LMFDB - Embedded newform 735.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [735,4,Mod(1,735)] 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("735.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	735.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-7,6,17,10,-21,0,-63,18,-35,-26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$43.3664038542$
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 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	735.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.43845 q^{2} +3.00000 q^{3} -5.93087 q^{4} +5.00000 q^{5} -4.31534 q^{6} +20.0388 q^{8} +9.00000 q^{9} -7.19224 q^{10} +7.61553 q^{11} -17.7926 q^{12} -52.3542 q^{13} +15.0000 q^{15} +18.6222 q^{16} +49.7235 q^{17} -12.9460 q^{18} -140.600 q^{19} -29.6543 q^{20} -10.9545 q^{22} -23.4470 q^{23} +60.1165 q^{24} +25.0000 q^{25} +75.3087 q^{26} +27.0000 q^{27} +157.170 q^{29} -21.5767 q^{30} -127.892 q^{31} -187.098 q^{32} +22.8466 q^{33} -71.5246 q^{34} -53.3778 q^{36} -115.477 q^{37} +202.246 q^{38} -157.062 q^{39} +100.194 q^{40} +188.617 q^{41} +322.186 q^{43} -45.1667 q^{44} +45.0000 q^{45} +33.7272 q^{46} +76.6477 q^{47} +55.8665 q^{48} -35.9612 q^{50} +149.170 q^{51} +310.506 q^{52} -424.172 q^{53} -38.8381 q^{54} +38.0776 q^{55} -421.801 q^{57} -226.081 q^{58} -107.784 q^{59} -88.9630 q^{60} -915.511 q^{61} +183.966 q^{62} +120.153 q^{64} -261.771 q^{65} -32.8636 q^{66} -451.723 q^{67} -294.903 q^{68} -70.3409 q^{69} +907.312 q^{71} +180.349 q^{72} -755.956 q^{73} +166.108 q^{74} +75.0000 q^{75} +833.882 q^{76} +225.926 q^{78} +22.5834 q^{79} +93.1109 q^{80} +81.0000 q^{81} -271.316 q^{82} -1112.25 q^{83} +248.617 q^{85} -463.447 q^{86} +471.511 q^{87} +152.606 q^{88} -1518.81 q^{89} -64.7301 q^{90} +139.061 q^{92} -383.676 q^{93} -110.254 q^{94} -703.002 q^{95} -561.293 q^{96} +549.987 q^{97} +68.5398 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 7 q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} - 21 q^{6} - 63 q^{8} + 18 q^{9} - 35 q^{10} - 26 q^{11} + 51 q^{12} - 14 q^{13} + 30 q^{15} + 297 q^{16} - 16 q^{17} - 63 q^{18} - 174 q^{19} + 85 q^{20} + 176 q^{22}+ \cdots - 234 q^{99}+O(q^{100}) $ 2 * q - 7 * q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 - 21 * q^6 - 63 * q^8 + 18 * q^9 - 35 * q^10 - 26 * q^11 + 51 * q^12 - 14 * q^13 + 30 * q^15 + 297 * q^16 - 16 * q^17 - 63 * q^18 - 174 * q^19 + 85 * q^20 + 176 * q^22 + 184 * q^23 - 189 * q^24 + 50 * q^25 - 138 * q^26 + 54 * q^27 - 32 * q^29 - 105 * q^30 - 330 * q^31 - 1071 * q^32 - 78 * q^33 + 294 * q^34 + 153 * q^36 - 132 * q^37 + 388 * q^38 - 42 * q^39 - 315 * q^40 - 200 * q^41 + 364 * q^43 - 816 * q^44 + 90 * q^45 - 1120 * q^46 - 292 * q^47 + 891 * q^48 - 175 * q^50 - 48 * q^51 + 1190 * q^52 + 34 * q^53 - 189 * q^54 - 130 * q^55 - 522 * q^57 + 826 * q^58 - 364 * q^59 + 255 * q^60 - 792 * q^61 + 1308 * q^62 + 2809 * q^64 - 70 * q^65 + 528 * q^66 - 788 * q^67 - 1802 * q^68 + 552 * q^69 + 454 * q^71 - 567 * q^72 - 778 * q^73 + 258 * q^74 + 150 * q^75 + 68 * q^76 - 414 * q^78 + 408 * q^79 + 1485 * q^80 + 162 * q^81 + 1890 * q^82 - 1136 * q^83 - 80 * q^85 - 696 * q^86 - 96 * q^87 + 2944 * q^88 - 36 * q^89 - 315 * q^90 + 4896 * q^92 - 990 * q^93 + 1940 * q^94 - 870 * q^95 - 3213 * q^96 + 498 * q^97 - 234 * 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 $

$2$	−1.43845	−0.508568	−0.254284	−	0.967130i	$-0.581840\pi$
$2$	−1.43845	−0.508568	−0.254284	+	0.967130i	$0.581840\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.93087	−0.741359
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−4.31534	−0.293622
$7$	0	0
$7$	0	0
$8$	20.0388	0.885599
$9$	9.00000	0.333333
$10$	−7.19224	−0.227438
$11$	7.61553	0.208743	0.104371	−	0.994538i	$-0.466717\pi$
$11$	7.61553	0.208743	0.104371	+	0.994538i	$0.466717\pi$
$12$	−17.7926	−0.428024
$13$	−52.3542	−1.11696	−0.558478	−	0.829519i	$-0.688615\pi$
$13$	−52.3542	−1.11696	−0.558478	+	0.829519i	$0.688615\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	18.6222	0.290971
$17$	49.7235	0.709395	0.354697	−	0.934981i	$-0.384584\pi$
$17$	49.7235	0.709395	0.354697	+	0.934981i	$0.384584\pi$
$18$	−12.9460	−0.169523
$19$	−140.600	−1.69768	−0.848840	−	0.528649i	$-0.822699\pi$
$19$	−140.600	−1.69768	−0.848840	+	0.528649i	$0.822699\pi$
$20$	−29.6543	−0.331546
$21$	0	0
$22$	−10.9545	−0.106160
$23$	−23.4470	−0.212566	−0.106283	−	0.994336i	$-0.533895\pi$
$23$	−23.4470	−0.212566	−0.106283	+	0.994336i	$0.533895\pi$
$24$	60.1165	0.511301
$25$	25.0000	0.200000
$26$	75.3087	0.568048
$27$	27.0000	0.192450
$28$	0	0
$29$	157.170	1.00641	0.503204	−	0.864168i	$-0.332155\pi$
$29$	157.170	1.00641	0.503204	+	0.864168i	$0.332155\pi$
$30$	−21.5767	−0.131312
$31$	−127.892	−0.740971	−0.370485	−	0.928838i	$-0.620809\pi$
$31$	−127.892	−0.740971	−0.370485	+	0.928838i	$0.620809\pi$
$32$	−187.098	−1.03358
$33$	22.8466	0.120518
$34$	−71.5246	−0.360776
$35$	0	0
$36$	−53.3778	−0.247120
$37$	−115.477	−0.513090	−0.256545	−	0.966532i	$-0.582584\pi$
$37$	−115.477	−0.513090	−0.256545	+	0.966532i	$0.582584\pi$
$38$	202.246	0.863386
$39$	−157.062	−0.644875
$40$	100.194	0.396052
$41$	188.617	0.718466	0.359233	−	0.933248i	$-0.383038\pi$
$41$	188.617	0.718466	0.359233	+	0.933248i	$0.383038\pi$
$42$	0	0
$43$	322.186	1.14262	0.571312	−	0.820733i	$-0.306435\pi$
$43$	322.186	1.14262	0.571312	+	0.820733i	$0.306435\pi$
$44$	−45.1667	−0.154753
$45$	45.0000	0.149071
$46$	33.7272	0.108104
$47$	76.6477	0.237877	0.118938	−	0.992902i	$-0.462051\pi$
$47$	76.6477	0.237877	0.118938	+	0.992902i	$0.462051\pi$
$48$	55.8665	0.167992
$49$	0	0
$50$	−35.9612	−0.101714
$51$	149.170	0.409569
$52$	310.506	0.828065
