LMFDB - Embedded newform 175.4.b.f.99.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [175,4,Mod(99,175)] 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("175.99"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-72,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 64x^{6} + 1264x^{4} + 8905x^{2} + 14400 $ x^8 + 64x^6 + 1264x^4 + 8905*x^2 + 14400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.8
Root			$5.87199i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.f.99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.87199i q^{2} -4.14916i q^{3} -15.7363 q^{4} +20.2147 q^{6} -7.00000i q^{7} -37.6910i q^{8} +9.78444 q^{9} +36.9922 q^{11} +65.2924i q^{12} -61.3165i q^{13} +34.1039 q^{14} +57.7401 q^{16} +44.8345i q^{17} +47.6697i q^{18} +139.701 q^{19} -29.0441 q^{21} +180.226i q^{22} +217.580i q^{23} -156.386 q^{24} +298.733 q^{26} -152.625i q^{27} +110.154i q^{28} +33.8226 q^{29} +124.437 q^{31} -20.2192i q^{32} -153.487i q^{33} -218.433 q^{34} -153.971 q^{36} -237.270i q^{37} +680.620i q^{38} -254.412 q^{39} +195.117 q^{41} -141.503i q^{42} -343.725i q^{43} -582.119 q^{44} -1060.05 q^{46} -16.8224i q^{47} -239.573i q^{48} -49.0000 q^{49} +186.026 q^{51} +964.894i q^{52} +346.965i q^{53} +743.586 q^{54} -263.837 q^{56} -579.641i q^{57} +164.783i q^{58} -135.340 q^{59} +490.414 q^{61} +606.255i q^{62} -68.4911i q^{63} +560.428 q^{64} +747.785 q^{66} -477.969i q^{67} -705.529i q^{68} +902.777 q^{69} +45.2557 q^{71} -368.786i q^{72} +100.781i q^{73} +1155.98 q^{74} -2198.37 q^{76} -258.945i q^{77} -1239.49i q^{78} -880.534 q^{79} -369.085 q^{81} +950.609i q^{82} -1155.03i q^{83} +457.047 q^{84} +1674.62 q^{86} -140.336i q^{87} -1394.27i q^{88} -619.374 q^{89} -429.216 q^{91} -3423.90i q^{92} -516.309i q^{93} +81.9585 q^{94} -83.8929 q^{96} +231.195i q^{97} -238.727i q^{98} +361.948 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9} + 200 q^{11} - 56 q^{14} + 320 q^{16} + 58 q^{19} - 42 q^{21} + 42 q^{24} + 1400 q^{26} - 258 q^{29} + 228 q^{31} - 406 q^{34} + 2202 q^{36} - 1348 q^{39} + 1342 q^{41}+ \cdots - 4766 q^{99}+O(q^{100}) $ 8 * q - 72 * q^4 + 2 * q^6 - 122 * q^9 + 200 * q^11 - 56 * q^14 + 320 * q^16 + 58 * q^19 - 42 * q^21 + 42 * q^24 + 1400 * q^26 - 258 * q^29 + 228 * q^31 - 406 * q^34 + 2202 * q^36 - 1348 * q^39 + 1342 * q^41 - 876 * q^44 - 1994 * q^46 - 392 * q^49 - 1770 * q^51 + 5554 * q^54 + 378 * q^56 - 2036 * q^59 + 100 * q^61 + 4842 * q^64 - 7682 * q^66 - 2160 * q^69 + 430 * q^71 - 1246 * q^74 - 6514 * q^76 + 1902 * q^79 + 56 * q^81 + 2310 * q^84 - 198 * q^86 - 5638 * q^89 + 616 * q^91 + 6112 * q^94 - 2690 * q^96 - 4766 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.87199i	1.72251i	0.508175	+	0.861254i	$0.330320\pi$
$2$	4.87199i	1.72251i	−0.508175	+	0.861254i	$0.669680\pi$
$3$	− 4.14916i	− 0.798507i	−0.916841	−	0.399253i	$-0.869269\pi$
$3$	− 4.14916i	− 0.798507i	0.916841	−	0.399253i	$-0.130731\pi$
$4$	−15.7363	−1.96703
$5$	0	0
$5$	0	0
$6$	20.2147	1.37543
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	− 37.6910i	− 1.66572i
$9$	9.78444	0.362387
$10$	0	0
$11$	36.9922	1.01396	0.506980	−	0.861958i	$-0.330762\pi$
$11$	36.9922	1.01396	0.506980	+	0.861958i	$0.330762\pi$
$12$	65.2924i	1.57069i
$13$	− 61.3165i	− 1.30817i	−0.756423	−	0.654083i	$-0.773055\pi$
$13$	− 61.3165i	− 1.30817i	0.756423	−	0.654083i	$-0.226945\pi$
$14$	34.1039	0.651047
$15$	0	0
$16$	57.7401	0.902189
$17$	44.8345i	0.639646i	0.947477	+	0.319823i	$0.103623\pi$
$17$	44.8345i	0.639646i	−0.947477	+	0.319823i	$0.896377\pi$
$18$	47.6697i	0.624214i
$19$	139.701	1.68682	0.843408	−	0.537273i	$-0.180545\pi$
$19$	139.701	1.68682	0.843408	+	0.537273i	$0.180545\pi$
$20$	0	0
$21$	−29.0441	−0.301807
$22$	180.226i	1.74656i
$23$	217.580i	1.97255i	0.165110	+	0.986275i	$0.447202\pi$
$23$	217.580i	1.97255i	−0.165110	+	0.986275i	$0.552798\pi$
$24$	−156.386	−1.33009
$25$	0	0
$26$	298.733	2.25333
$27$	− 152.625i	− 1.08788i
$28$	110.154i	0.743469i
$29$	33.8226	0.216576	0.108288	−	0.994120i	$-0.465463\pi$
$29$	33.8226	0.216576	0.108288	+	0.994120i	$0.465463\pi$
$30$	0	0
$31$	124.437	0.720952	0.360476	−	0.932769i	$-0.382614\pi$
$31$	124.437	0.720952	0.360476	+	0.932769i	$0.382614\pi$
$32$	− 20.2192i	− 0.111697i
$33$	− 153.487i	− 0.809655i
$34$	−218.433	−1.10179
$35$	0	0
$36$	−153.971	−0.712827
$37$	− 237.270i	− 1.05424i	−0.849790	−	0.527121i	$-0.823271\pi$
$37$	− 237.270i	− 1.05424i	0.849790	−	0.527121i	$-0.176729\pi$
$38$	680.620i	2.90556i
$39$	−254.412	−1.04458
$40$	0	0
$41$	195.117	0.743224	0.371612	−	0.928388i	$-0.378805\pi$
$41$	195.117	0.743224	0.371612	+	0.928388i	$0.378805\pi$
$42$	− 141.503i	− 0.519865i
$43$	− 343.725i	− 1.21901i	−0.792781	−	0.609506i	$-0.791368\pi$
$43$	− 343.725i	− 1.21901i	0.792781	−	0.609506i	$-0.208632\pi$
$44$	−582.119	−1.99450
$45$	0	0
$46$	−1060.05	−3.39773
$47$	− 16.8224i	− 0.0522085i	−0.999659	−	0.0261042i	$-0.991690\pi$
$47$	− 16.8224i	− 0.0522085i	0.999659	−	0.0261042i	$-0.00831018\pi$
$48$	− 239.573i	− 0.720404i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	186.026	0.510761
$52$	964.894i	2.57321i
$53$	346.965i	0.899231i	0.893222	+	0.449616i	$0.148439\pi$
$53$	346.965i	0.899231i	−0.893222	+	0.449616i	$0.851561\pi$
$54$	743.586	1.87387
$55$	0	0
$56$	−263.837	−0.629584
$57$	− 579.641i	− 1.34694i
$58$	164.783i	0.373054i
$59$	−135.340	−0.298640	−0.149320	−	0.988789i	$-0.547708\pi$
$59$	−135.340	−0.298640	−0.149320	+	0.988789i	$0.547708\pi$
$60$	0	0
$61$	490.414	1.02936	0.514681	−	0.857382i	$-0.327911\pi$
$61$	490.414	1.02936	0.514681	+	0.857382i	$0.327911\pi$
