LMFDB - Embedded newform 1449.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.509552$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.509552 q^{2} -1.74036 q^{4} -4.41546 q^{5} +1.00000 q^{7} -1.90591 q^{8} +O(q^{10})$	\(q+0.509552 q^{2} -1.74036 q^{4} -4.41546 q^{5} +1.00000 q^{7} -1.90591 q^{8} -2.24991 q^{10} +1.67510 q^{11} -4.66537 q^{13} +0.509552 q^{14} +2.50955 q^{16} -6.24991 q^{17} -0.694209 q^{19} +7.68447 q^{20} +0.853553 q^{22} +1.00000 q^{23} +14.4963 q^{25} -2.37725 q^{26} -1.74036 q^{28} -5.60012 q^{29} +4.24991 q^{31} +5.09056 q^{32} -3.18466 q^{34} -4.41546 q^{35} +9.26901 q^{37} -0.353736 q^{38} +8.41546 q^{40} +5.15582 q^{41} +4.20376 q^{43} -2.91528 q^{44} +0.509552 q^{46} +1.92501 q^{47} +1.00000 q^{49} +7.38662 q^{50} +8.11940 q^{52} +1.84066 q^{53} -7.39636 q^{55} -1.90591 q^{56} -2.85355 q^{58} -9.39779 q^{59} +7.72125 q^{61} +2.16555 q^{62} -2.42520 q^{64} +20.5998 q^{65} -8.22728 q^{67} +10.8771 q^{68} -2.24991 q^{70} +10.6654 q^{71} -11.9117 q^{73} +4.72305 q^{74} +1.20817 q^{76} +1.67510 q^{77} +0.581012 q^{79} -11.0808 q^{80} +2.62716 q^{82} -6.95032 q^{83} +27.5962 q^{85} +2.14204 q^{86} -3.19259 q^{88} +13.4963 q^{89} -4.66537 q^{91} -1.74036 q^{92} +0.980895 q^{94} +3.06525 q^{95} +10.0999 q^{97} +0.509552 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8	$ 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8} + 4 q^{10} + 5 q^{11} + 7 q^{13} + 8 q^{16} - 12 q^{17} + 3 q^{19} + q^{20} - q^{22} + 4 q^{23} + 7 q^{25} - 5 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} + 6 q^{32} - 9 q^{34} - 5 q^{35} + 20 q^{37} - 23 q^{38} + 21 q^{40} - 3 q^{41} + 9 q^{43} + 27 q^{44} - 7 q^{47} + 4 q^{49} - 3 q^{50} + 38 q^{52} + 6 q^{53} - 21 q^{55} + 3 q^{56} - 7 q^{58} + 2 q^{59} + 24 q^{61} + 9 q^{62} - 21 q^{64} + 14 q^{65} + q^{67} + 13 q^{68} + 4 q^{70} + 17 q^{71} + 16 q^{73} + 33 q^{74} - 25 q^{76} + 5 q^{77} - 10 q^{79} - 6 q^{80} - 7 q^{82} - 8 q^{83} + 17 q^{85} + 35 q^{86} - 12 q^{88} + 3 q^{89} + 7 q^{91} + 4 q^{92} + 8 q^{94} + 3 q^{95} - 2 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8 + 4 * q^10 + 5 * q^11 + 7 * q^13 + 8 * q^16 - 12 * q^17 + 3 * q^19 + q^20 - q^22 + 4 * q^23 + 7 * q^25 - 5 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 + 6 * q^32 - 9 * q^34 - 5 * q^35 + 20 * q^37 - 23 * q^38 + 21 * q^40 - 3 * q^41 + 9 * q^43 + 27 * q^44 - 7 * q^47 + 4 * q^49 - 3 * q^50 + 38 * q^52 + 6 * q^53 - 21 * q^55 + 3 * q^56 - 7 * q^58 + 2 * q^59 + 24 * q^61 + 9 * q^62 - 21 * q^64 + 14 * q^65 + q^67 + 13 * q^68 + 4 * q^70 + 17 * q^71 + 16 * q^73 + 33 * q^74 - 25 * q^76 + 5 * q^77 - 10 * q^79 - 6 * q^80 - 7 * q^82 - 8 * q^83 + 17 * q^85 + 35 * q^86 - 12 * q^88 + 3 * q^89 + 7 * q^91 + 4 * q^92 + 8 * q^94 + 3 * q^95 - 2 * 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 $

$2$	0.509552	0.360308	0.180154	−	0.983638i	$-0.442340\pi$
$2$	0.509552	0.360308	0.180154	+	0.983638i	$0.442340\pi$
$3$	0	0
$3$	0	0
$4$	−1.74036	−0.870178
$5$	−4.41546	−1.97465	−0.987327	−	0.158699i	$-0.949270\pi$
$5$	−4.41546	−1.97465	−0.987327	+	0.158699i	$0.949270\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.90591	−0.673840
$9$	0	0
$10$	−2.24991	−0.711484
$11$	1.67510	0.505063	0.252531	−	0.967589i	$-0.418737\pi$
$11$	1.67510	0.505063	0.252531	+	0.967589i	$0.418737\pi$
$12$	0	0
$13$	−4.66537	−1.29394	−0.646970	−	0.762515i	$-0.723964\pi$
$13$	−4.66537	−1.29394	−0.646970	+	0.762515i	$0.723964\pi$
$14$	0.509552	0.136184
$15$	0	0
$16$	2.50955	0.627388
$17$	−6.24991	−1.51583	−0.757913	−	0.652356i	$-0.773781\pi$
$17$	−6.24991	−1.51583	−0.757913	+	0.652356i	$0.773781\pi$
$18$	0	0
$19$	−0.694209	−0.159262	−0.0796312	−	0.996824i	$-0.525374\pi$
$19$	−0.694209	−0.159262	−0.0796312	+	0.996824i	$0.525374\pi$
$20$	7.68447	1.71830
$21$	0	0
$22$	0.853553	0.181978
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	14.4963	2.89926
$26$	−2.37725	−0.466217
$27$	0	0
$28$	−1.74036	−0.328896
$29$	−5.60012	−1.03992	−0.519958	−	0.854192i	$-0.674052\pi$
$29$	−5.60012	−1.03992	−0.519958	+	0.854192i	$0.674052\pi$
$30$	0	0
$31$	4.24991	0.763306	0.381653	−	0.924306i	$-0.375355\pi$
$31$	4.24991	0.763306	0.381653	+	0.924306i	$0.375355\pi$
$32$	5.09056	0.899893
$33$	0	0
$34$	−3.18466	−0.546164
$35$	−4.41546	−0.746349
$36$	0	0
$37$	9.26901	1.52382	0.761908	−	0.647685i	$-0.224263\pi$
$37$	9.26901	1.52382	0.761908	+	0.647685i	$0.224263\pi$
$38$	−0.353736	−0.0573835
$39$	0	0
$40$	8.41546	1.33060
$41$	5.15582	0.805203	0.402602	−	0.915375i	$-0.368106\pi$
$41$	5.15582	0.805203	0.402602	+	0.915375i	$0.368106\pi$
$42$	0	0
$43$	4.20376	0.641068	0.320534	−	0.947237i	$-0.396138\pi$
$43$	4.20376	0.641068	0.320534	+	0.947237i	$0.396138\pi$
$44$	−2.91528	−0.439495
$45$	0	0
$46$	0.509552	0.0751294
$47$	1.92501	0.280792	0.140396	−	0.990095i	$-0.455162\pi$
$47$	1.92501	0.280792	0.140396	+	0.990095i	$0.455162\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.38662	1.04463
$51$	0	0
$52$	8.11940	1.12596
$53$	1.84066	0.252833	0.126417	−	0.991977i	$-0.459652\pi$
$53$	1.84066	0.252833	0.126417	+	0.991977i	$0.459652\pi$
$54$	0	0
$55$	−7.39636	−0.997324
$56$	−1.90591	−0.254688
$57$	0	0
$58$	−2.85355	−0.374690
