LMFDB - Embedded newform 693.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [693,4,Mod(1,693)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("693.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(693, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-3,0,-3,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.561553 q^{2} -7.68466 q^{4} -6.68466 q^{5} +7.00000 q^{7} -8.80776 q^{8} -3.75379 q^{10} +11.0000 q^{11} +14.3002 q^{13} +3.93087 q^{14} +56.5312 q^{16} +47.7538 q^{17} +11.9157 q^{19} +51.3693 q^{20} +6.17708 q^{22} +44.4924 q^{23} -80.3153 q^{25} +8.03031 q^{26} -53.7926 q^{28} +139.501 q^{29} -208.600 q^{31} +102.207 q^{32} +26.8163 q^{34} -46.7926 q^{35} -253.086 q^{37} +6.69130 q^{38} +58.8769 q^{40} +156.294 q^{41} -263.386 q^{43} -84.5312 q^{44} +24.9848 q^{46} -386.533 q^{47} +49.0000 q^{49} -45.1013 q^{50} -109.892 q^{52} +36.5701 q^{53} -73.5312 q^{55} -61.6543 q^{56} +78.3371 q^{58} +114.762 q^{59} -53.0758 q^{61} -117.140 q^{62} -394.855 q^{64} -95.5919 q^{65} +132.348 q^{67} -366.972 q^{68} -26.2765 q^{70} -583.447 q^{71} -817.012 q^{73} -142.121 q^{74} -91.5682 q^{76} +77.0000 q^{77} -369.633 q^{79} -377.892 q^{80} +87.7671 q^{82} +69.1534 q^{83} -319.218 q^{85} -147.905 q^{86} -96.8854 q^{88} -467.295 q^{89} +100.101 q^{91} -341.909 q^{92} -217.059 q^{94} -79.6525 q^{95} -1170.11 q^{97} +27.5161 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - 3 q^{4} - q^{5} + 14 q^{7} + 3 q^{8} - 24 q^{10} + 22 q^{11} - 25 q^{13} - 21 q^{14} - 23 q^{16} + 112 q^{17} - 71 q^{19} + 78 q^{20} - 33 q^{22} + 56 q^{23} - 173 q^{25} + 148 q^{26} - 21 q^{28}+ \cdots - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 - 3 * q^4 - q^5 + 14 * q^7 + 3 * q^8 - 24 * q^10 + 22 * q^11 - 25 * q^13 - 21 * q^14 - 23 * q^16 + 112 * q^17 - 71 * q^19 + 78 * q^20 - 33 * q^22 + 56 * q^23 - 173 * q^25 + 148 * q^26 - 21 * q^28 + 11 * q^29 - 310 * q^31 + 291 * q^32 - 202 * q^34 - 7 * q^35 - 65 * q^37 + 302 * q^38 + 126 * q^40 - 42 * q^41 - 32 * q^43 - 33 * q^44 - 16 * q^46 - 101 * q^47 + 98 * q^49 + 285 * q^50 - 294 * q^52 - 166 * q^53 - 11 * q^55 + 21 * q^56 + 536 * q^58 + 11 * q^59 - 436 * q^61 + 244 * q^62 - 431 * q^64 - 319 * q^65 - 127 * q^67 - 66 * q^68 - 168 * q^70 - 936 * q^71 - 327 * q^73 - 812 * q^74 - 480 * q^76 + 154 * q^77 - 228 * q^79 - 830 * q^80 + 794 * q^82 + 262 * q^83 + 46 * q^85 - 972 * q^86 + 33 * q^88 - 44 * q^89 - 175 * q^91 - 288 * q^92 - 1234 * q^94 - 551 * q^95 - 2266 * q^97 - 147 * q^98

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 $

$2$	0.561553	0.198539	0.0992695	−	0.995061i	$-0.468349\pi$
$2$	0.561553	0.198539	0.0992695	+	0.995061i	$0.468349\pi$
$3$	0	0
$3$	0	0
$4$	−7.68466	−0.960582
$5$	−6.68466	−0.597894	−0.298947	−	0.954270i	$-0.596635\pi$
$5$	−6.68466	−0.597894	−0.298947	+	0.954270i	$0.596635\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.80776	−0.389252
$9$	0	0
$10$	−3.75379	−0.118705
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	14.3002	0.305089	0.152545	−	0.988297i	$-0.451253\pi$
$13$	14.3002	0.305089	0.152545	+	0.988297i	$0.451253\pi$
$14$	3.93087	0.0750407
$15$	0	0
$16$	56.5312	0.883301
$17$	47.7538	0.681294	0.340647	−	0.940191i	$-0.389354\pi$
$17$	47.7538	0.681294	0.340647	+	0.940191i	$0.389354\pi$
$18$	0	0
$19$	11.9157	0.143876	0.0719382	−	0.997409i	$-0.477082\pi$
$19$	11.9157	0.143876	0.0719382	+	0.997409i	$0.477082\pi$
$20$	51.3693	0.574326
$21$	0	0
$22$	6.17708	0.0598617
$23$	44.4924	0.403361	0.201681	−	0.979451i	$-0.435360\pi$
$23$	44.4924	0.403361	0.201681	+	0.979451i	$0.435360\pi$
$24$	0	0
$25$	−80.3153	−0.642523
$26$	8.03031	0.0605721
$27$	0	0
$28$	−53.7926	−0.363066
$29$	139.501	0.893265	0.446632	−	0.894718i	$-0.352623\pi$
$29$	139.501	0.893265	0.446632	+	0.894718i	$0.352623\pi$
$30$	0	0
$31$	−208.600	−1.20857	−0.604286	−	0.796767i	$-0.706542\pi$
$31$	−208.600	−1.20857	−0.604286	+	0.796767i	$0.706542\pi$
$32$	102.207	0.564621
$33$	0	0
$34$	26.8163	0.135263
$35$	−46.7926	−0.225983
$36$	0	0
$37$	−253.086	−1.12452	−0.562258	−	0.826962i	$-0.690067\pi$
$37$	−253.086	−1.12452	−0.562258	+	0.826962i	$0.690067\pi$
$38$	6.69130	0.0285651
$39$	0	0
$40$	58.8769	0.232731
$41$	156.294	0.595340	0.297670	−	0.954669i	$-0.403790\pi$
$41$	156.294	0.595340	0.297670	+	0.954669i	$0.403790\pi$
$42$	0	0
$43$	−263.386	−0.934094	−0.467047	−	0.884233i	$-0.654682\pi$
$43$	−263.386	−0.934094	−0.467047	+	0.884233i	$0.654682\pi$
$44$	−84.5312	−0.289626
$45$	0	0
$46$	24.9848	0.0800829
$47$	−386.533	−1.19961	−0.599805	−	0.800146i	$-0.704755\pi$
$47$	−386.533	−1.19961	−0.599805	+	0.800146i	$0.704755\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−45.1013	−0.127566
$51$	0	0
$52$	−109.892	−0.293063
$53$	36.5701	0.0947790	0.0473895	−	0.998876i	$-0.484910\pi$
$53$	36.5701	0.0947790	0.0473895	+	0.998876i	$0.484910\pi$
$54$	0	0
$55$	−73.5312	−0.180272
$56$	−61.6543	−0.147123
$57$	0	0
$58$	78.3371	0.177348
$59$	114.762	0.253234	0.126617	−	0.991952i	$-0.459588\pi$
$59$	114.762	0.253234	0.126617	+	0.991952i	$0.459588\pi$
$60$	0	0
$61$	−53.0758	−0.111404	−0.0557021	−	0.998447i	$-0.517740\pi$
