LMFDB - Embedded newform 175.4.b.d.99.1

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:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
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			99.1
Root			$-3.70156i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.d.99.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.70156i q^{2} -5.70156i q^{3} -5.70156 q^{4} -21.1047 q^{6} -7.00000i q^{7} -8.50781i q^{8} -5.50781 q^{9} -60.0156 q^{11} +32.5078i q^{12} +0.387503i q^{13} -25.9109 q^{14} -77.1047 q^{16} +35.4922i q^{17} +20.3875i q^{18} +6.08907 q^{19} -39.9109 q^{21} +222.152i q^{22} -31.5078i q^{23} -48.5078 q^{24} +1.43437 q^{26} -122.539i q^{27} +39.9109i q^{28} +292.942 q^{29} +130.303 q^{31} +217.345i q^{32} +342.183i q^{33} +131.377 q^{34} +31.4031 q^{36} -219.989i q^{37} -22.5391i q^{38} +2.20937 q^{39} -447.795 q^{41} +147.733i q^{42} +210.020i q^{43} +342.183 q^{44} -116.628 q^{46} -457.769i q^{47} +439.617i q^{48} -49.0000 q^{49} +202.361 q^{51} -2.20937i q^{52} +144.334i q^{53} -453.586 q^{54} -59.5547 q^{56} -34.7172i q^{57} -1084.34i q^{58} -767.328 q^{59} +667.884 q^{61} -482.325i q^{62} +38.5547i q^{63} +187.680 q^{64} +1266.61 q^{66} +77.4593i q^{67} -202.361i q^{68} -179.644 q^{69} -906.573 q^{71} +46.8594i q^{72} -1029.72i q^{73} -814.303 q^{74} -34.7172 q^{76} +420.109i q^{77} -8.17813i q^{78} +690.764 q^{79} -847.375 q^{81} +1657.54i q^{82} -979.408i q^{83} +227.555 q^{84} +777.403 q^{86} -1670.23i q^{87} +510.602i q^{88} +910.927 q^{89} +2.71252 q^{91} +179.644i q^{92} -742.931i q^{93} -1694.46 q^{94} +1239.21 q^{96} -11.1751i q^{97} +181.377i q^{98} +330.555 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4} - 46 q^{6} + 42 q^{9} - 112 q^{11} - 14 q^{14} - 270 q^{16} + 114 q^{19} - 70 q^{21} - 130 q^{24} - 276 q^{26} + 826 q^{29} - 324 q^{31} - 102 q^{34} + 100 q^{36} - 68 q^{39} - 1010 q^{41}+ \cdots + 874 q^{99}+O(q^{100}) $ 4 * q - 10 * q^4 - 46 * q^6 + 42 * q^9 - 112 * q^11 - 14 * q^14 - 270 * q^16 + 114 * q^19 - 70 * q^21 - 130 * q^24 - 276 * q^26 + 826 * q^29 - 324 * q^31 - 102 * q^34 + 100 * q^36 - 68 * q^39 - 1010 * q^41 + 690 * q^44 - 236 * q^46 - 196 * q^49 + 310 * q^51 - 1110 * q^54 + 210 * q^56 - 380 * q^59 + 1980 * q^61 - 722 * q^64 + 2518 * q^66 - 360 * q^69 + 254 * q^71 - 2412 * q^74 + 2 * q^76 + 2238 * q^79 - 316 * q^81 + 462 * q^84 + 3084 * q^86 + 3426 * q^89 + 728 * q^91 - 4652 * q^94 + 2434 * q^96 + 874 * 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 $

$2$	− 3.70156i	− 1.30870i	−0.756192	−	0.654350i	$-0.772942\pi$
$2$	− 3.70156i	− 1.30870i	0.756192	−	0.654350i	$-0.227058\pi$
$3$	− 5.70156i	− 1.09727i	−0.836063	−	0.548633i	$-0.815148\pi$
$3$	− 5.70156i	− 1.09727i	0.836063	−	0.548633i	$-0.184852\pi$
$4$	−5.70156	−0.712695
$5$	0	0
$5$	0	0
$6$	−21.1047	−1.43599
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	− 8.50781i	− 0.375996i
$9$	−5.50781	−0.203993
$10$	0	0
$11$	−60.0156	−1.64504	−0.822518	−	0.568739i	$-0.807431\pi$
$11$	−60.0156	−1.64504	−0.822518	+	0.568739i	$0.807431\pi$
$12$	32.5078i	0.782016i
$13$	0.387503i	0.00826723i	0.999991	+	0.00413362i	$0.00131577\pi$
$13$	0.387503i	0.00826723i	−0.999991	+	0.00413362i	$0.998684\pi$
$14$	−25.9109	−0.494642
$15$	0	0
$16$	−77.1047	−1.20476
$17$	35.4922i	0.506360i	0.967419	+	0.253180i	$0.0814765\pi$
$17$	35.4922i	0.506360i	−0.967419	+	0.253180i	$0.918524\pi$
$18$	20.3875i	0.266966i
$19$	6.08907	0.0735225	0.0367612	−	0.999324i	$-0.488296\pi$
$19$	6.08907	0.0735225	0.0367612	+	0.999324i	$0.488296\pi$
$20$	0	0
$21$	−39.9109	−0.414728
$22$	222.152i	2.15286i
$23$	− 31.5078i	− 0.285645i	−0.989748	−	0.142822i	$-0.954382\pi$
$23$	− 31.5078i	− 0.285645i	0.989748	−	0.142822i	$-0.0456178\pi$
$24$	−48.5078	−0.412567
$25$	0	0
$26$	1.43437	0.0108193
$27$	− 122.539i	− 0.873432i
$28$	39.9109i	0.269373i
$29$	292.942	1.87579	0.937896	−	0.346915i	$-0.112771\pi$
$29$	292.942	1.87579	0.937896	+	0.346915i	$0.112771\pi$
$30$	0	0
$31$	130.303	0.754940	0.377470	−	0.926022i	$-0.376794\pi$
$31$	130.303	0.754940	0.377470	+	0.926022i	$0.376794\pi$
$32$	217.345i	1.20067i
$33$	342.183i	1.80504i
$34$	131.377	0.662673
$35$	0	0
$36$	31.4031	0.145385
$37$	− 219.989i	− 0.977459i	−0.872435	−	0.488729i	$-0.837461\pi$
$37$	− 219.989i	− 0.977459i	0.872435	−	0.488729i	$-0.162539\pi$
$38$	− 22.5391i	− 0.0962189i
$39$	2.20937	0.00907135
$40$	0	0
$41$	−447.795	−1.70570	−0.852852	−	0.522153i	$-0.825129\pi$
$41$	−447.795	−1.70570	−0.852852	+	0.522153i	$0.825129\pi$
$42$	147.733i	0.542754i
$43$	210.020i	0.744832i	0.928066	+	0.372416i	$0.121471\pi$
$43$	210.020i	0.744832i	−0.928066	+	0.372416i	$0.878529\pi$
$44$	342.183	1.17241
$45$	0	0
$46$	−116.628	−0.373823
$47$	− 457.769i	− 1.42069i	−0.703854	−	0.710345i	$-0.748539\pi$
$47$	− 457.769i	− 1.42069i	0.703854	−	0.710345i	$-0.251461\pi$
$48$	439.617i	1.32194i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	202.361	0.555612
$52$	− 2.20937i	− 0.00589202i
$53$	144.334i	0.374073i	0.982353	+	0.187036i	$0.0598882\pi$
$53$	144.334i	0.374073i	−0.982353	+	0.187036i	$0.940112\pi$
$54$	−453.586	−1.14306
$55$	0	0
$56$	−59.5547	−0.142113
$57$	− 34.7172i	− 0.0806737i
$58$	− 1084.34i	− 2.45485i
$59$	−767.328	−1.69318	−0.846590	−	0.532246i	$-0.821348\pi$
$59$	−767.328	−1.69318	−0.846590	+	0.532246i	$0.821348\pi$
$60$	0	0
$61$	667.884	1.40187	0.700933	−	0.713227i	$-0.252767\pi$
$61$	667.884	1.40187	0.700933	+	0.713227i	$0.252767\pi$
$62$	− 482.325i	− 0.987989i
$63$	38.5547i	0.0771021i
$64$	187.680	0.366562
$65$	0	0
$66$	1266.61	2.36226
$67$	77.4593i	0.141241i	0.997503	+	0.0706206i	$0.0224980\pi$