$62$	606.255i	1.24185i
$63$	− 68.4911i	− 0.136969i
$64$	560.428	1.09459
$65$	0	0
$66$	747.785	1.39464
$67$	− 477.969i	− 0.871540i	−0.900058	−	0.435770i	$-0.856476\pi$
$67$	− 477.969i	− 0.871540i	0.900058	−	0.435770i	$-0.143524\pi$
$68$	− 705.529i	− 1.25820i
$69$	902.777	1.57510
$70$	0	0
$71$	45.2557	0.0756460	0.0378230	−	0.999284i	$-0.487958\pi$
$71$	45.2557	0.0756460	0.0378230	+	0.999284i	$0.487958\pi$
$72$	− 368.786i	− 0.603636i
$73$	100.781i	0.161583i	0.996731	+	0.0807913i	$0.0257447\pi$
$73$	100.781i	0.161583i	−0.996731	+	0.0807913i	$0.974255\pi$
$74$	1155.98	1.81594
$75$	0	0
$76$	−2198.37	−3.31803
$77$	− 258.945i	− 0.383241i
$78$	− 1239.49i	− 1.79930i
$79$	−880.534	−1.25402	−0.627012	−	0.779010i	$-0.715722\pi$
$79$	−880.534	−1.25402	−0.627012	+	0.779010i	$0.715722\pi$
$80$	0	0
$81$	−369.085	−0.506289
$82$	950.609i	1.28021i
$83$	− 1155.03i	− 1.52748i	−0.645525	−	0.763739i	$-0.723362\pi$
$83$	− 1155.03i	− 1.52748i	0.645525	−	0.763739i	$-0.276638\pi$
$84$	457.047	0.593665
$85$	0	0
$86$	1674.62	2.09976
$87$	− 140.336i	− 0.172937i
$88$	− 1394.27i	− 1.68898i
$89$	−619.374	−0.737680	−0.368840	−	0.929493i	$-0.620245\pi$
$89$	−619.374	−0.737680	−0.368840	+	0.929493i	$0.620245\pi$
$90$	0	0
$91$	−429.216	−0.494440
$92$	− 3423.90i	− 3.88007i
$93$	− 516.309i	− 0.575685i
$94$	81.9585	0.0899295
$95$	0	0
$96$	−83.8929	−0.0891904
$97$	231.195i	0.242003i	0.992652	+	0.121001i	$0.0386105\pi$
$97$	231.195i	0.242003i	−0.992652	+	0.121001i	$0.961389\pi$
$98$	− 238.727i	− 0.246073i
$99$	361.948	0.367446
$100$	0	0
$101$	1875.99	1.84820	0.924101	−	0.382149i	$-0.124816\pi$
$101$	1875.99	1.84820	0.924101	+	0.382149i	$0.124816\pi$
$102$	906.316i	0.879791i
$103$	1855.94i	1.77545i	0.460377	+	0.887723i	$0.347714\pi$
$103$	1855.94i	1.77545i	−0.460377	+	0.887723i	$0.652286\pi$
$104$	−2311.08	−2.17904
$105$	0	0
$106$	−1690.41	−1.54893
$107$	218.016i	0.196976i	0.995138	+	0.0984879i	$0.0314006\pi$
$107$	218.016i	0.196976i	−0.995138	+	0.0984879i	$0.968599\pi$
$108$	2401.74i	2.13989i
$109$	−847.217	−0.744484	−0.372242	−	0.928136i	$-0.621411\pi$
$109$	−847.217	−0.744484	−0.372242	+	0.928136i	$0.621411\pi$
$110$	0	0
$111$	−984.473	−0.841820
$112$	− 404.181i	− 0.340995i
$113$	1430.05i	1.19051i	0.803537	+	0.595254i	$0.202949\pi$
$113$	1430.05i	1.19051i	−0.803537	+	0.595254i	$0.797051\pi$
$114$	2824.00	2.32011
$115$	0	0
$116$	−532.242	−0.426012
$117$	− 599.948i	− 0.474062i
$118$	− 659.374i	− 0.514409i
$119$	313.842	0.241763
$120$	0	0
$121$	37.4229	0.0281164
$122$	2389.29i	1.77308i
$123$	− 809.573i	− 0.593470i
$124$	−1958.17	−1.41814
$125$	0	0
$126$	333.688	0.235931
$127$	1732.84i	1.21075i	0.795941	+	0.605374i	$0.206976\pi$
$127$	1732.84i	1.21075i	−0.795941	+	0.605374i	$0.793024\pi$
$128$	2568.65i	1.77374i
$129$	−1426.17	−0.973390
$130$	0	0
$131$	−2429.76	−1.62052	−0.810262	−	0.586068i	$-0.800675\pi$
$131$	−2429.76	−1.62052	−0.810262	+	0.586068i	$0.800675\pi$
$132$	2415.31i	1.59262i
$133$	− 977.904i	− 0.637557i
$134$	2328.66	1.50123
$135$	0	0
$136$	1689.86	1.06547
$137$	− 1452.84i	− 0.906018i	−0.891506	−	0.453009i	$-0.850351\pi$
$137$	− 1452.84i	− 0.906018i	0.891506	−	0.453009i	$-0.149649\pi$
$138$	4398.32i	2.71311i
$139$	−2927.47	−1.78637	−0.893183	−	0.449693i	$-0.851533\pi$
$139$	−2927.47	−1.78637	−0.893183	+	0.449693i	$0.851533\pi$
$140$	0	0
$141$	−69.7989	−0.0416888
$142$	220.485i	0.130301i
$143$	− 2268.23i	− 1.32643i
$144$	564.954	0.326941
$145$	0	0
$146$	−491.004	−0.278327
$147$	203.309i	0.114072i
$148$	3733.75i	2.07373i
$149$	−901.724	−0.495786	−0.247893	−	0.968787i	$-0.579738\pi$
$149$	−901.724	−0.495786	−0.247893	+	0.968787i	$0.579738\pi$
$150$	0	0
$151$	−1357.63	−0.731669	−0.365834	−	0.930680i	$-0.619216\pi$
$151$	−1357.63	−0.731669	−0.365834	+	0.930680i	$0.619216\pi$
$152$	− 5265.46i	− 2.80977i
$153$	438.681i	0.231799i
$154$	1261.58	0.660136
$155$	0	0
$156$	4003.50	2.05472
$157$	2318.72i	1.17869i	0.807883	+	0.589343i	$0.200613\pi$
$157$	2318.72i	1.17869i	−0.807883	+	0.589343i	$0.799387\pi$
$158$	− 4289.95i	− 2.16006i
$159$	1439.61	0.718042
$160$	0	0
$161$	1523.06	0.745554
$162$	− 1798.18i	− 0.872087i
$163$	− 1577.42i	− 0.757996i	−0.925397	−	0.378998i	$-0.876269\pi$
$163$	− 1577.42i	− 0.757996i	0.925397	−	0.378998i	$-0.123731\pi$
$164$	−3070.42	−1.46195
$165$	0	0
$166$	5627.28	2.63109
$167$	− 1038.02i	− 0.480982i	−0.970651	−	0.240491i	$-0.922691\pi$
$167$	− 1038.02i	− 0.480982i	0.970651	−	0.240491i	$-0.0773085\pi$
$168$	1094.70i	0.502728i
$169$	−1562.72	−0.711296
$170$	0	0
$171$	1366.89	0.611280
$172$	5408.95i	2.39784i
$173$	− 482.641i	− 0.212107i	−0.994360	−	0.106054i	$-0.966178\pi$
$173$	− 482.641i	− 0.212107i	0.994360	−	0.106054i	$-0.0338215\pi$
$174$	683.714	0.297886
$175$	0	0
$176$	2135.93	0.914784
$177$	561.547i	0.238466i
$178$	− 3017.58i	− 1.27066i
$179$	−2407.19	−1.00515	−0.502575	−	0.864534i	$-0.667614\pi$
$179$	−2407.19	−1.00515	−0.502575	+	0.864534i	$0.667614\pi$
$180$	0	0
$181$	532.600	0.218717	0.109359	−	0.994002i	$-0.465120\pi$
$181$	532.600	0.218717	0.109359	+	0.994002i	$0.465120\pi$
$182$	− 2091.13i	− 0.851677i
$183$	− 2034.81i	− 0.821952i
$184$	8200.83	3.28572
$185$	0	0
$186$	2515.45	0.991622
$187$	1658.53i	0.648575i
$188$	264.722i	0.102696i
$189$	−1068.37	−0.411178
$190$	0	0
$191$	336.675	0.127544	0.0637721	−	0.997964i	$-0.479687\pi$
$191$	336.675	0.127544	0.0637721	+	0.997964i	$0.479687\pi$
$192$	− 2325.31i	− 0.874035i
$193$	22.7366i	0.00847988i	0.999991	+	0.00423994i	$0.00134962\pi$
$193$	22.7366i	0.00847988i	−0.999991	+	0.00423994i	$0.998650\pi$