$61$	−53.0758	−0.111404	−0.0557021	+	0.998447i	$0.517740\pi$
$62$	−117.140	−0.239949
$63$	0	0
$64$	−394.855	−0.771201
$65$	−95.5919	−0.182411
$66$	0	0
$67$	132.348	0.241326	0.120663	−	0.992694i	$-0.461498\pi$
$67$	132.348	0.241326	0.120663	+	0.992694i	$0.461498\pi$
$68$	−366.972	−0.654439
$69$	0	0
$70$	−26.2765	−0.0448664
$71$	−583.447	−0.975245	−0.487623	−	0.873055i	$-0.662136\pi$
$71$	−583.447	−0.975245	−0.487623	+	0.873055i	$0.662136\pi$
$72$	0	0
$73$	−817.012	−1.30992	−0.654959	−	0.755664i	$-0.727314\pi$
$73$	−817.012	−1.30992	−0.654959	+	0.755664i	$0.727314\pi$
$74$	−142.121	−0.223260
$75$	0	0
$76$	−91.5682	−0.138205
$77$	77.0000	0.113961
$78$	0	0
$79$	−369.633	−0.526417	−0.263208	−	0.964739i	$-0.584781\pi$
$79$	−369.633	−0.526417	−0.263208	+	0.964739i	$0.584781\pi$
$80$	−377.892	−0.528120
$81$	0	0
$82$	87.7671	0.118198
$83$	69.1534	0.0914527	0.0457263	−	0.998954i	$-0.485440\pi$
$83$	69.1534	0.0914527	0.0457263	+	0.998954i	$0.485440\pi$
$84$	0	0
$85$	−319.218	−0.407342
$86$	−147.905	−0.185454
$87$	0	0
$88$	−96.8854	−0.117364
$89$	−467.295	−0.556553	−0.278276	−	0.960501i	$-0.589763\pi$
$89$	−467.295	−0.556553	−0.278276	+	0.960501i	$0.589763\pi$
$90$	0	0
$91$	100.101	0.115313
$92$	−341.909	−0.387462
$93$	0	0
$94$	−217.059	−0.238169
$95$	−79.6525	−0.0860229
$96$	0	0
$97$	−1170.11	−1.22481	−0.612404	−	0.790545i	$-0.709798\pi$
$97$	−1170.11	−1.22481	−0.612404	+	0.790545i	$0.709798\pi$
$98$	27.5161	0.0283627
$99$	0	0
$100$	617.196	0.617196
$101$	−181.299	−0.178613	−0.0893066	−	0.996004i	$-0.528465\pi$
$101$	−181.299	−0.178613	−0.0893066	+	0.996004i	$0.528465\pi$
$102$	0	0
$103$	−212.324	−0.203115	−0.101558	−	0.994830i	$-0.532383\pi$
$103$	−212.324	−0.203115	−0.101558	+	0.994830i	$0.532383\pi$
$104$	−125.953	−0.118756
$105$	0	0
$106$	20.5360	0.0188173
$107$	−220.111	−0.198868	−0.0994342	−	0.995044i	$-0.531703\pi$
$107$	−220.111	−0.198868	−0.0994342	+	0.995044i	$0.531703\pi$
$108$	0	0
$109$	−247.072	−0.217112	−0.108556	−	0.994090i	$-0.534623\pi$
$109$	−247.072	−0.217112	−0.108556	+	0.994090i	$0.534623\pi$
$110$	−41.2917	−0.0357910
$111$	0	0
$112$	395.719	0.333856
$113$	139.261	0.115935	0.0579673	−	0.998318i	$-0.481538\pi$
$113$	139.261	0.115935	0.0579673	+	0.998318i	$0.481538\pi$
$114$	0	0
$115$	−297.417	−0.241167
$116$	−1072.02	−0.858054
$117$	0	0
$118$	64.4451	0.0502767
$119$	334.277	0.257505
$120$	0	0
$121$	121.000	0.0909091
$122$	−29.8049	−0.0221181
$123$	0	0
$124$	1603.02	1.16093
$125$	1372.46	0.982055
$126$	0	0
$127$	676.918	0.472967	0.236483	−	0.971636i	$-0.424005\pi$
$127$	676.918	0.472967	0.236483	+	0.971636i	$0.424005\pi$
$128$	−1039.39	−0.717735
$129$	0	0
$130$	−53.6799	−0.0362157
$131$	−291.042	−0.194110	−0.0970551	−	0.995279i	$-0.530942\pi$
$131$	−291.042	−0.194110	−0.0970551	+	0.995279i	$0.530942\pi$
$132$	0	0
$133$	83.4100	0.0543802
$134$	74.3201	0.0479125
$135$	0	0
$136$	−420.604	−0.265195
$137$	−2322.41	−1.44830	−0.724148	−	0.689645i	$-0.757767\pi$
$137$	−2322.41	−1.44830	−0.724148	+	0.689645i	$0.757767\pi$
$138$	0	0
$139$	210.436	0.128409	0.0642047	−	0.997937i	$-0.479549\pi$
$139$	210.436	0.128409	0.0642047	+	0.997937i	$0.479549\pi$
$140$	359.585	0.217075
$141$	0	0
$142$	−327.636	−0.193624
$143$	157.302	0.0919878
$144$	0	0
$145$	−932.516	−0.534078
$146$	−458.796	−0.260070
$147$	0	0
$148$	1944.88	1.08019
$149$	929.815	0.511231	0.255616	−	0.966779i	$-0.417722\pi$
$149$	929.815	0.511231	0.255616	+	0.966779i	$0.417722\pi$
$150$	0	0
$151$	2300.58	1.23986	0.619929	−	0.784658i	$-0.287161\pi$
$151$	2300.58	1.23986	0.619929	+	0.784658i	$0.287161\pi$
$152$	−104.951	−0.0560042
$153$	0	0
$154$	43.2396	0.0226256
$155$	1394.42	0.722598
$156$	0	0
$157$	−1947.83	−0.990150	−0.495075	−	0.868850i	$-0.664859\pi$
$157$	−1947.83	−0.990150	−0.495075	+	0.868850i	$0.664859\pi$
$158$	−207.568	−0.104514
$159$	0	0
$160$	−683.221	−0.337584
$161$	311.447	0.152456
$162$	0	0
$163$	−2753.64	−1.32320	−0.661601	−	0.749856i	$-0.730123\pi$
$163$	−2753.64	−1.32320	−0.661601	+	0.749856i	$0.730123\pi$
$164$	−1201.06	−0.571873
$165$	0	0
$166$	38.8333	0.0181569
$167$	3883.50	1.79948	0.899742	−	0.436422i	$-0.143754\pi$
$167$	3883.50	1.79948	0.899742	+	0.436422i	$0.143754\pi$
$168$	0	0
$169$	−1992.50	−0.906921
$170$	−179.258	−0.0808731
$171$	0	0
$172$	2024.03	0.897274
$173$	3123.33	1.37262	0.686308	−	0.727311i	$-0.259230\pi$
$173$	3123.33	1.37262	0.686308	+	0.727311i	$0.259230\pi$
$174$	0	0
$175$	−562.207	−0.242851
$176$	621.844	0.266325
$177$	0	0
$178$	−262.411	−0.110497
$179$	−1631.29	−0.681165	−0.340582	−	0.940215i	$-0.610624\pi$
$179$	−1631.29	−0.681165	−0.340582	+	0.940215i	$0.610624\pi$
$180$	0	0
$181$	−853.165	−0.350360	−0.175180	−	0.984536i	$-0.556051\pi$
$181$	−853.165	−0.350360	−0.175180	+	0.984536i	$0.556051\pi$
$182$	56.2122	0.0228941
$183$	0	0
$184$	−391.879	−0.157009
$185$	1691.79	0.672342
$186$	0	0
$187$	525.292	0.205418
$188$	2970.37	1.15232
$189$	0	0
$190$	−44.7291	−0.0170789
$191$	−2826.32	−1.07071	−0.535354	−	0.844628i	$-0.679822\pi$
$191$	−2826.32	−1.07071	−0.535354	+	0.844628i	$0.679822\pi$
$192$	0	0
$193$	2442.22	0.910856	0.455428	−	0.890273i	$-0.349486\pi$
