LMFDB - Embedded newform 175.4.b.f.99.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(99,175)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			99.4
Root			$-1.50478i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.f.99.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.504784i q^{2} -4.26379i q^{3} +7.74519 q^{4} -2.15229 q^{6} +7.00000i q^{7} -7.94792i q^{8} +8.82008 q^{9} +54.8800 q^{11} -33.0239i q^{12} -16.0073i q^{13} +3.53349 q^{14} +57.9496 q^{16} -0.422056i q^{17} -4.45223i q^{18} -127.501 q^{19} +29.8465 q^{21} -27.7025i q^{22} -51.1101i q^{23} -33.8883 q^{24} -8.08024 q^{26} -152.729i q^{27} +54.2164i q^{28} -41.4750 q^{29} +192.354 q^{31} -92.8353i q^{32} -233.997i q^{33} -0.213047 q^{34} +68.3132 q^{36} -189.232i q^{37} +64.3605i q^{38} -68.2519 q^{39} -76.3187 q^{41} -15.0660i q^{42} -294.499i q^{43} +425.056 q^{44} -25.7996 q^{46} +540.297i q^{47} -247.085i q^{48} -49.0000 q^{49} -1.79956 q^{51} -123.980i q^{52} +661.316i q^{53} -77.0953 q^{54} +55.6354 q^{56} +543.639i q^{57} +20.9359i q^{58} -410.312 q^{59} +46.0495 q^{61} -97.0974i q^{62} +61.7405i q^{63} +416.735 q^{64} -118.118 q^{66} +10.4074i q^{67} -3.26890i q^{68} -217.923 q^{69} -491.117 q^{71} -70.1012i q^{72} +814.540i q^{73} -95.5215 q^{74} -987.522 q^{76} +384.160i q^{77} +34.4525i q^{78} +858.725 q^{79} -413.064 q^{81} +38.5244i q^{82} +1055.80i q^{83} +231.167 q^{84} -148.658 q^{86} +176.841i q^{87} -436.182i q^{88} -341.567 q^{89} +112.051 q^{91} -395.858i q^{92} -820.159i q^{93} +272.733 q^{94} -395.831 q^{96} -1417.21i q^{97} +24.7344i q^{98} +484.046 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 0.504784i	− 0.178468i	−0.996011	−	0.0892340i	$-0.971558\pi$
$2$	− 0.504784i	− 0.178468i	0.996011	−	0.0892340i	$-0.0284419\pi$
$3$	− 4.26379i	− 0.820567i	−0.911958	−	0.410284i	$-0.865430\pi$
$3$	− 4.26379i	− 0.820567i	0.911958	−	0.410284i	$-0.134570\pi$
$4$	7.74519	0.968149
$5$	0	0
$5$	0	0
$6$	−2.15229	−0.146445
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	− 7.94792i	− 0.351252i
$9$	8.82008	0.326670
$10$	0	0
$11$	54.8800	1.50427	0.752134	−	0.659010i	$-0.229025\pi$
$11$	54.8800	1.50427	0.752134	+	0.659010i	$0.229025\pi$
$12$	− 33.0239i	− 0.794431i
$13$	− 16.0073i	− 0.341510i	−0.985313	−	0.170755i	$-0.945379\pi$
$13$	− 16.0073i	− 0.341510i	0.985313	−	0.170755i	$-0.0546207\pi$
$14$	3.53349	0.0674546
$15$	0	0
$16$	57.9496	0.905462
$17$	− 0.422056i	− 0.00602139i	−0.999995	−	0.00301069i	$-0.999042\pi$
$17$	− 0.422056i	− 0.00602139i	0.999995	−	0.00301069i	$-0.000958335\pi$
$18$	− 4.45223i	− 0.0583000i
$19$	−127.501	−1.53952	−0.769758	−	0.638336i	$-0.779623\pi$
$19$	−127.501	−1.53952	−0.769758	+	0.638336i	$0.779623\pi$
$20$	0	0
$21$	29.8465	0.310145
$22$	− 27.7025i	− 0.268464i
$23$	− 51.1101i	− 0.463357i	−0.972792	−	0.231678i	$-0.925578\pi$
$23$	− 51.1101i	− 0.463357i	0.972792	−	0.231678i	$-0.0744217\pi$
$24$	−33.8883	−0.288226
$25$	0	0
$26$	−8.08024	−0.0609487
$27$	− 152.729i	− 1.08862i
$28$	54.2164i	0.365926i
$29$	−41.4750	−0.265576	−0.132788	−	0.991144i	$-0.542393\pi$
$29$	−41.4750	−0.265576	−0.132788	+	0.991144i	$0.542393\pi$
$30$	0	0
$31$	192.354	1.11445	0.557224	−	0.830362i	$-0.311866\pi$
$31$	192.354	1.11445	0.557224	+	0.830362i	$0.311866\pi$
$32$	− 92.8353i	− 0.512848i
$33$	− 233.997i	− 1.23435i
$34$	−0.213047	−0.00107462
$35$	0	0
$36$	68.3132	0.316265
$37$	− 189.232i	− 0.840801i	−0.907339	−	0.420400i	$-0.861890\pi$
$37$	− 189.232i	− 0.840801i	0.907339	−	0.420400i	$-0.138110\pi$
$38$	64.3605i	0.274754i
$39$	−68.2519	−0.280232
$40$	0	0
$41$	−76.3187	−0.290707	−0.145353	−	0.989380i	$-0.546432\pi$
$41$	−76.3187	−0.290707	−0.145353	+	0.989380i	$0.546432\pi$
$42$	− 15.0660i	− 0.0553510i
$43$	− 294.499i	− 1.04443i	−0.852813	−	0.522216i	$-0.825105\pi$
$43$	− 294.499i	− 1.04443i	0.852813	−	0.522216i	$-0.174895\pi$
$44$	425.056	1.45636
$45$	0	0
$46$	−25.7996	−0.0826943
$47$	540.297i	1.67682i	0.545043	+	0.838408i	$0.316513\pi$
$47$	540.297i	1.67682i	−0.545043	+	0.838408i	$0.683487\pi$
$48$	− 247.085i	− 0.742992i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−1.79956	−0.00494095
$52$	− 123.980i	− 0.330633i
$53$	661.316i	1.71394i	0.515368	+	0.856969i	$0.327655\pi$
$53$	661.316i	1.71394i	−0.515368	+	0.856969i	$0.672345\pi$
$54$	−77.0953	−0.194284
$55$	0	0
$56$	55.6354	0.132761
$57$	543.639i	1.26328i
$58$	20.9359i	0.0473969i
$59$	−410.312	−0.905390	−0.452695	−	0.891665i	$-0.649537\pi$
$59$	−410.312	−0.905390	−0.452695	+	0.891665i	$0.649537\pi$
$60$	0	0
$61$	46.0495	0.0966563	0.0483281	−	0.998832i	$-0.484611\pi$
$61$	46.0495	0.0966563	0.0483281	+	0.998832i	$0.484611\pi$
$62$	− 97.0974i	− 0.198893i
$63$	61.7405i	0.123469i
$64$	416.735	0.813935
$65$	0	0
$66$	−118.118	−0.220292
$67$	10.4074i	0.0189771i	0.999955	+	0.00948854i	$0.00302034\pi$
$67$	10.4074i	0.0189771i	−0.999955	+	0.00948854i	$0.996980\pi$
$68$	− 3.26890i	− 0.00582960i
$69$	−217.923	−0.380215
$70$	0	0
$71$	−491.117	−0.820913	−0.410456	−	0.911880i	$-0.634631\pi$
$71$	−491.117	−0.820913	−0.410456	+	0.911880i	$0.634631\pi$
$72$	− 70.1012i	− 0.114743i
$73$	814.540i	1.30595i	0.757378	+	0.652977i	$0.226480\pi$
$73$	814.540i	1.30595i	−0.757378	+	0.652977i	$0.773520\pi$
$74$	−95.5215	−0.150056
$75$	0	0
$76$	−987.522	−1.49048
$77$	384.160i	0.568560i
$78$	34.4525i	0.0500125i
$79$	858.725	1.22296	0.611482	−	0.791258i	$-0.290574\pi$
$79$	858.725	1.22296	0.611482	+	0.791258i	$0.290574\pi$
