LMFDB - Embedded newform 525.4.d.l.274.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,4,Mod(274,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.274"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,12,0,0,-36,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.4.d.l.274.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.82843i q^{2} +3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} -7.00000i q^{7} -5.14214i q^{8} -9.00000 q^{9} +48.5685 q^{11} -19.9706i q^{12} +43.6569i q^{13} -26.7990 q^{14} -72.9411 q^{16} -67.6569i q^{17} +34.4558i q^{18} +93.2548 q^{19} +21.0000 q^{21} -185.941i q^{22} +104.167i q^{23} +15.4264 q^{24} +167.137 q^{26} -27.0000i q^{27} +46.5980i q^{28} +58.7351 q^{29} -9.08831 q^{31} +238.113i q^{32} +145.706i q^{33} -259.019 q^{34} +59.9117 q^{36} -252.676i q^{37} -357.019i q^{38} -130.971 q^{39} +276.274 q^{41} -80.3970i q^{42} +92.6375i q^{43} -323.314 q^{44} +398.794 q^{46} -582.794i q^{47} -218.823i q^{48} -49.0000 q^{49} +202.971 q^{51} -290.617i q^{52} -623.019i q^{53} -103.368 q^{54} -35.9949 q^{56} +279.765i q^{57} -224.863i q^{58} +524.999 q^{59} -352.794 q^{61} +34.7939i q^{62} +63.0000i q^{63} +328.068 q^{64} +557.823 q^{66} -736.520i q^{67} +450.382i q^{68} -312.500 q^{69} -492.264 q^{71} +46.2792i q^{72} -1164.75i q^{73} -967.352 q^{74} -620.784 q^{76} -339.980i q^{77} +501.411i q^{78} +872.195 q^{79} +81.0000 q^{81} -1057.70i q^{82} +529.588i q^{83} -139.794 q^{84} +354.656 q^{86} +176.205i q^{87} -249.746i q^{88} +385.216 q^{89} +305.598 q^{91} -693.421i q^{92} -27.2649i q^{93} -2231.18 q^{94} -714.338 q^{96} -463.892i q^{97} +187.593i q^{98} -437.117 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 12 q^{6} - 36 q^{9} - 32 q^{11} - 28 q^{14} - 156 q^{16} + 192 q^{19} + 84 q^{21} - 108 q^{24} + 216 q^{26} - 376 q^{29} - 240 q^{31} - 312 q^{34} + 36 q^{36} - 456 q^{39} + 200 q^{41} - 1248 q^{44}+ \cdots + 288 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 12 * q^6 - 36 * q^9 - 32 * q^11 - 28 * q^14 - 156 * q^16 + 192 * q^19 + 84 * q^21 - 108 * q^24 + 216 * q^26 - 376 * q^29 - 240 * q^31 - 312 * q^34 + 36 * q^36 - 456 * q^39 + 200 * q^41 - 1248 * q^44 + 1120 * q^46 - 196 * q^49 + 744 * q^51 - 108 * q^54 + 252 * q^56 - 208 * q^59 - 936 * q^61 - 68 * q^64 + 1824 * q^66 - 96 * q^69 - 272 * q^71 - 2376 * q^74 - 1216 * q^76 + 864 * q^79 + 324 * q^81 - 84 * q^84 - 912 * q^86 + 2808 * q^89 + 1064 * q^91 - 3200 * q^94 - 2484 * q^96 + 288 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 3.82843i	− 1.35355i	−0.736188	−	0.676777i	$-0.763376\pi$
$2$	− 3.82843i	− 1.35355i	0.736188	−	0.676777i	$-0.236624\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−6.65685	−0.832107
$5$	0	0
$5$	0	0
$6$	11.4853	0.781474
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	− 5.14214i	− 0.227252i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	48.5685	1.33127	0.665635	−	0.746278i	$-0.268161\pi$
$11$	48.5685	1.33127	0.665635	+	0.746278i	$0.268161\pi$
$12$	− 19.9706i	− 0.480417i
$13$	43.6569i	0.931403i	0.884942	+	0.465701i	$0.154198\pi$
$13$	43.6569i	0.931403i	−0.884942	+	0.465701i	$0.845802\pi$
$14$	−26.7990	−0.511595
$15$	0	0
$16$	−72.9411	−1.13971
$17$	− 67.6569i	− 0.965247i	−0.875828	−	0.482623i	$-0.839684\pi$
$17$	− 67.6569i	− 0.965247i	0.875828	−	0.482623i	$-0.160316\pi$
$18$	34.4558i	0.451184i
$19$	93.2548	1.12601	0.563003	−	0.826455i	$-0.309646\pi$
$19$	93.2548	1.12601	0.563003	+	0.826455i	$0.309646\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	− 185.941i	− 1.80194i
$23$	104.167i	0.944357i	0.881503	+	0.472179i	$0.156532\pi$
$23$	104.167i	0.944357i	−0.881503	+	0.472179i	$0.843468\pi$
$24$	15.4264	0.131204
$25$	0	0
$26$	167.137	1.26070
$27$	− 27.0000i	− 0.192450i
$28$	46.5980i	0.314507i
$29$	58.7351	0.376098	0.188049	−	0.982160i	$-0.439784\pi$
$29$	58.7351	0.376098	0.188049	+	0.982160i	$0.439784\pi$
$30$	0	0
$31$	−9.08831	−0.0526551	−0.0263276	−	0.999653i	$-0.508381\pi$
$31$	−9.08831	−0.0526551	−0.0263276	+	0.999653i	$0.508381\pi$
$32$	238.113i	1.31540i
$33$	145.706i	0.768609i
$34$	−259.019	−1.30651
$35$	0	0
$36$	59.9117	0.277369
$37$	− 252.676i	− 1.12269i	−0.827580	−	0.561347i	$-0.810283\pi$
$37$	− 252.676i	− 1.12269i	0.827580	−	0.561347i	$-0.189717\pi$
$38$	− 357.019i	− 1.52411i
$39$	−130.971	−0.537745
$40$	0	0
$41$	276.274	1.05236	0.526180	−	0.850373i	$-0.323624\pi$
$41$	276.274	1.05236	0.526180	+	0.850373i	$0.323624\pi$
$42$	− 80.3970i	− 0.295370i
$43$	92.6375i	0.328537i	0.986416	+	0.164268i	$0.0525263\pi$
$43$	92.6375i	0.328537i	−0.986416	+	0.164268i	$0.947474\pi$
$44$	−323.314	−1.10776
$45$	0	0
$46$	398.794	1.27824
$47$	− 582.794i	− 1.80871i	−0.426784	−	0.904354i	$-0.640354\pi$
$47$	− 582.794i	− 1.80871i	0.426784	−	0.904354i	$-0.359646\pi$
$48$	− 218.823i	− 0.658009i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	202.971	0.557286
$52$	− 290.617i	− 0.775026i
$53$	− 623.019i	− 1.61468i	−0.590083	−	0.807342i	$-0.700905\pi$
$53$	− 623.019i	− 1.61468i	0.590083	−	0.807342i	$-0.299095\pi$
$54$	−103.368	−0.260491
$55$	0	0
$56$	−35.9949	−0.0858933
$57$	279.765i	0.650100i
$58$	− 224.863i	− 0.509068i
$59$	524.999	1.15846	0.579229	−	0.815165i	$-0.303354\pi$
$59$	524.999	1.15846	0.579229	+	0.815165i	$0.303354\pi$
$60$	0	0
$61$	−352.794	−0.740502	−0.370251	−	0.928932i	$-0.620728\pi$
$61$	−352.794	−0.740502	−0.370251	+	0.928932i	$0.620728\pi$
$62$	34.7939i	0.0712715i
$63$	63.0000i	0.125988i
$64$	328.068	0.640758
$65$	0	0
$66$	557.823	1.04035
$67$	− 736.520i	− 1.34299i	−0.741010	−	0.671494i	$-0.765653\pi$