$53$	−424.172	−1.09933	−0.549666	−	0.835385i	$-0.685245\pi$
$53$	−424.172	−1.09933	−0.549666	+	0.835385i	$0.685245\pi$
$54$	−38.8381	−0.0978739
$55$	38.0776	0.0933525
$56$	0	0
$57$	−421.801	−0.980157
$58$	−226.081	−0.511827
$59$	−107.784	−0.237835	−0.118918	−	0.992904i	$-0.537942\pi$
$59$	−107.784	−0.237835	−0.118918	+	0.992904i	$0.537942\pi$
$60$	−88.9630	−0.191418
$61$	−915.511	−1.92163	−0.960813	−	0.277197i	$-0.910595\pi$
$61$	−915.511	−1.92163	−0.960813	+	0.277197i	$0.910595\pi$
$62$	183.966	0.376834
$63$	0	0
$64$	120.153	0.234673
$65$	−261.771	−0.499518
$66$	−32.8636	−0.0612914
$67$	−451.723	−0.823684	−0.411842	−	0.911255i	$-0.635114\pi$
$67$	−451.723	−0.823684	−0.411842	+	0.911255i	$0.635114\pi$
$68$	−294.903	−0.525916
$69$	−70.3409	−0.122725
$70$	0	0
$71$	907.312	1.51659	0.758297	−	0.651909i	$-0.226032\pi$
$71$	907.312	1.51659	0.758297	+	0.651909i	$0.226032\pi$
$72$	180.349	0.295200
$73$	−755.956	−1.21203	−0.606014	−	0.795454i	$-0.707232\pi$
$73$	−755.956	−1.21203	−0.606014	+	0.795454i	$0.707232\pi$
$74$	166.108	0.260941
$75$	75.0000	0.115470
$76$	833.882	1.25859
$77$	0	0
$78$	225.926	0.327963
$79$	22.5834	0.0321624	0.0160812	−	0.999871i	$-0.494881\pi$
$79$	22.5834	0.0321624	0.0160812	+	0.999871i	$0.494881\pi$
$80$	93.1109	0.130126
$81$	81.0000	0.111111
$82$	−271.316	−0.365389
$83$	−1112.25	−1.47091	−0.735454	−	0.677575i	$-0.763031\pi$
$83$	−1112.25	−1.47091	−0.735454	+	0.677575i	$0.763031\pi$
$84$	0	0
$85$	248.617	0.317251
$86$	−463.447	−0.581102
$87$	471.511	0.581050
$88$	152.606	0.184862
$89$	−1518.81	−1.80892	−0.904458	−	0.426562i	$-0.859725\pi$
$89$	−1518.81	−1.80892	−0.904458	+	0.426562i	$0.859725\pi$
$90$	−64.7301	−0.0758128
$91$	0	0
$92$	139.061	0.157588
$93$	−383.676	−0.427800
$94$	−110.254	−0.120977
$95$	−703.002	−0.759226
$96$	−561.293	−0.596736
$97$	549.987	0.575698	0.287849	−	0.957676i	$-0.407060\pi$
$97$	549.987	0.575698	0.287849	+	0.957676i	$0.407060\pi$
$98$	0	0
$99$	68.5398	0.0695809
$100$	−148.272	−0.148272
$101$	533.299	0.525398	0.262699	−	0.964878i	$-0.415387\pi$
$101$	533.299	0.525398	0.262699	+	0.964878i	$0.415387\pi$
$102$	−214.574	−0.208294
$103$	−1357.79	−1.29890	−0.649451	−	0.760404i	$-0.725001\pi$
$103$	−1357.79	−1.29890	−0.649451	+	0.760404i	$0.725001\pi$
$104$	−1049.12	−0.989176
$105$	0	0
$106$	610.149	0.559084
$107$	913.693	0.825515	0.412757	−	0.910841i	$-0.364566\pi$
$107$	913.693	0.825515	0.412757	+	0.910841i	$0.364566\pi$
$108$	−160.133	−0.142675
$109$	−160.489	−0.141028	−0.0705139	−	0.997511i	$-0.522464\pi$
$109$	−160.489	−0.141028	−0.0705139	+	0.997511i	$0.522464\pi$
$110$	−54.7727	−0.0474761
$111$	−346.432	−0.296233
$112$	0	0
$113$	−1788.48	−1.48891	−0.744453	−	0.667675i	$-0.767290\pi$
$113$	−1788.48	−1.48891	−0.744453	+	0.667675i	$0.767290\pi$
$114$	606.739	0.498476
$115$	−117.235	−0.0950626
$116$	−932.157	−0.746109
$117$	−471.187	−0.372319
$118$	155.042	0.120955
$119$	0	0
$120$	300.582	0.228661
$121$	−1273.00	−0.956427
$122$	1316.91	0.977277
$123$	565.852	0.414806
$124$	758.511	0.549325
$125$	125.000	0.0894427
$126$	0	0
$127$	−271.015	−0.189360	−0.0946799	−	0.995508i	$-0.530183\pi$
$127$	−271.015	−0.189360	−0.0946799	+	0.995508i	$0.530183\pi$
$128$	1323.95	0.914231
$129$	966.557	0.659694
$130$	376.543	0.254039
$131$	763.151	0.508984	0.254492	−	0.967075i	$-0.418092\pi$
$131$	763.151	0.508984	0.254492	+	0.967075i	$0.418092\pi$
$132$	−135.500	−0.0893468
$133$	0	0
$134$	649.780	0.418899
$135$	135.000	0.0860663
$136$	996.400	0.628240
$137$	−240.934	−0.150251	−0.0751254	−	0.997174i	$-0.523936\pi$
$137$	−240.934	−0.150251	−0.0751254	+	0.997174i	$0.523936\pi$
$138$	101.182	0.0624141
$139$	103.150	0.0629427	0.0314714	−	0.999505i	$-0.489981\pi$
$139$	103.150	0.0629427	0.0314714	+	0.999505i	$0.489981\pi$
$140$	0	0
$141$	229.943	0.137338
$142$	−1305.12	−0.771291
$143$	−398.705	−0.233156
$144$	167.600	0.0969905
$145$	785.852	0.450079
$146$	1087.40	0.616398
$147$	0	0
$148$	684.881	0.380384
$149$	−3256.71	−1.79061	−0.895303	−	0.445458i	$-0.853041\pi$
$149$	−3256.71	−1.79061	−0.895303	+	0.445458i	$0.853041\pi$
$150$	−107.884	−0.0587244
$151$	1471.04	0.792793	0.396396	−	0.918080i	$-0.370261\pi$
$151$	1471.04	0.792793	0.396396	+	0.918080i	$0.370261\pi$
$152$	−2817.47	−1.50346
$153$	447.511	0.236465
$154$	0	0
$155$	−639.460	−0.331372
$156$	931.517	0.478084
$157$	1394.22	0.708730	0.354365	−	0.935107i	$-0.384697\pi$
$157$	1394.22	0.708730	0.354365	+	0.935107i	$0.384697\pi$
$158$	−32.4850	−0.0163567
$159$	−1272.52	−0.634699
$160$	−935.488	−0.462230
$161$	0	0
$162$	−116.514	−0.0565075
$163$	−3674.27	−1.76559	−0.882794	−	0.469760i	$-0.844341\pi$
$163$	−3674.27	−1.76559	−0.882794	+	0.469760i	$0.844341\pi$
$164$	−1118.67	−0.532641
$165$	114.233	0.0538971
$166$	1599.91	0.748056
$167$	−4041.09	−1.87251	−0.936254	−	0.351325i	$-0.885731\pi$
$167$	−4041.09	−1.87251	−0.936254	+	0.351325i	$0.885731\pi$
$168$	0	0
$169$	543.958	0.247591
$170$	−357.623	−0.161344
$171$	−1265.40	−0.565894
$172$	−1910.84	−0.847094
$173$	−59.4582	−0.0261302	−0.0130651	−	0.999915i	$-0.504159\pi$
$173$	−59.4582	−0.0261302	−0.0130651	+	0.999915i	$0.504159\pi$
$174$	−678.244	−0.295503
$175$	0	0
$176$	141.818	0.0607381
$177$	−323.352	−0.137314
$178$	2184.73	0.919957
$179$	−2973.32	−1.24155	−0.620773	−	0.783991i	$-0.713181\pi$
$179$	−2973.32	−1.24155	−0.620773	+	0.783991i	$0.713181\pi$
$180$	−266.889	−0.110515
$181$	676.220	0.277696	0.138848	−	0.990314i	$-0.455660\pi$
$181$	676.220	0.277696	0.138848	+	0.990314i	$0.455660\pi$
$182$	0	0
$183$	−2746.53	−1.10945
$184$	−469.849	−0.188249
$185$	−577.386	−0.229461
$186$	551.898	0.217565
$187$	378.671	0.148081
$188$	−454.588	−0.176352
$189$	0	0