$194$	−1126.38	−0.416852
$195$	0	0
$196$	771.077	0.281005
$197$	1085.33i	0.392520i	0.980552	+	0.196260i	$0.0628796\pi$
$197$	1085.33i	0.392520i	−0.980552	+	0.196260i	$0.937120\pi$
$198$	1763.41i	0.632928i
$199$	2630.28	0.936963	0.468482	−	0.883473i	$-0.344801\pi$
$199$	2630.28	0.936963	0.468482	+	0.883473i	$0.344801\pi$
$200$	0	0
$201$	−1983.17	−0.695931
$202$	9139.82i	3.18354i
$203$	− 236.758i	− 0.0818580i
$204$	−2927.35	−1.00469
$205$	0	0
$206$	−9042.11	−3.05822
$207$	2128.90i	0.714826i
$208$	− 3540.42i	− 1.18021i
$209$	5167.83	1.71037
$210$	0	0
$211$	614.178	0.200388	0.100194	−	0.994968i	$-0.468054\pi$
$211$	614.178	0.200388	0.100194	+	0.994968i	$0.468054\pi$
$212$	− 5459.93i	− 1.76882i
$213$	− 187.773i	− 0.0604039i
$214$	−1062.17	−0.339292
$215$	0	0
$216$	−5752.58	−1.81210
$217$	− 871.057i	− 0.272494i
$218$	− 4127.63i	− 1.28238i
$219$	418.157	0.129025
$220$	0	0
$221$	2749.10	0.836762
$222$	− 4796.34i	− 1.45004i
$223$	1930.30i	0.579651i	0.957079	+	0.289826i	$0.0935973\pi$
$223$	1930.30i	0.579651i	−0.957079	+	0.289826i	$0.906403\pi$
$224$	−141.535	−0.0422173
$225$	0	0
$226$	−6967.17	−2.05066
$227$	− 765.365i	− 0.223784i	−0.993720	−	0.111892i	$-0.964309\pi$
$227$	− 765.365i	− 0.223784i	0.993720	−	0.111892i	$-0.0356911\pi$
$228$	9121.39i	2.64947i
$229$	5712.66	1.64849	0.824243	−	0.566237i	$-0.191601\pi$
$229$	5712.66	1.64849	0.824243	+	0.566237i	$0.191601\pi$
$230$	0	0
$231$	−1074.41	−0.306021
$232$	− 1274.81i	− 0.360756i
$233$	− 864.418i	− 0.243047i	−0.992589	−	0.121523i	$-0.961222\pi$
$233$	− 864.418i	− 0.243047i	0.992589	−	0.121523i	$-0.0387779\pi$
$234$	2922.94	0.816575
$235$	0	0
$236$	2129.74	0.587435
$237$	3653.48i	1.00135i
$238$	1529.03i	0.416439i
$239$	1816.64	0.491669	0.245834	−	0.969312i	$-0.420938\pi$
$239$	1816.64	0.491669	0.245834	+	0.969312i	$0.420938\pi$
$240$	0	0
$241$	−2354.53	−0.629330	−0.314665	−	0.949203i	$-0.601892\pi$
$241$	−2354.53	−0.629330	−0.314665	+	0.949203i	$0.601892\pi$
$242$	182.324i	0.0484307i
$243$	− 2589.47i	− 0.683600i
$244$	−7717.28	−2.02479
$245$	0	0
$246$	3944.23	1.02226
$247$	− 8565.96i	− 2.20664i
$248$	− 4690.15i	− 1.20091i
$249$	−4792.39	−1.21970
$250$	0	0
$251$	−2983.66	−0.750306	−0.375153	−	0.926963i	$-0.622410\pi$
$251$	−2983.66	−0.750306	−0.375153	+	0.926963i	$0.622410\pi$
$252$	1077.79i	0.269423i
$253$	8048.78i	2.00009i
$254$	−8442.39	−2.08552
$255$	0	0
$256$	−8030.99	−1.96069
$257$	5121.50i	1.24307i	0.783385	+	0.621537i	$0.213492\pi$
$257$	5121.50i	1.24307i	−0.783385	+	0.621537i	$0.786508\pi$
$258$	− 6948.29i	− 1.67667i
$259$	−1660.89	−0.398466
$260$	0	0
$261$	330.935	0.0784843
$262$	− 11837.7i	− 2.79137i
$263$	4663.66i	1.09344i	0.837317	+	0.546718i	$0.184123\pi$
$263$	4663.66i	1.09344i	−0.837317	+	0.546718i	$0.815877\pi$
$264$	−5785.07	−1.34866
$265$	0	0
$266$	4764.34	1.09820
$267$	2569.88i	0.589043i
$268$	7521.45i	1.71435i
$269$	−5018.86	−1.13757	−0.568783	−	0.822488i	$-0.692586\pi$
$269$	−5018.86	−1.13757	−0.568783	+	0.822488i	$0.692586\pi$
$270$	0	0
$271$	−2512.66	−0.563222	−0.281611	−	0.959529i	$-0.590869\pi$
$271$	−2512.66	−0.563222	−0.281611	+	0.959529i	$0.590869\pi$
$272$	2588.75i	0.577081i
$273$	1780.89i	0.394814i
$274$	7078.21	1.56062
$275$	0	0
$276$	−14206.3	−3.09827
$277$	− 6286.82i	− 1.36368i	−0.731503	−	0.681839i	$-0.761181\pi$
$277$	− 6286.82i	− 1.36368i	0.731503	−	0.681839i	$-0.238819\pi$
$278$	− 14262.6i	− 3.07703i
$279$	1217.54	0.261263
$280$	0	0
$281$	8804.33	1.86912	0.934560	−	0.355807i	$-0.115794\pi$
$281$	8804.33	1.86912	0.934560	+	0.355807i	$0.115794\pi$
$282$	− 340.059i	− 0.0718093i
$283$	485.298i	0.101936i	0.998700	+	0.0509681i	$0.0162307\pi$
$283$	485.298i	0.101936i	−0.998700	+	0.0509681i	$0.983769\pi$
$284$	−712.157	−0.148798
$285$	0	0
$286$	11050.8	2.28478
$287$	− 1365.82i	− 0.280912i
$288$	− 197.834i	− 0.0404773i
$289$	2902.86	0.590854
$290$	0	0
$291$	959.265	0.193241
$292$	− 1585.92i	− 0.317838i
$293$	4004.48i	0.798444i	0.916854	+	0.399222i	$0.130720\pi$
$293$	4004.48i	0.798444i	−0.916854	+	0.399222i	$0.869280\pi$
$294$	−990.519	−0.196491
$295$	0	0
$296$	−8942.96	−1.75608
$297$	− 5645.92i	− 1.10306i
$298$	− 4393.19i	− 0.853995i
$299$	13341.3	2.58042
$300$	0	0
$301$	−2406.07	−0.460743
$302$	− 6614.33i	− 1.26031i
$303$	− 7783.81i	− 1.47580i
$304$	8066.32	1.52183
$305$	0	0
$306$	−2137.25	−0.399276
$307$	− 3154.13i	− 0.586370i	−0.956056	−	0.293185i	$-0.905285\pi$
$307$	− 3154.13i	− 0.586370i	0.956056	−	0.293185i	$-0.0947152\pi$
$308$	4074.84i	0.753848i
$309$	7700.59	1.41771
$310$	0	0
$311$	−1738.29	−0.316944	−0.158472	−	0.987363i	$-0.550657\pi$
$311$	−1738.29	−0.316944	−0.158472	+	0.987363i	$0.550657\pi$
$312$	9589.06i	1.73998i
$313$	5092.30i	0.919597i	0.888023	+	0.459799i	$0.152078\pi$
$313$	5092.30i	0.919597i	−0.888023	+	0.459799i	$0.847922\pi$
$314$	−11296.8	−2.03030
$315$	0	0
$316$	13856.3	2.46671
$317$	4874.67i	0.863686i	0.901949	+	0.431843i	$0.142137\pi$
$317$	4874.67i	0.863686i	−0.901949	+	0.431843i	$0.857863\pi$
$318$	7013.78i	1.23683i
$319$	1251.17	0.219600
$320$	0	0
$321$	904.584	0.157287
$322$	7420.35i	1.28422i
$323$	6263.41i	1.07896i
$324$	5808.02	0.995888
$325$	0	0
$326$	7685.19	1.30565
$327$	3515.24i	0.594475i
$328$	− 7354.17i	− 1.23801i
$329$	−117.757	−0.0197329
$330$	0	0
$331$	−2448.46	−0.406585	−0.203292	−	0.979118i	$-0.565164\pi$
$331$	−2448.46	−0.406585	−0.203292	+	0.979118i	$0.565164\pi$
$332$	18175.8i	3.00460i
$333$	− 2321.56i	− 0.382043i
$334$	5057.20	0.828496
$335$	0	0
$336$	−1677.01	−0.272287
$337$	6271.50i	1.01374i	0.862023	+	0.506870i	$0.169197\pi$
