LMFDB - Embedded newform 1617.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.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:	$95.4060884793$
Analytic rank:	$0$
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			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.561553 q^{2} +3.00000 q^{3} -7.68466 q^{4} -6.68466 q^{5} -1.68466 q^{6} +8.80776 q^{8} +9.00000 q^{9} +3.75379 q^{10} -11.0000 q^{11} -23.0540 q^{12} -14.3002 q^{13} -20.0540 q^{15} +56.5312 q^{16} +47.7538 q^{17} -5.05398 q^{18} -11.9157 q^{19} +51.3693 q^{20} +6.17708 q^{22} -44.4924 q^{23} +26.4233 q^{24} -80.3153 q^{25} +8.03031 q^{26} +27.0000 q^{27} -139.501 q^{29} +11.2614 q^{30} +208.600 q^{31} -102.207 q^{32} -33.0000 q^{33} -26.8163 q^{34} -69.1619 q^{36} -253.086 q^{37} +6.69130 q^{38} -42.9006 q^{39} -58.8769 q^{40} +156.294 q^{41} -263.386 q^{43} +84.5312 q^{44} -60.1619 q^{45} +24.9848 q^{46} -386.533 q^{47} +169.594 q^{48} +45.1013 q^{50} +143.261 q^{51} +109.892 q^{52} -36.5701 q^{53} -15.1619 q^{54} +73.5312 q^{55} -35.7471 q^{57} +78.3371 q^{58} +114.762 q^{59} +154.108 q^{60} +53.0758 q^{61} -117.140 q^{62} -394.855 q^{64} +95.5919 q^{65} +18.5312 q^{66} +132.348 q^{67} -366.972 q^{68} -133.477 q^{69} +583.447 q^{71} +79.2699 q^{72} +817.012 q^{73} +142.121 q^{74} -240.946 q^{75} +91.5682 q^{76} +24.0909 q^{78} -369.633 q^{79} -377.892 q^{80} +81.0000 q^{81} -87.7671 q^{82} +69.1534 q^{83} -319.218 q^{85} +147.905 q^{86} -418.503 q^{87} -96.8854 q^{88} -467.295 q^{89} +33.7841 q^{90} +341.909 q^{92} +625.801 q^{93} +217.059 q^{94} +79.6525 q^{95} -306.622 q^{96} +1170.11 q^{97} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 6 q^{3} - 3 q^{4} - q^{5} + 9 q^{6} - 3 q^{8} + 18 q^{9} + 24 q^{10} - 22 q^{11} - 9 q^{12} + 25 q^{13} - 3 q^{15} - 23 q^{16} + 112 q^{17} + 27 q^{18} + 71 q^{19} + 78 q^{20} - 33 q^{22}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q + 3 * q^2 + 6 * q^3 - 3 * q^4 - q^5 + 9 * q^6 - 3 * q^8 + 18 * q^9 + 24 * q^10 - 22 * q^11 - 9 * q^12 + 25 * q^13 - 3 * q^15 - 23 * q^16 + 112 * q^17 + 27 * q^18 + 71 * q^19 + 78 * q^20 - 33 * q^22 - 56 * q^23 - 9 * q^24 - 173 * q^25 + 148 * q^26 + 54 * q^27 - 11 * q^29 + 72 * q^30 + 310 * q^31 - 291 * q^32 - 66 * q^33 + 202 * q^34 - 27 * q^36 - 65 * q^37 + 302 * q^38 + 75 * q^39 - 126 * q^40 - 42 * q^41 - 32 * q^43 + 33 * q^44 - 9 * q^45 - 16 * q^46 - 101 * q^47 - 69 * q^48 - 285 * q^50 + 336 * q^51 + 294 * q^52 + 166 * q^53 + 81 * q^54 + 11 * q^55 + 213 * q^57 + 536 * q^58 + 11 * q^59 + 234 * q^60 + 436 * q^61 + 244 * q^62 - 431 * q^64 + 319 * q^65 - 99 * q^66 - 127 * q^67 - 66 * q^68 - 168 * q^69 + 936 * q^71 - 27 * q^72 + 327 * q^73 + 812 * q^74 - 519 * q^75 + 480 * q^76 + 444 * q^78 - 228 * q^79 - 830 * q^80 + 162 * q^81 - 794 * q^82 + 262 * q^83 + 46 * q^85 + 972 * q^86 - 33 * q^87 + 33 * q^88 - 44 * q^89 + 216 * q^90 + 288 * q^92 + 930 * q^93 + 1234 * q^94 + 551 * q^95 - 873 * q^96 + 2266 * q^97 - 198 * 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 $

$2$	−0.561553	−0.198539	−0.0992695	−	0.995061i	$-0.531651\pi$
$2$	−0.561553	−0.198539	−0.0992695	+	0.995061i	$0.531651\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−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	$0.596635\pi$
$6$	−1.68466	−0.114626
$7$	0	0
$7$	0	0
$8$	8.80776	0.389252
$9$	9.00000	0.333333
$10$	3.75379	0.118705
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−23.0540	−0.554592
$13$	−14.3002	−0.305089	−0.152545	−	0.988297i	$-0.548747\pi$
$13$	−14.3002	−0.305089	−0.152545	+	0.988297i	$0.548747\pi$
$14$	0	0
$15$	−20.0540	−0.345194
$16$	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	$0.389354\pi$
$18$	−5.05398	−0.0661796
$19$	−11.9157	−0.143876	−0.0719382	−	0.997409i	$-0.522918\pi$
$19$	−11.9157	−0.143876	−0.0719382	+	0.997409i	$0.522918\pi$
$20$	51.3693	0.574326
$21$	0	0
$22$	6.17708	0.0598617
$23$	−44.4924	−0.403361	−0.201681	−	0.979451i	$-0.564640\pi$
$23$	−44.4924	−0.403361	−0.201681	+	0.979451i	$0.564640\pi$
$24$	26.4233	0.224735
$25$	−80.3153	−0.642523
$26$	8.03031	0.0605721
$27$	27.0000	0.192450
$28$	0	0
$29$	−139.501	−0.893265	−0.446632	−	0.894718i	$-0.647377\pi$
$29$	−139.501	−0.893265	−0.446632	+	0.894718i	$0.647377\pi$
$30$	11.2614	0.0685345
$31$	208.600	1.20857	0.604286	−	0.796767i	$-0.293458\pi$
$31$	208.600	1.20857	0.604286	+	0.796767i	$0.293458\pi$
$32$	−102.207	−0.564621
$33$	−33.0000	−0.174078
$34$	−26.8163	−0.135263
$35$	0	0
$36$	−69.1619	−0.320194
$37$	−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
$39$	−42.9006	−0.176143
