	19.3085	0.776075	0.388038	+	0.921644i	$0.373153\pi$
$620$	−4.83026	−0.193988
$621$	0	0
$622$	84.3164	3.38078
$623$	−12.2108	−0.489217
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.1074	−0.803654
$627$	0	0
$628$	29.6252	1.18217
$629$	16.8561	0.672099
$630$	0	0
$631$	−7.71341	−0.307066	−0.153533	−	0.988144i	$-0.549065\pi$
$631$	−7.71341	−0.307066	−0.153533	+	0.988144i	$0.549065\pi$
$632$	−10.8599	−0.431983
$633$	0	0
$634$	−50.3876	−2.00115
$635$	−11.1509	−0.442510
$636$	0	0
$637$	1.38453	0.0548573
$638$	84.3164	3.33812
$639$	0	0
$640$	18.8245	0.744105
$641$	31.9677	1.26265	0.631325	−	0.775519i	$-0.282512\pi$
$641$	31.9677	1.26265	0.631325	+	0.775519i	$0.282512\pi$
$642$	0	0
$643$	−11.1634	−0.440240	−0.220120	−	0.975473i	$-0.570645\pi$
$643$	−11.1634	−0.440240	−0.220120	+	0.975473i	$0.570645\pi$
$644$	−23.4636	−0.924597
$645$	0	0
$646$	−31.2045	−1.22772
$647$	−12.2120	−0.480102	−0.240051	−	0.970760i	$-0.577164\pi$
$647$	−12.2120	−0.480102	−0.240051	+	0.970760i	$0.577164\pi$
$648$	0	0
$649$	−27.3618	−1.07404
$650$	3.41437	0.133923
$651$	0	0
$652$	52.4729	2.05500
$653$	4.93327	0.193054	0.0965270	−	0.995330i	$-0.469227\pi$
$653$	4.93327	0.193054	0.0965270	+	0.995330i	$0.469227\pi$
$654$	0	0
$655$	3.29904	0.128904
$656$	11.6248	0.453871
$657$	0	0
$658$	−32.7463	−1.27658
$659$	−6.30181	−0.245484	−0.122742	−	0.992439i	$-0.539169\pi$
$659$	−6.30181	−0.245484	−0.122742	+	0.992439i	$0.539169\pi$
$660$	0	0
$661$	13.0952	0.509346	0.254673	−	0.967027i	$-0.418032\pi$
$661$	13.0952	0.509346	0.254673	+	0.967027i	$0.418032\pi$
$662$	−21.4378	−0.833202
$663$	0	0
$664$	−39.8806	−1.54767
$665$	−2.20109	−0.0853548
$666$	0	0
$667$	−35.4299	−1.37185
$668$	85.9073	3.32385
$669$	0	0
$670$	17.0451	0.658508
$671$	42.9870	1.65949
$672$	0	0
$673$	−20.4540	−0.788442	−0.394221	−	0.919016i	$-0.628986\pi$
$673$	−20.4540	−0.788442	−0.394221	+	0.919016i	$0.628986\pi$
$674$	8.95956	0.345109
$675$	0	0
$676$	−45.2360	−1.73985
$677$	24.1715	0.928987	0.464494	−	0.885577i	$-0.346236\pi$
$677$	24.1715	0.928987	0.464494	+	0.885577i	$0.346236\pi$
$678$	0	0
$679$	−14.4296	−0.553757
$680$	29.5096	1.13164
$681$	0	0
$682$	16.1905	0.619966
$683$	−22.9443	−0.877941	−0.438970	−	0.898502i	$-0.644657\pi$
$683$	−22.9443	−0.877941	−0.438970	+	0.898502i	$0.644657\pi$
$684$	0	0
$685$	11.2963	0.431611
$686$	2.46608	0.0941553
$687$	0	0
$688$	18.8084	0.717063
$689$	15.6402	0.595843
$690$	0	0
$691$	7.19597	0.273747	0.136874	−	0.990589i	$-0.456295\pi$
$691$	7.19597	0.273747	0.136874	+	0.990589i	$0.456295\pi$
$692$	28.4877	1.08294
$693$	0	0
$694$	22.0950	0.838714
$695$	−13.4974	−0.511987
$696$	0	0
$697$	−14.8641	−0.563016