$67$	77.4593i	0.141241i	−0.997503	+	0.0706206i	$0.977502\pi$
$68$	− 202.361i	− 0.360880i
$69$	−179.644	−0.313428
$70$	0	0
$71$	−906.573	−1.51536	−0.757679	−	0.652627i	$-0.773667\pi$
$71$	−906.573	−1.51536	−0.757679	+	0.652627i	$0.773667\pi$
$72$	46.8594i	0.0767005i
$73$	− 1029.72i	− 1.65095i	−0.564436	−	0.825477i	$-0.690906\pi$
$73$	− 1029.72i	− 1.65095i	0.564436	−	0.825477i	$-0.309094\pi$
$74$	−814.303	−1.27920
$75$	0	0
$76$	−34.7172	−0.0523991
$77$	420.109i	0.621765i
$78$	− 8.17813i	− 0.0118717i
$79$	690.764	0.983760	0.491880	−	0.870663i	$-0.336310\pi$
$79$	690.764	0.983760	0.491880	+	0.870663i	$0.336310\pi$
$80$	0	0
$81$	−847.375	−1.16238
$82$	1657.54i	2.23225i
$83$	− 979.408i	− 1.29523i	−0.761968	−	0.647614i	$-0.775767\pi$
$83$	− 979.408i	− 1.29523i	0.761968	−	0.647614i	$-0.224233\pi$
$84$	227.555	0.295574
$85$	0	0
$86$	777.403	0.974762
$87$	− 1670.23i	− 2.05824i
$88$	510.602i	0.618526i
$89$	910.927	1.08492	0.542461	−	0.840081i	$-0.317493\pi$
$89$	910.927	1.08492	0.542461	+	0.840081i	$0.317493\pi$
$90$	0	0
$91$	2.71252	0.00312472
$92$	179.644i	0.203578i
$93$	− 742.931i	− 0.828370i
$94$	−1694.46	−1.85926
$95$	0	0
$96$	1239.21	1.31746
$97$	− 11.1751i	− 0.0116975i	−0.999983	−	0.00584876i	$-0.998138\pi$
$97$	− 11.1751i	− 0.0116975i	0.999983	−	0.00584876i	$-0.00186173\pi$
$98$	181.377i	0.186957i
$99$	330.555	0.335576
$100$	0	0
$101$	675.850	0.665837	0.332919	−	0.942956i	$-0.391967\pi$
$101$	675.850	0.665837	0.332919	+	0.942956i	$0.391967\pi$
$102$	− 749.052i	− 0.727129i
$103$	− 1528.88i	− 1.46257i	−0.682071	−	0.731286i	$-0.738921\pi$
$103$	− 1528.88i	− 1.46257i	0.682071	−	0.731286i	$-0.261079\pi$
$104$	3.29680	0.00310844
$105$	0	0
$106$	534.263	0.489549
$107$	− 701.595i	− 0.633886i	−0.948445	−	0.316943i	$-0.897344\pi$
$107$	− 701.595i	− 0.633886i	0.948445	−	0.316943i	$-0.102656\pi$
$108$	698.664i	0.622491i
$109$	597.130	0.524722	0.262361	−	0.964970i	$-0.415499\pi$
$109$	597.130	0.524722	0.262361	+	0.964970i	$0.415499\pi$
$110$	0	0
$111$	−1254.28	−1.07253
$112$	539.733i	0.455357i
$113$	88.1187i	0.0733585i	0.999327	+	0.0366792i	$0.0116780\pi$
$113$	88.1187i	0.0733585i	−0.999327	+	0.0366792i	$0.988322\pi$
$114$	−128.508	−0.105578
$115$	0	0
$116$	−1670.23	−1.33687
$117$	− 2.13429i	− 0.00168646i
$118$	2840.31i	2.21586i
$119$	248.445	0.191386
$120$	0	0
$121$	2270.87	1.70614
$122$	− 2472.22i	− 1.83462i
$123$	2553.13i	1.87161i
$124$	−742.931	−0.538042
$125$	0	0
$126$	142.713	0.100904
$127$	2119.35i	1.48080i	0.672166	+	0.740401i	$0.265364\pi$
$127$	2119.35i	1.48080i	−0.672166	+	0.740401i	$0.734636\pi$
$128$	1044.05i	0.720955i
$129$	1197.44	0.817279
$130$	0	0
$131$	−264.294	−0.176271	−0.0881353	−	0.996109i	$-0.528091\pi$
$131$	−264.294	−0.176271	−0.0881353	+	0.996109i	$0.528091\pi$
$132$	− 1950.98i	− 1.28644i
$133$	− 42.6235i	− 0.0277889i
$134$	286.720	0.184842
$135$	0	0
$136$	301.961	0.190389
$137$	− 468.227i	− 0.291995i	−0.989285	−	0.145997i	$-0.953361\pi$
$137$	− 468.227i	− 0.291995i	0.989285	−	0.145997i	$-0.0466391\pi$
$138$	664.962i	0.410184i
$139$	−956.908	−0.583913	−0.291956	−	0.956432i	$-0.594306\pi$
$139$	−956.908	−0.583913	−0.291956	+	0.956432i	$0.594306\pi$
$140$	0	0
$141$	−2610.00	−1.55888
$142$	3355.74i	1.98315i
$143$	− 23.2562i	− 0.0135999i
$144$	424.678	0.245763
$145$	0	0
$146$	−3811.57	−2.16060
$147$	279.377i	0.156752i
$148$	1254.28i	0.696630i
$149$	382.967	0.210563	0.105282	−	0.994442i	$-0.466426\pi$
$149$	382.967	0.210563	0.105282	+	0.994442i	$0.466426\pi$
$150$	0	0
$151$	1302.98	0.702218	0.351109	−	0.936335i	$-0.385805\pi$
$151$	1302.98	0.702218	0.351109	+	0.936335i	$0.385805\pi$
$152$	− 51.8046i	− 0.0276441i
$153$	− 195.484i	− 0.103294i
$154$	1555.06	0.813704
$155$	0	0
$156$	−12.5969	−0.00646511
$157$	3203.32i	1.62836i	0.580610	+	0.814182i	$0.302814\pi$
$157$	3203.32i	1.62836i	−0.580610	+	0.814182i	$0.697186\pi$
$158$	− 2556.91i	− 1.28745i
$159$	822.931	0.410457
$160$	0	0
$161$	−220.555	−0.107964
$162$	3136.61i	1.52121i
$163$	1368.81i	0.657752i	0.944373	+	0.328876i	$0.106670\pi$
$163$	1368.81i	0.657752i	−0.944373	+	0.328876i	$0.893330\pi$
$164$	2553.13	1.21565
$165$	0	0
$166$	−3625.34	−1.69507
$167$	− 3073.34i	− 1.42409i	−0.702136	−	0.712043i	$-0.747770\pi$
$167$	− 3073.34i	− 1.42409i	0.702136	−	0.712043i	$-0.252230\pi$
$168$	339.555i	0.155936i
$169$	2196.85	0.999932
$170$	0	0
$171$	−33.5374	−0.0149981
$172$	− 1197.44i	− 0.530839i
$173$	− 2402.16i	− 1.05568i	−0.849343	−	0.527841i	$-0.823002\pi$
$173$	− 2402.16i	− 1.05568i	0.849343	−	0.527841i	$-0.176998\pi$
$174$	−6182.45	−2.69362
$175$	0	0
$176$	4627.49	1.98187
$177$	4374.97i	1.85787i
$178$	− 3371.85i	− 1.41984i
$179$	1146.25	0.478628	0.239314	−	0.970942i	$-0.423077\pi$
$179$	1146.25	0.478628	0.239314	+	0.970942i	$0.423077\pi$
$180$	0	0
$181$	−475.847	−0.195411	−0.0977056	−	0.995215i	$-0.531150\pi$
$181$	−475.847	−0.195411	−0.0977056	+	0.995215i	$0.531150\pi$
$182$	− 10.0406i	− 0.00408932i
$183$	− 3807.98i	− 1.53822i
$184$	−268.062	−0.107401
$185$	0	0
$186$	−2750.01	−1.08409
$187$	− 2130.09i	− 0.832980i
$188$	2610.00i	1.01252i
$189$	−857.773	−0.330126
$190$	0	0
$191$	990.003	0.375048	0.187524	−	0.982260i	$-0.439954\pi$
$191$	990.003	0.375048	0.187524	+	0.982260i	$0.439954\pi$
$192$	− 1070.07i	− 0.402216i
$193$	1392.88i	0.519492i	0.965677	+	0.259746i	$0.0836389\pi$
$193$	1392.88i	0.519492i	−0.965677	+	0.259746i	$0.916361\pi$
$194$	−41.3653	−0.0153085
$195$	0	0
$196$	279.377	0.101814
$197$	3583.39i	1.29597i	0.761653	+	0.647985i	$0.224388\pi$
$197$	3583.39i	1.29597i	−0.761653	+	0.647985i	$0.775612\pi$