$190$	1011.23	0.386118
$191$	16.5589	0.00627308	0.00313654	−	0.999995i	$-0.499002\pi$
$191$	16.5589	0.00627308	0.00313654	+	0.999995i	$0.499002\pi$
$192$	360.458	0.135489
$193$	2694.68	1.00501	0.502506	−	0.864574i	$-0.332411\pi$
$193$	2694.68	1.00501	0.502506	+	0.864574i	$0.332411\pi$
$194$	−791.127	−0.292781
$195$	−785.312	−0.288397
$196$	0	0
$197$	1027.38	0.371561	0.185781	−	0.982591i	$-0.440519\pi$
$197$	1027.38	0.371561	0.185781	+	0.982591i	$0.440519\pi$
$198$	−98.5908	−0.0353866
$199$	−2823.77	−1.00589	−0.502944	−	0.864319i	$-0.667750\pi$
$199$	−2823.77	−1.00589	−0.502944	+	0.864319i	$0.667750\pi$
$200$	500.971	0.177120
$201$	−1355.17	−0.475554
$202$	−767.123	−0.267201
$203$	0	0
$204$	−884.710	−0.303638
$205$	943.087	0.321308
$206$	1953.11	0.660579
$207$	−211.023	−0.0708555
$208$	−974.948	−0.325002
$209$	−1070.75	−0.354378
$210$	0	0
$211$	5151.16	1.68067	0.840333	−	0.542071i	$-0.182360\pi$
$211$	5151.16	1.68067	0.840333	+	0.542071i	$0.182360\pi$
$212$	2515.71	0.814999
$213$	2721.94	0.875606
$214$	−1314.30	−0.419830
$215$	1610.93	0.510997
$216$	541.048	0.170434
$217$	0	0
$218$	230.855	0.0717222
$219$	−2267.87	−0.699764
$220$	−225.834	−0.0692077
$221$	−2603.23	−0.792363
$222$	498.324	0.150655
$223$	114.496	0.0343822	0.0171911	−	0.999852i	$-0.494528\pi$
$223$	114.496	0.0343822	0.0171911	+	0.999852i	$0.494528\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	2572.64	0.757209
$227$	−4744.75	−1.38731	−0.693657	−	0.720306i	$-0.744001\pi$
$227$	−4744.75	−1.38731	−0.693657	+	0.720306i	$0.744001\pi$
$228$	2501.65	0.726648
$229$	5384.47	1.55378	0.776891	−	0.629635i	$-0.216796\pi$
$229$	5384.47	1.55378	0.776891	+	0.629635i	$0.216796\pi$
$230$	168.636	0.0483458
$231$	0	0
$232$	3149.51	0.891274
$233$	−1608.91	−0.452373	−0.226187	−	0.974084i	$-0.572626\pi$
$233$	−1608.91	−0.452373	−0.226187	+	0.974084i	$0.572626\pi$
$234$	677.778	0.189349
$235$	383.239	0.106382
$236$	639.253	0.176321
$237$	67.7501	0.0185689
$238$	0	0
$239$	−3113.11	−0.842554	−0.421277	−	0.906932i	$-0.638418\pi$
$239$	−3113.11	−0.842554	−0.421277	+	0.906932i	$0.638418\pi$
$240$	279.333	0.0751285
$241$	−7136.38	−1.90745	−0.953724	−	0.300685i	$-0.902785\pi$
$241$	−7136.38	−1.90745	−0.953724	+	0.300685i	$0.902785\pi$
$242$	1831.15	0.486408
$243$	243.000	0.0641500
$244$	5429.78	1.42461
$245$	0	0
$246$	−813.948	−0.210957
$247$	7361.01	1.89624
$248$	−2562.81	−0.656203
$249$	−3336.75	−0.849229
$250$	−179.806	−0.0454877
$251$	225.504	0.0567079	0.0283539	−	0.999598i	$-0.490973\pi$
$251$	225.504	0.0567079	0.0283539	+	0.999598i	$0.490973\pi$
$252$	0	0
$253$	−178.561	−0.0443717
$254$	389.841	0.0963024
$255$	745.852	0.183165
$256$	−2865.65	−0.699621
$257$	4423.05	1.07355	0.536775	−	0.843725i	$-0.319642\pi$
$257$	4423.05	1.07355	0.536775	+	0.843725i	$0.319642\pi$
$258$	−1390.34	−0.335499
$259$	0	0
$260$	1552.53	0.370322
$261$	1414.53	0.335469
$262$	−1097.75	−0.258853
$263$	6540.56	1.53349	0.766746	−	0.641950i	$-0.221875\pi$
$263$	6540.56	1.53349	0.766746	+	0.641950i	$0.221875\pi$
$264$	457.819	0.106730
$265$	−2120.86	−0.491636
$266$	0	0
$267$	−4556.43	−1.04438
$268$	2679.11	0.610645
$269$	−2262.97	−0.512920	−0.256460	−	0.966555i	$-0.582556\pi$
$269$	−2262.97	−0.512920	−0.256460	+	0.966555i	$0.582556\pi$
$270$	−194.190	−0.0437706
$271$	−1615.68	−0.362160	−0.181080	−	0.983468i	$-0.557959\pi$
$271$	−1615.68	−0.362160	−0.181080	+	0.983468i	$0.557959\pi$
$272$	925.959	0.206414
$273$	0	0
$274$	346.571	0.0764127
$275$	190.388	0.0417485
$276$	417.183	0.0909835
$277$	4691.55	1.01765	0.508823	−	0.860871i	$-0.330081\pi$
$277$	4691.55	1.01765	0.508823	+	0.860871i	$0.330081\pi$
$278$	−148.375	−0.0320107
$279$	−1151.03	−0.246990
$280$	0	0
$281$	−8119.00	−1.72363	−0.861813	−	0.507226i	$-0.830671\pi$
$281$	−8119.00	−1.72363	−0.861813	+	0.507226i	$0.830671\pi$
$282$	−330.761	−0.0698459
$283$	3633.75	0.763265	0.381632	−	0.924314i	$-0.375362\pi$
$283$	3633.75	0.763265	0.381632	+	0.924314i	$0.375362\pi$
$284$	−5381.15	−1.12434
$285$	−2109.01	−0.438339
$286$	573.515	0.118576
$287$	0	0
$288$	−1683.88	−0.344526
$289$	−2440.58	−0.496759
$290$	−1130.41	−0.228896
$291$	1649.96	0.332379
$292$	4483.48	0.898547
$293$	7981.99	1.59151	0.795756	−	0.605618i	$-0.207074\pi$
$293$	7981.99	1.59151	0.795756	+	0.605618i	$0.207074\pi$
$294$	0	0
$295$	−538.920	−0.106363
$296$	−2314.03	−0.454392
$297$	205.619	0.0401725
$298$	4684.61	0.910645
$299$	1227.55	0.237427
$300$	−444.815	−0.0856047
$301$	0	0
$302$	−2116.02	−0.403189
$303$	1599.90	0.303339
$304$	−2618.28	−0.493977
$305$	−4577.56	−0.859377
$306$	−643.721	−0.120259
$307$	7118.15	1.32330	0.661652	−	0.749811i	$-0.269856\pi$
$307$	7118.15	1.32330	0.661652	+	0.749811i	$0.269856\pi$
$308$	0	0
$309$	−4073.36	−0.749921
$310$	919.830	0.168525
$311$	9155.92	1.66940	0.834702	−	0.550703i	$-0.185640\pi$
$311$	9155.92	1.66940	0.834702	+	0.550703i	$0.185640\pi$
$312$	−3147.35	−0.571101
$313$	6163.44	1.11303	0.556515	−	0.830838i	$-0.312138\pi$
$313$	6163.44	1.11303	0.556515	+	0.830838i	$0.312138\pi$
$314$	−2005.51	−0.360438
$315$	0	0
$316$	−133.939	−0.0238438
$317$	8658.37	1.53408	0.767038	−	0.641601i	$-0.221730\pi$
$317$	8658.37	1.53408	0.767038	+	0.641601i	$0.221730\pi$
$318$	1830.45	0.322788
$319$	1196.94	0.210080
$320$	600.763	0.104949
$321$	2741.08	0.476611
$322$	0	0
$323$	−6991.14	−1.20433
$324$	−480.400	−0.0823732
$325$	−1308.85	−0.223391
$326$	5285.24	0.897922
$327$	−481.466	−0.0814224
$328$	3779.67	0.636272
$329$	0	0
$330$	−164.318	−0.0274103
$331$	−128.477	−0.0213346	−0.0106673	−	0.999943i	$-0.503396\pi$
$331$	−128.477	−0.0213346	−0.0106673	+	0.999943i	$0.503396\pi$
$332$	6596.61	1.09047
$333$	−1039.30	−0.171030
$334$	5812.89	0.952297