$80$	0	0
$81$	−413.064	−0.566617
$82$	38.5244i	0.0518818i
$83$	1055.80i	1.39626i	0.715972	+	0.698129i	$0.245984\pi$
$83$	1055.80i	1.39626i	−0.715972	+	0.698129i	$0.754016\pi$
$84$	231.167	0.300267
$85$	0	0
$86$	−148.658	−0.186398
$87$	176.841i	0.217923i
$88$	− 436.182i	− 0.528376i
$89$	−341.567	−0.406809	−0.203405	−	0.979095i	$-0.565201\pi$
$89$	−341.567	−0.406809	−0.203405	+	0.979095i	$0.565201\pi$
$90$	0	0
$91$	112.051	0.129079
$92$	− 395.858i	− 0.448598i
$93$	− 820.159i	− 0.914479i
$94$	272.733	0.299258
$95$	0	0
$96$	−395.831	−0.420826
$97$	− 1417.21i	− 1.48346i	−0.670697	−	0.741731i	$-0.734005\pi$
$97$	− 1417.21i	− 1.48346i	0.670697	−	0.741731i	$-0.265995\pi$
$98$	24.7344i	0.0254954i
$99$	484.046	0.491398
$100$	0	0
$101$	−121.051	−0.119258	−0.0596289	−	0.998221i	$-0.518992\pi$
$101$	−121.051	−0.119258	−0.0596289	+	0.998221i	$0.518992\pi$
$102$	0.908388i	0 0.000881802i
$103$	655.594i	0.627161i	0.949562	+	0.313581i	$0.101529\pi$
$103$	655.594i	0.627161i	−0.949562	+	0.313581i	$0.898471\pi$
$104$	−127.225	−0.119956
$105$	0	0
$106$	333.821	0.305883
$107$	2069.81i	1.87006i	0.354567	+	0.935031i	$0.384628\pi$
$107$	2069.81i	1.87006i	−0.354567	+	0.935031i	$0.615372\pi$
$108$	− 1182.92i	− 1.05395i
$109$	−904.218	−0.794573	−0.397286	−	0.917695i	$-0.630048\pi$
$109$	−904.218	−0.794573	−0.397286	+	0.917695i	$0.630048\pi$
$110$	0	0
$111$	−806.848	−0.689934
$112$	405.647i	0.342232i
$113$	843.350i	0.702086i	0.936359	+	0.351043i	$0.114173\pi$
$113$	843.350i	0.702086i	−0.936359	+	0.351043i	$0.885827\pi$
$114$	274.420	0.225454
$115$	0	0
$116$	−321.232	−0.257117
$117$	− 141.186i	− 0.111561i
$118$	207.119i	0.161583i
$119$	2.95439	0.00227587
$120$	0	0
$121$	1680.81	1.26282
$122$	− 23.2450i	− 0.0172501i
$123$	325.407i	0.238544i
$124$	1489.82	1.07895
$125$	0	0
$126$	31.1656	0.0220353
$127$	2030.50i	1.41872i	0.704845	+	0.709362i	$0.251017\pi$
$127$	2030.50i	1.41872i	−0.704845	+	0.709362i	$0.748983\pi$
$128$	− 953.043i	− 0.658109i
$129$	−1255.68	−0.857027
$130$	0	0
$131$	2872.47	1.91579	0.957897	−	0.287113i	$-0.0926954\pi$
$131$	2872.47	1.91579	0.957897	+	0.287113i	$0.0926954\pi$
$132$	− 1812.35i	− 1.19504i
$133$	− 892.509i	− 0.581882i
$134$	5.25348	0.00338680
$135$	0	0
$136$	−3.35446	−0.00211502
$137$	− 1621.62i	− 1.01127i	−0.862747	−	0.505636i	$-0.831258\pi$
$137$	− 1621.62i	− 1.01127i	0.862747	−	0.505636i	$-0.168742\pi$
$138$	110.004i	0.0678562i
$139$	−2617.72	−1.59735	−0.798677	−	0.601760i	$-0.794466\pi$
$139$	−2617.72	−1.59735	−0.798677	+	0.601760i	$0.794466\pi$
$140$	0	0
$141$	2303.71	1.37594
$142$	247.908i	0.146507i
$143$	− 878.482i	− 0.513723i
$144$	511.120	0.295787
$145$	0	0
$146$	411.166	0.233071
$147$	208.926i	0.117224i
$148$	− 1465.64i	− 0.814021i
$149$	−2252.63	−1.23854	−0.619270	−	0.785178i	$-0.712571\pi$
$149$	−2252.63	−1.23854	−0.619270	+	0.785178i	$0.712571\pi$
$150$	0	0
$151$	−1849.61	−0.996813	−0.498407	−	0.866943i	$-0.666081\pi$
$151$	−1849.61	−0.996813	−0.498407	+	0.866943i	$0.666081\pi$
$152$	1013.37i	0.540757i
$153$	− 3.72257i	− 0.00196700i
$154$	193.918	0.101470
$155$	0	0
$156$	−528.624	−0.271307
$157$	− 3337.59i	− 1.69661i	−0.529505	−	0.848307i	$-0.677622\pi$
$157$	− 3337.59i	− 1.69661i	0.529505	−	0.848307i	$-0.322378\pi$
$158$	− 433.471i	− 0.218260i
$159$	2819.71	1.40640
$160$	0	0
$161$	357.771	0.175132
$162$	208.508i	0.101123i
$163$	471.099i	0.226376i	0.993574	+	0.113188i	$0.0361063\pi$
$163$	471.099i	0.226376i	−0.993574	+	0.113188i	$0.963894\pi$
$164$	−591.103	−0.281447
$165$	0	0
$166$	532.952	0.249187
$167$	− 3676.72i	− 1.70367i	−0.523809	−	0.851836i	$-0.675489\pi$
$167$	− 3676.72i	− 1.70367i	0.523809	−	0.851836i	$-0.324511\pi$
$168$	− 237.218i	− 0.108939i
$169$	1940.77	0.883371
$170$	0	0
$171$	−1124.57	−0.502913
$172$	− 2280.95i	− 1.01117i
$173$	− 1009.11i	− 0.443475i	−0.975106	−	0.221737i	$-0.928827\pi$
$173$	− 1009.11i	− 0.443475i	0.975106	−	0.221737i	$-0.0711728\pi$
$174$	89.2663	0.0388923
$175$	0	0
$176$	3180.27	1.36206
$177$	1749.48i	0.742934i
$178$	172.417i	0.0726024i
$179$	−3201.27	−1.33673	−0.668364	−	0.743835i	$-0.733005\pi$
$179$	−3201.27	−1.33673	−0.668364	+	0.743835i	$0.733005\pi$
$180$	0	0
$181$	−1970.14	−0.809056	−0.404528	−	0.914526i	$-0.632564\pi$
$181$	−1970.14	−0.809056	−0.404528	+	0.914526i	$0.632564\pi$
$182$	− 56.5617i	− 0.0230364i
$183$	− 196.345i	− 0.0793130i
$184$	−406.219	−0.162755
$185$	0	0
$186$	−414.003	−0.163205
$187$	− 23.1624i	− 0.00905778i
$188$	4184.70i	1.62341i
$189$	1069.11	0.411460
$190$	0	0
$191$	3953.39	1.49768	0.748842	−	0.662749i	$-0.230610\pi$
$191$	3953.39	1.49768	0.748842	+	0.662749i	$0.230610\pi$
$192$	− 1776.87i	− 0.667888i
$193$	− 2425.45i	− 0.904598i	−0.891866	−	0.452299i	$-0.850604\pi$
$193$	− 2425.45i	− 0.904598i	0.891866	−	0.452299i	$-0.149396\pi$
$194$	−715.385	−0.264751
$195$	0	0
$196$	−379.514	−0.138307
$197$	3126.50i	1.13073i	0.824840	+	0.565366i	$0.191265\pi$
$197$	3126.50i	1.13073i	−0.824840	+	0.565366i	$0.808735\pi$
$198$	− 244.338i	− 0.0876989i
$199$	3566.13	1.27033	0.635166	−	0.772376i	$-0.280932\pi$
$199$	3566.13	1.27033	0.635166	+	0.772376i	$0.280932\pi$
$200$	0	0
$201$	44.3749	0.0155720
$202$	61.1046i	0.0212837i
$203$	− 290.325i	− 0.100378i
$204$	−13.9379	−0.00478358
$205$	0	0
$206$	330.933	0.111928
$207$	− 450.795i	− 0.151364i
$208$	− 927.618i	− 0.309225i
$209$	−6997.27	−2.31584
$210$	0	0
$211$	−3699.18	−1.20693	−0.603465	−	0.797389i	$-0.706214\pi$