$40$	−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
$45$	−60.1619	−0.199298
$46$	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	$0.704755\pi$
$48$	169.594	0.509974
$49$	0	0
$50$	45.1013	0.127566
$51$	143.261	0.393345
$52$	109.892	0.293063
$53$	−36.5701	−0.0947790	−0.0473895	−	0.998876i	$-0.515090\pi$
$53$	−36.5701	−0.0947790	−0.0473895	+	0.998876i	$0.515090\pi$
$54$	−15.1619	−0.0382088
$55$	73.5312	0.180272
$56$	0	0
$57$	−35.7471	−0.0830671
$58$	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	$0.459588\pi$
$60$	154.108	0.331588
$61$	53.0758	0.111404	0.0557021	−	0.998447i	$-0.482260\pi$
$61$	53.0758	0.111404	0.0557021	+	0.998447i	$0.482260\pi$
$62$	−117.140	−0.239949
$63$	0	0
$64$	−394.855	−0.771201
$65$	95.5919	0.182411
$66$	18.5312	0.0345612
$67$	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
$69$	−133.477	−0.232881
$70$	0	0
$71$	583.447	0.975245	0.487623	−	0.873055i	$-0.337864\pi$
$71$	583.447	0.975245	0.487623	+	0.873055i	$0.337864\pi$
$72$	79.2699	0.129751
$73$	817.012	1.30992	0.654959	−	0.755664i	$-0.272686\pi$
$73$	817.012	1.30992	0.654959	+	0.755664i	$0.272686\pi$
$74$	142.121	0.223260
$75$	−240.946	−0.370961
$76$	91.5682	0.138205
$77$	0	0
$78$	24.0909	0.0349713
$79$	−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
$81$	81.0000	0.111111
$82$	−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
$87$	−418.503	−0.515727
$88$	−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	$0.589763\pi$
$90$	33.7841	0.0395684
$91$	0	0
$92$	341.909	0.387462
$93$	625.801	0.697769
$94$	217.059	0.238169
$95$	79.6525	0.0860229
$96$	−306.622	−0.325984
$97$	1170.11	1.22481	0.612404	−	0.790545i	$-0.290202\pi$
$97$	1170.11	1.22481	0.612404	+	0.790545i	$0.290202\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	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	$0.528465\pi$
$102$	−80.4488	−0.0780943
$103$	212.324	0.203115	0.101558	−	0.994830i	$-0.467617\pi$
$103$	212.324	0.203115	0.101558	+	0.994830i	$0.467617\pi$
$104$	−125.953	−0.118756
$105$	0	0
$106$	20.5360	0.0188173
$107$	220.111	0.198868	0.0994342	−	0.995044i	$-0.468297\pi$
$107$	220.111	0.198868	0.0994342	+	0.995044i	$0.468297\pi$
$108$	−207.486	−0.184864
$109$	−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
$111$	−759.258	−0.649240
$112$	0	0
$113$	−139.261	−0.115935	−0.0579673	−	0.998318i	$-0.518462\pi$
$113$	−139.261	−0.115935	−0.0579673	+	0.998318i	$0.518462\pi$
$114$	20.0739	0.0164921
$115$	297.417	0.241167
$116$	1072.02	0.858054
$117$	−128.702	−0.101696
$118$	−64.4451	−0.0502767
$119$	0	0
$120$	−176.631	−0.134368
$121$	121.000	0.0909091
$122$	−29.8049	−0.0221181
$123$	468.881	0.343720
$124$	−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
$129$	−790.159	−0.539299
$130$	−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	$0.530942\pi$
$132$	253.594	0.167216
$133$	0	0
$134$	−74.3201	−0.0479125
$135$	−180.486	−0.115065
$136$	420.604	0.265195
$137$	2322.41	1.44830	0.724148	−	0.689645i	$-0.242233\pi$
$137$	2322.41	1.44830	0.724148	+	0.689645i	$0.242233\pi$
$138$	74.9545	0.0462359
$139$	−210.436	−0.128409	−0.0642047	−	0.997937i	$-0.520451\pi$
$139$	−210.436	−0.128409	−0.0642047	+	0.997937i	$0.520451\pi$
$140$	0	0
$141$	−1159.60	−0.692595
$142$	−327.636	−0.193624
$143$	157.302	0.0919878
$144$	508.781	0.294434
$145$	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	$-0.582278\pi$
$149$	−929.815	−0.511231	−0.255616	+	0.966779i	$0.582278\pi$
$150$	135.304	0.0736501
$151$	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
$153$	429.784	0.227098
$154$	0	0
$155$	−1394.42	−0.722598
$156$	329.676	0.169200
$157$	1947.83	0.990150	0.495075	−	0.868850i	$-0.335141\pi$
$157$	1947.83	0.990150	0.495075	+	0.868850i	$0.335141\pi$
$158$	207.568	0.104514
$159$	−109.710	−0.0547207
$160$	683.221	0.337584
$161$	0	0
$162$	−45.4858	−0.0220599
$163$	−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
$165$	220.594	0.104080
$166$	−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
$171$	−107.241	−0.0479588
$172$	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	$0.259230\pi$
$174$	235.011	0.102392
$175$	0	0
$176$	−621.844	−0.266325
$177$	344.287	0.146204
$178$	262.411	0.110497
$179$	1631.29	0.681165	0.340582	−	0.940215i	$-0.389376\pi$
$179$	1631.29	0.681165	0.340582	+	0.940215i	$0.389376\pi$
$180$	462.324	0.191442
$181$	853.165	0.350360	0.175180	−	0.984536i	$-0.443949\pi$
$181$	853.165	0.350360	0.175180	+	0.984536i	$0.443949\pi$
$182$	0	0
$183$	159.227	0.0643192
$184$	−391.879	−0.157009
$185$	1691.79	0.672342