$193$	2442.22	0.910856	0.455428	+	0.890273i	$0.349486\pi$
$194$	−657.077	−0.243172
$195$	0	0
$196$	−376.548	−0.137226
$197$	−2772.18	−1.00259	−0.501294	−	0.865277i	$-0.667142\pi$
$197$	−2772.18	−1.00259	−0.501294	+	0.865277i	$0.667142\pi$
$198$	0	0
$199$	−5579.98	−1.98771	−0.993855	−	0.110688i	$-0.964694\pi$
$199$	−5579.98	−1.98771	−0.993855	+	0.110688i	$0.964694\pi$
$200$	707.399	0.250103
$201$	0	0
$202$	−101.809	−0.0354617
$203$	976.507	0.337622
$204$	0	0
$205$	−1044.77	−0.355950
$206$	−119.231	−0.0403263
$207$	0	0
$208$	808.407	0.269485
$209$	131.073	0.0433804
$210$	0	0
$211$	1573.05	0.513237	0.256619	−	0.966513i	$-0.417392\pi$
$211$	1573.05	0.513237	0.256619	+	0.966513i	$0.417392\pi$
$212$	−281.028	−0.0910430
$213$	0	0
$214$	−123.604	−0.0394831
$215$	1760.65	0.558489
$216$	0	0
$217$	−1460.20	−0.456797
$218$	−138.744	−0.0431052
$219$	0	0
$220$	565.062	0.173166
$221$	682.888	0.207855
$222$	0	0
$223$	732.945	0.220097	0.110048	−	0.993926i	$-0.464899\pi$
$223$	732.945	0.220097	0.110048	+	0.993926i	$0.464899\pi$
$224$	715.452	0.213407
$225$	0	0
$226$	78.2026	0.0230175
$227$	−6548.97	−1.91485	−0.957423	−	0.288687i	$-0.906781\pi$
$227$	−6548.97	−1.91485	−0.957423	+	0.288687i	$0.906781\pi$
$228$	0	0
$229$	−2326.27	−0.671285	−0.335643	−	0.941989i	$-0.608953\pi$
$229$	−2326.27	−0.671285	−0.335643	+	0.941989i	$0.608953\pi$
$230$	−167.015	−0.0478811
$231$	0	0
$232$	−1228.69	−0.347705
$233$	3859.26	1.08510	0.542551	−	0.840023i	$-0.317459\pi$
$233$	3859.26	1.08510	0.542551	+	0.840023i	$0.317459\pi$
$234$	0	0
$235$	2583.84	0.717239
$236$	−881.909	−0.243252
$237$	0	0
$238$	187.714	0.0511247
$239$	3688.00	0.998147	0.499074	−	0.866560i	$-0.333674\pi$
$239$	3688.00	0.998147	0.499074	+	0.866560i	$0.333674\pi$
$240$	0	0
$241$	−1818.30	−0.486004	−0.243002	−	0.970026i	$-0.578132\pi$
$241$	−1818.30	−0.486004	−0.243002	+	0.970026i	$0.578132\pi$
$242$	67.9479	0.0180490
$243$	0	0
$244$	407.869	0.107013
$245$	−327.548	−0.0854134
$246$	0	0
$247$	170.397	0.0438951
$248$	1837.30	0.470439
$249$	0	0
$250$	770.710	0.194976
$251$	−215.497	−0.0541915	−0.0270957	−	0.999633i	$-0.508626\pi$
$251$	−215.497	−0.0541915	−0.0270957	+	0.999633i	$0.508626\pi$
$252$	0	0
$253$	489.417	0.121618
$254$	380.125	0.0939023
$255$	0	0
$256$	2575.17	0.628703
$257$	−4699.31	−1.14060	−0.570301	−	0.821436i	$-0.693174\pi$
$257$	−4699.31	−1.14060	−0.570301	+	0.821436i	$0.693174\pi$
$258$	0	0
$259$	−1771.60	−0.425027
$260$	734.591	0.175221
$261$	0	0
$262$	−163.435	−0.0385384
$263$	−7504.38	−1.75947	−0.879734	−	0.475466i	$-0.842279\pi$
$263$	−7504.38	−1.75947	−0.879734	+	0.475466i	$0.842279\pi$
$264$	0	0
$265$	−244.458	−0.0566678
$266$	46.8391	0.0107966
$267$	0	0
$268$	−1017.05	−0.231813
$269$	2513.64	0.569736	0.284868	−	0.958567i	$-0.408050\pi$
$269$	2513.64	0.569736	0.284868	+	0.958567i	$0.408050\pi$
$270$	0	0
$271$	6385.39	1.43131	0.715654	−	0.698455i	$-0.246129\pi$
$271$	6385.39	1.43131	0.715654	+	0.698455i	$0.246129\pi$
$272$	2699.58	0.601787
$273$	0	0
$274$	−1304.15	−0.287543
$275$	−883.469	−0.193728
$276$	0	0
$277$	120.650	0.0261702	0.0130851	−	0.999914i	$-0.495835\pi$
$277$	120.650	0.0261702	0.0130851	+	0.999914i	$0.495835\pi$
$278$	118.171	0.0254943
$279$	0	0
$280$	412.138	0.0879642
$281$	2387.75	0.506908	0.253454	−	0.967347i	$-0.418433\pi$
$281$	2387.75	0.506908	0.253454	+	0.967347i	$0.418433\pi$
$282$	0	0
$283$	781.711	0.164198	0.0820988	−	0.996624i	$-0.473838\pi$
$283$	781.711	0.164198	0.0820988	+	0.996624i	$0.473838\pi$
$284$	4483.59	0.936803
$285$	0	0
$286$	88.3334	0.0182632
$287$	1094.05	0.225017
$288$	0	0
$289$	−2632.58	−0.535839
$290$	−523.657	−0.106035
$291$	0	0
$292$	6278.46	1.25828
$293$	−2168.03	−0.432278	−0.216139	−	0.976363i	$-0.569347\pi$
$293$	−2168.03	−0.432278	−0.216139	+	0.976363i	$0.569347\pi$
$294$	0	0
$295$	−767.147	−0.151407
$296$	2229.12	0.437720
$297$	0	0
$298$	522.140	0.101499
$299$	636.250	0.123061
$300$	0	0
$301$	−1843.70	−0.353054
$302$	1291.90	0.246160
$303$	0	0
$304$	673.610	0.127086
$305$	354.793	0.0666079
$306$	0	0
$307$	10309.9	1.91668	0.958338	−	0.285636i	$-0.0922047\pi$
$307$	10309.9	1.91668	0.958338	+	0.285636i	$0.0922047\pi$
$308$	−591.719	−0.109469
$309$	0	0
$310$	783.042	0.143464
$311$	6686.61	1.21917	0.609587	−	0.792719i	$-0.291335\pi$
$311$	6686.61	1.21917	0.609587	+	0.792719i	$0.291335\pi$
$312$	0	0
$313$	6457.33	1.16610	0.583051	−	0.812436i	$-0.301859\pi$
$313$	6457.33	1.16610	0.583051	+	0.812436i	$0.301859\pi$
$314$	−1093.81	−0.196583
$315$	0	0
$316$	2840.50	0.505666
$317$	−7170.30	−1.27042	−0.635212	−	0.772338i	$-0.719087\pi$
$317$	−7170.30	−1.27042	−0.635212	+	0.772338i	$0.719087\pi$
$318$	0	0
$319$	1534.51	0.269329
$320$	2639.47	0.461097
$321$	0	0
$322$	174.894	0.0302685
$323$	569.021	0.0980221
$324$	0	0
$325$	−1148.52	−0.196027
$326$	−1546.32	−0.262707
$327$	0	0
$328$	−1376.60	−0.231737
$329$	−2705.73	−0.453410
$330$	0	0
$331$	5849.52	0.971355	0.485678	−	0.874138i	$-0.338573\pi$
$331$	5849.52	0.971355	0.485678	+	0.874138i	$0.338573\pi$
$332$	−531.420	−0.0878478
$333$	0	0
$334$	2180.79	0.357268
$335$	−884.698	−0.144287