$211$	−3699.18	−1.20693	−0.603465	+	0.797389i	$0.706214\pi$
$212$	5122.02i	1.65935i
$213$	2094.02i	0.673614i
$214$	1044.81	0.333746
$215$	0	0
$216$	−1213.88	−0.382380
$217$	1346.48i	0.421222i
$218$	456.435i	0.141806i
$219$	3473.03	1.07162
$220$	0	0
$221$	−6.75599	−0.00205637
$222$	407.284i	0.123131i
$223$	102.041i	0.0306419i	0.999883	+	0.0153210i	$0.00487701\pi$
$223$	102.041i	0.0306419i	−0.999883	+	0.0153210i	$0.995123\pi$
$224$	649.847	0.193838
$225$	0	0
$226$	425.709	0.125300
$227$	5739.91i	1.67829i	0.543909	+	0.839144i	$0.316943\pi$
$227$	5739.91i	1.67829i	−0.543909	+	0.839144i	$0.683057\pi$
$228$	4210.59i	1.22304i
$229$	843.125	0.243298	0.121649	−	0.992573i	$-0.461182\pi$
$229$	843.125	0.243298	0.121649	+	0.992573i	$0.461182\pi$
$230$	0	0
$231$	1637.98	0.466541
$232$	329.640i	0.0932841i
$233$	2017.65i	0.567300i	0.958928	+	0.283650i	$0.0915454\pi$
$233$	2017.65i	0.567300i	−0.958928	+	0.283650i	$0.908455\pi$
$234$	−71.2683	−0.0199101
$235$	0	0
$236$	−3177.94	−0.876553
$237$	− 3661.43i	− 1.00352i
$238$	− 1.49133i	0 0.000406170i
$239$	−243.685	−0.0659526	−0.0329763	−	0.999456i	$-0.510499\pi$
$239$	−243.685	−0.0659526	−0.0329763	+	0.999456i	$0.510499\pi$
$240$	0	0
$241$	−3060.58	−0.818048	−0.409024	−	0.912524i	$-0.634131\pi$
$241$	−3060.58	−0.818048	−0.409024	+	0.912524i	$0.634131\pi$
$242$	− 848.448i	− 0.225373i
$243$	− 2362.47i	− 0.623674i
$244$	356.662	0.0935777
$245$	0	0
$246$	164.260	0.0425725
$247$	2040.95i	0.525760i
$248$	− 1528.82i	− 0.391452i
$249$	4501.72	1.14572
$250$	0	0
$251$	−6594.14	−1.65824	−0.829120	−	0.559070i	$-0.811158\pi$
$251$	−6594.14	−1.65824	−0.829120	+	0.559070i	$0.811158\pi$
$252$	478.192i	0.119537i
$253$	− 2804.92i	− 0.697012i
$254$	1024.96	0.253197
$255$	0	0
$256$	2852.80	0.696484
$257$	512.646i	0.124428i	0.998063	+	0.0622139i	$0.0198161\pi$
$257$	512.646i	0.124428i	−0.998063	+	0.0622139i	$0.980184\pi$
$258$	633.847i	0.152952i
$259$	1324.63	0.317793
$260$	0	0
$261$	−365.813	−0.0867557
$262$	− 1449.98i	− 0.341908i
$263$	3560.04i	0.834681i	0.908750	+	0.417341i	$0.137038\pi$
$263$	3560.04i	0.834681i	−0.908750	+	0.417341i	$0.862962\pi$
$264$	−1859.79	−0.433568
$265$	0	0
$266$	−450.524	−0.103847
$267$	1456.37i	0.333814i
$268$	80.6072i	0.0183726i
$269$	549.744	0.124604	0.0623020	−	0.998057i	$-0.480156\pi$
$269$	549.744	0.124604	0.0623020	+	0.998057i	$0.480156\pi$
$270$	0	0
$271$	−3944.91	−0.884267	−0.442134	−	0.896949i	$-0.645778\pi$
$271$	−3944.91	−0.884267	−0.442134	+	0.896949i	$0.645778\pi$
$272$	− 24.4580i	− 0.00545214i
$273$	− 477.764i	− 0.105918i
$274$	−818.567	−0.180480
$275$	0	0
$276$	−1687.86	−0.368105
$277$	− 1254.67i	− 0.272150i	−0.990699	−	0.136075i	$-0.956551\pi$
$277$	− 1254.67i	− 0.272150i	0.990699	−	0.136075i	$-0.0434488\pi$
$278$	1321.38i	0.285077i
$279$	1696.58	0.364056
$280$	0	0
$281$	−681.496	−0.144679	−0.0723393	−	0.997380i	$-0.523046\pi$
$281$	−681.496	−0.144679	−0.0723393	+	0.997380i	$0.523046\pi$
$282$	− 1162.88i	− 0.245561i
$283$	5946.88i	1.24914i	0.780970	+	0.624568i	$0.214725\pi$
$283$	5946.88i	1.24914i	−0.780970	+	0.624568i	$0.785275\pi$
$284$	−3803.79	−0.794766
$285$	0	0
$286$	−443.444	−0.0916831
$287$	− 534.231i	− 0.109877i
$288$	− 818.815i	− 0.167532i
$289$	4912.82	0.999964
$290$	0	0
$291$	−6042.69	−1.21728
$292$	6308.77i	1.26436i
$293$	− 2171.03i	− 0.432877i	−0.976296	−	0.216438i	$-0.930556\pi$
$293$	− 2171.03i	− 0.432877i	0.976296	−	0.216438i	$-0.0694441\pi$
$294$	105.462	0.0209207
$295$	0	0
$296$	−1504.00	−0.295333
$297$	− 8381.79i	− 1.63758i
$298$	1137.09i	0.221040i
$299$	−818.137	−0.158241
$300$	0	0
$301$	2061.49	0.394759
$302$	933.651i	0.177899i
$303$	516.137i	0.0978590i
$304$	−7388.64	−1.39397
$305$	0	0
$306$	−1.87909	−0.000351047 0
$307$	− 3644.59i	− 0.677549i	−0.940868	−	0.338775i	$-0.889988\pi$
$307$	− 3644.59i	− 0.677549i	0.940868	−	0.338775i	$-0.110012\pi$
$308$	2975.39i	0.550451i
$309$	2795.32	0.514628
$310$	0	0
$311$	2584.19	0.471177	0.235589	−	0.971853i	$-0.424298\pi$
$311$	2584.19	0.471177	0.235589	+	0.971853i	$0.424298\pi$
$312$	542.461i	0.0984320i
$313$	− 6693.63i	− 1.20878i	−0.796690	−	0.604388i	$-0.793418\pi$
$313$	− 6693.63i	− 1.20878i	0.796690	−	0.604388i	$-0.206582\pi$
$314$	−1684.76	−0.302791
$315$	0	0
$316$	6650.99	1.18401
$317$	− 3843.36i	− 0.680962i	−0.940251	−	0.340481i	$-0.889410\pi$
$317$	− 3843.36i	− 0.680962i	0.940251	−	0.340481i	$-0.110590\pi$
$318$	− 1423.35i	− 0.250998i
$319$	−2276.15	−0.399498
$320$	0	0
$321$	8825.26	1.53451
$322$	− 180.597i	− 0.0312555i
$323$	53.8126i	0.00927002i
$324$	−3199.26	−0.548570
$325$	0	0
$326$	237.803	0.0404009
$327$	3855.40i	0.652000i
$328$	606.575i	0.102111i
$329$	−3782.08	−0.633777
$330$	0	0
$331$	−821.922	−0.136486	−0.0682431	−	0.997669i	$-0.521739\pi$
$331$	−821.922	−0.136486	−0.0682431	+	0.997669i	$0.521739\pi$
$332$	8177.39i	1.35179i
$333$	− 1669.05i	− 0.274664i
$334$	−1855.95	−0.304051
$335$	0	0
$336$	1729.59	0.280825
$337$	1032.55i	0.166904i	0.996512	+	0.0834520i	$0.0265945\pi$
$337$	1032.55i	0.166904i	−0.996512	+	0.0834520i	$0.973405\pi$
$338$	− 979.667i	− 0.157653i
$339$	3595.87	0.576108
$340$	0	0
$341$	10556.4	1.67643
$342$	567.665i	0.0897538i
$343$	− 343.000i	− 0.0539949i
$344$	−2340.65	−0.366859
$345$	0	0
$346$	−509.382	−0.0791461
$347$	− 2587.70i	− 0.400332i	−0.979762	−	0.200166i	$-0.935852\pi$
$347$	− 2587.70i	− 0.400332i	0.979762	−	0.200166i	$-0.0641482\pi$
$348$	1369.67i	0.210982i
