LMFDB - Embedded newform 3381.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
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			1.2
Root			$0.509552$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.509552 q^{2} +1.00000 q^{3} -1.74036 q^{4} -4.41546 q^{5} -0.509552 q^{6} +1.90591 q^{8} +1.00000 q^{9} +2.24991 q^{10} -1.67510 q^{11} -1.74036 q^{12} +4.66537 q^{13} -4.41546 q^{15} +2.50955 q^{16} -6.24991 q^{17} -0.509552 q^{18} +0.694209 q^{19} +7.68447 q^{20} +0.853553 q^{22} -1.00000 q^{23} +1.90591 q^{24} +14.4963 q^{25} -2.37725 q^{26} +1.00000 q^{27} +5.60012 q^{29} +2.24991 q^{30} -4.24991 q^{31} -5.09056 q^{32} -1.67510 q^{33} +3.18466 q^{34} -1.74036 q^{36} +9.26901 q^{37} -0.353736 q^{38} +4.66537 q^{39} -8.41546 q^{40} +5.15582 q^{41} +4.20376 q^{43} +2.91528 q^{44} -4.41546 q^{45} +0.509552 q^{46} +1.92501 q^{47} +2.50955 q^{48} -7.38662 q^{50} -6.24991 q^{51} -8.11940 q^{52} -1.84066 q^{53} -0.509552 q^{54} +7.39636 q^{55} +0.694209 q^{57} -2.85355 q^{58} -9.39779 q^{59} +7.68447 q^{60} -7.72125 q^{61} +2.16555 q^{62} -2.42520 q^{64} -20.5998 q^{65} +0.853553 q^{66} -8.22728 q^{67} +10.8771 q^{68} -1.00000 q^{69} -10.6654 q^{71} +1.90591 q^{72} +11.9117 q^{73} -4.72305 q^{74} +14.4963 q^{75} -1.20817 q^{76} -2.37725 q^{78} +0.581012 q^{79} -11.0808 q^{80} +1.00000 q^{81} -2.62716 q^{82} -6.95032 q^{83} +27.5962 q^{85} -2.14204 q^{86} +5.60012 q^{87} -3.19259 q^{88} +13.4963 q^{89} +2.24991 q^{90} +1.74036 q^{92} -4.24991 q^{93} -0.980895 q^{94} -3.06525 q^{95} -5.09056 q^{96} -10.0999 q^{97} -1.67510 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{8} + 4 q^{9} - 4 q^{10} - 5 q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 12 q^{17} - 3 q^{19} + q^{20} - q^{22} - 4 q^{23} - 3 q^{24} + 7 q^{25} - 5 q^{26}+ \cdots - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^8 + 4 * q^9 - 4 * q^10 - 5 * q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 12 * q^17 - 3 * q^19 + q^20 - q^22 - 4 * q^23 - 3 * q^24 + 7 * q^25 - 5 * q^26 + 4 * q^27 + 6 * q^29 - 4 * q^30 - 4 * q^31 - 6 * q^32 - 5 * q^33 + 9 * q^34 + 4 * q^36 + 20 * q^37 - 23 * q^38 - 7 * q^39 - 21 * q^40 - 3 * q^41 + 9 * q^43 - 27 * q^44 - 5 * q^45 - 7 * q^47 + 8 * q^48 + 3 * q^50 - 12 * q^51 - 38 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^57 - 7 * q^58 + 2 * q^59 + q^60 - 24 * q^61 + 9 * q^62 - 21 * q^64 - 14 * q^65 - q^66 + q^67 + 13 * q^68 - 4 * q^69 - 17 * q^71 - 3 * q^72 - 16 * q^73 - 33 * q^74 + 7 * q^75 + 25 * q^76 - 5 * q^78 - 10 * q^79 - 6 * q^80 + 4 * q^81 + 7 * q^82 - 8 * q^83 + 17 * q^85 - 35 * q^86 + 6 * q^87 - 12 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 8 * q^94 - 3 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^99

Coefficient data

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

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

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

$2$	−0.509552	−0.360308	−0.180154	−	0.983638i	$-0.557660\pi$
$2$	−0.509552	−0.360308	−0.180154	+	0.983638i	$0.557660\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−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	$0.949270\pi$
$6$	−0.509552	−0.208024
$7$	0	0
$7$	0	0
$8$	1.90591	0.673840
$9$	1.00000	0.333333
$10$	2.24991	0.711484
$11$	−1.67510	−0.505063	−0.252531	−	0.967589i	$-0.581263\pi$
$11$	−1.67510	−0.505063	−0.252531	+	0.967589i	$0.581263\pi$
$12$	−1.74036	−0.502398
$13$	4.66537	1.29394	0.646970	−	0.762515i	$-0.276036\pi$
$13$	4.66537	1.29394	0.646970	+	0.762515i	$0.276036\pi$
$14$	0	0
$15$	−4.41546	−1.14007
$16$	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	$0.773781\pi$
$18$	−0.509552	−0.120103
$19$	0.694209	0.159262	0.0796312	−	0.996824i	$-0.474626\pi$
$19$	0.694209	0.159262	0.0796312	+	0.996824i	$0.474626\pi$
$20$	7.68447	1.71830
$21$	0	0
$22$	0.853553	0.181978
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.90591	0.389042
$25$	14.4963	2.89926
$26$	−2.37725	−0.466217
$27$	1.00000	0.192450
$28$	0	0
$29$	5.60012	1.03992	0.519958	−	0.854192i	$-0.325948\pi$
$29$	5.60012	1.03992	0.519958	+	0.854192i	$0.325948\pi$
$30$	2.24991	0.410775
$31$	−4.24991	−0.763306	−0.381653	−	0.924306i	$-0.624645\pi$
$31$	−4.24991	−0.763306	−0.381653	+	0.924306i	$0.624645\pi$
$32$	−5.09056	−0.899893
$33$	−1.67510	−0.291598
$34$	3.18466	0.546164
$35$	0	0
$36$	−1.74036	−0.290059
$37$	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
$39$	4.66537	0.747057
$40$	−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
