LMFDB - Embedded newform 575.4.b.k.24.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(24,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, 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("575.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.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:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 102 x^{14} + 4025 x^{12} + 79249 x^{10} + 832798 x^{8} + 4596761 x^{6} + 12272424 x^{4} + \cdots + 5760000 $ x^16 + 102x^14 + 4025x^12 + 79249x^10 + 832798x^8 + 4596761x^6 + 12272424x^4 + 14266000*x^2 + 5760000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.13
Root			$3.05234i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.4.b.k.24.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.05234i q^{2} +8.94300i q^{3} -8.42146 q^{4} -36.2401 q^{6} +32.7645i q^{7} -1.70790i q^{8} -52.9772 q^{9} +45.7496 q^{11} -75.3131i q^{12} -45.5638i q^{13} -132.773 q^{14} -60.4507 q^{16} -76.5649i q^{17} -214.681i q^{18} +32.0225 q^{19} -293.013 q^{21} +185.393i q^{22} -23.0000i q^{23} +15.2738 q^{24} +184.640 q^{26} -232.314i q^{27} -275.925i q^{28} -287.046 q^{29} -161.587 q^{31} -258.630i q^{32} +409.139i q^{33} +310.267 q^{34} +446.145 q^{36} +7.53926i q^{37} +129.766i q^{38} +407.476 q^{39} +216.246 q^{41} -1187.39i q^{42} +469.729i q^{43} -385.279 q^{44} +93.2038 q^{46} -210.315i q^{47} -540.610i q^{48} -730.514 q^{49} +684.719 q^{51} +383.713i q^{52} -58.7860i q^{53} +941.414 q^{54} +55.9587 q^{56} +286.377i q^{57} -1163.21i q^{58} +376.609 q^{59} -52.7319 q^{61} -654.805i q^{62} -1735.77i q^{63} +564.451 q^{64} -1657.97 q^{66} -16.7678i q^{67} +644.788i q^{68} +205.689 q^{69} -209.441 q^{71} +90.4800i q^{72} +811.910i q^{73} -30.5516 q^{74} -269.676 q^{76} +1498.96i q^{77} +1651.23i q^{78} -63.0408 q^{79} +647.197 q^{81} +876.300i q^{82} +1062.27i q^{83} +2467.60 q^{84} -1903.50 q^{86} -2567.05i q^{87} -78.1360i q^{88} -436.069 q^{89} +1492.87 q^{91} +193.694i q^{92} -1445.07i q^{93} +852.269 q^{94} +2312.93 q^{96} +655.840i q^{97} -2960.29i q^{98} -2423.68 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 84 q^{4} - 316 q^{9} + 82 q^{11} - 322 q^{14} + 196 q^{16} - 354 q^{19} + 584 q^{21} - 354 q^{24} + 792 q^{26} - 450 q^{29} - 72 q^{31} + 2102 q^{34} - 792 q^{36} + 2154 q^{39} + 1240 q^{41} + 282 q^{44}+ \cdots - 2638 q^{99}+O(q^{100}) $ 16 * q - 84 * q^4 - 316 * q^9 + 82 * q^11 - 322 * q^14 + 196 * q^16 - 354 * q^19 + 584 * q^21 - 354 * q^24 + 792 * q^26 - 450 * q^29 - 72 * q^31 + 2102 * q^34 - 792 * q^36 + 2154 * q^39 + 1240 * q^41 + 282 * q^44 + 276 * q^46 - 1762 * q^49 + 1914 * q^51 + 5898 * q^54 + 2062 * q^56 + 1352 * q^59 + 2894 * q^61 + 5766 * q^64 - 238 * q^66 + 2792 * q^71 + 1928 * q^74 + 4542 * q^76 - 1416 * q^79 + 8632 * q^81 + 7940 * q^84 + 7140 * q^86 - 2720 * q^89 + 5386 * q^91 + 6414 * q^94 + 9912 * q^96 - 2638 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.05234i	1.43272i	0.697732	+	0.716359i	$0.254193\pi$
$2$	4.05234i	1.43272i	−0.697732	+	0.716359i	$0.745807\pi$
$3$	8.94300i	1.72108i	0.509383	+	0.860540i	$0.329874\pi$
$3$	8.94300i	1.72108i	−0.509383	+	0.860540i	$0.670126\pi$
$4$	−8.42146	−1.05268
$5$	0	0
$5$	0	0
$6$	−36.2401	−2.46582
$7$	32.7645i	1.76912i	0.466429	+	0.884559i	$0.345540\pi$
$7$	32.7645i	1.76912i	−0.466429	+	0.884559i	$0.654460\pi$
$8$	− 1.70790i	− 0.0754794i
$9$	−52.9772	−1.96212
$10$	0	0
$11$	45.7496	1.25400	0.627001	−	0.779018i	$-0.284282\pi$
$11$	45.7496	1.25400	0.627001	+	0.779018i	$0.284282\pi$
$12$	− 75.3131i	− 1.81175i
$13$	− 45.5638i	− 0.972086i	−0.873935	−	0.486043i	$-0.838440\pi$
$13$	− 45.5638i	− 0.972086i	0.873935	−	0.486043i	$-0.161560\pi$
$14$	−132.773	−2.53465
$15$	0	0
$16$	−60.4507	−0.944542
$17$	− 76.5649i	− 1.09234i	−0.837676	−	0.546168i	$-0.816086\pi$
$17$	− 76.5649i	− 1.09234i	0.837676	−	0.546168i	$-0.183914\pi$
$18$	− 214.681i	− 2.81116i
$19$	32.0225	0.386656	0.193328	−	0.981134i	$-0.438072\pi$
$19$	32.0225	0.386656	0.193328	+	0.981134i	$0.438072\pi$
$20$	0	0
$21$	−293.013	−3.04479
$22$	185.393i	1.79663i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	15.2738	0.129906
$25$	0	0
$26$	184.640	1.39273
$27$	− 232.314i	− 1.65588i
$28$	− 275.925i	− 1.86232i
$29$	−287.046	−1.83804	−0.919019	−	0.394213i	$-0.871017\pi$
$29$	−287.046	−1.83804	−0.919019	+	0.394213i	$0.871017\pi$
$30$	0	0
$31$	−161.587	−0.936189	−0.468095	−	0.883678i	$-0.655059\pi$
$31$	−161.587	−0.936189	−0.468095	+	0.883678i	$0.655059\pi$
$32$	− 258.630i	− 1.42874i
$33$	409.139i	2.15824i
$34$	310.267	1.56501
$35$	0	0
$36$	446.145	2.06549
$37$	7.53926i	0.0334986i	0.999860	+	0.0167493i	$0.00533171\pi$
$37$	7.53926i	0.0334986i	−0.999860	+	0.0167493i	$0.994668\pi$
$38$	129.766i	0.553969i
$39$	407.476	1.67304
$40$	0	0
$41$	216.246	0.823704	0.411852	−	0.911251i	$-0.364882\pi$
$41$	216.246	0.823704	0.411852	+	0.911251i	$0.364882\pi$
$42$	− 1187.39i	− 4.36233i
$43$	469.729i	1.66589i	0.553359	+	0.832943i	$0.313346\pi$
$43$	469.729i	1.66589i	−0.553359	+	0.832943i	$0.686654\pi$
$44$	−385.279	−1.32007
$45$	0	0
$46$	93.2038	0.298742
$47$	− 210.315i	− 0.652715i	−0.945246	−	0.326358i	$-0.894179\pi$
$47$	− 210.315i	− 0.652715i	0.945246	−	0.326358i	$-0.105821\pi$
$48$	− 540.610i	− 1.62563i
$49$	−730.514	−2.12978
$50$	0	0
$51$	684.719	1.88000
$52$	383.713i	1.02330i
$53$	− 58.7860i	− 0.152356i	−0.997094	−	0.0761781i	$-0.975728\pi$
$53$	− 58.7860i	− 0.152356i	0.997094	−	0.0761781i	$-0.0242718\pi$
$54$	941.414	2.37241
$55$	0	0
$56$	55.9587	0.133532
$57$	286.377i	0.665466i
$58$	− 1163.21i	− 2.63339i