$67$	− 736.520i	− 1.34299i	0.741010	−	0.671494i	$-0.234347\pi$
$68$	450.382i	0.803188i
$69$	−312.500	−0.545225
$70$	0	0
$71$	−492.264	−0.822831	−0.411415	−	0.911448i	$-0.634965\pi$
$71$	−492.264	−0.822831	−0.411415	+	0.911448i	$0.634965\pi$
$72$	46.2792i	0.0757508i
$73$	− 1164.75i	− 1.86745i	−0.357987	−	0.933727i	$-0.616537\pi$
$73$	− 1164.75i	− 1.86745i	0.357987	−	0.933727i	$-0.383463\pi$
$74$	−967.352	−1.51963
$75$	0	0
$76$	−620.784	−0.936958
$77$	− 339.980i	− 0.503173i
$78$	501.411i	0.727867i
$79$	872.195	1.24215	0.621074	−	0.783752i	$-0.286697\pi$
$79$	872.195	1.24215	0.621074	+	0.783752i	$0.286697\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 1057.70i	− 1.42443i
$83$	529.588i	0.700359i	0.936683	+	0.350180i	$0.113879\pi$
$83$	529.588i	0.700359i	−0.936683	+	0.350180i	$0.886121\pi$
$84$	−139.794	−0.181581
$85$	0	0
$86$	354.656	0.444692
$87$	176.205i	0.217140i
$88$	− 249.746i	− 0.302534i
$89$	385.216	0.458796	0.229398	−	0.973333i	$-0.426324\pi$
$89$	385.216	0.458796	0.229398	+	0.973333i	$0.426324\pi$
$90$	0	0
$91$	305.598	0.352037
$92$	− 693.421i	− 0.785806i
$93$	− 27.2649i	− 0.0304005i
$94$	−2231.18	−2.44818
$95$	0	0
$96$	−714.338	−0.759446
$97$	− 463.892i	− 0.485579i	−0.970079	−	0.242789i	$-0.921938\pi$
$97$	− 463.892i	− 0.485579i	0.970079	−	0.242789i	$-0.0780624\pi$
$98$	187.593i	0.193365i
$99$	−437.117	−0.443757
$100$	0	0
$101$	−432.725	−0.426314	−0.213157	−	0.977018i	$-0.568375\pi$
$101$	−432.725	−0.426314	−0.213157	+	0.977018i	$0.568375\pi$
$102$	− 777.058i	− 0.754316i
$103$	− 512.626i	− 0.490393i	−0.969473	−	0.245197i	$-0.921147\pi$
$103$	− 512.626i	− 0.490393i	0.969473	−	0.245197i	$-0.0788525\pi$
$104$	224.489	0.211663
$105$	0	0
$106$	−2385.18	−2.18556
$107$	1963.09i	1.77363i	0.462122	+	0.886817i	$0.347088\pi$
$107$	1963.09i	1.77363i	−0.462122	+	0.886817i	$0.652912\pi$
$108$	179.735i	0.160139i
$109$	−545.176	−0.479068	−0.239534	−	0.970888i	$-0.576995\pi$
$109$	−545.176	−0.479068	−0.239534	+	0.970888i	$0.576995\pi$
$110$	0	0
$111$	758.029	0.648188
$112$	510.588i	0.430768i
$113$	231.823i	0.192992i	0.995333	+	0.0964961i	$0.0307635\pi$
$113$	231.823i	0.192992i	−0.995333	+	0.0964961i	$0.969236\pi$
$114$	1071.06	0.879945
$115$	0	0
$116$	−390.991	−0.312953
$117$	− 392.912i	− 0.310468i
$118$	− 2009.92i	− 1.56804i
$119$	−473.598	−0.364829
$120$	0	0
$121$	1027.90	0.772279
$122$	1350.65i	1.00231i
$123$	828.823i	0.607581i
$124$	60.4996	0.0438147
$125$	0	0
$126$	241.191	0.170532
$127$	2372.90i	1.65796i	0.559280	+	0.828979i	$0.311078\pi$
$127$	2372.90i	1.65796i	−0.559280	+	0.828979i	$0.688922\pi$
$128$	648.917i	0.448099i
$129$	−277.913	−0.189681
$130$	0	0
$131$	1200.04	0.800364	0.400182	−	0.916436i	$-0.368947\pi$
$131$	1200.04	0.800364	0.400182	+	0.916436i	$0.368947\pi$
$132$	− 969.941i	− 0.639565i
$133$	− 652.784i	− 0.425591i
$134$	−2819.71	−1.81781
$135$	0	0
$136$	−347.901	−0.219355
$137$	2781.25i	1.73444i	0.497924	+	0.867221i	$0.334096\pi$
$137$	2781.25i	1.73444i	−0.497924	+	0.867221i	$0.665904\pi$
$138$	1196.38i	0.737991i
$139$	1245.60	0.760078	0.380039	−	0.924971i	$-0.375911\pi$
$139$	1245.60	0.760078	0.380039	+	0.924971i	$0.375911\pi$
$140$	0	0
$141$	1748.38	1.04426
$142$	1884.60i	1.11375i
$143$	2120.35i	1.23995i
$144$	656.470	0.379902
$145$	0	0
$146$	−4459.17	−2.52770
$147$	− 147.000i	− 0.0824786i
$148$	1682.03i	0.934202i
$149$	−19.4046	−0.0106690	−0.00533452	−	0.999986i	$-0.501698\pi$
$149$	−19.4046	−0.0106690	−0.00533452	+	0.999986i	$0.501698\pi$
$150$	0	0
$151$	−2349.80	−1.26638	−0.633192	−	0.773995i	$-0.718256\pi$
$151$	−2349.80	−1.26638	−0.633192	+	0.773995i	$0.718256\pi$
$152$	− 479.529i	− 0.255888i
$153$	608.912i	0.321749i
$154$	−1301.59	−0.681071
$155$	0	0
$156$	871.852	0.447462
$157$	− 3898.46i	− 1.98172i	−0.134880	−	0.990862i	$-0.543065\pi$
$157$	− 3898.46i	− 1.98172i	0.134880	−	0.990862i	$-0.456935\pi$
$158$	− 3339.14i	− 1.68131i
$159$	1869.06	0.932239
$160$	0	0
$161$	729.166	0.356934
$162$	− 310.103i	− 0.150395i
$163$	− 1527.54i	− 0.734024i	−0.930216	−	0.367012i	$-0.880381\pi$
$163$	− 1527.54i	− 0.734024i	0.930216	−	0.367012i	$-0.119619\pi$
$164$	−1839.12	−0.875676
$165$	0	0
$166$	2027.49	0.947974
$167$	− 998.518i	− 0.462681i	−0.972873	−	0.231340i	$-0.925689\pi$
$167$	− 998.518i	− 0.462681i	0.972873	−	0.231340i	$-0.0743111\pi$
$168$	− 107.985i	− 0.0495905i
$169$	291.079	0.132489
$170$	0	0
$171$	−839.294	−0.375336
$172$	− 616.674i	− 0.273378i
$173$	− 685.253i	− 0.301149i	−0.988599	−	0.150575i	$-0.951888\pi$
$173$	− 685.253i	− 0.301149i	0.988599	−	0.150575i	$-0.0481124\pi$
$174$	674.589	0.293911
$175$	0	0
$176$	−3542.64	−1.51725
$177$	1575.00i	0.668836i
$178$	− 1474.77i	− 0.621005i
$179$	−1025.58	−0.428245	−0.214122	−	0.976807i	$-0.568689\pi$
$179$	−1025.58	−0.428245	−0.214122	+	0.976807i	$0.568689\pi$
$180$	0	0
$181$	2899.40	1.19067	0.595333	−	0.803479i	$-0.297020\pi$
$181$	2899.40	1.19067	0.595333	+	0.803479i	$0.297020\pi$
$182$	− 1169.96i	− 0.476501i
$183$	− 1058.38i	− 0.427529i
$184$	535.638	0.214608
$185$	0	0
$186$	−104.382	−0.0411486
$187$	− 3285.99i	− 1.28500i
$188$	3879.57i	1.50504i
$189$	−189.000	−0.0727393
$190$	0	0
$191$	1074.18	0.406939	0.203469	−	0.979081i	$-0.434778\pi$
$191$	1074.18	0.406939	0.203469	+	0.979081i	$0.434778\pi$
$192$	984.204i	0.369942i
$193$	− 898.999i	− 0.335292i	−0.985847	−	0.167646i	$-0.946383\pi$
$193$	− 898.999i	− 0.335292i	0.985847	−	0.167646i	$-0.0536166\pi$
$194$	−1775.98	−0.657257
$195$	0	0
$196$	326.186	0.118872
$197$	3063.63i	1.10799i	0.832519	+	0.553996i	$0.186898\pi$