$349$	8454.24	1.29669	0.648345	−	0.761346i	$-0.275461\pi$
$349$	8454.24	1.29669	0.648345	+	0.761346i	$0.275461\pi$
$350$	0	0
$351$	−2444.79	−0.371776
$352$	− 5094.80i	− 0.771460i
$353$	4129.63i	0.622658i	0.950302	+	0.311329i	$0.100774\pi$
$353$	4129.63i	0.622658i	−0.950302	+	0.311329i	$0.899226\pi$
$354$	883.111	0.132590
$355$	0	0
$356$	−2645.50	−0.393852
$357$	− 12.5969i	− 0.00186750i
$358$	1615.95i	0.238563i
$359$	4536.69	0.666957	0.333478	−	0.942758i	$-0.391778\pi$
$359$	4536.69	0.666957	0.333478	+	0.942758i	$0.391778\pi$
$360$	0	0
$361$	9397.56	1.37011
$362$	994.492i	0.144390i
$363$	− 7166.65i	− 1.03623i
$364$	867.859	0.124968
$365$	0	0
$366$	−99.1120	−0.0141548
$367$	1643.00i	0.233689i	0.993150	+	0.116844i	$0.0372779\pi$
$367$	1643.00i	0.233689i	−0.993150	+	0.116844i	$0.962722\pi$
$368$	− 2961.81i	− 0.419552i
$369$	−673.137	−0.0949650
$370$	0	0
$371$	−4629.21	−0.647808
$372$	− 6352.29i	− 0.885352i
$373$	− 9019.53i	− 1.25205i	−0.779804	−	0.626024i	$-0.784681\pi$
$373$	− 9019.53i	− 1.25205i	0.779804	−	0.626024i	$-0.215319\pi$
$374$	−11.6920	−0.00161652
$375$	0	0
$376$	4294.23	0.588984
$377$	663.904i	0.0906971i
$378$	− 539.667i	− 0.0734325i
$379$	2463.63	0.333900	0.166950	−	0.985965i	$-0.446608\pi$
$379$	2463.63	0.333900	0.166950	+	0.985965i	$0.446608\pi$
$380$	0	0
$381$	8657.63	1.16416
$382$	− 1995.61i	− 0.267289i
$383$	11007.2i	1.46852i	0.678867	+	0.734261i	$0.262471\pi$
$383$	11007.2i	1.46852i	−0.678867	+	0.734261i	$0.737529\pi$
$384$	−4063.58	−0.540023
$385$	0	0
$386$	−1224.33	−0.161442
$387$	− 2597.50i	− 0.341184i
$388$	− 10976.6i	− 1.43621i
$389$	−6162.79	−0.803254	−0.401627	−	0.915803i	$-0.631555\pi$
$389$	−6162.79	−0.803254	−0.401627	+	0.915803i	$0.631555\pi$
$390$	0	0
$391$	−21.5713	−0.00279005
$392$	389.448i	0.0501788i
$393$	− 12247.6i	− 1.57204i
$394$	1578.21	0.201799
$395$	0	0
$396$	3749.03	0.475747
$397$	− 15137.9i	− 1.91373i	−0.290537	−	0.956864i	$-0.593834\pi$
$397$	− 15137.9i	− 1.91373i	0.290537	−	0.956864i	$-0.406166\pi$
$398$	− 1800.12i	− 0.226714i
$399$	−3805.47	−0.477473
$400$	0	0
$401$	3302.29	0.411243	0.205622	−	0.978632i	$-0.434078\pi$
$401$	3302.29	0.411243	0.205622	+	0.978632i	$0.434078\pi$
$402$	− 22.3997i	− 0.00277910i
$403$	− 3079.08i	− 0.380596i
$404$	−937.564	−0.115459
$405$	0	0
$406$	−146.551	−0.0179143
$407$	− 10385.1i	− 1.26479i
$408$	14.3027i	0.00173552i
$409$	−4576.40	−0.553273	−0.276636	−	0.960975i	$-0.589220\pi$
$409$	−4576.40	−0.553273	−0.276636	+	0.960975i	$0.589220\pi$
$410$	0	0
$411$	−6914.25	−0.829817
$412$	5077.70i	0.607186i
$413$	− 2872.18i	− 0.342205i
$414$	−227.554	−0.0270137
$415$	0	0
$416$	−1486.05	−0.175143
$417$	11161.4i	1.31074i
$418$	3532.11i	0.413304i
$419$	6778.05	0.790285	0.395142	−	0.918620i	$-0.370695\pi$
$419$	6778.05	0.790285	0.395142	+	0.918620i	$0.370695\pi$
$420$	0	0
$421$	−3952.68	−0.457582	−0.228791	−	0.973476i	$-0.573477\pi$
$421$	−3952.68	−0.457582	−0.228791	+	0.973476i	$0.573477\pi$
$422$	1867.29i	0.215398i
$423$	4765.46i	0.547765i
$424$	5256.08	0.602024
$425$	0	0
$426$	1057.03	0.120219
$427$	322.346i	0.0365326i
$428$	16031.1i	1.81050i
$429$	−3745.67	−0.421544
$430$	0	0
$431$	−3435.10	−0.383905	−0.191952	−	0.981404i	$-0.561482\pi$
$431$	−3435.10	−0.383905	−0.191952	+	0.981404i	$0.561482\pi$
$432$	− 8850.60i	− 0.985705i
$433$	5457.82i	0.605742i	0.953032	+	0.302871i	$0.0979451\pi$
$433$	5457.82i	0.605742i	−0.953032	+	0.302871i	$0.902055\pi$
$434$	679.682	0.0751746
$435$	0	0
$436$	−7003.35	−0.769265
$437$	6516.61i	0.713344i
$438$	− 1753.13i	− 0.191250i
$439$	−4188.10	−0.455323	−0.227662	−	0.973740i	$-0.573108\pi$
$439$	−4188.10	−0.455323	−0.227662	+	0.973740i	$0.573108\pi$
$440$	0	0
$441$	−432.184	−0.0466671
$442$	3.41031i	0 0.000366996i
$443$	− 2097.50i	− 0.224955i	−0.993654	−	0.112478i	$-0.964121\pi$
$443$	− 2097.50i	− 0.224955i	0.993654	−	0.112478i	$-0.0358787\pi$
$444$	−6249.19	−0.667959
$445$	0	0
$446$	51.5085	0.00546860
$447$	9604.74i	1.01631i
$448$	2917.14i	0.307639i
$449$	−11318.2	−1.18962	−0.594809	−	0.803867i	$-0.702772\pi$
$449$	−11318.2	−1.18962	−0.594809	+	0.803867i	$0.702772\pi$
$450$	0	0
$451$	−4188.37	−0.437301
$452$	6531.91i	0.679724i
$453$	7886.34i	0.817952i
$454$	2897.41	0.299521
$455$	0	0
$456$	4320.79	0.443728
$457$	− 11696.4i	− 1.19723i	−0.801036	−	0.598617i	$-0.795717\pi$
$457$	− 11696.4i	− 1.19723i	0.801036	−	0.598617i	$-0.204283\pi$
$458$	− 425.596i	− 0.0434209i
$459$	−64.4603	−0.00655501
$460$	0	0
$461$	10685.0	1.07951	0.539753	−	0.841823i	$-0.318518\pi$
$461$	10685.0	1.07951	0.539753	+	0.841823i	$0.318518\pi$
$462$	− 826.825i	− 0.0832627i
$463$	10407.2i	1.04463i	0.852752	+	0.522315i	$0.174932\pi$
$463$	10407.2i	1.04463i	−0.852752	+	0.522315i	$0.825068\pi$
$464$	−2403.46	−0.240469
$465$	0	0
$466$	1018.48	0.101245
$467$	− 2978.24i	− 0.295111i	−0.989054	−	0.147555i	$-0.952860\pi$
$467$	− 2978.24i	− 0.295111i	0.989054	−	0.147555i	$-0.0471404\pi$
$468$	− 1093.51i	− 0.108008i
$469$	−72.8517	−0.00717266
$470$	0	0
$471$	−14230.8	−1.39219
$472$	3261.12i	0.318020i
$473$	− 16162.1i	− 1.57111i
$474$	−1848.23	−0.179097
$475$	0	0
$476$	22.8823	0.00220338
$477$	5832.86i	0.559892i
$478$	123.008i	0.0117704i
$479$	14214.3	1.35588	0.677940	−	0.735117i	$-0.262873\pi$
$479$	14214.3	1.35588	0.677940	+	0.735117i	$0.262873\pi$
$480$	0	0
$481$	−3029.11	−0.287142
$482$	1544.93i	0.145995i
$483$	− 1525.46i	− 0.143708i
$484$	13018.2	1.22260
$485$	0	0
$486$	−1192.54	−0.111306