$59$	376.609	0.831022	0.415511	−	0.909588i	$-0.363603\pi$
$59$	376.609	0.831022	0.415511	+	0.909588i	$0.363603\pi$
$60$	0	0
$61$	−52.7319	−0.110682	−0.0553412	−	0.998467i	$-0.517625\pi$
$61$	−52.7319	−0.110682	−0.0553412	+	0.998467i	$0.517625\pi$
$62$	− 654.805i	− 1.34130i
$63$	− 1735.77i	− 3.47122i
$64$	564.451	1.10244
$65$	0	0
$66$	−1657.97	−3.09215
$67$	− 16.7678i	− 0.0305748i	−0.999883	−	0.0152874i	$-0.995134\pi$
$67$	− 16.7678i	− 0.0305748i	0.999883	−	0.0152874i	$-0.00486633\pi$
$68$	644.788i	1.14988i
$69$	205.689	0.358870
$70$	0	0
$71$	−209.441	−0.350086	−0.175043	−	0.984561i	$-0.556006\pi$
$71$	−209.441	−0.350086	−0.175043	+	0.984561i	$0.556006\pi$
$72$	90.4800i	0.148100i
$73$	811.910i	1.30174i	0.759190	+	0.650869i	$0.225595\pi$
$73$	811.910i	1.30174i	−0.759190	+	0.650869i	$0.774405\pi$
$74$	−30.5516	−0.0479940
$75$	0	0
$76$	−269.676	−0.407026
$77$	1498.96i	2.21848i
$78$	1651.23i	2.39699i
$79$	−63.0408	−0.0897803	−0.0448902	−	0.998992i	$-0.514294\pi$
$79$	−63.0408	−0.0897803	−0.0448902	+	0.998992i	$0.514294\pi$
$80$	0	0
$81$	647.197	0.887787
$82$	876.300i	1.18014i
$83$	1062.27i	1.40481i	0.711779	+	0.702404i	$0.247890\pi$
$83$	1062.27i	1.40481i	−0.711779	+	0.702404i	$0.752110\pi$
$84$	2467.60	3.20520
$85$	0	0
$86$	−1903.50	−2.38674
$87$	− 2567.05i	− 3.16341i
$88$	− 78.1360i	− 0.0946514i
$89$	−436.069	−0.519362	−0.259681	−	0.965694i	$-0.583617\pi$
$89$	−436.069	−0.519362	−0.259681	+	0.965694i	$0.583617\pi$
$90$	0	0
$91$	1492.87	1.71973
$92$	193.694i	0.219500i
$93$	− 1445.07i	− 1.61126i
$94$	852.269	0.935158
$95$	0	0
$96$	2312.93	2.45898
$97$	655.840i	0.686499i	0.939244	+	0.343250i	$0.111528\pi$
$97$	655.840i	0.686499i	−0.939244	+	0.343250i	$0.888472\pi$
$98$	− 2960.29i	− 3.05137i
$99$	−2423.68	−2.46050
$100$	0	0
$101$	831.308	0.818993	0.409496	−	0.912312i	$-0.365704\pi$
$101$	831.308	0.818993	0.409496	+	0.912312i	$0.365704\pi$
$102$	2774.72i	2.69351i
$103$	801.275i	0.766524i	0.923640	+	0.383262i	$0.125199\pi$
$103$	801.275i	0.766524i	−0.923640	+	0.383262i	$0.874801\pi$
$104$	−77.8186	−0.0733725
$105$	0	0
$106$	238.221	0.218284
$107$	901.921i	0.814879i	0.913232	+	0.407439i	$0.133578\pi$
$107$	901.921i	0.814879i	−0.913232	+	0.407439i	$0.866422\pi$
$108$	1956.42i	1.74312i
$109$	−328.880	−0.289000	−0.144500	−	0.989505i	$-0.546157\pi$
$109$	−328.880	−0.289000	−0.144500	+	0.989505i	$0.546157\pi$
$110$	0	0
$111$	−67.4236	−0.0576537
$112$	− 1980.64i	− 1.67101i
$113$	2209.02i	1.83900i	0.393090	+	0.919500i	$0.371406\pi$
$113$	2209.02i	1.83900i	−0.393090	+	0.919500i	$0.628594\pi$
$114$	−1160.50	−0.953426
$115$	0	0
$116$	2417.35	1.93487
$117$	2413.84i	1.90735i
$118$	1526.15i	1.19062i
$119$	2508.61	1.93247
$120$	0	0
$121$	762.027	0.572522
$122$	− 213.688i	− 0.158577i
$123$	1933.88i	1.41766i
$124$	1360.80	0.985510
$125$	0	0
$126$	7033.94	4.97328
$127$	107.788i	0.0753124i	0.999291	+	0.0376562i	$0.0119892\pi$
$127$	107.788i	0.0753124i	−0.999291	+	0.0376562i	$0.988011\pi$
$128$	218.308i	0.150749i
$129$	−4200.79	−2.86712
$130$	0	0
$131$	−985.156	−0.657050	−0.328525	−	0.944495i	$-0.606551\pi$
$131$	−985.156	−0.657050	−0.328525	+	0.944495i	$0.606551\pi$
$132$	− 3445.54i	− 2.27194i
$133$	1049.20i	0.684040i
$134$	67.9489	0.0438051
$135$	0	0
$136$	−130.766	−0.0824489
$137$	65.3753i	0.0407692i	0.999792	+	0.0203846i	$0.00648908\pi$
$137$	65.3753i	0.0407692i	−0.999792	+	0.0203846i	$0.993511\pi$
$138$	833.521i	0.514160i
$139$	547.498	0.334088	0.167044	−	0.985949i	$-0.446578\pi$
$139$	547.498	0.334088	0.167044	+	0.985949i	$0.446578\pi$
$140$	0	0
$141$	1880.85	1.12338
$142$	− 848.727i	− 0.501574i
$143$	− 2084.52i	− 1.21900i
$144$	3202.51	1.85330
$145$	0	0
$146$	−3290.14	−1.86503
$147$	− 6532.98i	− 3.66552i
$148$	− 63.4916i	− 0.0352633i
$149$	−3316.98	−1.82374	−0.911871	−	0.410476i	$-0.865363\pi$
$149$	−3316.98	−1.82374	−0.911871	+	0.410476i	$0.865363\pi$
$150$	0	0
$151$	−1645.53	−0.886831	−0.443415	−	0.896316i	$-0.646233\pi$
$151$	−1645.53	−0.886831	−0.443415	+	0.896316i	$0.646233\pi$
$152$	− 54.6914i	− 0.0291846i
$153$	4056.19i	2.14329i
$154$	−6074.31	−3.17845
$155$	0	0
$156$	−3431.55	−1.76118
$157$	− 2088.75i	− 1.06179i	−0.847439	−	0.530893i	$-0.821857\pi$
$157$	− 2088.75i	− 1.06179i	0.847439	−	0.530893i	$-0.178143\pi$
$158$	− 255.463i	− 0.128630i
$159$	525.723	0.262217
$160$	0	0
$161$	753.584	0.368887
$162$	2622.66i	1.27195i
$163$	− 3160.08i	− 1.51851i	−0.650796	−	0.759253i	$-0.725565\pi$
$163$	− 3160.08i	− 1.51851i	0.650796	−	0.759253i	$-0.274435\pi$
$164$	−1821.10	−0.867099
$165$	0	0
$166$	−4304.67	−2.01269
$167$	− 1886.30i	− 0.874047i	−0.899450	−	0.437024i	$-0.856033\pi$
$167$	− 1886.30i	− 0.874047i	0.899450	−	0.437024i	$-0.143967\pi$
$168$	500.438i	0.229819i
$169$	120.944	0.0550495
$170$	0	0
$171$	−1696.46	−0.758664
$172$	− 3955.81i	− 1.75365i
$173$	425.628i	0.187051i	0.995617	+	0.0935256i	$0.0298137\pi$
$173$	425.628i	0.187051i	−0.995617	+	0.0935256i	$0.970186\pi$
$174$	10402.6	4.53228
$175$	0	0
$176$	−2765.59	−1.18446
$177$	3368.01i	1.43026i
$178$	− 1767.10i	− 0.744100i
$179$	−4680.83	−1.95453	−0.977266	−	0.212016i	$-0.931997\pi$
$179$	−4680.83	−1.95453	−0.977266	+	0.212016i	$0.931997\pi$
$180$	0	0
$181$	−380.264	−0.156159	−0.0780796	−	0.996947i	$-0.524879\pi$
$181$	−380.264	−0.156159	−0.0780796	+	0.996947i	$0.524879\pi$
$182$	6049.64i	2.46389i
$183$	− 471.581i	− 0.190493i
$184$	−39.2818	−0.0157386
$185$	0	0
$186$	5855.92	2.30848
$187$	− 3502.81i	− 1.36979i
$188$	1771.16i	0.687102i
$189$	7611.64	2.92945
$190$	0	0
$191$	1466.03	0.555383	0.277691	−	0.960670i	$-0.410431\pi$