$197$	3063.63i	1.10799i	−0.832519	+	0.553996i	$0.813102\pi$
$198$	1673.47i	0.600648i
$199$	949.522	0.338240	0.169120	−	0.985595i	$-0.445907\pi$
$199$	949.522	0.338240	0.169120	+	0.985595i	$0.445907\pi$
$200$	0	0
$201$	2209.56	0.775375
$202$	1656.66i	0.577039i
$203$	− 411.145i	− 0.142151i
$204$	−1351.15	−0.463721
$205$	0	0
$206$	−1962.55	−0.663774
$207$	− 937.499i	− 0.314786i
$208$	− 3184.38i	− 1.06152i
$209$	4529.25	1.49902
$210$	0	0
$211$	2306.64	0.752587	0.376294	−	0.926500i	$-0.377198\pi$
$211$	2306.64	0.752587	0.376294	+	0.926500i	$0.377198\pi$
$212$	4147.35i	1.34359i
$213$	− 1476.79i	− 0.475062i
$214$	7515.53	2.40071
$215$	0	0
$216$	−138.838	−0.0437348
$217$	63.6182i	0.0199018i
$218$	2087.17i	0.648444i
$219$	3494.26	1.07817
$220$	0	0
$221$	2953.69	0.899033
$222$	− 2902.06i	− 0.877357i
$223$	3227.61i	0.969222i	0.874730	+	0.484611i	$0.161039\pi$
$223$	3227.61i	0.969222i	−0.874730	+	0.484611i	$0.838961\pi$
$224$	1666.79	0.497174
$225$	0	0
$226$	887.519	0.261225
$227$	− 637.820i	− 0.186492i	−0.995643	−	0.0932458i	$-0.970276\pi$
$227$	− 637.820i	− 0.186492i	0.995643	−	0.0932458i	$-0.0297242\pi$
$228$	− 1862.35i	− 0.540953i
$229$	−544.774	−0.157204	−0.0786019	−	0.996906i	$-0.525046\pi$
$229$	−544.774	−0.157204	−0.0786019	+	0.996906i	$0.525046\pi$
$230$	0	0
$231$	1019.94	0.290507
$232$	− 302.024i	− 0.0854691i
$233$	5748.54i	1.61631i	0.588972	+	0.808154i	$0.299533\pi$
$233$	5748.54i	1.61631i	−0.588972	+	0.808154i	$0.700467\pi$
$234$	−1504.23	−0.420234
$235$	0	0
$236$	−3494.84	−0.963961
$237$	2616.59i	0.717154i
$238$	1813.14i	0.493816i
$239$	2678.10	0.724820	0.362410	−	0.932019i	$-0.381954\pi$
$239$	2678.10	0.724820	0.362410	+	0.932019i	$0.381954\pi$
$240$	0	0
$241$	−2202.16	−0.588604	−0.294302	−	0.955713i	$-0.595087\pi$
$241$	−2202.16	−0.588604	−0.294302	+	0.955713i	$0.595087\pi$
$242$	− 3935.25i	− 1.04532i
$243$	243.000i	0.0641500i
$244$	2348.50	0.616177
$245$	0	0
$246$	3173.09	0.822393
$247$	4071.21i	1.04877i
$248$	46.7333i	0.0119660i
$249$	−1588.76	−0.404353
$250$	0	0
$251$	−5716.90	−1.43764	−0.718820	−	0.695196i	$-0.755317\pi$
$251$	−5716.90	−1.43764	−0.718820	+	0.695196i	$0.755317\pi$
$252$	− 419.382i	− 0.104836i
$253$	5059.22i	1.25719i
$254$	9084.47	2.24413
$255$	0	0
$256$	5108.88	1.24728
$257$	4724.29i	1.14666i	0.819323	+	0.573332i	$0.194350\pi$
$257$	4724.29i	1.14666i	−0.819323	+	0.573332i	$0.805650\pi$
$258$	1063.97i	0.256743i
$259$	−1768.73	−0.424339
$260$	0	0
$261$	−528.616	−0.125366
$262$	− 4594.25i	− 1.08334i
$263$	− 5975.36i	− 1.40097i	−0.713665	−	0.700487i	$-0.752966\pi$
$263$	− 5975.36i	− 1.40097i	0.713665	−	0.700487i	$-0.247034\pi$
$264$	749.238	0.174668
$265$	0	0
$266$	−2499.14	−0.576059
$267$	1155.65i	0.264886i
$268$	4902.90i	1.11751i
$269$	−4486.11	−1.01681	−0.508407	−	0.861117i	$-0.669766\pi$
$269$	−4486.11	−1.01681	−0.508407	+	0.861117i	$0.669766\pi$
$270$	0	0
$271$	3827.68	0.857989	0.428994	−	0.903307i	$-0.358868\pi$
$271$	3827.68	0.857989	0.428994	+	0.903307i	$0.358868\pi$
$272$	4934.97i	1.10010i
$273$	916.794i	0.203249i
$274$	10647.8	2.34766
$275$	0	0
$276$	2080.26	0.453685
$277$	− 3420.54i	− 0.741950i	−0.928643	−	0.370975i	$-0.879024\pi$
$277$	− 3420.54i	− 0.741950i	0.928643	−	0.370975i	$-0.120976\pi$
$278$	− 4768.71i	− 1.02881i
$279$	81.7948	0.0175517
$280$	0	0
$281$	5235.92	1.11156	0.555781	−	0.831329i	$-0.312419\pi$
$281$	5235.92	1.11156	0.555781	+	0.831329i	$0.312419\pi$
$282$	− 6693.55i	− 1.41346i
$283$	6985.88i	1.46738i	0.679486	+	0.733688i	$0.262203\pi$
$283$	6985.88i	1.46738i	−0.679486	+	0.733688i	$0.737797\pi$
$284$	3276.93	0.684683
$285$	0	0
$286$	8117.60	1.67834
$287$	− 1933.92i	− 0.397755i
$288$	− 2143.01i	− 0.438466i
$289$	335.550	0.0682984
$290$	0	0
$291$	1391.68	0.280349
$292$	7753.59i	1.55392i
$293$	− 7399.70i	− 1.47541i	−0.675123	−	0.737705i	$-0.735910\pi$
$293$	− 7399.70i	− 1.47541i	0.675123	−	0.737705i	$-0.264090\pi$
$294$	−562.779	−0.111639
$295$	0	0
$296$	−1299.30	−0.255135
$297$	− 1311.35i	− 0.256203i
$298$	74.2892i	0.0144411i
$299$	−4547.58	−0.879577
$300$	0	0
$301$	648.463	0.124175
$302$	8996.04i	1.71412i
$303$	− 1298.17i	− 0.246133i
$304$	−6802.11	−1.28332
$305$	0	0
$306$	2331.17	0.435504
$307$	2668.64i	0.496116i	0.968745	+	0.248058i	$0.0797923\pi$
$307$	2668.64i	0.496116i	−0.968745	+	0.248058i	$0.920208\pi$
$308$	2263.20i	0.418693i
$309$	1537.88	0.283129
$310$	0	0
$311$	6189.25	1.12849	0.564244	−	0.825608i	$-0.309168\pi$
$311$	6189.25	1.12849	0.564244	+	0.825608i	$0.309168\pi$
$312$	673.468i	0.122204i
$313$	2921.59i	0.527598i	0.964578	+	0.263799i	$0.0849755\pi$
$313$	2921.59i	0.527598i	−0.964578	+	0.263799i	$0.915024\pi$
$314$	−14925.0	−2.68237
$315$	0	0
$316$	−5806.08	−1.03360
$317$	9825.56i	1.74088i	0.492276	+	0.870439i	$0.336165\pi$
$317$	9825.56i	1.74088i	−0.492276	+	0.870439i	$0.663835\pi$
$318$	− 7155.55i	− 1.26183i
$319$	2852.68	0.500687
$320$	0	0
$321$	−5889.26	−1.02401
$322$	− 2791.56i	− 0.483129i
$323$	− 6309.33i	− 1.08687i
$324$	−539.205	−0.0924563
$325$	0	0
$326$	−5848.07	−0.993541
$327$	− 1635.53i	− 0.276590i
$328$	− 1420.64i	− 0.239151i
$329$	−4079.56	−0.683627
$330$	0	0
$331$	−9258.17	−1.53739	−0.768693	−	0.639618i	$-0.779093\pi$
$331$	−9258.17	−1.53739	−0.768693	+	0.639618i	$0.779093\pi$
$332$	− 3525.39i	− 0.582774i
$333$	2274.09i	0.374232i
$334$	−3822.75	−0.626263
$335$	0	0
$336$	−1531.76	−0.248704
$337$	− 3693.98i	− 0.597103i	−0.954394	−	0.298552i	$-0.903497\pi$
$337$	− 3693.98i	− 0.597103i	0.954394	−	0.298552i	$-0.0965035\pi$
$338$	− 1114.38i	− 0.179331i
$339$	−695.470	−0.111424
$340$	0	0
$341$	−441.406	−0.0700982