$487$	18097.0i	1.68389i	0.539562	+	0.841946i	$0.318590\pi$
$487$	18097.0i	1.68389i	−0.539562	+	0.841946i	$0.681410\pi$
$488$	− 365.998i	− 0.0339507i
$489$	2008.67	0.185757
$490$	0	0
$491$	−13750.8	−1.26388	−0.631940	−	0.775017i	$-0.717741\pi$
$491$	−13750.8	−1.26388	−0.631940	+	0.775017i	$0.717741\pi$
$492$	2520.34i	0.230947i
$493$	17.5048i	0.00159914i
$494$	1030.24	0.0938314
$495$	0	0
$496$	11146.9	1.00909
$497$	− 3437.82i	− 0.310276i
$498$	− 2272.40i	− 0.204475i
$499$	−15684.2	−1.40706	−0.703528	−	0.710667i	$-0.748393\pi$
$499$	−15684.2	−1.40706	−0.703528	+	0.710667i	$0.748393\pi$
$500$	0	0
$501$	−15676.8	−1.39798
$502$	3328.61i	0.295943i
$503$	− 977.937i	− 0.0866880i	−0.999060	−	0.0433440i	$-0.986199\pi$
$503$	− 977.937i	− 0.0866880i	0.999060	−	0.0433440i	$-0.0138011\pi$
$504$	490.709	0.0433689
$505$	0	0
$506$	−1415.88	−0.124394
$507$	− 8275.02i	− 0.724865i
$508$	15726.6i	1.37354i
$509$	19664.6	1.71242	0.856208	−	0.516632i	$-0.172814\pi$
$509$	19664.6	1.71242	0.856208	+	0.516632i	$0.172814\pi$
$510$	0	0
$511$	−5701.78	−0.493604
$512$	− 9064.39i	− 0.782409i
$513$	19473.2i	1.67595i
$514$	258.775	0.0222064
$515$	0	0
$516$	−9725.49	−0.829730
$517$	29651.5i	2.52238i
$518$	− 668.650i	− 0.0567158i
$519$	−4302.63	−0.363901
$520$	0	0
$521$	12522.8	1.05304	0.526519	−	0.850163i	$-0.323497\pi$
$521$	12522.8	1.05304	0.526519	+	0.850163i	$0.323497\pi$
$522$	184.656i	0.0154831i
$523$	− 1104.42i	− 0.0923382i	−0.998934	−	0.0461691i	$-0.985299\pi$
$523$	− 1104.42i	− 0.0923382i	0.998934	−	0.0461691i	$-0.0147013\pi$
$524$	22247.8	1.85477
$525$	0	0
$526$	1797.05	0.148964
$527$	− 81.1843i	− 0.00671052i
$528$	− 13560.0i	− 1.11766i
$529$	9554.75	0.785301
$530$	0	0
$531$	−3618.98	−0.295763
$532$	− 6912.65i	− 0.563349i
$533$	1221.66i	0.0992794i
$534$	735.152	0.0595751
$535$	0	0
$536$	82.7170	0.00666573
$537$	13649.5i	1.09687i
$538$	− 277.502i	− 0.0222378i
$539$	−2689.12	−0.214895
$540$	0	0
$541$	−2126.45	−0.168989	−0.0844945	−	0.996424i	$-0.526928\pi$
$541$	−2126.45	−0.168989	−0.0844945	+	0.996424i	$0.526928\pi$
$542$	1991.33i	0.157813i
$543$	8400.25i	0.663884i
$544$	−39.1817	−0.00308805
$545$	0	0
$546$	−241.167	−0.0189029
$547$	− 2137.04i	− 0.167044i	−0.996506	−	0.0835220i	$-0.973383\pi$
$547$	− 2137.04i	− 0.167044i	0.996506	−	0.0835220i	$-0.0266169\pi$
$548$	− 12559.8i	− 0.979063i
$549$	406.160	0.0315747
$550$	0	0
$551$	5288.11	0.408859
$552$	1732.03i	0.133551i
$553$	6011.08i	0.462237i
$554$	−633.335	−0.0485701
$555$	0	0
$556$	−20274.8	−1.54648
$557$	− 19435.1i	− 1.47844i	−0.673465	−	0.739219i	$-0.735195\pi$
$557$	− 19435.1i	− 1.47844i	0.673465	−	0.739219i	$-0.264805\pi$
$558$	− 856.407i	− 0.0649724i
$559$	−4714.14	−0.356685
$560$	0	0
$561$	−98.7598	−0.00743252
$562$	344.008i	0.0258205i
$563$	− 1467.52i	− 0.109855i	−0.998490	−	0.0549276i	$-0.982507\pi$
$563$	− 1467.52i	− 0.109855i	0.998490	−	0.0549276i	$-0.0174928\pi$
$564$	17842.7	1.33212
$565$	0	0
$566$	3001.89	0.222931
$567$	− 2891.45i	− 0.214161i
$568$	3903.35i	0.288347i
$569$	−15080.3	−1.11107	−0.555536	−	0.831493i	$-0.687487\pi$
$569$	−15080.3	−1.11107	−0.555536	+	0.831493i	$0.687487\pi$
$570$	0	0
$571$	15800.8	1.15804	0.579022	−	0.815312i	$-0.303435\pi$
$571$	15800.8	1.15804	0.579022	+	0.815312i	$0.303435\pi$
$572$	− 6804.02i	− 0.497361i
$573$	− 16856.5i	− 1.22895i
$574$	−269.671	−0.0196095
$575$	0	0
$576$	3675.63	0.265888
$577$	6165.64i	0.444851i	0.974950	+	0.222425i	$0.0713973\pi$
$577$	6165.64i	0.444851i	−0.974950	+	0.222425i	$0.928603\pi$
$578$	− 2479.91i	− 0.178461i
$579$	−10341.6	−0.742283
$580$	0	0
$581$	−7390.62	−0.527736
$582$	3050.25i	0.217246i
$583$	36293.0i	2.57822i
$584$	6473.89	0.458719
$585$	0	0
$586$	−1095.90	−0.0772547
$587$	23104.0i	1.62454i	0.583285	+	0.812268i	$0.301767\pi$
$587$	23104.0i	1.62454i	−0.583285	+	0.812268i	$0.698233\pi$
$588$	1618.17i	0.113490i
$589$	−24525.4	−1.71571
$590$	0	0
$591$	13330.8	0.927841
$592$	− 10965.9i	− 0.761313i
$593$	− 6544.43i	− 0.453200i	−0.973988	−	0.226600i	$-0.927239\pi$
$593$	− 6544.43i	− 0.453200i	0.973988	−	0.226600i	$-0.0727610\pi$
$594$	−4230.99	−0.292255
$595$	0	0
$596$	−17447.0	−1.19909
$597$	− 15205.2i	− 1.04239i
$598$	412.982i	0.0282410i
$599$	−3215.71	−0.219349	−0.109675	−	0.993968i	$-0.534981\pi$
$599$	−3215.71	−0.219349	−0.109675	+	0.993968i	$0.534981\pi$
$600$	0	0
$601$	−234.653	−0.0159263	−0.00796313	−	0.999968i	$-0.502535\pi$
$601$	−234.653	−0.0159263	−0.00796313	+	0.999968i	$0.502535\pi$
$602$	− 1040.61i	− 0.0704517i
$603$	91.7939i	0.00619923i
$604$	−14325.6	−0.965064
$605$	0	0
$606$	260.537	0.0174647
$607$	− 14528.7i	− 0.971501i	−0.874098	−	0.485750i	$-0.838546\pi$
$607$	− 14528.7i	− 0.971501i	0.874098	−	0.485750i	$-0.161454\pi$
$608$	11836.6i	0.789537i
$609$	−1237.89	−0.0823672
$610$	0	0
$611$	8648.71	0.572650
$612$	− 28.8320i	− 0.00190435i
$613$	− 11332.7i	− 0.746696i	−0.927691	−	0.373348i	$-0.878210\pi$
$613$	− 11332.7i	− 0.746696i	0.927691	−	0.373348i	$-0.121790\pi$
$614$	−1839.73	−0.120921
$615$	0	0
$616$	3053.27	0.199708
$617$	− 17400.2i	− 1.13534i	−0.823255	−	0.567672i	$-0.807844\pi$
$617$	− 17400.2i	− 1.13534i	0.823255	−	0.567672i	$-0.192156\pi$
$618$	− 1411.03i	− 0.0918446i
$619$	11987.8	0.778402	0.389201	−	0.921153i	$-0.372751\pi$
$619$	11987.8	0.778402	0.389201	+	0.921153i	$0.372751\pi$
$620$	0	0
$621$	−7806.02	−0.504420
$622$	− 1304.46i	− 0.0840900i
$623$	− 2390.97i	− 0.153759i
$624$	−3955.17	−0.253740
$625$	0	0
$626$	−3378.84	−0.215728