$191$	1466.03	0.555383	0.277691	+	0.960670i	$0.410431\pi$
$192$	5047.88i	1.89739i
$193$	2129.58i	0.794253i	0.917764	+	0.397126i	$0.129993\pi$
$193$	2129.58i	0.794253i	−0.917764	+	0.397126i	$0.870007\pi$
$194$	−2657.69	−0.983561
$195$	0	0
$196$	6151.99	2.24198
$197$	− 1146.95i	− 0.414807i	−0.978256	−	0.207403i	$-0.933499\pi$
$197$	− 1146.95i	− 0.414807i	0.978256	−	0.207403i	$-0.0665013\pi$
$198$	− 9821.59i	− 3.52520i
$199$	3857.88	1.37426	0.687130	−	0.726535i	$-0.258870\pi$
$199$	3857.88	1.37426	0.687130	+	0.726535i	$0.258870\pi$
$200$	0	0
$201$	149.954	0.0526217
$202$	3368.74i	1.17339i
$203$	− 9404.93i	− 3.25171i
$204$	−5766.34	−1.97904
$205$	0	0
$206$	−3247.04	−1.09821
$207$	1218.47i	0.409130i
$208$	2754.36i	0.918176i
$209$	1465.02	0.484868
$210$	0	0
$211$	1880.22	0.613458	0.306729	−	0.951797i	$-0.400765\pi$
$211$	1880.22	0.613458	0.306729	+	0.951797i	$0.400765\pi$
$212$	495.064i	0.160383i
$213$	− 1873.03i	− 0.602526i
$214$	−3654.89	−1.16749
$215$	0	0
$216$	−396.770	−0.124985
$217$	− 5294.32i	− 1.65623i
$218$	− 1332.73i	− 0.414056i
$219$	−7260.91	−2.24040
$220$	0	0
$221$	−3488.58	−1.06184
$222$	− 273.223i	− 0.0826015i
$223$	3181.06i	0.955246i	0.878565	+	0.477623i	$0.158501\pi$
$223$	3181.06i	0.955246i	−0.878565	+	0.477623i	$0.841499\pi$
$224$	8473.89	2.52761
$225$	0	0
$226$	−8951.70	−2.63477
$227$	4817.24i	1.40851i	0.709947	+	0.704255i	$0.248719\pi$
$227$	4817.24i	1.40851i	−0.709947	+	0.704255i	$0.751281\pi$
$228$	− 2411.71i	− 0.700525i
$229$	1440.61	0.415714	0.207857	−	0.978159i	$-0.433351\pi$
$229$	1440.61	0.415714	0.207857	+	0.978159i	$0.433351\pi$
$230$	0	0
$231$	−13405.2	−3.81818
$232$	490.247i	0.138734i
$233$	− 2313.68i	− 0.650533i	−0.945622	−	0.325266i	$-0.894546\pi$
$233$	− 2313.68i	− 0.650533i	0.945622	−	0.325266i	$-0.105454\pi$
$234$	−9781.70	−2.73269
$235$	0	0
$236$	−3171.60	−0.874803
$237$	− 563.774i	− 0.154519i
$238$	10165.7i	2.76869i
$239$	2798.09	0.757293	0.378647	−	0.925541i	$-0.376390\pi$
$239$	2798.09	0.757293	0.378647	+	0.925541i	$0.376390\pi$
$240$	0	0
$241$	−1809.31	−0.483600	−0.241800	−	0.970326i	$-0.577738\pi$
$241$	−1809.31	−0.483600	−0.241800	+	0.970326i	$0.577738\pi$
$242$	3087.99i	0.820263i
$243$	− 484.593i	− 0.127929i
$244$	444.080	0.116514
$245$	0	0
$246$	−7836.75	−2.03111
$247$	− 1459.07i	− 0.375863i
$248$	275.975i	0.0706630i
$249$	−9499.86	−2.41779
$250$	0	0
$251$	5814.80	1.46226	0.731129	−	0.682239i	$-0.238994\pi$
$251$	5814.80	1.46226	0.731129	+	0.682239i	$0.238994\pi$
$252$	14617.7i	3.65409i
$253$	− 1052.24i	− 0.261478i
$254$	−436.795	−0.107901
$255$	0	0
$256$	3630.95	0.886462
$257$	1886.38i	0.457856i	0.973443	+	0.228928i	$0.0735220\pi$
$257$	1886.38i	0.457856i	−0.973443	+	0.228928i	$0.926478\pi$
$258$	− 17023.0i	− 4.10778i
$259$	−247.020	−0.0592629
$260$	0	0
$261$	15206.9	3.60645
$262$	− 3992.19i	− 0.941367i
$263$	6017.91i	1.41095i	0.708734	+	0.705476i	$0.249267\pi$
$263$	6017.91i	1.41095i	−0.708734	+	0.705476i	$0.750733\pi$
$264$	698.770	0.162903
$265$	0	0
$266$	−4251.72	−0.980037
$267$	− 3899.77i	− 0.893864i
$268$	141.209i	0.0321856i
$269$	3189.16	0.722848	0.361424	−	0.932401i	$-0.382291\pi$
$269$	3189.16	0.722848	0.361424	+	0.932401i	$0.382291\pi$
$270$	0	0
$271$	1167.43	0.261683	0.130841	−	0.991403i	$-0.458232\pi$
$271$	1167.43	0.261683	0.130841	+	0.991403i	$0.458232\pi$
$272$	4628.40i	1.03176i
$273$	13350.8i	2.95980i
$274$	−264.923	−0.0584109
$275$	0	0
$276$	−1732.20	−0.377776
$277$	− 4164.36i	− 0.903293i	−0.892197	−	0.451646i	$-0.850837\pi$
$277$	− 4164.36i	− 0.903293i	0.892197	−	0.451646i	$-0.149163\pi$
$278$	2218.65i	0.478654i
$279$	8560.42	1.83691
$280$	0	0
$281$	4958.59	1.05269	0.526343	−	0.850273i	$-0.323563\pi$
$281$	4958.59	1.05269	0.526343	+	0.850273i	$0.323563\pi$
$282$	7621.84i	1.60948i
$283$	− 2171.54i	− 0.456129i	−0.973646	−	0.228065i	$-0.926760\pi$
$283$	− 2171.54i	− 0.456129i	0.973646	−	0.228065i	$-0.0732397\pi$
$284$	1763.80	0.368529
$285$	0	0
$286$	8447.20	1.74648
$287$	7085.18i	1.45723i
$288$	13701.5i	2.80336i
$289$	−949.183	−0.193198
$290$	0	0
$291$	−5865.17	−1.18152
$292$	− 6837.47i	− 1.37032i
$293$	− 5170.00i	− 1.03083i	−0.856939	−	0.515417i	$-0.827637\pi$
$293$	− 5170.00i	− 1.03083i	0.856939	−	0.515417i	$-0.172363\pi$
$294$	26473.9	5.25166
$295$	0	0
$296$	12.8763	0.00252845
$297$	− 10628.3i	− 2.07648i
$298$	− 13441.5i	− 2.61291i
$299$	−1047.97	−0.202694
$300$	0	0
$301$	−15390.5	−2.94715
$302$	− 6668.25i	− 1.27058i
$303$	7434.39i	1.40955i
$304$	−1935.78	−0.365213
$305$	0	0
$306$	−16437.1	−3.07073
$307$	9189.38i	1.70836i	0.519980	+	0.854178i	$0.325939\pi$
$307$	9189.38i	1.70836i	−0.519980	+	0.854178i	$0.674061\pi$
$308$	− 12623.5i	− 2.33535i
$309$	−7165.80	−1.31925
$310$	0	0
$311$	2264.59	0.412903	0.206452	−	0.978457i	$-0.433808\pi$
$311$	2264.59	0.412903	0.206452	+	0.978457i	$0.433808\pi$
$312$	− 695.931i	− 0.126280i
$313$	− 2467.42i	− 0.445581i	−0.974866	−	0.222791i	$-0.928483\pi$
$313$	− 2467.42i	− 0.445581i	0.974866	−	0.222791i	$-0.0715166\pi$
$314$	8464.32	1.52124
$315$	0	0
$316$	530.896	0.0945102
$317$	7178.08i	1.27180i	0.771771	+	0.635901i	$0.219371\pi$
$317$	7178.08i	1.27180i	−0.771771	+	0.635901i	$0.780629\pi$
$318$	2130.41i	0.375684i
$319$	−13132.2	−2.30490
$320$	0	0
$321$	−8065.88	−1.40247
$322$	3053.78i	0.528511i
$323$	− 2451.80i	− 0.422358i
$324$	−5450.34	−0.934558
$325$	0	0
$326$	12805.7	2.17559
$327$	− 2941.17i	− 0.497392i
$328$	− 369.327i	− 0.0621727i
$329$	6890.88	1.15473
$330$	0	0
$331$	−10304.2	−1.71109	−0.855545	−	0.517728i	$-0.826778\pi$
$331$	−10304.2	−1.71109	−0.855545	+	0.517728i	$0.826778\pi$