$59$	−9.39779	−1.22349	−0.611744	−	0.791056i	$-0.709532\pi$
$59$	−9.39779	−1.22349	−0.611744	+	0.791056i	$0.709532\pi$
$60$	0	0
$61$	7.72125	0.988605	0.494302	−	0.869290i	$-0.335424\pi$
$61$	7.72125	0.988605	0.494302	+	0.869290i	$0.335424\pi$
$62$	2.16555	0.275025
$63$	0	0
$64$	−2.42520	−0.303149
$65$	20.5998	2.55508
$66$	0	0
$67$	−8.22728	−1.00512	−0.502561	−	0.864542i	$-0.667609\pi$
$67$	−8.22728	−1.00512	−0.502561	+	0.864542i	$0.667609\pi$
$68$	10.8771	1.31904
$69$	0	0
$70$	−2.24991	−0.268916
$71$	10.6654	1.26575	0.632873	−	0.774255i	$-0.281875\pi$
$71$	10.6654	1.26575	0.632873	+	0.774255i	$0.281875\pi$
$72$	0	0
$73$	−11.9117	−1.39416	−0.697082	−	0.716991i	$-0.745519\pi$
$73$	−11.9117	−1.39416	−0.697082	+	0.716991i	$0.745519\pi$
$74$	4.72305	0.549043
$75$	0	0
$76$	1.20817	0.138587
$77$	1.67510	0.190896
$78$	0	0
$79$	0.581012	0.0653689	0.0326845	−	0.999466i	$-0.489594\pi$
$79$	0.581012	0.0653689	0.0326845	+	0.999466i	$0.489594\pi$
$80$	−11.0808	−1.23887
$81$	0	0
$82$	2.62716	0.290121
$83$	−6.95032	−0.762897	−0.381449	−	0.924390i	$-0.624575\pi$
$83$	−6.95032	−0.762897	−0.381449	+	0.924390i	$0.624575\pi$
$84$	0	0
$85$	27.5962	2.99323
$86$	2.14204	0.230982
$87$	0	0
$88$	−3.19259	−0.340332
$89$	13.4963	1.43060	0.715302	−	0.698816i	$-0.246289\pi$
$89$	13.4963	1.43060	0.715302	+	0.698816i	$0.246289\pi$
$90$	0	0
$91$	−4.66537	−0.489064
$92$	−1.74036	−0.181445
$93$	0	0
$94$	0.980895	0.101172
$95$	3.06525	0.314488
$96$	0	0
$97$	10.0999	1.02549	0.512746	−	0.858540i	$-0.328628\pi$
$97$	10.0999	1.02549	0.512746	+	0.858540i	$0.328628\pi$
$98$	0.509552	0.0514726
$99$	0	0
$100$	−25.2287	−2.52287
$101$	13.7830	1.37146	0.685729	−	0.727857i	$-0.259484\pi$
$101$	13.7830	1.37146	0.685729	+	0.727857i	$0.259484\pi$
$102$	0	0
$103$	16.5553	1.63125	0.815623	−	0.578584i	$-0.196395\pi$
$103$	16.5553	1.63125	0.815623	+	0.578584i	$0.196395\pi$
$104$	8.89176	0.871909
$105$	0	0
$106$	0.937911	0.0910979
$107$	1.33463	0.129024	0.0645118	−	0.997917i	$-0.479451\pi$
$107$	1.33463	0.129024	0.0645118	+	0.997917i	$0.479451\pi$
$108$	0	0
$109$	4.00794	0.383891	0.191945	−	0.981406i	$-0.438520\pi$
$109$	4.00794	0.383891	0.191945	+	0.981406i	$0.438520\pi$
$110$	−3.76883	−0.359344
$111$	0	0
$112$	2.50955	0.237130
$113$	9.06525	0.852787	0.426394	−	0.904538i	$-0.359784\pi$
$113$	9.06525	0.852787	0.426394	+	0.904538i	$0.359784\pi$
$114$	0	0
$115$	−4.41546	−0.411744
$116$	9.74620	0.904912
$117$	0	0
$118$	−4.78867	−0.440832
$119$	−6.24991	−0.572928
$120$	0	0
$121$	−8.19403	−0.744911
$122$	3.93438	0.356202
$123$	0	0
$124$	−7.39636	−0.664212
$125$	−41.9305	−3.75038
$126$	0	0
$127$	−21.4977	−1.90761	−0.953807	−	0.300419i	$-0.902873\pi$
$127$	−21.4977	−1.90761	−0.953807	+	0.300419i	$0.902873\pi$
$128$	−11.4169	−1.00912
$129$	0	0
$130$	10.4967	0.920618
$131$	−1.17529	−0.102685	−0.0513426	−	0.998681i	$-0.516350\pi$
$131$	−1.17529	−0.102685	−0.0513426	+	0.998681i	$0.516350\pi$
$132$	0	0
$133$	−0.694209	−0.0601956
$134$	−4.19223	−0.362153
$135$	0	0
$136$	11.9117	1.02142
$137$	−15.8562	−1.35469	−0.677345	−	0.735666i	$-0.736869\pi$
$137$	−15.8562	−1.35469	−0.677345	+	0.735666i	$0.736869\pi$
$138$	0	0
$139$	−1.93438	−0.164072	−0.0820361	−	0.996629i	$-0.526142\pi$
$139$	−1.93438	−0.164072	−0.0820361	+	0.996629i	$0.526142\pi$
$140$	7.68447	0.649457
$141$	0	0
$142$	5.43457	0.456059
$143$	−7.81498	−0.653521
$144$	0	0
$145$	24.7271	2.05347
$146$	−6.06966	−0.502329
$147$	0	0
$148$	−16.1314	−1.32599
$149$	−2.61301	−0.214067	−0.107033	−	0.994255i	$-0.534135\pi$
$149$	−2.61301	−0.214067	−0.107033	+	0.994255i	$0.534135\pi$
$150$	0	0
$151$	−9.26281	−0.753797	−0.376898	−	0.926255i	$-0.623009\pi$
$151$	−9.26281	−0.753797	−0.376898	+	0.926255i	$0.623009\pi$
$152$	1.32310	0.107317
$153$	0	0
$154$	0.853553	0.0687813
$155$	−18.7653	−1.50727
$156$	0	0
$157$	−7.77504	−0.620516	−0.310258	−	0.950652i	$-0.600415\pi$
$157$	−7.77504	−0.620516	−0.310258	+	0.950652i	$0.600415\pi$
$158$	0.296056	0.0235530
$159$	0	0
$160$	−22.4772	−1.77698
$161$	1.00000	0.0788110
$162$	0	0
$163$	−7.12077	−0.557742	−0.278871	−	0.960329i	$-0.589960\pi$
$163$	−7.12077	−0.557742	−0.278871	+	0.960329i	$0.589960\pi$
$164$	−8.97296	−0.700670
$165$	0	0
$166$	−3.54156	−0.274878
$167$	11.9867	0.927562	0.463781	−	0.885950i	$-0.346493\pi$
$167$	11.9867	0.927562	0.463781	+	0.885950i	$0.346493\pi$
$168$	0	0
$169$	8.76567	0.674282
$170$	14.0617	1.07849
$171$	0	0
$172$	−7.31604	−0.557843
$173$	2.09993	0.159655	0.0798275	−	0.996809i	$-0.474563\pi$
$173$	2.09993	0.159655	0.0798275	+	0.996809i	$0.474563\pi$
$174$	0	0
$175$	14.4963	1.09582
$176$	4.20376	0.316870
$177$	0	0
$178$	6.87707	0.515458
$179$	15.9726	1.19385	0.596924	−	0.802298i	$-0.296389\pi$
$179$	15.9726	1.19385	0.596924	+	0.802298i	$0.296389\pi$
$180$	0	0
$181$	−2.89970	−0.215533	−0.107767	−	0.994176i	$-0.534370\pi$
$181$	−2.89970	−0.215533	−0.107767	+	0.994176i	$0.534370\pi$
$182$	−2.37725	−0.176214
$183$	0	0
$184$	−1.90591	−0.140505
$185$	−40.9270	−3.00901
$186$	0	0
$187$	−10.4692	−0.765587
$188$	−3.35021	−0.244339
$189$	0	0
$190$	1.56191	0.113313
$191$	21.5486	1.55921	0.779603	−	0.626275i	$-0.215421\pi$
$191$	21.5486	1.55921	0.779603	+	0.626275i	$0.215421\pi$