$198$	− 1223.57i	− 0.439168i
$199$	−623.947	−0.222263	−0.111132	−	0.993806i	$-0.535448\pi$
$199$	−623.947	−0.222263	−0.111132	+	0.993806i	$0.535448\pi$
$200$	0	0
$201$	441.639	0.154979
$202$	− 2501.70i	− 0.871381i
$203$	− 2050.60i	− 0.708983i
$204$	−1153.77	−0.395982
$205$	0	0
$206$	−5659.24	−1.91407
$207$	173.539i	0.0582696i
$208$	− 29.8783i	− 0.00996004i
$209$	−365.439	−0.120947
$210$	0	0
$211$	2266.48	0.739482	0.369741	−	0.929135i	$-0.379446\pi$
$211$	2266.48	0.739482	0.369741	+	0.929135i	$0.379446\pi$
$212$	− 822.931i	− 0.266600i
$213$	5168.88i	1.66275i
$214$	−2597.00	−0.829566
$215$	0	0
$216$	−1042.54	−0.328406
$217$	− 912.122i	− 0.285340i
$218$	− 2210.31i	− 0.686703i
$219$	−5871.01	−1.81154
$220$	0	0
$221$	−13.7533	−0.00418620
$222$	4642.80i	1.40362i
$223$	2118.67i	0.636217i	0.948054	+	0.318108i	$0.103048\pi$
$223$	2118.67i	0.636217i	−0.948054	+	0.318108i	$0.896952\pi$
$224$	1521.42	0.453812
$225$	0	0
$226$	326.177	0.0960042
$227$	− 2790.62i	− 0.815948i	−0.912994	−	0.407974i	$-0.866235\pi$
$227$	− 2790.62i	− 0.815948i	0.912994	−	0.407974i	$-0.133765\pi$
$228$	197.942i	0.0574958i
$229$	5813.77	1.67766	0.838832	−	0.544390i	$-0.183239\pi$
$229$	5813.77	1.67766	0.838832	+	0.544390i	$0.183239\pi$
$230$	0	0
$231$	2395.28	0.682242
$232$	− 2492.30i	− 0.705290i
$233$	1936.18i	0.544392i	0.962242	+	0.272196i	$0.0877499\pi$
$233$	1936.18i	0.544392i	−0.962242	+	0.272196i	$0.912250\pi$
$234$	−7.90022	−0.00220707
$235$	0	0
$236$	4374.97	1.20672
$237$	− 3938.43i	− 1.07945i
$238$	− 919.636i	− 0.250467i
$239$	2755.79	0.745845	0.372923	−	0.927862i	$-0.378356\pi$
$239$	2755.79	0.745845	0.372923	+	0.927862i	$0.378356\pi$
$240$	0	0
$241$	−4025.23	−1.07588	−0.537942	−	0.842982i	$-0.680798\pi$
$241$	−4025.23	−1.07588	−0.537942	+	0.842982i	$0.680798\pi$
$242$	− 8405.78i	− 2.23283i
$243$	1522.81i	0.402009i
$244$	−3807.98	−0.999103
$245$	0	0
$246$	9450.58	2.44938
$247$	2.35953i	0 0.000607827i
$248$	− 1108.59i	− 0.283854i
$249$	−5584.15	−1.42121
$250$	0	0
$251$	7166.64	1.80221	0.901104	−	0.433602i	$-0.142758\pi$
$251$	7166.64	1.80221	0.901104	+	0.433602i	$0.142758\pi$
$252$	− 219.822i	− 0.0549503i
$253$	1890.96i	0.469896i
$254$	7844.90	1.93792
$255$	0	0
$256$	5366.07	1.31008
$257$	− 647.756i	− 0.157221i	−0.996905	−	0.0786107i	$-0.974952\pi$
$257$	− 647.756i	− 0.157221i	0.996905	−	0.0786107i	$-0.0250484\pi$
$258$	− 4432.41i	− 1.06957i
$259$	−1539.92	−0.369445
$260$	0	0
$261$	−1613.47	−0.382649
$262$	978.300i	0.230685i
$263$	− 6131.33i	− 1.43754i	−0.695246	−	0.718771i	$-0.744705\pi$
$263$	− 6131.33i	− 1.43754i	0.695246	−	0.718771i	$-0.255295\pi$
$264$	2911.23	0.678688
$265$	0	0
$266$	−157.773	−0.0363673
$267$	− 5193.70i	− 1.19045i
$268$	− 441.639i	− 0.100662i
$269$	975.631	0.221135	0.110567	−	0.993869i	$-0.464733\pi$
$269$	975.631	0.221135	0.110567	+	0.993869i	$0.464733\pi$
$270$	0	0
$271$	4672.83	1.04743	0.523716	−	0.851893i	$-0.324545\pi$
$271$	4672.83	1.04743	0.523716	+	0.851893i	$0.324545\pi$
$272$	− 2736.61i	− 0.610043i
$273$	− 15.4656i	− 0.00342865i
$274$	−1733.17	−0.382134
$275$	0	0
$276$	1024.25	0.223379
$277$	− 2390.16i	− 0.518451i	−0.965817	−	0.259226i	$-0.916533\pi$
$277$	− 2390.16i	− 0.518451i	0.965817	−	0.259226i	$-0.0834673\pi$
$278$	3542.05i	0.764166i
$279$	−717.685	−0.154002
$280$	0	0
$281$	−3321.28	−0.705092	−0.352546	−	0.935794i	$-0.614684\pi$
$281$	−3321.28	−0.705092	−0.352546	+	0.935794i	$0.614684\pi$
$282$	9661.07i	2.04010i
$283$	8260.15i	1.73504i	0.497406	+	0.867518i	$0.334286\pi$
$283$	8260.15i	1.73504i	−0.497406	+	0.867518i	$0.665714\pi$
$284$	5168.88	1.07999
$285$	0	0
$286$	−86.0844	−0.0177982
$287$	3134.57i	0.644696i
$288$	− 1197.10i	− 0.244929i
$289$	3653.30	0.743600
$290$	0	0
$291$	−63.7155	−0.0128353
$292$	5871.01i	1.17663i
$293$	− 628.249i	− 0.125265i	−0.998037	−	0.0626326i	$-0.980050\pi$
$293$	− 628.249i	− 0.125265i	0.998037	−	0.0626326i	$-0.0199496\pi$
$294$	1034.13	0.205142
$295$	0	0
$296$	−1871.63	−0.367520
$297$	7354.26i	1.43683i
$298$	− 1417.58i	− 0.275564i
$299$	12.2094	0.00236149
$300$	0	0
$301$	1470.14	0.281520
$302$	− 4823.06i	− 0.918993i
$303$	− 3853.40i	− 0.730601i
$304$	−469.495	−0.0885770
$305$	0	0
$306$	−723.597	−0.135181
$307$	− 25.2045i	− 0.00468565i	−0.999997	−	0.00234283i	$-0.999254\pi$
$307$	− 25.2045i	− 0.00468565i	0.999997	−	0.00234283i	$-0.000745745\pi$
$308$	− 2395.28i	− 0.443129i
$309$	−8716.99	−1.60483
$310$	0	0
$311$	−9436.18	−1.72050	−0.860252	−	0.509870i	$-0.829694\pi$
$311$	−9436.18	−1.72050	−0.860252	+	0.509870i	$0.829694\pi$
$312$	− 18.7969i	− 0.00341079i
$313$	− 2778.60i	− 0.501776i	−0.968016	−	0.250888i	$-0.919277\pi$
$313$	− 2778.60i	− 0.501776i	0.968016	−	0.250888i	$-0.0807225\pi$
$314$	11857.3	2.13104
$315$	0	0
$316$	−3938.43	−0.701121
$317$	2980.21i	0.528030i	0.964519	+	0.264015i	$0.0850468\pi$
$317$	2980.21i	0.528030i	−0.964519	+	0.264015i	$0.914953\pi$
$318$	− 3046.13i	− 0.537165i
$319$	−17581.1	−3.08575
$320$	0	0
$321$	−4000.19	−0.695541
$322$	816.397i	0.141292i
$323$	216.114i	0.0372289i
$324$	4831.36	0.828423
$325$	0	0
$326$	5066.74	0.860800
$327$	− 3404.57i	− 0.575759i
$328$	3809.76i	0.641337i
$329$	−3204.38	−0.536970
$330$	0	0
$331$	2588.16	0.429782	0.214891	−	0.976638i	$-0.431060\pi$
$331$	2588.16	0.429782	0.214891	+	0.976638i	$0.431060\pi$
$332$	5584.15i	0.923103i
$333$	1211.66i	0.199395i
$334$	−11376.2	−1.86370
$335$	0	0
$336$	3077.32	0.499648
$337$	− 5284.64i	− 0.854221i	−0.904199	−	0.427110i	$-0.859532\pi$
$337$	− 5284.64i	− 0.854221i	0.904199	−	0.427110i	$-0.140468\pi$
$338$	− 8131.78i	− 1.30861i
$339$	502.414	0.0804938