$186$	−351.420	−0.138534
$187$	−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	$-0.320178\pi$
$191$	2826.32	1.07071	0.535354	+	0.844628i	$0.320178\pi$
$192$	−1184.57	−0.445253
$193$	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
$195$	286.776	0.105315
$196$	0	0
$197$	2772.18	1.00259	0.501294	−	0.865277i	$-0.332858\pi$
$197$	2772.18	1.00259	0.501294	+	0.865277i	$0.332858\pi$
$198$	55.5937	0.0199539
$199$	5579.98	1.98771	0.993855	−	0.110688i	$-0.0353055\pi$
$199$	5579.98	1.98771	0.993855	+	0.110688i	$0.0353055\pi$
$200$	−707.399	−0.250103
$201$	397.043	0.139329
$202$	101.809	0.0354617
$203$	0	0
$204$	−1100.91	−0.377840
$205$	−1044.77	−0.355950
$206$	−119.231	−0.0403263
$207$	−400.432	−0.134454
$208$	−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
$213$	1750.34	0.563058
$214$	−123.604	−0.0394831
$215$	1760.65	0.558489
$216$	237.810	0.0749116
$217$	0	0
$218$	138.744	0.0431052
$219$	2451.04	0.756282
$220$	−565.062	−0.173166
$221$	−682.888	−0.207855
$222$	426.364	0.128899
$223$	−732.945	−0.220097	−0.110048	−	0.993926i	$-0.535101\pi$
$223$	−732.945	−0.220097	−0.110048	+	0.993926i	$0.535101\pi$
$224$	0	0
$225$	−722.838	−0.214174
$226$	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	$0.906781\pi$
$228$	274.705	0.0797928
$229$	2326.27	0.671285	0.335643	−	0.941989i	$-0.391047\pi$
$229$	2326.27	0.671285	0.335643	+	0.941989i	$0.391047\pi$
$230$	−167.015	−0.0478811
$231$	0	0
$232$	−1228.69	−0.347705
$233$	−3859.26	−1.08510	−0.542551	−	0.840023i	$-0.682541\pi$
$233$	−3859.26	−1.08510	−0.542551	+	0.840023i	$0.682541\pi$
$234$	72.2728	0.0201907
$235$	2583.84	0.717239
$236$	−881.909	−0.243252
$237$	−1108.90	−0.303927
$238$	0	0
$239$	−3688.00	−0.998147	−0.499074	−	0.866560i	$-0.666326\pi$
$239$	−3688.00	−0.998147	−0.499074	+	0.866560i	$0.666326\pi$
$240$	−1133.68	−0.304910
$241$	1818.30	0.486004	0.243002	−	0.970026i	$-0.421868\pi$
$241$	1818.30	0.486004	0.243002	+	0.970026i	$0.421868\pi$
$242$	−67.9479	−0.0180490
$243$	243.000	0.0641500
$244$	−407.869	−0.107013
$245$	0	0
$246$	−263.301	−0.0682418
$247$	170.397	0.0438951
$248$	1837.30	0.470439
$249$	207.460	0.0528002
$250$	−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
$255$	−957.653	−0.235179
$256$	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	$0.693174\pi$
$258$	443.716	0.107072
$259$	0	0
$260$	−734.591	−0.175221
$261$	−1255.51	−0.297755
$262$	163.435	0.0385384
$263$	7504.38	1.75947	0.879734	−	0.475466i	$-0.157721\pi$
$263$	7504.38	1.75947	0.879734	+	0.475466i	$0.157721\pi$
$264$	−290.656	−0.0677601
$265$	244.458	0.0566678
$266$	0	0
$267$	−1401.89	−0.321326
$268$	−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	$0.408050\pi$
$270$	101.352	0.0228448
$271$	−6385.39	−1.43131	−0.715654	−	0.698455i	$-0.753871\pi$
$271$	−6385.39	−1.43131	−0.715654	+	0.698455i	$0.753871\pi$
$272$	2699.58	0.601787
$273$	0	0
$274$	−1304.15	−0.287543
$275$	883.469	0.193728
$276$	1025.73	0.223701
$277$	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
$279$	1877.40	0.402857
$280$	0	0
$281$	−2387.75	−0.506908	−0.253454	−	0.967347i	$-0.581567\pi$
$281$	−2387.75	−0.506908	−0.253454	+	0.967347i	$0.581567\pi$
$282$	651.176	0.137507
$283$	−781.711	−0.164198	−0.0820988	−	0.996624i	$-0.526162\pi$
$283$	−781.711	−0.164198	−0.0820988	+	0.996624i	$0.526162\pi$
$284$	−4483.59	−0.936803
$285$	238.957	0.0496653
$286$	−88.3334	−0.0182632
$287$	0	0
$288$	−919.867	−0.188207
$289$	−2632.58	−0.535839
$290$	−523.657	−0.106035
$291$	3510.32	0.707144
$292$	−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
$297$	−297.000	−0.0580259
$298$	522.140	0.101499
$299$	636.250	0.123061
$300$	1851.59	0.356338
$301$	0	0
$302$	−1291.90	−0.246160
$303$	−543.897	−0.103122
$304$	−673.610	−0.127086
$305$	−354.793	−0.0666079
$306$	−241.346	−0.0450878
$307$	−10309.9	−1.91668	−0.958338	−	0.285636i	$-0.907795\pi$
$307$	−10309.9	−1.91668	−0.958338	+	0.285636i	$0.907795\pi$
$308$	0	0
$309$	636.972	0.117269
$310$	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	$0.291335\pi$
$312$	−377.858	−0.0685641
$313$	−6457.33	−1.16610	−0.583051	−	0.812436i	$-0.698141\pi$
$313$	−6457.33	−1.16610	−0.583051	+	0.812436i	$0.698141\pi$
$314$	−1093.81	−0.196583
$315$	0	0
$316$	2840.50	0.505666
$317$	7170.30	1.27042	0.635212	−	0.772338i	$-0.280913\pi$
$317$	7170.30	1.27042	0.635212	+	0.772338i	$0.280913\pi$
$318$	61.6081	0.0108642
$319$	1534.51	0.269329
$320$	2639.47	0.461097
$321$	660.333	0.114817
$322$	0	0
$323$	−569.021	−0.0980221
$324$	−622.457	−0.106731
$325$	1148.52	0.196027
$326$	1546.32	0.262707