$192$	0	0
$193$	4.58722	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$193$	4.58722	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$194$	5.14645	0.369493
$195$	0	0
$196$	−1.74036	−0.124311
$197$	11.8656	0.845389	0.422695	−	0.906272i	$-0.361084\pi$
$197$	11.8656	0.845389	0.422695	+	0.906272i	$0.361084\pi$
$198$	0	0
$199$	−2.52866	−0.179252	−0.0896259	−	0.995976i	$-0.528567\pi$
$199$	−2.52866	−0.179252	−0.0896259	+	0.995976i	$0.528567\pi$
$200$	−27.6286	−1.95364
$201$	0	0
$202$	7.02315	0.494147
$203$	−5.60012	−0.393051
$204$	0	0
$205$	−22.7653	−1.59000
$206$	8.43581	0.587751
$207$	0	0
$208$	−11.7080	−0.811803
$209$	−1.16287	−0.0804376
$210$	0	0
$211$	−21.9797	−1.51314	−0.756572	−	0.653911i	$-0.773127\pi$
$211$	−21.9797	−1.51314	−0.756572	+	0.653911i	$0.773127\pi$
$212$	−3.20340	−0.220010
$213$	0	0
$214$	0.680065	0.0464883
$215$	−18.5615	−1.26589
$216$	0	0
$217$	4.24991	0.288503
$218$	2.04225	0.138319
$219$	0	0
$220$	12.8723	0.867850
$221$	29.1581	1.96139
$222$	0	0
$223$	0.0302726	0.00202720	0.00101360	−	0.999999i	$-0.499677\pi$
$223$	0.0302726	0.00202720	0.00101360	+	0.999999i	$0.499677\pi$
$224$	5.09056	0.340128
$225$	0	0
$226$	4.61922	0.307266
$227$	−21.2393	−1.40970	−0.704852	−	0.709355i	$-0.748986\pi$
$227$	−21.2393	−1.40970	−0.704852	+	0.709355i	$0.748986\pi$
$228$	0	0
$229$	17.4360	1.15220	0.576102	−	0.817378i	$-0.304573\pi$
$229$	17.4360	1.15220	0.576102	+	0.817378i	$0.304573\pi$
$230$	−2.24991	−0.148355
$231$	0	0
$232$	10.6733	0.700737
$233$	11.2082	0.734272	0.367136	−	0.930167i	$-0.380338\pi$
$233$	11.2082	0.734272	0.367136	+	0.930167i	$0.380338\pi$
$234$	0	0
$235$	−8.49982	−0.554467
$236$	16.3555	1.06465
$237$	0	0
$238$	−3.18466	−0.206431
$239$	18.5654	1.20090	0.600449	−	0.799663i	$-0.294989\pi$
$239$	18.5654	1.20090	0.600449	+	0.799663i	$0.294989\pi$
$240$	0	0
$241$	−13.6556	−0.879637	−0.439818	−	0.898087i	$-0.644957\pi$
$241$	−13.6556	−0.879637	−0.439818	+	0.898087i	$0.644957\pi$
$242$	−4.17529	−0.268398
$243$	0	0
$244$	−13.4377	−0.860262
$245$	−4.41546	−0.282093
$246$	0	0
$247$	3.23874	0.206076
$248$	−8.09993	−0.514346
$249$	0	0
$250$	−21.3658	−1.35129
$251$	−0.604526	−0.0381574	−0.0190787	−	0.999818i	$-0.506073\pi$
$251$	−0.604526	−0.0381574	−0.0190787	+	0.999818i	$0.506073\pi$
$252$	0	0
$253$	1.67510	0.105313
$254$	−10.9542	−0.687329
$255$	0	0
$256$	−0.967116	−0.0604447
$257$	−14.1602	−0.883291	−0.441645	−	0.897190i	$-0.645605\pi$
$257$	−14.1602	−0.883291	−0.441645	+	0.897190i	$0.645605\pi$
$258$	0	0
$259$	9.26901	0.575948
$260$	−35.8509	−2.22338
$261$	0	0
$262$	−0.598870	−0.0369983
$263$	−20.5788	−1.26895	−0.634473	−	0.772945i	$-0.718783\pi$
$263$	−20.5788	−1.26895	−0.634473	+	0.772945i	$0.718783\pi$
$264$	0	0
$265$	−8.12734	−0.499259
$266$	−0.353736	−0.0216889
$267$	0	0
$268$	14.3184	0.874635
$269$	0.496655	0.0302816	0.0151408	−	0.999885i	$-0.495180\pi$
$269$	0.496655	0.0302816	0.0151408	+	0.999885i	$0.495180\pi$
$270$	0	0
$271$	24.0547	1.46122	0.730609	−	0.682797i	$-0.239236\pi$
$271$	24.0547	1.46122	0.730609	+	0.682797i	$0.239236\pi$
$272$	−15.6845	−0.951011
$273$	0	0
$274$	−8.07958	−0.488105
$275$	24.2828	1.46431
$276$	0	0
$277$	23.2402	1.39637	0.698183	−	0.715919i	$-0.253992\pi$
$277$	23.2402	1.39637	0.698183	+	0.715919i	$0.253992\pi$
$278$	−0.985670	−0.0591165
$279$	0	0
$280$	8.41546	0.502920
$281$	15.0697	0.898985	0.449492	−	0.893284i	$-0.351605\pi$
$281$	15.0697	0.898985	0.449492	+	0.893284i	$0.351605\pi$
$282$	0	0
$283$	9.22691	0.548483	0.274241	−	0.961661i	$-0.411573\pi$
$283$	9.22691	0.548483	0.274241	+	0.961661i	$0.411573\pi$
$284$	−18.5615	−1.10142
$285$	0	0
$286$	−3.98214	−0.235469
$287$	5.15582	0.304338
$288$	0	0
$289$	22.0614	1.29773
$290$	12.5998	0.739883
$291$	0	0
$292$	20.7307	1.21317
$293$	12.3881	0.723718	0.361859	−	0.932233i	$-0.382142\pi$
$293$	12.3881	0.723718	0.361859	+	0.932233i	$0.382142\pi$
$294$	0	0
$295$	41.4956	2.41596
$296$	−17.6659	−1.02681
$297$	0	0
$298$	−1.33147	−0.0771299
$299$	−4.66537	−0.269805
$300$	0	0
$301$	4.20376	0.242301
$302$	−4.71989	−0.271599
$303$	0	0
$304$	−1.74215	−0.0999194
$305$	−34.0929	−1.95215
$306$	0	0
$307$	0.823281	0.0469871	0.0234936	−	0.999724i	$-0.492521\pi$
$307$	0.823281	0.0469871	0.0234936	+	0.999724i	$0.492521\pi$
$308$	−2.91528	−0.166113
$309$	0	0
$310$	−9.56191	−0.543080
$311$	30.6632	1.73875	0.869375	−	0.494152i	$-0.164521\pi$
$311$	30.6632	1.73875	0.869375	+	0.494152i	$0.164521\pi$
$312$	0	0
$313$	13.1323	0.742282	0.371141	−	0.928577i	$-0.378967\pi$
$313$	13.1323	0.742282	0.371141	+	0.928577i	$0.378967\pi$
$314$	−3.96179	−0.223577
$315$	0	0
$316$	−1.01117	−0.0568826
$317$	15.2269	0.855229	0.427614	−	0.903961i	$-0.359354\pi$
$317$	15.2269	0.855229	0.427614	+	0.903961i	$0.359354\pi$
$318$	0	0
$319$	−9.38078	−0.525223
$320$	10.7084	0.598615
$321$	0	0
$322$	0.509552	0.0283962
$323$	4.33874	0.241414
$324$	0	0
$325$	−67.6305	−3.75147
$326$	−3.62841	−0.200959
$327$	0	0
$328$	−9.82651	−0.542578
$329$	1.92501	0.106129
$330$	0	0
$331$	25.5674	1.40531	0.702655	−	0.711530i	$-0.251998\pi$
$331$	25.5674	1.40531	0.702655	+	0.711530i	$0.251998\pi$
$332$	12.0960	0.663857
$333$	0	0
$334$	6.10787	0.334208
$335$	36.3272	1.98477