$332$	− 8945.85i	− 1.47882i
$333$	− 399.409i	− 0.0657281i
$334$	7643.91	1.25226
$335$	0	0
$336$	17712.8	2.87593
$337$	− 1582.19i	− 0.255749i	−0.991790	−	0.127874i	$-0.959185\pi$
$337$	− 1582.19i	− 0.255749i	0.991790	−	0.127874i	$-0.0408154\pi$
$338$	490.105i	0.0788705i
$339$	−19755.2	−3.16507
$340$	0	0
$341$	−7392.54	−1.17398
$342$	− 6874.64i	− 1.08695i
$343$	− 12696.7i	− 1.99871i
$344$	802.253	0.125740
$345$	0	0
$346$	−1724.79	−0.267992
$347$	2566.92i	0.397117i	0.980089	+	0.198559i	$0.0636260\pi$
$347$	2566.92i	0.397117i	−0.980089	+	0.198559i	$0.936374\pi$
$348$	21618.3i	3.33007i
$349$	215.226	0.0330108	0.0165054	−	0.999864i	$-0.494746\pi$
$349$	215.226	0.0330108	0.0165054	+	0.999864i	$0.494746\pi$
$350$	0	0
$351$	−10585.1	−1.60966
$352$	− 11832.2i	− 1.79165i
$353$	5111.32i	0.770675i	0.922776	+	0.385338i	$0.125915\pi$
$353$	5111.32i	0.770675i	−0.922776	+	0.385338i	$0.874085\pi$
$354$	−13648.3	−2.04915
$355$	0	0
$356$	3672.34	0.546724
$357$	22434.5i	3.32594i
$358$	− 18968.3i	− 2.80029i
$359$	4317.14	0.634680	0.317340	−	0.948312i	$-0.397210\pi$
$359$	4317.14	0.634680	0.317340	+	0.948312i	$0.397210\pi$
$360$	0	0
$361$	−5833.56	−0.850497
$362$	− 1540.96i	− 0.223732i
$363$	6814.80i	0.985356i
$364$	−12572.2	−1.81033
$365$	0	0
$366$	1911.01	0.272923
$367$	− 7628.24i	− 1.08499i	−0.840059	−	0.542495i	$-0.817480\pi$
$367$	− 7628.24i	− 1.08499i	0.840059	−	0.542495i	$-0.182520\pi$
$368$	1390.37i	0.196951i
$369$	−11456.1	−1.61620
$370$	0	0
$371$	1926.10	0.269536
$372$	12169.6i	1.69614i
$373$	− 8487.60i	− 1.17821i	−0.808057	−	0.589104i	$-0.799481\pi$
$373$	− 8487.60i	− 1.17821i	0.808057	−	0.589104i	$-0.200519\pi$
$374$	14194.6	1.96253
$375$	0	0
$376$	−359.198	−0.0492666
$377$	13078.9i	1.78673i
$378$	30845.0i	4.19707i
$379$	1227.48	0.166363	0.0831815	−	0.996534i	$-0.473492\pi$
$379$	1227.48	0.166363	0.0831815	+	0.996534i	$0.473492\pi$
$380$	0	0
$381$	−963.951	−0.129619
$382$	5940.85i	0.795707i
$383$	4106.30i	0.547838i	0.961753	+	0.273919i	$0.0883201\pi$
$383$	4106.30i	0.547838i	−0.961753	+	0.273919i	$0.911680\pi$
$384$	−1952.33	−0.259452
$385$	0	0
$386$	−8629.80	−1.13794
$387$	− 24884.9i	− 3.26866i
$388$	− 5523.13i	− 0.722666i
$389$	11206.6	1.46066	0.730328	−	0.683097i	$-0.239367\pi$
$389$	11206.6	1.46066	0.730328	+	0.683097i	$0.239367\pi$
$390$	0	0
$391$	−1760.99	−0.227768
$392$	1247.65i	0.160754i
$393$	− 8810.25i	− 1.13084i
$394$	4647.84	0.594301
$395$	0	0
$396$	20411.0	2.59013
$397$	7629.81i	0.964557i	0.876018	+	0.482278i	$0.160191\pi$
$397$	7629.81i	0.964557i	−0.876018	+	0.482278i	$0.839809\pi$
$398$	15633.4i	1.96893i
$399$	−9383.01	−1.17729
$400$	0	0
$401$	1221.46	0.152112	0.0760560	−	0.997104i	$-0.475767\pi$
$401$	1221.46	0.152112	0.0760560	+	0.997104i	$0.475767\pi$
$402$	607.666i	0.0753921i
$403$	7362.51i	0.910056i
$404$	−7000.83	−0.862140
$405$	0	0
$406$	38112.0	4.65878
$407$	344.918i	0.0420073i
$408$	− 1169.44i	− 0.141901i
$409$	−16112.2	−1.94791	−0.973957	−	0.226731i	$-0.927196\pi$
$409$	−16112.2	−1.94791	−0.973957	+	0.226731i	$0.927196\pi$
$410$	0	0
$411$	−584.651	−0.0701672
$412$	− 6747.91i	− 0.806907i
$413$	12339.4i	1.47018i
$414$	−4937.67	−0.586168
$415$	0	0
$416$	−11784.2	−1.38886
$417$	4896.27i	0.574992i
$418$	5936.75i	0.694679i
$419$	−1997.98	−0.232954	−0.116477	−	0.993193i	$-0.537160\pi$
$419$	−1997.98	−0.232954	−0.116477	+	0.993193i	$0.537160\pi$
$420$	0	0
$421$	−4849.75	−0.561431	−0.280715	−	0.959791i	$-0.590572\pi$
$421$	−4849.75	−0.561431	−0.280715	+	0.959791i	$0.590572\pi$
$422$	7619.29i	0.878913i
$423$	11141.9i	1.28070i
$424$	−100.401	−0.0114998
$425$	0	0
$426$	7590.16	0.863250
$427$	− 1727.74i	− 0.195810i
$428$	− 7595.49i	− 0.857809i
$429$	18641.9	2.09799
$430$	0	0
$431$	13262.3	1.48219	0.741094	−	0.671402i	$-0.234307\pi$
$431$	13262.3	1.48219	0.741094	+	0.671402i	$0.234307\pi$
$432$	14043.5i	1.56405i
$433$	− 13656.0i	− 1.51562i	−0.652473	−	0.757812i	$-0.726269\pi$
$433$	− 13656.0i	− 1.51562i	0.652473	−	0.757812i	$-0.273731\pi$
$434$	21454.4	2.37291
$435$	0	0
$436$	2769.65	0.304225
$437$	− 736.517i	− 0.0806234i
$438$	− 29423.7i	− 3.20986i
$439$	1132.68	0.123143	0.0615717	−	0.998103i	$-0.480389\pi$
$439$	1132.68	0.123143	0.0615717	+	0.998103i	$0.480389\pi$
$440$	0	0
$441$	38700.5	4.17887
$442$	− 14136.9i	− 1.52132i
$443$	− 6583.81i	− 0.706109i	−0.935603	−	0.353054i	$-0.885143\pi$
$443$	− 6583.81i	− 0.706109i	0.935603	−	0.353054i	$-0.114857\pi$
$444$	567.805	0.0606910
$445$	0	0
$446$	−12890.8	−1.36860
$447$	− 29663.7i	− 3.13881i
$448$	18494.0i	1.95035i
$449$	12407.9	1.30416	0.652078	−	0.758152i	$-0.273897\pi$
$449$	12407.9	1.30416	0.652078	+	0.758152i	$0.273897\pi$
$450$	0	0
$451$	9893.15	1.03293
$452$	− 18603.2i	− 1.93588i
$453$	− 14716.0i	− 1.52631i
$454$	−19521.1	−2.01800
$455$	0	0
$456$	489.105	0.0502290
$457$	17336.1i	1.77450i	0.461286	+	0.887251i	$0.347388\pi$
$457$	17336.1i	1.77450i	−0.461286	+	0.887251i	$0.652612\pi$
$458$	5837.86i	0.595601i
$459$	−17787.1	−1.80878
$460$	0	0
$461$	9940.64	1.00430	0.502150	−	0.864781i	$-0.332543\pi$
$461$	9940.64	1.00430	0.502150	+	0.864781i	$0.332543\pi$
$462$	− 54322.5i	− 5.47037i
$463$	− 6459.56i	− 0.648383i	−0.945991	−	0.324191i	$-0.894908\pi$
$463$	− 6459.56i	− 0.648383i	0.945991	−	0.324191i	$-0.105092\pi$
$464$	17352.1	1.73610
$465$	0	0
$466$	9375.82	0.932031
$467$	8151.66i	0.807739i	0.914817	+	0.403869i	$0.132335\pi$
$467$	8151.66i	0.807739i	−0.914817	+	0.403869i	$0.867665\pi$
$468$	− 20328.1i	− 2.00783i
$469$	549.389	0.0540905
$470$	0	0
$471$	18679.7	1.82742
$472$	− 643.212i	− 0.0627251i