$45$	−4.41546	−0.658218
$46$	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	$0.455162\pi$
$48$	2.50955	0.362223
$49$	0	0
$50$	−7.38662	−1.04463
$51$	−6.24991	−0.875162
$52$	−8.11940	−1.12596
$53$	−1.84066	−0.252833	−0.126417	−	0.991977i	$-0.540348\pi$
$53$	−1.84066	−0.252833	−0.126417	+	0.991977i	$0.540348\pi$
$54$	−0.509552	−0.0693413
$55$	7.39636	0.997324
$56$	0	0
$57$	0.694209	0.0919502
$58$	−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	$0.709532\pi$
$60$	7.68447	0.992061
$61$	−7.72125	−0.988605	−0.494302	−	0.869290i	$-0.664576\pi$
$61$	−7.72125	−0.988605	−0.494302	+	0.869290i	$0.664576\pi$
$62$	2.16555	0.275025
$63$	0	0
$64$	−2.42520	−0.303149
$65$	−20.5998	−2.55508
$66$	0.853553	0.105065
$67$	−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
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−10.6654	−1.26575	−0.632873	−	0.774255i	$-0.718125\pi$
$71$	−10.6654	−1.26575	−0.632873	+	0.774255i	$0.718125\pi$
$72$	1.90591	0.224613
$73$	11.9117	1.39416	0.697082	−	0.716991i	$-0.254481\pi$
$73$	11.9117	1.39416	0.697082	+	0.716991i	$0.254481\pi$
$74$	−4.72305	−0.549043
$75$	14.4963	1.67389
$76$	−1.20817	−0.138587
$77$	0	0
$78$	−2.37725	−0.269171
$79$	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
$81$	1.00000	0.111111
$82$	−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
$87$	5.60012	0.600396
$88$	−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	$0.246289\pi$
$90$	2.24991	0.237161
$91$	0	0
$92$	1.74036	0.181445
$93$	−4.24991	−0.440695
$94$	−0.980895	−0.101172
$95$	−3.06525	−0.314488
$96$	−5.09056	−0.519554
$97$	−10.0999	−1.02549	−0.512746	−	0.858540i	$-0.671372\pi$
$97$	−10.0999	−1.02549	−0.512746	+	0.858540i	$0.671372\pi$
$98$	0	0
$99$	−1.67510	−0.168354
$100$	−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	$0.259484\pi$
$102$	3.18466	0.315328
$103$	−16.5553	−1.63125	−0.815623	−	0.578584i	$-0.803605\pi$
$103$	−16.5553	−1.63125	−0.815623	+	0.578584i	$0.803605\pi$
$104$	8.89176	0.871909
$105$	0	0
$106$	0.937911	0.0910979
$107$	−1.33463	−0.129024	−0.0645118	−	0.997917i	$-0.520549\pi$
$107$	−1.33463	−0.129024	−0.0645118	+	0.997917i	$0.520549\pi$
$108$	−1.74036	−0.167466
$109$	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
$111$	9.26901	0.879776
$112$	0	0
$113$	−9.06525	−0.852787	−0.426394	−	0.904538i	$-0.640216\pi$
$113$	−9.06525	−0.852787	−0.426394	+	0.904538i	$0.640216\pi$
$114$	−0.353736	−0.0331304
$115$	4.41546	0.411744
$116$	−9.74620	−0.904912
$117$	4.66537	0.431314
$118$	4.78867	0.440832
$119$	0	0
$120$	−8.41546	−0.768223
$121$	−8.19403	−0.744911
$122$	3.93438	0.356202
$123$	5.15582	0.464884
$124$	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
$129$	4.20376	0.370121
$130$	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	$0.516350\pi$
$132$	2.91528	0.253742
$133$	0	0
$134$	4.19223	0.362153
$135$	−4.41546	−0.380022
$136$	−11.9117	−1.02142
$137$	15.8562	1.35469	0.677345	−	0.735666i	$-0.263131\pi$
$137$	15.8562	1.35469	0.677345	+	0.735666i	$0.263131\pi$
$138$	0.509552	0.0433760
$139$	1.93438	0.164072	0.0820361	−	0.996629i	$-0.473858\pi$
$139$	1.93438	0.164072	0.0820361	+	0.996629i	$0.473858\pi$
$140$	0	0
$141$	1.92501	0.162115
$142$	5.43457	0.456059
$143$	−7.81498	−0.653521
$144$	2.50955	0.209129
$145$	−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	$-0.465865\pi$
$149$	2.61301	0.214067	0.107033	+	0.994255i	$0.465865\pi$
$150$	−7.38662	−0.603115
$151$	−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
$153$	−6.24991	−0.505275
$154$	0	0
$155$	18.7653	1.50727
$156$	−8.11940	−0.650073
$157$	7.77504	0.620516	0.310258	−	0.950652i	$-0.399585\pi$
$157$	7.77504	0.620516	0.310258	+	0.950652i	$0.399585\pi$
$158$	−0.296056	−0.0235530
$159$	−1.84066	−0.145973
$160$	22.4772	1.77698
$161$	0	0
$162$	−0.509552	−0.0400342
$163$	−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
$165$	7.39636	0.575805
$166$	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
$171$	0.694209	0.0530875
$172$	−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	$0.474563\pi$
$174$	−2.85355	−0.216327
$175$	0	0
$176$	−4.20376	−0.316870
$177$	−9.39779	−0.706381
$178$	−6.87707	−0.515458
$179$	−15.9726	−1.19385	−0.596924	−	0.802298i	$-0.703611\pi$
$179$	−15.9726	−1.19385	−0.596924	+	0.802298i	$0.703611\pi$
$180$	7.68447	0.572767
$181$	2.89970	0.215533	0.107767	−	0.994176i	$-0.465630\pi$
$181$	2.89970	0.215533	0.107767	+	0.994176i	$0.465630\pi$
$182$	0	0
$183$	−7.72125	−0.570771
$184$	−1.90591	−0.140505
$185$	−40.9270	−3.00901