$337$	6271.50i	1.01374i	−0.862023	+	0.506870i	$0.830803\pi$
$338$	− 7613.54i	− 1.22521i
$339$	5933.50	0.950629
$340$	0	0
$341$	4603.19	0.731017
$342$	6659.48i	1.05293i
$343$	343.000i	0.0539949i
$344$	−12955.3	−2.03054
$345$	0	0
$346$	2351.42	0.365356
$347$	− 417.338i	− 0.0645645i	−0.999479	−	0.0322822i	$-0.989722\pi$
$347$	− 417.338i	− 0.0645645i	0.999479	−	0.0322822i	$-0.0102775\pi$
$348$	2208.36i	0.340174i
$349$	−5971.08	−0.915829	−0.457915	−	0.888996i	$-0.651403\pi$
$349$	−5971.08	−0.915829	−0.457915	+	0.888996i	$0.651403\pi$
$350$	0	0
$351$	−9358.42	−1.42312
$352$	− 747.954i	− 0.113256i
$353$	− 12012.6i	− 1.81124i	−0.424089	−	0.905620i	$-0.639406\pi$
$353$	− 12012.6i	− 1.81124i	0.424089	−	0.905620i	$-0.360594\pi$
$354$	−2735.85	−0.410759
$355$	0	0
$356$	9746.64	1.45104
$357$	− 1302.18i	− 0.193050i
$358$	− 11727.8i	− 1.73138i
$359$	−6312.46	−0.928020	−0.464010	−	0.885830i	$-0.653590\pi$
$359$	−6312.46	−0.928020	−0.464010	+	0.885830i	$0.653590\pi$
$360$	0	0
$361$	12657.3	1.84535
$362$	2594.82i	0.376743i
$363$	− 155.274i	− 0.0224511i
$364$	6754.26	0.972580
$365$	0	0
$366$	9913.55	1.41582
$367$	− 12228.6i	− 1.73932i	−0.493652	−	0.869659i	$-0.664338\pi$
$367$	− 12228.6i	− 1.73932i	0.493652	−	0.869659i	$-0.335662\pi$
$368$	12563.1i	1.77961i
$369$	1909.11	0.269335
$370$	0	0
$371$	2428.75	0.339877
$372$	8124.77i	1.13239i
$373$	7876.24i	1.09334i	0.837348	+	0.546671i	$0.184105\pi$
$373$	7876.24i	1.09334i	−0.837348	+	0.546671i	$0.815895\pi$
$374$	−8080.33	−1.11718
$375$	0	0
$376$	−634.053	−0.0869649
$377$	− 2073.89i	− 0.283317i
$378$	− 5205.10i	− 0.708258i
$379$	−5992.79	−0.812214	−0.406107	−	0.913826i	$-0.633114\pi$
$379$	−5992.79	−0.812214	−0.406107	+	0.913826i	$0.633114\pi$
$380$	0	0
$381$	7189.85	0.966790
$382$	1640.28i	0.219696i
$383$	7373.61i	0.983743i	0.870668	+	0.491872i	$0.163687\pi$
$383$	7373.61i	0.983743i	−0.870668	+	0.491872i	$0.836313\pi$
$384$	10657.7	1.41634
$385$	0	0
$386$	−110.772	−0.0146067
$387$	− 3363.15i	− 0.441754i
$388$	− 3638.14i	− 0.476028i
$389$	−1896.88	−0.247238	−0.123619	−	0.992330i	$-0.539450\pi$
$389$	−1896.88	−0.247238	−0.123619	+	0.992330i	$0.539450\pi$
$390$	0	0
$391$	−9755.12	−1.26173
$392$	1846.86i	0.237961i
$393$	10081.5i	1.29400i
$394$	−5287.71	−0.676119
$395$	0	0
$396$	−5695.71	−0.722778
$397$	791.156i	0.100018i	0.998749	+	0.0500088i	$0.0159249\pi$
$397$	791.156i	0.100018i	−0.998749	+	0.0500088i	$0.984075\pi$
$398$	12814.7i	1.61393i
$399$	−4057.49	−0.509094
$400$	0	0
$401$	4410.55	0.549258	0.274629	−	0.961550i	$-0.411445\pi$
$401$	4410.55	0.549258	0.274629	+	0.961550i	$0.411445\pi$
$402$	− 9661.99i	− 1.19875i
$403$	− 7630.03i	− 0.943124i
$404$	−29521.1	−3.63548
$405$	0	0
$406$	1153.48	0.141001
$407$	− 8777.14i	− 1.06896i
$408$	− 7011.51i	− 0.850787i
$409$	4403.73	0.532397	0.266199	−	0.963918i	$-0.414232\pi$
$409$	4403.73	0.532397	0.266199	+	0.963918i	$0.414232\pi$
$410$	0	0
$411$	−6028.07	−0.723461
$412$	− 29205.5i	− 3.49236i
$413$	947.379i	0.112875i
$414$	−10372.0	−1.23129
$415$	0	0
$416$	−1239.77	−0.146117
$417$	12146.6i	1.42643i
$418$	25177.6i	2.94612i
$419$	14272.6	1.66411	0.832057	−	0.554691i	$-0.187163\pi$
$419$	14272.6	1.66411	0.832057	+	0.554691i	$0.187163\pi$
$420$	0	0
$421$	−15530.4	−1.79788	−0.898939	−	0.438074i	$-0.855661\pi$
$421$	−15530.4	−1.79788	−0.898939	+	0.438074i	$0.855661\pi$
$422$	2992.27i	0.345169i
$423$	− 164.598i	− 0.0189197i
$424$	13077.5	1.49787
$425$	0	0
$426$	914.830	0.104046
$427$	− 3432.90i	− 0.389062i
$428$	− 3430.76i	− 0.387458i
$429$	−9411.27	−1.05916
$430$	0	0
$431$	−5260.75	−0.587938	−0.293969	−	0.955815i	$-0.594976\pi$
$431$	−5260.75	−0.587938	−0.293969	+	0.955815i	$0.594976\pi$
$432$	− 8812.56i	− 0.981469i
$433$	6610.65i	0.733690i	0.930282	+	0.366845i	$0.119562\pi$
$433$	6610.65i	0.733690i	−0.930282	+	0.366845i	$0.880438\pi$
$434$	4243.78	0.469373
$435$	0	0
$436$	13332.0	1.46442
$437$	30396.1i	3.32733i
$438$	2037.26i	0.222246i
$439$	8870.46	0.964383	0.482191	−	0.876066i	$-0.339841\pi$
$439$	8870.46	0.964383	0.482191	+	0.876066i	$0.339841\pi$
$440$	0	0
$441$	−479.438	−0.0517695
$442$	13393.6i	1.44133i
$443$	− 13221.8i	− 1.41802i	−0.705196	−	0.709012i	$-0.749141\pi$
$443$	− 13221.8i	− 1.41802i	0.705196	−	0.709012i	$-0.250859\pi$
$444$	15491.9	1.65589
$445$	0	0
$446$	−9404.38	−0.998454
$447$	3741.40i	0.395889i
$448$	− 3923.00i	− 0.413715i
$449$	−4192.70	−0.440681	−0.220341	−	0.975423i	$-0.570717\pi$
$449$	−4192.70	−0.440681	−0.220341	+	0.975423i	$0.570717\pi$
$450$	0	0
$451$	7217.82	0.753600
$452$	− 22503.6i	− 2.34177i
$453$	5633.01i	0.584243i
$454$	3728.85	0.385470
$455$	0	0
$456$	−21847.3	−2.24362
$457$	4701.78i	0.481270i	0.970616	+	0.240635i	$0.0773556\pi$
$457$	4701.78i	0.481270i	−0.970616	+	0.240635i	$0.922644\pi$
$458$	27832.0i	2.83953i
$459$	6842.86	0.695855
$460$	0	0
$461$	−1949.82	−0.196989	−0.0984946	−	0.995138i	$-0.531403\pi$
$461$	−1949.82	−0.196989	−0.0984946	+	0.995138i	$0.531403\pi$
$462$	− 5234.50i	− 0.527123i
$463$	7315.85i	0.734333i	0.930155	+	0.367166i	$0.119672\pi$
$463$	7315.85i	0.734333i	−0.930155	+	0.367166i	$0.880328\pi$
$464$	1952.92	0.195392
$465$	0	0
$466$	4211.43	0.418650
$467$	1897.54i	0.188025i	0.995571	+	0.0940126i	$0.0299694\pi$
$467$	1897.54i	0.188025i	−0.995571	+	0.0940126i	$0.970031\pi$
$468$	9440.94i	0.932495i
$469$	−3345.78	−0.329411
$470$	0	0
$471$	9620.74	0.941189
$472$	5101.10i	0.497451i
$473$	− 12715.1i	− 1.23603i
$474$	−17799.7	−1.72483
$475$	0	0
$476$	−4938.70	−0.475557
$477$	3394.85i	0.325869i
$478$	8850.66i	0.846903i
$479$	−10534.4	−1.00486	−0.502429	−	0.864619i	$-0.667560\pi$