$627$	29834.9i	1.90030i
$628$	− 25850.3i	− 1.64258i
$629$	−79.8667	−0.00506279
$630$	0	0
$631$	−3800.38	−0.239764	−0.119882	−	0.992788i	$-0.538252\pi$
$631$	−3800.38	−0.239764	−0.119882	+	0.992788i	$0.538252\pi$
$632$	− 6825.08i	− 0.429568i
$633$	15772.6i	0.990368i
$634$	−1940.07	−0.121530
$635$	0	0
$636$	21839.2	1.36161
$637$	784.359i	0.0487872i
$638$	1148.96i	0.0712976i
$639$	−4331.69	−0.268167
$640$	0	0
$641$	12294.2	0.757552	0.378776	−	0.925488i	$-0.376345\pi$
$641$	12294.2	0.757552	0.378776	+	0.925488i	$0.376345\pi$
$642$	− 4454.85i	− 0.273861i
$643$	− 14615.4i	− 0.896382i	−0.893938	−	0.448191i	$-0.852068\pi$
$643$	− 14615.4i	− 0.896382i	0.893938	−	0.448191i	$-0.147932\pi$
$644$	2771.01	0.169554
$645$	0	0
$646$	27.1637	0.00165440
$647$	− 16047.0i	− 0.975071i	−0.873103	−	0.487536i	$-0.837896\pi$
$647$	− 16047.0i	− 0.975071i	0.873103	−	0.487536i	$-0.162104\pi$
$648$	3283.00i	0.199025i
$649$	−22517.9	−1.36195
$650$	0	0
$651$	5741.12	0.345641
$652$	3648.75i	0.219166i
$653$	− 623.425i	− 0.0373607i	−0.999826	−	0.0186803i	$-0.994054\pi$
$653$	− 623.425i	− 0.0373607i	0.999826	−	0.0186803i	$-0.00594648\pi$
$654$	1946.14	0.116361
$655$	0	0
$656$	−4422.63	−0.263224
$657$	7184.30i	0.426616i
$658$	1909.13i	0.113109i
$659$	−963.459	−0.0569515	−0.0284758	−	0.999594i	$-0.509065\pi$
$659$	−963.459	−0.0569515	−0.0284758	+	0.999594i	$0.509065\pi$
$660$	0	0
$661$	24183.6	1.42304	0.711522	−	0.702664i	$-0.248006\pi$
$661$	24183.6	1.42304	0.711522	+	0.702664i	$0.248006\pi$
$662$	414.893i	0.0243584i
$663$	28.8061i	0.00168739i
$664$	8391.43	0.490438
$665$	0	0
$666$	−842.507	−0.0490187
$667$	2119.79i	0.123057i
$668$	− 28476.9i	− 1.64941i
$669$	435.080	0.0251438
$670$	0	0
$671$	2527.20	0.145397
$672$	− 2770.81i	− 0.159057i
$673$	− 14167.0i	− 0.811437i	−0.913998	−	0.405718i	$-0.867021\pi$
$673$	− 14167.0i	− 0.811437i	0.913998	−	0.405718i	$-0.132979\pi$
$674$	521.215	0.0297870
$675$	0	0
$676$	15031.6	0.855235
$677$	− 19397.1i	− 1.10117i	−0.834780	−	0.550584i	$-0.814405\pi$
$677$	− 19397.1i	− 1.10117i	0.834780	−	0.550584i	$-0.185595\pi$
$678$	− 1815.14i	− 0.102817i
$679$	9920.47	0.560696
$680$	0	0
$681$	24473.8	1.37715
$682$	− 5328.71i	− 0.299189i
$683$	− 20984.8i	− 1.17564i	−0.808992	−	0.587820i	$-0.799986\pi$
$683$	− 20984.8i	− 1.17564i	0.808992	−	0.587820i	$-0.200014\pi$
$684$	−8710.02	−0.486895
$685$	0	0
$686$	−173.141	−0.00963636
$687$	− 3594.91i	− 0.199643i
$688$	− 17066.1i	− 0.945694i
$689$	10585.9	0.585328
$690$	0	0
$691$	−18079.2	−0.995320	−0.497660	−	0.867372i	$-0.665807\pi$
$691$	−18079.2	−0.995320	−0.497660	+	0.867372i	$0.665807\pi$
$692$	− 7815.75i	− 0.429350i
$693$	3388.32i	0.185731i
$694$	−1306.23	−0.0714465
$695$	0	0
$696$	1405.52	0.0765459
$697$	32.2107i	0.00175046i
$698$	− 4267.56i	− 0.231418i
$699$	8602.86	0.465508
$700$	0	0
$701$	18660.6	1.00542	0.502711	−	0.864455i	$-0.332336\pi$
$701$	18660.6	1.00542	0.502711	+	0.864455i	$0.332336\pi$
$702$	1234.09i	0.0663500i
$703$	24127.4i	1.29443i
$704$	22870.4	1.22438
$705$	0	0
$706$	2084.57	0.111125
$707$	− 847.358i	− 0.0450752i
$708$	13550.1i	0.719271i
$709$	24545.4	1.30017	0.650085	−	0.759862i	$-0.274733\pi$
$709$	24545.4	1.30017	0.650085	+	0.759862i	$0.274733\pi$
$710$	0	0
$711$	7574.02	0.399505
$712$	2714.74i	0.142892i
$713$	− 9831.26i	− 0.516387i
$714$	−6.35871	−0.000333290 0
$715$	0	0
$716$	−24794.5	−1.29415
$717$	1039.02i	0.0541185i
$718$	− 2290.05i	− 0.119030i
$719$	7300.32	0.378659	0.189330	−	0.981914i	$-0.439369\pi$
$719$	7300.32	0.378659	0.189330	+	0.981914i	$0.439369\pi$
$720$	0	0
$721$	−4589.16	−0.237045
$722$	− 4743.74i	− 0.244520i
$723$	13049.7i	0.671263i
$724$	−15259.1	−0.783286
$725$	0	0
$726$	−3617.61	−0.184934
$727$	15466.0i	0.788998i	0.918896	+	0.394499i	$0.129082\pi$
$727$	15466.0i	0.788998i	−0.918896	+	0.394499i	$0.870918\pi$
$728$	− 890.574i	− 0.0453391i
$729$	−21225.8	−1.07838
$730$	0	0
$731$	−124.295	−0.00628893
$732$	− 1520.73i	− 0.0767868i
$733$	13830.5i	0.696917i	0.937324	+	0.348459i	$0.113295\pi$
$733$	13830.5i	0.696917i	−0.937324	+	0.348459i	$0.886705\pi$
$734$	829.358	0.0417060
$735$	0	0
$736$	−4744.83	−0.237631
$737$	571.157i	0.0285466i
$738$	339.788i	0.0169482i
$739$	−11714.5	−0.583118	−0.291559	−	0.956553i	$-0.594174\pi$
$739$	−11714.5	−0.583118	−0.291559	+	0.956553i	$0.594174\pi$
$740$	0	0
$741$	8702.20	0.431422
$742$	2336.75i	0.115613i
$743$	− 28963.4i	− 1.43010i	−0.699074	−	0.715050i	$-0.746404\pi$
$743$	− 28963.4i	− 1.43010i	0.699074	−	0.715050i	$-0.253596\pi$
$744$	−6518.56	−0.321212
$745$	0	0
$746$	−4552.91	−0.223450
$747$	9312.26i	0.456115i
$748$	− 179.397i	− 0.00876928i
$749$	−14488.7	−0.706817
$750$	0	0
$751$	6331.99	0.307667	0.153833	−	0.988097i	$-0.450838\pi$
$751$	6331.99	0.307667	0.153833	+	0.988097i	$0.450838\pi$
$752$	31310.0i	1.51829i
$753$	28116.0i	1.36070i
$754$	335.128	0.0161865
$755$	0	0
$756$	8280.43	0.398355
$757$	− 6256.80i	− 0.300406i	−0.988655	−	0.150203i	$-0.952007\pi$
$757$	− 6256.80i	− 0.300406i	0.988655	−	0.150203i	$-0.0479927\pi$
$758$	− 1243.60i	− 0.0595905i
$759$	−11959.6	−0.571945
$760$	0	0
$761$	13953.7	0.664679	0.332340	−	0.943160i	$-0.392162\pi$
$761$	13953.7	0.664679	0.332340	+	0.943160i	$0.392162\pi$
$762$	− 4370.23i	− 0.207765i
$763$	− 6329.53i	− 0.300320i
$764$	30619.8	1.44998
$765$	0	0
$766$	5556.28	0.262084
$767$	6568.00i	0.309200i
$768$	− 12163.7i	− 0.571512i
$769$	−34278.1	−1.60741	−0.803707	−	0.595025i	$-0.797142\pi$