$473$	21489.9i	2.08902i
$474$	2284.60	0.221382
$475$	0	0
$476$	−21126.2	−2.03428
$477$	3114.32i	0.298941i
$478$	11338.8i	1.08499i
$479$	214.878	0.0204969	0.0102485	−	0.999947i	$-0.496738\pi$
$479$	214.878	0.0204969	0.0102485	+	0.999947i	$0.496738\pi$
$480$	0	0
$481$	343.517	0.0325635
$482$	− 7331.92i	− 0.692863i
$483$	6739.30i	0.634883i
$484$	−6417.38	−0.602684
$485$	0	0
$486$	1963.74	0.183286
$487$	6216.61i	0.578442i	0.957262	+	0.289221i	$0.0933963\pi$
$487$	6216.61i	0.578442i	−0.957262	+	0.289221i	$0.906604\pi$
$488$	90.0611i	0.00835425i
$489$	28260.6	2.61347
$490$	0	0
$491$	−17480.8	−1.60671	−0.803356	−	0.595499i	$-0.796954\pi$
$491$	−17480.8	−1.60671	−0.803356	+	0.595499i	$0.796954\pi$
$492$	− 16286.1i	− 1.49235i
$493$	21977.7i	2.00776i
$494$	5912.63	0.538506
$495$	0	0
$496$	9768.04	0.884270
$497$	− 6862.24i	− 0.619343i
$498$	− 38496.7i	− 3.46401i
$499$	17884.0	1.60440	0.802200	−	0.597055i	$-0.203663\pi$
$499$	17884.0	1.60440	0.802200	+	0.597055i	$0.203663\pi$
$500$	0	0
$501$	16869.1	1.50431
$502$	23563.5i	2.09501i
$503$	− 3215.45i	− 0.285029i	−0.989793	−	0.142515i	$-0.954481\pi$
$503$	− 3215.45i	− 0.285029i	0.989793	−	0.142515i	$-0.0455188\pi$
$504$	−2964.53	−0.262005
$505$	0	0
$506$	4264.04	0.374624
$507$	1081.60i	0.0947446i
$508$	− 907.736i	− 0.0792801i
$509$	−6737.65	−0.586721	−0.293360	−	0.956002i	$-0.594774\pi$
$509$	−6737.65	−0.586721	−0.293360	+	0.956002i	$0.594774\pi$
$510$	0	0
$511$	−26601.9	−2.30293
$512$	16460.3i	1.42080i
$513$	− 7439.26i	− 0.640256i
$514$	−7644.24	−0.655979
$515$	0	0
$516$	35376.8	3.01817
$517$	− 9621.84i	− 0.818507i
$518$	− 1001.01i	− 0.0849070i
$519$	−3806.39	−0.321930
$520$	0	0
$521$	13464.5	1.13223	0.566114	−	0.824327i	$-0.308446\pi$
$521$	13464.5	1.13223	0.566114	+	0.824327i	$0.308446\pi$
$522$	61623.5i	5.16702i
$523$	21370.5i	1.78674i	0.449317	+	0.893372i	$0.351667\pi$
$523$	21370.5i	1.78674i	−0.449317	+	0.893372i	$0.648333\pi$
$524$	8296.46	0.691665
$525$	0	0
$526$	−24386.6	−2.02150
$527$	12371.9i	1.02263i
$528$	− 24732.7i	− 2.03855i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−19951.7	−1.63056
$532$	− 8835.81i	− 0.720077i
$533$	− 9852.96i	− 0.800711i
$534$	15803.2	1.28066
$535$	0	0
$536$	−28.6378	−0.00230777
$537$	− 41860.6i	− 3.36391i
$538$	12923.5i	1.03564i
$539$	−33420.7	−2.67075
$540$	0	0
$541$	−14485.2	−1.15114	−0.575572	−	0.817751i	$-0.695221\pi$
$541$	−14485.2	−1.15114	−0.575572	+	0.817751i	$0.695221\pi$
$542$	4730.81i	0.374918i
$543$	− 3400.70i	− 0.268763i
$544$	−19802.0	−1.56067
$545$	0	0
$546$	−54101.9	−4.24056
$547$	− 18926.7i	− 1.47943i	−0.672922	−	0.739714i	$-0.734961\pi$
$547$	− 18926.7i	− 1.47943i	0.672922	−	0.739714i	$-0.265039\pi$
$548$	− 550.555i	− 0.0429171i
$549$	2793.59	0.217172
$550$	0	0
$551$	−9191.93	−0.710689
$552$	− 351.297i	− 0.0270873i
$553$	− 2065.50i	− 0.158832i
$554$	16875.4	1.29416
$555$	0	0
$556$	−4610.73	−0.351688
$557$	2146.15i	0.163259i	0.996663	+	0.0816294i	$0.0260124\pi$
$557$	2146.15i	0.163259i	−0.996663	+	0.0816294i	$0.973988\pi$
$558$	34689.7i	2.63178i
$559$	21402.6	1.61938
$560$	0	0
$561$	31325.6	2.35752
$562$	20093.9i	1.50820i
$563$	14066.3i	1.05297i	0.850183	+	0.526487i	$0.176491\pi$
$563$	14066.3i	1.05297i	−0.850183	+	0.526487i	$0.823509\pi$
$564$	−15839.5	−1.18256
$565$	0	0
$566$	8799.81	0.653505
$567$	21205.1i	1.57060i
$568$	357.706i	0.0264243i
$569$	1253.22	0.0923337	0.0461668	−	0.998934i	$-0.485299\pi$
$569$	1253.22	0.0923337	0.0461668	+	0.998934i	$0.485299\pi$
$570$	0	0
$571$	12499.6	0.916097	0.458048	−	0.888927i	$-0.348549\pi$
$571$	12499.6	0.916097	0.458048	+	0.888927i	$0.348549\pi$
$572$	17554.7i	1.28322i
$573$	13110.7i	0.955858i
$574$	−28711.6	−2.08780
$575$	0	0
$576$	−29903.0	−2.16312
$577$	− 7260.56i	− 0.523849i	−0.965088	−	0.261925i	$-0.915643\pi$
$577$	− 7260.56i	− 0.523849i	0.965088	−	0.261925i	$-0.0843572\pi$
$578$	− 3846.41i	− 0.276799i
$579$	−19044.9	−1.36697
$580$	0	0
$581$	−34804.7	−2.48527
$582$	− 23767.7i	− 1.69279i
$583$	− 2689.44i	− 0.191055i
$584$	1386.67	0.0982545
$585$	0	0
$586$	20950.6	1.47690
$587$	14411.6i	1.01334i	0.862139	+	0.506671i	$0.169124\pi$
$587$	14411.6i	1.01334i	−0.862139	+	0.506671i	$0.830876\pi$
$588$	55017.2i	3.85863i
$589$	−5174.42	−0.361983
$590$	0	0
$591$	10257.2	0.713916
$592$	− 455.753i	− 0.0316408i
$593$	− 21665.4i	− 1.50032i	−0.661256	−	0.750160i	$-0.729976\pi$
$593$	− 21665.4i	− 1.50032i	0.661256	−	0.750160i	$-0.270024\pi$
$594$	43069.3	2.97501
$595$	0	0
$596$	27933.8	1.91982
$597$	34501.0i	2.36521i
$598$	− 4246.72i	− 0.290403i
$599$	23174.4	1.58077	0.790384	−	0.612612i	$-0.209881\pi$
$599$	23174.4	1.58077	0.790384	+	0.612612i	$0.209881\pi$
$600$	0	0
$601$	−22969.1	−1.55895	−0.779475	−	0.626433i	$-0.784514\pi$
$601$	−22969.1	−1.55895	−0.779475	+	0.626433i	$0.784514\pi$
$602$	− 62367.4i	− 4.22243i
$603$	888.311i	0.0599914i
$604$	13857.8	0.933551
$605$	0	0
$606$	−30126.7	−2.01949
$607$	− 4606.78i	− 0.308045i	−0.988067	−	0.154023i	$-0.950777\pi$
$607$	− 4606.78i	− 0.308045i	0.988067	−	0.154023i	$-0.0492229\pi$
$608$	− 8281.98i	− 0.552432i
$609$	84108.2	5.59645
$610$	0	0
$611$	−9582.75	−0.634495
$612$	− 34159.1i	− 2.25621i
$613$	− 7780.72i	− 0.512660i	−0.966589	−	0.256330i	$-0.917487\pi$
$613$	− 7780.72i	− 0.512660i	0.966589	−	0.256330i	$-0.0825133\pi$
$614$	−37238.5	−2.44759
$615$	0	0
$616$	2560.09	0.167449
$617$	2438.43i	0.159105i	0.996831	+	0.0795523i	$0.0253491\pi$
$617$	2438.43i	0.159105i	−0.996831	+	0.0795523i	$0.974651\pi$
$618$	− 29038.3i	− 1.89011i
$619$	−6821.13	−0.442915	−0.221458	−	0.975170i	$-0.571081\pi$
