LMFDB - Embedded newform 1150.4.b.r.599.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(599,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.599");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 214x^{8} + 15751x^{6} + 460323x^{4} + 4609305x^{2} + 8503056 $ x^10 + 214x^8 + 15751x^6 + 460323x^4 + 4609305x^2 + 8503056
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			599.10
Root			$9.27140i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.r.599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +8.27140i q^{3} -4.00000 q^{4} -16.5428 q^{6} +22.8621i q^{7} -8.00000i q^{8} -41.4161 q^{9} -0.987996 q^{11} -33.0856i q^{12} +4.22276i q^{13} -45.7241 q^{14} +16.0000 q^{16} +73.8741i q^{17} -82.8322i q^{18} -71.1586 q^{19} -189.101 q^{21} -1.97599i q^{22} -23.0000i q^{23} +66.1712 q^{24} -8.44552 q^{26} -119.241i q^{27} -91.4483i q^{28} -27.8984 q^{29} -136.861 q^{31} +32.0000i q^{32} -8.17211i q^{33} -147.748 q^{34} +165.664 q^{36} -201.718i q^{37} -142.317i q^{38} -34.9281 q^{39} -204.140 q^{41} -378.203i q^{42} +54.1024i q^{43} +3.95198 q^{44} +46.0000 q^{46} -41.4490i q^{47} +132.342i q^{48} -179.674 q^{49} -611.042 q^{51} -16.8910i q^{52} -428.612i q^{53} +238.483 q^{54} +182.897 q^{56} -588.581i q^{57} -55.7969i q^{58} +164.858 q^{59} +188.248 q^{61} -273.722i q^{62} -946.857i q^{63} -64.0000 q^{64} +16.3442 q^{66} +932.167i q^{67} -295.496i q^{68} +190.242 q^{69} -263.336 q^{71} +331.329i q^{72} -900.401i q^{73} +403.437 q^{74} +284.634 q^{76} -22.5876i q^{77} -69.8563i q^{78} +956.182 q^{79} -131.942 q^{81} -408.279i q^{82} -194.877i q^{83} +756.405 q^{84} -108.205 q^{86} -230.759i q^{87} +7.90397i q^{88} +213.751 q^{89} -96.5410 q^{91} +92.0000i q^{92} -1132.03i q^{93} +82.8981 q^{94} -264.685 q^{96} +628.762i q^{97} -359.348i q^{98} +40.9189 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 40 q^{4} + 20 q^{6} - 168 q^{9} - 52 q^{11} + 12 q^{14} + 160 q^{16} - 148 q^{19} - 176 q^{21} - 80 q^{24} - 244 q^{26} - 74 q^{29} + 440 q^{31} - 924 q^{34} + 672 q^{36} - 390 q^{39} + 224 q^{41}+ \cdots - 4794 q^{99}+O(q^{100}) $ 10 * q - 40 * q^4 + 20 * q^6 - 168 * q^9 - 52 * q^11 + 12 * q^14 + 160 * q^16 - 148 * q^19 - 176 * q^21 - 80 * q^24 - 244 * q^26 - 74 * q^29 + 440 * q^31 - 924 * q^34 + 672 * q^36 - 390 * q^39 + 224 * q^41 + 208 * q^44 + 460 * q^46 - 364 * q^49 + 1126 * q^51 - 1472 * q^54 - 48 * q^56 + 362 * q^59 + 3148 * q^61 - 640 * q^64 + 1564 * q^66 - 230 * q^69 - 60 * q^71 - 880 * q^74 + 592 * q^76 - 1218 * q^79 + 2906 * q^81 + 704 * q^84 + 388 * q^86 + 3326 * q^89 + 722 * q^91 - 1480 * q^94 + 320 * q^96 - 4794 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

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

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	8.27140i	1.59183i	0.605407	+	0.795916i	$0.293010\pi$
$3$	8.27140i	1.59183i	−0.605407	+	0.795916i	$0.706990\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−16.5428	−1.12560
$7$	22.8621i	1.23444i	0.786792	+	0.617218i	$0.211740\pi$
$7$	22.8621i	1.23444i	−0.786792	+	0.617218i	$0.788260\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−41.4161	−1.53393
$10$	0	0
$11$	−0.987996	−0.0270811	−0.0135405	−	0.999908i	$-0.504310\pi$
$11$	−0.987996	−0.0270811	−0.0135405	+	0.999908i	$0.504310\pi$
$12$	− 33.0856i	− 0.795916i
$13$	4.22276i	0.0900910i	0.998985	+	0.0450455i	$0.0143433\pi$
$13$	4.22276i	0.0900910i	−0.998985	+	0.0450455i	$0.985657\pi$
$14$	−45.7241	−0.872878
$15$	0	0
$16$	16.0000	0.250000
$17$	73.8741i	1.05395i	0.849882	+	0.526973i	$0.176673\pi$
$17$	73.8741i	1.05395i	−0.849882	+	0.526973i	$0.823327\pi$
$18$	− 82.8322i	− 1.08465i
$19$	−71.1586	−0.859205	−0.429602	−	0.903018i	$-0.641346\pi$
$19$	−71.1586	−0.859205	−0.429602	+	0.903018i	$0.641346\pi$
$20$	0	0
$21$	−189.101	−1.96501
$22$	− 1.97599i	− 0.0191492i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	66.1712	0.562798
$25$	0	0
$26$	−8.44552	−0.0637039
$27$	− 119.241i	− 0.849926i
$28$	− 91.4483i	− 0.617218i
$29$	−27.8984	−0.178642	−0.0893209	−	0.996003i	$-0.528470\pi$
$29$	−27.8984	−0.178642	−0.0893209	+	0.996003i	$0.528470\pi$
$30$	0	0
$31$	−136.861	−0.792935	−0.396468	−	0.918049i	$-0.629764\pi$
$31$	−136.861	−0.792935	−0.396468	+	0.918049i	$0.629764\pi$
$32$	32.0000i	0.176777i
$33$	− 8.17211i	− 0.0431085i
$34$	−147.748	−0.745253
$35$	0	0
$36$	165.664	0.766965
$37$	− 201.718i	− 0.896278i	−0.893964	−	0.448139i	$-0.852087\pi$
$37$	− 201.718i	− 0.896278i	0.893964	−	0.448139i	$-0.147913\pi$
$38$	− 142.317i	− 0.607550i
$39$	−34.9281	−0.143410
$40$	0	0
$41$	−204.140	−0.777592	−0.388796	−	0.921324i	$-0.627109\pi$
$41$	−204.140	−0.777592	−0.388796	+	0.921324i	$0.627109\pi$
$42$	− 378.203i	− 1.38947i
$43$	54.1024i	0.191873i	0.995387	+	0.0959365i	$0.0305846\pi$
$43$	54.1024i	0.191873i	−0.995387	+	0.0959365i	$0.969415\pi$
$44$	3.95198	0.0135405
$45$	0	0
$46$	46.0000	0.147442
$47$	− 41.4490i	− 0.128638i	−0.997929	−	0.0643188i	$-0.979513\pi$
$47$	− 41.4490i	− 0.128638i	0.997929	−	0.0643188i	$-0.0204874\pi$
$48$	132.342i	0.397958i
$49$	−179.674	−0.523831
$50$	0	0
$51$	−611.042	−1.67771
$52$	− 16.8910i	− 0.0450455i
$53$	− 428.612i	− 1.11084i	−0.831571	−	0.555418i	$-0.812558\pi$
$53$	− 428.612i	− 1.11084i	0.831571	−	0.555418i	$-0.187442\pi$
$54$	238.483	0.600988
$55$	0	0
$56$	182.897	0.436439
$57$	− 588.581i	− 1.36771i
$58$	− 55.7969i	− 0.126319i
$59$	164.858	0.363774	0.181887	−	0.983319i	$-0.441779\pi$
$59$	164.858	0.363774	0.181887	+	0.983319i	$0.441779\pi$
$60$	0	0
$61$	188.248	0.395125	0.197563	−	0.980290i	$-0.436697\pi$