$698$	−22.0574	−0.834887
$699$	0	0
$700$	4.08154	0.154268
$701$	−51.9897	−1.96363	−0.981813	−	0.189852i	$-0.939199\pi$
$701$	−51.9897	−1.96363	−0.981813	+	0.189852i	$0.939199\pi$
$702$	0	0
$703$	−6.45396	−0.243416
$704$	−38.6543	−1.45684
$705$	0	0
$706$	17.0277	0.640845
$707$	−2.74872	−0.103376
$708$	0	0
$709$	34.3332	1.28941	0.644706	−	0.764431i	$-0.276980\pi$
$709$	34.3332	1.28941	0.644706	+	0.764431i	$0.276980\pi$
$710$	39.9270	1.49843
$711$	0	0
$712$	−62.6813	−2.34908
$713$	−6.80326	−0.254784
$714$	0	0
$715$	−7.68088	−0.287248
$716$	107.525	4.01838
$717$	0	0
$718$	59.2769	2.21219
$719$	−2.95956	−0.110373	−0.0551865	−	0.998476i	$-0.517575\pi$
$719$	−2.95956	−0.110373	−0.0551865	+	0.998476i	$0.517575\pi$
$720$	0	0
$721$	17.6252	0.656397
$722$	−34.9078	−1.29913
$723$	0	0
$724$	86.0192	3.19688
$725$	6.16309	0.228891
$726$	0	0
$727$	39.8589	1.47829	0.739143	−	0.673549i	$-0.235231\pi$
$727$	39.8589	1.47829	0.739143	+	0.673549i	$0.235231\pi$
$728$	7.10716	0.263409
$729$	0	0
$730$	8.34653	0.308919
$731$	−24.0494	−0.889500
$732$	0	0
$733$	−2.22118	−0.0820412	−0.0410206	−	0.999158i	$-0.513061\pi$
$733$	−2.22118	−0.0820412	−0.0410206	+	0.999158i	$0.513061\pi$
$734$	60.8950	2.24768
$735$	0	0
$736$	−4.71841	−0.173923
$737$	−38.3441	−1.41242
$738$	0	0
$739$	−26.5373	−0.976190	−0.488095	−	0.872791i	$-0.662308\pi$
$739$	−26.5373	−0.976190	−0.488095	+	0.872791i	$0.662308\pi$
$740$	11.9677	0.439943
$741$	0	0
$742$	27.8577	1.02269
$743$	17.1713	0.629952	0.314976	−	0.949100i	$-0.398003\pi$
$743$	17.1713	0.629952	0.314976	+	0.949100i	$0.398003\pi$
$744$	0	0
$745$	−12.4022	−0.454381
$746$	−84.2161	−3.08337
$747$	0	0
$748$	−130.167	−4.75939
$749$	−11.0952	−0.405411
$750$	0	0
$751$	24.1156	0.879991	0.439995	−	0.898000i	$-0.354980\pi$
$751$	24.1156	0.879991	0.439995	+	0.898000i	$0.354980\pi$
$752$	−59.6998	−2.17703
$753$	0	0
$754$	21.0431	0.766343
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−26.2665	−0.954672	−0.477336	−	0.878721i	$-0.658398\pi$
$757$	−26.2665	−0.954672	−0.477336	+	0.878721i	$0.658398\pi$
$758$	15.2690	0.554595
$759$	0	0
$760$	−11.2988	−0.409850
$761$	−14.3832	−0.521391	−0.260696	−	0.965421i	$-0.583952\pi$
$761$	−14.3832	−0.521391	−0.260696	+	0.965421i	$0.583952\pi$
$762$	0	0
$763$	−15.9596	−0.577775
$764$	−3.48795	−0.126190
$765$	0	0
$766$	−24.3562	−0.880026
$767$	−6.82874	−0.246572
$768$	0	0
$769$	−9.09636	−0.328023	−0.164012	−	0.986458i	$-0.552443\pi$
$769$	−9.09636	−0.328023	−0.164012	+	0.986458i	$0.552443\pi$
$770$	−13.6809	−0.493025
$771$	0	0
$772$	67.8899	2.44341
$773$	26.4581	0.951632	0.475816	−	0.879545i	$-0.342153\pi$
$773$	26.4581	0.951632	0.475816	+	0.879545i	$0.342153\pi$
