LMFDB - Embedded newform 2175.4.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,4,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

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:	yes
Analytic conductor:	$128.329154262$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 $ x^6 - x^5 - 49x^4 + 27x^3 + 692x^2 - 82x - 2588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.54861$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.54861 q^{2} -3.00000 q^{3} +4.59262 q^{4} -10.6458 q^{6} +21.2667 q^{7} -12.0915 q^{8} +9.00000 q^{9} -40.9431 q^{11} -13.7779 q^{12} +8.89857 q^{13} +75.4671 q^{14} -79.6488 q^{16} +7.68473 q^{17} +31.9375 q^{18} +40.7787 q^{19} -63.8000 q^{21} -145.291 q^{22} +80.7285 q^{23} +36.2744 q^{24} +31.5775 q^{26} -27.0000 q^{27} +97.6698 q^{28} +29.0000 q^{29} +82.1098 q^{31} -185.911 q^{32} +122.829 q^{33} +27.2701 q^{34} +41.3336 q^{36} -223.283 q^{37} +144.708 q^{38} -26.6957 q^{39} -274.422 q^{41} -226.401 q^{42} -53.7872 q^{43} -188.036 q^{44} +286.474 q^{46} -17.0236 q^{47} +238.946 q^{48} +109.272 q^{49} -23.0542 q^{51} +40.8678 q^{52} +23.0475 q^{53} -95.8124 q^{54} -257.145 q^{56} -122.336 q^{57} +102.910 q^{58} -399.574 q^{59} +388.308 q^{61} +291.376 q^{62} +191.400 q^{63} -22.5340 q^{64} +435.873 q^{66} -399.434 q^{67} +35.2930 q^{68} -242.186 q^{69} +92.2884 q^{71} -108.823 q^{72} -979.407 q^{73} -792.343 q^{74} +187.281 q^{76} -870.724 q^{77} -94.7326 q^{78} +62.2586 q^{79} +81.0000 q^{81} -973.817 q^{82} -163.597 q^{83} -293.009 q^{84} -190.870 q^{86} -87.0000 q^{87} +495.062 q^{88} +1371.94 q^{89} +189.243 q^{91} +370.755 q^{92} -246.330 q^{93} -60.4101 q^{94} +557.732 q^{96} -1263.64 q^{97} +387.763 q^{98} -368.488 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) $ 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * q^99

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 $

$2$	3.54861	1.25462	0.627311	−	0.778769i	$-0.284155\pi$
$2$	3.54861	1.25462	0.627311	+	0.778769i	$0.284155\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.59262	0.574078
$5$	0	0
$5$	0	0
$6$	−10.6458	−0.724357
$7$	21.2667	1.14829	0.574146	−	0.818753i	$-0.305334\pi$
$7$	21.2667	1.14829	0.574146	+	0.818753i	$0.305334\pi$
$8$	−12.0915	−0.534372
$9$	9.00000	0.333333
$10$	0	0
$11$	−40.9431	−1.12226	−0.561128	−	0.827729i	$-0.689632\pi$
$11$	−40.9431	−1.12226	−0.561128	+	0.827729i	$0.689632\pi$
$12$	−13.7779	−0.331444
$13$	8.89857	0.189848	0.0949238	−	0.995485i	$-0.469739\pi$
$13$	8.89857	0.189848	0.0949238	+	0.995485i	$0.469739\pi$
$14$	75.4671	1.44067
$15$	0	0
$16$	−79.6488	−1.24451
$17$	7.68473	0.109636	0.0548182	−	0.998496i	$-0.482542\pi$
$17$	7.68473	0.109636	0.0548182	+	0.998496i	$0.482542\pi$
$18$	31.9375	0.418207
$19$	40.7787	0.492383	0.246192	−	0.969221i	$-0.420821\pi$
$19$	40.7787	0.492383	0.246192	+	0.969221i	$0.420821\pi$
$20$	0	0