$336$	0	0
$337$	16.2890	0.887318	0.443659	−	0.896196i	$-0.353680\pi$
$337$	16.2890	0.887318	0.443659	+	0.896196i	$0.353680\pi$
$338$	4.46657	0.242949
$339$	0	0
$340$	−48.0273	−2.60464
$341$	7.11904	0.385518
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.01198	−0.431977
$345$	0	0
$346$	1.07003	0.0575250
$347$	−21.6729	−1.16346	−0.581732	−	0.813380i	$-0.697625\pi$
$347$	−21.6729	−1.16346	−0.581732	+	0.813380i	$0.697625\pi$
$348$	0	0
$349$	−21.5198	−1.15193	−0.575964	−	0.817475i	$-0.695373\pi$
$349$	−21.5198	−1.15193	−0.575964	+	0.817475i	$0.695373\pi$
$350$	7.38662	0.394831
$351$	0	0
$352$	8.52722	0.454503
$353$	20.1852	1.07435	0.537174	−	0.843471i	$-0.319492\pi$
$353$	20.1852	1.07435	0.537174	+	0.843471i	$0.319492\pi$
$354$	0	0
$355$	−47.0925	−2.49941
$356$	−23.4884	−1.24488
$357$	0	0
$358$	8.13887	0.430153
$359$	−23.2492	−1.22705	−0.613524	−	0.789676i	$-0.710249\pi$
$359$	−23.2492	−1.22705	−0.613524	+	0.789676i	$0.710249\pi$
$360$	0	0
$361$	−18.5181	−0.974635
$362$	−1.47755	−0.0776583
$363$	0	0
$364$	8.11940	0.425572
$365$	52.5959	2.75299
$366$	0	0
$367$	27.7171	1.44682	0.723409	−	0.690419i	$-0.242574\pi$
$367$	27.7171	1.44682	0.723409	+	0.690419i	$0.242574\pi$
$368$	2.50955	0.130819
$369$	0	0
$370$	−20.8544	−1.08417
$371$	1.84066	0.0955621
$372$	0	0
$373$	−24.7303	−1.28048	−0.640242	−	0.768173i	$-0.721166\pi$
$373$	−24.7303	−1.28048	−0.640242	+	0.768173i	$0.721166\pi$
$374$	−5.33463	−0.275847
$375$	0	0
$376$	−3.66890	−0.189209
$377$	26.1266	1.34559
$378$	0	0
$379$	−3.11140	−0.159822	−0.0799109	−	0.996802i	$-0.525464\pi$
$379$	−3.11140	−0.159822	−0.0799109	+	0.996802i	$0.525464\pi$
$380$	−5.33463	−0.273661
$381$	0	0
$382$	10.9802	0.561794
$383$	18.9503	0.968316	0.484158	−	0.874980i	$-0.339126\pi$
$383$	18.9503	0.968316	0.484158	+	0.874980i	$0.339126\pi$
$384$	0	0
$385$	−7.39636	−0.376953
$386$	2.33743	0.118972
$387$	0	0
$388$	−17.5775	−0.892362
$389$	20.1693	1.02262	0.511312	−	0.859395i	$-0.329160\pi$
$389$	20.1693	1.02262	0.511312	+	0.859395i	$0.329160\pi$
$390$	0	0
$391$	−6.24991	−0.316071
$392$	−1.90591	−0.0962629
$393$	0	0
$394$	6.04615	0.304600
$395$	−2.56543	−0.129081
$396$	0	0
$397$	8.41634	0.422404	0.211202	−	0.977442i	$-0.432262\pi$
$397$	8.41634	0.422404	0.211202	+	0.977442i	$0.432262\pi$
$398$	−1.28848	−0.0645859
$399$	0	0
$400$	36.3792	1.81896
$401$	5.86329	0.292799	0.146399	−	0.989226i	$-0.453232\pi$
$401$	5.86329	0.292799	0.146399	+	0.989226i	$0.453232\pi$
$402$	0	0
$403$	−19.8274	−0.987673
$404$	−23.9873	−1.19341
$405$	0	0
$406$	−2.85355	−0.141619
$407$	15.5266	0.769623
$408$	0	0
$409$	−26.2686	−1.29890	−0.649451	−	0.760404i	$-0.725001\pi$
$409$	−26.2686	−1.29890	−0.649451	+	0.760404i	$0.725001\pi$
$410$	−11.6001	−0.572889
$411$	0	0
$412$	−28.8122	−1.41947
$413$	−9.39779	−0.462435
$414$	0	0
$415$	30.6889	1.50646
$416$	−23.7494	−1.16441
$417$	0	0
$418$	−0.592544	−0.0289823
$419$	0.800648	0.0391142	0.0195571	−	0.999809i	$-0.493774\pi$
$419$	0.800648	0.0391142	0.0195571	+	0.999809i	$0.493774\pi$
$420$	0	0
$421$	23.3845	1.13969	0.569846	−	0.821751i	$-0.307003\pi$
$421$	23.3845	1.13969	0.569846	+	0.821751i	$0.307003\pi$
$422$	−11.1998	−0.545198
$423$	0	0
$424$	−3.50812	−0.170369
$425$	−90.6005	−4.39477
$426$	0	0
$427$	7.72125	0.373658
$428$	−2.32273	−0.112274
$429$	0	0
$430$	−9.45808	−0.456109
$431$	33.4853	1.61293	0.806465	−	0.591281i	$-0.201378\pi$
$431$	33.4853	1.61293	0.806465	+	0.591281i	$0.201378\pi$
$432$	0	0
$433$	24.1373	1.15996	0.579982	−	0.814629i	$-0.303059\pi$
$433$	24.1373	1.15996	0.579982	+	0.814629i	$0.303059\pi$
$434$	2.16555	0.103950
$435$	0	0
$436$	−6.97524	−0.334053
$437$	−0.694209	−0.0332085
$438$	0	0
$439$	−16.4469	−0.784968	−0.392484	−	0.919759i	$-0.628384\pi$
$439$	−16.4469	−0.784968	−0.392484	+	0.919759i	$0.628384\pi$
$440$	14.0968	0.672037
$441$	0	0
$442$	14.8576	0.706704
$443$	−9.58542	−0.455417	−0.227709	−	0.973729i	$-0.573123\pi$
$443$	−9.58542	−0.455417	−0.227709	+	0.973729i	$0.573123\pi$
$444$	0	0
$445$	−59.5923	−2.82495
$446$	0.0154255	0.000730417 0
$447$	0	0
$448$	−2.42520	−0.114580
$449$	−4.17319	−0.196945	−0.0984725	−	0.995140i	$-0.531396\pi$
$449$	−4.17319	−0.196945	−0.0984725	+	0.995140i	$0.531396\pi$
$450$	0	0
$451$	8.63653	0.406678
$452$	−15.7768	−0.742077
$453$	0	0
$454$	−10.8226	−0.507927
$455$	20.5998	0.965731
$456$	0	0
$457$	8.02668	0.375472	0.187736	−	0.982220i	$-0.439885\pi$
$457$	8.02668	0.375472	0.187736	+	0.982220i	$0.439885\pi$
$458$	8.88456	0.415148
$459$	0	0
$460$	7.68447	0.358290
$461$	−11.6192	−0.541158	−0.270579	−	0.962698i	$-0.587215\pi$
$461$	−11.6192	−0.541158	−0.270579	+	0.962698i	$0.587215\pi$
$462$	0	0
$463$	−35.8714	−1.66708	−0.833542	−	0.552456i	$-0.813691\pi$
$463$	−35.8714	−1.66708	−0.833542	+	0.552456i	$0.813691\pi$
$464$	−14.0538	−0.652431
$465$	0	0
$466$	5.71115	0.264564
$467$	13.1731	0.609579	0.304790	−	0.952420i	$-0.401414\pi$
$467$	13.1731	0.609579	0.304790	+	0.952420i	$0.401414\pi$
$468$	0	0
$469$	−8.22728	−0.379900
$470$	−4.33110	−0.199779
$471$	0	0
$472$	17.9113	0.824435
$473$	7.04174	0.323779
$474$	0	0
$475$	−10.0635	−0.461743
$476$	10.8771	0.498550
$477$	0	0
$478$	9.46006	0.432693