$340$	0	0
$341$	−7820.22	−1.24190
$342$	124.141i	0.0196280i
$343$	343.000i	0.0539949i
$344$	1786.81	0.280054
$345$	0	0
$346$	−8891.75	−1.38157
$347$	11787.7i	1.82363i	0.410606	+	0.911813i	$0.365317\pi$
$347$	11787.7i	1.82363i	−0.410606	+	0.911813i	$0.634683\pi$
$348$	9522.91i	1.46690i
$349$	−8765.61	−1.34445	−0.672224	−	0.740347i	$-0.734661\pi$
$349$	−8765.61	−1.34445	−0.672224	+	0.740347i	$0.734661\pi$
$350$	0	0
$351$	47.4843	0.00722086
$352$	− 13044.1i	− 1.97515i
$353$	8578.29i	1.29342i	0.762737	+	0.646709i	$0.223855\pi$
$353$	8578.29i	1.29342i	−0.762737	+	0.646709i	$0.776145\pi$
$354$	16194.2	2.43139
$355$	0	0
$356$	−5193.70	−0.773218
$357$	− 1416.53i	− 0.210001i
$358$	− 4242.90i	− 0.626380i
$359$	−3730.38	−0.548417	−0.274209	−	0.961670i	$-0.588416\pi$
$359$	−3730.38	−0.548417	−0.274209	+	0.961670i	$0.588416\pi$
$360$	0	0
$361$	−6821.92	−0.994594
$362$	1761.38i	0.255735i
$363$	− 12947.5i	− 1.87209i
$364$	−15.4656	−0.00222697
$365$	0	0
$366$	−14095.5	−2.01307
$367$	515.769i	0.0733594i	0.999327	+	0.0366797i	$0.0116781\pi$
$367$	515.769i	0.0733594i	−0.999327	+	0.0366797i	$0.988322\pi$
$368$	2429.40i	0.344134i
$369$	2466.37	0.347952
$370$	0	0
$371$	1010.34	0.141386
$372$	4235.87i	0.590375i
$373$	10922.0i	1.51614i	0.652173	+	0.758071i	$0.273858\pi$
$373$	10922.0i	1.51614i	−0.652173	+	0.758071i	$0.726142\pi$
$374$	−7884.64	−1.09012
$375$	0	0
$376$	−3894.61	−0.534173
$377$	113.516i	0.0155076i
$378$	3175.10i	0.432036i
$379$	8403.14	1.13889	0.569446	−	0.822029i	$-0.307158\pi$
$379$	8403.14	1.13889	0.569446	+	0.822029i	$0.307158\pi$
$380$	0	0
$381$	12083.6	1.62483
$382$	− 3664.56i	− 0.490825i
$383$	− 3030.68i	− 0.404336i	−0.979351	−	0.202168i	$-0.935201\pi$
$383$	− 3030.68i	− 0.404336i	0.979351	−	0.202168i	$-0.0647986\pi$
$384$	5952.74	0.791080
$385$	0	0
$386$	5155.85	0.679859
$387$	− 1156.75i	− 0.151941i
$388$	63.7155i	0.00833677i
$389$	−1403.25	−0.182899	−0.0914497	−	0.995810i	$-0.529150\pi$
$389$	−1403.25	−0.182899	−0.0914497	+	0.995810i	$0.529150\pi$
$390$	0	0
$391$	1118.28	0.144639
$392$	416.883i	0.0537137i
$393$	1506.89i	0.193416i
$394$	13264.1	1.69604
$395$	0	0
$396$	−1884.68	−0.239163
$397$	− 55.9118i	− 0.00706835i	−0.999994	−	0.00353417i	$-0.998875\pi$
$397$	− 55.9118i	− 0.00706835i	0.999994	−	0.00353417i	$-0.00112497\pi$
$398$	2309.58i	0.290876i
$399$	−243.020	−0.0304918
$400$	0	0
$401$	−3730.18	−0.464529	−0.232264	−	0.972653i	$-0.574613\pi$
$401$	−3730.18	−0.464529	−0.232264	+	0.972653i	$0.574613\pi$
$402$	− 1634.75i	− 0.202821i
$403$	50.4928i	0.00624126i
$404$	−3853.40	−0.474539
$405$	0	0
$406$	−7590.41	−0.927846
$407$	13202.8i	1.60795i
$408$	− 1721.65i	− 0.208908i
$409$	1968.50	0.237985	0.118993	−	0.992895i	$-0.462033\pi$
$409$	1968.50	0.237985	0.118993	+	0.992895i	$0.462033\pi$
$410$	0	0
$411$	−2669.62	−0.320396
$412$	8716.99i	1.04237i
$413$	5371.30i	0.639962i
$414$	642.366	0.0762574
$415$	0	0
$416$	−84.2220	−0.00992625
$417$	5455.87i	0.640708i
$418$	1352.70i	0.158283i
$419$	13208.4	1.54003	0.770015	−	0.638026i	$-0.220249\pi$
$419$	13208.4	1.54003	0.770015	+	0.638026i	$0.220249\pi$
$420$	0	0
$421$	−7485.74	−0.866586	−0.433293	−	0.901253i	$-0.642648\pi$
$421$	−7485.74	−0.866586	−0.433293	+	0.901253i	$0.642648\pi$
$422$	− 8389.50i	− 0.967760i
$423$	2521.30i	0.289811i
$424$	1227.97	0.140650
$425$	0	0
$426$	19132.9	2.17604
$427$	− 4675.19i	− 0.529856i
$428$	4000.19i	0.451767i
$429$	−132.597	−0.0149227
$430$	0	0
$431$	10623.4	1.18726	0.593631	−	0.804737i	$-0.297694\pi$
$431$	10623.4	1.18726	0.593631	+	0.804737i	$0.297694\pi$
$432$	9448.34i	1.05228i
$433$	7268.73i	0.806727i	0.915040	+	0.403364i	$0.132159\pi$
$433$	7268.73i	0.806727i	−0.915040	+	0.403364i	$0.867841\pi$
$434$	−3376.28	−0.373425
$435$	0	0
$436$	−3404.57	−0.373967
$437$	− 191.853i	− 0.0210013i
$438$	21731.9i	2.37076i
$439$	743.352	0.0808161	0.0404080	−	0.999183i	$-0.487134\pi$
$439$	743.352	0.0808161	0.0404080	+	0.999183i	$0.487134\pi$
$440$	0	0
$441$	269.883	0.0291419
$442$	50.9088i	0.00547847i
$443$	10857.8i	1.16449i	0.813013	+	0.582246i	$0.197826\pi$
$443$	10857.8i	1.16449i	−0.813013	+	0.582246i	$0.802174\pi$
$444$	7151.36	0.764389
$445$	0	0
$446$	7842.37	0.832617
$447$	− 2183.51i	− 0.231044i
$448$	− 1313.76i	− 0.138547i
$449$	11162.9	1.17329	0.586645	−	0.809844i	$-0.300448\pi$
$449$	11162.9	1.17329	0.586645	+	0.809844i	$0.300448\pi$
$450$	0	0
$451$	26874.7	2.80594
$452$	− 502.414i	− 0.0522822i
$453$	− 7429.02i	− 0.770520i
$454$	−10329.7	−1.06783
$455$	0	0
$456$	−295.367	−0.0303330
$457$	− 13451.0i	− 1.37683i	−0.725318	−	0.688414i	$-0.758307\pi$
$457$	− 13451.0i	− 1.37683i	0.725318	−	0.688414i	$-0.241693\pi$
$458$	− 21520.0i	− 2.19556i
$459$	4349.18	0.442271
$460$	0	0
$461$	−9139.46	−0.923356	−0.461678	−	0.887048i	$-0.652752\pi$
$461$	−9139.46	−0.923356	−0.461678	+	0.887048i	$0.652752\pi$
$462$	− 8866.28i	− 0.892850i
$463$	− 13122.5i	− 1.31718i	−0.752504	−	0.658588i	$-0.771154\pi$
$463$	− 13122.5i	− 1.31718i	0.752504	−	0.658588i	$-0.228846\pi$
$464$	−22587.2	−2.25988
$465$	0	0
$466$	7166.89	0.712446
$467$	− 11921.2i	− 1.18126i	−0.806943	−	0.590629i	$-0.798880\pi$
$467$	− 11921.2i	− 1.18126i	0.806943	−	0.590629i	$-0.201120\pi$
$468$	12.1688i	0.00120193i
$469$	542.215	0.0533842
$470$	0	0
$471$	18264.0	1.78675
$472$	6528.28i	0.636628i
$473$	− 12604.5i	− 1.22528i
$474$	−14578.4	−1.41267
$475$	0	0
$476$	−1416.53	−0.136400
$477$	− 794.966i	− 0.0763082i
$478$	− 10200.7i	− 0.976088i
$479$	−825.281	−0.0787224	−0.0393612	−	0.999225i	$-0.512532\pi$
$479$	−825.281	−0.0787224	−0.0393612	+	0.999225i	$0.512532\pi$