$619$	−6821.13	−0.442915	−0.221458	+	0.975170i	$0.571081\pi$
$620$	0	0
$621$	−5343.21	−0.345275
$622$	9176.88i	0.591574i
$623$	− 14287.6i	− 0.918813i
$624$	−24632.2	−1.58025
$625$	0	0
$626$	9998.83	0.638393
$627$	13101.6i	0.834496i
$628$	17590.3i	1.11772i
$629$	577.242	0.0365917
$630$	0	0
$631$	−15698.7	−0.990420	−0.495210	−	0.868773i	$-0.664909\pi$
$631$	−15698.7	−0.990420	−0.495210	+	0.868773i	$0.664909\pi$
$632$	107.668i	0.00677657i
$633$	16814.8i	1.05581i
$634$	−29088.0	−1.82213
$635$	0	0
$636$	−4427.36	−0.276032
$637$	33284.9i	2.07033i
$638$	− 53216.3i	− 3.30228i
$639$	11095.6	0.686909
$640$	0	0
$641$	15360.1	0.946472	0.473236	−	0.880936i	$-0.343086\pi$
$641$	15360.1	0.946472	0.473236	+	0.880936i	$0.343086\pi$
$642$	− 32685.7i	− 2.00935i
$643$	14107.3i	0.865221i	0.901581	+	0.432610i	$0.142407\pi$
$643$	14107.3i	0.865221i	−0.901581	+	0.432610i	$0.857593\pi$
$644$	−6346.28	−0.388320
$645$	0	0
$646$	9935.52	0.605121
$647$	8107.22i	0.492624i	0.969191	+	0.246312i	$0.0792188\pi$
$647$	8107.22i	0.492624i	−0.969191	+	0.246312i	$0.920781\pi$
$648$	− 1105.35i	− 0.0670097i
$649$	17229.7	1.04210
$650$	0	0
$651$	47347.0	2.85050
$652$	26612.5i	1.59850i
$653$	25123.5i	1.50560i	0.658247	+	0.752802i	$0.271298\pi$
$653$	25123.5i	1.50560i	−0.658247	+	0.752802i	$0.728702\pi$
$654$	11918.6	0.712623
$655$	0	0
$656$	−13072.2	−0.778023
$657$	− 43012.7i	− 2.55416i
$658$	27924.2i	1.65440i
$659$	809.773	0.0478669	0.0239334	−	0.999714i	$-0.492381\pi$
$659$	809.773	0.0478669	0.0239334	+	0.999714i	$0.492381\pi$
$660$	0	0
$661$	4086.36	0.240455	0.120228	−	0.992746i	$-0.461638\pi$
$661$	4086.36	0.240455	0.120228	+	0.992746i	$0.461638\pi$
$662$	− 41756.2i	− 2.45151i
$663$	− 31198.4i	− 1.82752i
$664$	1814.25	0.106034
$665$	0	0
$666$	1618.54	0.0941699
$667$	6602.06i	0.383258i
$668$	15885.4i	0.920095i
$669$	−28448.2	−1.64405
$670$	0	0
$671$	−2412.47	−0.138796
$672$	75781.9i	4.35022i
$673$	8985.56i	0.514662i	0.966323	+	0.257331i	$0.0828431\pi$
$673$	8985.56i	0.514662i	−0.966323	+	0.257331i	$0.917157\pi$
$674$	6411.56	0.366416
$675$	0	0
$676$	−1018.52	−0.0579497
$677$	10328.0i	0.586316i	0.956064	+	0.293158i	$0.0947061\pi$
$677$	10328.0i	0.586316i	−0.956064	+	0.293158i	$0.905294\pi$
$678$	− 80055.0i	− 4.53465i
$679$	−21488.3	−1.21450
$680$	0	0
$681$	−43080.6	−2.42416
$682$	− 29957.1i	− 1.68199i
$683$	− 4516.74i	− 0.253043i	−0.991964	−	0.126521i	$-0.959619\pi$
$683$	− 4516.74i	− 0.253043i	0.991964	−	0.126521i	$-0.0403812\pi$
$684$	14286.7	0.798633
$685$	0	0
$686$	51451.3	2.86359
$687$	12883.4i	0.715476i
$688$	− 28395.5i	− 1.57350i
$689$	−2678.51	−0.148103
$690$	0	0
$691$	7957.06	0.438062	0.219031	−	0.975718i	$-0.429710\pi$
$691$	7957.06	0.438062	0.219031	+	0.975718i	$0.429710\pi$
$692$	− 3584.41i	− 0.196906i
$693$	− 79410.9i	− 4.35291i
$694$	−10402.0	−0.568957
$695$	0	0
$696$	−4384.28	−0.238773
$697$	− 16556.8i	− 0.899762i
$698$	872.167i	0.0472951i
$699$	20691.2	1.11962
$700$	0	0
$701$	7970.58	0.429450	0.214725	−	0.976675i	$-0.431114\pi$
$701$	7970.58	0.429450	0.214725	+	0.976675i	$0.431114\pi$
$702$	− 42894.4i	− 2.30619i
$703$	241.426i	0.0129524i
$704$	25823.4	1.38247
$705$	0	0
$706$	−20712.8	−1.10416
$707$	27237.4i	1.44889i
$708$	− 28363.6i	− 1.50561i
$709$	−2079.18	−0.110134	−0.0550672	−	0.998483i	$-0.517537\pi$
$709$	−2079.18	−0.110134	−0.0550672	+	0.998483i	$0.517537\pi$
$710$	0	0
$711$	3339.72	0.176159
$712$	744.765i	0.0392012i
$713$	3716.50i	0.195209i
$714$	−90912.2	−4.76513
$715$	0	0
$716$	39419.4	2.05750
$717$	25023.3i	1.30336i
$718$	17494.5i	0.909318i
$719$	−29046.5	−1.50661	−0.753305	−	0.657671i	$-0.771542\pi$
$719$	−29046.5	−1.50661	−0.753305	+	0.657671i	$0.771542\pi$
$720$	0	0
$721$	−26253.4	−1.35607
$722$	− 23639.6i	− 1.21852i
$723$	− 16180.6i	− 0.832315i
$724$	3202.38	0.164386
$725$	0	0
$726$	−27615.9	−1.41174
$727$	14037.7i	0.716133i	0.933696	+	0.358067i	$0.116564\pi$
$727$	14037.7i	0.716133i	−0.933696	+	0.358067i	$0.883436\pi$
$728$	− 2549.69i	− 0.129805i
$729$	21808.0	1.10796
$730$	0	0
$731$	35964.8	1.81971
$732$	3971.40i	0.200529i
$733$	− 10862.5i	− 0.547358i	−0.961821	−	0.273679i	$-0.911759\pi$
$733$	− 10862.5i	− 0.547358i	0.961821	−	0.273679i	$-0.0882407\pi$
$734$	30912.2	1.55448
$735$	0	0
$736$	−5948.49	−0.297913
$737$	− 767.121i	− 0.0383409i
$738$	− 46423.9i	− 2.31557i
$739$	8660.65	0.431106	0.215553	−	0.976492i	$-0.430845\pi$
$739$	8660.65	0.431106	0.215553	+	0.976492i	$0.430845\pi$
$740$	0	0
$741$	13048.4	0.646890
$742$	7805.20i	0.386170i
$743$	33484.3i	1.65332i	0.562700	+	0.826661i	$0.309762\pi$
$743$	33484.3i	1.65332i	−0.562700	+	0.826661i	$0.690238\pi$
$744$	−2468.04	−0.121617
$745$	0	0
$746$	34394.7	1.68804
$747$	− 56275.9i	− 2.75640i
$748$	29498.8i	1.44196i
$749$	−29551.0	−1.44162
$750$	0	0
$751$	29209.3	1.41926	0.709628	−	0.704577i	$-0.248863\pi$
$751$	29209.3	1.41926	0.709628	+	0.704577i	$0.248863\pi$
$752$	12713.7i	0.616517i
$753$	52001.7i	2.51666i
$754$	−53000.1	−2.55988
$755$	0	0
$756$	−64101.2	−3.08378
$757$	3283.09i	0.157630i	0.996889	+	0.0788151i	$0.0251137\pi$
$757$	3283.09i	0.157630i	−0.996889	+	0.0788151i	$0.974886\pi$
$758$	4974.18i	0.238351i
$759$	9410.19	0.450024
$760$	0	0
$761$	22957.1	1.09355	0.546776	−	0.837279i	$-0.315855\pi$
$761$	22957.1	1.09355	0.546776	+	0.837279i	$0.315855\pi$
$762$	− 3906.26i	− 0.185707i
$763$	− 10775.6i	− 0.511275i
$764$	−12346.1	−0.584642
$765$	0	0
$766$	−16640.1	−0.784898
$767$	− 17159.7i	− 0.807825i
$768$	32471.6i	1.52567i
$769$	2510.87	0.117743	0.0588714	−	0.998266i	$-0.481250\pi$
$769$	2510.87	0.117743	0.0588714	+	0.998266i	$0.481250\pi$