$327$	−741.216	−0.125350
$328$	1376.60	0.231737
$329$	0	0
$330$	−123.875	−0.0206639
$331$	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
$333$	−2277.78	−0.374839
$334$	−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
$339$	−417.784	−0.0669349
$340$	2453.08	0.391285
$341$	−2294.60	−0.364398
$342$	60.2217	0.00952169
$343$	0	0
$344$	−2319.84	−0.363598
$345$	892.250	0.139238
$346$	−1753.92	−0.272518
$347$	9431.05	1.45904	0.729518	−	0.683962i	$-0.239745\pi$
$347$	9431.05	1.45904	0.729518	+	0.683962i	$0.239745\pi$
$348$	3216.05	0.495398
$349$	−3933.02	−0.603238	−0.301619	−	0.953429i	$-0.597527\pi$
$349$	−3933.02	−0.603238	−0.301619	+	0.953429i	$0.597527\pi$
$350$	0	0
$351$	−386.105	−0.0587144
$352$	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	$0.335760\pi$
$354$	−193.335	−0.0290273
$355$	−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	$-0.199712\pi$
$359$	11013.2	1.61910	0.809548	+	0.587053i	$0.199712\pi$
$360$	−529.892	−0.0775771
$361$	−6717.02	−0.979300
$362$	−479.097	−0.0695602
$363$	363.000	0.0524864
$364$	0	0
$365$	−5461.45	−0.783192
$366$	−89.4146	−0.0127699
$367$	7733.24	1.09992	0.549962	−	0.835190i	$-0.314642\pi$
$367$	7733.24	1.09992	0.549962	+	0.835190i	$0.314642\pi$
$368$	−2515.21	−0.356289
$369$	1406.64	0.198447
$370$	−950.032	−0.133486
$371$	0	0
$372$	−4809.07	−0.670265
$373$	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
$375$	4117.39	0.566989
$376$	−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
$381$	2030.76	0.273068
$382$	−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	$0.381665\pi$
$384$	3118.17	0.414384
$385$	0	0
$386$	−1371.44	−0.180840
$387$	−2370.48	−0.311365
$388$	−8991.88	−1.17653
$389$	7194.64	0.937745	0.468872	−	0.883266i	$-0.344660\pi$
$389$	7194.64	0.937745	0.468872	+	0.883266i	$0.344660\pi$
$390$	−161.040	−0.0209091
$391$	−2124.68	−0.274808
$392$	0	0
$393$	−873.125	−0.112070
$394$	−1556.73	−0.199053
$395$	2470.87	0.314741
$396$	760.781	0.0965422
$397$	−846.201	−0.106976	−0.0534882	−	0.998568i	$-0.517034\pi$
$397$	−846.201	−0.106976	−0.0534882	+	0.998568i	$0.517034\pi$
$398$	−3133.45	−0.394638
$399$	0	0
$400$	−4540.33	−0.567541
$401$	4219.37	0.525450	0.262725	−	0.964871i	$-0.415379\pi$
$401$	4219.37	0.525450	0.262725	+	0.964871i	$0.415379\pi$
$402$	−222.960	−0.0276623
$403$	−2983.02	−0.368722
$404$	1393.22	0.171573
$405$	−541.457	−0.0664327
$406$	0	0
$407$	2783.95	0.339054
$408$	1261.81	0.153110
$409$	8028.62	0.970636	0.485318	−	0.874338i	$-0.338704\pi$
$409$	8028.62	0.970636	0.485318	+	0.874338i	$0.338704\pi$
$410$	586.693	0.0706700
$411$	6967.22	0.836174
$412$	−1631.64	−0.195109
$413$	0	0
$414$	224.864	0.0266943
$415$	−462.267	−0.0546790
$416$	1461.58	0.172260
$417$	−631.307	−0.0741372
$418$	−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
$423$	−3478.80	−0.399870
$424$	−322.100	−0.0368929
$425$	−3835.36	−0.437747
$426$	−982.909	−0.111789
$427$	0	0
$428$	−1691.48	−0.191029
$429$	471.906	0.0531092
$430$	−988.697	−0.110882
$431$	−9698.52	−1.08390	−0.541951	−	0.840410i	$-0.682314\pi$
$431$	−9698.52	−1.08390	−0.541951	+	0.840410i	$0.682314\pi$
$432$	1526.34	0.169991
$433$	14385.7	1.59661	0.798307	−	0.602251i	$-0.205729\pi$
$433$	14385.7	1.59661	0.798307	+	0.602251i	$0.205729\pi$
$434$	0	0
$435$	2797.55	0.308350
$436$	1898.66	0.208554
$437$	530.159	0.0580342
$438$	−1376.39	−0.150151
$439$	806.402	0.0876708	0.0438354	−	0.999039i	$-0.486042\pi$
$439$	806.402	0.0876708	0.0438354	+	0.999039i	$0.486042\pi$
$440$	647.646	0.0701711
$441$	0	0
$442$	383.478	0.0412674
$443$	4748.19	0.509240	0.254620	−	0.967041i	$-0.418050\pi$
$443$	4748.19	0.509240	0.254620	+	0.967041i	$0.418050\pi$
$444$	5834.64	0.623648
$445$	3123.71	0.332760
$446$	411.587	0.0436978
$447$	−2789.45	−0.295159
$448$	0	0
$449$	−7104.43	−0.746724	−0.373362	−	0.927686i	$-0.621795\pi$
$449$	−7104.43	−0.746724	−0.373362	+	0.927686i	$0.621795\pi$
$450$	405.912	0.0425219
$451$	−1719.23	−0.179502
$452$	1070.18	0.111365
$453$	6901.74	0.715833
$454$	3677.59	0.380172
$455$	0	0
$456$	−314.852	−0.0323340
$457$	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
$459$	1289.35	0.131115
$460$	−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
$465$	−4183.27	−0.417192
$466$	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	$0.504023\pi$
$468$	989.028	0.0976877
$469$	0	0
$470$	−1450.96	−0.142400
$471$	5843.48	0.571663
$472$	1010.80	0.0985716
$473$	2897.25	0.281640
$474$	622.705	0.0603413
$475$	957.015	0.0924439
$476$	0	0
$477$	−329.131	−0.0315930