$479$	−5.23080	−0.239002	−0.119501	−	0.992834i	$-0.538129\pi$
$479$	−5.23080	−0.239002	−0.119501	+	0.992834i	$0.538129\pi$
$480$	0	0
$481$	−43.2434	−1.97173
$482$	−6.95826	−0.316940
$483$	0	0
$484$	14.2605	0.648206
$485$	−44.5959	−2.02499
$486$	0	0
$487$	28.9779	1.31311	0.656557	−	0.754277i	$-0.272012\pi$
$487$	28.9779	1.31311	0.656557	+	0.754277i	$0.272012\pi$
$488$	−14.7160	−0.666162
$489$	0	0
$490$	−2.24991	−0.101641
$491$	16.6912	0.753262	0.376631	−	0.926363i	$-0.377083\pi$
$491$	16.6912	0.753262	0.376631	+	0.926363i	$0.377083\pi$
$492$	0	0
$493$	35.0002	1.57633
$494$	1.65031	0.0742509
$495$	0	0
$496$	10.6654	0.478889
$497$	10.6654	0.478407
$498$	0	0
$499$	9.57213	0.428507	0.214254	−	0.976778i	$-0.431268\pi$
$499$	9.57213	0.428507	0.214254	+	0.976778i	$0.431268\pi$
$500$	72.9740	3.26350
$501$	0	0
$502$	−0.308038	−0.0137484
$503$	20.6348	0.920060	0.460030	−	0.887903i	$-0.347839\pi$
$503$	20.6348	0.920060	0.460030	+	0.887903i	$0.347839\pi$
$504$	0	0
$505$	−60.8582	−2.70815
$506$	0.853553	0.0379451
$507$	0	0
$508$	37.4137	1.65996
$509$	−6.23665	−0.276434	−0.138217	−	0.990402i	$-0.544137\pi$
$509$	−6.23665	−0.276434	−0.138217	+	0.990402i	$0.544137\pi$
$510$	0	0
$511$	−11.9117	−0.526945
$512$	22.3410	0.987342
$513$	0	0
$514$	−7.21538	−0.318257
$515$	−73.0994	−3.22115
$516$	0	0
$517$	3.22460	0.141818
$518$	4.72305	0.207519
$519$	0	0
$520$	−39.2612	−1.72172
$521$	−12.6615	−0.554709	−0.277355	−	0.960768i	$-0.589458\pi$
$521$	−12.6615	−0.554709	−0.277355	+	0.960768i	$0.589458\pi$
$522$	0	0
$523$	43.3446	1.89533	0.947663	−	0.319272i	$-0.103438\pi$
$523$	43.3446	1.89533	0.947663	+	0.319272i	$0.103438\pi$
$524$	2.04542	0.0893545
$525$	0	0
$526$	−10.4860	−0.457211
$527$	−26.5615	−1.15704
$528$	0	0
$529$	1.00000	0.0434783
$530$	−4.14131	−0.179887
$531$	0	0
$532$	1.20817	0.0523809
$533$	−24.0538	−1.04189
$534$	0	0
$535$	−5.89301	−0.254777
$536$	15.6804	0.677291
$537$	0	0
$538$	0.253072	0.0109107
$539$	1.67510	0.0721518
$540$	0	0
$541$	5.87439	0.252560	0.126280	−	0.991995i	$-0.459696\pi$
$541$	5.87439	0.252560	0.126280	+	0.991995i	$0.459696\pi$
$542$	12.2571	0.526488
$543$	0	0
$544$	−31.8156	−1.36408
$545$	−17.6969	−0.758051
$546$	0	0
$547$	13.8580	0.592524	0.296262	−	0.955107i	$-0.404260\pi$
$547$	13.8580	0.592524	0.296262	+	0.955107i	$0.404260\pi$
$548$	27.5955	1.17882
$549$	0	0
$550$	12.3734	0.527602
$551$	3.88765	0.165620
$552$	0	0
$553$	0.581012	0.0247071
$554$	11.8421	0.503122
$555$	0	0
$556$	3.36652	0.142772
$557$	18.5921	0.787773	0.393887	−	0.919159i	$-0.371130\pi$
$557$	18.5921	0.787773	0.393887	+	0.919159i	$0.371130\pi$
$558$	0	0
$559$	−19.6121	−0.829503
$560$	−11.0808	−0.468251
$561$	0	0
$562$	7.67882	0.323911
$563$	−5.09582	−0.214763	−0.107382	−	0.994218i	$-0.534247\pi$
$563$	−5.09582	−0.214763	−0.107382	+	0.994218i	$0.534247\pi$
$564$	0	0
$565$	−40.0273	−1.68396
$566$	4.70160	0.197623
$567$	0	0
$568$	−20.3272	−0.852911
$569$	29.3028	1.22844	0.614218	−	0.789136i	$-0.289471\pi$
$569$	29.3028	1.22844	0.614218	+	0.789136i	$0.289471\pi$
$570$	0	0
$571$	−5.10736	−0.213736	−0.106868	−	0.994273i	$-0.534082\pi$
$571$	−5.10736	−0.213736	−0.106868	+	0.994273i	$0.534082\pi$
$572$	13.6008	0.568680
$573$	0	0
$574$	2.62716	0.109656
$575$	14.4963	0.604537
$576$	0	0
$577$	24.7379	1.02985	0.514926	−	0.857235i	$-0.327819\pi$
$577$	24.7379	1.02985	0.514926	+	0.857235i	$0.327819\pi$
$578$	11.2414	0.467581
$579$	0	0
$580$	−43.0340	−1.78689
$581$	−6.95032	−0.288348
$582$	0	0
$583$	3.08329	0.127697
$584$	22.7027	0.939444
$585$	0	0
$586$	6.31236	0.260761
$587$	−38.8319	−1.60276	−0.801382	−	0.598152i	$-0.795902\pi$
$587$	−38.8319	−1.60276	−0.801382	+	0.598152i	$0.795902\pi$
$588$	0	0
$589$	−2.95032	−0.121566
$590$	21.1442	0.870492
$591$	0	0
$592$	23.2611	0.956024
$593$	−36.9162	−1.51596	−0.757982	−	0.652275i	$-0.773815\pi$
$593$	−36.9162	−1.51596	−0.757982	+	0.652275i	$0.773815\pi$
$594$	0	0
$595$	27.5962	1.13133
$596$	4.54758	0.186276
$597$	0	0
$598$	−2.37725	−0.0972130
$599$	−40.4224	−1.65161	−0.825807	−	0.563953i	$-0.809280\pi$
$599$	−40.4224	−1.65161	−0.825807	+	0.563953i	$0.809280\pi$
$600$	0	0
$601$	2.98464	0.121746	0.0608730	−	0.998146i	$-0.480612\pi$
$601$	2.98464	0.121746	0.0608730	+	0.998146i	$0.480612\pi$
$602$	2.14204	0.0873029
$603$	0	0
$604$	16.1206	0.655937
$605$	36.1804	1.47094
$606$	0	0
$607$	−29.1893	−1.18476	−0.592378	−	0.805660i	$-0.701811\pi$
$607$	−29.1893	−1.18476	−0.592378	+	0.805660i	$0.701811\pi$
$608$	−3.53392	−0.143319
$609$	0	0
$610$	−17.3721	−0.703376
$611$	−8.98090	−0.363328
$612$	0	0
$613$	−5.88595	−0.237731	−0.118866	−	0.992910i	$-0.537926\pi$
$613$	−5.88595	−0.237731	−0.118866	+	0.992910i	$0.537926\pi$
$614$	0.419505	0.0169298
$615$	0	0
$616$	−3.19259	−0.128633
$617$	8.07326	0.325017	0.162509	−	0.986707i	$-0.448041\pi$
$617$	8.07326	0.325017	0.162509	+	0.986707i	$0.448041\pi$
$618$	0	0
$619$	10.8589	0.436457	0.218228	−	0.975898i	$-0.429972\pi$
$619$	10.8589	0.436457	0.218228	+	0.975898i	$0.429972\pi$
$620$	32.6583	1.31159
$621$	0	0
$622$	15.6245	0.626486
$623$	13.4963	0.540717
$624$	0	0
$625$	112.661	4.50644
$626$	6.69160	0.267450
$627$	0	0
$628$	13.5313	0.539959
$629$	−57.9305	−2.30984
$630$	0	0