$186$	2.16555	0.158786
$187$	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	$-0.784579\pi$
$191$	−21.5486	−1.55921	−0.779603	+	0.626275i	$0.784579\pi$
$192$	−2.42520	−0.175023
$193$	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
$195$	−20.5998	−1.47518
$196$	0	0
$197$	−11.8656	−0.845389	−0.422695	−	0.906272i	$-0.638916\pi$
$197$	−11.8656	−0.845389	−0.422695	+	0.906272i	$0.638916\pi$
$198$	0.853553	0.0606594
$199$	2.52866	0.179252	0.0896259	−	0.995976i	$-0.471433\pi$
$199$	2.52866	0.179252	0.0896259	+	0.995976i	$0.471433\pi$
$200$	27.6286	1.95364
$201$	−8.22728	−0.580307
$202$	−7.02315	−0.494147
$203$	0	0
$204$	10.8771	0.761547
$205$	−22.7653	−1.59000
$206$	8.43581	0.587751
$207$	−1.00000	−0.0695048
$208$	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
$213$	−10.6654	−0.730779
$214$	0.680065	0.0464883
$215$	−18.5615	−1.26589
$216$	1.90591	0.129681
$217$	0	0
$218$	−2.04225	−0.138319
$219$	11.9117	0.804921
$220$	−12.8723	−0.867850
$221$	−29.1581	−1.96139
$222$	−4.72305	−0.316990
$223$	−0.0302726	−0.00202720	−0.00101360	−	0.999999i	$-0.500323\pi$
$223$	−0.0302726	−0.00202720	−0.00101360	+	0.999999i	$0.500323\pi$
$224$	0	0
$225$	14.4963	0.966419
$226$	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	$0.748986\pi$
$228$	−1.20817	−0.0800131
$229$	−17.4360	−1.15220	−0.576102	−	0.817378i	$-0.695427\pi$
$229$	−17.4360	−1.15220	−0.576102	+	0.817378i	$0.695427\pi$
$230$	−2.24991	−0.148355
$231$	0	0
$232$	10.6733	0.700737
$233$	−11.2082	−0.734272	−0.367136	−	0.930167i	$-0.619662\pi$
$233$	−11.2082	−0.734272	−0.367136	+	0.930167i	$0.619662\pi$
$234$	−2.37725	−0.155406
$235$	−8.49982	−0.554467
$236$	16.3555	1.06465
$237$	0.581012	0.0377408
$238$	0	0
$239$	−18.5654	−1.20090	−0.600449	−	0.799663i	$-0.705011\pi$
$239$	−18.5654	−1.20090	−0.600449	+	0.799663i	$0.705011\pi$
$240$	−11.0808	−0.715264
$241$	13.6556	0.879637	0.439818	−	0.898087i	$-0.355043\pi$
$241$	13.6556	0.879637	0.439818	+	0.898087i	$0.355043\pi$
$242$	4.17529	0.268398
$243$	1.00000	0.0641500
$244$	13.4377	0.860262
$245$	0	0
$246$	−2.62716	−0.167502
$247$	3.23874	0.206076
$248$	−8.09993	−0.514346
$249$	−6.95032	−0.440459
$250$	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
$255$	27.5962	1.72814
$256$	−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	$0.645605\pi$
$258$	−2.14204	−0.133357
$259$	0	0
$260$	35.8509	2.22338
$261$	5.60012	0.346639
$262$	0.598870	0.0369983
$263$	20.5788	1.26895	0.634473	−	0.772945i	$-0.281217\pi$
$263$	20.5788	1.26895	0.634473	+	0.772945i	$0.281217\pi$
$264$	−3.19259	−0.196491
$265$	8.12734	0.499259
$266$	0	0
$267$	13.4963	0.825960
$268$	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	$0.495180\pi$
$270$	2.24991	0.136925
$271$	−24.0547	−1.46122	−0.730609	−	0.682797i	$-0.760764\pi$
$271$	−24.0547	−1.46122	−0.730609	+	0.682797i	$0.760764\pi$
$272$	−15.6845	−0.951011
$273$	0	0
$274$	−8.07958	−0.488105
$275$	−24.2828	−1.46431
$276$	1.74036	0.104757
$277$	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
$279$	−4.24991	−0.254435
$280$	0	0
$281$	−15.0697	−0.898985	−0.449492	−	0.893284i	$-0.648395\pi$
$281$	−15.0697	−0.898985	−0.449492	+	0.893284i	$0.648395\pi$
$282$	−0.980895	−0.0584114
$283$	−9.22691	−0.548483	−0.274241	−	0.961661i	$-0.588427\pi$
$283$	−9.22691	−0.548483	−0.274241	+	0.961661i	$0.588427\pi$
$284$	18.5615	1.10142
$285$	−3.06525	−0.181570
$286$	3.98214	0.235469
$287$	0	0
$288$	−5.09056	−0.299964
$289$	22.0614	1.29773
$290$	12.5998	0.739883
$291$	−10.0999	−0.592069
$292$	−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
$297$	−1.67510	−0.0971994
$298$	−1.33147	−0.0771299
$299$	−4.66537	−0.269805
$300$	−25.2287	−1.45658
$301$	0	0
$302$	4.71989	0.271599
$303$	13.7830	0.791811
$304$	1.74215	0.0999194
$305$	34.0929	1.95215
$306$	3.18466	0.182055
$307$	−0.823281	−0.0469871	−0.0234936	−	0.999724i	$-0.507479\pi$
$307$	−0.823281	−0.0469871	−0.0234936	+	0.999724i	$0.507479\pi$
$308$	0	0
$309$	−16.5553	−0.941800
$310$	−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	$0.164521\pi$
$312$	8.89176	0.503397
$313$	−13.1323	−0.742282	−0.371141	−	0.928577i	$-0.621033\pi$
$313$	−13.1323	−0.742282	−0.371141	+	0.928577i	$0.621033\pi$
$314$	−3.96179	−0.223577
$315$	0	0
$316$	−1.01117	−0.0568826
$317$	−15.2269	−0.855229	−0.427614	−	0.903961i	$-0.640646\pi$
$317$	−15.2269	−0.855229	−0.427614	+	0.903961i	$0.640646\pi$
$318$	0.937911	0.0525954
$319$	−9.38078	−0.525223
$320$	10.7084	0.598615
$321$	−1.33463	−0.0744918
$322$	0	0
$323$	−4.33874	−0.241414