$335$	−2258.62	−0.368363
$336$	0	0
$337$	7784.57	1.25832	0.629158	−	0.777277i	$-0.283400\pi$
$337$	7784.57	1.25832	0.629158	+	0.777277i	$0.283400\pi$
$338$	−782.455	−0.125917
$339$	−5365.45	−0.859620
$340$	−1474.52	−0.235197
$341$	−973.965	−0.154672
$342$	1820.22	0.287795
$343$	0	0
$344$	6456.22	1.01191
$345$	−351.704	−0.0548844
$346$	85.5274	0.0132890
$347$	3740.26	0.578638	0.289319	−	0.957233i	$-0.406571\pi$
$347$	3740.26	0.578638	0.289319	+	0.957233i	$0.406571\pi$
$348$	−2796.47	−0.430766
$349$	−5676.86	−0.870703	−0.435351	−	0.900261i	$-0.643376\pi$
$349$	−5676.86	−0.870703	−0.435351	+	0.900261i	$0.643376\pi$
$350$	0	0
$351$	−1413.56	−0.214958
$352$	−1424.85	−0.215752
$353$	−909.564	−0.137142	−0.0685711	−	0.997646i	$-0.521844\pi$
$353$	−909.564	−0.137142	−0.0685711	+	0.997646i	$0.521844\pi$
$354$	465.125	0.0698337
$355$	4536.56	0.678241
$356$	9007.87	1.34106
$357$	0	0
$358$	4276.97	0.631410
$359$	2678.57	0.393788	0.196894	−	0.980425i	$-0.436915\pi$
$359$	2678.57	0.393788	0.196894	+	0.980425i	$0.436915\pi$
$360$	901.747	0.132017
$361$	12909.5	1.88212
$362$	−972.706	−0.141227
$363$	−3819.01	−0.552193
$364$	0	0
$365$	−3779.78	−0.542035
$366$	3950.74	0.564231
$367$	716.898	0.101967	0.0509833	−	0.998700i	$-0.483764\pi$
$367$	716.898	0.101967	0.0509833	+	0.998700i	$0.483764\pi$
$368$	−436.633	−0.0618508
$369$	1697.56	0.239489
$370$	830.540	0.116697
$371$	0	0
$372$	2275.53	0.317153
$373$	−2006.15	−0.278484	−0.139242	−	0.990258i	$-0.544467\pi$
$373$	−2006.15	−0.278484	−0.139242	+	0.990258i	$0.544467\pi$
$374$	−544.698	−0.0753092
$375$	375.000	0.0516398
$376$	1535.93	0.210664
$377$	−8228.53	−1.12411
$378$	0	0
$379$	7277.53	0.986336	0.493168	−	0.869934i	$-0.335839\pi$
$379$	7277.53	0.986336	0.493168	+	0.869934i	$0.335839\pi$
$380$	4169.41	0.562859
$381$	−813.045	−0.109327
$382$	−23.8191	−0.00319029
$383$	−5953.94	−0.794339	−0.397170	−	0.917745i	$-0.630008\pi$
$383$	−5953.94	−0.794339	−0.397170	+	0.917745i	$0.630008\pi$
$384$	3971.84	0.527831
$385$	0	0
$386$	−3876.16	−0.511117
$387$	2899.67	0.380875
$388$	−3261.90	−0.426799
$389$	1867.98	0.243472	0.121736	−	0.992563i	$-0.461154\pi$
$389$	1867.98	0.243472	0.121736	+	0.992563i	$0.461154\pi$
$390$	1129.63	0.146669
$391$	−1165.86	−0.150794
$392$	0	0
$393$	2289.45	0.293862
$394$	−1477.83	−0.188964
$395$	112.917	0.0143834
$396$	−406.500	−0.0515844
$397$	2160.07	0.273075	0.136538	−	0.990635i	$-0.456403\pi$
$397$	2160.07	0.273075	0.136538	+	0.990635i	$0.456403\pi$
$398$	4061.85	0.511563
$399$	0	0
$400$	465.554	0.0581943
$401$	1954.81	0.243438	0.121719	−	0.992565i	$-0.461159\pi$
$401$	1954.81	0.243438	0.121719	+	0.992565i	$0.461159\pi$
$402$	1949.34	0.241851
$403$	6695.68	0.827632
$404$	−3162.93	−0.389509
$405$	405.000	0.0496904
$406$	0	0
$407$	−879.420	−0.107104
$408$	2989.20	0.362714
$409$	−14895.1	−1.80077	−0.900384	−	0.435096i	$-0.856714\pi$
$409$	−14895.1	−1.80077	−0.900384	+	0.435096i	$0.856714\pi$
$410$	−1356.58	−0.163407
$411$	−722.801	−0.0867474
$412$	8052.86	0.962952
$413$	0	0
$414$	303.545	0.0360348
$415$	−5561.25	−0.657810
$416$	9795.34	1.15446
$417$	309.449	0.0363400
$418$	1540.21	0.180225
$419$	−12608.9	−1.47013	−0.735067	−	0.677994i	$-0.762849\pi$
$419$	−12608.9	−1.47013	−0.735067	+	0.677994i	$0.762849\pi$
$420$	0	0
$421$	−7862.86	−0.910243	−0.455122	−	0.890429i	$-0.650404\pi$
$421$	−7862.86	−0.910243	−0.455122	+	0.890429i	$0.650404\pi$
$422$	−7409.67	−0.854732
$423$	689.829	0.0792923
$424$	−8499.91	−0.973567
$425$	1243.09	0.141879
$426$	−3915.36	−0.445305
$427$	0	0
$428$	−5419.00	−0.612002
$429$	−1196.11	−0.134613
$430$	−2317.23	−0.259877
$431$	14291.0	1.59715	0.798575	−	0.601896i	$-0.205588\pi$
$431$	14291.0	1.59715	0.798575	+	0.601896i	$0.205588\pi$
$432$	502.799	0.0559975
$433$	13759.0	1.52705	0.763527	−	0.645776i	$-0.223466\pi$
$433$	13759.0	1.52705	0.763527	+	0.645776i	$0.223466\pi$
$434$	0	0
$435$	2357.56	0.259853
$436$	951.838	0.104552
$437$	3296.65	0.360870
$438$	3262.21	0.355878
$439$	6093.13	0.662436	0.331218	−	0.943554i	$-0.392541\pi$
$439$	6093.13	0.662436	0.331218	+	0.943554i	$0.392541\pi$
$440$	763.031	0.0826729
$441$	0	0
$442$	3744.61	0.402970
$443$	13449.5	1.44244	0.721222	−	0.692704i	$-0.243581\pi$
$443$	13449.5	1.44244	0.721222	+	0.692704i	$0.243581\pi$
$444$	2054.64	0.219615
$445$	−7594.05	−0.808972
$446$	−164.697	−0.0174857
$447$	−9770.14	−1.03381
$448$	0	0
$449$	6893.83	0.724588	0.362294	−	0.932064i	$-0.381994\pi$
$449$	6893.83	0.724588	0.362294	+	0.932064i	$0.381994\pi$
$450$	−323.651	−0.0339045
$451$	1436.42	0.149974
$452$	10607.3	1.10381
$453$	4413.13	0.457719
$454$	6825.07	0.705543
$455$	0	0
$456$	−8452.40	−0.868026
$457$	−11820.1	−1.20990	−0.604948	−	0.796265i	$-0.706806\pi$
$457$	−11820.1	−1.20990	−0.604948	+	0.796265i	$0.706806\pi$
$458$	−7745.28	−0.790203
$459$	1342.53	0.136523
$460$	695.304	0.0704755
$461$	−8443.38	−0.853031	−0.426516	−	0.904480i	$-0.640259\pi$
$461$	−8443.38	−0.853031	−0.426516	+	0.904480i	$0.640259\pi$
$462$	0	0
$463$	−1269.74	−0.127451	−0.0637257	−	0.997967i	$-0.520298\pi$
$463$	−1269.74	−0.127451	−0.0637257	+	0.997967i	$0.520298\pi$
$464$	2926.86	0.292836
$465$	−1918.38	−0.191318
$466$	2314.33	0.230063
$467$	16481.8	1.63316	0.816579	−	0.577233i	$-0.195868\pi$
$467$	16481.8	1.63316	0.816579	+	0.577233i	$0.195868\pi$
$468$	2794.55	0.276022
$469$	0	0
$470$	−551.268	−0.0541024
$471$	4182.65	0.409186
$472$	−2159.87	−0.210627
$473$	2453.61	0.238514
$474$	−97.4549	−0.00944357
$475$	−3515.01	−0.339536
$476$	0	0
$477$	−3817.55	−0.366444
$478$	4478.05	0.428496
$479$	1400.21	0.133564	0.0667822	−	0.997768i	$-0.478727\pi$
$479$	1400.21	0.133564	0.0667822	+	0.997768i	$0.478727\pi$
$480$	−2806.46	−0.266869