$769$	−34278.1	−1.60741	−0.803707	+	0.595025i	$0.797142\pi$
$770$	0	0
$771$	2185.82	0.102101
$772$	− 18785.5i	− 0.875786i
$773$	− 8769.31i	− 0.408034i	−0.978967	−	0.204017i	$-0.934600\pi$
$773$	− 8769.31i	− 0.408034i	0.978967	−	0.204017i	$-0.0653997\pi$
$774$	−1311.18	−0.0608905
$775$	0	0
$776$	−11263.9	−0.521069
$777$	− 5647.94i	− 0.260770i
$778$	3110.88i	0.143355i
$779$	9730.73	0.447547
$780$	0	0
$781$	−26952.5	−1.23487
$782$	10.8889i	0 0.000497934i
$783$	6334.45i	0.289112i
$784$	−2839.53	−0.129352
$785$	0	0
$786$	−6182.40	−0.280558
$787$	− 3546.15i	− 0.160618i	−0.996770	−	0.0803092i	$-0.974409\pi$
$787$	− 3546.15i	− 0.160618i	0.996770	−	0.0803092i	$-0.0255908\pi$
$788$	24215.4i	1.09472i
$789$	15179.3	0.684912
$790$	0	0
$791$	−5903.45	−0.265363
$792$	− 3847.16i	− 0.172604i
$793$	− 737.130i	− 0.0330091i
$794$	−7641.37	−0.341539
$795$	0	0
$796$	27620.3	1.22987
$797$	− 18169.4i	− 0.807520i	−0.914865	−	0.403760i	$-0.867703\pi$
$797$	− 18169.4i	− 0.807520i	0.914865	−	0.403760i	$-0.132297\pi$
$798$	1920.94i	0.0852137i
$799$	228.035	0.0100968
$800$	0	0
$801$	−3012.65	−0.132892
$802$	− 1666.94i	− 0.0733938i
$803$	44702.0i	1.96451i
$804$	343.692	0.0150760
$805$	0	0
$806$	−1554.27	−0.0679241
$807$	− 2343.99i	− 0.102246i
$808$	962.104i	0.0418895i
$809$	−11853.0	−0.515115	−0.257557	−	0.966263i	$-0.582918\pi$
$809$	−11853.0	−0.515115	−0.257557	+	0.966263i	$0.582918\pi$
$810$	0	0
$811$	−11921.1	−0.516161	−0.258080	−	0.966123i	$-0.583090\pi$
$811$	−11921.1	−0.516161	−0.258080	+	0.966123i	$0.583090\pi$
$812$	− 2248.62i	− 0.0971813i
$813$	16820.3i	0.725601i
$814$	−5242.22	−0.225724
$815$	0	0
$816$	−104.284	−0.00447384
$817$	37548.9i	1.60792i
$818$	2310.09i	0.0987415i
$819$	988.301	0.0421661
$820$	0	0
$821$	−42094.1	−1.78940	−0.894698	−	0.446671i	$-0.852610\pi$
$821$	−42094.1	−1.78940	−0.894698	+	0.446671i	$0.852610\pi$
$822$	3490.20i	0.148096i
$823$	− 1622.17i	− 0.0687064i	−0.999410	−	0.0343532i	$-0.989063\pi$
$823$	− 1622.17i	− 0.0687064i	0.999410	−	0.0343532i	$-0.0109371\pi$
$824$	5210.61	0.220291
$825$	0	0
$826$	−1449.83	−0.0610727
$827$	28171.7i	1.18455i	0.805735	+	0.592277i	$0.201771\pi$
$827$	28171.7i	1.18455i	−0.805735	+	0.592277i	$0.798229\pi$
$828$	− 3491.50i	− 0.146543i
$829$	9244.23	0.387293	0.193646	−	0.981071i	$-0.437969\pi$
$829$	9244.23	0.387293	0.193646	+	0.981071i	$0.437969\pi$
$830$	0	0
$831$	−5349.63	−0.223317
$832$	− 6670.81i	− 0.277967i
$833$	20.6807i	0 0.000860198i
$834$	5634.10	0.233924
$835$	0	0
$836$	−54195.2	−2.24208
$837$	− 29378.2i	− 1.21321i
$838$	− 3421.45i	− 0.141041i
$839$	43012.5	1.76991	0.884956	−	0.465674i	$-0.154188\pi$
$839$	43012.5	1.76991	0.884956	+	0.465674i	$0.154188\pi$
$840$	0	0
$841$	−22668.8	−0.929469
$842$	1995.25i	0.0816637i
$843$	2905.76i	0.118718i
$844$	−28650.9	−1.16849
$845$	0	0
$846$	2405.53	0.0977585
$847$	11765.7i	0.477302i
$848$	38323.0i	1.55191i
$849$	25356.3	1.02500
$850$	0	0
$851$	−9671.70	−0.389591
$852$	16218.6i	0.652159i
$853$	19756.9i	0.793041i	0.918026	+	0.396521i	$0.129783\pi$
$853$	19756.9i	0.793041i	−0.918026	+	0.396521i	$0.870217\pi$
$854$	162.715	0.00651991
$855$	0	0
$856$	16450.7	0.656862
$857$	325.658i	0.0129805i	0.999979	+	0.00649024i	$0.00206592\pi$
$857$	325.658i	0.0129805i	−0.999979	+	0.00649024i	$0.997934\pi$
$858$	1890.75i	0.0752321i
$859$	−3275.22	−0.130092	−0.0650461	−	0.997882i	$-0.520719\pi$
$859$	−3275.22	−0.130092	−0.0650461	+	0.997882i	$0.520719\pi$
$860$	0	0
$861$	−2277.85	−0.0901613
$862$	1733.98i	0.0685147i
$863$	− 44585.0i	− 1.75862i	−0.476247	−	0.879311i	$-0.658003\pi$
$863$	− 44585.0i	− 1.75862i	0.476247	−	0.879311i	$-0.341997\pi$
$864$	−14178.7	−0.558297
$865$	0	0
$866$	2755.02	0.108106
$867$	− 20947.3i	− 0.820537i
$868$	10428.8i	0.407805i
$869$	47126.9	1.83967
$870$	0	0
$871$	166.594	0.00648087
$872$	7186.65i	0.279095i
$873$	− 12499.9i	− 0.484602i
$874$	3289.48	0.127309
$875$	0	0
$876$	26899.3	1.03749
$877$	16433.8i	0.632761i	0.948632	+	0.316380i	$0.102468\pi$
$877$	16433.8i	0.632761i	−0.948632	+	0.316380i	$0.897532\pi$
$878$	2114.08i	0.0812607i
$879$	−9256.82	−0.355205
$880$	0	0
$881$	−39080.2	−1.49449	−0.747245	−	0.664549i	$-0.768624\pi$
$881$	−39080.2	−1.49449	−0.747245	+	0.664549i	$0.768624\pi$
$882$	218.159i	0.00832858i
$883$	− 48548.2i	− 1.85026i	−0.379652	−	0.925129i	$-0.623956\pi$
$883$	− 48548.2i	− 1.85026i	0.379652	−	0.925129i	$-0.376044\pi$
$884$	−52.3264	−0.00199087
$885$	0	0
$886$	−1058.78	−0.0401473
$887$	− 13447.1i	− 0.509029i	−0.967069	−	0.254514i	$-0.918084\pi$
$887$	− 13447.1i	− 0.509029i	0.967069	−	0.254514i	$-0.0819156\pi$
$888$	6412.76i	0.242340i
$889$	−14213.5	−0.536227
$890$	0	0
$891$	−22669.0	−0.852344
$892$	790.325i	0.0296660i
$893$	− 68888.5i	− 2.58148i
$894$	4848.32	0.181378
$895$	0	0
$896$	6671.30	0.248742
$897$	3488.37i	0.129847i
$898$	5713.24i	0.212309i
$899$	−7977.90	−0.295971
$900$	0	0
$901$	279.112	0.0103203
$902$	2114.22i	0.0780442i
$903$	− 8789.76i	− 0.323926i
$904$	6702.87	0.246609
$905$	0	0
$906$	3980.89	0.145978
$907$	15750.3i	0.576604i	0.957539	+	0.288302i	$0.0930908\pi$
$907$	15750.3i	0.576604i	−0.957539	+	0.288302i	$0.906909\pi$
$908$	44456.7i	1.62483i
$909$	−1067.68	−0.0389579
$910$	0	0
$911$	31449.7	1.14377	0.571886	−	0.820333i	$-0.306212\pi$
$911$	31449.7	1.14377	0.571886	+	0.820333i	$0.306212\pi$
$912$	31503.6i	1.14385i
$913$	57942.4i	2.10034i
$914$	−5904.16	−0.213668
$915$	0	0
$916$	6530.17	0.235549