$342$	3213.17i	0.508037i
$343$	343.000i	0.0539949i
$344$	476.355	0.0746608
$345$	0	0
$346$	−2623.44	−0.407622
$347$	3832.83i	0.592960i	0.955039	+	0.296480i	$0.0958128\pi$
$347$	3832.83i	0.592960i	−0.955039	+	0.296480i	$0.904187\pi$
$348$	− 1172.97i	− 0.180684i
$349$	−8325.22	−1.27690	−0.638451	−	0.769662i	$-0.720425\pi$
$349$	−8325.22	−1.27690	−0.638451	+	0.769662i	$0.720425\pi$
$350$	0	0
$351$	1178.74	0.179248
$352$	11564.8i	1.75115i
$353$	8991.52i	1.35572i	0.735189	+	0.677862i	$0.237093\pi$
$353$	8991.52i	1.35572i	−0.735189	+	0.677862i	$0.762907\pi$
$354$	6029.76	0.905306
$355$	0	0
$356$	−2564.33	−0.381767
$357$	− 1420.79i	− 0.210634i
$358$	3926.38i	0.579652i
$359$	12893.8	1.89557	0.947783	−	0.318917i	$-0.103319\pi$
$359$	12893.8	1.89557	0.947783	+	0.318917i	$0.103319\pi$
$360$	0	0
$361$	1837.46	0.267891
$362$	− 11100.1i	− 1.61163i
$363$	3083.71i	0.445875i
$364$	−2034.32	−0.292932
$365$	0	0
$366$	−4051.94	−0.578684
$367$	− 7480.17i	− 1.06393i	−0.846767	−	0.531964i	$-0.821454\pi$
$367$	− 7480.17i	− 1.06393i	0.846767	−	0.531964i	$-0.178546\pi$
$368$	− 7598.02i	− 1.07629i
$369$	−2486.47	−0.350787
$370$	0	0
$371$	−4361.14	−0.610293
$372$	181.499i	0.0252964i
$373$	3523.32i	0.489090i	0.969638	+	0.244545i	$0.0786386\pi$
$373$	3523.32i	0.489090i	−0.969638	+	0.244545i	$0.921361\pi$
$374$	−12580.2	−1.73932
$375$	0	0
$376$	−2996.81	−0.411033
$377$	2564.19i	0.350298i
$378$	723.573i	0.0984565i
$379$	−13515.4	−1.83177	−0.915886	−	0.401438i	$-0.868510\pi$
$379$	−13515.4	−1.83177	−0.915886	+	0.401438i	$0.868510\pi$
$380$	0	0
$381$	−7118.69	−0.957222
$382$	− 4112.44i	− 0.550813i
$383$	657.182i	0.0876774i	0.999039	+	0.0438387i	$0.0139588\pi$
$383$	657.182i	0.0876774i	−0.999039	+	0.0438387i	$0.986041\pi$
$384$	−1946.75	−0.258710
$385$	0	0
$386$	−3441.75	−0.453836
$387$	− 833.738i	− 0.109512i
$388$	3088.06i	0.404053i
$389$	9741.87	1.26975	0.634875	−	0.772615i	$-0.281052\pi$
$389$	9741.87	1.26975	0.634875	+	0.772615i	$0.281052\pi$
$390$	0	0
$391$	7047.58	0.911538
$392$	251.965i	0.0324646i
$393$	3600.11i	0.462091i
$394$	11728.9	1.49973
$395$	0	0
$396$	2909.82	0.369253
$397$	4407.42i	0.557184i	0.960410	+	0.278592i	$0.0898678\pi$
$397$	4407.42i	0.557184i	−0.960410	+	0.278592i	$0.910132\pi$
$398$	− 3635.18i	− 0.457827i
$399$	1958.35	0.245715
$400$	0	0
$401$	−11569.5	−1.44078	−0.720391	−	0.693568i	$-0.756037\pi$
$401$	−11569.5	−1.44078	−0.720391	+	0.693568i	$0.756037\pi$
$402$	− 8459.14i	− 1.04951i
$403$	− 396.767i	− 0.0490431i
$404$	2880.59	0.354739
$405$	0	0
$406$	−1574.04	−0.192410
$407$	− 12272.1i	− 1.49461i
$408$	− 1043.70i	− 0.126645i
$409$	3083.03	0.372729	0.186364	−	0.982481i	$-0.440330\pi$
$409$	3083.03	0.372729	0.186364	+	0.982481i	$0.440330\pi$
$410$	0	0
$411$	−8343.76	−1.00138
$412$	3412.47i	0.408060i
$413$	− 3674.99i	− 0.437856i
$414$	−3589.15	−0.426079
$415$	0	0
$416$	−10395.3	−1.22517
$417$	3736.81i	0.438831i
$418$	− 17339.9i	− 2.02900i
$419$	5415.21	0.631385	0.315692	−	0.948862i	$-0.397763\pi$
$419$	5415.21	0.631385	0.315692	+	0.948862i	$0.397763\pi$
$420$	0	0
$421$	4188.34	0.484863	0.242432	−	0.970168i	$-0.422055\pi$
$421$	4188.34	0.484863	0.242432	+	0.970168i	$0.422055\pi$
$422$	− 8830.82i	− 1.01867i
$423$	5245.15i	0.602902i
$424$	−3203.65	−0.366941
$425$	0	0
$426$	−5653.79	−0.643021
$427$	2469.56i	0.279884i
$428$	− 13068.0i	− 1.47585i
$429$	−6361.05	−0.715884
$430$	0	0
$431$	−9108.41	−1.01795	−0.508975	−	0.860781i	$-0.669976\pi$
$431$	−9108.41	−1.01795	−0.508975	+	0.860781i	$0.669976\pi$
$432$	1969.41i	0.219336i
$433$	16847.0i	1.86979i	0.354930	+	0.934893i	$0.384505\pi$
$433$	16847.0i	1.86979i	−0.354930	+	0.934893i	$0.615495\pi$
$434$	243.558	0.0269381
$435$	0	0
$436$	3629.16	0.398635
$437$	9714.03i	1.06335i
$438$	− 13377.5i	− 1.45937i
$439$	−8434.14	−0.916946	−0.458473	−	0.888708i	$-0.651603\pi$
$439$	−8434.14	−0.916946	−0.458473	+	0.888708i	$0.651603\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	− 11308.0i	− 1.21689i
$443$	4298.49i	0.461010i	0.973071	+	0.230505i	$0.0740378\pi$
$443$	4298.49i	0.461010i	−0.973071	+	0.230505i	$0.925962\pi$
$444$	−5046.09	−0.539362
$445$	0	0
$446$	12356.7	1.31189
$447$	− 58.2139i	− 0.00615978i
$448$	− 2296.48i	− 0.242184i
$449$	−10545.0	−1.10835	−0.554173	−	0.832402i	$-0.686965\pi$
$449$	−10545.0	−1.10835	−0.554173	+	0.832402i	$0.686965\pi$
$450$	0	0
$451$	13418.2	1.40098
$452$	− 1543.21i	− 0.160590i
$453$	− 7049.40i	− 0.731147i
$454$	−2441.85	−0.252426
$455$	0	0
$456$	1438.59	0.147737
$457$	11952.4i	1.22344i	0.791075	+	0.611719i	$0.209522\pi$
$457$	11952.4i	1.22344i	−0.791075	+	0.611719i	$0.790478\pi$
$458$	2085.63i	0.212784i
$459$	−1826.74	−0.185762
$460$	0	0
$461$	−17200.9	−1.73780	−0.868900	−	0.494988i	$-0.835173\pi$
$461$	−17200.9	−1.73780	−0.868900	+	0.494988i	$0.835173\pi$
$462$	− 3904.76i	− 0.393217i
$463$	10368.7i	1.04076i	0.853934	+	0.520381i	$0.174210\pi$
$463$	10368.7i	1.04076i	−0.853934	+	0.520381i	$0.825790\pi$
$464$	−4284.20	−0.428640
$465$	0	0
$466$	22007.9	2.18776
$467$	− 16879.5i	− 1.67257i	−0.548296	−	0.836284i	$-0.684723\pi$
$467$	− 16879.5i	− 1.67257i	0.548296	−	0.836284i	$-0.315277\pi$
$468$	2615.56i	0.258342i
$469$	−5155.64	−0.507602
$470$	0	0
$471$	11695.4	1.14415
$472$	− 2699.62i	− 0.263263i
$473$	4499.27i	0.437371i
$474$	10017.4	0.970706
$475$	0	0
$476$	3152.67	0.303577
$477$	5607.17i	0.538228i
$478$	− 10252.9i	− 0.981082i
$479$	−7329.12	−0.699115	−0.349558	−	0.936915i	$-0.613668\pi$
$479$	−7329.12	−0.699115	−0.349558	+	0.936915i	$0.613668\pi$
$480$	0	0
$481$	11031.0	1.04568
$482$	8430.80i	0.796706i