$481$	6045.72	0.573100
$482$	10265.3	0.970066
$483$	0	0
$484$	7550.02	0.709055
$485$	2749.93	0.257460
$486$	−349.543	−0.0326246
$487$	−14165.9	−1.31811	−0.659055	−	0.752094i	$-0.729044\pi$
$487$	−14165.9	−1.31811	−0.659055	+	0.752094i	$0.729044\pi$
$488$	−18345.8	−1.70179
$489$	−11022.8	−1.01936
$490$	0	0
$491$	4739.28	0.435603	0.217801	−	0.975993i	$-0.430112\pi$
$491$	4739.28	0.435603	0.217801	+	0.975993i	$0.430112\pi$
$492$	−3356.00	−0.307520
$493$	7815.06	0.713940
$494$	−10588.4	−0.964364
$495$	342.699	0.0311175
$496$	−2381.63	−0.215601
$497$	0	0
$498$	4799.74	0.431890
$499$	−11370.0	−1.02003	−0.510013	−	0.860167i	$-0.670360\pi$
$499$	−11370.0	−1.02003	−0.510013	+	0.860167i	$0.670360\pi$
$500$	−741.359	−0.0663091
$501$	−12123.3	−1.08109
$502$	−324.375	−0.0288398
$503$	−9212.48	−0.816629	−0.408314	−	0.912841i	$-0.633883\pi$
$503$	−9212.48	−0.816629	−0.408314	+	0.912841i	$0.633883\pi$
$504$	0	0
$505$	2666.50	0.234965
$506$	256.851	0.0225660
$507$	1631.87	0.142947
$508$	1607.36	0.140384
$509$	15938.3	1.38792	0.693960	−	0.720014i	$-0.255864\pi$
$509$	15938.3	1.38792	0.693960	+	0.720014i	$0.255864\pi$
$510$	−1072.87	−0.0931518
$511$	0	0
$512$	−6469.49	−0.558426
$513$	−3796.21	−0.326719
$514$	−6362.33	−0.545973
$515$	−6788.94	−0.580886
$516$	−5732.52	−0.489070
$517$	583.713	0.0496550
$518$	0	0
$519$	−178.374	−0.0150863
$520$	−5245.58	−0.442373
$521$	−6442.99	−0.541790	−0.270895	−	0.962609i	$-0.587320\pi$
$521$	−6442.99	−0.541790	−0.270895	+	0.962609i	$0.587320\pi$
$522$	−2034.73	−0.170609
$523$	−986.655	−0.0824922	−0.0412461	−	0.999149i	$-0.513133\pi$
$523$	−986.655	−0.0824922	−0.0412461	+	0.999149i	$0.513133\pi$
$524$	−4526.15	−0.377339
$525$	0	0
$526$	−9408.26	−0.779885
$527$	−6359.24	−0.525641
$528$	425.453	0.0350672
$529$	−11617.2	−0.954815
$530$	3050.75	0.250030
$531$	−970.057	−0.0792785
$532$	0	0
$533$	−9874.91	−0.802495
$534$	6554.19	0.531137
$535$	4568.47	0.369181
$536$	−9052.01	−0.729454
$537$	−8919.97	−0.716807
$538$	3255.16	0.260855
$539$	0	0
$540$	−800.667	−0.0638060
$541$	−12681.1	−1.00777	−0.503885	−	0.863771i	$-0.668097\pi$
$541$	−12681.1	−1.00777	−0.503885	+	0.863771i	$0.668097\pi$
$542$	2324.06	0.184183
$543$	2028.66	0.160328
$544$	−9303.14	−0.733215
$545$	−802.443	−0.0630695
$546$	0	0
$547$	−14826.4	−1.15892	−0.579462	−	0.814999i	$-0.696737\pi$
$547$	−14826.4	−1.15892	−0.579462	+	0.814999i	$0.696737\pi$
$548$	1428.95	0.111390
$549$	−8239.60	−0.640542
$550$	−273.863	−0.0212320
$551$	−22098.2	−1.70856
$552$	−1409.55	−0.108685
$553$	0	0
$554$	−6748.55	−0.517542
$555$	−1732.16	−0.132479
$556$	−611.767	−0.0466632
$557$	1926.46	0.146547	0.0732737	−	0.997312i	$-0.476655\pi$
$557$	1926.46	0.146547	0.0732737	+	0.997312i	$0.476655\pi$
$558$	1655.69	0.125611
$559$	−16867.8	−1.27626
$560$	0	0
$561$	1136.01	0.0854946
$562$	11678.8	0.876581
$563$	18624.8	1.39422	0.697108	−	0.716966i	$-0.254470\pi$
$563$	18624.8	1.39422	0.697108	+	0.716966i	$0.254470\pi$
$564$	−1363.76	−0.101817
$565$	−8942.41	−0.665859
$566$	−5226.96	−0.388172
$567$	0	0
$568$	18181.5	1.34309
$569$	−20093.9	−1.48045	−0.740227	−	0.672357i	$-0.765282\pi$
$569$	−20093.9	−1.48045	−0.740227	+	0.672357i	$0.765282\pi$
$570$	3033.69	0.222925
$571$	4535.25	0.332389	0.166195	−	0.986093i	$-0.446852\pi$
$571$	4535.25	0.332389	0.166195	+	0.986093i	$0.446852\pi$
$572$	2364.66	0.172852
$573$	49.6766	0.00362176
$574$	0	0
$575$	−586.174	−0.0425133
$576$	1081.37	0.0782243
$577$	−10034.6	−0.723994	−0.361997	−	0.932179i	$-0.617905\pi$
$577$	−10034.6	−0.723994	−0.361997	+	0.932179i	$0.617905\pi$
$578$	3510.64	0.252636
$579$	8084.05	0.580244
$580$	−4660.79	−0.333670
$581$	0	0
$582$	−2373.38	−0.169037
$583$	−3230.30	−0.229477
$584$	−15148.5	−1.07337
$585$	−2355.94	−0.166506
$586$	−11481.7	−0.809391
$587$	11192.6	0.786999	0.393499	−	0.919325i	$-0.371264\pi$
$587$	11192.6	0.786999	0.393499	+	0.919325i	$0.371264\pi$
$588$	0	0
$589$	17981.7	1.25793
$590$	775.209	0.0540929
$591$	3082.13	0.214521
$592$	−2150.44	−0.149295
$593$	−20317.6	−1.40699	−0.703493	−	0.710703i	$-0.748377\pi$
$593$	−20317.6	−1.40699	−0.703493	+	0.710703i	$0.748377\pi$
$594$	−295.772	−0.0204305
$595$	0	0
$596$	19315.1	1.32748
$597$	−8471.31	−0.580750
$598$	−1765.76	−0.120748
$599$	−26376.5	−1.79919	−0.899594	−	0.436727i	$-0.856138\pi$
$599$	−26376.5	−1.79919	−0.899594	+	0.436727i	$0.856138\pi$
$600$	1502.91	0.102260
$601$	−9266.13	−0.628907	−0.314454	−	0.949273i	$-0.601821\pi$
$601$	−9266.13	−0.628907	−0.314454	+	0.949273i	$0.601821\pi$
$602$	0	0
$603$	−4065.51	−0.274561
$604$	−8724.56	−0.587744
$605$	−6365.02	−0.427727
$606$	−2301.37	−0.154268
$607$	−11338.0	−0.758149	−0.379075	−	0.925366i	$-0.623758\pi$
$607$	−11338.0	−0.758149	−0.379075	+	0.925366i	$0.623758\pi$
$608$	26306.0	1.75469
$609$	0	0
$610$	6584.57	0.437052
$611$	−4012.83	−0.265698
$612$	−2654.13	−0.175305
$613$	25712.5	1.69416	0.847078	−	0.531469i	$-0.178360\pi$
$613$	25712.5	1.69416	0.847078	+	0.531469i	$0.178360\pi$
$614$	−10239.1	−0.672990
$615$	2829.26	0.185507
$616$	0	0
$617$	663.465	0.0432903	0.0216451	−	0.999766i	$-0.493110\pi$
$617$	663.465	0.0432903	0.0216451	+	0.999766i	$0.493110\pi$
$618$	5859.32	0.381386
$619$	12768.7	0.829108	0.414554	−	0.910025i	$-0.363938\pi$
$619$	12768.7	0.829108	0.414554	+	0.910025i	$0.363938\pi$
$620$	3792.56	0.245666
$621$	−633.068	−0.0409084
$622$	−13170.3	−0.849005
$623$	0	0
$624$	−2924.84	−0.187640
$625$	625.000	0.0400000
$626$	−8865.78	−0.566051
$627$	−3212.24	−0.204600
$628$	−8268.92	−0.525423
$629$	−5741.93	−0.363984
$630$	0	0
$631$	−14937.6	−0.942400	−0.471200	−	0.882026i	$-0.656179\pi$
$631$	−14937.6	−0.942400	−0.471200	+	0.882026i	$0.656179\pi$
$632$	452.544	0.0284829