$61$	188.248	0.395125	0.197563	+	0.980290i	$0.436697\pi$
$62$	− 273.722i	− 0.560690i
$63$	− 946.857i	− 1.89354i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	16.3442	0.0304823
$67$	932.167i	1.69974i	0.526995	+	0.849868i	$0.323318\pi$
$67$	932.167i	1.69974i	−0.526995	+	0.849868i	$0.676682\pi$
$68$	− 295.496i	− 0.526973i
$69$	190.242	0.331920
$70$	0	0
$71$	−263.336	−0.440173	−0.220086	−	0.975480i	$-0.570634\pi$
$71$	−263.336	−0.440173	−0.220086	+	0.975480i	$0.570634\pi$
$72$	331.329i	0.542326i
$73$	− 900.401i	− 1.44362i	−0.692094	−	0.721808i	$-0.743311\pi$
$73$	− 900.401i	− 1.44362i	0.692094	−	0.721808i	$-0.256689\pi$
$74$	403.437	0.633764
$75$	0	0
$76$	284.634	0.429602
$77$	− 22.5876i	− 0.0334299i
$78$	− 69.8563i	− 0.101406i
$79$	956.182	1.36176	0.680879	−	0.732396i	$-0.261598\pi$
$79$	956.182	1.36176	0.680879	+	0.732396i	$0.261598\pi$
$80$	0	0
$81$	−131.942	−0.180990
$82$	− 408.279i	− 0.549841i
$83$	− 194.877i	− 0.257717i	−0.991663	−	0.128858i	$-0.958869\pi$
$83$	− 194.877i	− 0.257717i	0.991663	−	0.128858i	$-0.0411313\pi$
$84$	756.405	0.982507
$85$	0	0
$86$	−108.205	−0.135675
$87$	− 230.759i	− 0.284368i
$88$	7.90397i	0.00957461i
$89$	213.751	0.254580	0.127290	−	0.991866i	$-0.459372\pi$
$89$	213.751	0.254580	0.127290	+	0.991866i	$0.459372\pi$
$90$	0	0
$91$	−96.5410	−0.111211
$92$	92.0000i	0.104257i
$93$	− 1132.03i	− 1.26222i
$94$	82.8981	0.0909605
$95$	0	0
$96$	−264.685	−0.281399
$97$	628.762i	0.658156i	0.944303	+	0.329078i	$0.106738\pi$
$97$	628.762i	0.658156i	−0.944303	+	0.329078i	$0.893262\pi$
$98$	− 359.348i	− 0.370404i
$99$	40.9189	0.0415405
$100$	0	0
$101$	−51.1371	−0.0503796	−0.0251898	−	0.999683i	$-0.508019\pi$
$101$	−51.1371	−0.0503796	−0.0251898	+	0.999683i	$0.508019\pi$
$102$	− 1222.08i	− 1.18632i
$103$	− 363.089i	− 0.347342i	−0.984804	−	0.173671i	$-0.944437\pi$
$103$	− 363.089i	− 0.347342i	0.984804	−	0.173671i	$-0.0555630\pi$
$104$	33.7821	0.0318520
$105$	0	0
$106$	857.223	0.785480
$107$	1514.53i	1.36837i	0.729310	+	0.684184i	$0.239841\pi$
$107$	1514.53i	1.36837i	−0.729310	+	0.684184i	$0.760159\pi$
$108$	476.965i	0.424963i
$109$	−811.356	−0.712971	−0.356485	−	0.934301i	$-0.616025\pi$
$109$	−811.356	−0.712971	−0.356485	+	0.934301i	$0.616025\pi$
$110$	0	0
$111$	1668.49	1.42672
$112$	365.793i	0.308609i
$113$	− 12.4496i	− 0.0103642i	−0.999987	−	0.00518211i	$-0.998350\pi$
$113$	− 12.4496i	− 0.0103642i	0.999987	−	0.00518211i	$-0.00164952\pi$
$114$	1177.16	0.967117
$115$	0	0
$116$	111.594	0.0893209
$117$	− 174.890i	− 0.138193i
$118$	329.716i	0.257227i
$119$	−1688.91	−1.30103
$120$	0	0
$121$	−1330.02	−0.999267
$122$	376.495i	0.279396i
$123$	− 1688.52i	− 1.23780i
$124$	547.445	0.396468
$125$	0	0
$126$	1893.71	1.33893
$127$	− 632.599i	− 0.442000i	−0.975274	−	0.221000i	$-0.929068\pi$
$127$	− 632.599i	− 0.442000i	0.975274	−	0.221000i	$-0.0709322\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−447.503	−0.305430
$130$	0	0
$131$	2243.16	1.49607	0.748036	−	0.663658i	$-0.230997\pi$
$131$	2243.16	1.49607	0.748036	+	0.663658i	$0.230997\pi$
$132$	32.6884i	0.0215543i
$133$	− 1626.83i	− 1.06063i
$134$	−1864.33	−1.20190
$135$	0	0
$136$	590.993	0.372626
$137$	1252.77i	0.781252i	0.920550	+	0.390626i	$0.127741\pi$
$137$	1252.77i	0.781252i	−0.920550	+	0.390626i	$0.872259\pi$
$138$	380.484i	0.234703i
$139$	−2815.19	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$139$	−2815.19	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$140$	0	0
$141$	342.842	0.204769
$142$	− 526.673i	− 0.311249i
$143$	− 4.17207i	− 0.00243976i
$144$	−662.657	−0.383482
$145$	0	0
$146$	1800.80	1.02079
$147$	− 1486.16i	− 0.833851i
$148$	806.873i	0.448139i
$149$	−2406.46	−1.32312	−0.661560	−	0.749892i	$-0.730105\pi$
$149$	−2406.46	−1.32312	−0.661560	+	0.749892i	$0.730105\pi$
$150$	0	0
$151$	2185.59	1.17788	0.588942	−	0.808175i	$-0.299545\pi$
$151$	2185.59	1.17788	0.588942	+	0.808175i	$0.299545\pi$
$152$	569.268i	0.303775i
$153$	− 3059.58i	− 1.61668i
$154$	45.1753	0.0236385
$155$	0	0
$156$	139.713	0.0717048
$157$	1074.18i	0.546044i	0.962008	+	0.273022i	$0.0880232\pi$
$157$	1074.18i	0.546044i	−0.962008	+	0.273022i	$0.911977\pi$
$158$	1912.36i	0.962908i
$159$	3545.22	1.76827
$160$	0	0
$161$	525.828	0.257398
$162$	− 263.884i	− 0.127979i
$163$	− 2706.30i	− 1.30045i	−0.759741	−	0.650226i	$-0.774674\pi$
$163$	− 2706.30i	− 1.30045i	0.759741	−	0.650226i	$-0.225326\pi$
$164$	816.559	0.388796
$165$	0	0
$166$	389.754	0.182233
$167$	455.547i	0.211086i	0.994415	+	0.105543i	$0.0336580\pi$
$167$	455.547i	0.211086i	−0.994415	+	0.105543i	$0.966342\pi$
$168$	1512.81i	0.694737i
$169$	2179.17	0.991884
$170$	0	0
$171$	2947.11	1.31796
$172$	− 216.410i	− 0.0959365i
$173$	1342.64i	0.590051i	0.955489	+	0.295026i	$0.0953282\pi$
$173$	1342.64i	0.590051i	−0.955489	+	0.295026i	$0.904672\pi$
$174$	461.518	0.201078
$175$	0	0
$176$	−15.8079	−0.00677027
$177$	1363.61i	0.579068i
$178$	427.503i	0.180015i
$179$	4655.64	1.94401	0.972007	−	0.234951i	$-0.0754929\pi$
$179$	4655.64	1.94401	0.972007	+	0.234951i	$0.0754929\pi$
$180$	0	0
$181$	−176.320	−0.0724074	−0.0362037	−	0.999344i	$-0.511527\pi$
$181$	−176.320	−0.0724074	−0.0362037	+	0.999344i	$0.511527\pi$
$182$	− 193.082i	− 0.0786384i
$183$	1557.07i	0.628973i
$184$	−184.000	−0.0737210
$185$	0	0
$186$	2264.07	0.892524
$187$	− 72.9873i	− 0.0285420i
$188$	165.796i	0.0643188i
$189$	2726.10	1.04918
$190$	0	0
$191$	1672.03	0.633422	0.316711	−	0.948522i	$-0.397421\pi$
$191$	1672.03	0.633422	0.316711	+	0.948522i	$0.397421\pi$
$192$	− 529.370i	− 0.198979i