$631$	23.1891	0.923145	0.461572	−	0.887103i	$-0.347285\pi$
$631$	23.1891	0.923145	0.461572	+	0.887103i	$0.347285\pi$
$632$	−1.10736	−0.0440482
$633$	0	0
$634$	7.75891	0.308146
$635$	94.9223	3.76688
$636$	0	0
$637$	−4.66537	−0.184849
$638$	−4.78000	−0.189242
$639$	0	0
$640$	50.4108	1.99266
$641$	46.7138	1.84509	0.922543	−	0.385895i	$-0.126107\pi$
$641$	46.7138	1.84509	0.922543	+	0.385895i	$0.126107\pi$
$642$	0	0
$643$	33.7524	1.33106	0.665532	−	0.746369i	$-0.268205\pi$
$643$	33.7524	1.33106	0.665532	+	0.746369i	$0.268205\pi$
$644$	−1.74036	−0.0685796
$645$	0	0
$646$	2.21082	0.0869834
$647$	−18.9277	−0.744124	−0.372062	−	0.928208i	$-0.621349\pi$
$647$	−18.9277	−0.744124	−0.372062	+	0.928208i	$0.621349\pi$
$648$	0	0
$649$	−15.7423	−0.617938
$650$	−34.4613	−1.35168
$651$	0	0
$652$	12.3927	0.485335
$653$	10.1696	0.397967	0.198984	−	0.980003i	$-0.436236\pi$
$653$	10.1696	0.397967	0.198984	+	0.980003i	$0.436236\pi$
$654$	0	0
$655$	5.18943	0.202768
$656$	12.9388	0.505175
$657$	0	0
$658$	0.980895	0.0382393
$659$	4.60754	0.179484	0.0897421	−	0.995965i	$-0.471396\pi$
$659$	4.60754	0.179484	0.0897421	+	0.995965i	$0.471396\pi$
$660$	0	0
$661$	−31.1205	−1.21045	−0.605223	−	0.796056i	$-0.706916\pi$
$661$	−31.1205	−1.21045	−0.605223	+	0.796056i	$0.706916\pi$
$662$	13.0279	0.506345
$663$	0	0
$664$	13.2467	0.514071
$665$	3.06525	0.118865
$666$	0	0
$667$	−5.60012	−0.216837
$668$	−20.8612	−0.807144
$669$	0	0
$670$	18.5106	0.715128
$671$	12.9339	0.499308
$672$	0	0
$673$	−36.9447	−1.42411	−0.712057	−	0.702122i	$-0.752236\pi$
$673$	−36.9447	−1.42411	−0.712057	+	0.702122i	$0.752236\pi$
$674$	8.30010	0.319708
$675$	0	0
$676$	−15.2554	−0.586746
$677$	−0.331401	−0.0127368	−0.00636838	−	0.999980i	$-0.502027\pi$
$677$	−0.331401	−0.0127368	−0.00636838	+	0.999980i	$0.502027\pi$
$678$	0	0
$679$	10.0999	0.387600
$680$	−52.5959	−2.01696
$681$	0	0
$682$	3.62752	0.138905
$683$	16.8856	0.646109	0.323055	−	0.946380i	$-0.395290\pi$
$683$	16.8856	0.646109	0.323055	+	0.946380i	$0.395290\pi$
$684$	0	0
$685$	70.0126	2.67504
$686$	0.509552	0.0194548
$687$	0	0
$688$	10.5496	0.402198
$689$	−8.58734	−0.327152
$690$	0	0
$691$	−11.1963	−0.425929	−0.212964	−	0.977060i	$-0.568312\pi$
$691$	−11.1963	−0.425929	−0.212964	+	0.977060i	$0.568312\pi$
$692$	−3.65463	−0.138928
$693$	0	0
$694$	−11.0435	−0.419206
$695$	8.54119	0.323986
$696$	0	0
$697$	−32.2234	−1.22055
$698$	−10.9655	−0.415049
$699$	0	0
$700$	−25.2287	−0.953556
$701$	−9.50609	−0.359040	−0.179520	−	0.983754i	$-0.557454\pi$
$701$	−9.50609	−0.359040	−0.179520	+	0.983754i	$0.557454\pi$
$702$	0	0
$703$	−6.43463	−0.242687
$704$	−4.06245	−0.153110
$705$	0	0
$706$	10.2854	0.387096
$707$	13.7830	0.518362
$708$	0	0
$709$	36.2890	1.36286	0.681431	−	0.731882i	$-0.261358\pi$
$709$	36.2890	1.36286	0.681431	+	0.731882i	$0.261358\pi$
$710$	−23.9961	−0.900558
$711$	0	0
$712$	−25.7227	−0.963998
$713$	4.24991	0.159160
$714$	0	0
$715$	34.5067	1.29048
$716$	−27.7980	−1.03886
$717$	0	0
$718$	−11.8467	−0.442115
$719$	33.5772	1.25222	0.626109	−	0.779736i	$-0.284647\pi$
$719$	33.5772	1.25222	0.626109	+	0.779736i	$0.284647\pi$
$720$	0	0
$721$	16.5553	0.616553
$722$	−9.43593	−0.351169
$723$	0	0
$724$	5.04651	0.187552
$725$	−81.1809	−3.01498
$726$	0	0
$727$	37.5410	1.39232	0.696159	−	0.717887i	$-0.254891\pi$
$727$	37.5410	1.39232	0.696159	+	0.717887i	$0.254891\pi$
$728$	8.89176	0.329551
$729$	0	0
$730$	26.8004	0.991925
$731$	−26.2731	−0.971747
$732$	0	0
$733$	24.2222	0.894668	0.447334	−	0.894367i	$-0.352374\pi$
$733$	24.2222	0.894668	0.447334	+	0.894367i	$0.352374\pi$
$734$	14.1233	0.521300
$735$	0	0
$736$	5.09056	0.187641
$737$	−13.7815	−0.507650
$738$	0	0
$739$	−19.4377	−0.715028	−0.357514	−	0.933908i	$-0.616376\pi$
$739$	−19.4377	−0.715028	−0.357514	+	0.933908i	$0.616376\pi$
$740$	71.2275	2.61837
$741$	0	0
$742$	0.937911	0.0344318
$743$	−35.7282	−1.31074	−0.655370	−	0.755308i	$-0.727487\pi$
$743$	−35.7282	−1.31074	−0.655370	+	0.755308i	$0.727487\pi$
$744$	0	0
$745$	11.5377	0.422707
$746$	−12.6014	−0.461369
$747$	0	0
$748$	18.2202	0.666197
$749$	1.33463	0.0487664
$750$	0	0
$751$	14.4795	0.528363	0.264182	−	0.964473i	$-0.414898\pi$
$751$	14.4795	0.528363	0.264182	+	0.964473i	$0.414898\pi$
$752$	4.83092	0.176166
$753$	0	0
$754$	13.3129	0.484826
$755$	40.8996	1.48849
$756$	0	0
$757$	21.8034	0.792460	0.396230	−	0.918151i	$-0.370318\pi$
$757$	21.8034	0.792460	0.396230	+	0.918151i	$0.370318\pi$
$758$	−1.58542	−0.0575851
$759$	0	0
$760$	−5.84209	−0.211915
$761$	5.98196	0.216846	0.108423	−	0.994105i	$-0.465420\pi$
$761$	5.98196	0.216846	0.108423	+	0.994105i	$0.465420\pi$
$762$	0	0
$763$	4.00794	0.145097
$764$	−37.5023	−1.35679
$765$	0	0
$766$	9.65619	0.348892
$767$	43.8441	1.58312
$768$	0	0
$769$	−16.0694	−0.579476	−0.289738	−	0.957106i	$-0.593568\pi$
$769$	−16.0694	−0.579476	−0.289738	+	0.957106i	$0.593568\pi$
$770$	−3.76883	−0.135819
$771$	0	0
$772$	−7.98340	−0.287329
$773$	−36.5039	−1.31295	−0.656476	−	0.754347i	$-0.727954\pi$
$773$	−36.5039	−1.31295	−0.656476	+	0.754347i	$0.727954\pi$
$774$	0	0
$775$	61.6079	2.21302
$776$	−19.2495	−0.691018
$777$	0	0
$778$	10.2773	0.368460
$779$	−3.57921	−0.128239
$780$	0	0
$781$	17.8656	0.639282
$782$	−3.18466	−0.113883