$21$	−63.8000	−0.662967
$22$	−145.291	−1.40801
$23$	80.7285	0.731872	0.365936	−	0.930640i	$-0.380749\pi$
$23$	80.7285	0.731872	0.365936	+	0.930640i	$0.380749\pi$
$24$	36.2744	0.308520
$25$	0	0
$26$	31.5775	0.238187
$27$	−27.0000	−0.192450
$28$	97.6698	0.659209
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	82.1098	0.475721	0.237861	−	0.971299i	$-0.423554\pi$
$31$	82.1098	0.475721	0.237861	+	0.971299i	$0.423554\pi$
$32$	−185.911	−1.02702
$33$	122.829	0.647935
$34$	27.2701	0.137552
$35$	0	0
$36$	41.3336	0.191359
$37$	−223.283	−0.992093	−0.496047	−	0.868296i	$-0.665215\pi$
$37$	−223.283	−0.992093	−0.496047	+	0.868296i	$0.665215\pi$
$38$	144.708	0.617755
$39$	−26.6957	−0.109609
$40$	0	0
$41$	−274.422	−1.04531	−0.522653	−	0.852545i	$-0.675058\pi$
$41$	−274.422	−1.04531	−0.522653	+	0.852545i	$0.675058\pi$
$42$	−226.401	−0.831774
$43$	−53.7872	−0.190755	−0.0953775	−	0.995441i	$-0.530406\pi$
$43$	−53.7872	−0.190755	−0.0953775	+	0.995441i	$0.530406\pi$
$44$	−188.036	−0.644262
$45$	0	0
$46$	286.474	0.918223
$47$	−17.0236	−0.0528329	−0.0264165	−	0.999651i	$-0.508410\pi$
$47$	−17.0236	−0.0528329	−0.0264165	+	0.999651i	$0.508410\pi$
$48$	238.946	0.718520
$49$	109.272	0.318577
$50$	0	0
$51$	−23.0542	−0.0632986
$52$	40.8678	0.108987
$53$	23.0475	0.0597325	0.0298662	−	0.999554i	$-0.490492\pi$
$53$	23.0475	0.0597325	0.0298662	+	0.999554i	$0.490492\pi$
$54$	−95.8124	−0.241452
$55$	0	0
$56$	−257.145	−0.613615
$57$	−122.336	−0.284278
$58$	102.910	0.232978
$59$	−399.574	−0.881697	−0.440848	−	0.897582i	$-0.645322\pi$
$59$	−399.574	−0.881697	−0.440848	+	0.897582i	$0.645322\pi$
$60$	0	0
$61$	388.308	0.815046	0.407523	−	0.913195i	$-0.366393\pi$
$61$	388.308	0.815046	0.407523	+	0.913195i	$0.366393\pi$
$62$	291.376	0.596851
$63$	191.400	0.382764
$64$	−22.5340	−0.0440118
$65$	0	0
$66$	435.873	0.812913
$67$	−399.434	−0.728337	−0.364169	−	0.931333i	$-0.618647\pi$
$67$	−399.434	−0.728337	−0.364169	+	0.931333i	$0.618647\pi$
$68$	35.2930	0.0629398
$69$	−242.186	−0.422547
$70$	0	0
$71$	92.2884	0.154262	0.0771311	−	0.997021i	$-0.475424\pi$
$71$	92.2884	0.154262	0.0771311	+	0.997021i	$0.475424\pi$
$72$	−108.823	−0.178124
$73$	−979.407	−1.57029	−0.785144	−	0.619314i	$-0.787411\pi$
$73$	−979.407	−1.57029	−0.785144	+	0.619314i	$0.787411\pi$
$74$	−792.343	−1.24470
$75$	0	0
$76$	187.281	0.282666
$77$	−870.724	−1.28868
$78$	−94.7326	−0.137517
$79$	62.2586	0.0886664	0.0443332	−	0.999017i	$-0.485884\pi$
$79$	62.2586	0.0886664	0.0443332	+	0.999017i	$0.485884\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−973.817	−1.31146
$83$	−163.597	−0.216350	−0.108175	−	0.994132i	$-0.534501\pi$
$83$	−163.597	−0.216350	−0.108175	+	0.994132i	$0.534501\pi$
$84$	−293.009	−0.380595
$85$	0	0
$86$	−190.870	−0.239326
$87$	−87.0000	−0.107211
$88$	495.062	0.599702
$89$	1371.94	1.63399	0.816994	−	0.576646i	$-0.195639\pi$