$480$	0	0
$481$	85.2464	0.00808088
$482$	14899.6i	1.40801i
$483$	1257.51i	0.118465i
$484$	−12947.5	−1.21596
$485$	0	0
$486$	5636.76	0.526108
$487$	6678.32i	0.621404i	0.950507	+	0.310702i	$0.100564\pi$
$487$	6678.32i	0.621404i	−0.950507	+	0.310702i	$0.899436\pi$
$488$	− 5682.23i	− 0.527096i
$489$	7804.36	0.721729
$490$	0	0
$491$	−3098.65	−0.284807	−0.142403	−	0.989809i	$-0.545483\pi$
$491$	−3098.65	−0.284807	−0.142403	+	0.989809i	$0.545483\pi$
$492$	− 14556.8i	− 1.33389i
$493$	10397.2i	0.949827i
$494$	8.73395	0.000795464 0
$495$	0	0
$496$	−10047.0	−0.909522
$497$	6346.01i	0.572752i
$498$	20670.1i	1.85994i
$499$	2265.10	0.203206	0.101603	−	0.994825i	$-0.467603\pi$
$499$	2265.10	0.203206	0.101603	+	0.994825i	$0.467603\pi$
$500$	0	0
$501$	−17522.8	−1.56260
$502$	− 26527.8i	− 2.35855i
$503$	− 9980.43i	− 0.884703i	−0.896842	−	0.442351i	$-0.854144\pi$
$503$	− 9980.43i	− 0.884703i	0.896842	−	0.442351i	$-0.145856\pi$
$504$	328.016	0.0289901
$505$	0	0
$506$	6999.51	0.614953
$507$	− 12525.5i	− 1.09719i
$508$	− 12083.6i	− 1.05536i
$509$	4956.16	0.431588	0.215794	−	0.976439i	$-0.430766\pi$
$509$	4956.16	0.431588	0.215794	+	0.976439i	$0.430766\pi$
$510$	0	0
$511$	−7208.04	−0.624002
$512$	− 11510.4i	− 0.993541i
$513$	− 746.148i	− 0.0642169i
$514$	−2397.71	−0.205756
$515$	0	0
$516$	−6827.30	−0.582471
$517$	27473.3i	2.33709i
$518$	5700.12i	0.483492i
$519$	−13696.1	−1.15836
$520$	0	0
$521$	3442.97	0.289519	0.144759	−	0.989467i	$-0.453759\pi$
$521$	3442.97	0.289519	0.144759	+	0.989467i	$0.453759\pi$
$522$	5972.36i	0.500772i
$523$	− 11109.9i	− 0.928878i	−0.885605	−	0.464439i	$-0.846256\pi$
$523$	− 11109.9i	− 0.928878i	0.885605	−	0.464439i	$-0.153744\pi$
$524$	1506.89	0.125627
$525$	0	0
$526$	−22695.5	−1.88131
$527$	4624.74i	0.382271i
$528$	− 26383.9i	− 2.17464i
$529$	11174.3	0.918407
$530$	0	0
$531$	4226.30	0.345397
$532$	243.020i	0.0198050i
$533$	− 173.522i	− 0.0141015i
$534$	−19224.8	−1.55794
$535$	0	0
$536$	659.009	0.0531061
$537$	− 6535.39i	− 0.525182i
$538$	− 3611.36i	− 0.289399i
$539$	2940.77	0.235005
$540$	0	0
$541$	3680.65	0.292502	0.146251	−	0.989248i	$-0.453279\pi$
$541$	3680.65	0.292502	0.146251	+	0.989248i	$0.453279\pi$
$542$	− 17296.8i	− 1.37078i
$543$	2713.07i	0.214418i
$544$	−7714.06	−0.607974
$545$	0	0
$546$	−57.2469	−0.00448707
$547$	− 14657.4i	− 1.14572i	−0.819655	−	0.572858i	$-0.805835\pi$
$547$	− 14657.4i	− 1.14572i	0.819655	−	0.572858i	$-0.194165\pi$
$548$	2669.62i	0.208103i
$549$	−3678.58	−0.285971
$550$	0	0
$551$	1783.74	0.137913
$552$	1528.37i	0.117848i
$553$	− 4835.35i	− 0.371826i
$554$	−8847.33	−0.678497
$555$	0	0
$556$	5455.87	0.416152
$557$	6547.61i	0.498081i	0.968493	+	0.249040i	$0.0801152\pi$
$557$	6547.61i	0.498081i	−0.968493	+	0.249040i	$0.919885\pi$
$558$	2656.55i	0.201543i
$559$	−81.3835	−0.00615770
$560$	0	0
$561$	−12144.8	−0.914001
$562$	12293.9i	0.922754i
$563$	− 11983.3i	− 0.897047i	−0.893771	−	0.448523i	$-0.851950\pi$
$563$	− 11983.3i	− 0.897047i	0.893771	−	0.448523i	$-0.148050\pi$
$564$	14881.1	1.11100
$565$	0	0
$566$	30575.5	2.27064
$567$	5931.62i	0.439338i
$568$	7712.95i	0.569768i
$569$	−19164.8	−1.41200	−0.706002	−	0.708210i	$-0.749503\pi$
$569$	−19164.8	−1.41200	−0.706002	+	0.708210i	$0.749503\pi$
$570$	0	0
$571$	14618.8	1.07142	0.535708	−	0.844403i	$-0.320045\pi$
$571$	14618.8	1.07142	0.535708	+	0.844403i	$0.320045\pi$
$572$	132.597i	0.00969258i
$573$	− 5644.57i	− 0.411527i
$574$	11602.8	0.843713
$575$	0	0
$576$	−1033.70	−0.0747760
$577$	− 12446.8i	− 0.898033i	−0.893523	−	0.449017i	$-0.851774\pi$
$577$	− 12446.8i	− 0.898033i	0.893523	−	0.449017i	$-0.148226\pi$
$578$	− 13522.9i	− 0.973149i
$579$	7941.62	0.570021
$580$	0	0
$581$	−6855.85	−0.489550
$582$	235.847i	0.0167976i
$583$	− 8662.32i	− 0.615363i
$584$	−8760.66	−0.620752
$585$	0	0
$586$	−2325.50	−0.163935
$587$	7467.53i	0.525073i	0.964922	+	0.262537i	$0.0845590\pi$
$587$	7467.53i	0.525073i	−0.964922	+	0.262537i	$0.915441\pi$
$588$	− 1592.88i	− 0.111717i
$589$	793.424	0.0555050
$590$	0	0
$591$	20430.9	1.42202
$592$	16962.2i	1.17760i
$593$	26703.9i	1.84924i	0.380892	+	0.924619i	$0.375617\pi$
$593$	26703.9i	1.84924i	−0.380892	+	0.924619i	$0.624383\pi$
$594$	27222.2	1.88037
$595$	0	0
$596$	−2183.51	−0.150067
$597$	3557.47i	0.243882i
$598$	− 45.1938i	− 0.00309048i
$599$	−6896.60	−0.470430	−0.235215	−	0.971943i	$-0.575579\pi$
$599$	−6896.60	−0.470430	−0.235215	+	0.971943i	$0.575579\pi$
$600$	0	0
$601$	18156.5	1.23231	0.616154	−	0.787626i	$-0.288690\pi$
$601$	18156.5	1.23231	0.616154	+	0.787626i	$0.288690\pi$
$602$	− 5441.82i	− 0.368425i
$603$	− 426.631i	− 0.0288122i
$604$	−7429.02	−0.500468
$605$	0	0
$606$	−14263.6	−0.956137
$607$	2074.50i	0.138717i	0.997592	+	0.0693587i	$0.0220953\pi$
$607$	2074.50i	0.138717i	−0.997592	+	0.0693587i	$0.977905\pi$
$608$	1323.43i	0.0882766i
$609$	−11691.6	−0.777943
$610$	0	0
$611$	177.387	0.0117452
$612$	1114.57i	0.0736171i
$613$	12159.4i	0.801162i	0.916261	+	0.400581i	$0.131192\pi$
$613$	12159.4i	0.801162i	−0.916261	+	0.400581i	$0.868808\pi$
$614$	−93.2959	−0.00613211
$615$	0	0
$616$	3574.21	0.233781
$617$	4359.84i	0.284474i	0.989833	+	0.142237i	$0.0454295\pi$
$617$	4359.84i	0.284474i	−0.989833	+	0.142237i	$0.954570\pi$
$618$	32266.5i	2.10024i
$619$	29046.6	1.88608	0.943038	−	0.332684i	$-0.107954\pi$
$619$	29046.6	1.88608	0.943038	+	0.332684i	$0.107954\pi$
$620$	0	0
$621$	−3860.94	−0.249491
$622$	34928.6i	2.25162i
$623$	− 6376.49i	− 0.410062i
$624$	−170.353	−0.0109288
$625$	0	0
$626$	−10285.2	−0.656674
$627$	2083.57i	0.132711i
$628$	− 18264.0i	− 1.16053i
$629$	7807.89	0.494946