$193$	2954.45i	1.10189i	0.834540	+	0.550947i	$0.185733\pi$
$193$	2954.45i	1.10189i	−0.834540	+	0.550947i	$0.814267\pi$
$194$	−1257.52	−0.465386
$195$	0	0
$196$	718.696	0.261915
$197$	1817.35i	0.657264i	0.944458	+	0.328632i	$0.106588\pi$
$197$	1817.35i	0.657264i	−0.944458	+	0.328632i	$0.893412\pi$
$198$	81.8379i	0.0293735i
$199$	−4780.99	−1.70309	−0.851546	−	0.524280i	$-0.824334\pi$
$199$	−4780.99	−1.70309	−0.851546	+	0.524280i	$0.824334\pi$
$200$	0	0
$201$	−7710.33	−2.70569
$202$	− 102.274i	− 0.0356237i
$203$	− 637.816i	− 0.220522i
$204$	2444.17	0.838853
$205$	0	0
$206$	726.179	0.245608
$207$	952.570i	0.319846i
$208$	67.5641i	0.0225227i
$209$	70.3044	0.0232682
$210$	0	0
$211$	3201.72	1.04462	0.522312	−	0.852755i	$-0.325070\pi$
$211$	3201.72	1.04462	0.522312	+	0.852755i	$0.325070\pi$
$212$	1714.45i	0.555418i
$213$	− 2178.16i	− 0.700681i
$214$	−3029.06	−0.967582
$215$	0	0
$216$	−953.930	−0.300494
$217$	− 3128.93i	− 0.978828i
$218$	− 1622.71i	− 0.504147i
$219$	7447.58	2.29799
$220$	0	0
$221$	−311.952	−0.0949511
$222$	3336.99i	1.00885i
$223$	1897.23i	0.569721i	0.958569	+	0.284860i	$0.0919473\pi$
$223$	1897.23i	0.569721i	−0.958569	+	0.284860i	$0.908053\pi$
$224$	−731.586	−0.218219
$225$	0	0
$226$	24.8991	0.00732861
$227$	4468.42i	1.30652i	0.757135	+	0.653258i	$0.226598\pi$
$227$	4468.42i	1.30652i	−0.757135	+	0.653258i	$0.773402\pi$
$228$	2354.32i	0.683855i
$229$	5345.88	1.54264	0.771322	−	0.636445i	$-0.219596\pi$
$229$	5345.88	1.54264	0.771322	+	0.636445i	$0.219596\pi$
$230$	0	0
$231$	186.831	0.0532147
$232$	223.187i	0.0631594i
$233$	4480.46i	1.25976i	0.776691	+	0.629882i	$0.216897\pi$
$233$	4480.46i	1.25976i	−0.776691	+	0.629882i	$0.783103\pi$
$234$	349.780	0.0977173
$235$	0	0
$236$	−659.432	−0.181887
$237$	7908.97i	2.16769i
$238$	− 3377.83i	− 0.919967i
$239$	−779.836	−0.211060	−0.105530	−	0.994416i	$-0.533654\pi$
$239$	−779.836	−0.211060	−0.105530	+	0.994416i	$0.533654\pi$
$240$	0	0
$241$	−4204.89	−1.12390	−0.561952	−	0.827170i	$-0.689950\pi$
$241$	−4204.89	−1.12390	−0.561952	+	0.827170i	$0.689950\pi$
$242$	− 2660.05i	− 0.706588i
$243$	− 4310.86i	− 1.13803i
$244$	−752.991	−0.197563
$245$	0	0
$246$	3377.04	0.875254
$247$	− 300.485i	− 0.0774066i
$248$	1094.89i	0.280345i
$249$	1611.90	0.410242
$250$	0	0
$251$	2761.48	0.694433	0.347217	−	0.937785i	$-0.387127\pi$
$251$	2761.48	0.694433	0.347217	+	0.937785i	$0.387127\pi$
$252$	3787.43i	0.946768i
$253$	22.7239i	0.00564680i
$254$	1265.20	0.312542
$255$	0	0
$256$	256.000	0.0625000
$257$	− 586.469i	− 0.142346i	−0.997464	−	0.0711730i	$-0.977326\pi$
$257$	− 586.469i	− 0.142346i	0.997464	−	0.0711730i	$-0.0226742\pi$
$258$	− 895.006i	− 0.215971i
$259$	4611.70	1.10640
$260$	0	0
$261$	1155.44	0.274024
$262$	4486.31i	1.05788i
$263$	− 5970.38i	− 1.39981i	−0.714238	−	0.699903i	$-0.753226\pi$
$263$	− 5970.38i	− 1.39981i	0.714238	−	0.699903i	$-0.246774\pi$
$264$	−65.3769	−0.0152412
$265$	0	0
$266$	3253.66	0.749981
$267$	1768.02i	0.405248i
$268$	− 3728.67i	− 0.849868i
$269$	−3914.34	−0.887217	−0.443608	−	0.896221i	$-0.646302\pi$
$269$	−3914.34	−0.887217	−0.443608	+	0.896221i	$0.646302\pi$
$270$	0	0
$271$	78.0865	0.0175034	0.00875170	−	0.999962i	$-0.497214\pi$
$271$	78.0865	0.0175034	0.00875170	+	0.999962i	$0.497214\pi$
$272$	1181.99i	0.263487i
$273$	− 798.529i	− 0.177030i
$274$	−2505.54	−0.552428
$275$	0	0
$276$	−760.969	−0.165960
$277$	− 76.0512i	− 0.0164963i	−0.999966	−	0.00824815i	$-0.997375\pi$
$277$	− 76.0512i	− 0.0164963i	0.999966	−	0.00824815i	$-0.00262550\pi$
$278$	− 5630.38i	− 1.21470i
$279$	5668.26	1.21631
$280$	0	0
$281$	−3852.67	−0.817905	−0.408952	−	0.912556i	$-0.634106\pi$
$281$	−3852.67	−0.817905	−0.408952	+	0.912556i	$0.634106\pi$
$282$	685.683i	0.144794i
$283$	1550.13i	0.325603i	0.986659	+	0.162802i	$0.0520531\pi$
$283$	1550.13i	0.325603i	−0.986659	+	0.162802i	$0.947947\pi$
$284$	1053.35	0.220086
$285$	0	0
$286$	8.34414	0.00172517
$287$	− 4667.06i	− 0.959887i
$288$	− 1325.31i	− 0.271163i
$289$	−544.378	−0.110804
$290$	0	0
$291$	−5200.74	−1.04767
$292$	3601.60i	0.721808i
$293$	− 4236.01i	− 0.844608i	−0.906454	−	0.422304i	$-0.861221\pi$
$293$	− 4236.01i	− 0.844608i	0.906454	−	0.422304i	$-0.138779\pi$
$294$	2972.31	0.589622
$295$	0	0
$296$	−1613.75	−0.316882
$297$	117.810i	0.0230169i
$298$	− 4812.92i	− 0.935587i
$299$	97.1234	0.0187853
$300$	0	0
$301$	−1236.89	−0.236855
$302$	4371.17i	0.832890i
$303$	− 422.976i	− 0.0801958i
$304$	−1138.54	−0.214801
$305$	0	0
$306$	6119.15	1.14317
$307$	− 10078.1i	− 1.87357i	−0.349903	−	0.936786i	$-0.613785\pi$
$307$	− 10078.1i	− 1.87357i	0.349903	−	0.936786i	$-0.386215\pi$
$308$	90.3505i	0.0167149i
$309$	3003.26	0.552911
$310$	0	0
$311$	−2652.97	−0.483717	−0.241859	−	0.970312i	$-0.577757\pi$
$311$	−2652.97	−0.483717	−0.241859	+	0.970312i	$0.577757\pi$
$312$	279.425i	0.0507030i
$313$	− 6710.68i	− 1.21185i	−0.795521	−	0.605926i	$-0.792803\pi$
$313$	− 6710.68i	− 1.21185i	0.795521	−	0.605926i	$-0.207197\pi$
$314$	−2148.36	−0.386111
$315$	0	0
$316$	−3824.73	−0.680879
$317$	− 3677.89i	− 0.651642i	−0.945431	−	0.325821i	$-0.894359\pi$
$317$	− 3677.89i	− 0.651642i	0.945431	−	0.325821i	$-0.105641\pi$
$318$	7090.44i	1.25035i
$319$	27.5635	0.00483781
$320$	0	0
$321$	−12527.3	−2.17821
$322$	1051.66i	0.182008i
$323$	− 5256.77i	− 0.905556i
$324$	527.768	0.0904952
$325$	0	0
$326$	5412.60	0.919558
$327$	− 6711.05i	− 1.13493i
$328$	1633.12i	0.274920i
$329$	947.611	0.158795
$330$	0	0
$331$	−9806.39	−1.62842	−0.814212	−	0.580568i	$-0.802831\pi$
$331$	−9806.39	−1.62842	−0.814212	+	0.580568i	$0.802831\pi$
$332$	779.507i	0.128858i