$89$	1371.94	1.63399	0.816994	+	0.576646i	$0.195639\pi$
$90$	0	0
$91$	189.243	0.218001
$92$	370.755	0.420151
$93$	−246.330	−0.274658
$94$	−60.4101	−0.0662853
$95$	0	0
$96$	557.732	0.592951
$97$	−1263.64	−1.32271	−0.661355	−	0.750073i	$-0.730018\pi$
$97$	−1263.64	−1.32271	−0.661355	+	0.750073i	$0.730018\pi$
$98$	387.763	0.399693
$99$	−368.488	−0.374085
$100$	0	0
$101$	148.533	0.146332	0.0731661	−	0.997320i	$-0.476690\pi$
$101$	148.533	0.146332	0.0731661	+	0.997320i	$0.476690\pi$
$102$	−81.8103	−0.0794159
$103$	−1485.59	−1.42116	−0.710580	−	0.703616i	$-0.751567\pi$
$103$	−1485.59	−1.42116	−0.710580	+	0.703616i	$0.751567\pi$
$104$	−107.597	−0.101449
$105$	0	0
$106$	81.7866	0.0749417
$107$	−461.476	−0.416940	−0.208470	−	0.978029i	$-0.566848\pi$
$107$	−461.476	−0.416940	−0.208470	+	0.978029i	$0.566848\pi$
$108$	−124.001	−0.110481
$109$	1516.06	1.33222	0.666112	−	0.745851i	$-0.267957\pi$
$109$	1516.06	1.33222	0.666112	+	0.745851i	$0.267957\pi$
$110$	0	0
$111$	669.848	0.572785
$112$	−1693.87	−1.42906
$113$	−160.406	−0.133537	−0.0667686	−	0.997768i	$-0.521269\pi$
$113$	−160.406	−0.133537	−0.0667686	+	0.997768i	$0.521269\pi$
$114$	−434.123	−0.356661
$115$	0	0
$116$	133.186	0.106604
$117$	80.0871	0.0632826
$118$	−1417.93	−1.10620
$119$	163.429	0.125895
$120$	0	0
$121$	345.339	0.259458
$122$	1377.95	1.02257
$123$	823.267	0.603508
$124$	377.099	0.273101
$125$	0	0
$126$	679.204	0.480225
$127$	−2583.64	−1.80520	−0.902601	−	0.430477i	$-0.858345\pi$
$127$	−2583.64	−1.80520	−0.902601	+	0.430477i	$0.858345\pi$
$128$	1407.32	0.971803
$129$	161.362	0.110132
$130$	0	0
$131$	−151.210	−0.100850	−0.0504249	−	0.998728i	$-0.516058\pi$
$131$	−151.210	−0.100850	−0.0504249	+	0.998728i	$0.516058\pi$
$132$	564.109	0.371965
$133$	867.228	0.565400
$134$	−1417.43	−0.913788
$135$	0	0
$136$	−92.9195	−0.0585866
$137$	257.407	0.160524	0.0802619	−	0.996774i	$-0.474424\pi$
$137$	257.407	0.160524	0.0802619	+	0.996774i	$0.474424\pi$
$138$	−859.422	−0.530136
$139$	−2294.42	−1.40007	−0.700037	−	0.714106i	$-0.746833\pi$
$139$	−2294.42	−1.40007	−0.700037	+	0.714106i	$0.746833\pi$
$140$	0	0
$141$	51.0708	0.0305031
$142$	327.495	0.193541
$143$	−364.335	−0.213058
$144$	−716.839	−0.414838
$145$	0	0
$146$	−3475.53	−1.97012
$147$	−327.815	−0.183930
$148$	−1025.45	−0.569538
$149$	−835.743	−0.459508	−0.229754	−	0.973249i	$-0.573792\pi$
$149$	−835.743	−0.459508	−0.229754	+	0.973249i	$0.573792\pi$
$150$	0	0
$151$	−3370.60	−1.81653	−0.908263	−	0.418400i	$-0.862591\pi$
$151$	−3370.60	−1.81653	−0.908263	+	0.418400i	$0.862591\pi$
$152$	−493.074	−0.263116
$153$	69.1625	0.0365455
$154$	−3089.86	−1.61680
$155$	0	0
$156$	−122.603	−0.0629238
$157$	137.311	0.0698002	0.0349001	−	0.999391i	$-0.488889\pi$
$157$	137.311	0.0698002	0.0349001	+	0.999391i	$0.488889\pi$
$158$	220.932	0.111243
$159$	−69.1426	−0.0344866
$160$	0	0
$161$	1716.83	0.840404
$162$	287.437	0.139402