$774$	0	0
$775$	1.18344	0.0425104
$776$	−74.0707	−2.65898
$777$	0	0
$778$	53.5169	1.91867
$779$	5.69121	0.203909
$780$	0	0
$781$	−89.8187	−3.21397
$782$	81.4984	2.91438
$783$	0	0
$784$	4.49592	0.160568
$785$	−7.25833	−0.259061
$786$	0	0
$787$	−50.2532	−1.79133	−0.895667	−	0.444726i	$-0.853301\pi$
$787$	−50.2532	−1.79133	−0.895667	+	0.444726i	$0.853301\pi$
$788$	26.6413	0.949059
$789$	0	0
$790$	5.21723	0.185621
$791$	−9.24885	−0.328851
$792$	0	0
$793$	10.7284	0.380975
$794$	13.8722	0.492307
$795$	0	0
$796$	15.6992	0.556445
$797$	−28.3414	−1.00390	−0.501952	−	0.864896i	$-0.667385\pi$
$797$	−28.3414	−1.00390	−0.501952	+	0.864896i	$0.667385\pi$
$798$	0	0
$799$	76.3354	2.70055
$800$	0.820777	0.0290188
$801$	0	0
$802$	−80.0415	−2.82636
$803$	−18.7761	−0.662595
$804$	0	0
$805$	5.74872	0.202616
$806$	4.04070	0.142328
$807$	0	0
$808$	−14.1099	−0.496383
$809$	−30.2258	−1.06268	−0.531341	−	0.847158i	$-0.678312\pi$
$809$	−30.2258	−1.06268	−0.531341	+	0.847158i	$0.678312\pi$
$810$	0	0
$811$	−1.55553	−0.0546220	−0.0273110	−	0.999627i	$-0.508694\pi$
$811$	−1.55553	−0.0546220	−0.0273110	+	0.999627i	$0.508694\pi$
$812$	25.1549	0.882765
$813$	0	0
$814$	−40.1145	−1.40601
$815$	−12.8561	−0.450331
$816$	0	0
$817$	9.20815	0.322152
$818$	−20.2012	−0.706317
$819$	0	0
$820$	−10.5534	−0.368539
$821$	−15.2120	−0.530901	−0.265451	−	0.964124i	$-0.585521\pi$
$821$	−15.2120	−0.530901	−0.265451	+	0.964124i	$0.585521\pi$
$822$	0	0
$823$	52.9313	1.84507	0.922535	−	0.385914i	$-0.126114\pi$
$823$	52.9313	1.84507	0.922535	+	0.385914i	$0.126114\pi$
$824$	90.4747	3.15183
$825$	0	0
$826$	−12.1631	−0.423208
$827$	−32.4581	−1.12868	−0.564340	−	0.825543i	$-0.690869\pi$
$827$	−32.4581	−1.12868	−0.564340	+	0.825543i	$0.690869\pi$
$828$	0	0
$829$	−11.5571	−0.401395	−0.200697	−	0.979653i	$-0.564321\pi$
$829$	−11.5571	−0.401395	−0.200697	+	0.979653i	$0.564321\pi$
$830$	19.1591	0.665023
$831$	0	0
$832$	−9.64706	−0.334452
$833$	−5.74872	−0.199181
$834$	0	0
$835$	−21.0478	−0.728388
$836$	49.8391	1.72372
$837$	0	0
$838$	38.9079	1.34405
$839$	50.1175	1.73025	0.865124	−	0.501557i	$-0.167239\pi$
$839$	50.1175	1.73025	0.865124	+	0.501557i	$0.167239\pi$
$840$	0	0
$841$	8.98366	0.309781
$842$	27.3624	0.942971
$843$	0	0
$844$	−0.859491	−0.0295849
$845$	11.0831	0.381269
$846$	0	0
$847$	19.7761	0.679516
$848$	50.7874	1.74405
$849$	0	0
$850$	−14.1768	−0.486260
$851$	16.8561	0.577821
$852$	0	0
$853$	−34.7284	−1.18908	−0.594538	−	0.804067i	$-0.702665\pi$
$853$	−34.7284	−1.18908	−0.594538	+	0.804067i	$0.702665\pi$
$854$	19.1089	0.653895
$855$	0	0
$856$	−56.9547	−1.94667
$857$	45.3606	1.54949	0.774745	−	0.632274i	$-0.217878\pi$