$630$	0	0
$631$	−14708.4	−0.927946	−0.463973	−	0.885849i	$-0.653577\pi$
$631$	−14708.4	−0.927946	−0.463973	+	0.885849i	$0.653577\pi$
$632$	− 5876.89i	− 0.369889i
$633$	− 12922.5i	− 0.811408i
$634$	11031.4	0.691033
$635$	0	0
$636$	−4691.99	−0.292531
$637$	− 18.9876i	− 0.00118103i
$638$	65077.6i	4.03832i
$639$	4993.23	0.309123
$640$	0	0
$641$	−24048.7	−1.48185	−0.740925	−	0.671588i	$-0.765612\pi$
$641$	−24048.7	−1.48185	−0.740925	+	0.671588i	$0.765612\pi$
$642$	14806.9i	0.910255i
$643$	19196.1i	1.17732i	0.808380	+	0.588661i	$0.200345\pi$
$643$	19196.1i	1.17732i	−0.808380	+	0.588661i	$0.799655\pi$
$644$	1257.51	0.0769452
$645$	0	0
$646$	799.960	0.0487214
$647$	− 7185.82i	− 0.436636i	−0.975878	−	0.218318i	$-0.929943\pi$
$647$	− 7185.82i	− 0.436636i	0.975878	−	0.218318i	$-0.0700571\pi$
$648$	7209.31i	0.437050i
$649$	46051.7	2.78534
$650$	0	0
$651$	−5200.52	−0.313094
$652$	− 7804.36i	− 0.468777i
$653$	403.329i	0.0241707i	0.999927	+	0.0120854i	$0.00384698\pi$
$653$	403.329i	0.0241707i	−0.999927	+	0.0120854i	$0.996153\pi$
$654$	−12602.2	−0.753496
$655$	0	0
$656$	34527.1	2.05497
$657$	5671.50i	0.336783i
$658$	11861.2i	0.702733i
$659$	−20527.9	−1.21343	−0.606716	−	0.794919i	$-0.707513\pi$
$659$	−20527.9	−1.21343	−0.606716	+	0.794919i	$0.707513\pi$
$660$	0	0
$661$	−4372.75	−0.257307	−0.128654	−	0.991690i	$-0.541066\pi$
$661$	−4372.75	−0.257307	−0.128654	+	0.991690i	$0.541066\pi$
$662$	− 9580.22i	− 0.562456i
$663$	78.4155i	0.00459337i
$664$	−8332.62	−0.487000
$665$	0	0
$666$	4485.03	0.260948
$667$	− 9229.97i	− 0.535811i
$668$	17522.8i	1.01494i
$669$	12079.7	0.698099
$670$	0	0
$671$	−40083.5	−2.30612
$672$	− 8674.45i	− 0.497953i
$673$	− 19902.7i	− 1.13996i	−0.821659	−	0.569980i	$-0.806951\pi$
$673$	− 19902.7i	− 1.13996i	0.821659	−	0.569980i	$-0.193049\pi$
$674$	−19561.4	−1.11792
$675$	0	0
$676$	−12525.5	−0.712647
$677$	− 9714.40i	− 0.551484i	−0.961232	−	0.275742i	$-0.911076\pi$
$677$	− 9714.40i	− 0.551484i	0.961232	−	0.275742i	$-0.0889235\pi$
$678$	− 1859.72i	− 0.105342i
$679$	−78.2257	−0.00442125
$680$	0	0
$681$	−15910.9	−0.895312
$682$	28947.0i	1.62528i
$683$	− 88.0227i	− 0.00493132i	−0.999997	−	0.00246566i	$-0.999215\pi$
$683$	− 88.0227i	− 0.00493132i	0.999997	−	0.00246566i	$-0.000784845\pi$
$684$	191.216	0.0106891
$685$	0	0
$686$	1269.64	0.0706631
$687$	− 33147.6i	− 1.84084i
$688$	− 16193.5i	− 0.897345i
$689$	−55.9300	−0.00309254
$690$	0	0
$691$	3722.48	0.204934	0.102467	−	0.994736i	$-0.467326\pi$
$691$	3722.48	0.204934	0.102467	+	0.994736i	$0.467326\pi$
$692$	13696.1i	0.752380i
$693$	− 2313.88i	− 0.126836i
$694$	43633.0	2.38658
$695$	0	0
$696$	−14210.0	−0.773891
$697$	− 15893.2i	− 0.863700i
$698$	32446.5i	1.75948i
$699$	11039.2	0.597343
$700$	0	0
$701$	13167.2	0.709443	0.354722	−	0.934972i	$-0.384576\pi$
$701$	13167.2	0.709443	0.354722	+	0.934972i	$0.384576\pi$
$702$	− 175.766i	− 0.00944994i
$703$	− 1339.53i	− 0.0718652i
$704$	−11263.7	−0.603007
$705$	0	0
$706$	31753.1	1.69270
$707$	− 4730.95i	− 0.251663i
$708$	− 24944.2i	− 1.32409i
$709$	14539.6	0.770166	0.385083	−	0.922882i	$-0.374173\pi$
$709$	14539.6	0.770166	0.385083	+	0.922882i	$0.374173\pi$
$710$	0	0
$711$	−3804.60	−0.200680
$712$	− 7749.99i	− 0.407926i
$713$	− 4105.57i	− 0.215645i
$714$	−5243.36	−0.274829
$715$	0	0
$716$	−6535.39	−0.341116
$717$	− 15712.3i	− 0.818391i
$718$	13808.2i	0.717713i
$719$	−26509.9	−1.37504	−0.687519	−	0.726166i	$-0.741300\pi$
$719$	−26509.9	−1.37504	−0.687519	+	0.726166i	$0.741300\pi$
$720$	0	0
$721$	−10702.1	−0.552800
$722$	25251.8i	1.30163i
$723$	22950.1i	1.18053i
$724$	2713.07	0.139269
$725$	0	0
$726$	−47926.1	−2.45001
$727$	− 11761.8i	− 0.600030i	−0.953934	−	0.300015i	$-0.903008\pi$
$727$	− 11761.8i	− 0.600030i	0.953934	−	0.300015i	$-0.0969918\pi$
$728$	− 23.0776i	− 0.00117488i
$729$	−14196.7	−0.721270
$730$	0	0
$731$	−7454.08	−0.377153
$732$	21711.5i	1.09628i
$733$	− 6458.47i	− 0.325442i	−0.986672	−	0.162721i	$-0.947973\pi$
$733$	− 6458.47i	− 0.325442i	0.986672	−	0.162721i	$-0.0520270\pi$
$734$	1909.15	0.0960055
$735$	0	0
$736$	6848.07	0.342967
$737$	− 4648.77i	− 0.232347i
$738$	− 9129.43i	− 0.455364i
$739$	−33663.0	−1.67566	−0.837830	−	0.545932i	$-0.816176\pi$
$739$	−33663.0	−1.67566	−0.837830	+	0.545932i	$0.816176\pi$
$740$	0	0
$741$	13.4530	0.000666948 0
$742$	− 3739.84i	− 0.185032i
$743$	− 19979.4i	− 0.986506i	−0.869886	−	0.493253i	$-0.835808\pi$
$743$	− 19979.4i	− 0.986506i	0.869886	−	0.493253i	$-0.164192\pi$
$744$	−6320.72	−0.311463
$745$	0	0
$746$	40428.5	1.98417
$747$	5394.39i	0.264218i
$748$	12144.8i	0.593661i
$749$	−4911.17	−0.239586
$750$	0	0
$751$	−31850.0	−1.54757	−0.773785	−	0.633449i	$-0.781639\pi$
$751$	−31850.0	−1.54757	−0.773785	+	0.633449i	$0.781639\pi$
$752$	35296.1i	1.71159i
$753$	− 40861.1i	− 1.97750i
$754$	420.186	0.0202948
$755$	0	0
$756$	4890.65	0.235279
$757$	39396.6i	1.89154i	0.324840	+	0.945769i	$0.394690\pi$
$757$	39396.6i	1.89154i	−0.324840	+	0.945769i	$0.605310\pi$
$758$	− 31104.7i	− 1.49047i
$759$	10781.4	0.515601
$760$	0	0
$761$	−7170.87	−0.341582	−0.170791	−	0.985307i	$-0.554632\pi$
$761$	−7170.87	−0.341582	−0.170791	+	0.985307i	$0.554632\pi$
$762$	− 44728.2i	− 2.12642i
$763$	− 4179.91i	− 0.198326i
$764$	−5644.57	−0.267295
$765$	0	0
$766$	−11218.3	−0.529154
$767$	− 297.342i	− 0.0139979i
$768$	− 30595.0i	− 1.43750i
$769$	−27056.7	−1.26878	−0.634388	−	0.773015i	$-0.718748\pi$
$769$	−27056.7	−1.26878	−0.634388	+	0.773015i	$0.718748\pi$
$770$	0	0
$771$	−3693.22	−0.172514
$772$	− 7941.62i	− 0.370240i
$773$	40087.8i	1.86527i	0.360815	+	0.932637i	$0.382499\pi$