$336$	0	0
$337$	9784.35	1.58157	0.790783	−	0.612097i	$-0.209674\pi$
$337$	9784.35	1.58157	0.790783	+	0.612097i	$0.209674\pi$
$338$	−1118.90	−0.180059
$339$	0	0
$340$	2453.08	0.391285
$341$	−2294.60	−0.364398
$342$	0	0
$343$	343.000	0.0539949
$344$	2319.84	0.363598
$345$	0	0
$346$	1753.92	0.272518
$347$	−9431.05	−1.45904	−0.729518	−	0.683962i	$-0.760255\pi$
$347$	−9431.05	−1.45904	−0.729518	+	0.683962i	$0.760255\pi$
$348$	0	0
$349$	3933.02	0.603238	0.301619	−	0.953429i	$-0.402473\pi$
$349$	3933.02	0.603238	0.301619	+	0.953429i	$0.402473\pi$
$350$	−315.709	−0.0482153
$351$	0	0
$352$	1124.28	0.170240
$353$	6544.48	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$353$	6544.48	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$354$	0	0
$355$	3900.14	0.583093
$356$	3591.01	0.534615
$357$	0	0
$358$	−916.056	−0.135238
$359$	−11013.2	−1.61910	−0.809548	−	0.587053i	$-0.800288\pi$
$359$	−11013.2	−1.61910	−0.809548	+	0.587053i	$0.800288\pi$
$360$	0	0
$361$	−6717.02	−0.979300
$362$	−479.097	−0.0695602
$363$	0	0
$364$	−769.244	−0.110767
$365$	5461.45	0.783192
$366$	0	0
$367$	−7733.24	−1.09992	−0.549962	−	0.835190i	$-0.685358\pi$
$367$	−7733.24	−1.09992	−0.549962	+	0.835190i	$0.685358\pi$
$368$	2515.21	0.356289
$369$	0	0
$370$	950.032	0.133486
$371$	255.990	0.0358231
$372$	0	0
$373$	2225.01	0.308865	0.154433	−	0.988003i	$-0.450645\pi$
$373$	2225.01	0.308865	0.154433	+	0.988003i	$0.450645\pi$
$374$	294.979	0.0407834
$375$	0	0
$376$	3404.49	0.466950
$377$	1994.89	0.272525
$378$	0	0
$379$	−1141.43	−0.154700	−0.0773500	−	0.997004i	$-0.524646\pi$
$379$	−1141.43	−0.154700	−0.0773500	+	0.997004i	$0.524646\pi$
$380$	612.102	0.0826320
$381$	0	0
$382$	−1587.13	−0.212577
$383$	5445.56	0.726515	0.363257	−	0.931689i	$-0.381665\pi$
$383$	5445.56	0.726515	0.363257	+	0.931689i	$0.381665\pi$
$384$	0	0
$385$	−514.719	−0.0681363
$386$	1371.44	0.180840
$387$	0	0
$388$	8991.88	1.17653
$389$	−7194.64	−0.937745	−0.468872	−	0.883266i	$-0.655340\pi$
$389$	−7194.64	−0.937745	−0.468872	+	0.883266i	$0.655340\pi$
$390$	0	0
$391$	2124.68	0.274808
$392$	−431.580	−0.0556074
$393$	0	0
$394$	−1556.73	−0.199053
$395$	2470.87	0.314741
$396$	0	0
$397$	846.201	0.106976	0.0534882	−	0.998568i	$-0.482966\pi$
$397$	846.201	0.106976	0.0534882	+	0.998568i	$0.482966\pi$
$398$	−3133.45	−0.394638
$399$	0	0
$400$	−4540.33	−0.567541
$401$	−4219.37	−0.525450	−0.262725	−	0.964871i	$-0.584621\pi$
$401$	−4219.37	−0.525450	−0.262725	+	0.964871i	$0.584621\pi$
$402$	0	0
$403$	−2983.02	−0.368722
$404$	1393.22	0.171573
$405$	0	0
$406$	548.360	0.0670312
$407$	−2783.95	−0.339054
$408$	0	0
$409$	−8028.62	−0.970636	−0.485318	−	0.874338i	$-0.661296\pi$
$409$	−8028.62	−0.970636	−0.485318	+	0.874338i	$0.661296\pi$
$410$	−586.693	−0.0706700
$411$	0	0
$412$	1631.64	0.195109
$413$	803.336	0.0957133
$414$	0	0
$415$	−462.267	−0.0546790
$416$	1461.58	0.172260
$417$	0	0
$418$	73.6043	0.00861269
$419$	11381.3	1.32699	0.663497	−	0.748179i	$-0.269071\pi$
$419$	11381.3	1.32699	0.663497	+	0.748179i	$0.269071\pi$
$420$	0	0
$421$	−2476.59	−0.286702	−0.143351	−	0.989672i	$-0.545788\pi$
$421$	−2476.59	−0.286702	−0.143351	+	0.989672i	$0.545788\pi$
$422$	883.349	0.101898
$423$	0	0
$424$	−322.100	−0.0368929
$425$	−3835.36	−0.437747
$426$	0	0
$427$	−371.530	−0.0421068
$428$	1691.48	0.191029
$429$	0	0
$430$	988.697	0.110882
$431$	9698.52	1.08390	0.541951	−	0.840410i	$-0.317686\pi$
$431$	9698.52	1.08390	0.541951	+	0.840410i	$0.317686\pi$
$432$	0	0
$433$	−14385.7	−1.59661	−0.798307	−	0.602251i	$-0.794271\pi$
$433$	−14385.7	−1.59661	−0.798307	+	0.602251i	$0.794271\pi$
$434$	−819.981	−0.0906920
$435$	0	0
$436$	1898.66	0.208554
$437$	530.159	0.0580342
$438$	0	0
$439$	−806.402	−0.0876708	−0.0438354	−	0.999039i	$-0.513958\pi$
$439$	−806.402	−0.0876708	−0.0438354	+	0.999039i	$0.513958\pi$
$440$	647.646	0.0701711
$441$	0	0
$442$	383.478	0.0412674
$443$	−4748.19	−0.509240	−0.254620	−	0.967041i	$-0.581950\pi$
$443$	−4748.19	−0.509240	−0.254620	+	0.967041i	$0.581950\pi$
$444$	0	0
$445$	3123.71	0.332760
$446$	411.587	0.0436978
$447$	0	0
$448$	−2763.99	−0.291487
$449$	7104.43	0.746724	0.373362	−	0.927686i	$-0.378205\pi$
$449$	7104.43	0.746724	0.373362	+	0.927686i	$0.378205\pi$
$450$	0	0
$451$	1719.23	0.179502
$452$	−1070.18	−0.111365
$453$	0	0
$454$	−3677.59	−0.380172
$455$	−669.143	−0.0689449
$456$	0	0
$457$	4600.05	0.470856	0.235428	−	0.971892i	$-0.424351\pi$
$457$	4600.05	0.470856	0.235428	+	0.971892i	$0.424351\pi$
$458$	−1306.32	−0.133276
$459$	0	0
$460$	2285.55	0.231661
$461$	−7098.21	−0.717130	−0.358565	−	0.933505i	$-0.616734\pi$
$461$	−7098.21	−0.717130	−0.358565	+	0.933505i	$0.616734\pi$
$462$	0	0
$463$	10036.0	1.00737	0.503684	−	0.863888i	$-0.331978\pi$
$463$	10036.0	1.00737	0.503684	+	0.863888i	$0.331978\pi$
$464$	7886.16	0.789021
$465$	0	0
$466$	2167.18	0.215435
$467$	−255.074	−0.0252750	−0.0126375	−	0.999920i	$-0.504023\pi$
$467$	−255.074	−0.0252750	−0.0126375	+	0.999920i	$0.504023\pi$
$468$	0	0
$469$	926.433	0.0912125
$470$	1450.96	0.142400
$471$	0	0
$472$	−1010.80	−0.0985716
$473$	−2897.25	−0.281640
$474$	0	0