$783$	0	0
$784$	2.50955	0.0896269
$785$	34.3304	1.22530
$786$	0	0
$787$	−44.7402	−1.59482	−0.797408	−	0.603440i	$-0.793796\pi$
$787$	−44.7402	−1.59482	−0.797408	+	0.603440i	$0.793796\pi$
$788$	−20.6504	−0.735639
$789$	0	0
$790$	−1.30722	−0.0465089
$791$	9.06525	0.322323
$792$	0	0
$793$	−36.0225	−1.27920
$794$	4.28857	0.152196
$795$	0	0
$796$	4.40076	0.155981
$797$	11.3919	0.403521	0.201761	−	0.979435i	$-0.435334\pi$
$797$	11.3919	0.403521	0.201761	+	0.979435i	$0.435334\pi$
$798$	0	0
$799$	−12.0312	−0.425632
$800$	73.7943	2.60902
$801$	0	0
$802$	2.98765	0.105498
$803$	−19.9534	−0.704141
$804$	0	0
$805$	−4.41546	−0.155625
$806$	−10.1031	−0.355866
$807$	0	0
$808$	−26.2691	−0.924143
$809$	23.4803	0.825523	0.412761	−	0.910839i	$-0.364564\pi$
$809$	23.4803	0.825523	0.412761	+	0.910839i	$0.364564\pi$
$810$	0	0
$811$	−0.320003	−0.0112368	−0.00561841	−	0.999984i	$-0.501788\pi$
$811$	−0.320003	−0.0112368	−0.00561841	+	0.999984i	$0.501788\pi$
$812$	9.74620	0.342025
$813$	0	0
$814$	7.91160	0.277301
$815$	31.4415	1.10135
$816$	0	0
$817$	−2.91829	−0.102098
$818$	−13.3853	−0.468004
$819$	0	0
$820$	39.6197	1.38358
$821$	−50.4892	−1.76208	−0.881042	−	0.473038i	$-0.843157\pi$
$821$	−50.4892	−1.76208	−0.881042	+	0.473038i	$0.843157\pi$
$822$	0	0
$823$	−29.6259	−1.03270	−0.516348	−	0.856379i	$-0.672709\pi$
$823$	−29.6259	−1.03270	−0.516348	+	0.856379i	$0.672709\pi$
$824$	−31.5529	−1.09920
$825$	0	0
$826$	−4.78867	−0.166619
$827$	3.13318	0.108951	0.0544757	−	0.998515i	$-0.482651\pi$
$827$	3.13318	0.108951	0.0544757	+	0.998515i	$0.482651\pi$
$828$	0	0
$829$	−20.1070	−0.698345	−0.349172	−	0.937059i	$-0.613537\pi$
$829$	−20.1070	−0.698345	−0.349172	+	0.937059i	$0.613537\pi$
$830$	15.6376	0.542789
$831$	0	0
$832$	11.3144	0.392257
$833$	−6.24991	−0.216547
$834$	0	0
$835$	−52.9270	−1.83161
$836$	2.02381	0.0699950
$837$	0	0
$838$	0.407972	0.0140932
$839$	6.63075	0.228919	0.114460	−	0.993428i	$-0.463486\pi$
$839$	6.63075	0.228919	0.114460	+	0.993428i	$0.463486\pi$
$840$	0	0
$841$	2.36131	0.0814244
$842$	11.9156	0.410640
$843$	0	0
$844$	38.2525	1.31670
$845$	−38.7045	−1.33147
$846$	0	0
$847$	−8.19403	−0.281550
$848$	4.61922	0.158625
$849$	0	0
$850$	−46.1657	−1.58347
$851$	9.26901	0.317738
$852$	0	0
$853$	−36.1804	−1.23879	−0.619397	−	0.785078i	$-0.712623\pi$
$853$	−36.1804	−1.23879	−0.619397	+	0.785078i	$0.712623\pi$
$854$	3.93438	0.134632
$855$	0	0
$856$	−2.54368	−0.0869413
$857$	−12.6983	−0.433764	−0.216882	−	0.976198i	$-0.569589\pi$
$857$	−12.6983	−0.433764	−0.216882	+	0.976198i	$0.569589\pi$
$858$	0	0
$859$	49.2759	1.68127	0.840636	−	0.541600i	$-0.182181\pi$
$859$	49.2759	1.68127	0.840636	+	0.541600i	$0.182181\pi$
$860$	32.3037	1.10155
$861$	0	0
$862$	17.0625	0.581152
$863$	3.28011	0.111656	0.0558282	−	0.998440i	$-0.482220\pi$
$863$	3.28011	0.111656	0.0558282	+	0.998440i	$0.482220\pi$
$864$	0	0
$865$	−9.27218	−0.315263
$866$	12.2992	0.417944
$867$	0	0
$868$	−7.39636	−0.251049
$869$	0.973255	0.0330154
$870$	0	0
$871$	38.3833	1.30057
$872$	−7.63876	−0.258681
$873$	0	0
$874$	−0.353736	−0.0119653
$875$	−41.9305	−1.41751
$876$	0	0
$877$	7.64380	0.258113	0.129056	−	0.991637i	$-0.458805\pi$
$877$	7.64380	0.258113	0.129056	+	0.991637i	$0.458805\pi$
$878$	−8.38056	−0.282830
$879$	0	0
$880$	−18.5615	−0.625709
$881$	−5.21611	−0.175735	−0.0878676	−	0.996132i	$-0.528005\pi$
$881$	−5.21611	−0.175735	−0.0878676	+	0.996132i	$0.528005\pi$
$882$	0	0
$883$	−45.2052	−1.52127	−0.760637	−	0.649177i	$-0.775113\pi$
$883$	−45.2052	−1.52127	−0.760637	+	0.649177i	$0.775113\pi$
$884$	−50.7455	−1.70676
$885$	0	0
$886$	−4.88428	−0.164090
$887$	8.59600	0.288626	0.144313	−	0.989532i	$-0.453903\pi$
$887$	8.59600	0.288626	0.144313	+	0.989532i	$0.453903\pi$
$888$	0	0
$889$	−21.4977	−0.721010
$890$	−30.3654	−1.01785
$891$	0	0
$892$	−0.0526851	−0.00176403
$893$	−1.33636	−0.0447196
$894$	0	0
$895$	−70.5263	−2.35744
$896$	−11.4169	−0.381412
$897$	0	0
$898$	−2.12646	−0.0709609
$899$	−23.8000	−0.793774
$900$	0	0
$901$	−11.5039	−0.383251
$902$	4.40076	0.146529
$903$	0	0
$904$	−17.2775	−0.574642
$905$	12.8035	0.425603
$906$	0	0
$907$	2.50429	0.0831537	0.0415769	−	0.999135i	$-0.486762\pi$
$907$	2.50429	0.0831537	0.0415769	+	0.999135i	$0.486762\pi$
$908$	36.9640	1.22669
$909$	0	0
$910$	10.4967	0.347961
$911$	−29.5776	−0.979951	−0.489975	−	0.871736i	$-0.662994\pi$
$911$	−29.5776	−0.979951	−0.489975	+	0.871736i	$0.662994\pi$
$912$	0	0
$913$	−11.6425	−0.385311
$914$	4.09001	0.135286
$915$	0	0
$916$	−30.3448	−1.00262
$917$	−1.17529	−0.0388114
$918$	0	0
$919$	23.5583	0.777117	0.388558	−	0.921424i	$-0.372973\pi$
$919$	23.5583	0.777117	0.388558	+	0.921424i	$0.372973\pi$
$920$	8.41546	0.277450
$921$	0	0
$922$	−5.92057	−0.194984
$923$	−49.7579	−1.63780
$924$	0	0
$925$	134.366	4.41794
$926$	−18.2783	−0.600664
$927$	0	0
$928$	−28.5078	−0.935813
$929$	37.9337	1.24456	0.622281	−	0.782794i	$-0.286206\pi$
$929$	37.9337	1.24456	0.622281	+	0.782794i	$0.286206\pi$
$930$	0	0
$931$	−0.694209	−0.0227518
$932$	−19.5062	−0.638947
$933$	0	0
$934$	6.71240	0.219636
$935$	46.2265	1.51177
$936$	0	0
$937$	−42.3490	−1.38348	−0.691741	−	0.722146i	$-0.743156\pi$
$937$	−42.3490	−1.38348	−0.691741	+	0.722146i	$0.743156\pi$