$479$	−10534.4	−1.00486	−0.502429	+	0.864619i	$0.667560\pi$
$480$	0	0
$481$	−14548.6	−1.37912
$482$	− 11471.2i	− 1.08403i
$483$	− 6319.44i	− 0.595330i
$484$	−588.897	−0.0553058
$485$	0	0
$486$	12615.9	1.17751
$487$	− 11937.0i	− 1.11071i	−0.831614	−	0.555355i	$-0.812583\pi$
$487$	− 11937.0i	− 1.11071i	0.831614	−	0.555355i	$-0.187417\pi$
$488$	− 18484.2i	− 1.71463i
$489$	−6544.99	−0.605265
$490$	0	0
$491$	100.060	0.00919683	0.00459841	−	0.999989i	$-0.498536\pi$
$491$	100.060	0.00919683	0.00459841	+	0.999989i	$0.498536\pi$
$492$	12739.7i	1.16737i
$493$	1516.42i	0.138532i
$494$	41733.3	3.80095
$495$	0	0
$496$	7184.99	0.650435
$497$	− 316.790i	− 0.0285915i
$498$	− 23348.5i	− 2.10095i
$499$	15611.7	1.40055	0.700275	−	0.713874i	$-0.253061\pi$
$499$	15611.7	1.40055	0.700275	+	0.713874i	$0.253061\pi$
$500$	0	0
$501$	−4306.90	−0.384068
$502$	− 14536.3i	− 1.29241i
$503$	2270.39i	0.201256i	0.994924	+	0.100628i	$0.0320852\pi$
$503$	2270.39i	0.201256i	−0.994924	+	0.100628i	$0.967915\pi$
$504$	−2581.50	−0.228153
$505$	0	0
$506$	−39213.6	−3.44517
$507$	6483.97i	0.567975i
$508$	− 27268.5i	− 2.38158i
$509$	2579.79	0.224651	0.112325	−	0.993671i	$-0.464170\pi$
$509$	2579.79	0.224651	0.112325	+	0.993671i	$0.464170\pi$
$510$	0	0
$511$	705.467	0.0610725
$512$	− 18577.7i	− 1.60357i
$513$	− 21321.8i	− 1.83505i
$514$	−24951.9	−2.14121
$515$	0	0
$516$	22442.6	1.91469
$517$	− 622.297i	− 0.0529373i
$518$	− 8091.84i	− 0.686361i
$519$	−2002.56	−0.169369
$520$	0	0
$521$	12793.9	1.07584	0.537918	−	0.842997i	$-0.319211\pi$
$521$	12793.9	1.07584	0.537918	+	0.842997i	$0.319211\pi$
$522$	1612.31i	0.135190i
$523$	− 11273.4i	− 0.942548i	−0.881987	−	0.471274i	$-0.843794\pi$
$523$	− 11273.4i	− 0.942548i	0.881987	−	0.471274i	$-0.156206\pi$
$524$	38235.3	3.18763
$525$	0	0
$526$	−22721.3	−1.88345
$527$	5579.07i	0.461154i
$528$	− 8862.33i	− 0.730461i
$529$	−35174.2	−2.89095
$530$	0	0
$531$	−1324.22	−0.108223
$532$	15388.6i	1.25410i
$533$	− 11963.9i	− 0.972260i
$534$	−12520.4	−1.01463
$535$	0	0
$536$	−18015.1	−1.45175
$537$	9987.83i	0.802619i
$538$	− 24451.8i	− 1.95947i
$539$	−1812.62	−0.144852
$540$	0	0
$541$	10282.9	0.817183	0.408591	−	0.912717i	$-0.366020\pi$
$541$	10282.9	0.817183	0.408591	+	0.912717i	$0.366020\pi$
$542$	− 12241.6i	− 0.970154i
$543$	− 2209.85i	− 0.174647i
$544$	906.520	0.0714462
$545$	0	0
$546$	−8676.46	−0.680070
$547$	2160.94i	0.168912i	0.996427	+	0.0844562i	$0.0269153\pi$
$547$	2160.94i	0.168912i	−0.996427	+	0.0844562i	$0.973085\pi$
$548$	22862.3i	1.78217i
$549$	4798.42	0.373027
$550$	0	0
$551$	4725.04	0.365324
$552$	− 34026.6i	− 2.62367i
$553$	6163.74i	0.473976i
$554$	30629.3	2.34895
$555$	0	0
$556$	46067.5	3.51384
$557$	− 15648.4i	− 1.19038i	−0.803585	−	0.595190i	$-0.797077\pi$
$557$	− 15648.4i	− 1.19038i	0.803585	−	0.595190i	$-0.202923\pi$
$558$	5931.86i	0.450028i
$559$	−21076.0	−1.59467
$560$	0	0
$561$	6881.51	0.517892
$562$	42894.6i	3.21957i
$563$	− 21951.6i	− 1.64325i	−0.570031	−	0.821623i	$-0.693069\pi$
$563$	− 21951.6i	− 1.64325i	0.570031	−	0.821623i	$-0.306931\pi$
$564$	1098.37	0.0820033
$565$	0	0
$566$	−2364.36	−0.175586
$567$	2583.59i	0.191359i
$568$	− 1705.74i	− 0.126005i
$569$	−14019.2	−1.03289	−0.516445	−	0.856321i	$-0.672745\pi$
$569$	−14019.2	−1.03289	−0.516445	+	0.856321i	$0.672745\pi$
$570$	0	0
$571$	−21706.3	−1.59086	−0.795430	−	0.606045i	$-0.792755\pi$
$571$	−21706.3	−1.59086	−0.795430	+	0.606045i	$0.792755\pi$
$572$	35693.5i	2.60913i
$573$	− 1396.92i	− 0.101845i
$574$	6654.26	0.483874
$575$	0	0
$576$	5483.48	0.396664
$577$	− 12569.0i	− 0.906853i	−0.891294	−	0.453427i	$-0.850201\pi$
$577$	− 12569.0i	− 0.906853i	0.891294	−	0.453427i	$-0.149799\pi$
$578$	14142.7i	1.01775i
$579$	94.3379	0.00677124
$580$	0	0
$581$	−8085.19	−0.577332
$582$	4673.53i	0.332859i
$583$	12835.0i	0.911785i
$584$	3798.54	0.269152
$585$	0	0
$586$	−19509.8	−1.37533
$587$	1330.20i	0.0935322i	0.998906	+	0.0467661i	$0.0148915\pi$
$587$	1330.20i	0.0935322i	−0.998906	+	0.0467661i	$0.985108\pi$
$588$	− 3199.33i	− 0.224384i
$589$	17383.9	1.21611
$590$	0	0
$591$	4503.20	0.313430
$592$	− 13700.0i	− 0.951126i
$593$	9866.91i	0.683281i	0.939831	+	0.341640i	$0.110982\pi$
$593$	9866.91i	0.683281i	−0.939831	+	0.341640i	$0.889018\pi$
$594$	27506.9	1.90003
$595$	0	0
$596$	14189.8	0.975228
$597$	− 10913.5i	− 0.748172i
$598$	64998.6i	4.44480i
$599$	−4185.17	−0.285478	−0.142739	−	0.989760i	$-0.545591\pi$
$599$	−4185.17	−0.285478	−0.142739	+	0.989760i	$0.545591\pi$
$600$	0	0
$601$	10599.8	0.719426	0.359713	−	0.933063i	$-0.382875\pi$
$601$	10599.8	0.719426	0.359713	+	0.933063i	$0.382875\pi$
$602$	− 11722.4i	− 0.793634i
$603$	− 4676.66i	− 0.315834i
$604$	21364.0	1.43922
$605$	0	0
$606$	37922.6	2.54208
$607$	13290.1i	0.888679i	0.895858	+	0.444340i	$0.146562\pi$
$607$	13290.1i	0.888679i	−0.895858	+	0.444340i	$0.853438\pi$
$608$	− 2824.64i	− 0.188412i
$609$	−982.349	−0.0653642
$610$	0	0
$611$	−1031.49	−0.0682973
$612$	− 6903.20i	− 0.455957i
$613$	21327.1i	1.40521i	0.711579	+	0.702606i	$0.247980\pi$
$613$	21327.1i	1.40521i	−0.711579	+	0.702606i	$0.752020\pi$
$614$	15366.9	1.01003
$615$	0	0
$616$	−9759.92	−0.638374
$617$	− 13741.6i	− 0.896622i	−0.893878	−	0.448311i	$-0.852026\pi$
$617$	− 13741.6i	− 0.896622i	0.893878	−	0.448311i	$-0.147974\pi$
$618$	37517.2i	2.44201i
$619$	−19444.3	−1.26257	−0.631285	−	0.775551i	$-0.717472\pi$
$619$	−19444.3	−1.26257	−0.631285	+	0.775551i	$0.717472\pi$
$620$	0	0
$621$	33208.1	2.14589
$622$	− 8468.94i	− 0.545938i
$623$	4335.62i	0.278817i
$624$	−14689.8	−0.942407
$625$	0	0
$626$	−24809.6	−1.58401