$475$	−957.015	−0.0924439
$476$	−2568.80	−0.247355
$477$	0	0
$478$	2071.01	0.198171
$479$	2390.02	0.227981	0.113991	−	0.993482i	$-0.463637\pi$
$479$	2390.02	0.227981	0.113991	+	0.993482i	$0.463637\pi$
$480$	0	0
$481$	−3619.18	−0.343078
$482$	−1021.07	−0.0964907
$483$	0	0
$484$	−929.844	−0.0873257
$485$	7821.77	0.732306
$486$	0	0
$487$	−16853.6	−1.56819	−0.784095	−	0.620641i	$-0.786872\pi$
$487$	−16853.6	−1.56819	−0.784095	+	0.620641i	$0.786872\pi$
$488$	467.479	0.0433643
$489$	0	0
$490$	−183.936	−0.0169579
$491$	−16965.7	−1.55938	−0.779688	−	0.626168i	$-0.784622\pi$
$491$	−16965.7	−1.55938	−0.779688	+	0.626168i	$0.784622\pi$
$492$	0	0
$493$	6661.70	0.608576
$494$	95.6869	0.00871489
$495$	0	0
$496$	−11792.4	−1.06753
$497$	−4084.13	−0.368608
$498$	0	0
$499$	−18186.3	−1.63153	−0.815763	−	0.578386i	$-0.803683\pi$
$499$	−18186.3	−1.63153	−0.815763	+	0.578386i	$0.803683\pi$
$500$	−10546.9	−0.943344
$501$	0	0
$502$	−121.013	−0.0107591
$503$	484.954	0.0429881	0.0214941	−	0.999769i	$-0.493158\pi$
$503$	484.954	0.0429881	0.0214941	+	0.999769i	$0.493158\pi$
$504$	0	0
$505$	1211.92	0.106792
$506$	274.833	0.0241459
$507$	0	0
$508$	−5201.89	−0.454324
$509$	7573.95	0.659547	0.329773	−	0.944060i	$-0.393028\pi$
$509$	7573.95	0.659547	0.329773	+	0.944060i	$0.393028\pi$
$510$	0	0
$511$	−5719.09	−0.495103
$512$	9761.22	0.842557
$513$	0	0
$514$	−2638.91	−0.226454
$515$	1419.31	0.121442
$516$	0	0
$517$	−4251.86	−0.361696
$518$	−994.849	−0.0843844
$519$	0	0
$520$	841.951	0.0710038
$521$	8619.11	0.724779	0.362390	−	0.932027i	$-0.381961\pi$
$521$	8619.11	0.724779	0.362390	+	0.932027i	$0.381961\pi$
$522$	0	0
$523$	10682.2	0.893116	0.446558	−	0.894755i	$-0.352650\pi$
$523$	10682.2	0.893116	0.446558	+	0.894755i	$0.352650\pi$
$524$	2236.56	0.186459
$525$	0	0
$526$	−4214.11	−0.349323
$527$	−9961.46	−0.823393
$528$	0	0
$529$	−10187.4	−0.837300
$530$	−137.276	−0.0112508
$531$	0	0
$532$	−640.977	−0.0522366
$533$	2235.03	0.181632
$534$	0	0
$535$	1471.37	0.118902
$536$	−1165.69	−0.0939365
$537$	0	0
$538$	1411.54	0.113115
$539$	539.000	0.0430730
$540$	0	0
$541$	−16632.0	−1.32175	−0.660873	−	0.750498i	$-0.729814\pi$
$541$	−16632.0	−1.32175	−0.660873	+	0.750498i	$0.729814\pi$
$542$	3585.73	0.284170
$543$	0	0
$544$	4880.79	0.384673
$545$	1651.59	0.129810
$546$	0	0
$547$	5814.76	0.454518	0.227259	−	0.973834i	$-0.427024\pi$
$547$	5814.76	0.454518	0.227259	+	0.973834i	$0.427024\pi$
$548$	17846.9	1.39121
$549$	0	0
$550$	−496.114	−0.0384625
$551$	1662.25	0.128520
$552$	0	0
$553$	−2587.43	−0.198967
$554$	67.7512	0.00519580
$555$	0	0
$556$	−1617.13	−0.123348
$557$	14323.2	1.08958	0.544788	−	0.838574i	$-0.316610\pi$
$557$	14323.2	1.08958	0.544788	+	0.838574i	$0.316610\pi$
$558$	0	0
$559$	−3766.47	−0.284982
$560$	−2645.24	−0.199611
$561$	0	0
$562$	1340.85	0.100641
$563$	−12389.8	−0.927476	−0.463738	−	0.885972i	$-0.653492\pi$
$563$	−12389.8	−0.927476	−0.463738	+	0.885972i	$0.653492\pi$
$564$	0	0
$565$	−930.915	−0.0693166
$566$	438.972	0.0325996
$567$	0	0
$568$	5138.86	0.379616
$569$	−11823.1	−0.871092	−0.435546	−	0.900166i	$-0.643445\pi$
$569$	−11823.1	−0.871092	−0.435546	+	0.900166i	$0.643445\pi$
$570$	0	0
$571$	−26117.3	−1.91414	−0.957069	−	0.289860i	$-0.906391\pi$
$571$	−26117.3	−1.91414	−0.957069	+	0.289860i	$0.906391\pi$
$572$	−1208.81	−0.0883619
$573$	0	0
$574$	614.370	0.0446747
$575$	−3573.42	−0.259169
$576$	0	0
$577$	11861.7	0.855821	0.427911	−	0.903821i	$-0.359250\pi$
$577$	11861.7	0.855821	0.427911	+	0.903821i	$0.359250\pi$
$578$	−1478.33	−0.106385
$579$	0	0
$580$	7166.07	0.513025
$581$	484.074	0.0345659
$582$	0	0
$583$	402.271	0.0285769
$584$	7196.05	0.509888
$585$	0	0
$586$	−1217.46	−0.0858241
$587$	−365.737	−0.0257165	−0.0128582	−	0.999917i	$-0.504093\pi$
$587$	−365.737	−0.0257165	−0.0128582	+	0.999917i	$0.504093\pi$
$588$	0	0
$589$	−2485.62	−0.173885
$590$	−430.793	−0.0300601
$591$	0	0
$592$	−14307.3	−0.993286
$593$	17264.3	1.19554	0.597772	−	0.801666i	$-0.296053\pi$
$593$	17264.3	1.19554	0.597772	+	0.801666i	$0.296053\pi$
$594$	0	0
$595$	−2234.52	−0.153961
$596$	−7145.31	−0.491080
$597$	0	0
$598$	357.288	0.0244324
$599$	53.8079	0.00367033	0.00183517	−	0.999998i	$-0.499416\pi$
$599$	53.8079	0.00367033	0.00183517	+	0.999998i	$0.499416\pi$
$600$	0	0
$601$	3018.63	0.204879	0.102440	−	0.994739i	$-0.467335\pi$
$601$	3018.63	0.204879	0.102440	+	0.994739i	$0.467335\pi$
$602$	−1035.34	−0.0700950
$603$	0	0
$604$	−17679.2	−1.19099
$605$	−808.844	−0.0543540
$606$	0	0
$607$	−8991.00	−0.601208	−0.300604	−	0.953749i	$-0.597188\pi$
$607$	−8991.00	−0.601208	−0.300604	+	0.953749i	$0.597188\pi$
$608$	1217.87	0.0812357
$609$	0	0
$610$	199.235	0.0132243
$611$	−5527.50	−0.365988
$612$	0	0
$613$	−16880.3	−1.11222	−0.556110	−	0.831109i	$-0.687707\pi$
$613$	−16880.3	−1.11222	−0.556110	+	0.831109i	$0.687707\pi$
$614$	5789.58	0.380535
$615$	0	0
$616$	−678.198	−0.0443594
$617$	−23828.4	−1.55477	−0.777387	−	0.629022i	$-0.783455\pi$
$617$	−23828.4	−1.55477	−0.777387	+	0.629022i	$0.783455\pi$
$618$	0	0
$619$	12588.4	0.817402	0.408701	−	0.912668i	$-0.365982\pi$
$619$	12588.4	0.817402	0.408701	+	0.912668i	$0.365982\pi$