$324$	−1.74036	−0.0966865
$325$	67.6305	3.75147
$326$	3.62841	0.200959
$327$	4.00794	0.221639
$328$	9.82651	0.542578
$329$	0	0
$330$	−3.76883	−0.207467
$331$	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
$333$	9.26901	0.507939
$334$	−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
$339$	−9.06525	−0.492357
$340$	−48.0273	−2.60464
$341$	7.11904	0.385518
$342$	−0.353736	−0.0191278
$343$	0	0
$344$	8.01198	0.431977
$345$	4.41546	0.237720
$346$	−1.07003	−0.0575250
$347$	21.6729	1.16346	0.581732	−	0.813380i	$-0.302375\pi$
$347$	21.6729	1.16346	0.581732	+	0.813380i	$0.302375\pi$
$348$	−9.74620	−0.522451
$349$	21.5198	1.15193	0.575964	−	0.817475i	$-0.304627\pi$
$349$	21.5198	1.15193	0.575964	+	0.817475i	$0.304627\pi$
$350$	0	0
$351$	4.66537	0.249019
$352$	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	$0.319492\pi$
$354$	4.78867	0.254515
$355$	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	$-0.289751\pi$
$359$	23.2492	1.22705	0.613524	+	0.789676i	$0.289751\pi$
$360$	−8.41546	−0.443534
$361$	−18.5181	−0.974635
$362$	−1.47755	−0.0776583
$363$	−8.19403	−0.430075
$364$	0	0
$365$	−52.5959	−2.75299
$366$	3.93438	0.205653
$367$	−27.7171	−1.44682	−0.723409	−	0.690419i	$-0.757426\pi$
$367$	−27.7171	−1.44682	−0.723409	+	0.690419i	$0.757426\pi$
$368$	−2.50955	−0.130819
$369$	5.15582	0.268401
$370$	20.8544	1.08417
$371$	0	0
$372$	7.39636	0.383483
$373$	−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
$375$	−41.9305	−2.16528
$376$	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
$381$	−21.4977	−1.10136
$382$	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	$0.339126\pi$
$384$	11.4169	0.582616
$385$	0	0
$386$	−2.33743	−0.118972
$387$	4.20376	0.213689
$388$	17.5775	0.892362
$389$	−20.1693	−1.02262	−0.511312	−	0.859395i	$-0.670840\pi$
$389$	−20.1693	−1.02262	−0.511312	+	0.859395i	$0.670840\pi$
$390$	10.4967	0.531519
$391$	6.24991	0.316071
$392$	0	0
$393$	−1.17529	−0.0592854
$394$	6.04615	0.304600
$395$	−2.56543	−0.129081
$396$	2.91528	0.146498
$397$	−8.41634	−0.422404	−0.211202	−	0.977442i	$-0.567738\pi$
$397$	−8.41634	−0.422404	−0.211202	+	0.977442i	$0.567738\pi$
$398$	−1.28848	−0.0645859
$399$	0	0
$400$	36.3792	1.81896
$401$	−5.86329	−0.292799	−0.146399	−	0.989226i	$-0.546768\pi$
$401$	−5.86329	−0.292799	−0.146399	+	0.989226i	$0.546768\pi$
$402$	4.19223	0.209089
$403$	−19.8274	−0.987673
$404$	−23.9873	−1.19341
$405$	−4.41546	−0.219406
$406$	0	0
$407$	−15.5266	−0.769623
$408$	−11.9117	−0.589719
$409$	26.2686	1.29890	0.649451	−	0.760404i	$-0.274999\pi$
$409$	26.2686	1.29890	0.649451	+	0.760404i	$0.274999\pi$
$410$	11.6001	0.572889
$411$	15.8562	0.782130
$412$	28.8122	1.41947
$413$	0	0
$414$	0.509552	0.0250431
$415$	30.6889	1.50646
$416$	−23.7494	−1.16441
$417$	1.93438	0.0947271
$418$	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
$423$	1.92501	0.0935973
$424$	−3.50812	−0.170369
$425$	−90.6005	−4.39477
$426$	5.43457	0.263306
$427$	0	0
$428$	2.32273	0.112274
$429$	−7.81498	−0.377311
$430$	9.45808	0.456109
$431$	−33.4853	−1.61293	−0.806465	−	0.591281i	$-0.798622\pi$
$431$	−33.4853	−1.61293	−0.806465	+	0.591281i	$0.798622\pi$
$432$	2.50955	0.120741
$433$	−24.1373	−1.15996	−0.579982	−	0.814629i	$-0.696941\pi$
$433$	−24.1373	−1.15996	−0.579982	+	0.814629i	$0.696941\pi$
$434$	0	0
$435$	−24.7271	−1.18557
$436$	−6.97524	−0.334053
$437$	−0.694209	−0.0332085
$438$	−6.06966	−0.290020
$439$	16.4469	0.784968	0.392484	−	0.919759i	$-0.371616\pi$
$439$	16.4469	0.784968	0.392484	+	0.919759i	$0.371616\pi$
$440$	14.0968	0.672037
$441$	0	0
$442$	14.8576	0.706704
$443$	9.58542	0.455417	0.227709	−	0.973729i	$-0.426877\pi$
$443$	9.58542	0.455417	0.227709	+	0.973729i	$0.426877\pi$
$444$	−16.1314	−0.765562
$445$	−59.5923	−2.82495
$446$	0.0154255	0.000730417 0
$447$	2.61301	0.123591
$448$	0	0
$449$	4.17319	0.196945	0.0984725	−	0.995140i	$-0.468604\pi$
$449$	4.17319	0.196945	0.0984725	+	0.995140i	$0.468604\pi$
$450$	−7.38662	−0.348209
$451$	−8.63653	−0.406678
$452$	15.7768	0.742077
$453$	−9.26281	−0.435205
$454$	10.8226	0.507927
$455$	0	0
$456$	1.32310	0.0619598
$457$	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
$459$	−6.24991	−0.291721
$460$	−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
$465$	18.7653	0.870220
$466$	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	$0.401414\pi$
$468$	−8.11940	−0.375320
$469$	0	0
$470$	4.33110	0.199779
$471$	7.77504	0.358255
$472$	−17.9113	−0.824435
$473$	−7.04174	−0.323779
$474$	−0.296056	−0.0135983