$773$	40087.8i	1.86527i	−0.360815	+	0.932637i	$0.617501\pi$
$774$	−4281.79	−0.198845
$775$	0	0
$776$	−95.0757	−0.00439822
$777$	8779.97i	0.405379i
$778$	5194.23i	0.239360i
$779$	−2726.65	−0.125408
$780$	0	0
$781$	54408.6	2.49282
$782$	− 4139.39i	− 0.189289i
$783$	− 35896.9i	− 1.63838i
$784$	3778.13	0.172109
$785$	0	0
$786$	5577.84	0.253123
$787$	23501.4i	1.06447i	0.846598	+	0.532233i	$0.178647\pi$
$787$	23501.4i	1.06447i	−0.846598	+	0.532233i	$0.821353\pi$
$788$	− 20430.9i	− 0.923632i
$789$	−34958.1	−1.57737
$790$	0	0
$791$	616.831	0.0277269
$792$	− 2812.30i	− 0.126175i
$793$	258.807i	0.0115896i
$794$	−206.961	−0.00925035
$795$	0	0
$796$	3557.47	0.158406
$797$	1635.92i	0.0727066i	0.999339	+	0.0363533i	$0.0115742\pi$
$797$	1635.92i	0.0727066i	−0.999339	+	0.0363533i	$0.988426\pi$
$798$	899.555i	0.0399046i
$799$	16247.2	0.719381
$800$	0	0
$801$	−5017.21	−0.221316
$802$	13807.5i	0.607929i
$803$	61799.3i	2.71588i
$804$	−2518.03	−0.110453
$805$	0	0
$806$	186.902	0.00816794
$807$	− 5562.62i	− 0.242644i
$808$	− 5750.00i	− 0.250352i
$809$	33824.6	1.46998	0.734988	−	0.678080i	$-0.237188\pi$
$809$	33824.6	1.46998	0.734988	+	0.678080i	$0.237188\pi$
$810$	0	0
$811$	−3636.30	−0.157445	−0.0787224	−	0.996897i	$-0.525084\pi$
$811$	−3636.30	−0.157445	−0.0787224	+	0.996897i	$0.525084\pi$
$812$	11691.6i	0.505289i
$813$	− 26642.4i	− 1.14931i
$814$	48870.9	2.10433
$815$	0	0
$816$	−15603.0	−0.669379
$817$	1278.83i	0.0547619i
$818$	− 7286.52i	− 0.311451i
$819$	−14.9401	−0.000637421 0
$820$	0	0
$821$	2417.39	0.102762	0.0513808	−	0.998679i	$-0.483638\pi$
$821$	2417.39	0.102762	0.0513808	+	0.998679i	$0.483638\pi$
$822$	9881.78i	0.419302i
$823$	15752.2i	0.667178i	0.942719	+	0.333589i	$0.108260\pi$
$823$	15752.2i	0.667178i	−0.942719	+	0.333589i	$0.891740\pi$
$824$	−13007.4	−0.549920
$825$	0	0
$826$	19882.2	0.837518
$827$	− 4850.49i	− 0.203952i	−0.994787	−	0.101976i	$-0.967484\pi$
$827$	− 4850.49i	− 0.203952i	0.994787	−	0.101976i	$-0.0325164\pi$
$828$	− 989.444i	− 0.0415284i
$829$	−33160.3	−1.38927	−0.694636	−	0.719362i	$-0.744434\pi$
$829$	−33160.3	−1.38927	−0.694636	+	0.719362i	$0.744434\pi$
$830$	0	0
$831$	−13627.7	−0.568879
$832$	72.7264i	0.00303045i
$833$	− 1739.12i	− 0.0723371i
$834$	20195.2	0.838494
$835$	0	0
$836$	2083.57	0.0861984
$837$	− 15967.2i	− 0.659388i
$838$	− 48891.7i	− 2.01544i
$839$	−14966.6	−0.615857	−0.307929	−	0.951409i	$-0.599636\pi$
$839$	−14966.6	−0.615857	−0.307929	+	0.951409i	$0.599636\pi$
$840$	0	0
$841$	61426.1	2.51860
$842$	27708.9i	1.13410i
$843$	18936.5i	0.773674i
$844$	−12922.5	−0.527025
$845$	0	0
$846$	9332.76	0.379275
$847$	− 15896.1i	− 0.644861i
$848$	− 11128.9i	− 0.450668i
$849$	47095.8	1.90380
$850$	0	0
$851$	−6931.37	−0.279206
$852$	− 29470.7i	− 1.18504i
$853$	− 5806.35i	− 0.233066i	−0.993187	−	0.116533i	$-0.962822\pi$
$853$	− 5806.35i	− 0.233066i	0.993187	−	0.116533i	$-0.0371781\pi$
$854$	−17305.5	−0.693422
$855$	0	0
$856$	−5969.04	−0.238338
$857$	9191.52i	0.366367i	0.983079	+	0.183183i	$0.0586402\pi$
$857$	9191.52i	0.366367i	−0.983079	+	0.183183i	$0.941360\pi$
$858$	490.816i	0.0195293i
$859$	−15029.2	−0.596962	−0.298481	−	0.954416i	$-0.596480\pi$
$859$	−15029.2	−0.596962	−0.298481	+	0.954416i	$0.596480\pi$
$860$	0	0
$861$	17871.9	0.707403
$862$	− 39323.1i	− 1.55377i
$863$	17147.5i	0.676369i	0.941080	+	0.338185i	$0.109813\pi$
$863$	17147.5i	0.676369i	−0.941080	+	0.338185i	$0.890187\pi$
$864$	26633.3	1.04871
$865$	0	0
$866$	26905.7	1.05576
$867$	− 20829.5i	− 0.815927i
$868$	5200.52i	0.203361i
$869$	−41456.6	−1.61832
$870$	0	0
$871$	−30.0157	−0.00116767
$872$	− 5080.27i	− 0.197293i
$873$	61.5504i	0.00238621i
$874$	−710.156	−0.0274844
$875$	0	0
$876$	33473.9	1.29107
$877$	− 33483.6i	− 1.28924i	−0.764505	−	0.644618i	$-0.777016\pi$
$877$	− 33483.6i	− 1.28924i	0.764505	−	0.644618i	$-0.222984\pi$
$878$	− 2751.56i	− 0.105764i
$879$	−3582.00	−0.137449
$880$	0	0
$881$	−10695.6	−0.409019	−0.204509	−	0.978865i	$-0.565560\pi$
$881$	−10695.6	−0.409019	−0.204509	+	0.978865i	$0.565560\pi$
$882$	− 998.988i	− 0.0381379i
$883$	− 31934.0i	− 1.21706i	−0.793530	−	0.608531i	$-0.791759\pi$
$883$	− 31934.0i	− 1.21706i	0.793530	−	0.608531i	$-0.208241\pi$
$884$	78.4155	0.00298348
$885$	0	0
$886$	40190.9	1.52397
$887$	10655.2i	0.403345i	0.979453	+	0.201672i	$0.0646376\pi$
$887$	10655.2i	0.403345i	−0.979453	+	0.201672i	$0.935362\pi$
$888$	10671.2i	0.403268i
$889$	14835.4	0.559690
$890$	0	0
$891$	50855.7	1.91216
$892$	− 12079.7i	− 0.453429i
$893$	− 2787.38i	− 0.104453i
$894$	−8082.41	−0.302367
$895$	0	0
$896$	7308.38	0.272495
$897$	− 69.6125i	− 0.00259119i
$898$	− 41320.0i	− 1.53549i
$899$	38171.3	1.41611
$900$	0	0
$901$	−5122.74	−0.189415
$902$	− 99478.4i	− 3.67214i
$903$	− 8382.11i	− 0.308903i
$904$	749.697	0.0275825
$905$	0	0
$906$	−27499.0	−1.00838
$907$	− 20737.2i	− 0.759171i	−0.925157	−	0.379585i	$-0.876067\pi$
$907$	− 20737.2i	− 0.759171i	0.925157	−	0.379585i	$-0.123933\pi$
$908$	15910.9i	0.581522i
$909$	−3722.45	−0.135826
$910$	0	0
$911$	11734.8	0.426774	0.213387	−	0.976968i	$-0.431550\pi$
$911$	11734.8	0.426774	0.213387	+	0.976968i	$0.431550\pi$
$912$	2676.86i	0.0971926i
$913$	58779.8i	2.13070i
$914$	−49789.7	−1.80185
$915$	0	0
$916$	−33147.6	−1.19566
$917$	1850.06i	0.0666240i
$918$	− 16098.8i	− 0.578800i
$919$	21922.8	0.786906	0.393453	−	0.919345i	$-0.371280\pi$
$919$	21922.8	0.786906	0.393453	+	0.919345i	$0.371280\pi$
$920$	0	0
$921$	−143.705	−0.00514141
$922$	33830.3i	1.20840i
$923$	− 351.300i	− 0.0125278i
$924$	−13656.8	−0.486230
$925$	0	0
$926$	−48573.6	−1.72379
$927$	8420.77i	0.298354i