$478$	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	$0.463637\pi$
$480$	2049.66	0.194904
$481$	3619.18	0.343078
$482$	−1021.07	−0.0964907
$483$	0	0
$484$	−929.844	−0.0873257
$485$	−7821.77	−0.732306
$486$	−136.457	−0.0127363
$487$	−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
$489$	−8260.93	−0.763952
$490$	0	0
$491$	16965.7	1.55938	0.779688	−	0.626168i	$-0.215378\pi$
$491$	16965.7	1.55938	0.779688	+	0.626168i	$0.215378\pi$
$492$	−3603.19	−0.330171
$493$	−6661.70	−0.608576
$494$	−95.6869	−0.00871489
$495$	661.781	0.0600906
$496$	11792.4	1.06753
$497$	0	0
$498$	−116.500	−0.0104829
$499$	−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
$501$	11650.5	1.03893
$502$	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
$507$	−5977.51	−0.523611
$508$	−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	$0.393028\pi$
$510$	537.773	0.0466921
$511$	0	0
$512$	−9761.22	−0.842557
$513$	−321.724	−0.0276890
$514$	2638.91	0.226454
$515$	−1419.31	−0.121442
$516$	6072.10	0.518041
$517$	4251.86	0.361696
$518$	0	0
$519$	9369.99	0.792480
$520$	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	$0.381961\pi$
$522$	705.034	0.0591159
$523$	−10682.2	−0.893116	−0.446558	−	0.894755i	$-0.647350\pi$
$523$	−10682.2	−0.893116	−0.446558	+	0.894755i	$0.647350\pi$
$524$	2236.56	0.186459
$525$	0	0
$526$	−4214.11	−0.349323
$527$	9961.46	0.823393
$528$	−1865.53	−0.153763
$529$	−10187.4	−0.837300
$530$	−137.276	−0.0112508
$531$	1032.86	0.0844112
$532$	0	0
$533$	−2235.03	−0.181632
$534$	787.233	0.0637957
$535$	−1471.37	−0.118902
$536$	1165.69	0.0939365
$537$	4893.88	0.393271
$538$	−1411.54	−0.113115
$539$	0	0
$540$	1386.97	0.110529
$541$	−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
$543$	2559.49	0.202281
$544$	−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
$549$	477.682	0.0371347
$550$	−496.114	−0.0384625
$551$	1662.25	0.128520
$552$	−1175.64	−0.0906493
$553$	0	0
$554$	−67.7512	−0.00519580
$555$	5075.38	0.388177
$556$	1617.13	0.123348
$557$	−14323.2	−1.08958	−0.544788	−	0.838574i	$-0.683390\pi$
$557$	−14323.2	−1.08958	−0.544788	+	0.838574i	$0.683390\pi$
$558$	−1054.26	−0.0799829
$559$	3766.47	0.284982
$560$	0	0
$561$	−1575.88	−0.118598
$562$	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	$0.653492\pi$
$564$	8911.12	0.665294
$565$	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	$-0.356555\pi$
$569$	11823.1	0.871092	0.435546	+	0.900166i	$0.356555\pi$
$570$	−134.187	−0.00986050
$571$	−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
$573$	8478.95	0.618173
$574$	0	0
$575$	3573.42	0.259169
$576$	−3553.70	−0.257067
$577$	−11861.7	−0.855821	−0.427911	−	0.903821i	$-0.640750\pi$
$577$	−11861.7	−0.855821	−0.427911	+	0.903821i	$0.640750\pi$
$578$	1478.33	0.106385
$579$	7326.67	0.525883
$580$	−7166.07	−0.513025
$581$	0	0
$582$	−1971.23	−0.140396
$583$	402.271	0.0285769
$584$	7196.05	0.509888
$585$	860.327	0.0608036
$586$	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
$591$	8316.54	0.578844
$592$	−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	$0.296053\pi$
$594$	166.781	0.0115204
$595$	0	0
$596$	7145.31	0.491080
$597$	16739.9	1.14761
$598$	−357.288	−0.0244324
$599$	−53.8079	−0.00367033	−0.00183517	−	0.999998i	$-0.500584\pi$
$599$	−53.8079	−0.00367033	−0.00183517	+	0.999998i	$0.500584\pi$
$600$	−2122.20	−0.144397
$601$	−3018.63	−0.204879	−0.102440	−	0.994739i	$-0.532665\pi$
$601$	−3018.63	−0.204879	−0.102440	+	0.994739i	$0.532665\pi$
$602$	0	0
$603$	1191.13	0.0804419
$604$	−17679.2	−1.19099
$605$	−808.844	−0.0543540
$606$	305.427	0.0204738
$607$	8991.00	0.601208	0.300604	−	0.953749i	$-0.402812\pi$
$607$	8991.00	0.601208	0.300604	+	0.953749i	$0.402812\pi$
$608$	1217.87	0.0812357
$609$	0	0
$610$	199.235	0.0132243
$611$	5527.50	0.365988
$612$	−3302.74	−0.218146
$613$	−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
$615$	−3134.31	−0.205508
$616$	0	0
$617$	23828.4	1.55477	0.777387	−	0.629022i	$-0.216545\pi$
$617$	23828.4	1.55477	0.777387	+	0.629022i	$0.216545\pi$
$618$	−357.693	−0.0232824
$619$	−12588.4	−0.817402	−0.408701	−	0.912668i	$-0.634018\pi$
$619$	−12588.4	−0.817402	−0.408701	+	0.912668i	$0.634018\pi$
$620$	10715.7	0.694115
$621$	−1201.30	−0.0776269
$622$	−3754.89	−0.242053
$623$	0	0
$624$	−2425.22	−0.155587
$625$	864.972	0.0553582
$626$	3626.13	0.231516
$627$	393.219	0.0250457
$628$	−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
$633$	4719.14	0.296318
$634$	−4026.50	−0.252228
$635$	−4524.97	−0.282784
$636$	843.085	0.0525637
$637$	0	0
$638$	−861.709	−0.0534724
$639$	5251.02	0.325082