$857$	45.3606	1.54949	0.774745	+	0.632274i	$0.217878\pi$
$858$	0	0
$859$	−36.0057	−1.22850	−0.614250	−	0.789112i	$-0.710541\pi$
$859$	−36.0057	−1.22850	−0.614250	+	0.789112i	$0.710541\pi$
$860$	−17.0749	−0.582249
$861$	0	0
$862$	−0.525956	−0.0179141
$863$	−30.2708	−1.03043	−0.515215	−	0.857061i	$-0.672288\pi$
$863$	−30.2708	−1.03043	−0.515215	+	0.857061i	$0.672288\pi$
$864$	0	0
$865$	−6.97965	−0.237315
$866$	45.7272	1.55388
$867$	0	0
$868$	4.83026	0.163950
$869$	−11.7365	−0.398135
$870$	0	0
$871$	−9.56963	−0.324255
$872$	−81.9244	−2.77431
$873$	0	0
$874$	−31.2045	−1.05551
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	22.5246	0.760601	0.380300	−	0.924863i	$-0.375821\pi$
$877$	22.5246	0.760601	0.380300	+	0.924863i	$0.375821\pi$
$878$	−89.4099	−3.01744
$879$	0	0
$880$	−24.9416	−0.840783
$881$	−8.87248	−0.298922	−0.149461	−	0.988768i	$-0.547754\pi$
$881$	−8.87248	−0.298922	−0.149461	+	0.988768i	$0.547754\pi$
$882$	0	0
$883$	10.5441	0.354836	0.177418	−	0.984136i	$-0.443225\pi$
$883$	10.5441	0.354836	0.177418	+	0.984136i	$0.443225\pi$
$884$	−32.4862	−1.09263
$885$	0	0
$886$	−67.1041	−2.25441
$887$	23.6877	0.795354	0.397677	−	0.917525i	$-0.369816\pi$
$887$	23.6877	0.795354	0.397677	+	0.917525i	$0.369816\pi$
$888$	0	0
$889$	11.1509	0.373989
$890$	30.1129	1.00939
$891$	0	0
$892$	−26.0985	−0.873844
$893$	−29.2276	−0.978066
$894$	0	0
$895$	−26.3441	−0.880586
$896$	−18.8245	−0.628883
$897$	0	0
$898$	−24.5737	−0.820035
$899$	7.29364	0.243257
$900$	0	0
$901$	−64.9395	−2.16345
$902$	35.3737	1.17781
$903$	0	0
$904$	−47.4767	−1.57905
$905$	−21.0752	−0.700562
$906$	0	0
$907$	32.9392	1.09373	0.546864	−	0.837221i	$-0.315821\pi$
$907$	32.9392	1.09373	0.546864	+	0.837221i	$0.315821\pi$
$908$	101.075	3.35430
$909$	0	0
$910$	−3.41437	−0.113185
$911$	29.8494	0.988956	0.494478	−	0.869190i	$-0.335359\pi$
$911$	29.8494	0.988956	0.494478	+	0.869190i	$0.335359\pi$
$912$	0	0
$913$	−43.0999	−1.42640
$914$	−57.9270	−1.91606
$915$	0	0
$916$	−87.1268	−2.87875
$917$	−3.29904	−0.108944
$918$	0	0
$919$	−0.385850	−0.0127280	−0.00636401	−	0.999980i	$-0.502026\pi$
$919$	−0.385850	−0.0127280	−0.00636401	+	0.999980i	$0.502026\pi$
$920$	29.5096	0.972903
$921$	0	0
$922$	83.1948	2.73988
$923$	−22.4163	−0.737841
$924$	0	0
$925$	−2.93216	−0.0964087
$926$	−80.3064	−2.63903
$927$	0	0
$928$	5.05852	0.166054
$929$	16.4822	0.540764	0.270382	−	0.962753i	$-0.412850\pi$
$929$	16.4822	0.540764	0.270382	+	0.962753i	$0.412850\pi$
$930$	0	0
$931$	2.20109	0.0721380
$932$	−28.6429	−0.938228
$933$	0	0
$934$	−84.2864	−2.75794
$935$	31.8917	1.04297
$936$	0	0
$937$	−28.2133	−0.921688	−0.460844	−	0.887481i	$-0.652453\pi$