$475$	10.0635	0.461743
$476$	0	0
$477$	−1.84066	−0.0842778
$478$	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	$0.538129\pi$
$480$	22.4772	1.02594
$481$	43.2434	1.97173
$482$	−6.95826	−0.316940
$483$	0	0
$484$	14.2605	0.648206
$485$	44.5959	2.02499
$486$	−0.509552	−0.0231138
$487$	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
$489$	−7.12077	−0.322012
$490$	0	0
$491$	−16.6912	−0.753262	−0.376631	−	0.926363i	$-0.622917\pi$
$491$	−16.6912	−0.753262	−0.376631	+	0.926363i	$0.622917\pi$
$492$	−8.97296	−0.404532
$493$	−35.0002	−1.57633
$494$	−1.65031	−0.0742509
$495$	7.39636	0.332441
$496$	−10.6654	−0.478889
$497$	0	0
$498$	3.54156	0.158701
$499$	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
$501$	11.9867	0.535528
$502$	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
$507$	8.76567	0.389297
$508$	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	$0.544137\pi$
$510$	−14.0617	−0.622664
$511$	0	0
$512$	−22.3410	−0.987342
$513$	0.694209	0.0306501
$514$	7.21538	0.318257
$515$	73.0994	3.22115
$516$	−7.31604	−0.322071
$517$	−3.22460	−0.141818
$518$	0	0
$519$	2.09993	0.0921769
$520$	−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	$0.589458\pi$
$522$	−2.85355	−0.124897
$523$	−43.3446	−1.89533	−0.947663	−	0.319272i	$-0.896562\pi$
$523$	−43.3446	−1.89533	−0.947663	+	0.319272i	$0.896562\pi$
$524$	2.04542	0.0893545
$525$	0	0
$526$	−10.4860	−0.457211
$527$	26.5615	1.15704
$528$	−4.20376	−0.182945
$529$	1.00000	0.0434783
$530$	−4.14131	−0.179887
$531$	−9.39779	−0.407829
$532$	0	0
$533$	24.0538	1.04189
$534$	−6.87707	−0.297600
$535$	5.89301	0.254777
$536$	−15.6804	−0.677291
$537$	−15.9726	−0.689268
$538$	−0.253072	−0.0109107
$539$	0	0
$540$	7.68447	0.330687
$541$	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
$543$	2.89970	0.124438
$544$	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
$549$	−7.72125	−0.329535
$550$	12.3734	0.527602
$551$	3.88765	0.165620
$552$	−1.90591	−0.0811208
$553$	0	0
$554$	−11.8421	−0.503122
$555$	−40.9270	−1.73725
$556$	−3.36652	−0.142772
$557$	−18.5921	−0.787773	−0.393887	−	0.919159i	$-0.628870\pi$
$557$	−18.5921	−0.787773	−0.393887	+	0.919159i	$0.628870\pi$
$558$	2.16555	0.0916751
$559$	19.6121	0.829503
$560$	0	0
$561$	10.4692	0.442012
$562$	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	$0.534247\pi$
$564$	−3.35021	−0.141069
$565$	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	$-0.710529\pi$
$569$	−29.3028	−1.22844	−0.614218	+	0.789136i	$0.710529\pi$
$570$	1.56191	0.0654211
$571$	−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
$573$	−21.5486	−0.900208
$574$	0	0
$575$	−14.4963	−0.604537
$576$	−2.42520	−0.101050
$577$	−24.7379	−1.02985	−0.514926	−	0.857235i	$-0.672181\pi$
$577$	−24.7379	−1.02985	−0.514926	+	0.857235i	$0.672181\pi$
$578$	−11.2414	−0.467581
$579$	4.58722	0.190638
$580$	43.0340	1.78689
$581$	0	0
$582$	5.14645	0.213327
$583$	3.08329	0.127697
$584$	22.7027	0.939444
$585$	−20.5998	−0.851695
$586$	−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
$591$	−11.8656	−0.488086
$592$	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	$0.773815\pi$
$594$	0.853553	0.0350217
$595$	0	0
$596$	−4.54758	−0.186276
$597$	2.52866	0.103491
$598$	2.37725	0.0972130
$599$	40.4224	1.65161	0.825807	−	0.563953i	$-0.190720\pi$
$599$	40.4224	1.65161	0.825807	+	0.563953i	$0.190720\pi$
$600$	27.6286	1.12793
$601$	−2.98464	−0.121746	−0.0608730	−	0.998146i	$-0.519388\pi$
$601$	−2.98464	−0.121746	−0.0608730	+	0.998146i	$0.519388\pi$
$602$	0	0
$603$	−8.22728	−0.335041
$604$	16.1206	0.655937
$605$	36.1804	1.47094
$606$	−7.02315	−0.285296
$607$	29.1893	1.18476	0.592378	−	0.805660i	$-0.298189\pi$
$607$	29.1893	1.18476	0.592378	+	0.805660i	$0.298189\pi$
$608$	−3.53392	−0.143319
$609$	0	0
$610$	−17.3721	−0.703376
$611$	8.98090	0.363328
$612$	10.8771	0.439679
$613$	−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
$615$	−22.7653	−0.917986
$616$	0	0
$617$	−8.07326	−0.325017	−0.162509	−	0.986707i	$-0.551959\pi$
$617$	−8.07326	−0.325017	−0.162509	+	0.986707i	$0.551959\pi$
$618$	8.43581	0.339338
$619$	−10.8589	−0.436457	−0.218228	−	0.975898i	$-0.570028\pi$
$619$	−10.8589	−0.436457	−0.218228	+	0.975898i	$0.570028\pi$
$620$	−32.6583	−1.31159
$621$	−1.00000	−0.0401286
$622$	−15.6245	−0.626486
$623$	0	0
$624$	11.7080	0.468695
$625$	112.661	4.50644
$626$	6.69160	0.267450
$627$	−1.16287	−0.0464406
$628$	−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
$633$	−21.9797	−0.873614
$634$	7.75891	0.308146