$770$	0	0
$771$	−16869.9	−0.788007
$772$	− 17934.2i	− 0.836096i
$773$	11566.1i	0.538167i	0.963117	+	0.269084i	$0.0867208\pi$
$773$	11566.1i	0.538167i	−0.963117	+	0.269084i	$0.913279\pi$
$774$	100842.	4.68307
$775$	0	0
$776$	1120.11	0.0518166
$777$	− 2209.10i	− 0.101996i
$778$	45412.8i	2.09271i
$779$	6924.72	0.318490
$780$	0	0
$781$	−9581.85	−0.439008
$782$	− 7136.14i	− 0.326327i
$783$	66684.7i	3.04357i
$784$	44160.0	2.01166
$785$	0	0
$786$	35702.1	1.62017
$787$	− 751.934i	− 0.0340579i	−0.999855	−	0.0170289i	$-0.994579\pi$
$787$	− 751.934i	− 0.0340579i	0.999855	−	0.0170289i	$-0.00542074\pi$
$788$	9659.01i	0.436660i
$789$	−53818.1	−2.42836
$790$	0	0
$791$	−72377.4	−3.25341
$792$	4139.42i	0.185717i
$793$	2402.66i	0.107593i
$794$	−30918.6	−1.38194
$795$	0	0
$796$	−32489.0	−1.44666
$797$	− 15898.3i	− 0.706585i	−0.935513	−	0.353292i	$-0.885062\pi$
$797$	− 15898.3i	− 0.706585i	0.935513	−	0.353292i	$-0.114938\pi$
$798$	− 38023.1i	− 1.68672i
$799$	−16102.8	−0.712985
$800$	0	0
$801$	23101.7	1.01905
$802$	4949.78i	0.217934i
$803$	37144.6i	1.63238i
$804$	−1262.84	−0.0553940
$805$	0	0
$806$	−29835.4	−1.30385
$807$	28520.6i	1.24408i
$808$	− 1419.80i	− 0.0618171i
$809$	30018.9	1.30458	0.652292	−	0.757967i	$-0.273807\pi$
$809$	30018.9	1.30458	0.652292	+	0.757967i	$0.273807\pi$
$810$	0	0
$811$	5012.47	0.217030	0.108515	−	0.994095i	$-0.465390\pi$
$811$	5012.47	0.217030	0.108515	+	0.994095i	$0.465390\pi$
$812$	79203.2i	3.42301i
$813$	10440.3i	0.450377i
$814$	−1397.73	−0.0601846
$815$	0	0
$816$	−41391.8	−1.77574
$817$	15041.9i	0.644125i
$818$	− 65292.1i	− 2.79081i
$819$	−79088.3	−3.37432
$820$	0	0
$821$	−2707.44	−0.115092	−0.0575459	−	0.998343i	$-0.518328\pi$
$821$	−2707.44	−0.115092	−0.0575459	+	0.998343i	$0.518328\pi$
$822$	− 2369.20i	− 0.100530i
$823$	− 15781.8i	− 0.668431i	−0.942497	−	0.334215i	$-0.891529\pi$
$823$	− 15781.8i	− 0.668431i	0.942497	−	0.334215i	$-0.108471\pi$
$824$	1368.50	0.0578568
$825$	0	0
$826$	−50003.5	−2.10635
$827$	− 21176.8i	− 0.890435i	−0.895422	−	0.445218i	$-0.853126\pi$
$827$	− 21176.8i	− 0.890435i	0.895422	−	0.445218i	$-0.146874\pi$
$828$	− 10261.3i	− 0.430684i
$829$	−20905.6	−0.875852	−0.437926	−	0.899011i	$-0.644287\pi$
$829$	−20905.6	−0.875852	−0.437926	+	0.899011i	$0.644287\pi$
$830$	0	0
$831$	37241.8	1.55464
$832$	− 25718.5i	− 1.07167i
$833$	55931.7i	2.32643i
$834$	−19841.4	−0.823801
$835$	0	0
$836$	−12337.6	−0.510412
$837$	37538.8i	1.55022i
$838$	− 8096.51i	− 0.333758i
$839$	−28724.8	−1.18199	−0.590995	−	0.806675i	$-0.701265\pi$
$839$	−28724.8	−1.18199	−0.590995	+	0.806675i	$0.701265\pi$
$840$	0	0
$841$	58006.4	2.37839
$842$	− 19652.8i	− 0.804372i
$843$	44344.6i	1.81176i
$844$	−15834.2	−0.645777
$845$	0	0
$846$	−45150.8	−1.83489
$847$	24967.4i	1.01286i
$848$	3553.66i	0.143907i
$849$	19420.1	0.785035
$850$	0	0
$851$	173.403	0.00698493
$852$	15773.7i	0.634268i
$853$	37511.5i	1.50571i	0.658188	+	0.752854i	$0.271323\pi$
$853$	37511.5i	1.50571i	−0.658188	+	0.752854i	$0.728677\pi$
$854$	7001.37	0.280541
$855$	0	0
$856$	1540.40	0.0615066
$857$	26541.3i	1.05792i	0.848648	+	0.528958i	$0.177417\pi$
$857$	26541.3i	1.05792i	−0.848648	+	0.528958i	$0.822583\pi$
$858$	75543.3i	3.00583i
$859$	−20535.4	−0.815667	−0.407833	−	0.913056i	$-0.633716\pi$
$859$	−20535.4	−0.815667	−0.407833	+	0.913056i	$0.633716\pi$
$860$	0	0
$861$	−63362.7	−2.50801
$862$	53743.4i	2.12356i
$863$	− 21337.9i	− 0.841660i	−0.907140	−	0.420830i	$-0.861739\pi$
$863$	− 21337.9i	− 0.841660i	0.907140	−	0.420830i	$-0.138261\pi$
$864$	−60083.3	−2.36583
$865$	0	0
$866$	55338.7	2.17146
$867$	− 8488.54i	− 0.332510i
$868$	44585.9i	1.74348i
$869$	−2884.09	−0.112585
$870$	0	0
$871$	−764.004	−0.0297214
$872$	561.696i	0.0218136i
$873$	− 34744.5i	− 1.34699i
$874$	2984.62	0.115511
$875$	0	0
$876$	61147.5	2.35843
$877$	− 42216.5i	− 1.62548i	−0.582624	−	0.812742i	$-0.697974\pi$
$877$	− 42216.5i	− 1.62548i	0.582624	−	0.812742i	$-0.302026\pi$
$878$	4590.01i	0.176430i
$879$	46235.3	1.77415
$880$	0	0
$881$	−33457.6	−1.27947	−0.639737	−	0.768594i	$-0.720957\pi$
$881$	−33457.6	−1.27947	−0.639737	+	0.768594i	$0.720957\pi$
$882$	156828.i	5.98715i
$883$	− 22192.2i	− 0.845785i	−0.906180	−	0.422893i	$-0.861015\pi$
$883$	− 22192.2i	− 0.845785i	0.906180	−	0.422893i	$-0.138985\pi$
$884$	29379.0	1.11779
$885$	0	0
$886$	26679.8	1.01165
$887$	17904.9i	0.677777i	0.940826	+	0.338889i	$0.110051\pi$
$887$	17904.9i	0.677777i	−0.940826	+	0.338889i	$0.889949\pi$
$888$	115.153i	0.00435167i
$889$	−3531.64	−0.133237
$890$	0	0
$891$	29609.0	1.11329
$892$	− 26789.2i	− 1.00557i
$893$	− 6734.82i	− 0.252376i
$894$	120208.	4.49703
$895$	0	0
$896$	−7152.77	−0.266693
$897$	− 9371.96i	− 0.348852i
$898$	50281.1i	1.86849i
$899$	46382.9	1.72075
$900$	0	0
$901$	−4500.95	−0.166424
$902$	40090.4i	1.47989i
$903$	− 137637.i	− 5.07228i
$904$	3772.79	0.138807
$905$	0	0
$906$	59634.1	2.18677
$907$	9482.20i	0.347135i	0.984822	+	0.173567i	$0.0555295\pi$
$907$	9482.20i	0.347135i	−0.984822	+	0.173567i	$0.944471\pi$
$908$	− 40568.2i	− 1.48271i
$909$	−44040.4	−1.60696
$910$	0	0
$911$	24018.2	0.873499	0.436750	−	0.899583i	$-0.356130\pi$
$911$	24018.2	0.873499	0.436750	+	0.899583i	$0.356130\pi$
$912$	− 17311.7i	− 0.628561i
$913$	48598.3i	1.76163i
$914$	−70251.7	−2.54236
$915$	0	0
$916$	−12132.1	−0.437615
$917$	− 32278.2i	− 1.16240i
$918$	− 72079.3i	− 2.59147i
$919$	39263.9	1.40935	0.704677	−	0.709528i	$-0.251092\pi$
$919$	39263.9	1.40935	0.704677	+	0.709528i	$0.251092\pi$
$920$	0	0
$921$	−82180.5	−2.94022
$922$	40282.9i	1.43888i
$923$	9542.92i	0.340313i