$633$	15453.5	0.970333
$634$	−12454.6	−0.780182
$635$	−1355.08	−0.0846843
$636$	7547.13	0.470540
$637$	0	0
$638$	−1721.73	−0.106840
$639$	8165.81	0.505531
$640$	6619.74	0.408856
$641$	10903.0	0.671829	0.335914	−	0.941893i	$-0.390955\pi$
$641$	10903.0	0.671829	0.335914	+	0.941893i	$0.390955\pi$
$642$	−3942.90	−0.242389
$643$	7623.47	0.467559	0.233779	−	0.972290i	$-0.424891\pi$
$643$	7623.47	0.467559	0.233779	+	0.972290i	$0.424891\pi$
$644$	0	0
$645$	4832.78	0.295024
$646$	10056.4	0.612482
$647$	−5384.84	−0.327202	−0.163601	−	0.986527i	$-0.552311\pi$
$647$	−5384.84	−0.327202	−0.163601	+	0.986527i	$0.552311\pi$
$648$	1623.14	0.0983999
$649$	−820.833	−0.0496464
$650$	1882.72	0.113610
$651$	0	0
$652$	21791.6	1.30893
$653$	297.318	0.0178177	0.00890883	−	0.999960i	$-0.497164\pi$
$653$	297.318	0.0178177	0.00890883	+	0.999960i	$0.497164\pi$
$654$	692.564	0.0414088
$655$	3815.76	0.227624
$656$	3512.47	0.209053
$657$	−6803.61	−0.404009
$658$	0	0
$659$	10324.7	0.610309	0.305155	−	0.952303i	$-0.401292\pi$
$659$	10324.7	0.610309	0.305155	+	0.952303i	$0.401292\pi$
$660$	−677.501	−0.0399571
$661$	−4272.98	−0.251437	−0.125718	−	0.992066i	$-0.540124\pi$
$661$	−4272.98	−0.251437	−0.125718	+	0.992066i	$0.540124\pi$
$662$	184.808	0.0108501
$663$	−7809.69	−0.457471
$664$	−22288.2	−1.30263
$665$	0	0
$666$	1494.97	0.0869804
$667$	−3685.17	−0.213928
$668$	23967.2	1.38820
$669$	343.488	0.0198506
$670$	3248.90	0.187337
$671$	−6972.10	−0.401125
$672$	0	0
$673$	−23033.4	−1.31927	−0.659637	−	0.751584i	$-0.729290\pi$
$673$	−23033.4	−1.31927	−0.659637	+	0.751584i	$0.729290\pi$
$674$	−11197.7	−0.639940
$675$	675.000	0.0384900
$676$	−3226.15	−0.183554
$677$	−5113.50	−0.290292	−0.145146	−	0.989410i	$-0.546365\pi$
$677$	−5113.50	−0.290292	−0.145146	+	0.989410i	$0.546365\pi$
$678$	7717.91	0.437175
$679$	0	0
$680$	4982.00	0.280957
$681$	−14234.2	−0.800966
$682$	1401.00	0.0786613
$683$	1341.02	0.0751286	0.0375643	−	0.999294i	$-0.488040\pi$
$683$	1341.02	0.0751286	0.0375643	+	0.999294i	$0.488040\pi$
$684$	7504.94	0.419530
$685$	−1204.67	−0.0671942
$686$	0	0
$687$	16153.4	0.897076
$688$	5999.80	0.332471
$689$	22207.2	1.22790
$690$	505.908	0.0279125
$691$	16809.3	0.925404	0.462702	−	0.886514i	$-0.346880\pi$
$691$	16809.3	0.925404	0.462702	+	0.886514i	$0.346880\pi$
$692$	352.639	0.0193718
$693$	0	0
$694$	−5380.16	−0.294277
$695$	515.748	0.0281489
$696$	9448.53	0.514577
$697$	9378.71	0.509676
$698$	8165.86	0.442812
$699$	−4826.72	−0.261178
$700$	0	0
$701$	13467.0	0.725592	0.362796	−	0.931869i	$-0.381822\pi$
$701$	13467.0	0.725592	0.362796	+	0.931869i	$0.381822\pi$
$702$	2033.33	0.109321
$703$	16236.1	0.871064
$704$	915.025	0.0489863
$705$	1149.72	0.0614196
$706$	1308.36	0.0697461
$707$	0	0
$708$	1917.76	0.101799
$709$	35514.3	1.88119	0.940597	−	0.339525i	$-0.110266\pi$
$709$	35514.3	1.88119	0.940597	+	0.339525i	$0.110266\pi$
$710$	−6525.61	−0.344932
$711$	203.250	0.0107208
$712$	−30435.2	−1.60198
$713$	2998.68	0.157506
$714$	0	0
$715$	−1993.52	−0.104271
$716$	17634.4	0.920431
$717$	−9339.34	−0.486449
$718$	−3852.99	−0.200268
$719$	−4993.61	−0.259013	−0.129506	−	0.991579i	$-0.541339\pi$
$719$	−4993.61	−0.259013	−0.129506	+	0.991579i	$0.541339\pi$
$720$	837.998	0.0433755
$721$	0	0
$722$	−18569.6	−0.957186
$723$	−21409.1	−1.10127
$724$	−4010.57	−0.205872
$725$	3929.26	0.201281
$726$	5493.45	0.280828
$727$	4223.35	0.215454	0.107727	−	0.994181i	$-0.465643\pi$
$727$	4223.35	0.215454	0.107727	+	0.994181i	$0.465643\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	5437.02	0.275662
$731$	16020.2	0.810572
$732$	16289.3	0.822502
$733$	−19030.9	−0.958968	−0.479484	−	0.877551i	$-0.659176\pi$
$733$	−19030.9	−0.958968	−0.479484	+	0.877551i	$0.659176\pi$
$734$	−1031.22	−0.0518570
$735$	0	0
$736$	4386.87	0.219704
$737$	−3440.11	−0.171938
$738$	−2441.85	−0.121796
$739$	−27772.5	−1.38245	−0.691224	−	0.722641i	$-0.742928\pi$
$739$	−27772.5	−1.38245	−0.691224	+	0.722641i	$0.742928\pi$
$740$	3424.40	0.170113
$741$	22083.0	1.09479
$742$	0	0
$743$	26880.7	1.32726	0.663631	−	0.748060i	$-0.269015\pi$
$743$	26880.7	1.32726	0.663631	+	0.748060i	$0.269015\pi$
$744$	−7688.42	−0.378859
$745$	−16283.6	−0.800783
$746$	2885.74	0.141628
$747$	−10010.2	−0.490302
$748$	−2245.85	−0.109781
$749$	0	0
$750$	−539.418	−0.0262623
$751$	−35166.6	−1.70872	−0.854360	−	0.519681i	$-0.826051\pi$
$751$	−35166.6	−1.70872	−0.854360	+	0.519681i	$0.826051\pi$
$752$	1427.35	0.0692154
$753$	676.512	0.0327403
$754$	11836.3	0.571688
$755$	7355.21	0.354548
$756$	0	0
$757$	14589.2	0.700467	0.350233	−	0.936662i	$-0.386102\pi$
$757$	14589.2	0.700467	0.350233	+	0.936662i	$0.386102\pi$
$758$	−10468.3	−0.501619
$759$	−535.683	−0.0256180
$760$	−14087.3	−0.672370
$761$	782.826	0.0372897	0.0186448	−	0.999826i	$-0.494065\pi$
$761$	782.826	0.0372897	0.0186448	+	0.999826i	$0.494065\pi$
$762$	1169.52	0.0556002
$763$	0	0
$764$	−98.2085	−0.00465060
$765$	2237.56	0.105750
$766$	8564.42	0.403975
$767$	5642.95	0.265652
$768$	−8596.95	−0.403927
$769$	16548.4	0.776008	0.388004	−	0.921658i	$-0.373165\pi$
$769$	16548.4	0.776008	0.388004	+	0.921658i	$0.373165\pi$
$770$	0	0
$771$	13269.2	0.619815
$772$	−15981.8	−0.745075
$773$	−5744.94	−0.267310	−0.133655	−	0.991028i	$-0.542671\pi$
$773$	−5744.94	−0.267310	−0.133655	+	0.991028i	$0.542671\pi$
$774$	−4171.02	−0.193701
$775$	−3197.30	−0.148194
$776$	11021.1	0.509837
$777$	0	0
$778$	−2687.00	−0.123822
$779$	−26519.7	−1.21973
$780$	4657.59	0.213806
$781$	6909.66	0.316578
$782$	1677.03	0.0766888
$783$	4243.60	0.193683
$784$	0	0
$785$	6971.09	0.316954
$786$	−3293.26	−0.149449
$787$	34744.3	1.57370	0.786848	−	0.617146i	$-0.211711\pi$