$627$	− 21442.2i	− 1.36574i
$628$	− 36488.0i	− 2.31852i
$629$	10637.9	0.674341
$630$	0	0
$631$	−27425.6	−1.73026	−0.865130	−	0.501547i	$-0.832764\pi$
$631$	−27425.6	−1.73026	−0.865130	+	0.501547i	$0.832764\pi$
$632$	33188.2i	2.08886i
$633$	− 2548.33i	− 0.160011i
$634$	−23749.3	−1.48771
$635$	0	0
$636$	−22654.1	−1.41241
$637$	3004.51i	0.186881i
$638$	6095.70i	0.378262i
$639$	442.802	0.0274131
$640$	0	0
$641$	−10053.8	−0.619502	−0.309751	−	0.950818i	$-0.600246\pi$
$641$	−10053.8	−0.619502	−0.309751	+	0.950818i	$0.600246\pi$
$642$	4407.12i	0.270927i
$643$	− 4044.18i	− 0.248036i	−0.992280	−	0.124018i	$-0.960422\pi$
$643$	− 4044.18i	− 0.248036i	0.992280	−	0.124018i	$-0.0395780\pi$
$644$	−23967.3	−1.46653
$645$	0	0
$646$	−30515.3	−1.85853
$647$	− 1486.90i	− 0.0903491i	−0.998979	−	0.0451746i	$-0.985616\pi$
$647$	− 1486.90i	− 0.0903491i	0.998979	−	0.0451746i	$-0.0143844\pi$
$648$	13911.2i	0.843338i
$649$	−5006.52	−0.302809
$650$	0	0
$651$	−3614.16	−0.217588
$652$	24822.8i	1.49100i
$653$	4269.30i	0.255851i	0.991784	+	0.127925i	$0.0408318\pi$
$653$	4269.30i	0.255851i	−0.991784	+	0.127925i	$0.959168\pi$
$654$	−17126.2	−1.02399
$655$	0	0
$656$	11266.1	0.670528
$657$	986.086i	0.0585554i
$658$	− 573.710i	− 0.0339902i
$659$	−1607.29	−0.0950094	−0.0475047	−	0.998871i	$-0.515127\pi$
$659$	−1607.29	−0.0950094	−0.0475047	+	0.998871i	$0.515127\pi$
$660$	0	0
$661$	8001.31	0.470824	0.235412	−	0.971896i	$-0.424356\pi$
$661$	8001.31	0.470824	0.235412	+	0.971896i	$0.424356\pi$
$662$	− 11928.9i	− 0.700345i
$663$	− 11406.5i	− 0.668160i
$664$	−43534.1	−2.54436
$665$	0	0
$666$	11310.6	0.658073
$667$	7359.14i	0.427207i
$668$	16334.5i	0.946109i
$669$	8009.12	0.462855
$670$	0	0
$671$	18141.5	1.04373
$672$	587.250i	0.0337108i
$673$	3347.96i	0.191760i	0.995393	+	0.0958800i	$0.0305665\pi$
$673$	3347.96i	0.191760i	−0.995393	+	0.0958800i	$0.969433\pi$
$674$	−30554.7	−1.74617
$675$	0	0
$676$	24591.4	1.39914
$677$	7.91810i	0 0.000449508i	1.00000		0.000224754i	$7.15415e-5\pi$
$677$	7.91810i	0 0.000449508i	−1.00000		0.000224754i	$0.999928\pi$
$678$	28907.9i	1.63747i
$679$	1618.36	0.0914684
$680$	0	0
$681$	−3175.62	−0.178693
$682$	22426.7i	1.25918i
$683$	32136.0i	1.80036i	0.435513	+	0.900182i	$0.356567\pi$
$683$	32136.0i	1.80036i	−0.435513	+	0.900182i	$0.643433\pi$
$684$	−21509.8	−1.20241
$685$	0	0
$686$	−1671.09	−0.0930067
$687$	− 23702.8i	− 1.31633i
$688$	− 19846.7i	− 1.09978i
$689$	21274.7	1.17634
$690$	0	0
$691$	−6196.84	−0.341156	−0.170578	−	0.985344i	$-0.554563\pi$
$691$	−6196.84	−0.341156	−0.170578	+	0.985344i	$0.554563\pi$
$692$	7594.98i	0.417222i
$693$	− 2533.64i	− 0.138881i
$694$	2033.27	0.111213
$695$	0	0
$696$	−5289.39	−0.288066
$697$	8747.99i	0.475400i
$698$	− 29091.0i	− 1.57752i
$699$	−3586.61	−0.194075
$700$	0	0
$701$	1651.38	0.0889755	0.0444878	−	0.999010i	$-0.485834\pi$
$701$	1651.38	0.0889755	0.0444878	+	0.999010i	$0.485834\pi$
$702$	− 45594.1i	− 2.45134i
$703$	− 33146.8i	− 1.77831i
$704$	20731.5	1.10987
$705$	0	0
$706$	58525.4	3.11988
$707$	− 13132.0i	− 0.698555i
$708$	− 8836.66i	− 0.469071i
$709$	−15399.8	−0.815728	−0.407864	−	0.913043i	$-0.633726\pi$
$709$	−15399.8	−0.815728	−0.407864	+	0.913043i	$0.633726\pi$
$710$	0	0
$711$	−8615.53	−0.454441
$712$	23344.8i	1.22877i
$713$	27075.0i	1.42211i
$714$	6344.21	0.332530
$715$	0	0
$716$	37880.2	1.97716
$717$	− 7537.54i	− 0.392601i
$718$	− 30754.3i	− 1.59852i
$719$	−3507.31	−0.181920	−0.0909600	−	0.995855i	$-0.528994\pi$
$719$	−3507.31	−0.181920	−0.0909600	+	0.995855i	$0.528994\pi$
$720$	0	0
$721$	12991.6	0.671056
$722$	61666.0i	3.17863i
$723$	9769.32i	0.502524i
$724$	−8381.14	−0.430225
$725$	0	0
$726$	756.492	0.0386722
$727$	− 22516.7i	− 1.14869i	−0.818613	−	0.574346i	$-0.805256\pi$
$727$	− 22516.7i	− 1.14869i	0.818613	−	0.574346i	$-0.194744\pi$
$728$	16177.6i	0.823600i
$729$	−20709.4	−1.05215
$730$	0	0
$731$	15410.7	0.779736
$732$	32020.3i	1.61681i
$733$	− 10964.7i	− 0.552509i	−0.961084	−	0.276255i	$-0.910907\pi$
$733$	− 10964.7i	− 0.552509i	0.961084	−	0.276255i	$-0.0890933\pi$
$734$	59577.8	2.99599
$735$	0	0
$736$	4399.31	0.220327
$737$	− 17681.1i	− 0.883707i
$738$	9301.17i	0.463931i
$739$	−3029.33	−0.150793	−0.0753963	−	0.997154i	$-0.524022\pi$
$739$	−3029.33	−0.150793	−0.0753963	+	0.997154i	$0.524022\pi$
$740$	0	0
$741$	−35541.6	−1.76201
$742$	11832.9i	0.585442i
$743$	− 8921.04i	− 0.440486i	−0.975445	−	0.220243i	$-0.929315\pi$
$743$	− 8921.04i	− 0.440486i	0.975445	−	0.220243i	$-0.0706850\pi$
$744$	−19460.2	−0.958932
$745$	0	0
$746$	−38373.0	−1.88329
$747$	− 11301.3i	− 0.553537i
$748$	− 26099.1i	− 1.27577i
$749$	1526.11	0.0744498
$750$	0	0
$751$	−6202.64	−0.301382	−0.150691	−	0.988581i	$-0.548150\pi$
$751$	−6202.64	−0.301382	−0.150691	+	0.988581i	$0.548150\pi$
$752$	− 971.326i	− 0.0471019i
$753$	12379.7i	0.599124i
$754$	10104.0	0.488016
$755$	0	0
$756$	16812.2	0.808801
$757$	12099.3i	0.580922i	0.956887	+	0.290461i	$0.0938086\pi$
$757$	12099.3i	0.580922i	−0.956887	+	0.290461i	$0.906191\pi$
$758$	− 29196.8i	− 1.39905i
$759$	33395.7	1.59708
$760$	0	0
$761$	29332.7	1.39725	0.698626	−	0.715487i	$-0.253795\pi$
$761$	29332.7	1.39725	0.698626	+	0.715487i	$0.253795\pi$
$762$	35028.9i	1.66530i
$763$	5930.52i	0.281388i
$764$	−5298.01	−0.250884
$765$	0	0
$766$	−35924.1	−1.69451
$767$	8298.57i	0.390670i
$768$	33321.9i	1.56563i
$769$	13280.1	0.622748	0.311374	−	0.950287i	$-0.399211\pi$
$769$	13280.1	0.622748	0.311374	+	0.950287i	$0.399211\pi$
$770$	0	0
$771$	21249.9	0.992604
$772$	− 357.789i	− 0.0166802i
$773$	11248.1i	0.523370i	0.965153	+	0.261685i	$0.0842781\pi$