$333$	8354.38i	1.37483i
$334$	−911.094	−0.149260
$335$	0	0
$336$	−3025.62	−0.491253
$337$	− 7143.41i	− 1.15468i	−0.816505	−	0.577339i	$-0.804091\pi$
$337$	− 7143.41i	− 1.15468i	0.816505	−	0.577339i	$-0.195909\pi$
$338$	4358.34i	0.701368i
$339$	102.975	0.0164981
$340$	0	0
$341$	135.218	0.0214736
$342$	5894.22i	0.931938i
$343$	3733.97i	0.587800i
$344$	432.819	0.0678374
$345$	0	0
$346$	−2685.28	−0.417229
$347$	− 5160.20i	− 0.798312i	−0.916883	−	0.399156i	$-0.869303\pi$
$347$	− 5160.20i	− 0.798312i	0.916883	−	0.399156i	$-0.130697\pi$
$348$	923.037i	0.142184i
$349$	−8712.04	−1.33623	−0.668115	−	0.744058i	$-0.732899\pi$
$349$	−8712.04	−1.33623	−0.668115	+	0.744058i	$0.732899\pi$
$350$	0	0
$351$	503.527	0.0765706
$352$	− 31.6159i	− 0.00478731i
$353$	7117.30i	1.07313i	0.843858	+	0.536566i	$0.180279\pi$
$353$	7117.30i	1.07313i	−0.843858	+	0.536566i	$0.819721\pi$
$354$	−2727.21	−0.409463
$355$	0	0
$356$	−855.005	−0.127290
$357$	− 13969.7i	− 2.07102i
$358$	9311.27i	1.37463i
$359$	−3548.73	−0.521712	−0.260856	−	0.965378i	$-0.584005\pi$
$359$	−3548.73	−0.521712	−0.260856	+	0.965378i	$0.584005\pi$
$360$	0	0
$361$	−1795.46	−0.261767
$362$	− 352.639i	− 0.0511998i
$363$	− 11001.2i	− 1.59066i
$364$	386.164	0.0556057
$365$	0	0
$366$	−3114.15	−0.444751
$367$	80.0110i	0.0113802i	0.999984	+	0.00569011i	$0.00181123\pi$
$367$	80.0110i	0.0113802i	−0.999984	+	0.00569011i	$0.998189\pi$
$368$	− 368.000i	− 0.0521286i
$369$	8454.67	1.19277
$370$	0	0
$371$	9798.95	1.37126
$372$	4528.14i	0.631110i
$373$	− 3211.09i	− 0.445747i	−0.974847	−	0.222874i	$-0.928456\pi$
$373$	− 3211.09i	− 0.445747i	0.974847	−	0.222874i	$-0.0715437\pi$
$374$	145.975	0.0201823
$375$	0	0
$376$	−331.592	−0.0454802
$377$	− 117.808i	− 0.0160940i
$378$	5452.20i	0.741881i
$379$	8177.57	1.10832	0.554160	−	0.832410i	$-0.313039\pi$
$379$	8177.57	1.10832	0.554160	+	0.832410i	$0.313039\pi$
$380$	0	0
$381$	5232.48	0.703591
$382$	3344.05i	0.447897i
$383$	− 12926.5i	− 1.72458i	−0.506413	−	0.862291i	$-0.669029\pi$
$383$	− 12926.5i	− 1.72458i	0.506413	−	0.862291i	$-0.330971\pi$
$384$	1058.74	0.140699
$385$	0	0
$386$	−5908.89	−0.779157
$387$	− 2240.71i	− 0.294320i
$388$	− 2515.05i	− 0.329078i
$389$	−4734.66	−0.617113	−0.308556	−	0.951206i	$-0.599846\pi$
$389$	−4734.66	−0.617113	−0.308556	+	0.951206i	$0.599846\pi$
$390$	0	0
$391$	1699.10	0.219763
$392$	1437.39i	0.185202i
$393$	18554.0i	2.38150i
$394$	−3634.71	−0.464756
$395$	0	0
$396$	−163.676	−0.0207702
$397$	− 9938.80i	− 1.25646i	−0.778028	−	0.628229i	$-0.783780\pi$
$397$	− 9938.80i	− 1.25646i	0.778028	−	0.628229i	$-0.216220\pi$
$398$	− 9561.98i	− 1.20427i
$399$	13456.2	1.68835
$400$	0	0
$401$	−335.296	−0.0417553	−0.0208777	−	0.999782i	$-0.506646\pi$
$401$	−335.296	−0.0417553	−0.0208777	+	0.999782i	$0.506646\pi$
$402$	− 15420.7i	− 1.91322i
$403$	− 577.932i	− 0.0714363i
$404$	204.549	0.0251898
$405$	0	0
$406$	1275.63	0.155932
$407$	199.297i	0.0242722i
$408$	4888.34i	0.593159i
$409$	−6560.85	−0.793186	−0.396593	−	0.917995i	$-0.629808\pi$
$409$	−6560.85	−0.793186	−0.396593	+	0.917995i	$0.629808\pi$
$410$	0	0
$411$	−10362.2	−1.24362
$412$	1452.36i	0.173671i
$413$	3769.00i	0.449056i
$414$	−1905.14	−0.226166
$415$	0	0
$416$	−135.128	−0.0159260
$417$	− 23285.6i	− 2.73453i
$418$	140.609i	0.0164531i
$419$	−7116.69	−0.829768	−0.414884	−	0.909874i	$-0.636178\pi$
$419$	−7116.69	−0.829768	−0.414884	+	0.909874i	$0.636178\pi$
$420$	0	0
$421$	9799.43	1.13443	0.567215	−	0.823570i	$-0.308021\pi$
$421$	9799.43	1.13443	0.567215	+	0.823570i	$0.308021\pi$
$422$	6403.44i	0.738660i
$423$	1716.66i	0.197321i
$424$	−3428.89	−0.392740
$425$	0	0
$426$	4356.32	0.495456
$427$	4303.73i	0.487757i
$428$	− 6058.13i	− 0.684184i
$429$	34.5089	0.00388369
$430$	0	0
$431$	−10523.3	−1.17608	−0.588039	−	0.808832i	$-0.700100\pi$
$431$	−10523.3	−1.17608	−0.588039	+	0.808832i	$0.700100\pi$
$432$	− 1907.86i	− 0.212481i
$433$	4770.97i	0.529511i	0.964316	+	0.264755i	$0.0852912\pi$
$433$	4770.97i	0.529511i	−0.964316	+	0.264755i	$0.914709\pi$
$434$	6257.86	0.692136
$435$	0	0
$436$	3245.42	0.356485
$437$	1636.65i	0.179157i
$438$	14895.2i	1.62493i
$439$	−10231.2	−1.11232	−0.556159	−	0.831076i	$-0.687726\pi$
$439$	−10231.2	−1.11232	−0.556159	+	0.831076i	$0.687726\pi$
$440$	0	0
$441$	7441.40	0.803520
$442$	− 623.905i	− 0.0671405i
$443$	− 10508.1i	− 1.12699i	−0.826121	−	0.563493i	$-0.809457\pi$
$443$	− 10508.1i	− 1.12699i	0.826121	−	0.563493i	$-0.190543\pi$
$444$	−6673.97	−0.713362
$445$	0	0
$446$	−3794.46	−0.402854
$447$	− 19904.8i	− 2.10618i
$448$	− 1463.17i	− 0.154304i
$449$	−7230.71	−0.759996	−0.379998	−	0.924987i	$-0.624075\pi$
$449$	−7230.71	−0.759996	−0.379998	+	0.924987i	$0.624075\pi$
$450$	0	0
$451$	201.689	0.0210580
$452$	49.7983i	0.00518211i
$453$	18077.9i	1.87499i
$454$	−8936.83	−0.923847
$455$	0	0
$456$	−4708.65	−0.483558
$457$	2418.74i	0.247580i	0.992308	+	0.123790i	$0.0395049\pi$
$457$	2418.74i	0.247580i	−0.992308	+	0.123790i	$0.960495\pi$
$458$	10691.8i	1.09081i
$459$	8808.84	0.895776
$460$	0	0
$461$	−11342.8	−1.14596	−0.572979	−	0.819570i	$-0.694212\pi$
$461$	−11342.8	−1.14596	−0.572979	+	0.819570i	$0.694212\pi$
$462$	373.663i	0.0376285i
$463$	11785.9i	1.18302i	0.806299	+	0.591508i	$0.201467\pi$
$463$	11785.9i	1.18302i	−0.806299	+	0.591508i	$0.798533\pi$
$464$	−446.375	−0.0446604
$465$	0	0
$466$	−8960.93	−0.890787
$467$	16294.5i	1.61461i	0.590137	+	0.807303i	$0.299074\pi$
$467$	16294.5i	1.61461i	−0.590137	+	0.807303i	$0.700926\pi$
$468$	699.561i	0.0690966i
$469$	−21311.3	−2.09821
$470$	0	0
$471$	−8884.97	−0.869210