$620$	−10715.7	−0.694115
$621$	0	0
$622$	3754.89	0.242053
$623$	−3271.07	−0.210357
$624$	0	0
$625$	864.972	0.0553582
$626$	3626.13	0.231516
$627$	0	0
$628$	14968.4	0.951121
$629$	−12085.8	−0.766126
$630$	0	0
$631$	−11576.5	−0.730354	−0.365177	−	0.930938i	$-0.618992\pi$
$631$	−11576.5	−0.730354	−0.365177	+	0.930938i	$0.618992\pi$
$632$	3255.64	0.204909
$633$	0	0
$634$	−4026.50	−0.252228
$635$	−4524.97	−0.282784
$636$	0	0
$637$	700.709	0.0435842
$638$	861.709	0.0534724
$639$	0	0
$640$	6947.97	0.429129
$641$	9004.11	0.554822	0.277411	−	0.960751i	$-0.410524\pi$
$641$	9004.11	0.554822	0.277411	+	0.960751i	$0.410524\pi$
$642$	0	0
$643$	19692.3	1.20776	0.603879	−	0.797076i	$-0.293621\pi$
$643$	19692.3	1.20776	0.603879	+	0.797076i	$0.293621\pi$
$644$	−2393.36	−0.146447
$645$	0	0
$646$	319.535	0.0194612
$647$	−25548.5	−1.55242	−0.776208	−	0.630477i	$-0.782859\pi$
$647$	−25548.5	−1.55242	−0.776208	+	0.630477i	$0.782859\pi$
$648$	0	0
$649$	1262.39	0.0763528
$650$	−644.957	−0.0389189
$651$	0	0
$652$	21160.8	1.27105
$653$	−15206.7	−0.911306	−0.455653	−	0.890157i	$-0.650594\pi$
$653$	−15206.7	−0.911306	−0.455653	+	0.890157i	$0.650594\pi$
$654$	0	0
$655$	1945.51	0.116057
$656$	8835.47	0.525864
$657$	0	0
$658$	−1519.41	−0.0900195
$659$	−30126.4	−1.78082	−0.890408	−	0.455162i	$-0.849581\pi$
$659$	−30126.4	−1.78082	−0.890408	+	0.455162i	$0.849581\pi$
$660$	0	0
$661$	−5122.44	−0.301422	−0.150711	−	0.988578i	$-0.548156\pi$
$661$	−5122.44	−0.301422	−0.150711	+	0.988578i	$0.548156\pi$
$662$	3284.81	0.192852
$663$	0	0
$664$	−609.087	−0.0355981
$665$	−557.567	−0.0325136
$666$	0	0
$667$	6206.73	0.360308
$668$	−29843.3	−1.72855
$669$	0	0
$670$	−496.805	−0.0286466
$671$	−583.834	−0.0335896
$672$	0	0
$673$	9238.65	0.529159	0.264579	−	0.964364i	$-0.414767\pi$
$673$	9238.65	0.529159	0.264579	+	0.964364i	$0.414767\pi$
$674$	5494.43	0.314002
$675$	0	0
$676$	15311.7	0.871172
$677$	12245.5	0.695171	0.347586	−	0.937648i	$-0.387002\pi$
$677$	12245.5	0.695171	0.347586	+	0.937648i	$0.387002\pi$
$678$	0	0
$679$	−8190.76	−0.462934
$680$	2811.59	0.158558
$681$	0	0
$682$	−1288.54	−0.0723472
$683$	3210.33	0.179853	0.0899267	−	0.995948i	$-0.471337\pi$
$683$	3210.33	0.179853	0.0899267	+	0.995948i	$0.471337\pi$
$684$	0	0
$685$	15524.5	0.865927
$686$	192.613	0.0107201
$687$	0	0
$688$	−14889.6	−0.825086
$689$	522.959	0.0289160
$690$	0	0
$691$	15819.4	0.870908	0.435454	−	0.900211i	$-0.356588\pi$
$691$	15819.4	0.870908	0.435454	+	0.900211i	$0.356588\pi$
$692$	−24001.7	−1.31851
$693$	0	0
$694$	−5296.03	−0.289675
$695$	−1406.69	−0.0767752
$696$	0	0
$697$	7463.61	0.405602
$698$	2208.60	0.119766
$699$	0	0
$700$	4320.37	0.233278
$701$	13417.1	0.722907	0.361454	−	0.932390i	$-0.382281\pi$
$701$	13417.1	0.722907	0.361454	+	0.932390i	$0.382281\pi$
$702$	0	0
$703$	−3015.70	−0.161791
$704$	−4343.41	−0.232526
$705$	0	0
$706$	3675.07	0.195911
$707$	−1269.09	−0.0675095
$708$	0	0
$709$	−25127.4	−1.33100	−0.665500	−	0.746398i	$-0.731782\pi$
$709$	−25127.4	−1.33100	−0.665500	+	0.746398i	$0.731782\pi$
$710$	2190.14	0.115767
$711$	0	0
$712$	4115.83	0.216639
$713$	−9281.14	−0.487491
$714$	0	0
$715$	−1051.51	−0.0549990
$716$	12535.9	0.654315
$717$	0	0
$718$	−6184.51	−0.321454
$719$	−4894.82	−0.253889	−0.126944	−	0.991910i	$-0.540517\pi$
$719$	−4894.82	−0.253889	−0.126944	+	0.991910i	$0.540517\pi$
$720$	0	0
$721$	−1486.27	−0.0767704
$722$	−3771.96	−0.194429
$723$	0	0
$724$	6556.28	0.336550
$725$	−11204.1	−0.573943
$726$	0	0
$727$	19968.3	1.01869	0.509343	−	0.860564i	$-0.329888\pi$
$727$	19968.3	1.01869	0.509343	+	0.860564i	$0.329888\pi$
$728$	−881.669	−0.0448857
$729$	0	0
$730$	3066.89	0.155494
$731$	−12577.7	−0.636392
$732$	0	0
$733$	−3961.01	−0.199595	−0.0997975	−	0.995008i	$-0.531820\pi$
$733$	−3961.01	−0.199595	−0.0997975	+	0.995008i	$0.531820\pi$
$734$	−4342.62	−0.218378
$735$	0	0
$736$	4547.45	0.227746
$737$	1455.82	0.0727624
$738$	0	0
$739$	−30962.0	−1.54121	−0.770605	−	0.637313i	$-0.780046\pi$
$739$	−30962.0	−1.54121	−0.770605	+	0.637313i	$0.780046\pi$
$740$	−13000.9	−0.645839
$741$	0	0
$742$	143.752	0.00711227
$743$	9800.99	0.483935	0.241967	−	0.970284i	$-0.422207\pi$
$743$	9800.99	0.483935	0.241967	+	0.970284i	$0.422207\pi$
$744$	0	0
$745$	−6215.50	−0.305662
$746$	1249.46	0.0613217
$747$	0	0
$748$	−4036.69	−0.197321
$749$	−1540.78	−0.0751652
$750$	0	0
$751$	6366.71	0.309354	0.154677	−	0.987965i	$-0.450566\pi$
$751$	6366.71	0.309354	0.154677	+	0.987965i	$0.450566\pi$
$752$	−21851.2	−1.05962
$753$	0	0
$754$	1120.24	0.0541069
$755$	−15378.6	−0.741304
$756$	0	0
$757$	18741.7	0.899838	0.449919	−	0.893069i	$-0.351453\pi$
$757$	18741.7	0.899838	0.449919	+	0.893069i	$0.351453\pi$
$758$	−640.973	−0.0307140
$759$	0	0
$760$	701.560	0.0334846
$761$	31269.0	1.48949	0.744745	−	0.667350i	$-0.232571\pi$
$761$	31269.0	1.48949	0.744745	+	0.667350i	$0.232571\pi$
$762$	0	0
$763$	−1729.50	−0.0820606
$764$	21719.3	1.02850
$765$	0	0
$766$	3057.97	0.144241
$767$	1641.12	0.0772588
$768$	0	0
$769$	6522.45	0.305859	0.152930	−	0.988237i	$-0.451129\pi$
$769$	6522.45	0.305859	0.152930	+	0.988237i	$0.451129\pi$