$924$	112892.	4.01933
$925$	0	0
$926$	26176.3	0.928950
$927$	− 42449.3i	− 1.50401i
$928$	74238.7i	2.62608i
$929$	−30536.7	−1.07845	−0.539223	−	0.842163i	$-0.681282\pi$
$929$	−30536.7	−1.07845	−0.539223	+	0.842163i	$0.681282\pi$
$930$	0	0
$931$	−23392.9	−0.823491
$932$	19484.6i	0.684805i
$933$	20252.2i	0.710640i
$934$	−33033.3	−1.15726
$935$	0	0
$936$	4122.61	0.143965
$937$	− 37724.9i	− 1.31528i	−0.753332	−	0.657640i	$-0.771555\pi$
$937$	− 37724.9i	− 1.31528i	0.753332	−	0.657640i	$-0.228445\pi$
$938$	2226.31i	0.0774964i
$939$	22066.1	0.766881
$940$	0	0
$941$	33414.9	1.15759	0.578797	−	0.815472i	$-0.303522\pi$
$941$	33414.9	1.15759	0.578797	+	0.815472i	$0.303522\pi$
$942$	75696.4i	2.61817i
$943$	− 4973.65i	− 0.171754i
$944$	−22766.3	−0.784935
$945$	0	0
$946$	−87084.5	−2.99298
$947$	− 11604.7i	− 0.398205i	−0.979979	−	0.199103i	$-0.936197\pi$
$947$	− 11604.7i	− 0.398205i	0.979979	−	0.199103i	$-0.0638027\pi$
$948$	4747.80i	0.162660i
$949$	36993.7	1.26540
$950$	0	0
$951$	−64193.5	−2.18887
$952$	− 4284.47i	− 0.145862i
$953$	− 33381.8i	− 1.13467i	−0.823487	−	0.567336i	$-0.807974\pi$
$953$	− 33381.8i	− 1.13467i	0.823487	−	0.567336i	$-0.192026\pi$
$954$	−12620.3	−0.428298
$955$	0	0
$956$	−23564.0	−0.797190
$957$	− 117442.i	− 3.96693i
$958$	870.758i	0.0293663i
$959$	−2141.99	−0.0721256
$960$	0	0
$961$	−3680.68	−0.123550
$962$	1392.05i	0.0466543i
$963$	− 47781.2i	− 1.59889i
$964$	15237.0	0.509078
$965$	0	0
$966$	−27309.9	−0.909609
$967$	44013.3i	1.46367i	0.681479	+	0.731837i	$0.261337\pi$
$967$	44013.3i	1.46367i	−0.681479	+	0.731837i	$0.738663\pi$
$968$	− 1301.47i	− 0.0432136i
$969$	21926.4	0.726913
$970$	0	0
$971$	54941.3	1.81581	0.907905	−	0.419176i	$-0.137681\pi$
$971$	54941.3	1.81581	0.907905	+	0.419176i	$0.137681\pi$
$972$	4080.98i	0.134668i
$973$	17938.5i	0.591040i
$974$	−25191.8	−0.828745
$975$	0	0
$976$	3187.68	0.104544
$977$	52377.7i	1.71516i	0.514351	+	0.857580i	$0.328033\pi$
$977$	52377.7i	1.71516i	−0.514351	+	0.857580i	$0.671967\pi$
$978$	114521.i	3.74437i
$979$	−19950.0	−0.651282
$980$	0	0
$981$	17423.1	0.567052
$982$	− 70838.0i	− 2.30197i
$983$	18746.9i	0.608275i	0.952628	+	0.304137i	$0.0983682\pi$
$983$	18746.9i	0.608275i	−0.952628	+	0.304137i	$0.901632\pi$
$984$	3302.89	0.107004
$985$	0	0
$986$	−89060.9	−2.87655
$987$	61625.1i	1.98738i
$988$	12287.5i	0.395664i
$989$	10803.8	0.347361
$990$	0	0
$991$	51116.6	1.63852	0.819260	−	0.573423i	$-0.194385\pi$
$991$	51116.6	1.63852	0.819260	+	0.573423i	$0.194385\pi$
$992$	41791.2i	1.33757i
$993$	− 92150.5i	− 2.94492i
$994$	27808.1	0.887344
$995$	0	0
$996$	80002.7	2.54516
$997$	− 36176.5i	− 1.14917i	−0.818445	−	0.574584i	$-0.805164\pi$
$997$	− 36176.5i	− 1.14917i	0.818445	−	0.574584i	$-0.194836\pi$
$998$	72471.9i	2.29865i
$999$	1751.47	0.0554696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.b.k.24.13		16
5.2	odd	4		575.4.a.n.1.3		8
5.3	odd	4		115.4.a.f.1.6	✓	8
5.4	even	2	inner	575.4.b.k.24.4		16
15.8	even	4		1035.4.a.r.1.3		8
20.3	even	4		1840.4.a.v.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.f.1.6	✓	8	5.3	odd	4
575.4.a.n.1.3		8	5.2	odd	4
575.4.b.k.24.4		16	5.4	even	2	inner
575.4.b.k.24.13		16	1.1	even	1	trivial
1035.4.a.r.1.3		8	15.8	even	4
1840.4.a.v.1.7		8	20.3	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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.05234i q^{2} +8.94300i q^{3} -8.42146 q^{4} -36.2401 q^{6} +32.7645i q^{7} -1.70790i q^{8} -52.9772 q^{9} +45.7496 q^{11} -75.3131i q^{12} -45.5638i q^{13} -132.773 q^{14} -60.4507 q^{16} -76.5649i q^{17} -214.681i q^{18} +32.0225 q^{19} -293.013 q^{21} +185.393i q^{22} -23.0000i q^{23} +15.2738 q^{24} +184.640 q^{26} -232.314i q^{27} -275.925i q^{28} -287.046 q^{29} -161.587 q^{31} -258.630i q^{32} +409.139i q^{33} +310.267 q^{34} +446.145 q^{36} +7.53926i q^{37} +129.766i q^{38} +407.476 q^{39} +216.246 q^{41} -1187.39i q^{42} +469.729i q^{43} -385.279 q^{44} +93.2038 q^{46} -210.315i q^{47} -540.610i q^{48} -730.514 q^{49} +684.719 q^{51} +383.713i q^{52} -58.7860i q^{53} +941.414 q^{54} +55.9587 q^{56} +286.377i q^{57} -1163.21i q^{58} +376.609 q^{59} -52.7319 q^{61} -654.805i q^{62} -1735.77i q^{63} +564.451 q^{64} -1657.97 q^{66} -16.7678i q^{67} +644.788i q^{68} +205.689 q^{69} -209.441 q^{71} +90.4800i q^{72} +811.910i q^{73} -30.5516 q^{74} -269.676 q^{76} +1498.96i q^{77} +1651.23i q^{78} -63.0408 q^{79} +647.197 q^{81} +876.300i q^{82} +1062.27i q^{83} +2467.60 q^{84} -1903.50 q^{86} -2567.05i q^{87} -78.1360i q^{88} -436.069 q^{89} +1492.87 q^{91} +193.694i q^{92} -1445.07i q^{93} +852.269 q^{94} +2312.93 q^{96} +655.840i q^{97} -2960.29i q^{98} -2423.68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 84 q^{4} - 316 q^{9} + 82 q^{11} - 322 q^{14} + 196 q^{16} - 354 q^{19} + 584 q^{21} - 354 q^{24} + 792 q^{26} - 450 q^{29} - 72 q^{31} + 2102 q^{34} - 792 q^{36} + 2154 q^{39} + 1240 q^{41} + 282 q^{44}+ \cdots - 2638 q^{99}+O(q^{100}) \) 16 * q - 84 * q^4 - 316 * q^9 + 82 * q^11 - 322 * q^14 + 196 * q^16 - 354 * q^19 + 584 * q^21 - 354 * q^24 + 792 * q^26 - 450 * q^29 - 72 * q^31 + 2102 * q^34 - 792 * q^36 + 2154 * q^39 + 1240 * q^41 + 282 * q^44 + 276 * q^46 - 1762 * q^49 + 1914 * q^51 + 5898 * q^54 + 2062 * q^56 + 1352 * q^59 + 2894 * q^61 + 5766 * q^64 - 238 * q^66 + 2792 * q^71 + 1928 * q^74 + 4542 * q^76 - 1416 * q^79 + 8632 * q^81 + 7940 * q^84 + 7140 * q^86 - 2720 * q^89 + 5386 * q^91 + 6414 * q^94 + 9912 * q^96 - 2638 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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