$483$	2187.50i	0.206076i
$484$	−6842.60	−0.642619
$485$	0	0
$486$	930.308	0.0868305
$487$	− 17209.5i	− 1.60131i	−0.599125	−	0.800655i	$-0.704485\pi$
$487$	− 17209.5i	− 1.60131i	0.599125	−	0.800655i	$-0.295515\pi$
$488$	1814.11i	0.168281i
$489$	4582.61	0.423789
$490$	0	0
$491$	11392.5	1.04712	0.523560	−	0.851989i	$-0.324604\pi$
$491$	11392.5	1.04712	0.523560	+	0.851989i	$0.324604\pi$
$492$	− 5517.35i	− 0.505572i
$493$	− 3973.83i	− 0.363027i
$494$	15586.3	1.41956
$495$	0	0
$496$	662.912	0.0600113
$497$	3445.85i	0.311001i
$498$	6082.47i	0.547313i
$499$	−19079.4	−1.71164	−0.855822	−	0.517271i	$-0.826948\pi$
$499$	−19079.4	−1.71164	−0.855822	+	0.517271i	$0.826948\pi$
$500$	0	0
$501$	2995.55	0.267129
$502$	21886.7i	1.94592i
$503$	13499.4i	1.19663i	0.801259	+	0.598317i	$0.204164\pi$
$503$	13499.4i	1.19663i	−0.801259	+	0.598317i	$0.795836\pi$
$504$	323.955	0.0286311
$505$	0	0
$506$	19368.8	1.70168
$507$	873.237i	0.0764928i
$508$	− 15796.0i	− 1.37960i
$509$	4328.85	0.376960	0.188480	−	0.982077i	$-0.439644\pi$
$509$	4328.85	0.376960	0.188480	+	0.982077i	$0.439644\pi$
$510$	0	0
$511$	−8153.27	−0.705831
$512$	− 14367.6i	− 1.24017i
$513$	− 2517.88i	− 0.216700i
$514$	18086.6	1.55207
$515$	0	0
$516$	1850.02	0.157835
$517$	− 28305.5i	− 2.40788i
$518$	6771.47i	0.574365i
$519$	2055.76	0.173869
$520$	0	0
$521$	19395.7	1.63098	0.815490	−	0.578771i	$-0.196468\pi$
$521$	19395.7	1.63098	0.815490	+	0.578771i	$0.196468\pi$
$522$	2023.77i	0.169689i
$523$	− 20413.8i	− 1.70675i	−0.521294	−	0.853377i	$-0.674550\pi$
$523$	− 20413.8i	− 1.70675i	0.521294	−	0.853377i	$-0.325450\pi$
$524$	−7988.47	−0.665989
$525$	0	0
$526$	−22876.2	−1.89629
$527$	614.887i	0.0508252i
$528$	− 10627.9i	− 0.875987i
$529$	1316.34	0.108189
$530$	0	0
$531$	−4724.99	−0.386153
$532$	4345.49i	0.354137i
$533$	12061.3i	0.980171i
$534$	4424.32	0.358537
$535$	0	0
$536$	−3787.28	−0.305197
$537$	− 3076.75i	− 0.247247i
$538$	17174.8i	1.37631i
$539$	−2379.86	−0.190181
$540$	0	0
$541$	−4919.18	−0.390928	−0.195464	−	0.980711i	$-0.562621\pi$
$541$	−4919.18	−0.390928	−0.195464	+	0.980711i	$0.562621\pi$
$542$	− 14654.0i	− 1.16133i
$543$	8698.19i	0.687431i
$544$	16110.0	1.26969
$545$	0	0
$546$	3509.88	0.275108
$547$	15334.2i	1.19862i	0.800518	+	0.599308i	$0.204558\pi$
$547$	15334.2i	1.19862i	−0.800518	+	0.599308i	$0.795442\pi$
$548$	− 18514.4i	− 1.44324i
$549$	3175.15	0.246834
$550$	0	0
$551$	5477.33	0.423488
$552$	1606.92i	0.123904i
$553$	− 6105.37i	− 0.469487i
$554$	−13095.3	−1.00427
$555$	0	0
$556$	−8291.81	−0.632466
$557$	− 8613.78i	− 0.655256i	−0.944807	−	0.327628i	$-0.893751\pi$
$557$	− 8613.78i	− 0.655256i	0.944807	−	0.327628i	$-0.106249\pi$
$558$	− 313.145i	− 0.0237572i
$559$	−4044.26	−0.306000
$560$	0	0
$561$	9857.98	0.741897
$562$	− 20045.3i	− 1.50456i
$563$	2320.81i	0.173731i	0.996220	+	0.0868654i	$0.0276850\pi$
$563$	2320.81i	0.173731i	−0.996220	+	0.0868654i	$0.972315\pi$
$564$	−11638.7	−0.868934
$565$	0	0
$566$	26744.9	1.98617
$567$	− 567.000i	− 0.0419961i
$568$	2531.29i	0.186990i
$569$	1736.04	0.127906	0.0639529	−	0.997953i	$-0.479629\pi$
$569$	1736.04	0.127906	0.0639529	+	0.997953i	$0.479629\pi$
$570$	0	0
$571$	23897.8	1.75148	0.875738	−	0.482786i	$-0.160375\pi$
$571$	23897.8	1.75148	0.875738	+	0.482786i	$0.160375\pi$
$572$	− 14114.9i	− 1.03177i
$573$	3222.55i	0.234946i
$574$	−7403.87	−0.538382
$575$	0	0
$576$	−2952.61	−0.213586
$577$	8029.26i	0.579311i	0.957131	+	0.289655i	$0.0935407\pi$
$577$	8029.26i	0.579311i	−0.957131	+	0.289655i	$0.906459\pi$
$578$	− 1284.63i	− 0.0924455i
$579$	2697.00	0.193581
$580$	0	0
$581$	3707.12	0.264711
$582$	− 5327.93i	− 0.379467i
$583$	− 30259.1i	− 2.14958i
$584$	−5989.32	−0.424383
$585$	0	0
$586$	−28329.2	−1.99705
$587$	− 8015.14i	− 0.563578i	−0.959476	−	0.281789i	$-0.909072\pi$
$587$	− 8015.14i	− 0.563578i	0.959476	−	0.281789i	$-0.0909278\pi$
$588$	978.558i	0.0686310i
$589$	−847.529	−0.0592900
$590$	0	0
$591$	−9190.88	−0.639700
$592$	18430.5i	1.27954i
$593$	− 12820.3i	− 0.887801i	−0.896076	−	0.443901i	$-0.853594\pi$
$593$	− 12820.3i	− 0.887801i	0.896076	−	0.443901i	$-0.146406\pi$
$594$	−5020.41	−0.346784
$595$	0	0
$596$	129.174	0.00887779
$597$	2848.57i	0.195283i
$598$	17410.1i	1.19055i
$599$	15330.5	1.04572	0.522860	−	0.852418i	$-0.324865\pi$
$599$	15330.5	1.04572	0.522860	+	0.852418i	$0.324865\pi$
$600$	0	0
$601$	107.658	0.00730694	0.00365347	−	0.999993i	$-0.498837\pi$
$601$	107.658	0.00730694	0.00365347	+	0.999993i	$0.498837\pi$
$602$	− 2482.59i	− 0.168078i
$603$	6628.68i	0.447663i
$604$	15642.3	1.05377
$605$	0	0
$606$	−4969.97	−0.333154
$607$	− 8213.06i	− 0.549189i	−0.961560	−	0.274595i	$-0.911456\pi$
$607$	− 8213.06i	− 0.549189i	0.961560	−	0.274595i	$-0.0885436\pi$
$608$	22205.2i	1.48115i
$609$	1233.44	0.0820712
$610$	0	0
$611$	25443.0	1.68463
$612$	− 4053.44i	− 0.267729i
$613$	− 1242.60i	− 0.0818732i	−0.999162	−	0.0409366i	$-0.986966\pi$
$613$	− 1242.60i	− 0.0818732i	0.999162	−	0.0409366i	$-0.0130342\pi$
$614$	10216.7	0.671519
$615$	0	0
$616$	−1748.22	−0.114347
$617$	− 13170.6i	− 0.859367i	−0.902980	−	0.429683i	$-0.858625\pi$
$617$	− 13170.6i	− 0.859367i	0.902980	−	0.429683i	$-0.141375\pi$
$618$	− 5887.65i	− 0.383230i
$619$	−19774.1	−1.28399	−0.641993	−	0.766711i	$-0.721892\pi$
$619$	−19774.1	−1.28399	−0.641993	+	0.766711i	$0.721892\pi$
$620$	0	0
$621$	2812.50	0.181742
$622$	− 23695.1i	− 1.52747i
$623$	− 2696.51i	− 0.173409i
$624$	9553.14	0.612871
$625$	0	0
$626$	11185.1	0.714132
$627$	13587.8i	0.865459i
$628$	25951.5i	1.64901i
$629$	−17095.3	−1.08368
$630$	0	0