$163$	859.793	0.413155	0.206577	−	0.978430i	$-0.433767\pi$
$163$	859.793	0.413155	0.206577	+	0.978430i	$0.433767\pi$
$164$	−1260.32	−0.600087
$165$	0	0
$166$	−580.541	−0.271438
$167$	900.368	0.417201	0.208601	−	0.978001i	$-0.433109\pi$
$167$	900.368	0.417201	0.208601	+	0.978001i	$0.433109\pi$
$168$	771.435	0.354271
$169$	−2117.82	−0.963958
$170$	0	0
$171$	367.008	0.164128
$172$	−247.024	−0.109508
$173$	465.780	0.204697	0.102348	−	0.994749i	$-0.467364\pi$
$173$	465.780	0.204697	0.102348	+	0.994749i	$0.467364\pi$
$174$	−308.729	−0.134510
$175$	0	0
$176$	3261.07	1.39666
$177$	1198.72	0.509048
$178$	4868.47	2.05004
$179$	1143.90	0.477650	0.238825	−	0.971063i	$-0.423238\pi$
$179$	1143.90	0.477650	0.238825	+	0.971063i	$0.423238\pi$
$180$	0	0
$181$	−345.618	−0.141931	−0.0709657	−	0.997479i	$-0.522608\pi$
$181$	−345.618	−0.141931	−0.0709657	+	0.997479i	$0.522608\pi$
$182$	671.550	0.273509
$183$	−1164.93	−0.470567
$184$	−976.125	−0.391092
$185$	0	0
$186$	−874.127	−0.344592
$187$	−314.637	−0.123040
$188$	−78.1829	−0.0303302
$189$	−574.200	−0.220989
$190$	0	0
$191$	3080.33	1.16694	0.583469	−	0.812135i	$-0.301695\pi$
$191$	3080.33	1.16694	0.583469	+	0.812135i	$0.301695\pi$
$192$	67.6021	0.0254102
$193$	−4561.76	−1.70136	−0.850680	−	0.525684i	$-0.823810\pi$
$193$	−4561.76	−1.70136	−0.850680	+	0.525684i	$0.823810\pi$
$194$	−4484.15	−1.65950
$195$	0	0
$196$	501.844	0.182888
$197$	−2521.13	−0.911794	−0.455897	−	0.890033i	$-0.650681\pi$
$197$	−2521.13	−0.911794	−0.455897	+	0.890033i	$0.650681\pi$
$198$	−1307.62	−0.469336
$199$	−3000.49	−1.06884	−0.534420	−	0.845219i	$-0.679470\pi$
$199$	−3000.49	−1.06884	−0.534420	+	0.845219i	$0.679470\pi$
$200$	0	0
$201$	1198.30	0.420506
$202$	527.084	0.183592
$203$	616.734	0.213233
$204$	−105.879	−0.0363383
$205$	0	0
$206$	−5271.78	−1.78302
$207$	726.557	0.243957
$208$	−708.761	−0.236268
$209$	−1669.61	−0.552580
$210$	0	0
$211$	843.401	0.275176	0.137588	−	0.990490i	$-0.456065\pi$
$211$	843.401	0.275176	0.137588	+	0.990490i	$0.456065\pi$
$212$	105.849	0.0342911
$213$	−276.865	−0.0890633
$214$	−1637.60	−0.523103
$215$	0	0
$216$	326.469	0.102840
$217$	1746.20	0.546267
$218$	5379.91	1.67144
$219$	2938.22	0.906606
$220$	0	0
$221$	68.3831	0.0208142
$222$	2377.03	0.718629
$223$	−1068.88	−0.320975	−0.160488	−	0.987038i	$-0.551307\pi$
$223$	−1068.88	−0.320975	−0.160488	+	0.987038i	$0.551307\pi$
$224$	−3953.70	−1.17932
$225$	0	0
$226$	−569.217	−0.167539
$227$	−1688.90	−0.493817	−0.246909	−	0.969039i	$-0.579415\pi$
$227$	−1688.90	−0.493817	−0.246909	+	0.969039i	$0.579415\pi$
$228$	−561.844	−0.163197
$229$	−3085.30	−0.890317	−0.445158	−	0.895452i	$-0.646853\pi$
$229$	−3085.30	−0.890317	−0.445158	+	0.895452i	$0.646853\pi$
$230$	0	0
$231$	2612.17	0.744019
$232$	−350.652	−0.0992304
$233$	4810.41	1.35254	0.676268	−	0.736656i	$-0.263596\pi$
$233$	4810.41	1.35254	0.676268	+	0.736656i	$0.263596\pi$