$635$	94.9223	3.76688
$636$	3.20340	0.127023
$637$	0	0
$638$	4.78000	0.189242
$639$	−10.6654	−0.421915
$640$	−50.4108	−1.99266
$641$	−46.7138	−1.84509	−0.922543	−	0.385895i	$-0.873893\pi$
$641$	−46.7138	−1.84509	−0.922543	+	0.385895i	$0.873893\pi$
$642$	0.680065	0.0268400
$643$	−33.7524	−1.33106	−0.665532	−	0.746369i	$-0.731795\pi$
$643$	−33.7524	−1.33106	−0.665532	+	0.746369i	$0.731795\pi$
$644$	0	0
$645$	−18.5615	−0.730860
$646$	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	$0.621349\pi$
$648$	1.90591	0.0748711
$649$	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	$-0.563764\pi$
$653$	−10.1696	−0.397967	−0.198984	+	0.980003i	$0.563764\pi$
$654$	−2.04225	−0.0798585
$655$	5.18943	0.202768
$656$	12.9388	0.505175
$657$	11.9117	0.464722
$658$	0	0
$659$	−4.60754	−0.179484	−0.0897421	−	0.995965i	$-0.528604\pi$
$659$	−4.60754	−0.179484	−0.0897421	+	0.995965i	$0.528604\pi$
$660$	−12.8723	−0.501053
$661$	31.1205	1.21045	0.605223	−	0.796056i	$-0.293084\pi$
$661$	31.1205	1.21045	0.605223	+	0.796056i	$0.293084\pi$
$662$	−13.0279	−0.506345
$663$	−29.1581	−1.13241
$664$	−13.2467	−0.514071
$665$	0	0
$666$	−4.72305	−0.183014
$667$	−5.60012	−0.216837
$668$	−20.8612	−0.807144
$669$	−0.0302726	−0.00117041
$670$	−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
$675$	14.4963	0.557962
$676$	−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	$0.502027\pi$
$678$	4.61922	0.177400
$679$	0	0
$680$	52.5959	2.01696
$681$	−21.2393	−0.813893
$682$	−3.62752	−0.138905
$683$	−16.8856	−0.646109	−0.323055	−	0.946380i	$-0.604710\pi$
$683$	−16.8856	−0.646109	−0.323055	+	0.946380i	$0.604710\pi$
$684$	−1.20817	−0.0461956
$685$	−70.0126	−2.67504
$686$	0	0
$687$	−17.4360	−0.665225
$688$	10.5496	0.402198
$689$	−8.58734	−0.327152
$690$	−2.24991	−0.0856526
$691$	11.1963	0.425929	0.212964	−	0.977060i	$-0.431688\pi$
$691$	11.1963	0.425929	0.212964	+	0.977060i	$0.431688\pi$
$692$	−3.65463	−0.138928
$693$	0	0
$694$	−11.0435	−0.419206
$695$	−8.54119	−0.323986
$696$	10.6733	0.404571
$697$	−32.2234	−1.22055
$698$	−10.9655	−0.415049
$699$	−11.2082	−0.423932
$700$	0	0
$701$	9.50609	0.359040	0.179520	−	0.983754i	$-0.442546\pi$
$701$	9.50609	0.359040	0.179520	+	0.983754i	$0.442546\pi$
$702$	−2.37725	−0.0897235
$703$	6.43463	0.242687
$704$	4.06245	0.153110
$705$	−8.49982	−0.320122
$706$	−10.2854	−0.387096
$707$	0	0
$708$	16.3555	0.614677
$709$	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
$711$	0.581012	0.0217896
$712$	25.7227	0.963998
$713$	4.24991	0.159160
$714$	0	0
$715$	34.5067	1.29048
$716$	27.7980	1.03886
$717$	−18.5654	−0.693339
$718$	−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	$0.284647\pi$
$720$	−11.0808	−0.412958
$721$	0	0
$722$	9.43593	0.351169
$723$	13.6556	0.507858
$724$	−5.04651	−0.187552
$725$	81.1809	3.01498
$726$	4.17529	0.154959
$727$	−37.5410	−1.39232	−0.696159	−	0.717887i	$-0.745109\pi$
$727$	−37.5410	−1.39232	−0.696159	+	0.717887i	$0.745109\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	26.8004	0.991925
$731$	−26.2731	−0.971747
$732$	13.4377	0.496673
$733$	−24.2222	−0.894668	−0.447334	−	0.894367i	$-0.647626\pi$
$733$	−24.2222	−0.894668	−0.447334	+	0.894367i	$0.647626\pi$
$734$	14.1233	0.521300
$735$	0	0
$736$	5.09056	0.187641
$737$	13.7815	0.507650
$738$	−2.62716	−0.0967071
$739$	−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
$741$	3.23874	0.118978
$742$	0	0
$743$	35.7282	1.31074	0.655370	−	0.755308i	$-0.272513\pi$
$743$	35.7282	1.31074	0.655370	+	0.755308i	$0.272513\pi$
$744$	−8.09993	−0.296958
$745$	−11.5377	−0.422707
$746$	12.6014	0.461369
$747$	−6.95032	−0.254299
$748$	−18.2202	−0.666197
$749$	0	0
$750$	21.3658	0.780168
$751$	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
$753$	−0.604526	−0.0220302
$754$	−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
$759$	1.67510	0.0608024
$760$	−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	$0.465420\pi$
$762$	10.9542	0.396829
$763$	0	0
$764$	37.5023	1.35679
$765$	27.5962	0.997744
$766$	−9.65619	−0.348892
$767$	−43.8441	−1.58312
$768$	−0.967116	−0.0348978
$769$	16.0694	0.579476	0.289738	−	0.957106i	$-0.406432\pi$
$769$	16.0694	0.579476	0.289738	+	0.957106i	$0.406432\pi$
$770$	0	0
$771$	−14.1602	−0.509968
$772$	−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	$0.727954\pi$
$774$	−2.14204	−0.0769939
$775$	−61.6079	−2.21302
$776$	−19.2495	−0.691018
$777$	0	0
$778$	10.2773	0.368460
$779$	3.57921	0.128239
$780$	35.8509	1.28367
$781$	17.8656	0.639282
$782$	−3.18466	−0.113883