$937$	−28.2133	−0.921688	−0.460844	+	0.887481i	$0.652453\pi$
$938$	−17.0451	−0.556541
$939$	0	0
$940$	54.1975	1.76773
$941$	−43.0396	−1.40305	−0.701525	−	0.712645i	$-0.747497\pi$
$941$	−43.0396	−1.40305	−0.701525	+	0.712645i	$0.747497\pi$
$942$	0	0
$943$	−14.8641	−0.484040
$944$	−22.1746	−0.721721
$945$	0	0
$946$	57.2331	1.86081
$947$	55.9533	1.81824	0.909119	−	0.416536i	$-0.136756\pi$
$947$	55.9533	1.81824	0.909119	+	0.416536i	$0.136756\pi$
$948$	0	0
$949$	−4.68600	−0.152114
$950$	5.42807	0.176110
$951$	0	0
$952$	−29.5096	−0.956412
$953$	26.9596	0.873306	0.436653	−	0.899630i	$-0.356164\pi$
$953$	26.9596	0.873306	0.436653	+	0.899630i	$0.356164\pi$
$954$	0	0
$955$	0.854566	0.0276531
$956$	−15.7724	−0.510115
$957$	0	0
$958$	26.9596	0.871024
$959$	−11.2963	−0.364778
$960$	0	0
$961$	−29.5995	−0.954822
$962$	−10.0115	−0.322783
$963$	0	0
$964$	26.2924	0.846821
$965$	−16.6334	−0.535448
$966$	0	0
$967$	−49.9748	−1.60708	−0.803540	−	0.595250i	$-0.797053\pi$
$967$	−49.9748	−1.60708	−0.803540	+	0.595250i	$0.797053\pi$
$968$	101.516	3.26284
$969$	0	0
$970$	35.5845	1.14255
$971$	7.07891	0.227173	0.113586	−	0.993528i	$-0.463766\pi$
$971$	7.07891	0.227173	0.113586	+	0.993528i	$0.463766\pi$
$972$	0	0
$973$	13.4974	0.432708
$974$	−102.991	−3.30005
$975$	0	0
$976$	34.8376	1.11512
$977$	9.37914	0.300065	0.150033	−	0.988681i	$-0.452062\pi$
$977$	9.37914	0.300065	0.150033	+	0.988681i	$0.452062\pi$
$978$	0	0
$979$	−67.7412	−2.16502
$980$	−4.08154	−0.130380
$981$	0	0
$982$	19.2589	0.614575
$983$	−45.5237	−1.45198	−0.725991	−	0.687705i	$-0.758618\pi$
$983$	−45.5237	−1.45198	−0.725991	+	0.687705i	$0.758618\pi$
$984$	0	0
$985$	−6.52727	−0.207976
$986$	−87.3728	−2.78252
$987$	0	0
$988$	12.4385	0.395720
$989$	−24.0494	−0.764727
$990$	0	0
$991$	27.3818	0.869813	0.434906	−	0.900476i	$-0.356781\pi$
$991$	27.3818	0.869813	0.434906	+	0.900476i	$0.356781\pi$
$992$	0.971340	0.0308401
$993$	0	0
$994$	−39.9270	−1.26641
$995$	−3.84640	−0.121939
$996$	0	0
$997$	51.9002	1.64369	0.821847	−	0.569708i	$-0.192944\pi$
$997$	51.9002	1.64369	0.821847	+	0.569708i	$0.192944\pi$
$998$	−46.8924	−1.48435
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.m.1.4	✓	4
3.2	odd	2		945.2.a.n.1.1	yes	4
5.4	even	2		4725.2.a.bx.1.1		4
7.6	odd	2		6615.2.a.be.1.4		4
15.14	odd	2		4725.2.a.bo.1.4		4
21.20	even	2		6615.2.a.bh.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.m.1.4	✓	4	1.1	even	1	trivial
945.2.a.n.1.1	yes	4	3.2	odd	2
4725.2.a.bo.1.4		4	15.14	odd	2
4725.2.a.bx.1.1		4	5.4	even	2
6615.2.a.be.1.4		4	7.6	odd	2