$631$	−14308.8	−0.902735	−0.451367	−	0.892338i	$-0.649064\pi$
$631$	−14308.8	−0.902735	−0.451367	+	0.892338i	$0.649064\pi$
$632$	− 4484.95i	− 0.282281i
$633$	6919.93i	0.434507i
$634$	37616.4	2.35637
$635$	0	0
$636$	−12442.0	−0.775722
$637$	− 2139.19i	− 0.133058i
$638$	− 10921.3i	− 0.677707i
$639$	4430.38	0.274277
$640$	0	0
$641$	11537.5	0.710925	0.355463	−	0.934691i	$-0.384323\pi$
$641$	11537.5	0.710925	0.355463	+	0.934691i	$0.384323\pi$
$642$	22546.6i	1.38605i
$643$	− 19603.0i	− 1.20228i	−0.799144	−	0.601139i	$-0.794714\pi$
$643$	− 19603.0i	− 1.20228i	0.799144	−	0.601139i	$-0.205286\pi$
$644$	−4853.95	−0.297007
$645$	0	0
$646$	−24154.8	−1.47114
$647$	21650.0i	1.31553i	0.753223	+	0.657765i	$0.228498\pi$
$647$	21650.0i	1.31553i	−0.753223	+	0.657765i	$0.771502\pi$
$648$	− 416.513i	− 0.0252503i
$649$	25498.4	1.54222
$650$	0	0
$651$	−190.855	−0.0114903
$652$	10168.6i	0.610787i
$653$	2927.33i	0.175429i	0.996146	+	0.0877145i	$0.0279563\pi$
$653$	2927.33i	0.175429i	−0.996146	+	0.0877145i	$0.972044\pi$
$654$	−6261.50	−0.374379
$655$	0	0
$656$	−20151.7	−1.19938
$657$	10482.8i	0.622484i
$658$	15618.3i	0.925326i
$659$	4778.76	0.282480	0.141240	−	0.989975i	$-0.454891\pi$
$659$	4778.76	0.282480	0.141240	+	0.989975i	$0.454891\pi$
$660$	0	0
$661$	−31510.3	−1.85417	−0.927086	−	0.374849i	$-0.877695\pi$
$661$	−31510.3	−1.85417	−0.927086	+	0.374849i	$0.877695\pi$
$662$	35444.2i	2.08093i
$663$	8861.06i	0.519057i
$664$	2723.21	0.159158
$665$	0	0
$666$	8706.17	0.506542
$667$	6118.23i	0.355170i
$668$	6646.99i	0.385000i
$669$	−9682.82	−0.559581
$670$	0	0
$671$	−17134.7	−0.985808
$672$	5000.37i	0.287044i
$673$	8992.15i	0.515040i	0.966273	+	0.257520i	$0.0829054\pi$
$673$	8992.15i	0.515040i	−0.966273	+	0.257520i	$0.917095\pi$
$674$	−14142.1	−0.808211
$675$	0	0
$676$	−1937.67	−0.110245
$677$	19340.8i	1.09797i	0.835832	+	0.548985i	$0.184986\pi$
$677$	19340.8i	1.09797i	−0.835832	+	0.548985i	$0.815014\pi$
$678$	2662.56i	0.150818i
$679$	−3247.25	−0.183531
$680$	0	0
$681$	1913.46	0.107671
$682$	1689.89i	0.0948816i
$683$	4255.14i	0.238387i	0.992871	+	0.119194i	$0.0380309\pi$
$683$	4255.14i	0.238387i	−0.992871	+	0.119194i	$0.961969\pi$
$684$	5587.05	0.312319
$685$	0	0
$686$	1313.15	0.0730850
$687$	− 1634.32i	− 0.0907616i
$688$	− 6757.08i	− 0.374435i
$689$	27199.1	1.50392
$690$	0	0
$691$	−17505.5	−0.963733	−0.481867	−	0.876245i	$-0.660041\pi$
$691$	−17505.5	−0.963733	−0.481867	+	0.876245i	$0.660041\pi$
$692$	4561.63i	0.250588i
$693$	3059.82i	0.167724i
$694$	14673.7	0.802603
$695$	0	0
$696$	906.071	0.0493456
$697$	− 18691.8i	− 1.01579i
$698$	31872.5i	1.72836i
$699$	−17245.6	−0.933175
$700$	0	0
$701$	−3240.77	−0.174611	−0.0873054	−	0.996182i	$-0.527826\pi$
$701$	−3240.77	−0.174611	−0.0873054	+	0.996182i	$0.527826\pi$
$702$	− 4512.70i	− 0.242622i
$703$	− 23563.3i	− 1.26416i
$704$	15933.8	0.853022
$705$	0	0
$706$	34423.4	1.83504
$707$	3029.07i	0.161132i
$708$	− 10484.5i	− 0.556543i
$709$	−19949.3	−1.05672	−0.528358	−	0.849022i	$-0.677192\pi$
$709$	−19949.3	−1.05672	−0.528358	+	0.849022i	$0.677192\pi$
$710$	0	0
$711$	−7849.76	−0.414049
$712$	− 1980.83i	− 0.104262i
$713$	− 946.698i	− 0.0497253i
$714$	−5439.41	−0.285105
$715$	0	0
$716$	6827.17	0.356345
$717$	8034.30i	0.418475i
$718$	− 49362.9i	− 2.56575i
$719$	11259.4	0.584011	0.292006	−	0.956417i	$-0.405677\pi$
$719$	11259.4	0.584011	0.292006	+	0.956417i	$0.405677\pi$
$720$	0	0
$721$	−3588.38	−0.185351
$722$	− 7034.60i	− 0.362605i
$723$	− 6606.47i	− 0.339830i
$724$	−19300.9	−0.990761
$725$	0	0
$726$	11805.8	0.603516
$727$	− 12228.5i	− 0.623840i	−0.950108	−	0.311920i	$-0.899028\pi$
$727$	− 12228.5i	− 0.623840i	0.950108	−	0.311920i	$-0.100972\pi$
$728$	− 1571.43i	− 0.0800013i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	6267.56	0.317119
$732$	7045.49i	0.355750i
$733$	26635.1i	1.34214i	0.741392	+	0.671072i	$0.234166\pi$
$733$	26635.1i	1.34214i	−0.741392	+	0.671072i	$0.765834\pi$
$734$	−28637.3	−1.44008
$735$	0	0
$736$	−24803.4	−1.24221
$737$	− 35771.7i	− 1.78788i
$738$	9519.26i	0.474809i
$739$	6074.00	0.302349	0.151174	−	0.988507i	$-0.451694\pi$
$739$	6074.00	0.302349	0.151174	+	0.988507i	$0.451694\pi$
$740$	0	0
$741$	−12213.6	−0.605505
$742$	16696.3i	0.826065i
$743$	4016.87i	0.198337i	0.995071	+	0.0991686i	$0.0316183\pi$
$743$	4016.87i	0.198337i	−0.995071	+	0.0991686i	$0.968382\pi$
$744$	−140.200	−0.00690858
$745$	0	0
$746$	13488.8	0.662010
$747$	− 4766.29i	− 0.233453i
$748$	21874.4i	1.06926i
$749$	13741.6	0.670370
$750$	0	0
$751$	−23913.2	−1.16192	−0.580962	−	0.813931i	$-0.697324\pi$
$751$	−23913.2	−1.16192	−0.580962	+	0.813931i	$0.697324\pi$
$752$	42509.6i	2.06139i
$753$	− 17150.7i	− 0.830022i
$754$	9816.81	0.474147
$755$	0	0
$756$	1258.15	0.0605269
$757$	31044.9i	1.49055i	0.666758	+	0.745275i	$0.267682\pi$
$757$	31044.9i	1.49055i	−0.666758	+	0.745275i	$0.732318\pi$
$758$	51742.9i	2.47940i
$759$	−15177.6	−0.725842
$760$	0	0
$761$	14011.8	0.667446	0.333723	−	0.942671i	$-0.391695\pi$
$761$	14011.8	0.667446	0.333723	+	0.942671i	$0.391695\pi$
$762$	27253.4i	1.29565i
$763$	3816.23i	0.181071i
$764$	−7150.69	−0.338616
$765$	0	0
$766$	2515.97	0.118676
$767$	22919.8i	1.07899i
$768$	15326.6i	0.720120i
$769$	−3342.49	−0.156740	−0.0783701	−	0.996924i	$-0.524972\pi$
$769$	−3342.49	−0.156740	−0.0783701	+	0.996924i	$0.524972\pi$
$770$	0	0
$771$	−14172.9	−0.662027
$772$	5984.51i	0.278999i
$773$	21074.6i	0.980594i	0.871555	+	0.490297i	$0.163112\pi$
$773$	21074.6i	0.980594i	−0.871555	+	0.490297i	$0.836888\pi$
$774$	−3191.90	−0.148231
$775$	0	0
$776$	−2385.40	−0.110349