$783$	5.60012	0.200132
$784$	0	0
$785$	−34.3304	−1.22530
$786$	0.598870	0.0213610
$787$	44.7402	1.59482	0.797408	−	0.603440i	$-0.206204\pi$
$787$	44.7402	1.59482	0.797408	+	0.603440i	$0.206204\pi$
$788$	20.6504	0.735639
$789$	20.5788	0.732626
$790$	1.30722	0.0465089
$791$	0	0
$792$	−3.19259	−0.113444
$793$	−36.0225	−1.27920
$794$	4.28857	0.152196
$795$	8.12734	0.288247
$796$	−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
$801$	13.4963	0.476868
$802$	2.98765	0.105498
$803$	−19.9534	−0.704141
$804$	14.3184	0.504971
$805$	0	0
$806$	10.1031	0.355866
$807$	0.496655	0.0174831
$808$	26.2691	0.924143
$809$	−23.4803	−0.825523	−0.412761	−	0.910839i	$-0.635436\pi$
$809$	−23.4803	−0.825523	−0.412761	+	0.910839i	$0.635436\pi$
$810$	2.24991	0.0790537
$811$	0.320003	0.0112368	0.00561841	−	0.999984i	$-0.498212\pi$
$811$	0.320003	0.0112368	0.00561841	+	0.999984i	$0.498212\pi$
$812$	0	0
$813$	−24.0547	−0.843634
$814$	7.91160	0.277301
$815$	31.4415	1.10135
$816$	−15.6845	−0.549066
$817$	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	$-0.156843\pi$
$821$	50.4892	1.76208	0.881042	+	0.473038i	$0.156843\pi$
$822$	−8.07958	−0.281808
$823$	−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
$825$	−24.2828	−0.845418
$826$	0	0
$827$	−3.13318	−0.108951	−0.0544757	−	0.998515i	$-0.517349\pi$
$827$	−3.13318	−0.108951	−0.0544757	+	0.998515i	$0.517349\pi$
$828$	1.74036	0.0604816
$829$	20.1070	0.698345	0.349172	−	0.937059i	$-0.386463\pi$
$829$	20.1070	0.698345	0.349172	+	0.937059i	$0.386463\pi$
$830$	−15.6376	−0.542789
$831$	23.2402	0.806193
$832$	−11.3144	−0.392257
$833$	0	0
$834$	−0.985670	−0.0341309
$835$	−52.9270	−1.83161
$836$	2.02381	0.0699950
$837$	−4.24991	−0.146898
$838$	−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
$843$	−15.0697	−0.519029
$844$	38.2525	1.31670
$845$	−38.7045	−1.33147
$846$	−0.980895	−0.0337239
$847$	0	0
$848$	−4.61922	−0.158625
$849$	−9.22691	−0.316667
$850$	46.1657	1.58347
$851$	−9.26901	−0.317738
$852$	18.5615	0.635908
$853$	36.1804	1.23879	0.619397	−	0.785078i	$-0.287377\pi$
$853$	36.1804	1.23879	0.619397	+	0.785078i	$0.287377\pi$
$854$	0	0
$855$	−3.06525	−0.104829
$856$	−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	$0.569589\pi$
$858$	3.98214	0.135948
$859$	−49.2759	−1.68127	−0.840636	−	0.541600i	$-0.817819\pi$
$859$	−49.2759	−1.68127	−0.840636	+	0.541600i	$0.817819\pi$
$860$	32.3037	1.10155
$861$	0	0
$862$	17.0625	0.581152
$863$	−3.28011	−0.111656	−0.0558282	−	0.998440i	$-0.517780\pi$
$863$	−3.28011	−0.111656	−0.0558282	+	0.998440i	$0.517780\pi$
$864$	−5.09056	−0.173185
$865$	−9.27218	−0.315263
$866$	12.2992	0.417944
$867$	22.0614	0.749243
$868$	0	0
$869$	−0.973255	−0.0330154
$870$	12.5998	0.427172
$871$	−38.3833	−1.30057
$872$	7.63876	0.258681
$873$	−10.0999	−0.341831
$874$	0.353736	0.0119653
$875$	0	0
$876$	−20.7307	−0.700425
$877$	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
$879$	12.3881	0.417839
$880$	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
$885$	41.4956	1.39486
$886$	−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	$0.453903\pi$
$888$	17.6659	0.592828
$889$	0	0
$890$	30.3654	1.01785
$891$	−1.67510	−0.0561181
$892$	0.0526851	0.00176403
$893$	1.33636	0.0447196
$894$	−1.33147	−0.0445310
$895$	70.5263	2.35744
$896$	0	0
$897$	−4.66537	−0.155772
$898$	−2.12646	−0.0709609
$899$	−23.8000	−0.793774
$900$	−25.2287	−0.840957
$901$	11.5039	0.383251
$902$	4.40076	0.146529
$903$	0	0
$904$	−17.2775	−0.574642
$905$	−12.8035	−0.425603
$906$	4.71989	0.156808
$907$	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
$909$	13.7830	0.457152
$910$	0	0
$911$	29.5776	0.979951	0.489975	−	0.871736i	$-0.337006\pi$
$911$	29.5776	0.979951	0.489975	+	0.871736i	$0.337006\pi$
$912$	1.74215	0.0576885
$913$	11.6425	0.385311
$914$	−4.09001	−0.135286
$915$	34.0929	1.12708
$916$	30.3448	1.00262
$917$	0	0
$918$	3.18466	0.105109
$919$	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
$921$	−0.823281	−0.0271280
$922$	5.92057	0.194984
$923$	−49.7579	−1.63780
$924$	0	0
$925$	134.366	4.41794
$926$	18.2783	0.600664
$927$	−16.5553	−0.543749
$928$	−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	$0.286206\pi$
$930$	−9.56191	−0.313547
$931$	0	0
$932$	19.5062	0.638947
$933$	30.6632	1.00387
$934$	−6.71240	−0.219636
$935$	−46.2265	−1.51177
$936$	8.89176	0.290636
$937$	42.3490	1.38348	0.691741	−	0.722146i	$-0.256844\pi$
$937$	42.3490	1.38348	0.691741	+	0.722146i	$0.256844\pi$
$938$	0	0
$939$	−13.1323	−0.428556