$787$	34744.3	1.57370	0.786848	+	0.617146i	$0.211711\pi$
$788$	−6093.24	−0.275460
$789$	19621.7	0.885362
$790$	−162.425	−0.00731496
$791$	0	0
$792$	1373.46	0.0616207
$793$	47930.8	2.14637
$794$	−3107.15	−0.138877
$795$	−6362.58	−0.283846
$796$	16747.4	0.745724
$797$	−21748.7	−0.966600	−0.483300	−	0.875455i	$-0.660562\pi$
$797$	−21748.7	−0.966600	−0.483300	+	0.875455i	$0.660562\pi$
$798$	0	0
$799$	3811.19	0.168749
$800$	−4677.44	−0.206716
$801$	−13669.3	−0.602972
$802$	−2811.89	−0.123805
$803$	−5757.01	−0.253002
$804$	8037.34	0.352556
$805$	0	0
$806$	−9631.38	−0.420907
$807$	−6788.90	−0.296134
$808$	10686.7	0.465292
$809$	−42350.6	−1.84050	−0.920252	−	0.391325i	$-0.872017\pi$
$809$	−42350.6	−1.84050	−0.920252	+	0.391325i	$0.872017\pi$
$810$	−582.571	−0.0252709
$811$	18910.7	0.818796	0.409398	−	0.912356i	$-0.365739\pi$
$811$	18910.7	0.818796	0.409398	+	0.912356i	$0.365739\pi$
$812$	0	0
$813$	−4847.03	−0.209093
$814$	1265.00	0.0544696
$815$	−18371.3	−0.789595
$816$	2777.88	0.119173
$817$	−45299.4	−1.93981
$818$	21425.8	0.915813
$819$	0	0
$820$	−5593.33	−0.238204
$821$	6593.42	0.280283	0.140141	−	0.990132i	$-0.455244\pi$
$821$	6593.42	0.280283	0.140141	+	0.990132i	$0.455244\pi$
$822$	1039.71	0.0441169
$823$	26762.4	1.13351	0.566755	−	0.823886i	$-0.308199\pi$
$823$	26762.4	1.13351	0.566755	+	0.823886i	$0.308199\pi$
$824$	−27208.5	−1.15031
$825$	571.165	0.0241035
$826$	0	0
$827$	−24016.7	−1.00985	−0.504924	−	0.863164i	$-0.668479\pi$
$827$	−24016.7	−1.00985	−0.504924	+	0.863164i	$0.668479\pi$
$828$	1251.55	0.0525293
$829$	28422.3	1.19077	0.595383	−	0.803442i	$-0.297000\pi$
$829$	28422.3	1.19077	0.595383	+	0.803442i	$0.297000\pi$
$830$	7999.56	0.334541
$831$	14074.7	0.587538
$832$	−6290.49	−0.262120
$833$	0	0
$834$	−445.126	−0.0184814
$835$	−20205.4	−0.837411
$836$	6350.46	0.262721
$837$	−3453.09	−0.142600
$838$	18137.3	0.747663
$839$	−6637.09	−0.273108	−0.136554	−	0.990633i	$-0.543603\pi$
$839$	−6637.09	−0.273108	−0.136554	+	0.990633i	$0.543603\pi$
$840$	0	0
$841$	313.546	0.0128560
$842$	11310.3	0.462921
$843$	−24357.0	−0.995136
$844$	−30550.9	−1.24598
$845$	2719.79	0.110726
$846$	−992.283	−0.0403255
$847$	0	0
$848$	−7899.01	−0.319874
$849$	10901.2	0.440671
$850$	−1788.11	−0.0721551
$851$	2707.59	0.109066
$852$	−16143.5	−0.649138
$853$	1406.88	0.0564720	0.0282360	−	0.999601i	$-0.491011\pi$
$853$	1406.88	0.0564720	0.0282360	+	0.999601i	$0.491011\pi$
$854$	0	0
$855$	−6327.02	−0.253075
$856$	18309.3	0.731075
$857$	−27943.4	−1.11380	−0.556901	−	0.830579i	$-0.688010\pi$
$857$	−27943.4	−1.11380	−0.556901	+	0.830579i	$0.688010\pi$
$858$	1720.55	0.0684598
$859$	−1936.63	−0.0769233	−0.0384616	−	0.999260i	$-0.512246\pi$
$859$	−1936.63	−0.0769233	−0.0384616	+	0.999260i	$0.512246\pi$
$860$	−9554.20	−0.378832
$861$	0	0
$862$	−20556.8	−0.812259
$863$	7947.70	0.313491	0.156746	−	0.987639i	$-0.449900\pi$
$863$	7947.70	0.313491	0.156746	+	0.987639i	$0.449900\pi$
$864$	−5051.63	−0.198912
$865$	−297.291	−0.0116858
$866$	−19791.6	−0.776611
$867$	−7321.73	−0.286804
$868$	0	0
$869$	171.984	0.00671365
$870$	−3391.22	−0.132153
$871$	23649.6	0.920019
$872$	−3216.00	−0.124894
$873$	4949.88	0.191899
$874$	−4742.06	−0.183527
$875$	0	0
$876$	13450.4	0.518776
$877$	38655.0	1.48835	0.744177	−	0.667983i	$-0.232842\pi$
$877$	38655.0	1.48835	0.744177	+	0.667983i	$0.232842\pi$
$878$	−8764.65	−0.336893
$879$	23946.0	0.918859
$880$	709.088	0.0271629
$881$	18879.4	0.721978	0.360989	−	0.932570i	$-0.382439\pi$
$881$	18879.4	0.721978	0.360989	+	0.932570i	$0.382439\pi$
$882$	0	0
$883$	−35098.1	−1.33765	−0.668825	−	0.743420i	$-0.733202\pi$
$883$	−35098.1	−1.33765	−0.668825	+	0.743420i	$0.733202\pi$
$884$	15439.4	0.587425
$885$	−1616.76	−0.0614089
$886$	−19346.3	−0.733581
$887$	48816.4	1.84791	0.923954	−	0.382504i	$-0.124938\pi$
$887$	48816.4	1.84791	0.923954	+	0.382504i	$0.124938\pi$
$888$	−6942.08	−0.262344
$889$	0	0
$890$	10923.6	0.411417
$891$	616.858	0.0231936
$892$	−679.062	−0.0254895
$893$	−10776.7	−0.403839
$894$	14053.8	0.525761
$895$	−14866.6	−0.555236
$896$	0	0
$897$	3682.64	0.137079
$898$	−9916.41	−0.368502
$899$	−20100.8	−0.745718
$900$	−1334.45	−0.0494239
$901$	−21091.3	−0.779860
$902$	−2066.22	−0.0762721
$903$	0	0
$904$	−35839.1	−1.31857
$905$	3381.10	0.124190
$906$	−6348.05	−0.232781
$907$	−6010.83	−0.220051	−0.110026	−	0.993929i	$-0.535093\pi$
$907$	−6010.83	−0.220051	−0.110026	+	0.993929i	$0.535093\pi$
$908$	28140.5	1.02850
$909$	4799.69	0.175133
$910$	0	0
$911$	25780.1	0.937576	0.468788	−	0.883311i	$-0.344691\pi$
$911$	25780.1	0.937576	0.468788	+	0.883311i	$0.344691\pi$
$912$	−7854.85	−0.285198
$913$	−8470.37	−0.307041
$914$	17002.6	0.615314
$915$	−13732.7	−0.496162
$916$	−31934.6	−1.15191
$917$	0	0
$918$	−1931.16	−0.0694313
$919$	26731.8	0.959522	0.479761	−	0.877399i	$-0.340723\pi$
$919$	26731.8	0.959522	0.479761	+	0.877399i	$0.340723\pi$
$920$	−2349.25	−0.0841874
$921$	21354.4	0.764010
$922$	12145.4	0.433824
$923$	−47501.6	−1.69397
$924$	0	0
$925$	−2886.93	−0.102618
$926$	1826.46	0.0648177
$927$	−12220.1	−0.432967
$928$	−29406.2	−1.04020
$929$	30464.6	1.07590	0.537949	−	0.842977i	$-0.319199\pi$
$929$	30464.6	1.07590	0.537949	+	0.842977i	$0.319199\pi$
$930$	2759.49	0.0972981
$931$	0	0
$932$	9542.22	0.335371
$933$	27467.7	0.963830
$934$	−23708.1	−0.830572
$935$	1893.35	0.0662238
$936$	−9442.04	−0.329725
$937$	28533.4	0.994819	0.497409	−	0.867516i	$-0.334285\pi$
$937$	28533.4	0.994819	0.497409	+	0.867516i	$0.334285\pi$
$938$	0	0
$939$	18490.3	0.642608
$940$	−2272.94	−0.0788671
$941$	−34455.8	−1.19365	−0.596827	−	0.802370i	$-0.703572\pi$
$941$	−34455.8	−1.19365	−0.596827	+	0.802370i	$0.703572\pi$
$942$	−6016.52	−0.208099