$773$	11248.1i	0.523370i	−0.965153	+	0.261685i	$0.915722\pi$
$774$	16385.3	0.760925
$775$	0	0
$776$	8713.96	0.403110
$777$	6891.31i	0.318178i
$778$	− 9241.57i	− 0.425869i
$779$	27258.0	1.25368
$780$	0	0
$781$	1674.11	0.0767021
$782$	− 47526.8i	− 2.17335i
$783$	− 5162.17i	− 0.235608i
$784$	−2829.26	−0.128884
$785$	0	0
$786$	−49116.7	−2.22893
$787$	436.193i	0.0197568i	0.999951	+	0.00987839i	$0.00314444\pi$
$787$	436.193i	0.0197568i	−0.999951	+	0.00987839i	$0.996856\pi$
$788$	− 17079.0i	− 0.772100i
$789$	19350.3	0.873116
$790$	0	0
$791$	10010.3	0.449970
$792$	− 13642.2i	− 0.612063i
$793$	− 30070.5i	− 1.34657i
$794$	−3854.50	−0.172281
$795$	0	0
$796$	−41390.8	−1.84304
$797$	24034.0i	1.06816i	0.845433	+	0.534082i	$0.179343\pi$
$797$	24034.0i	1.06816i	−0.845433	+	0.534082i	$0.820657\pi$
$798$	− 19768.0i	− 0.876918i
$799$	754.224	0.0333949
$800$	0	0
$801$	−6060.23	−0.267325
$802$	21488.1i	0.946101i
$803$	3728.11i	0.163838i
$804$	31207.7	1.36892
$805$	0	0
$806$	37173.4	1.62454
$807$	20824.1i	0.908354i
$808$	− 70708.1i	− 3.07859i
$809$	−16188.0	−0.703509	−0.351754	−	0.936092i	$-0.614415\pi$
$809$	−16188.0	−0.703509	−0.351754	+	0.936092i	$0.614415\pi$
$810$	0	0
$811$	3732.42	0.161607	0.0808033	−	0.996730i	$-0.474251\pi$
$811$	3732.42	0.161607	0.0808033	+	0.996730i	$0.474251\pi$
$812$	3725.69i	0.161018i
$813$	10425.4i	0.449737i
$814$	42762.1	1.84129
$815$	0	0
$816$	10741.1	0.460803
$817$	− 48018.6i	− 2.05625i
$818$	21454.9i	0.917058i
$819$	−4199.64	−0.179178
$820$	0	0
$821$	37226.4	1.58247	0.791237	−	0.611510i	$-0.209438\pi$
$821$	37226.4	1.58247	0.791237	+	0.611510i	$0.209438\pi$
$822$	− 29368.7i	− 1.24617i
$823$	21355.7i	0.904511i	0.891888	+	0.452256i	$0.149380\pi$
$823$	21355.7i	0.904511i	−0.891888	+	0.452256i	$0.850620\pi$
$824$	69952.2	2.95740
$825$	0	0
$826$	−4615.62	−0.194428
$827$	18852.9i	0.792718i	0.918096	+	0.396359i	$0.129726\pi$
$827$	18852.9i	0.792718i	−0.918096	+	0.396359i	$0.870274\pi$
$828$	− 33501.0i	− 1.40609i
$829$	43227.9	1.81106	0.905528	−	0.424286i	$-0.139475\pi$
$829$	43227.9	1.81106	0.905528	+	0.424286i	$0.139475\pi$
$830$	0	0
$831$	−26085.1	−1.08891
$832$	− 34363.5i	− 1.43190i
$833$	− 2196.89i	− 0.0913779i
$834$	−59177.9	−2.45703
$835$	0	0
$836$	−81322.4	−3.36435
$837$	− 18992.1i	− 0.784306i
$838$	69536.1i	2.86645i
$839$	−30311.3	−1.24727	−0.623637	−	0.781714i	$-0.714346\pi$
$839$	−30311.3	−1.24727	−0.623637	+	0.781714i	$0.714346\pi$
$840$	0	0
$841$	−23245.0	−0.953095
$842$	− 75664.0i	− 3.09686i
$843$	− 36530.6i	− 1.49250i
$844$	−9664.88	−0.394169
$845$	0	0
$846$	801.918	0.0325893
$847$	− 261.960i	− 0.0106270i
$848$	20033.8i	0.811276i
$849$	2013.58	0.0813968
$850$	0	0
$851$	51625.3	2.07955
$852$	2954.85i	0.118816i
$853$	− 22797.0i	− 0.915071i	−0.889191	−	0.457535i	$-0.848732\pi$
$853$	− 22797.0i	− 0.915071i	0.889191	−	0.457535i	$-0.151268\pi$
$854$	16725.0	0.670162
$855$	0	0
$856$	8217.25	0.328107
$857$	11280.3i	0.449625i	0.974402	+	0.224813i	$0.0721770\pi$
$857$	11280.3i	0.449625i	−0.974402	+	0.224813i	$0.927823\pi$
$858$	− 45851.6i	− 1.82442i
$859$	−43369.3	−1.72263	−0.861315	−	0.508071i	$-0.830359\pi$
$859$	−43369.3	−1.72263	−0.861315	+	0.508071i	$0.830359\pi$
$860$	0	0
$861$	−5667.01	−0.224310
$862$	− 25630.3i	− 1.01273i
$863$	9846.22i	0.388377i	0.980964	+	0.194189i	$0.0622074\pi$
$863$	9846.22i	0.388377i	−0.980964	+	0.194189i	$0.937793\pi$
$864$	−3085.95	−0.121512
$865$	0	0
$866$	−32207.0	−1.26379
$867$	− 12044.5i	− 0.471801i
$868$	13707.2i	0.536005i
$869$	−32572.9	−1.27153
$870$	0	0
$871$	−29307.4	−1.14012
$872$	31932.5i	1.24010i
$873$	2262.11i	0.0876985i
$874$	−148090.	−5.73136
$875$	0	0
$876$	−6580.23	−0.253796
$877$	15229.8i	0.586400i	0.956051	+	0.293200i	$0.0947202\pi$
$877$	15229.8i	0.586400i	−0.956051	+	0.293200i	$0.905280\pi$
$878$	43216.8i	1.66116i
$879$	16615.2	0.637563
$880$	0	0
$881$	27891.7	1.06662	0.533311	−	0.845919i	$-0.320948\pi$
$881$	27891.7	1.06662	0.533311	+	0.845919i	$0.320948\pi$
$882$	− 2335.81i	− 0.0891734i
$883$	18300.3i	0.697456i	0.937224	+	0.348728i	$0.113386\pi$
$883$	18300.3i	0.697456i	−0.937224	+	0.348728i	$0.886614\pi$
$884$	−43260.6	−1.64594
$885$	0	0
$886$	64416.3	2.44256
$887$	19104.8i	0.723197i	0.932334	+	0.361598i	$0.117769\pi$
$887$	19104.8i	0.723197i	−0.932334	+	0.361598i	$0.882231\pi$
$888$	37105.8i	1.40224i
$889$	12129.9	0.457620
$890$	0	0
$891$	−13653.3	−0.513357
$892$	− 30375.7i	− 1.14019i
$893$	− 2350.10i	− 0.0880661i
$894$	−18228.1	−0.681921
$895$	0	0
$896$	17980.5	0.670410
$897$	− 55355.2i	− 2.06048i
$898$	− 20426.8i	− 0.759077i
$899$	4208.78	0.156141
$900$	0	0
$901$	−15556.0	−0.575189
$902$	35165.1i	1.29808i
$903$	9983.19i	0.367907i
$904$	53899.9	1.98306
$905$	0	0
$906$	−27444.0	−1.00636
$907$	29518.8i	1.08066i	0.841455	+	0.540328i	$0.181700\pi$
$907$	29518.8i	1.08066i	−0.841455	+	0.540328i	$0.818300\pi$
$908$	12044.0i	0.440191i
$909$	18355.5	0.669764
$910$	0	0
$911$	−30322.2	−1.10277	−0.551383	−	0.834253i	$-0.685900\pi$
$911$	−30322.2	−1.10277	−0.551383	+	0.834253i	$0.685900\pi$
$912$	− 33468.5i	− 1.21519i
$913$	− 42727.0i	− 1.54880i
$914$	−22907.0	−0.828991
$915$	0	0
$916$	−89895.9	−3.24263
$917$	17008.3i	0.612501i
$918$	33338.3i	1.19862i
$919$	37780.9	1.35612	0.678062	−	0.735005i	$-0.262820\pi$
$919$	37780.9	1.35612	0.678062	+	0.735005i	$0.262820\pi$
$920$	0	0
$921$	−13087.0	−0.468220
$922$	− 9499.48i	− 0.339315i
$923$	− 2774.93i	− 0.0989575i
$924$	16907.2	0.601953
$925$	0	0
$926$	−35642.7	−1.26489
$927$	18159.3i	0.643398i
$928$	− 683.867i	− 0.0241908i
$929$	−48493.3	−1.71261	−0.856305	−	0.516470i	$-0.827246\pi$