$770$	−289.042	−0.0135277
$771$	0	0
$772$	−18767.7	−0.874952
$773$	27552.5	1.28201	0.641007	−	0.767535i	$-0.278517\pi$
$773$	27552.5	1.28201	0.641007	+	0.767535i	$0.278517\pi$
$774$	0	0
$775$	16753.8	0.776535
$776$	10306.0	0.476759
$777$	0	0
$778$	−4040.17	−0.186179
$779$	1862.35	0.0856554
$780$	0	0
$781$	−6417.92	−0.294048
$782$	1193.12	0.0545600
$783$	0	0
$784$	2770.03	0.126186
$785$	13020.6	0.592005
$786$	0	0
$787$	−31775.6	−1.43923	−0.719616	−	0.694372i	$-0.755682\pi$
$787$	−31775.6	−1.43923	−0.719616	+	0.694372i	$0.755682\pi$
$788$	21303.3	0.963068
$789$	0	0
$790$	1387.52	0.0624884
$791$	974.830	0.0438192
$792$	0	0
$793$	−758.993	−0.0339882
$794$	475.187	0.0212390
$795$	0	0
$796$	42880.2	1.90936
$797$	26583.5	1.18147	0.590737	−	0.806864i	$-0.298837\pi$
$797$	26583.5	1.18147	0.590737	+	0.806864i	$0.298837\pi$
$798$	0	0
$799$	−18458.4	−0.817287
$800$	−8208.82	−0.362782
$801$	0	0
$802$	−2369.40	−0.104322
$803$	−8987.13	−0.394955
$804$	0	0
$805$	−2081.92	−0.0911527
$806$	−1675.13	−0.0732057
$807$	0	0
$808$	1596.84	0.0695255
$809$	4580.99	0.199084	0.0995421	−	0.995033i	$-0.468262\pi$
$809$	4580.99	0.199084	0.0995421	+	0.995033i	$0.468262\pi$
$810$	0	0
$811$	−11867.9	−0.513855	−0.256928	−	0.966431i	$-0.582710\pi$
$811$	−11867.9	−0.513855	−0.256928	+	0.966431i	$0.582710\pi$
$812$	−7504.12	−0.324314
$813$	0	0
$814$	−1563.33	−0.0673155
$815$	18407.2	0.791135
$816$	0	0
$817$	−3138.44	−0.134394
$818$	−4508.50	−0.192709
$819$	0	0
$820$	8028.69	0.341920
$821$	−6389.94	−0.271633	−0.135816	−	0.990734i	$-0.543366\pi$
$821$	−6389.94	−0.271633	−0.135816	+	0.990734i	$0.543366\pi$
$822$	0	0
$823$	−35488.9	−1.50312	−0.751558	−	0.659667i	$-0.770697\pi$
$823$	−35488.9	−1.50312	−0.751558	+	0.659667i	$0.770697\pi$
$824$	1870.10	0.0790631
$825$	0	0
$826$	451.116	0.0190028
$827$	506.928	0.0213151	0.0106576	−	0.999943i	$-0.496608\pi$
$827$	506.928	0.0213151	0.0106576	+	0.999943i	$0.496608\pi$
$828$	0	0
$829$	−18129.2	−0.759535	−0.379768	−	0.925082i	$-0.623996\pi$
$829$	−18129.2	−0.759535	−0.379768	+	0.925082i	$0.623996\pi$
$830$	−259.587	−0.0108559
$831$	0	0
$832$	−5646.50	−0.235285
$833$	2339.94	0.0973277
$834$	0	0
$835$	−25959.8	−1.07590
$836$	−1007.25	−0.0416704
$837$	0	0
$838$	6391.18	0.263460
$839$	4031.98	0.165911	0.0829556	−	0.996553i	$-0.473564\pi$
$839$	4031.98	0.165911	0.0829556	+	0.996553i	$0.473564\pi$
$840$	0	0
$841$	−4928.49	−0.202078
$842$	−1390.74	−0.0569216
$843$	0	0
$844$	−12088.3	−0.493006
$845$	13319.2	0.542242
$846$	0	0
$847$	847.000	0.0343604
$848$	2067.35	0.0837183
$849$	0	0
$850$	−2153.76	−0.0869098
$851$	−11260.4	−0.453586
$852$	0	0
$853$	20432.0	0.820140	0.410070	−	0.912054i	$-0.365504\pi$
$853$	20432.0	0.820140	0.410070	+	0.912054i	$0.365504\pi$
$854$	−208.634	−0.00835984
$855$	0	0
$856$	1938.68	0.0774099
$857$	−27874.2	−1.11105	−0.555523	−	0.831501i	$-0.687482\pi$
$857$	−27874.2	−1.11105	−0.555523	+	0.831501i	$0.687482\pi$
$858$	0	0
$859$	−7972.63	−0.316674	−0.158337	−	0.987385i	$-0.550613\pi$
$859$	−7972.63	−0.316674	−0.158337	+	0.987385i	$0.550613\pi$
$860$	−13530.0	−0.536475
$861$	0	0
$862$	5446.23	0.215197
$863$	45524.5	1.79568	0.897840	−	0.440322i	$-0.145136\pi$
$863$	45524.5	1.79568	0.897840	+	0.440322i	$0.145136\pi$
$864$	0	0
$865$	−20878.4	−0.820679
$866$	−8078.34	−0.316990
$867$	0	0
$868$	11221.2	0.438791
$869$	−4065.96	−0.158721
$870$	0	0
$871$	1892.59	0.0736258
$872$	2176.15	0.0845113
$873$	0	0
$874$	297.712	0.0115220
$875$	9607.24	0.371182
$876$	0	0
$877$	24262.3	0.934186	0.467093	−	0.884208i	$-0.345301\pi$
$877$	24262.3	0.934186	0.467093	+	0.884208i	$0.345301\pi$
$878$	−452.837	−0.0174061
$879$	0	0
$880$	−4156.81	−0.159234
$881$	6944.67	0.265575	0.132788	−	0.991145i	$-0.457607\pi$
$881$	6944.67	0.265575	0.132788	+	0.991145i	$0.457607\pi$
$882$	0	0
$883$	−11539.6	−0.439794	−0.219897	−	0.975523i	$-0.570572\pi$
$883$	−11539.6	−0.439794	−0.219897	+	0.975523i	$0.570572\pi$
$884$	−5247.76	−0.199662
$885$	0	0
$886$	−2666.36	−0.101104
$887$	259.602	0.00982702	0.00491351	−	0.999988i	$-0.498436\pi$
$887$	259.602	0.00982702	0.00491351	+	0.999988i	$0.498436\pi$
$888$	0	0
$889$	4738.43	0.178765
$890$	1754.13	0.0660657
$891$	0	0
$892$	−5632.43	−0.211421
$893$	−4605.82	−0.172596
$894$	0	0
$895$	10904.6	0.407264
$896$	−7275.74	−0.271278
$897$	0	0
$898$	3989.51	0.148254
$899$	−29099.9	−1.07957
$900$	0	0
$901$	1746.36	0.0645723
$902$	965.438	0.0356381
$903$	0	0
$904$	−1226.58	−0.0451277
$905$	5703.11	0.209478
$906$	0	0
$907$	−20655.5	−0.756179	−0.378089	−	0.925769i	$-0.623419\pi$
$907$	−20655.5	−0.756179	−0.378089	+	0.925769i	$0.623419\pi$
$908$	50326.6	1.83937
$909$	0	0
$910$	−375.759	−0.0136882
$911$	14371.0	0.522648	0.261324	−	0.965251i	$-0.415841\pi$
$911$	14371.0	0.522648	0.261324	+	0.965251i	$0.415841\pi$
$912$	0	0
$913$	760.688	0.0275740
$914$	2583.17	0.0934833
$915$	0	0
$916$	17876.6	0.644825
$917$	−2037.29	−0.0733668
$918$	0	0
$919$	12066.1	0.433107	0.216553	−	0.976271i	$-0.430518\pi$
$919$	12066.1	0.433107	0.216553	+	0.976271i	$0.430518\pi$
$920$	2619.58	0.0938748
$921$	0	0
$922$	−3986.02	−0.142378
$923$	−8343.40	−0.297537