$777$	− 5306.20i	− 0.244992i
$778$	− 37296.0i	− 1.71867i
$779$	25763.9	1.18496
$780$	0	0
$781$	−23908.5	−1.09541
$782$	− 26981.1i	− 1.23382i
$783$	− 1585.85i	− 0.0723800i
$784$	3574.12	0.162815
$785$	0	0
$786$	13782.8	0.625464
$787$	21394.8i	0.969048i	0.874778	+	0.484524i	$0.161007\pi$
$787$	21394.8i	0.969048i	−0.874778	+	0.484524i	$0.838993\pi$
$788$	− 20394.1i	− 0.921968i
$789$	17926.1	0.808853
$790$	0	0
$791$	1622.76	0.0729442
$792$	2247.71i	0.100845i
$793$	− 15401.9i	− 0.689706i
$794$	16873.5	0.754179
$795$	0	0
$796$	−6320.83	−0.281452
$797$	20645.0i	0.917547i	0.888553	+	0.458773i	$0.151711\pi$
$797$	20645.0i	0.917547i	−0.888553	+	0.458773i	$0.848289\pi$
$798$	− 7497.41i	− 0.332588i
$799$	−39430.0	−1.74585
$800$	0	0
$801$	−3466.95	−0.152932
$802$	44293.0i	1.95017i
$803$	− 56570.4i	− 2.48608i
$804$	−14708.7	−0.645194
$805$	0	0
$806$	−1518.99	−0.0663825
$807$	− 13458.3i	− 0.587058i
$808$	2225.13i	0.0968810i
$809$	15939.0	0.692688	0.346344	−	0.938108i	$-0.387423\pi$
$809$	15939.0	0.692688	0.346344	+	0.938108i	$0.387423\pi$
$810$	0	0
$811$	22829.2	0.988460	0.494230	−	0.869331i	$-0.335450\pi$
$811$	22829.2	0.988460	0.494230	+	0.869331i	$0.335450\pi$
$812$	2736.94i	0.118285i
$813$	11483.0i	0.495360i
$814$	−46982.9	−2.02303
$815$	0	0
$816$	−14804.9	−0.635141
$817$	8638.90i	0.369935i
$818$	− 11803.2i	− 0.504508i
$819$	−2750.38	−0.117346
$820$	0	0
$821$	−5700.22	−0.242313	−0.121157	−	0.992633i	$-0.538660\pi$
$821$	−5700.22	−0.242313	−0.121157	+	0.992633i	$0.538660\pi$
$822$	31943.5i	1.35542i
$823$	32438.0i	1.37390i	0.726707	+	0.686948i	$0.241050\pi$
$823$	32438.0i	1.37390i	−0.726707	+	0.686948i	$0.758950\pi$
$824$	−2635.99	−0.111443
$825$	0	0
$826$	−14069.4	−0.592662
$827$	− 12762.6i	− 0.536638i	−0.963330	−	0.268319i	$-0.913532\pi$
$827$	− 12762.6i	− 0.536638i	0.963330	−	0.268319i	$-0.0864681\pi$
$828$	6240.79i	0.261935i
$829$	30766.7	1.28899	0.644494	−	0.764609i	$-0.277068\pi$
$829$	30766.7	1.28899	0.644494	+	0.764609i	$0.277068\pi$
$830$	0	0
$831$	10261.6	0.428365
$832$	14322.4i	0.596804i
$833$	3315.19i	0.137892i
$834$	14306.1	0.593981
$835$	0	0
$836$	−30150.6	−1.24734
$837$	245.384i	0.0101335i
$838$	− 20731.7i	− 0.854613i
$839$	9779.71	0.402423	0.201212	−	0.979548i	$-0.435512\pi$
$839$	9779.71	0.402423	0.201212	+	0.979548i	$0.435512\pi$
$840$	0	0
$841$	−20939.2	−0.858551
$842$	− 16034.8i	− 0.656288i
$843$	15707.8i	0.641760i
$844$	−15355.0	−0.626233
$845$	0	0
$846$	20080.7	0.816061
$847$	− 7195.32i	− 0.291894i
$848$	45443.7i	1.84026i
$849$	−20957.6	−0.847190
$850$	0	0
$851$	26320.4	1.06023
$852$	9830.79i	0.395302i
$853$	− 24201.5i	− 0.971445i	−0.874113	−	0.485723i	$-0.838556\pi$
$853$	− 24201.5i	− 0.971445i	0.874113	−	0.485723i	$-0.161444\pi$
$854$	9454.52	0.378837
$855$	0	0
$856$	10094.5	0.403062
$857$	21036.7i	0.838507i	0.907869	+	0.419254i	$0.137708\pi$
$857$	21036.7i	0.838507i	−0.907869	+	0.419254i	$0.862292\pi$
$858$	24352.8i	0.968988i
$859$	6179.19	0.245438	0.122719	−	0.992441i	$-0.460839\pi$
$859$	6179.19	0.245438	0.122719	+	0.992441i	$0.460839\pi$
$860$	0	0
$861$	5801.76	0.229644
$862$	34870.9i	1.37785i
$863$	− 50256.2i	− 1.98232i	−0.132671	−	0.991160i	$-0.542356\pi$
$863$	− 50256.2i	− 1.98232i	0.132671	−	0.991160i	$-0.457644\pi$
$864$	6429.04	0.253149
$865$	0	0
$866$	64497.7	2.53085
$867$	1006.65i	0.0394321i
$868$	− 423.497i	− 0.0165604i
$869$	42361.2	1.65363
$870$	0	0
$871$	32154.1	1.25086
$872$	2803.37i	0.108869i
$873$	4175.03i	0.161860i
$874$	37189.5	1.43930
$875$	0	0
$876$	−23260.8	−0.897156
$877$	− 9175.95i	− 0.353306i	−0.984273	−	0.176653i	$-0.943473\pi$
$877$	− 9175.95i	− 0.353306i	0.984273	−	0.176653i	$-0.0565271\pi$
$878$	32289.5i	1.24114i
$879$	22199.1	0.851828
$880$	0	0
$881$	−26172.7	−1.00089	−0.500444	−	0.865769i	$-0.666830\pi$
$881$	−26172.7	−1.00089	−0.500444	+	0.865769i	$0.666830\pi$
$882$	− 1688.34i	− 0.0644549i
$883$	− 18615.5i	− 0.709471i	−0.934967	−	0.354736i	$-0.884571\pi$
$883$	− 18615.5i	− 0.709471i	0.934967	−	0.354736i	$-0.115429\pi$
$884$	−19662.3	−0.748092
$885$	0	0
$886$	16456.4	0.624001
$887$	− 12837.8i	− 0.485964i	−0.970031	−	0.242982i	$-0.921874\pi$
$887$	− 12837.8i	− 0.485964i	0.970031	−	0.242982i	$-0.0781256\pi$
$888$	− 3897.89i	− 0.147302i
$889$	16610.3	0.626649
$890$	0	0
$891$	3934.05	0.147919
$892$	− 21485.7i	− 0.806496i
$893$	− 54348.4i	− 2.03662i
$894$	−222.868	−0.00833759
$895$	0	0
$896$	4542.42	0.169366
$897$	− 13642.7i	− 0.507824i
$898$	40370.6i	1.50020i
$899$	−533.803	−0.0198035
$900$	0	0
$901$	−42151.5	−1.55857
$902$	− 51370.7i	− 1.89630i
$903$	1945.39i	0.0716926i
$904$	1192.07	0.0438579
$905$	0	0
$906$	−26988.1	−0.989647
$907$	− 26766.1i	− 0.979885i	−0.871755	−	0.489942i	$-0.837018\pi$
$907$	− 26766.1i	− 0.979885i	0.871755	−	0.489942i	$-0.162982\pi$
$908$	4245.87i	0.155181i
$909$	3894.52	0.142105
$910$	0	0
$911$	5022.67	0.182666	0.0913328	−	0.995820i	$-0.470887\pi$
$911$	5022.67	0.182666	0.0913328	+	0.995820i	$0.470887\pi$
$912$	− 20406.3i	− 0.740923i
$913$	25721.3i	0.932367i
$914$	45759.0	1.65599
$915$	0	0
$916$	3626.48	0.130810
$917$	− 8400.26i	− 0.302509i
$918$	6993.52i	0.251439i
$919$	−9541.79	−0.342497	−0.171248	−	0.985228i	$-0.554780\pi$
$919$	−9541.79	−0.342497	−0.171248	+	0.985228i	$0.554780\pi$
$920$	0	0
$921$	−8005.93	−0.286432
$922$	65852.4i	2.35220i
$923$	− 21490.7i	− 0.766387i
$924$	−6789.59	−0.241733
$925$	0	0
$926$	39695.7	1.40873
$927$	4613.63i	0.163464i
$928$	13985.6i	0.494718i
$929$	25479.6	0.899846	0.449923	−	0.893067i	$-0.351451\pi$
$929$	25479.6	0.899846	0.449923	+	0.893067i	$0.351451\pi$