$928$	63669.6i	2.25222i
$929$	36647.7	1.29427	0.647133	−	0.762377i	$-0.275968\pi$
$929$	36647.7	1.29427	0.647133	+	0.762377i	$0.275968\pi$
$930$	0	0
$931$	−298.364	−0.0105032
$932$	− 11039.2i	− 0.387986i
$933$	53801.0i	1.88785i
$934$	−44127.1	−1.54591
$935$	0	0
$936$	−18.1582	−0.000634101 0
$937$	24959.7i	0.870222i	0.900377	+	0.435111i	$0.143291\pi$
$937$	24959.7i	0.870222i	−0.900377	+	0.435111i	$0.856709\pi$
$938$	− 2007.04i	− 0.0698638i
$939$	−15842.4	−0.550581
$940$	0	0
$941$	17086.7	0.591935	0.295968	−	0.955198i	$-0.404358\pi$
$941$	17086.7	0.591935	0.295968	+	0.955198i	$0.404358\pi$
$942$	− 67605.2i	− 2.33832i
$943$	14109.0i	0.487226i
$944$	59164.6	2.03988
$945$	0	0
$946$	−46656.3	−1.60352
$947$	54034.9i	1.85417i	0.374852	+	0.927085i	$0.377693\pi$
$947$	54034.9i	1.85417i	−0.374852	+	0.927085i	$0.622307\pi$
$948$	22455.2i	0.769316i
$949$	399.020	0.0136488
$950$	0	0
$951$	16991.9	0.579389
$952$	− 2113.73i	− 0.0719603i
$953$	24377.9i	0.828622i	0.910135	+	0.414311i	$0.135977\pi$
$953$	24377.9i	0.828622i	−0.910135	+	0.414311i	$0.864023\pi$
$954$	−2942.62	−0.0998645
$955$	0	0
$956$	−15712.3	−0.531561
$957$	100240.i	3.38588i
$958$	3054.83i	0.103024i
$959$	−3277.59	−0.110364
$960$	0	0
$961$	−12812.1	−0.430066
$962$	− 315.545i	− 0.0105754i
$963$	3864.25i	0.129308i
$964$	22950.1	0.766777
$965$	0	0
$966$	4654.74	0.155035
$967$	− 32668.1i	− 1.08639i	−0.839607	−	0.543194i	$-0.817215\pi$
$967$	− 32668.1i	− 1.08639i	0.839607	−	0.543194i	$-0.182785\pi$
$968$	− 19320.2i	− 0.641502i
$969$	1232.19	0.0408500
$970$	0	0
$971$	24623.2	0.813795	0.406898	−	0.913474i	$-0.366611\pi$
$971$	24623.2	0.813795	0.406898	+	0.913474i	$0.366611\pi$
$972$	− 8682.37i	− 0.286510i
$973$	6698.36i	0.220698i
$974$	24720.2	0.813231
$975$	0	0
$976$	−51497.0	−1.68891
$977$	− 18320.6i	− 0.599926i	−0.953951	−	0.299963i	$-0.903026\pi$
$977$	− 18320.6i	− 0.599926i	0.953951	−	0.299963i	$-0.0969743\pi$
$978$	− 28888.3i	− 0.944526i
$979$	−54669.8	−1.78473
$980$	0	0
$981$	−3288.88	−0.107040
$982$	11469.8i	0.372727i
$983$	− 2176.33i	− 0.0706145i	−0.999377	−	0.0353072i	$-0.988759\pi$
$983$	− 2176.33i	− 0.0706145i	0.999377	−	0.0353072i	$-0.0112410\pi$
$984$	21721.6	0.703718
$985$	0	0
$986$	38485.7	1.24304
$987$	18270.0i	0.589199i
$988$	− 13.4530i	0 0.000433196i
$989$	6617.28	0.212758
$990$	0	0
$991$	−21487.2	−0.688761	−0.344381	−	0.938830i	$-0.611911\pi$
$991$	−21487.2	−0.688761	−0.344381	+	0.938830i	$0.611911\pi$
$992$	28320.8i	0.906437i
$993$	− 14756.5i	− 0.471586i
$994$	23490.2	0.749560
$995$	0	0
$996$	31838.4	1.01289
$997$	− 19083.0i	− 0.606182i	−0.952962	−	0.303091i	$-0.901981\pi$
$997$	− 19083.0i	− 0.606182i	0.952962	−	0.303091i	$-0.0980187\pi$
$998$	− 8384.41i	− 0.265936i
$999$	−26957.2	−0.853743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.d.99.1		4
5.2	odd	4		175.4.a.e.1.2	yes	2
5.3	odd	4		175.4.a.d.1.1	✓	2
5.4	even	2	inner	175.4.b.d.99.4		4
15.2	even	4		1575.4.a.s.1.1		2
15.8	even	4		1575.4.a.v.1.2		2
35.13	even	4		1225.4.a.r.1.1		2
35.27	even	4		1225.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.d.1.1	✓	2	5.3	odd	4
175.4.a.e.1.2	yes	2	5.2	odd	4
175.4.b.d.99.1		4	1.1	even	1	trivial
175.4.b.d.99.4		4	5.4	even	2	inner
1225.4.a.r.1.1		2	35.13	even	4
1225.4.a.t.1.2		2	35.27	even	4
1575.4.a.s.1.1		2	15.2	even	4
1575.4.a.v.1.2		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.70156i q^{2} -5.70156i q^{3} -5.70156 q^{4} -21.1047 q^{6} -7.00000i q^{7} -8.50781i q^{8} -5.50781 q^{9} -60.0156 q^{11} +32.5078i q^{12} +0.387503i q^{13} -25.9109 q^{14} -77.1047 q^{16} +35.4922i q^{17} +20.3875i q^{18} +6.08907 q^{19} -39.9109 q^{21} +222.152i q^{22} -31.5078i q^{23} -48.5078 q^{24} +1.43437 q^{26} -122.539i q^{27} +39.9109i q^{28} +292.942 q^{29} +130.303 q^{31} +217.345i q^{32} +342.183i q^{33} +131.377 q^{34} +31.4031 q^{36} -219.989i q^{37} -22.5391i q^{38} +2.20937 q^{39} -447.795 q^{41} +147.733i q^{42} +210.020i q^{43} +342.183 q^{44} -116.628 q^{46} -457.769i q^{47} +439.617i q^{48} -49.0000 q^{49} +202.361 q^{51} -2.20937i q^{52} +144.334i q^{53} -453.586 q^{54} -59.5547 q^{56} -34.7172i q^{57} -1084.34i q^{58} -767.328 q^{59} +667.884 q^{61} -482.325i q^{62} +38.5547i q^{63} +187.680 q^{64} +1266.61 q^{66} +77.4593i q^{67} -202.361i q^{68} -179.644 q^{69} -906.573 q^{71} +46.8594i q^{72} -1029.72i q^{73} -814.303 q^{74} -34.7172 q^{76} +420.109i q^{77} -8.17813i q^{78} +690.764 q^{79} -847.375 q^{81} +1657.54i q^{82} -979.408i q^{83} +227.555 q^{84} +777.403 q^{86} -1670.23i q^{87} +510.602i q^{88} +910.927 q^{89} +2.71252 q^{91} +179.644i q^{92} -742.931i q^{93} -1694.46 q^{94} +1239.21 q^{96} -11.1751i q^{97} +181.377i q^{98} +330.555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} - 46 q^{6} + 42 q^{9} - 112 q^{11} - 14 q^{14} - 270 q^{16} + 114 q^{19} - 70 q^{21} - 130 q^{24} - 276 q^{26} + 826 q^{29} - 324 q^{31} - 102 q^{34} + 100 q^{36} - 68 q^{39} - 1010 q^{41}+ \cdots + 874 q^{99}+O(q^{100}) \) 4 * q - 10 * q^4 - 46 * q^6 + 42 * q^9 - 112 * q^11 - 14 * q^14 - 270 * q^16 + 114 * q^19 - 70 * q^21 - 130 * q^24 - 276 * q^26 + 826 * q^29 - 324 * q^31 - 102 * q^34 + 100 * q^36 - 68 * q^39 - 1010 * q^41 + 690 * q^44 - 236 * q^46 - 196 * q^49 + 310 * q^51 - 1110 * q^54 + 210 * q^56 - 380 * q^59 + 1980 * q^61 - 722 * q^64 + 2518 * q^66 - 360 * q^69 + 254 * q^71 - 2412 * q^74 + 2 * q^76 + 2238 * q^79 - 316 * q^81 + 462 * q^84 + 3084 * q^86 + 3426 * q^89 + 728 * q^91 - 4652 * q^94 + 2434 * q^96 + 874 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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