6615.2.a.bh.1.1		4	21.20	even	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46608 q^{2} +4.08154 q^{4} -1.00000 q^{5} +1.00000 q^{7} +5.13325 q^{8} +O(q^{10})\)	\(q+2.46608 q^{2} +4.08154 q^{4} -1.00000 q^{5} +1.00000 q^{7} +5.13325 q^{8} -2.46608 q^{10} +5.54762 q^{11} +1.38453 q^{13} +2.46608 q^{14} +4.49592 q^{16} -5.74872 q^{17} +2.20109 q^{19} -4.08154 q^{20} +13.6809 q^{22} -5.74872 q^{23} +1.00000 q^{25} +3.41437 q^{26} +4.08154 q^{28} +6.16309 q^{29} +1.18344 q^{31} +0.820777 q^{32} -14.1768 q^{34} -1.00000 q^{35} -2.93216 q^{37} +5.42807 q^{38} -5.13325 q^{40} +2.58563 q^{41} +4.18344 q^{43} +22.6429 q^{44} -14.1768 q^{46} -13.2787 q^{47} +1.00000 q^{49} +2.46608 q^{50} +5.65104 q^{52} +11.2963 q^{53} -5.54762 q^{55} +5.13325 q^{56} +15.1987 q^{58} -4.93216 q^{59} +7.74872 q^{61} +2.91846 q^{62} -6.96773 q^{64} -1.38453 q^{65} -6.91181 q^{67} -23.4636 q^{68} -2.46608 q^{70} -16.1905 q^{71} -3.38453 q^{73} -7.23093 q^{74} +8.98387 q^{76} +5.54762 q^{77} -2.11560 q^{79} -4.49592 q^{80} +6.37636 q^{82} -7.76907 q^{83} +5.74872 q^{85} +10.3167 q^{86} +28.4773 q^{88} -12.2108 q^{89} +1.38453 q^{91} -23.4636 q^{92} -32.7463 q^{94} -2.20109 q^{95} -14.4296 q^{97} +2.46608 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	\( 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{14} + 19 q^{16} + 4 q^{19} - 9 q^{20} + 10 q^{22} + 4 q^{25} + 22 q^{26} + 9 q^{28} + 10 q^{29} + 6 q^{31} - 23 q^{32} - 13 q^{34} - 4 q^{35} + 10 q^{37} + q^{38} + 6 q^{40} + 2 q^{41} + 18 q^{43} + 36 q^{44} - 13 q^{46} - 18 q^{47} + 4 q^{49} - q^{50} - 34 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{56} - 14 q^{58} + 2 q^{59} + 8 q^{61} + 19 q^{62} + 54 q^{64} - 2 q^{65} + 10 q^{67} - 13 q^{68} + q^{70} + 8 q^{71} - 10 q^{73} - 36 q^{74} - 5 q^{76} + 4 q^{77} + 12 q^{79} - 19 q^{80} + 24 q^{82} - 24 q^{83} + 16 q^{86} + 4 q^{88} + 8 q^{89} + 2 q^{91} - 13 q^{92} - 38 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) \) 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + q^10 + 4 * q^11 + 2 * q^13 - q^14 + 19 * q^16 + 4 * q^19 - 9 * q^20 + 10 * q^22 + 4 * q^25 + 22 * q^26 + 9 * q^28 + 10 * q^29 + 6 * q^31 - 23 * q^32 - 13 * q^34 - 4 * q^35 + 10 * q^37 + q^38 + 6 * q^40 + 2 * q^41 + 18 * q^43 + 36 * q^44 - 13 * q^46 - 18 * q^47 + 4 * q^49 - q^50 - 34 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^56 - 14 * q^58 + 2 * q^59 + 8 * q^61 + 19 * q^62 + 54 * q^64 - 2 * q^65 + 10 * q^67 - 13 * q^68 + q^70 + 8 * q^71 - 10 * q^73 - 36 * q^74 - 5 * q^76 + 4 * q^77 + 12 * q^79 - 19 * q^80 + 24 * q^82 - 24 * q^83 + 16 * q^86 + 4 * q^88 + 8 * q^89 + 2 * q^91 - 13 * q^92 - 38 * q^94 - 4 * q^95 + 10 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more