$640$	−6947.97	−0.429129
$641$	−9004.11	−0.554822	−0.277411	−	0.960751i	$-0.589476\pi$
$641$	−9004.11	−0.554822	−0.277411	+	0.960751i	$0.589476\pi$
$642$	−370.812	−0.0227956
$643$	−19692.3	−1.20776	−0.603879	−	0.797076i	$-0.706379\pi$
$643$	−19692.3	−1.20776	−0.603879	+	0.797076i	$0.706379\pi$
$644$	0	0
$645$	5281.94	0.322444
$646$	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	$0.782859\pi$
$648$	713.429	0.0432502
$649$	−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	$-0.349406\pi$
$653$	15206.7	0.911306	0.455653	+	0.890157i	$0.349406\pi$
$654$	416.232	0.0248868
$655$	1945.51	0.116057
$656$	8835.47	0.525864
$657$	7353.11	0.436639
$658$	0	0
$659$	30126.4	1.78082	0.890408	−	0.455162i	$-0.150419\pi$
$659$	30126.4	1.78082	0.890408	+	0.455162i	$0.150419\pi$
$660$	−1695.19	−0.0999774
$661$	5122.44	0.301422	0.150711	−	0.988578i	$-0.451844\pi$
$661$	5122.44	0.301422	0.150711	+	0.988578i	$0.451844\pi$
$662$	−3284.81	−0.192852
$663$	−2048.66	−0.120005
$664$	609.087	0.0355981
$665$	0	0
$666$	1279.09	0.0744201
$667$	6206.73	0.360308
$668$	−29843.3	−1.72855
$669$	−2198.83	−0.127073
$670$	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
$675$	−2168.51	−0.123654
$676$	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	$0.387002\pi$
$678$	234.608	0.0132892
$679$	0	0
$680$	−2811.59	−0.158558
$681$	−19646.9	−1.10554
$682$	1288.54	0.0723472
$683$	−3210.33	−0.179853	−0.0899267	−	0.995948i	$-0.528663\pi$
$683$	−3210.33	−0.179853	−0.0899267	+	0.995948i	$0.528663\pi$
$684$	824.114	0.0460684
$685$	−15524.5	−0.865927
$686$	0	0
$687$	6978.81	0.387567
$688$	−14889.6	−0.825086
$689$	522.959	0.0289160
$690$	−501.045	−0.0276442
$691$	−15819.4	−0.870908	−0.435454	−	0.900211i	$-0.643412\pi$
$691$	−15819.4	−0.870908	−0.435454	+	0.900211i	$0.643412\pi$
$692$	−24001.7	−1.31851
$693$	0	0
$694$	−5296.03	−0.289675
$695$	1406.69	0.0767752
$696$	−3686.07	−0.200748
$697$	7463.61	0.405602
$698$	2208.60	0.119766
$699$	−11577.8	−0.626484
$700$	0	0
$701$	−13417.1	−0.722907	−0.361454	−	0.932390i	$-0.617719\pi$
$701$	−13417.1	−0.722907	−0.361454	+	0.932390i	$0.617719\pi$
$702$	216.818	0.0116571
$703$	3015.70	0.161791
$704$	4343.41	0.232526
$705$	7751.53	0.414098
$706$	−3675.07	−0.195911
$707$	0	0
$708$	−2645.73	−0.140441
$709$	−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
$711$	−3326.69	−0.175472
$712$	−4115.83	−0.216639
$713$	−9281.14	−0.487491
$714$	0	0
$715$	−1051.51	−0.0549990
$716$	−12535.9	−0.654315
$717$	−11064.0	−0.576281
$718$	−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	$0.540517\pi$
$720$	−3401.03	−0.176040
$721$	0	0
$722$	3771.96	0.194429
$723$	5454.90	0.280594
$724$	−6556.28	−0.336550
$725$	11204.1	0.573943
$726$	−203.844	−0.0104206
$727$	−19968.3	−1.01869	−0.509343	−	0.860564i	$-0.670112\pi$
$727$	−19968.3	−1.01869	−0.509343	+	0.860564i	$0.670112\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	3066.89	0.155494
$731$	−12577.7	−0.636392
$732$	−1223.61	−0.0617839
$733$	3961.01	0.199595	0.0997975	−	0.995008i	$-0.468180\pi$
$733$	3961.01	0.199595	0.0997975	+	0.995008i	$0.468180\pi$
$734$	−4342.62	−0.218378
$735$	0	0
$736$	4547.45	0.227746
$737$	−1455.82	−0.0727624
$738$	−789.904	−0.0393994
$739$	−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
$741$	511.191	0.0253429
$742$	0	0
$743$	−9800.99	−0.483935	−0.241967	−	0.970284i	$-0.577793\pi$
$743$	−9800.99	−0.483935	−0.241967	+	0.970284i	$0.577793\pi$
$744$	5511.91	0.271608
$745$	6215.50	0.305662
$746$	−1249.46	−0.0613217
$747$	622.381	0.0304842
$748$	4036.69	0.197321
$749$	0	0
$750$	−2312.13	−0.112569
$751$	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
$753$	−646.492	−0.0312875
$754$	−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
$759$	1468.25	0.0702162
$760$	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	$0.232571\pi$
$762$	−1140.38	−0.0542145
$763$	0	0
$764$	−21719.3	−1.02850
$765$	−2872.96	−0.135781
$766$	−3057.97	−0.144241
$767$	−1641.12	−0.0772588
$768$	7725.50	0.362982
$769$	−6522.45	−0.305859	−0.152930	−	0.988237i	$-0.548871\pi$
$769$	−6522.45	−0.305859	−0.152930	+	0.988237i	$0.548871\pi$
$770$	0	0
$771$	−14097.9	−0.658527
$772$	−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	$0.278517\pi$
$774$	1331.15	0.0618180
$775$	−16753.8	−0.776535
$776$	10306.0	0.476759
$777$	0	0
$778$	−4040.17	−0.186179
$779$	−1862.35	−0.0856554
$780$	−2203.77	−0.101164
$781$	−6417.92	−0.294048
$782$	1193.12	0.0545600
$783$	−3766.53	−0.171909
$784$	0	0
$785$	−13020.6	−0.592005
$786$	490.306	0.0222502
$787$	31775.6	1.43923	0.719616	−	0.694372i	$-0.244318\pi$