$940$	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	$0.313196\pi$
$942$	−3.96179	−0.129082
$943$	−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	$-0.648872\pi$
$947$	−27.7471	−0.901659	−0.450829	+	0.892610i	$0.648872\pi$
$948$	−1.01117	−0.0328412
$949$	55.5727	1.80397
$950$	−5.12786	−0.166370
$951$	−15.2269	−0.493766
$952$	0	0
$953$	−6.25582	−0.202646	−0.101323	−	0.994854i	$-0.532308\pi$
$953$	−6.25582	−0.202646	−0.101323	+	0.994854i	$0.532308\pi$
$954$	0.937911	0.0303660
$955$	95.1472	3.07889
$956$	32.3105	1.04500
$957$	−9.38078	−0.303237
$958$	2.66537	0.0861142
$959$	0	0
$960$	10.7084	0.345611
$961$	−12.9383	−0.417364
$962$	−22.0348	−0.710429
$963$	−1.33463	−0.0430079
$964$	−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
$969$	−4.33874	−0.139381
$970$	−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	$0.534245\pi$
$972$	−1.74036	−0.0558220
$973$	0	0
$974$	−14.7658	−0.473125
$975$	67.6305	2.16591
$976$	−19.3769	−0.620239
$977$	−47.1871	−1.50965	−0.754825	−	0.655926i	$-0.772278\pi$
$977$	−47.1871	−1.50965	−0.754825	+	0.655926i	$0.772278\pi$
$978$	3.62841	0.116024
$979$	−22.6077	−0.722545
$980$	0	0
$981$	4.00794	0.127964
$982$	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	$0.595066\pi$
$984$	9.82651	0.313258
$985$	52.3921	1.66935
$986$	17.8344	0.567965
$987$	0	0
$988$	−5.63656	−0.179323
$989$	−4.20376	−0.133672
$990$	−3.76883	−0.119781
$991$	−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
$993$	25.5674	0.811356
$994$	0	0
$995$	−11.1652	−0.353960
$996$	12.0960	0.383278
$997$	−59.5570	−1.88619	−0.943094	−	0.332525i	$-0.892099\pi$
$997$	−59.5570	−1.88619	−0.943094	+	0.332525i	$0.892099\pi$
$998$	−4.87750	−0.154395
$999$	9.26901	0.293259

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.2	✓	4	7.6	odd	2
1449.2.a.p.1.3		4	21.20	even	2
3381.2.a.w.1.2		4	1.1	even	1	trivial
7728.2.a.cd.1.4		4	28.27	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-0.509552 q^{2} +1.00000 q^{3} -1.74036 q^{4} -4.41546 q^{5} -0.509552 q^{6} +1.90591 q^{8} +1.00000 q^{9} +2.24991 q^{10} -1.67510 q^{11} -1.74036 q^{12} +4.66537 q^{13} -4.41546 q^{15} +2.50955 q^{16} -6.24991 q^{17} -0.509552 q^{18} +0.694209 q^{19} +7.68447 q^{20} +0.853553 q^{22} -1.00000 q^{23} +1.90591 q^{24} +14.4963 q^{25} -2.37725 q^{26} +1.00000 q^{27} +5.60012 q^{29} +2.24991 q^{30} -4.24991 q^{31} -5.09056 q^{32} -1.67510 q^{33} +3.18466 q^{34} -1.74036 q^{36} +9.26901 q^{37} -0.353736 q^{38} +4.66537 q^{39} -8.41546 q^{40} +5.15582 q^{41} +4.20376 q^{43} +2.91528 q^{44} -4.41546 q^{45} +0.509552 q^{46} +1.92501 q^{47} +2.50955 q^{48} -7.38662 q^{50} -6.24991 q^{51} -8.11940 q^{52} -1.84066 q^{53} -0.509552 q^{54} +7.39636 q^{55} +0.694209 q^{57} -2.85355 q^{58} -9.39779 q^{59} +7.68447 q^{60} -7.72125 q^{61} +2.16555 q^{62} -2.42520 q^{64} -20.5998 q^{65} +0.853553 q^{66} -8.22728 q^{67} +10.8771 q^{68} -1.00000 q^{69} -10.6654 q^{71} +1.90591 q^{72} +11.9117 q^{73} -4.72305 q^{74} +14.4963 q^{75} -1.20817 q^{76} -2.37725 q^{78} +0.581012 q^{79} -11.0808 q^{80} +1.00000 q^{81} -2.62716 q^{82} -6.95032 q^{83} +27.5962 q^{85} -2.14204 q^{86} +5.60012 q^{87} -3.19259 q^{88} +13.4963 q^{89} +2.24991 q^{90} +1.74036 q^{92} -4.24991 q^{93} -0.980895 q^{94} -3.06525 q^{95} -5.09056 q^{96} -10.0999 q^{97} -1.67510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{8} + 4 q^{9} - 4 q^{10} - 5 q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 12 q^{17} - 3 q^{19} + q^{20} - q^{22} - 4 q^{23} - 3 q^{24} + 7 q^{25} - 5 q^{26}+ \cdots - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^8 + 4 * q^9 - 4 * q^10 - 5 * q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 12 * q^17 - 3 * q^19 + q^20 - q^22 - 4 * q^23 - 3 * q^24 + 7 * q^25 - 5 * q^26 + 4 * q^27 + 6 * q^29 - 4 * q^30 - 4 * q^31 - 6 * q^32 - 5 * q^33 + 9 * q^34 + 4 * q^36 + 20 * q^37 - 23 * q^38 - 7 * q^39 - 21 * q^40 - 3 * q^41 + 9 * q^43 - 27 * q^44 - 5 * q^45 - 7 * q^47 + 8 * q^48 + 3 * q^50 - 12 * q^51 - 38 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^57 - 7 * q^58 + 2 * q^59 + q^60 - 24 * q^61 + 9 * q^62 - 21 * q^64 - 14 * q^65 - q^66 + q^67 + 13 * q^68 - 4 * q^69 - 17 * q^71 - 3 * q^72 - 16 * q^73 - 33 * q^74 + 7 * q^75 + 25 * q^76 - 5 * q^78 - 10 * q^79 - 6 * q^80 + 4 * q^81 + 7 * q^82 - 8 * q^83 + 17 * q^85 - 35 * q^86 + 6 * q^87 - 12 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 8 * q^94 - 3 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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