$924$	0	0
$925$	20326.7	0.722527
$926$	5635.73	0.200002
$927$	0	0
$928$	14258.0	0.504356
$929$	53667.0	1.89532	0.947662	−	0.319275i	$-0.103439\pi$
$929$	53667.0	1.89532	0.947662	+	0.319275i	$0.103439\pi$
$930$	0	0
$931$	583.870	0.0205538
$932$	−29657.1	−1.04233
$933$	0	0
$934$	−143.238	−0.00501808
$935$	−3511.40	−0.122818
$936$	0	0
$937$	26617.1	0.928007	0.464004	−	0.885833i	$-0.346412\pi$
$937$	26617.1	0.928007	0.464004	+	0.885833i	$0.346412\pi$
$938$	520.241	0.0181092
$939$	0	0
$940$	−19855.9	−0.688967
$941$	39750.7	1.37709	0.688543	−	0.725196i	$-0.258251\pi$
$941$	39750.7	1.37709	0.688543	+	0.725196i	$0.258251\pi$
$942$	0	0
$943$	6953.88	0.240137
$944$	6487.66	0.223681
$945$	0	0
$946$	−1626.96	−0.0559165
$947$	−18606.6	−0.638471	−0.319235	−	0.947675i	$-0.603426\pi$
$947$	−18606.6	−0.638471	−0.319235	+	0.947675i	$0.603426\pi$
$948$	0	0
$949$	−11683.4	−0.399642
$950$	−537.414	−0.0183537
$951$	0	0
$952$	−2944.23	−0.100234
$953$	46514.2	1.58105	0.790526	−	0.612428i	$-0.209807\pi$
$953$	46514.2	1.58105	0.790526	+	0.612428i	$0.209807\pi$
$954$	0	0
$955$	18893.0	0.640169
$956$	−28341.1	−0.958803
$957$	0	0
$958$	1342.12	0.0452631
$959$	−16256.8	−0.547404
$960$	0	0
$961$	13723.1	0.460646
$962$	−2032.36	−0.0681143
$963$	0	0
$964$	13973.0	0.466847
$965$	−16325.4	−0.544595
$966$	0	0
$967$	34299.6	1.14064	0.570321	−	0.821422i	$-0.306819\pi$
$967$	34299.6	1.14064	0.570321	+	0.821422i	$0.306819\pi$
$968$	−1065.74	−0.0353865
$969$	0	0
$970$	4392.34	0.145391
$971$	27751.4	0.917182	0.458591	−	0.888648i	$-0.348354\pi$
$971$	27751.4	0.917182	0.458591	+	0.888648i	$0.348354\pi$
$972$	0	0
$973$	1473.05	0.0485342
$974$	−9464.17	−0.311347
$975$	0	0
$976$	−3000.44	−0.0984034
$977$	4009.25	0.131287	0.0656434	−	0.997843i	$-0.479090\pi$
$977$	4009.25	0.131287	0.0656434	+	0.997843i	$0.479090\pi$
$978$	0	0
$979$	−5140.25	−0.167807
$980$	2517.10	0.0820466
$981$	0	0
$982$	−9527.16	−0.309597
$983$	−9792.68	−0.317739	−0.158870	−	0.987300i	$-0.550785\pi$
$983$	−9792.68	−0.317739	−0.158870	+	0.987300i	$0.550785\pi$
$984$	0	0
$985$	18531.1	0.599441
$986$	3740.90	0.120826
$987$	0	0
$988$	−1309.44	−0.0421649
$989$	−11718.7	−0.376777
$990$	0	0
$991$	51763.7	1.65926	0.829631	−	0.558312i	$-0.188551\pi$
$991$	51763.7	1.65926	0.829631	+	0.558312i	$0.188551\pi$
$992$	−21320.5	−0.682386
$993$	0	0
$994$	−2293.45	−0.0731830
$995$	37300.3	1.18844
$996$	0	0
$997$	−17369.9	−0.551767	−0.275884	−	0.961191i	$-0.588970\pi$
$997$	−17369.9	−0.551767	−0.275884	+	0.961191i	$0.588970\pi$
$998$	−10212.6	−0.323921
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.g.1.2		2
3.2	odd	2		231.4.a.h.1.1	✓	2
21.20	even	2		1617.4.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.h.1.1	✓	2	3.2	odd	2
693.4.a.g.1.2		2	1.1	even	1	trivial
1617.4.a.m.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+0.561553 q^{2} -7.68466 q^{4} -6.68466 q^{5} +7.00000 q^{7} -8.80776 q^{8} -3.75379 q^{10} +11.0000 q^{11} +14.3002 q^{13} +3.93087 q^{14} +56.5312 q^{16} +47.7538 q^{17} +11.9157 q^{19} +51.3693 q^{20} +6.17708 q^{22} +44.4924 q^{23} -80.3153 q^{25} +8.03031 q^{26} -53.7926 q^{28} +139.501 q^{29} -208.600 q^{31} +102.207 q^{32} +26.8163 q^{34} -46.7926 q^{35} -253.086 q^{37} +6.69130 q^{38} +58.8769 q^{40} +156.294 q^{41} -263.386 q^{43} -84.5312 q^{44} +24.9848 q^{46} -386.533 q^{47} +49.0000 q^{49} -45.1013 q^{50} -109.892 q^{52} +36.5701 q^{53} -73.5312 q^{55} -61.6543 q^{56} +78.3371 q^{58} +114.762 q^{59} -53.0758 q^{61} -117.140 q^{62} -394.855 q^{64} -95.5919 q^{65} +132.348 q^{67} -366.972 q^{68} -26.2765 q^{70} -583.447 q^{71} -817.012 q^{73} -142.121 q^{74} -91.5682 q^{76} +77.0000 q^{77} -369.633 q^{79} -377.892 q^{80} +87.7671 q^{82} +69.1534 q^{83} -319.218 q^{85} -147.905 q^{86} -96.8854 q^{88} -467.295 q^{89} +100.101 q^{91} -341.909 q^{92} -217.059 q^{94} -79.6525 q^{95} -1170.11 q^{97} +27.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 3 q^{4} - q^{5} + 14 q^{7} + 3 q^{8} - 24 q^{10} + 22 q^{11} - 25 q^{13} - 21 q^{14} - 23 q^{16} + 112 q^{17} - 71 q^{19} + 78 q^{20} - 33 q^{22} + 56 q^{23} - 173 q^{25} + 148 q^{26} - 21 q^{28}+ \cdots - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 - 3 * q^4 - q^5 + 14 * q^7 + 3 * q^8 - 24 * q^10 + 22 * q^11 - 25 * q^13 - 21 * q^14 - 23 * q^16 + 112 * q^17 - 71 * q^19 + 78 * q^20 - 33 * q^22 + 56 * q^23 - 173 * q^25 + 148 * q^26 - 21 * q^28 + 11 * q^29 - 310 * q^31 + 291 * q^32 - 202 * q^34 - 7 * q^35 - 65 * q^37 + 302 * q^38 + 126 * q^40 - 42 * q^41 - 32 * q^43 - 33 * q^44 - 16 * q^46 - 101 * q^47 + 98 * q^49 + 285 * q^50 - 294 * q^52 - 166 * q^53 - 11 * q^55 + 21 * q^56 + 536 * q^58 + 11 * q^59 - 436 * q^61 + 244 * q^62 - 431 * q^64 - 319 * q^65 - 127 * q^67 - 66 * q^68 - 168 * q^70 - 936 * q^71 - 327 * q^73 - 812 * q^74 - 480 * q^76 + 154 * q^77 - 228 * q^79 - 830 * q^80 + 794 * q^82 + 262 * q^83 + 46 * q^85 - 972 * q^86 + 33 * q^88 - 44 * q^89 - 175 * q^91 - 288 * q^92 - 1234 * q^94 - 551 * q^95 - 2266 * q^97 - 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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