$472$	− 1318.86i	− 0.128614i
$473$	− 53.4530i	− 0.00519613i
$474$	−15817.9	−1.53279
$475$	0	0
$476$	6755.66	0.650515
$477$	17751.4i	1.70394i
$478$	− 1559.67i	− 0.149242i
$479$	785.274	0.0749062	0.0374531	−	0.999298i	$-0.488076\pi$
$479$	785.274	0.0749062	0.0374531	+	0.999298i	$0.488076\pi$
$480$	0	0
$481$	851.808	0.0807466
$482$	− 8409.79i	− 0.794721i
$483$	4349.33i	0.409734i
$484$	5320.10	0.499633
$485$	0	0
$486$	8621.72	0.804710
$487$	8714.29i	0.810846i	0.914129	+	0.405423i	$0.132876\pi$
$487$	8714.29i	0.810846i	−0.914129	+	0.405423i	$0.867124\pi$
$488$	− 1505.98i	− 0.139698i
$489$	22384.9	2.07010
$490$	0	0
$491$	−10943.1	−1.00582	−0.502909	−	0.864339i	$-0.667737\pi$
$491$	−10943.1	−1.00582	−0.502909	+	0.864339i	$0.667737\pi$
$492$	6754.09i	0.618898i
$493$	− 2060.97i	− 0.188279i
$494$	600.971	0.0547347
$495$	0	0
$496$	−2189.78	−0.198234
$497$	− 6020.41i	− 0.543365i
$498$	3223.81i	0.290085i
$499$	5601.85	0.502552	0.251276	−	0.967916i	$-0.419150\pi$
$499$	5601.85	0.502552	0.251276	+	0.967916i	$0.419150\pi$
$500$	0	0
$501$	−3768.01	−0.336013
$502$	5522.95i	0.491039i
$503$	− 21689.8i	− 1.92266i	−0.275395	−	0.961331i	$-0.588809\pi$
$503$	− 21689.8i	− 1.92266i	0.275395	−	0.961331i	$-0.411191\pi$
$504$	−7574.86	−0.669466
$505$	0	0
$506$	−45.4478	−0.00399289
$507$	18024.8i	1.57891i
$508$	2530.39i	0.221000i
$509$	−13826.0	−1.20399	−0.601993	−	0.798501i	$-0.705627\pi$
$509$	−13826.0	−1.20399	−0.601993	+	0.798501i	$0.705627\pi$
$510$	0	0
$511$	20585.0	1.78205
$512$	512.000i	0.0441942i
$513$	8485.04i	0.730260i
$514$	1172.94	0.100654
$515$	0	0
$516$	1790.01	0.152715
$517$	40.9515i	0.00348364i
$518$	9223.40i	0.782341i
$519$	−11105.5	−0.939263
$520$	0	0
$521$	−11971.1	−1.00664	−0.503322	−	0.864099i	$-0.667889\pi$
$521$	−11971.1	−1.00664	−0.503322	+	0.864099i	$0.667889\pi$
$522$	2310.89i	0.193764i
$523$	9668.82i	0.808390i	0.914673	+	0.404195i	$0.132448\pi$
$523$	9668.82i	0.808390i	−0.914673	+	0.404195i	$0.867552\pi$
$524$	−8972.62	−0.748036
$525$	0	0
$526$	11940.8	0.989813
$527$	− 10110.5i	− 0.835712i
$528$	− 130.754i	− 0.0107771i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−6827.78	−0.558004
$532$	6507.33i	0.530317i
$533$	− 862.033i	− 0.0700540i
$534$	−3536.05	−0.286554
$535$	0	0
$536$	7457.34	0.600948
$537$	38508.6i	3.09454i
$538$	− 7828.67i	− 0.627357i
$539$	177.517	0.0141859
$540$	0	0
$541$	6793.28	0.539863	0.269932	−	0.962880i	$-0.412999\pi$
$541$	6793.28	0.539863	0.269932	+	0.962880i	$0.412999\pi$
$542$	156.173i	0.0123768i
$543$	− 1458.41i	− 0.115260i
$544$	−2363.97	−0.186313
$545$	0	0
$546$	1597.06	0.125179
$547$	− 5584.91i	− 0.436551i	−0.975887	−	0.218276i	$-0.929957\pi$
$547$	− 5584.91i	− 0.436551i	0.975887	−	0.218276i	$-0.0700432\pi$
$548$	− 5011.08i	− 0.390626i
$549$	−7796.49	−0.606095
$550$	0	0
$551$	1985.21	0.153490
$552$	− 1521.94i	− 0.117351i
$553$	21860.3i	1.68100i
$554$	152.102	0.0116646
$555$	0	0
$556$	11260.8	0.858925
$557$	4172.24i	0.317385i	0.987328	+	0.158693i	$0.0507279\pi$
$557$	4172.24i	0.317385i	−0.987328	+	0.158693i	$0.949272\pi$
$558$	11336.5i	0.860059i
$559$	−228.461	−0.0172860
$560$	0	0
$561$	603.707	0.0454341
$562$	− 7705.34i	− 0.578346i
$563$	− 7734.17i	− 0.578963i	−0.957184	−	0.289482i	$-0.906517\pi$
$563$	− 7734.17i	− 0.578963i	0.957184	−	0.289482i	$-0.0934829\pi$
$564$	−1371.37	−0.102385
$565$	0	0
$566$	−3100.26	−0.230236
$567$	− 3016.47i	− 0.223421i
$568$	2106.69i	0.155625i
$569$	−1392.55	−0.102599	−0.0512995	−	0.998683i	$-0.516336\pi$
$569$	−1392.55	−0.102599	−0.0512995	+	0.998683i	$0.516336\pi$
$570$	0	0
$571$	16996.1	1.24565	0.622823	−	0.782363i	$-0.285986\pi$
$571$	16996.1	1.24565	0.622823	+	0.782363i	$0.285986\pi$
$572$	16.6883i	0.00121988i
$573$	13830.0i	1.00830i
$574$	9334.11	0.678743
$575$	0	0
$576$	2650.63	0.191741
$577$	− 6849.23i	− 0.494172i	−0.968994	−	0.247086i	$-0.920527\pi$
$577$	− 6849.23i	− 0.494172i	0.968994	−	0.247086i	$-0.0794730\pi$
$578$	− 1088.76i	− 0.0783500i
$579$	−24437.4	−1.75403
$580$	0	0
$581$	4455.29	0.318135
$582$	− 10401.5i	− 0.740817i
$583$	423.467i	0.0300827i
$584$	−7203.21	−0.510395
$585$	0	0
$586$	8472.01	0.597228
$587$	6701.08i	0.471181i	0.971852	+	0.235590i	$0.0757024\pi$
$587$	6701.08i	0.471181i	−0.971852	+	0.235590i	$0.924298\pi$
$588$	5944.62i	0.416925i
$589$	9738.85	0.681294
$590$	0	0
$591$	−15032.1	−1.04625
$592$	− 3227.49i	− 0.224070i
$593$	− 9898.21i	− 0.685448i	−0.939436	−	0.342724i	$-0.888650\pi$
$593$	− 9898.21i	− 0.685448i	0.939436	−	0.342724i	$-0.111350\pi$
$594$	−235.620	−0.0162754
$595$	0	0
$596$	9625.84	0.661560
$597$	− 39545.5i	− 2.71104i
$598$	194.247i	0.0132832i
$599$	−13149.2	−0.896934	−0.448467	−	0.893799i	$-0.648030\pi$
$599$	−13149.2	−0.896934	−0.448467	+	0.893799i	$0.648030\pi$
$600$	0	0
$601$	27142.4	1.84220	0.921100	−	0.389326i	$-0.127292\pi$
$601$	27142.4	1.84220	0.921100	+	0.389326i	$0.127292\pi$
$602$	− 2473.79i	− 0.167482i
$603$	− 38606.7i	− 2.60728i
$604$	−8742.34	−0.588942
$605$	0	0
$606$	845.952	0.0567070
$607$	− 6708.40i	− 0.448576i	−0.974523	−	0.224288i	$-0.927994\pi$
$607$	− 6708.40i	− 0.448576i	0.974523	−	0.224288i	$-0.0720056\pi$
$608$	− 2277.07i	− 0.151887i
$609$	5275.63	0.351033
$610$	0	0
$611$	175.029	0.0115891
$612$	12238.3i	0.808340i
$613$	17755.6i	1.16989i	0.811074	+	0.584943i	$0.198883\pi$
$613$	17755.6i	1.16989i	−0.811074	+	0.584943i	$0.801117\pi$
$614$	20156.2	1.32482
$615$	0	0
$616$	−180.701	−0.0118192
$617$	14555.4i	0.949722i	0.880061	+	0.474861i	$0.157502\pi$
$617$	14555.4i	0.949722i	−0.880061	+	0.474861i	$0.842498\pi$
$618$	6006.52i	0.390967i