$930$	0	0
$931$	−4569.49	−0.160858
$932$	− 38267.2i	− 1.34494i
$933$	18567.7i	0.651533i
$934$	−64621.9	−2.26391
$935$	0	0
$936$	−2020.41	−0.0705545
$937$	− 33608.3i	− 1.17176i	−0.810399	−	0.585878i	$-0.800750\pi$
$937$	− 33608.3i	− 1.17176i	0.810399	−	0.585878i	$-0.199250\pi$
$938$	19738.0i	0.687066i
$939$	−8764.77	−0.304609
$940$	0	0
$941$	−19173.6	−0.664232	−0.332116	−	0.943239i	$-0.607762\pi$
$941$	−19173.6	−0.664232	−0.332116	+	0.943239i	$0.607762\pi$
$942$	− 44774.9i	− 1.54867i
$943$	28778.5i	0.993804i
$944$	−38294.0	−1.32030
$945$	0	0
$946$	17225.1	0.592005
$947$	− 979.315i	− 0.0336045i	−0.999859	−	0.0168023i	$-0.994651\pi$
$947$	− 979.315i	− 0.0336045i	0.999859	−	0.0168023i	$-0.00534857\pi$
$948$	− 17418.2i	− 0.596749i
$949$	50849.5	1.73935
$950$	0	0
$951$	−29476.7	−1.00510
$952$	2435.31i	0.0829083i
$953$	− 3048.61i	− 0.103624i	−0.998657	−	0.0518122i	$-0.983500\pi$
$953$	− 3048.61i	− 0.103624i	0.998657	−	0.0518122i	$-0.0164997\pi$
$954$	21466.7	0.728521
$955$	0	0
$956$	−17827.7	−0.603127
$957$	8558.03i	0.289072i
$958$	28059.0i	0.946290i
$959$	19468.8	0.655557
$960$	0	0
$961$	−29708.4	−0.997227
$962$	− 42231.6i	− 1.41538i
$963$	− 17667.8i	− 0.591211i
$964$	14659.4	0.489781
$965$	0	0
$966$	8374.67	0.278934
$967$	− 14467.9i	− 0.481133i	−0.970633	−	0.240567i	$-0.922667\pi$
$967$	− 14467.9i	− 0.481133i	0.970633	−	0.240567i	$-0.0773332\pi$
$968$	− 5285.62i	− 0.175502i
$969$	18928.0	0.627507
$970$	0	0
$971$	12952.8	0.428090	0.214045	−	0.976824i	$-0.431336\pi$
$971$	12952.8	0.428090	0.214045	+	0.976824i	$0.431336\pi$
$972$	− 1617.62i	− 0.0533797i
$973$	− 8719.23i	− 0.287282i
$974$	−65885.4	−2.16746
$975$	0	0
$976$	25733.2	0.843954
$977$	47244.2i	1.54706i	0.633760	+	0.773529i	$0.281511\pi$
$977$	47244.2i	1.54706i	−0.633760	+	0.773529i	$0.718489\pi$
$978$	− 17544.2i	− 0.573621i
$979$	18709.4	0.610781
$980$	0	0
$981$	4906.58	0.159689
$982$	− 43615.3i	− 1.41733i
$983$	1536.84i	0.0498651i	0.999689	+	0.0249326i	$0.00793711\pi$
$983$	1536.84i	0.0498651i	−0.999689	+	0.0249326i	$0.992063\pi$
$984$	4261.92	0.138074
$985$	0	0
$986$	−15213.5	−0.491376
$987$	− 12238.7i	− 0.394692i
$988$	− 27101.5i	− 0.872685i
$989$	−9649.73	−0.310256
$990$	0	0
$991$	3785.22	0.121334	0.0606668	−	0.998158i	$-0.480677\pi$
$991$	3785.22	0.121334	0.0606668	+	0.998158i	$0.480677\pi$
$992$	− 2164.04i	− 0.0692625i
$993$	− 27774.5i	− 0.887610i
$994$	13192.2	0.420956
$995$	0	0
$996$	10576.2	0.336465
$997$	− 25894.9i	− 0.822566i	−0.911508	−	0.411283i	$-0.865081\pi$
$997$	− 25894.9i	− 0.822566i	0.911508	−	0.411283i	$-0.134919\pi$
$998$	73044.0i	2.31680i
$999$	−6822.26	−0.216063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.d.l.274.1		4
5.2	odd	4		525.4.a.l.1.2		2
5.3	odd	4		105.4.a.e.1.1	✓	2
5.4	even	2	inner	525.4.d.l.274.4		4
15.2	even	4		1575.4.a.q.1.1		2
15.8	even	4		315.4.a.k.1.2		2
20.3	even	4		1680.4.a.bo.1.1		2
35.13	even	4		735.4.a.o.1.1		2
105.83	odd	4		2205.4.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.1	✓	2	5.3	odd	4
315.4.a.k.1.2		2	15.8	even	4
525.4.a.l.1.2		2	5.2	odd	4
525.4.d.l.274.1		4	1.1	even	1	trivial
525.4.d.l.274.4		4	5.4	even	2	inner
735.4.a.o.1.1		2	35.13	even	4
1575.4.a.q.1.1		2	15.2	even	4
1680.4.a.bo.1.1		2	20.3	even	4
2205.4.a.bb.1.2		2	105.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.82843i q^{2} +3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} -7.00000i q^{7} -5.14214i q^{8} -9.00000 q^{9} +48.5685 q^{11} -19.9706i q^{12} +43.6569i q^{13} -26.7990 q^{14} -72.9411 q^{16} -67.6569i q^{17} +34.4558i q^{18} +93.2548 q^{19} +21.0000 q^{21} -185.941i q^{22} +104.167i q^{23} +15.4264 q^{24} +167.137 q^{26} -27.0000i q^{27} +46.5980i q^{28} +58.7351 q^{29} -9.08831 q^{31} +238.113i q^{32} +145.706i q^{33} -259.019 q^{34} +59.9117 q^{36} -252.676i q^{37} -357.019i q^{38} -130.971 q^{39} +276.274 q^{41} -80.3970i q^{42} +92.6375i q^{43} -323.314 q^{44} +398.794 q^{46} -582.794i q^{47} -218.823i q^{48} -49.0000 q^{49} +202.971 q^{51} -290.617i q^{52} -623.019i q^{53} -103.368 q^{54} -35.9949 q^{56} +279.765i q^{57} -224.863i q^{58} +524.999 q^{59} -352.794 q^{61} +34.7939i q^{62} +63.0000i q^{63} +328.068 q^{64} +557.823 q^{66} -736.520i q^{67} +450.382i q^{68} -312.500 q^{69} -492.264 q^{71} +46.2792i q^{72} -1164.75i q^{73} -967.352 q^{74} -620.784 q^{76} -339.980i q^{77} +501.411i q^{78} +872.195 q^{79} +81.0000 q^{81} -1057.70i q^{82} +529.588i q^{83} -139.794 q^{84} +354.656 q^{86} +176.205i q^{87} -249.746i q^{88} +385.216 q^{89} +305.598 q^{91} -693.421i q^{92} -27.2649i q^{93} -2231.18 q^{94} -714.338 q^{96} -463.892i q^{97} +187.593i q^{98} -437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 12 q^{6} - 36 q^{9} - 32 q^{11} - 28 q^{14} - 156 q^{16} + 192 q^{19} + 84 q^{21} - 108 q^{24} + 216 q^{26} - 376 q^{29} - 240 q^{31} - 312 q^{34} + 36 q^{36} - 456 q^{39} + 200 q^{41} - 1248 q^{44}+ \cdots + 288 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 12 * q^6 - 36 * q^9 - 32 * q^11 - 28 * q^14 - 156 * q^16 + 192 * q^19 + 84 * q^21 - 108 * q^24 + 216 * q^26 - 376 * q^29 - 240 * q^31 - 312 * q^34 + 36 * q^36 - 456 * q^39 + 200 * q^41 - 1248 * q^44 + 1120 * q^46 - 196 * q^49 + 744 * q^51 - 108 * q^54 + 252 * q^56 - 208 * q^59 - 936 * q^61 - 68 * q^64 + 1824 * q^66 - 96 * q^69 - 272 * q^71 - 2376 * q^74 - 1216 * q^76 + 864 * q^79 + 324 * q^81 - 84 * q^84 - 912 * q^86 + 2808 * q^89 + 1064 * q^91 - 3200 * q^94 - 2484 * q^96 + 288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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