$787$	31775.6	1.43923	0.719616	+	0.694372i	$0.244318\pi$
$788$	−21303.3	−0.963068
$789$	22513.1	1.01583
$790$	−1387.52	−0.0624884
$791$	0	0
$792$	−871.969	−0.0391213
$793$	−758.993	−0.0339882
$794$	475.187	0.0212390
$795$	733.375	0.0327172
$796$	−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
$801$	−4205.66	−0.185518
$802$	−2369.40	−0.104322
$803$	−8987.13	−0.394955
$804$	−3051.14	−0.133837
$805$	0	0
$806$	1675.13	0.0732057
$807$	7540.91	0.328938
$808$	−1596.84	−0.0695255
$809$	−4580.99	−0.199084	−0.0995421	−	0.995033i	$-0.531738\pi$
$809$	−4580.99	−0.199084	−0.0995421	+	0.995033i	$0.531738\pi$
$810$	304.057	0.0131895
$811$	11867.9	0.513855	0.256928	−	0.966431i	$-0.417290\pi$
$811$	11867.9	0.513855	0.256928	+	0.966431i	$0.417290\pi$
$812$	0	0
$813$	−19156.2	−0.826366
$814$	−1563.33	−0.0673155
$815$	18407.2	0.791135
$816$	8098.74	0.347442
$817$	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	$-0.456634\pi$
$821$	6389.94	0.271633	0.135816	+	0.990734i	$0.456634\pi$
$822$	−3912.46	−0.166013
$823$	−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
$825$	2650.41	0.111849
$826$	0	0
$827$	−506.928	−0.0213151	−0.0106576	−	0.999943i	$-0.503392\pi$
$827$	−506.928	−0.0213151	−0.0106576	+	0.999943i	$0.503392\pi$
$828$	3077.18	0.129154
$829$	18129.2	0.759535	0.379768	−	0.925082i	$-0.376004\pi$
$829$	18129.2	0.759535	0.379768	+	0.925082i	$0.376004\pi$
$830$	259.587	0.0108559
$831$	361.949	0.0151094
$832$	5646.50	0.235285
$833$	0	0
$834$	354.512	0.0147191
$835$	−25959.8	−1.07590
$836$	−1007.25	−0.0416704
$837$	5632.21	0.232590
$838$	−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
$843$	−7163.25	−0.292664
$844$	−12088.3	−0.493006
$845$	13319.2	0.542242
$846$	1953.53	0.0793897
$847$	0	0
$848$	−2067.35	−0.0837183
$849$	−2345.13	−0.0947995
$850$	2153.76	0.0869098
$851$	11260.4	0.453586
$852$	−13450.8	−0.540864
$853$	−20432.0	−0.820140	−0.410070	−	0.912054i	$-0.634496\pi$
$853$	−20432.0	−0.820140	−0.410070	+	0.912054i	$0.634496\pi$
$854$	0	0
$855$	716.872	0.0286743
$856$	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	$0.687482\pi$
$858$	−265.000	−0.0105442
$859$	7972.63	0.316674	0.158337	−	0.987385i	$-0.449387\pi$
$859$	7972.63	0.316674	0.158337	+	0.987385i	$0.449387\pi$
$860$	−13530.0	−0.536475
$861$	0	0
$862$	5446.23	0.215197
$863$	−45524.5	−1.79568	−0.897840	−	0.440322i	$-0.854864\pi$
$863$	−45524.5	−1.79568	−0.897840	+	0.440322i	$0.854864\pi$
$864$	−2759.60	−0.108661
$865$	−20878.4	−0.820679
$866$	−8078.34	−0.316990
$867$	−7897.73	−0.309367
$868$	0	0
$869$	4065.96	0.158721
$870$	−1570.97	−0.0612194
$871$	−1892.59	−0.0736258
$872$	−2176.15	−0.0845113
$873$	10531.0	0.408270
$874$	−297.712	−0.0115220
$875$	0	0
$876$	−18835.4	−0.726471
$877$	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
$879$	−6504.09	−0.249576
$880$	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
$885$	−2301.44	−0.0874148
$886$	−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	$0.498436\pi$
$888$	−6687.37	−0.252718
$889$	0	0
$890$	−1754.13	−0.0660657
$891$	−891.000	−0.0335013
$892$	5632.43	0.211421
$893$	4605.82	0.172596
$894$	1566.42	0.0586006
$895$	−10904.6	−0.407264
$896$	0	0
$897$	1908.75	0.0710494
$898$	3989.51	0.148254
$899$	−29099.9	−1.07957
$900$	5554.76	0.205732
$901$	−1746.36	−0.0645723
$902$	965.438	0.0356381
$903$	0	0
$904$	−1226.58	−0.0451277
$905$	−5703.11	−0.209478
$906$	−3875.69	−0.142121
$907$	−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
$909$	−1631.69	−0.0595378
$910$	0	0
$911$	−14371.0	−0.522648	−0.261324	−	0.965251i	$-0.584159\pi$
$911$	−14371.0	−0.522648	−0.261324	+	0.965251i	$0.584159\pi$
$912$	−2020.83	−0.0733732
$913$	−760.688	−0.0275740
$914$	−2583.17	−0.0934833
$915$	−1064.38	−0.0384561
$916$	−17876.6	−0.644825
$917$	0	0
$918$	−724.039	−0.0260314
$919$	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
$921$	−30929.8	−1.10659
$922$	3986.02	0.142378
$923$	−8343.40	−0.297537
$924$	0	0
$925$	20326.7	0.722527
$926$	−5635.73	−0.200002
$927$	1910.91	0.0677051
$928$	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	$0.103439\pi$
$930$	2349.13	0.0828289
$931$	0	0
$932$	29657.1	1.04233
$933$	20059.8	0.703890
$934$	143.238	0.00501808
$935$	3511.40	0.122818
$936$	−1133.57	−0.0395855
$937$	−26617.1	−0.928007	−0.464004	−	0.885833i	$-0.653588\pi$
$937$	−26617.1	−0.928007	−0.464004	+	0.885833i	$0.653588\pi$
$938$	0	0
$939$	−19372.0	−0.673249
$940$	−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	$0.258251\pi$
$942$	−3281.42	−0.113497