$917$	20107.3i	0.724102i
$918$	32.5385i	0.00116986i
$919$	30355.6	1.08960	0.544798	−	0.838567i	$-0.316606\pi$
$919$	30355.6	1.08960	0.544798	+	0.838567i	$0.316606\pi$
$920$	0	0
$921$	−15539.8	−0.555975
$922$	− 5393.64i	− 0.192657i
$923$	7861.46i	0.280350i
$924$	12686.5	0.451682
$925$	0	0
$926$	5253.39	0.186433
$927$	5782.39i	0.204874i
$928$	3850.34i	0.136200i
$929$	28533.8	1.00771	0.503856	−	0.863788i	$-0.331914\pi$
$929$	28533.8	1.00771	0.503856	+	0.863788i	$0.331914\pi$
$930$	0	0
$931$	6247.56	0.219931
$932$	15627.1i	0.549231i
$933$	− 11018.5i	− 0.386632i
$934$	−1503.37	−0.0526678
$935$	0	0
$936$	−1122.13	−0.0391860
$937$	− 19269.1i	− 0.671817i	−0.941895	−	0.335908i	$-0.890957\pi$
$937$	− 19269.1i	− 0.671817i	0.941895	−	0.335908i	$-0.109043\pi$
$938$	36.7743i	0.00128009i
$939$	−28540.3	−0.991881
$940$	0	0
$941$	−2097.84	−0.0726756	−0.0363378	−	0.999340i	$-0.511569\pi$
$941$	−2097.84	−0.0726756	−0.0363378	+	0.999340i	$0.511569\pi$
$942$	7183.46i	0.248461i
$943$	3900.66i	0.134701i
$944$	−23777.4	−0.819797
$945$	0	0
$946$	−8158.36	−0.280392
$947$	5072.14i	0.174047i	0.996206	+	0.0870235i	$0.0277355\pi$
$947$	5072.14i	0.174047i	−0.996206	+	0.0870235i	$0.972264\pi$
$948$	− 28358.5i	− 0.971561i
$949$	13038.6	0.445997
$950$	0	0
$951$	−16387.3	−0.558775
$952$	− 23.4813i	0 0.000799403i
$953$	23464.1i	0.797564i	0.917046	+	0.398782i	$0.130567\pi$
$953$	23464.1i	0.797564i	−0.917046	+	0.398782i	$0.869433\pi$
$954$	2944.33	0.0999227
$955$	0	0
$956$	−1887.39	−0.0638519
$957$	9705.02i	0.327815i
$958$	− 7175.13i	− 0.241981i
$959$	11351.3	0.382225
$960$	0	0
$961$	7209.24	0.241994
$962$	1529.04i	0.0512457i
$963$	18255.9i	0.610892i
$964$	−23704.8	−0.791992
$965$	0	0
$966$	−770.028	−0.0256472
$967$	− 6994.27i	− 0.232596i	−0.993214	−	0.116298i	$-0.962897\pi$
$967$	− 6994.27i	− 0.232596i	0.993214	−	0.116298i	$-0.0371028\pi$
$968$	− 13359.0i	− 0.443568i
$969$	229.446	0.00760667
$970$	0	0
$971$	37752.5	1.24772	0.623860	−	0.781536i	$-0.285564\pi$
$971$	37752.5	1.24772	0.623860	+	0.781536i	$0.285564\pi$
$972$	− 18297.8i	− 0.603809i
$973$	− 18324.0i	− 0.603743i
$974$	9135.09	0.300521
$975$	0	0
$976$	2668.55	0.0875186
$977$	− 6380.21i	− 0.208926i	−0.994529	−	0.104463i	$-0.966688\pi$
$977$	− 6380.21i	− 0.208926i	0.994529	−	0.104463i	$-0.0333124\pi$
$978$	− 1013.94i	− 0.0331517i
$979$	−18745.2	−0.611950
$980$	0	0
$981$	−7975.28	−0.259563
$982$	6941.18i	0.225562i
$983$	− 17330.0i	− 0.562299i	−0.959664	−	0.281150i	$-0.909284\pi$
$983$	− 17330.0i	− 0.562299i	0.959664	−	0.281150i	$-0.0907157\pi$
$984$	2586.31	0.0837891
$985$	0	0
$986$	8.83612	0.000285395 0
$987$	16126.0i	0.520057i
$988$	15807.6i	0.509014i
$989$	−15051.9	−0.483945
$990$	0	0
$991$	32191.8	1.03189	0.515947	−	0.856621i	$-0.327440\pi$
$991$	32191.8	1.03189	0.515947	+	0.856621i	$0.327440\pi$
$992$	− 17857.3i	− 0.571542i
$993$	3504.50i	0.111996i
$994$	−1735.35	−0.0553743
$995$	0	0
$996$	34866.7	1.10923
$997$	− 2678.20i	− 0.0850748i	−0.999095	−	0.0425374i	$-0.986456\pi$
$997$	− 2678.20i	− 0.0850748i	0.999095	−	0.0425374i	$-0.0135442\pi$
$998$	7917.13i	0.251114i
$999$	−28901.4	−0.915314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.f.99.4		8
5.2	odd	4		175.4.a.g.1.3	✓	4
5.3	odd	4		175.4.a.h.1.2	yes	4
5.4	even	2	inner	175.4.b.f.99.5		8
15.2	even	4		1575.4.a.bl.1.2		4
15.8	even	4		1575.4.a.bg.1.3		4
35.13	even	4		1225.4.a.bd.1.2		4
35.27	even	4		1225.4.a.z.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.3	✓	4	5.2	odd	4
175.4.a.h.1.2	yes	4	5.3	odd	4
175.4.b.f.99.4		8	1.1	even	1	trivial
175.4.b.f.99.5		8	5.4	even	2	inner
1225.4.a.z.1.3		4	35.27	even	4
1225.4.a.bd.1.2		4	35.13	even	4
1575.4.a.bg.1.3		4	15.8	even	4
1575.4.a.bl.1.2		4	15.2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.504784i q^{2} -4.26379i q^{3} +7.74519 q^{4} -2.15229 q^{6} +7.00000i q^{7} -7.94792i q^{8} +8.82008 q^{9} +54.8800 q^{11} -33.0239i q^{12} -16.0073i q^{13} +3.53349 q^{14} +57.9496 q^{16} -0.422056i q^{17} -4.45223i q^{18} -127.501 q^{19} +29.8465 q^{21} -27.7025i q^{22} -51.1101i q^{23} -33.8883 q^{24} -8.08024 q^{26} -152.729i q^{27} +54.2164i q^{28} -41.4750 q^{29} +192.354 q^{31} -92.8353i q^{32} -233.997i q^{33} -0.213047 q^{34} +68.3132 q^{36} -189.232i q^{37} +64.3605i q^{38} -68.2519 q^{39} -76.3187 q^{41} -15.0660i q^{42} -294.499i q^{43} +425.056 q^{44} -25.7996 q^{46} +540.297i q^{47} -247.085i q^{48} -49.0000 q^{49} -1.79956 q^{51} -123.980i q^{52} +661.316i q^{53} -77.0953 q^{54} +55.6354 q^{56} +543.639i q^{57} +20.9359i q^{58} -410.312 q^{59} +46.0495 q^{61} -97.0974i q^{62} +61.7405i q^{63} +416.735 q^{64} -118.118 q^{66} +10.4074i q^{67} -3.26890i q^{68} -217.923 q^{69} -491.117 q^{71} -70.1012i q^{72} +814.540i q^{73} -95.5215 q^{74} -987.522 q^{76} +384.160i q^{77} +34.4525i q^{78} +858.725 q^{79} -413.064 q^{81} +38.5244i q^{82} +1055.80i q^{83} +231.167 q^{84} -148.658 q^{86} +176.841i q^{87} -436.182i q^{88} -341.567 q^{89} +112.051 q^{91} -395.858i q^{92} -820.159i q^{93} +272.733 q^{94} -395.831 q^{96} -1417.21i q^{97} +24.7344i q^{98} +484.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9} + 200 q^{11} - 56 q^{14} + 320 q^{16} + 58 q^{19} - 42 q^{21} + 42 q^{24} + 1400 q^{26} - 258 q^{29} + 228 q^{31} - 406 q^{34} + 2202 q^{36} - 1348 q^{39} + 1342 q^{41}+ \cdots - 4766 q^{99}+O(q^{100}) \) 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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