$929$	−48493.3	−1.71261	−0.856305	+	0.516470i	$0.827246\pi$
$930$	0	0
$931$	−6845.33	−0.240974
$932$	13602.7i	0.478081i
$933$	7212.46i	0.253082i
$934$	−9244.81	−0.323875
$935$	0	0
$936$	−22612.7	−0.789656
$937$	− 18661.7i	− 0.650641i	−0.945604	−	0.325320i	$-0.894528\pi$
$937$	− 18661.7i	− 0.650641i	0.945604	−	0.325320i	$-0.105472\pi$
$938$	− 16300.6i	− 0.567413i
$939$	21128.8	0.734305
$940$	0	0
$941$	−31270.1	−1.08329	−0.541644	−	0.840608i	$-0.682198\pi$
$941$	−31270.1	−1.08329	−0.541644	+	0.840608i	$0.682198\pi$
$942$	46872.1i	1.62121i
$943$	42453.7i	1.46605i
$944$	−7814.53	−0.269429
$945$	0	0
$946$	61948.0	2.12907
$947$	22405.3i	0.768823i	0.923162	+	0.384412i	$0.125596\pi$
$947$	22405.3i	0.768823i	−0.923162	+	0.384412i	$0.874404\pi$
$948$	− 57492.2i	− 1.96968i
$949$	6179.55	0.211377
$950$	0	0
$951$	20225.8	0.689660
$952$	− 11829.0i	− 0.402711i
$953$	21777.9i	0.740247i	0.928983	+	0.370123i	$0.120685\pi$
$953$	21777.9i	0.740247i	−0.928983	+	0.370123i	$0.879315\pi$
$954$	−16539.7	−0.561313
$955$	0	0
$956$	−28587.2	−0.967129
$957$	− 5191.32i	− 0.175352i
$958$	− 51323.3i	− 1.73088i
$959$	−10169.9	−0.342442
$960$	0	0
$961$	−14306.5	−0.480229
$962$	− 70880.5i	− 2.37555i
$963$	2133.16i	0.0713814i
$964$	37051.5	1.23791
$965$	0	0
$966$	30788.2	1.02546
$967$	14432.2i	0.479945i	0.970780	+	0.239973i	$0.0771384\pi$
$967$	14432.2i	0.479945i	−0.970780	+	0.239973i	$0.922862\pi$
$968$	− 1410.51i	− 0.0468341i
$969$	25987.9	0.861561
$970$	0	0
$971$	6672.57	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$971$	6672.57	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$972$	40748.6i	1.34466i
$973$	20492.3i	0.675183i
$974$	58156.7	1.91321
$975$	0	0
$976$	28316.5	0.928678
$977$	33300.2i	1.09045i	0.838290	+	0.545224i	$0.183556\pi$
$977$	33300.2i	1.09045i	−0.838290	+	0.545224i	$0.816444\pi$
$978$	− 31887.1i	− 1.04257i
$979$	−22912.0	−0.747979
$980$	0	0
$981$	−8289.55	−0.269791
$982$	487.491i	0.0158416i
$983$	17243.3i	0.559486i	0.960075	+	0.279743i	$0.0902492\pi$
$983$	17243.3i	0.559486i	−0.960075	+	0.279743i	$0.909751\pi$
$984$	−30513.6	−0.988557
$985$	0	0
$986$	−7387.99	−0.238622
$987$	488.592i	0.0157569i
$988$	134796.i	4.34053i
$989$	74787.8	2.40456
$990$	0	0
$991$	−32266.7	−1.03429	−0.517147	−	0.855897i	$-0.673006\pi$
$991$	−32266.7	−1.03429	−0.517147	+	0.855897i	$0.673006\pi$
$992$	− 2516.02i	− 0.0805278i
$993$	10159.1i	0.324661i
$994$	1543.40	0.0492491
$995$	0	0
$996$	75414.4	2.39919
$997$	14339.8i	0.455514i	0.973718	+	0.227757i	$0.0731391\pi$
$997$	14339.8i	0.455514i	−0.973718	+	0.227757i	$0.926861\pi$
$998$	76059.8i	2.41246i
$999$	−36213.3	−1.14688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.f.99.8		8
5.2	odd	4		175.4.a.h.1.1	yes	4
5.3	odd	4		175.4.a.g.1.4	✓	4
5.4	even	2	inner	175.4.b.f.99.1		8
15.2	even	4		1575.4.a.bg.1.4		4
15.8	even	4		1575.4.a.bl.1.1		4
35.13	even	4		1225.4.a.z.1.4		4
35.27	even	4		1225.4.a.bd.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.4	✓	4	5.3	odd	4
175.4.a.h.1.1	yes	4	5.2	odd	4
175.4.b.f.99.1		8	5.4	even	2	inner
175.4.b.f.99.8		8	1.1	even	1	trivial
1225.4.a.z.1.4		4	35.13	even	4
1225.4.a.bd.1.1		4	35.27	even	4
1575.4.a.bg.1.4		4	15.2	even	4
1575.4.a.bl.1.1		4	15.8	even	4

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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.87199i q^{2} -4.14916i q^{3} -15.7363 q^{4} +20.2147 q^{6} -7.00000i q^{7} -37.6910i q^{8} +9.78444 q^{9} +36.9922 q^{11} +65.2924i q^{12} -61.3165i q^{13} +34.1039 q^{14} +57.7401 q^{16} +44.8345i q^{17} +47.6697i q^{18} +139.701 q^{19} -29.0441 q^{21} +180.226i q^{22} +217.580i q^{23} -156.386 q^{24} +298.733 q^{26} -152.625i q^{27} +110.154i q^{28} +33.8226 q^{29} +124.437 q^{31} -20.2192i q^{32} -153.487i q^{33} -218.433 q^{34} -153.971 q^{36} -237.270i q^{37} +680.620i q^{38} -254.412 q^{39} +195.117 q^{41} -141.503i q^{42} -343.725i q^{43} -582.119 q^{44} -1060.05 q^{46} -16.8224i q^{47} -239.573i q^{48} -49.0000 q^{49} +186.026 q^{51} +964.894i q^{52} +346.965i q^{53} +743.586 q^{54} -263.837 q^{56} -579.641i q^{57} +164.783i q^{58} -135.340 q^{59} +490.414 q^{61} +606.255i q^{62} -68.4911i q^{63} +560.428 q^{64} +747.785 q^{66} -477.969i q^{67} -705.529i q^{68} +902.777 q^{69} +45.2557 q^{71} -368.786i q^{72} +100.781i q^{73} +1155.98 q^{74} -2198.37 q^{76} -258.945i q^{77} -1239.49i q^{78} -880.534 q^{79} -369.085 q^{81} +950.609i q^{82} -1155.03i q^{83} +457.047 q^{84} +1674.62 q^{86} -140.336i q^{87} -1394.27i q^{88} -619.374 q^{89} -429.216 q^{91} -3423.90i q^{92} -516.309i q^{93} +81.9585 q^{94} -83.8929 q^{96} +231.195i q^{97} -238.727i q^{98} +361.948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9} + 200 q^{11} - 56 q^{14} + 320 q^{16} + 58 q^{19} - 42 q^{21} + 42 q^{24} + 1400 q^{26} - 258 q^{29} + 228 q^{31} - 406 q^{34} + 2202 q^{36} - 1348 q^{39} + 1342 q^{41}+ \cdots - 4766 q^{99}+O(q^{100}) \) 8 * q - 72 * q^4 + 2 * q^6 - 122 * q^9 + 200 * q^11 - 56 * q^14 + 320 * q^16 + 58 * q^19 - 42 * q^21 + 42 * q^24 + 1400 * q^26 - 258 * q^29 + 228 * q^31 - 406 * q^34 + 2202 * q^36 - 1348 * q^39 + 1342 * q^41 - 876 * q^44 - 1994 * q^46 - 392 * q^49 - 1770 * q^51 + 5554 * q^54 + 378 * q^56 - 2036 * q^59 + 100 * q^61 + 4842 * q^64 - 7682 * q^66 - 2160 * q^69 + 430 * q^71 - 1246 * q^74 - 6514 * q^76 + 1902 * q^79 + 56 * q^81 + 2310 * q^84 - 198 * q^86 - 5638 * q^89 + 616 * q^91 + 6112 * q^94 - 2690 * q^96 - 4766 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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