$619$	−29452.5	−1.91243	−0.956215	−	0.292665i	$-0.905458\pi$
$619$	−29452.5	−1.91243	−0.956215	+	0.292665i	$0.905458\pi$
$620$	0	0
$621$	−2742.55	−0.177222
$622$	− 5305.94i	− 0.342040i
$623$	4886.80i	0.314262i
$624$	−558.850	−0.0358524
$625$	0	0
$626$	13421.4	0.856909
$627$	581.516i	0.0370391i
$628$	− 4296.72i	− 0.273022i
$629$	14901.8	0.944629
$630$	0	0
$631$	9911.19	0.625290	0.312645	−	0.949870i	$-0.398785\pi$
$631$	9911.19	0.625290	0.312645	+	0.949870i	$0.398785\pi$
$632$	− 7649.46i	− 0.481454i
$633$	26482.7i	1.66286i
$634$	7355.77	0.460781
$635$	0	0
$636$	−14180.9	−0.884133
$637$	− 758.720i	− 0.0471924i
$638$	55.1271i	0.00342085i
$639$	10906.4	0.675194
$640$	0	0
$641$	−13548.5	−0.834844	−0.417422	−	0.908713i	$-0.637066\pi$
$641$	−13548.5	−0.834844	−0.417422	+	0.908713i	$0.637066\pi$
$642$	− 25054.6i	− 1.54023i
$643$	− 2254.31i	− 0.138260i	−0.997608	−	0.0691302i	$-0.977978\pi$
$643$	− 2254.31i	− 0.138260i	0.997608	−	0.0691302i	$-0.0220224\pi$
$644$	−2103.31	−0.128699
$645$	0	0
$646$	10513.5	0.640325
$647$	− 17967.1i	− 1.09175i	−0.837868	−	0.545874i	$-0.816198\pi$
$647$	− 17967.1i	− 1.09175i	0.837868	−	0.545874i	$-0.183802\pi$
$648$	1055.54i	0.0639897i
$649$	−162.879	−0.00985141
$650$	0	0
$651$	25880.6	1.55813
$652$	10825.2i	0.650226i
$653$	− 365.239i	− 0.0218881i	−0.999940	−	0.0109440i	$-0.996516\pi$
$653$	− 365.239i	− 0.0218881i	0.999940	−	0.0109440i	$-0.00348367\pi$
$654$	13422.1	0.802517
$655$	0	0
$656$	−3266.24	−0.194398
$657$	37291.1i	2.21440i
$658$	1895.22i	0.112285i
$659$	−29229.1	−1.72778	−0.863889	−	0.503683i	$-0.831978\pi$
$659$	−29229.1	−1.72778	−0.863889	+	0.503683i	$0.831978\pi$
$660$	0	0
$661$	−23681.1	−1.39348	−0.696740	−	0.717324i	$-0.745367\pi$
$661$	−23681.1	−1.39348	−0.696740	+	0.717324i	$0.745367\pi$
$662$	− 19612.8i	− 1.15147i
$663$	− 2580.28i	− 0.151146i
$664$	−1559.01	−0.0911167
$665$	0	0
$666$	−16708.8	−0.972150
$667$	641.664i	0.0372494i
$668$	− 1822.19i	− 0.105543i
$669$	−15692.7	−0.906900
$670$	0	0
$671$	−185.988	−0.0107004
$672$	− 6051.24i	− 0.347369i
$673$	16016.2i	0.917351i	0.888604	+	0.458676i	$0.151676\pi$
$673$	16016.2i	0.917351i	−0.888604	+	0.458676i	$0.848324\pi$
$674$	14286.8	0.816480
$675$	0	0
$676$	−8716.67	−0.495942
$677$	− 24082.5i	− 1.36715i	−0.729878	−	0.683577i	$-0.760423\pi$
$677$	− 24082.5i	− 1.36715i	0.729878	−	0.683577i	$-0.239577\pi$
$678$	205.951i	0.0116659i
$679$	−14374.8	−0.812451
$680$	0	0
$681$	−36960.1	−2.07975
$682$	270.437i	0.0151841i
$683$	24224.1i	1.35711i	0.734549	+	0.678556i	$0.237394\pi$
$683$	24224.1i	1.35711i	−0.734549	+	0.678556i	$0.762606\pi$
$684$	−11788.4	−0.658980
$685$	0	0
$686$	−7467.94	−0.415637
$687$	44217.9i	2.45563i
$688$	865.639i	0.0479683i
$689$	1809.92	0.100076
$690$	0	0
$691$	13948.7	0.767919	0.383960	−	0.923350i	$-0.374560\pi$
$691$	13948.7	0.767919	0.383960	+	0.923350i	$0.374560\pi$
$692$	− 5370.55i	− 0.295026i
$693$	935.491i	0.0512790i
$694$	10320.4	0.564492
$695$	0	0
$696$	−1846.07	−0.100539
$697$	− 15080.6i	− 0.819540i
$698$	− 17424.1i	− 0.944858i
$699$	−37059.7	−2.00533
$700$	0	0
$701$	−33334.1	−1.79602	−0.898011	−	0.439973i	$-0.854988\pi$
$701$	−33334.1	−1.79602	−0.898011	+	0.439973i	$0.854988\pi$
$702$	1007.05i	0.0541436i
$703$	14354.0i	0.770087i
$704$	63.2317	0.00338514
$705$	0	0
$706$	−14234.6	−0.758819
$707$	− 1169.10i	− 0.0621903i
$708$	− 5454.43i	− 0.289534i
$709$	7107.92	0.376507	0.188253	−	0.982120i	$-0.439717\pi$
$709$	7107.92	0.376507	0.188253	+	0.982120i	$0.439717\pi$
$710$	0	0
$711$	−39601.3	−2.08884
$712$	− 1710.01i	− 0.0900075i
$713$	3147.81i	0.165338i
$714$	27939.4	1.46443
$715$	0	0
$716$	−18622.5	−0.972007
$717$	− 6450.34i	− 0.335972i
$718$	− 7097.45i	− 0.368906i
$719$	13003.1	0.674456	0.337228	−	0.941423i	$-0.390511\pi$
$719$	13003.1	0.674456	0.337228	+	0.941423i	$0.390511\pi$
$720$	0	0
$721$	8300.98	0.428772
$722$	− 3590.92i	− 0.185097i
$723$	− 34780.4i	− 1.78907i
$724$	705.279	0.0362037
$725$	0	0
$726$	22002.3	1.12477
$727$	5524.86i	0.281851i	0.990020	+	0.140926i	$0.0450078\pi$
$727$	5524.86i	0.281851i	−0.990020	+	0.140926i	$0.954992\pi$
$728$	772.328i	0.0393192i
$729$	32094.4	1.63057
$730$	0	0
$731$	−3996.77	−0.202224
$732$	− 6228.29i	− 0.314487i
$733$	− 22106.6i	− 1.11395i	−0.830529	−	0.556976i	$-0.811962\pi$
$733$	− 22106.6i	− 1.11395i	0.830529	−	0.556976i	$-0.188038\pi$
$734$	−160.022	−0.00804703
$735$	0	0
$736$	736.000	0.0368605
$737$	− 920.977i	− 0.0460307i
$738$	16909.3i	0.843416i
$739$	6619.82	0.329518	0.164759	−	0.986334i	$-0.447315\pi$
$739$	6619.82	0.329518	0.164759	+	0.986334i	$0.447315\pi$
$740$	0	0
$741$	2485.44	0.123218
$742$	19597.9i	0.969625i
$743$	− 18894.1i	− 0.932919i	−0.884543	−	0.466460i	$-0.845529\pi$
$743$	− 18894.1i	− 0.932919i	0.884543	−	0.466460i	$-0.154471\pi$
$744$	−9056.27	−0.446262
$745$	0	0
$746$	6422.17	0.315191
$747$	8071.03i	0.395320i
$748$	291.949i	0.0142710i
$749$	−34625.3	−1.68916
$750$	0	0
$751$	4931.32	0.239609	0.119805	−	0.992797i	$-0.461773\pi$
$751$	4931.32	0.239609	0.119805	+	0.992797i	$0.461773\pi$
$752$	− 663.185i	− 0.0321594i
$753$	22841.3i	1.10542i
$754$	235.617	0.0113802
$755$	0	0
$756$	−10904.4	−0.524589
$757$	6248.31i	0.299998i	0.988686	+	0.149999i	$0.0479271\pi$
$757$	6248.31i	0.299998i	−0.988686	+	0.149999i	$0.952073\pi$
$758$	16355.1i	0.783701i
$759$	−187.959	−0.00898875
$760$	0	0
$761$	−18867.0	−0.898723	−0.449361	−	0.893350i	$-0.648348\pi$
$761$	−18867.0	−0.898723	−0.449361	+	0.893350i	$0.648348\pi$
$762$	10465.0i	0.497514i
$763$	− 18549.3i	− 0.880117i
$764$	−6688.11	−0.316711
$765$	0	0
$766$	25853.1	1.21946