$938$	−4.19223	−0.136881
$939$	0	0
$940$	14.7927	0.482485
$941$	33.9735	1.10750	0.553752	−	0.832682i	$-0.313196\pi$
$941$	33.9735	1.10750	0.553752	+	0.832682i	$0.313196\pi$
$942$	0	0
$943$	5.15582	0.167896
$944$	−23.5842	−0.767602
$945$	0	0
$946$	3.58813	0.116660
$947$	27.7471	0.901659	0.450829	−	0.892610i	$-0.351128\pi$
$947$	27.7471	0.901659	0.450829	+	0.892610i	$0.351128\pi$
$948$	0	0
$949$	55.5727	1.80397
$950$	−5.12786	−0.166370
$951$	0	0
$952$	11.9117	0.386062
$953$	6.25582	0.202646	0.101323	−	0.994854i	$-0.467692\pi$
$953$	6.25582	0.202646	0.101323	+	0.994854i	$0.467692\pi$
$954$	0	0
$955$	−95.1472	−3.07889
$956$	−32.3105	−1.04500
$957$	0	0
$958$	−2.66537	−0.0861142
$959$	−15.8562	−0.512024
$960$	0	0
$961$	−12.9383	−0.417364
$962$	−22.0348	−0.710429
$963$	0	0
$964$	23.7657	0.765441
$965$	−20.2547	−0.652021
$966$	0	0
$967$	43.1613	1.38797	0.693987	−	0.719988i	$-0.255853\pi$
$967$	43.1613	1.38797	0.693987	+	0.719988i	$0.255853\pi$
$968$	15.6171	0.501951
$969$	0	0
$970$	−22.7239	−0.729621
$971$	−6.69189	−0.214753	−0.107377	−	0.994218i	$-0.534245\pi$
$971$	−6.69189	−0.214753	−0.107377	+	0.994218i	$0.534245\pi$
$972$	0	0
$973$	−1.93438	−0.0620135
$974$	14.7658	0.473125
$975$	0	0
$976$	19.3769	0.620239
$977$	47.1871	1.50965	0.754825	−	0.655926i	$-0.227722\pi$
$977$	47.1871	1.50965	0.754825	+	0.655926i	$0.227722\pi$
$978$	0	0
$979$	22.6077	0.722545
$980$	7.68447	0.245472
$981$	0	0
$982$	8.50502	0.271406
$983$	−18.4505	−0.588480	−0.294240	−	0.955732i	$-0.595066\pi$
$983$	−18.4505	−0.588480	−0.294240	+	0.955732i	$0.595066\pi$
$984$	0	0
$985$	−52.3921	−1.66935
$986$	17.8344	0.567965
$987$	0	0
$988$	−5.63656	−0.179323
$989$	4.20376	0.133672
$990$	0	0
$991$	−24.3764	−0.774342	−0.387171	−	0.922008i	$-0.626548\pi$
$991$	−24.3764	−0.774342	−0.387171	+	0.922008i	$0.626548\pi$
$992$	21.6344	0.686894
$993$	0	0
$994$	5.43457	0.172374
$995$	11.1652	0.353960
$996$	0	0
$997$	59.5570	1.88619	0.943094	−	0.332525i	$-0.107901\pi$
$997$	59.5570	1.88619	0.943094	+	0.332525i	$0.107901\pi$
$998$	4.87750	0.154395
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.p.1.3		4
3.2	odd	2		483.2.a.i.1.2	✓	4
12.11	even	2		7728.2.a.cd.1.4		4
21.20	even	2		3381.2.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.2	✓	4	3.2	odd	2
1449.2.a.p.1.3		4	1.1	even	1	trivial
3381.2.a.w.1.2		4	21.20	even	2
7728.2.a.cd.1.4		4	12.11	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+0.509552 q^{2} -1.74036 q^{4} -4.41546 q^{5} +1.00000 q^{7} -1.90591 q^{8} +O(q^{10})\)	\(q+0.509552 q^{2} -1.74036 q^{4} -4.41546 q^{5} +1.00000 q^{7} -1.90591 q^{8} -2.24991 q^{10} +1.67510 q^{11} -4.66537 q^{13} +0.509552 q^{14} +2.50955 q^{16} -6.24991 q^{17} -0.694209 q^{19} +7.68447 q^{20} +0.853553 q^{22} +1.00000 q^{23} +14.4963 q^{25} -2.37725 q^{26} -1.74036 q^{28} -5.60012 q^{29} +4.24991 q^{31} +5.09056 q^{32} -3.18466 q^{34} -4.41546 q^{35} +9.26901 q^{37} -0.353736 q^{38} +8.41546 q^{40} +5.15582 q^{41} +4.20376 q^{43} -2.91528 q^{44} +0.509552 q^{46} +1.92501 q^{47} +1.00000 q^{49} +7.38662 q^{50} +8.11940 q^{52} +1.84066 q^{53} -7.39636 q^{55} -1.90591 q^{56} -2.85355 q^{58} -9.39779 q^{59} +7.72125 q^{61} +2.16555 q^{62} -2.42520 q^{64} +20.5998 q^{65} -8.22728 q^{67} +10.8771 q^{68} -2.24991 q^{70} +10.6654 q^{71} -11.9117 q^{73} +4.72305 q^{74} +1.20817 q^{76} +1.67510 q^{77} +0.581012 q^{79} -11.0808 q^{80} +2.62716 q^{82} -6.95032 q^{83} +27.5962 q^{85} +2.14204 q^{86} -3.19259 q^{88} +13.4963 q^{89} -4.66537 q^{91} -1.74036 q^{92} +0.980895 q^{94} +3.06525 q^{95} +10.0999 q^{97} +0.509552 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8	\( 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8} + 4 q^{10} + 5 q^{11} + 7 q^{13} + 8 q^{16} - 12 q^{17} + 3 q^{19} + q^{20} - q^{22} + 4 q^{23} + 7 q^{25} - 5 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} + 6 q^{32} - 9 q^{34} - 5 q^{35} + 20 q^{37} - 23 q^{38} + 21 q^{40} - 3 q^{41} + 9 q^{43} + 27 q^{44} - 7 q^{47} + 4 q^{49} - 3 q^{50} + 38 q^{52} + 6 q^{53} - 21 q^{55} + 3 q^{56} - 7 q^{58} + 2 q^{59} + 24 q^{61} + 9 q^{62} - 21 q^{64} + 14 q^{65} + q^{67} + 13 q^{68} + 4 q^{70} + 17 q^{71} + 16 q^{73} + 33 q^{74} - 25 q^{76} + 5 q^{77} - 10 q^{79} - 6 q^{80} - 7 q^{82} - 8 q^{83} + 17 q^{85} + 35 q^{86} - 12 q^{88} + 3 q^{89} + 7 q^{91} + 4 q^{92} + 8 q^{94} + 3 q^{95} - 2 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8 + 4 * q^10 + 5 * q^11 + 7 * q^13 + 8 * q^16 - 12 * q^17 + 3 * q^19 + q^20 - q^22 + 4 * q^23 + 7 * q^25 - 5 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 + 6 * q^32 - 9 * q^34 - 5 * q^35 + 20 * q^37 - 23 * q^38 + 21 * q^40 - 3 * q^41 + 9 * q^43 + 27 * q^44 - 7 * q^47 + 4 * q^49 - 3 * q^50 + 38 * q^52 + 6 * q^53 - 21 * q^55 + 3 * q^56 - 7 * q^58 + 2 * q^59 + 24 * q^61 + 9 * q^62 - 21 * q^64 + 14 * q^65 + q^67 + 13 * q^68 + 4 * q^70 + 17 * q^71 + 16 * q^73 + 33 * q^74 - 25 * q^76 + 5 * q^77 - 10 * q^79 - 6 * q^80 - 7 * q^82 - 8 * q^83 + 17 * q^85 + 35 * q^86 - 12 * q^88 + 3 * q^89 + 7 * q^91 + 4 * q^92 + 8 * q^94 + 3 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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