$943$	−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	$-0.396574\pi$
$947$	18606.6	0.638471	0.319235	+	0.947675i	$0.396574\pi$
$948$	8521.50	0.291947
$949$	−11683.4	−0.399642
$950$	−537.414	−0.0183537
$951$	21510.9	0.733479
$952$	0	0
$953$	−46514.2	−1.58105	−0.790526	−	0.612428i	$-0.790193\pi$
$953$	−46514.2	−1.58105	−0.790526	+	0.612428i	$0.790193\pi$
$954$	184.824	0.00627244
$955$	−18893.0	−0.640169
$956$	28341.1	0.958803
$957$	4603.53	0.155497
$958$	−1342.12	−0.0452631
$959$	0	0
$960$	7918.41	0.266214
$961$	13723.1	0.460646
$962$	−2032.36	−0.0681143
$963$	1981.00	0.0662895
$964$	−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
$969$	−1707.06	−0.0565931
$970$	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	$0.348354\pi$
$972$	−1867.37	−0.0616214
$973$	0	0
$974$	9464.17	0.311347
$975$	3445.57	0.113176
$976$	3000.44	0.0984034
$977$	−4009.25	−0.131287	−0.0656434	−	0.997843i	$-0.520910\pi$
$977$	−4009.25	−0.131287	−0.0656434	+	0.997843i	$0.520910\pi$
$978$	4638.95	0.151674
$979$	5140.25	0.167807
$980$	0	0
$981$	−2223.65	−0.0723707
$982$	−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	$0.550785\pi$
$984$	4129.79	0.133794
$985$	−18531.1	−0.599441
$986$	3740.90	0.120826
$987$	0	0
$988$	−1309.44	−0.0421649
$989$	11718.7	0.376777
$990$	−371.625	−0.0119303
$991$	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
$993$	17548.6	0.560812
$994$	0	0
$995$	−37300.3	−1.18844
$996$	−1594.26	−0.0507190
$997$	17369.9	0.551767	0.275884	−	0.961191i	$-0.411030\pi$
$997$	17369.9	0.551767	0.275884	+	0.961191i	$0.411030\pi$
$998$	10212.6	0.323921
$999$	−6833.33	−0.216413

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.h.1.1	✓	2	7.6	odd	2
693.4.a.g.1.2		2	21.20	even	2
1617.4.a.m.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q-0.561553 q^{2} +3.00000 q^{3} -7.68466 q^{4} -6.68466 q^{5} -1.68466 q^{6} +8.80776 q^{8} +9.00000 q^{9} +3.75379 q^{10} -11.0000 q^{11} -23.0540 q^{12} -14.3002 q^{13} -20.0540 q^{15} +56.5312 q^{16} +47.7538 q^{17} -5.05398 q^{18} -11.9157 q^{19} +51.3693 q^{20} +6.17708 q^{22} -44.4924 q^{23} +26.4233 q^{24} -80.3153 q^{25} +8.03031 q^{26} +27.0000 q^{27} -139.501 q^{29} +11.2614 q^{30} +208.600 q^{31} -102.207 q^{32} -33.0000 q^{33} -26.8163 q^{34} -69.1619 q^{36} -253.086 q^{37} +6.69130 q^{38} -42.9006 q^{39} -58.8769 q^{40} +156.294 q^{41} -263.386 q^{43} +84.5312 q^{44} -60.1619 q^{45} +24.9848 q^{46} -386.533 q^{47} +169.594 q^{48} +45.1013 q^{50} +143.261 q^{51} +109.892 q^{52} -36.5701 q^{53} -15.1619 q^{54} +73.5312 q^{55} -35.7471 q^{57} +78.3371 q^{58} +114.762 q^{59} +154.108 q^{60} +53.0758 q^{61} -117.140 q^{62} -394.855 q^{64} +95.5919 q^{65} +18.5312 q^{66} +132.348 q^{67} -366.972 q^{68} -133.477 q^{69} +583.447 q^{71} +79.2699 q^{72} +817.012 q^{73} +142.121 q^{74} -240.946 q^{75} +91.5682 q^{76} +24.0909 q^{78} -369.633 q^{79} -377.892 q^{80} +81.0000 q^{81} -87.7671 q^{82} +69.1534 q^{83} -319.218 q^{85} +147.905 q^{86} -418.503 q^{87} -96.8854 q^{88} -467.295 q^{89} +33.7841 q^{90} +341.909 q^{92} +625.801 q^{93} +217.059 q^{94} +79.6525 q^{95} -306.622 q^{96} +1170.11 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 6 q^{3} - 3 q^{4} - q^{5} + 9 q^{6} - 3 q^{8} + 18 q^{9} + 24 q^{10} - 22 q^{11} - 9 q^{12} + 25 q^{13} - 3 q^{15} - 23 q^{16} + 112 q^{17} + 27 q^{18} + 71 q^{19} + 78 q^{20} - 33 q^{22}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q + 3 * q^2 + 6 * q^3 - 3 * q^4 - q^5 + 9 * q^6 - 3 * q^8 + 18 * q^9 + 24 * q^10 - 22 * q^11 - 9 * q^12 + 25 * q^13 - 3 * q^15 - 23 * q^16 + 112 * q^17 + 27 * q^18 + 71 * q^19 + 78 * q^20 - 33 * q^22 - 56 * q^23 - 9 * q^24 - 173 * q^25 + 148 * q^26 + 54 * q^27 - 11 * q^29 + 72 * q^30 + 310 * q^31 - 291 * q^32 - 66 * q^33 + 202 * q^34 - 27 * q^36 - 65 * q^37 + 302 * q^38 + 75 * q^39 - 126 * q^40 - 42 * q^41 - 32 * q^43 + 33 * q^44 - 9 * q^45 - 16 * q^46 - 101 * q^47 - 69 * q^48 - 285 * q^50 + 336 * q^51 + 294 * q^52 + 166 * q^53 + 81 * q^54 + 11 * q^55 + 213 * q^57 + 536 * q^58 + 11 * q^59 + 234 * q^60 + 436 * q^61 + 244 * q^62 - 431 * q^64 + 319 * q^65 - 99 * q^66 - 127 * q^67 - 66 * q^68 - 168 * q^69 + 936 * q^71 - 27 * q^72 + 327 * q^73 + 812 * q^74 - 519 * q^75 + 480 * q^76 + 444 * q^78 - 228 * q^79 - 830 * q^80 + 162 * q^81 - 794 * q^82 + 262 * q^83 + 46 * q^85 + 972 * q^86 - 33 * q^87 + 33 * q^88 - 44 * q^89 + 216 * q^90 + 288 * q^92 + 930 * q^93 + 1234 * q^94 + 551 * q^95 - 873 * q^96 + 2266 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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