$767$	696.156i	0.0327728i
$768$	2117.48i	0.0994895i
$769$	4752.05	0.222839	0.111420	−	0.993773i	$-0.464460\pi$
$769$	4752.05	0.222839	0.111420	+	0.993773i	$0.464460\pi$
$770$	0	0
$771$	4850.92	0.226591
$772$	− 11817.8i	− 0.550947i
$773$	− 7050.12i	− 0.328040i	−0.986457	−	0.164020i	$-0.947554\pi$
$773$	− 7050.12i	− 0.328040i	0.986457	−	0.164020i	$-0.0524462\pi$
$774$	4481.42	0.208115
$775$	0	0
$776$	5030.10	0.232693
$777$	38145.2i	1.76120i
$778$	− 9469.32i	− 0.436364i
$779$	14526.3	0.668111
$780$	0	0
$781$	260.175	0.0119204
$782$	3398.21i	0.155396i
$783$	3326.64i	0.151832i
$784$	−2874.78	−0.130958
$785$	0	0
$786$	−37108.1	−1.68397
$787$	19524.7i	0.884345i	0.896930	+	0.442173i	$0.145792\pi$
$787$	19524.7i	0.884345i	−0.896930	+	0.442173i	$0.854208\pi$
$788$	− 7269.41i	− 0.328632i
$789$	49383.4	2.22826
$790$	0	0
$791$	284.623	0.0127940
$792$	− 327.351i	− 0.0146868i
$793$	794.925i	0.0355972i
$794$	19877.6	0.888450
$795$	0	0
$796$	19124.0	0.851546
$797$	− 25078.7i	− 1.11459i	−0.830313	−	0.557297i	$-0.811838\pi$
$797$	− 25078.7i	− 1.11459i	0.830313	−	0.557297i	$-0.188162\pi$
$798$	26912.4i	1.19384i
$799$	3062.01	0.135577
$800$	0	0
$801$	−8852.74	−0.390507
$802$	− 670.592i	− 0.0295255i
$803$	889.592i	0.0390947i
$804$	30841.3	1.35285
$805$	0	0
$806$	1155.86	0.0505131
$807$	− 32377.0i	− 1.41230i
$808$	409.097i	0.0178119i
$809$	−11722.0	−0.509425	−0.254712	−	0.967017i	$-0.581981\pi$
$809$	−11722.0	−0.509425	−0.254712	+	0.967017i	$0.581981\pi$
$810$	0	0
$811$	−3540.61	−0.153302	−0.0766509	−	0.997058i	$-0.524423\pi$
$811$	−3540.61	−0.153302	−0.0766509	+	0.997058i	$0.524423\pi$
$812$	2551.26i	0.110261i
$813$	645.885i	0.0278625i
$814$	−398.594	−0.0171630
$815$	0	0
$816$	−9776.67	−0.419427
$817$	− 3849.85i	− 0.164858i
$818$	− 13121.7i	− 0.560867i
$819$	3998.35	0.170591
$820$	0	0
$821$	−43675.7	−1.85663	−0.928315	−	0.371794i	$-0.878743\pi$
$821$	−43675.7	−1.85663	−0.928315	+	0.371794i	$0.878743\pi$
$822$	− 20724.3i	− 0.879373i
$823$	28732.5i	1.21695i	0.793572	+	0.608476i	$0.208219\pi$
$823$	28732.5i	1.21695i	−0.793572	+	0.608476i	$0.791781\pi$
$824$	−2904.72	−0.122804
$825$	0	0
$826$	−7537.99	−0.317531
$827$	33810.2i	1.42164i	0.703373	+	0.710821i	$0.251676\pi$
$827$	33810.2i	1.42164i	−0.703373	+	0.710821i	$0.748324\pi$
$828$	− 3810.28i	− 0.159923i
$829$	34664.8	1.45230	0.726150	−	0.687536i	$-0.241308\pi$
$829$	34664.8	1.45230	0.726150	+	0.687536i	$0.241308\pi$
$830$	0	0
$831$	629.050	0.0262593
$832$	− 270.257i	− 0.0112614i
$833$	− 13273.3i	− 0.552090i
$834$	46571.1	1.93360
$835$	0	0
$836$	−281.217	−0.0116341
$837$	16319.5i	0.673936i
$838$	− 14233.4i	− 0.586735i
$839$	−11115.4	−0.457384	−0.228692	−	0.973499i	$-0.573445\pi$
$839$	−11115.4	−0.457384	−0.228692	+	0.973499i	$0.573445\pi$
$840$	0	0
$841$	−23610.7	−0.968087
$842$	19598.9i	0.802163i
$843$	− 31867.0i	− 1.30197i
$844$	−12806.9	−0.522312
$845$	0	0
$846$	−3433.31	−0.139527
$847$	− 30407.1i	− 1.23353i
$848$	− 6857.79i	− 0.277709i
$849$	−12821.8	−0.518306
$850$	0	0
$851$	−4639.52	−0.186887
$852$	8712.64i	0.350341i
$853$	− 23695.9i	− 0.951153i	−0.879674	−	0.475577i	$-0.842239\pi$
$853$	− 23695.9i	− 0.951153i	0.879674	−	0.475577i	$-0.157761\pi$
$854$	−8607.46	−0.344896
$855$	0	0
$856$	12116.3	0.483791
$857$	− 20605.9i	− 0.821336i	−0.911785	−	0.410668i	$-0.865295\pi$
$857$	− 20605.9i	− 0.821336i	0.911785	−	0.410668i	$-0.134705\pi$
$858$	69.0177i	0.00274618i
$859$	43187.1	1.71540	0.857698	−	0.514154i	$-0.171894\pi$
$859$	43187.1	1.71540	0.857698	+	0.514154i	$0.171894\pi$
$860$	0	0
$861$	38603.1	1.52798
$862$	− 21046.6i	− 0.831613i
$863$	47551.7i	1.87564i	0.347117	+	0.937822i	$0.387161\pi$
$863$	47551.7i	1.87564i	−0.347117	+	0.937822i	$0.612839\pi$
$864$	3815.72	0.150247
$865$	0	0
$866$	−9541.94	−0.374421
$867$	− 4502.77i	− 0.176381i
$868$	12515.7i	0.489414i
$869$	−944.704	−0.0368779
$870$	0	0
$871$	−3936.32	−0.153131
$872$	6490.85i	0.252073i
$873$	− 26040.9i	− 1.00956i
$874$	−3273.29	−0.126683
$875$	0	0
$876$	−29790.3	−1.14900
$877$	8116.63i	0.312519i	0.987716	+	0.156260i	$0.0499436\pi$
$877$	8116.63i	0.312519i	−0.987716	+	0.156260i	$0.950056\pi$
$878$	− 20462.4i	− 0.786528i
$879$	35037.7	1.34447
$880$	0	0
$881$	24256.1	0.927594	0.463797	−	0.885942i	$-0.346487\pi$
$881$	24256.1	0.927594	0.463797	+	0.885942i	$0.346487\pi$
$882$	14882.8i	0.568174i
$883$	46859.3i	1.78589i	0.450167	+	0.892945i	$0.351365\pi$
$883$	46859.3i	1.78589i	−0.450167	+	0.892945i	$0.648635\pi$
$884$	1247.81	0.0474755
$885$	0	0
$886$	21016.2	0.796899
$887$	− 14372.4i	− 0.544057i	−0.962289	−	0.272028i	$-0.912306\pi$
$887$	− 14372.4i	− 0.544057i	0.962289	−	0.272028i	$-0.0876945\pi$
$888$	− 13347.9i	− 0.504423i
$889$	14462.5	0.545621
$890$	0	0
$891$	130.358	0.00490142
$892$	− 7588.91i	− 0.284860i
$893$	2949.45i	0.110526i
$894$	39809.6	1.48930
$895$	0	0
$896$	2926.34	0.109110
$897$	803.347i	0.0299030i
$898$	− 14461.4i	− 0.537398i
$899$	3818.21	0.141651
$900$	0	0
$901$	31663.3	1.17076
$902$	403.378i	0.0148903i
$903$	− 10230.8i	− 0.377033i
$904$	−99.5965	−0.00366431
$905$	0	0
$906$	−36155.7	−1.32582
$907$	− 17332.1i	− 0.634512i	−0.948340	−	0.317256i	$-0.897239\pi$
$907$	− 17332.1i	− 0.634512i	0.948340	−	0.317256i	$-0.102761\pi$
$908$	− 17873.7i	− 0.653258i
$909$	2117.90	0.0772787
$910$	0	0
$911$	44304.9	1.61129	0.805646	−	0.592397i	$-0.201818\pi$
$911$	44304.9	1.61129	0.805646	+	0.592397i	$0.201818\pi$
$912$	− 9417.30i	− 0.341927i
$913$	192.537i	0.00697926i
$914$	−4837.49	−0.175066
$915$	0	0
$916$	−21383.5	−0.771322
$917$	51283.2i	1.84680i
$918$	17617.7i	0.633410i