$943$	−4422.50	−0.152722
$944$	−2007.17	−0.0692033
$945$	0	0
$946$	−3529.39	−0.121301
$947$	2477.68	0.0850198	0.0425099	−	0.999096i	$-0.486465\pi$
$947$	2477.68	0.0850198	0.0425099	+	0.999096i	$0.486465\pi$
$948$	−401.817	−0.0137662
$949$	39577.5	1.35378
$950$	5056.16	0.172677
$951$	25975.1	0.885699
$952$	0	0
$953$	41690.6	1.41709	0.708547	−	0.705663i	$-0.249351\pi$
$953$	41690.6	1.41709	0.708547	+	0.705663i	$0.249351\pi$
$954$	5491.35	0.186361
$955$	82.7943	0.00280541
$956$	18463.5	0.624635
$957$	3590.81	0.121290
$958$	−2014.13	−0.0679266
$959$	0	0
$960$	1802.29	0.0605923
$961$	−13434.6	−0.450962
$962$	−8696.44	−0.291460
$963$	8223.24	0.275172
$964$	42325.0	1.41410
$965$	13473.4	0.449455
$966$	0	0
$967$	−20641.3	−0.686430	−0.343215	−	0.939257i	$-0.611516\pi$
$967$	−20641.3	−0.686430	−0.343215	+	0.939257i	$0.611516\pi$
$968$	−25509.5	−0.847010
$969$	−20973.4	−0.695318
$970$	−3955.63	−0.130936
$971$	−6626.17	−0.218995	−0.109497	−	0.993987i	$-0.534924\pi$
$971$	−6626.17	−0.218995	−0.109497	+	0.993987i	$0.534924\pi$
$972$	−1441.20	−0.0475582
$973$	0	0
$974$	20377.0	0.670349
$975$	−3926.56	−0.128975
$976$	−17048.8	−0.559138
$977$	41961.0	1.37405	0.687027	−	0.726632i	$-0.258915\pi$
$977$	41961.0	1.37405	0.687027	+	0.726632i	$0.258915\pi$
$978$	15855.7	0.518415
$979$	−11566.5	−0.377598
$980$	0	0
$981$	−1444.40	−0.0470093
$982$	−6817.21	−0.221533
$983$	16781.7	0.544510	0.272255	−	0.962225i	$-0.412231\pi$
$983$	16781.7	0.544510	0.272255	+	0.962225i	$0.412231\pi$
$984$	11339.0	0.367352
$985$	5136.88	0.166167
$986$	−11241.6	−0.363087
$987$	0	0
$988$	−43657.2	−1.40579
$989$	−7554.27	−0.242884
$990$	−492.954	−0.0158254
$991$	50319.8	1.61298	0.806489	−	0.591250i	$-0.201365\pi$
$991$	50319.8	1.61298	0.806489	+	0.591250i	$0.201365\pi$
$992$	23928.3	0.765851
$993$	−385.432	−0.0123176
$994$	0	0
$995$	−14118.9	−0.449847
$996$	19789.8	0.629583
$997$	−12949.5	−0.411348	−0.205674	−	0.978621i	$-0.565939\pi$
$997$	−12949.5	−0.411348	−0.205674	+	0.978621i	$0.565939\pi$
$998$	16355.2	0.518753
$999$	−3117.89	−0.0987443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.k.1.2		2
3.2	odd	2		2205.4.a.bh.1.1		2
7.6	odd	2		105.4.a.c.1.2	✓	2
21.20	even	2		315.4.a.m.1.1		2
28.27	even	2		1680.4.a.bk.1.1		2
35.13	even	4		525.4.d.i.274.3		4
35.27	even	4		525.4.d.i.274.2		4
35.34	odd	2		525.4.a.p.1.1		2
105.104	even	2		1575.4.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.c.1.2	✓	2	7.6	odd	2
315.4.a.m.1.1		2	21.20	even	2
525.4.a.p.1.1		2	35.34	odd	2
525.4.d.i.274.2		4	35.27	even	4
525.4.d.i.274.3		4	35.13	even	4
735.4.a.k.1.2		2	1.1	even	1	trivial
1575.4.a.m.1.2		2	105.104	even	2
1680.4.a.bk.1.1		2	28.27	even	2
2205.4.a.bh.1.1		2	3.2	odd	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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43845 q^{2} +3.00000 q^{3} -5.93087 q^{4} +5.00000 q^{5} -4.31534 q^{6} +20.0388 q^{8} +9.00000 q^{9} -7.19224 q^{10} +7.61553 q^{11} -17.7926 q^{12} -52.3542 q^{13} +15.0000 q^{15} +18.6222 q^{16} +49.7235 q^{17} -12.9460 q^{18} -140.600 q^{19} -29.6543 q^{20} -10.9545 q^{22} -23.4470 q^{23} +60.1165 q^{24} +25.0000 q^{25} +75.3087 q^{26} +27.0000 q^{27} +157.170 q^{29} -21.5767 q^{30} -127.892 q^{31} -187.098 q^{32} +22.8466 q^{33} -71.5246 q^{34} -53.3778 q^{36} -115.477 q^{37} +202.246 q^{38} -157.062 q^{39} +100.194 q^{40} +188.617 q^{41} +322.186 q^{43} -45.1667 q^{44} +45.0000 q^{45} +33.7272 q^{46} +76.6477 q^{47} +55.8665 q^{48} -35.9612 q^{50} +149.170 q^{51} +310.506 q^{52} -424.172 q^{53} -38.8381 q^{54} +38.0776 q^{55} -421.801 q^{57} -226.081 q^{58} -107.784 q^{59} -88.9630 q^{60} -915.511 q^{61} +183.966 q^{62} +120.153 q^{64} -261.771 q^{65} -32.8636 q^{66} -451.723 q^{67} -294.903 q^{68} -70.3409 q^{69} +907.312 q^{71} +180.349 q^{72} -755.956 q^{73} +166.108 q^{74} +75.0000 q^{75} +833.882 q^{76} +225.926 q^{78} +22.5834 q^{79} +93.1109 q^{80} +81.0000 q^{81} -271.316 q^{82} -1112.25 q^{83} +248.617 q^{85} -463.447 q^{86} +471.511 q^{87} +152.606 q^{88} -1518.81 q^{89} -64.7301 q^{90} +139.061 q^{92} -383.676 q^{93} -110.254 q^{94} -703.002 q^{95} -561.293 q^{96} +549.987 q^{97} +68.5398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 7 q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} - 21 q^{6} - 63 q^{8} + 18 q^{9} - 35 q^{10} - 26 q^{11} + 51 q^{12} - 14 q^{13} + 30 q^{15} + 297 q^{16} - 16 q^{17} - 63 q^{18} - 174 q^{19} + 85 q^{20} + 176 q^{22}+ \cdots - 234 q^{99}+O(q^{100}) \) 2 * q - 7 * q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 - 21 * q^6 - 63 * q^8 + 18 * q^9 - 35 * q^10 - 26 * q^11 + 51 * q^12 - 14 * q^13 + 30 * q^15 + 297 * q^16 - 16 * q^17 - 63 * q^18 - 174 * q^19 + 85 * q^20 + 176 * q^22 + 184 * q^23 - 189 * q^24 + 50 * q^25 - 138 * q^26 + 54 * q^27 - 32 * q^29 - 105 * q^30 - 330 * q^31 - 1071 * q^32 - 78 * q^33 + 294 * q^34 + 153 * q^36 - 132 * q^37 + 388 * q^38 - 42 * q^39 - 315 * q^40 - 200 * q^41 + 364 * q^43 - 816 * q^44 + 90 * q^45 - 1120 * q^46 - 292 * q^47 + 891 * q^48 - 175 * q^50 - 48 * q^51 + 1190 * q^52 + 34 * q^53 - 189 * q^54 - 130 * q^55 - 522 * q^57 + 826 * q^58 - 364 * q^59 + 255 * q^60 - 792 * q^61 + 1308 * q^62 + 2809 * q^64 - 70 * q^65 + 528 * q^66 - 788 * q^67 - 1802 * q^68 + 552 * q^69 + 454 * q^71 - 567 * q^72 - 778 * q^73 + 258 * q^74 + 150 * q^75 + 68 * q^76 - 414 * q^78 + 408 * q^79 + 1485 * q^80 + 162 * q^81 + 1890 * q^82 - 1136 * q^83 - 80 * q^85 - 696 * q^86 - 96 * q^87 + 2944 * q^88 - 36 * q^89 - 315 * q^90 + 4896 * q^92 - 990 * q^93 + 1940 * q^94 - 870 * q^95 - 3213 * q^96 + 498 * q^97 - 234 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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