$919$	4764.32	0.171013	0.0855063	−	0.996338i	$-0.472749\pi$
$919$	4764.32	0.171013	0.0855063	+	0.996338i	$0.472749\pi$
$920$	0	0
$921$	83359.9	2.98241
$922$	− 22685.6i	− 0.810315i
$923$	− 1112.01i	− 0.0396556i
$924$	−747.325	−0.0266074
$925$	0	0
$926$	−23571.8	−0.836519
$927$	15037.7i	0.532799i
$928$	− 892.750i	− 0.0315797i
$929$	23226.2	0.820267	0.410134	−	0.912025i	$-0.365482\pi$
$929$	23226.2	0.820267	0.410134	+	0.912025i	$0.365482\pi$
$930$	0	0
$931$	12785.3	0.450078
$932$	− 17921.9i	− 0.629882i
$933$	− 21943.8i	− 0.769996i
$934$	−32589.1	−1.14170
$935$	0	0
$936$	−1399.12	−0.0488587
$937$	− 23752.5i	− 0.828133i	−0.910247	−	0.414066i	$-0.864108\pi$
$937$	− 23752.5i	− 0.828133i	0.910247	−	0.414066i	$-0.135892\pi$
$938$	− 42622.5i	− 1.48366i
$939$	55506.7	1.92907
$940$	0	0
$941$	−23210.4	−0.804079	−0.402040	−	0.915622i	$-0.631699\pi$
$941$	−23210.4	−0.804079	−0.402040	+	0.915622i	$0.631699\pi$
$942$	− 17769.9i	− 0.614624i
$943$	4695.21i	0.162139i
$944$	2637.73	0.0909436
$945$	0	0
$946$	106.906	0.00367422
$947$	− 7778.59i	− 0.266917i	−0.991054	−	0.133458i	$-0.957392\pi$
$947$	− 7778.59i	− 0.266917i	0.991054	−	0.133458i	$-0.0426082\pi$
$948$	− 31635.9i	− 1.08385i
$949$	3802.18	0.130057
$950$	0	0
$951$	30421.3	1.03731
$952$	13511.3i	0.459983i
$953$	− 3979.52i	− 0.135267i	−0.997710	−	0.0676334i	$-0.978455\pi$
$953$	− 3979.52i	− 0.135267i	0.997710	−	0.0676334i	$-0.0215448\pi$
$954$	−35502.8	−1.20487
$955$	0	0
$956$	3119.34	0.105530
$957$	227.989i	0.00770098i
$958$	1570.55i	0.0529667i
$959$	−28640.9	−0.964405
$960$	0	0
$961$	−11060.0	−0.371254
$962$	1703.62i	0.0570964i
$963$	− 62726.0i	− 2.09898i
$964$	16819.6	0.561952
$965$	0	0
$966$	−8698.66	−0.289726
$967$	17325.9i	0.576178i	0.957604	+	0.288089i	$0.0930198\pi$
$967$	17325.9i	0.576178i	−0.957604	+	0.288089i	$0.906980\pi$
$968$	10640.2i	0.353294i
$969$	43480.9	1.44149
$970$	0	0
$971$	20183.0	0.667048	0.333524	−	0.942742i	$-0.391762\pi$
$971$	20183.0	0.667048	0.333524	+	0.942742i	$0.391762\pi$
$972$	17243.4i	0.569016i
$973$	− 64361.0i	− 2.12058i
$974$	−17428.6	−0.573355
$975$	0	0
$976$	3011.96	0.0987814
$977$	47185.7i	1.54514i	0.634928	+	0.772571i	$0.281030\pi$
$977$	47185.7i	1.54514i	−0.634928	+	0.772571i	$0.718970\pi$
$978$	44769.8i	1.46378i
$979$	−211.185	−0.00689430
$980$	0	0
$981$	33603.2	1.09365
$982$	− 21886.3i	− 0.711221i
$983$	7347.90i	0.238415i	0.992869	+	0.119207i	$0.0380353\pi$
$983$	7347.90i	0.238415i	−0.992869	+	0.119207i	$0.961965\pi$
$984$	−13508.2	−0.437627
$985$	0	0
$986$	4121.94	0.133133
$987$	7838.07i	0.252775i
$988$	1201.94i	0.0387033i
$989$	1244.36	0.0400083
$990$	0	0
$991$	34327.0	1.10034	0.550168	−	0.835054i	$-0.314564\pi$
$991$	34327.0	1.10034	0.550168	+	0.835054i	$0.314564\pi$
$992$	− 4379.56i	− 0.140172i
$993$	− 81112.6i	− 2.59218i
$994$	12040.8	0.384217
$995$	0	0
$996$	−6447.62	−0.205121
$997$	37611.0i	1.19474i	0.801967	+	0.597368i	$0.203787\pi$
$997$	37611.0i	1.19474i	−0.801967	+	0.597368i	$0.796213\pi$
$998$	11203.7i	0.355358i
$999$	−24053.1	−0.761770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.r.599.10		10
5.2	odd	4		1150.4.a.q.1.5	✓	5
5.3	odd	4		1150.4.a.v.1.1	yes	5
5.4	even	2	inner	1150.4.b.r.599.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.q.1.5	✓	5	5.2	odd	4
1150.4.a.v.1.1	yes	5	5.3	odd	4
1150.4.b.r.599.1		10	5.4	even	2	inner
1150.4.b.r.599.10		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +8.27140i q^{3} -4.00000 q^{4} -16.5428 q^{6} +22.8621i q^{7} -8.00000i q^{8} -41.4161 q^{9} -0.987996 q^{11} -33.0856i q^{12} +4.22276i q^{13} -45.7241 q^{14} +16.0000 q^{16} +73.8741i q^{17} -82.8322i q^{18} -71.1586 q^{19} -189.101 q^{21} -1.97599i q^{22} -23.0000i q^{23} +66.1712 q^{24} -8.44552 q^{26} -119.241i q^{27} -91.4483i q^{28} -27.8984 q^{29} -136.861 q^{31} +32.0000i q^{32} -8.17211i q^{33} -147.748 q^{34} +165.664 q^{36} -201.718i q^{37} -142.317i q^{38} -34.9281 q^{39} -204.140 q^{41} -378.203i q^{42} +54.1024i q^{43} +3.95198 q^{44} +46.0000 q^{46} -41.4490i q^{47} +132.342i q^{48} -179.674 q^{49} -611.042 q^{51} -16.8910i q^{52} -428.612i q^{53} +238.483 q^{54} +182.897 q^{56} -588.581i q^{57} -55.7969i q^{58} +164.858 q^{59} +188.248 q^{61} -273.722i q^{62} -946.857i q^{63} -64.0000 q^{64} +16.3442 q^{66} +932.167i q^{67} -295.496i q^{68} +190.242 q^{69} -263.336 q^{71} +331.329i q^{72} -900.401i q^{73} +403.437 q^{74} +284.634 q^{76} -22.5876i q^{77} -69.8563i q^{78} +956.182 q^{79} -131.942 q^{81} -408.279i q^{82} -194.877i q^{83} +756.405 q^{84} -108.205 q^{86} -230.759i q^{87} +7.90397i q^{88} +213.751 q^{89} -96.5410 q^{91} +92.0000i q^{92} -1132.03i q^{93} +82.8981 q^{94} -264.685 q^{96} +628.762i q^{97} -359.348i q^{98} +40.9189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 40 q^{4} + 20 q^{6} - 168 q^{9} - 52 q^{11} + 12 q^{14} + 160 q^{16} - 148 q^{19} - 176 q^{21} - 80 q^{24} - 244 q^{26} - 74 q^{29} + 440 q^{31} - 924 q^{34} + 672 q^{36} - 390 q^{39} + 224 q^{41}+ \cdots - 4794 q^{99}+O(q^{100}) \) 10 * q - 40 * q^4 + 20 * q^6 - 168 * q^9 - 52 * q^11 + 12 * q^14 + 160 * q^16 - 148 * q^19 - 176 * q^21 - 80 * q^24 - 244 * q^26 - 74 * q^29 + 440 * q^31 - 924 * q^34 + 672 * q^36 - 390 * q^39 + 224 * q^41 + 208 * q^44 + 460 * q^46 - 364 * q^49 + 1126 * q^51 - 1472 * q^54 - 48 * q^56 + 362 * q^59 + 3148 * q^61 - 640 * q^64 + 1564 * q^66 - 230 * q^69 - 60 * q^71 - 880 * q^74 + 592 * q^76 - 1218 * q^79 + 2906 * q^81 + 704 * q^84 + 388 * q^86 + 3326 * q^89 + 722 * q^91 - 1480 * q^94 + 320 * q^96 - 4794 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more