$234$	284.198	0.0793957
$235$	0	0
$236$	−1835.09	−0.506162
$237$	−186.776	−0.0511916
$238$	579.944	0.157950
$239$	−1730.66	−0.468399	−0.234200	−	0.972189i	$-0.575247\pi$
$239$	−1730.66	−0.468399	−0.234200	+	0.972189i	$0.575247\pi$
$240$	0	0
$241$	−1671.50	−0.446767	−0.223383	−	0.974731i	$-0.571710\pi$
$241$	−1671.50	−0.446767	−0.223383	+	0.974731i	$0.571710\pi$
$242$	1225.47	0.325522
$243$	−243.000	−0.0641500
$244$	1783.35	0.467900
$245$	0	0
$246$	2921.45	0.757175
$247$	362.872	0.0934778
$248$	−992.827	−0.254212
$249$	490.790	0.124910
$250$	0	0
$251$	−5324.14	−1.33887	−0.669436	−	0.742870i	$-0.733464\pi$
$251$	−5324.14	−1.33887	−0.669436	+	0.742870i	$0.733464\pi$
$252$	879.028	0.219736
$253$	−3305.28	−0.821348
$254$	−9168.32	−2.26485
$255$	0	0
$256$	5174.31	1.26326
$257$	−1994.20	−0.484026	−0.242013	−	0.970273i	$-0.577808\pi$
$257$	−1994.20	−0.484026	−0.242013	+	0.970273i	$0.577808\pi$
$258$	572.609	0.138175
$259$	−4748.48	−1.13921
$260$	0	0
$261$	261.000	0.0618984
$262$	−536.586	−0.126528
$263$	1671.01	0.391782	0.195891	−	0.980626i	$-0.437240\pi$
$263$	1671.01	0.391782	0.195891	+	0.980626i	$0.437240\pi$
$264$	−1485.19	−0.346238
$265$	0	0
$266$	3077.45	0.709364
$267$	−4115.81	−0.943384
$268$	−1834.45	−0.418122
$269$	890.546	0.201850	0.100925	−	0.994894i	$-0.467820\pi$
$269$	890.546	0.201850	0.100925	+	0.994894i	$0.467820\pi$
$270$	0	0
$271$	6649.82	1.49058	0.745292	−	0.666739i	$-0.232310\pi$
$271$	6649.82	1.49058	0.745292	+	0.666739i	$0.232310\pi$
$272$	−612.079	−0.136444
$273$	−567.729	−0.125863
$274$	913.436	0.201397
$275$	0	0
$276$	−1112.27	−0.242575
$277$	−6233.58	−1.35213	−0.676064	−	0.736843i	$-0.736316\pi$
$277$	−6233.58	−1.35213	−0.676064	+	0.736843i	$0.736316\pi$
$278$	−8142.01	−1.75657
$279$	738.989	0.158574
$280$	0	0
$281$	−3089.87	−0.655964	−0.327982	−	0.944684i	$-0.606369\pi$
$281$	−3089.87	−0.655964	−0.327982	+	0.944684i	$0.606369\pi$
$282$	181.230	0.0382699
$283$	−1644.01	−0.345323	−0.172662	−	0.984981i	$-0.555237\pi$
$283$	−1644.01	−0.345323	−0.172662	+	0.984981i	$0.555237\pi$
$284$	423.845	0.0885584
$285$	0	0
$286$	−1292.88	−0.267307
$287$	−5836.05	−1.20032
$288$	−1673.20	−0.342341
$289$	−4853.94	−0.987980
$290$	0	0
$291$	3790.91	0.763667
$292$	−4498.05	−0.901467
$293$	66.3144	0.0132223	0.00661114	−	0.999978i	$-0.497896\pi$
$293$	66.3144	0.0132223	0.00661114	+	0.999978i	$0.497896\pi$
$294$	−1163.29	−0.230763
$295$	0	0
$296$	2699.81	0.530147
$297$	1105.46	0.215978
$298$	−2965.72	−0.576510
$299$	718.369	0.138944
$300$	0	0
$301$	−1143.87	−0.219043
$302$	−11960.9	−2.27905
$303$	−445.598	−0.0844849
$304$	−3247.98	−0.612777
$305$	0	0
$306$	245.431	0.0458508
$307$	6736.52	1.25236	0.626179	−	0.779679i	$-0.284618\pi$
$307$	6736.52	1.25236	0.626179	+	0.779679i	$0.284618\pi$
$308$	−3998.91	−0.739801
$309$	4456.77	0.820507
$310$	0	0
$311$	−5470.56	−0.997450	−0.498725	−	0.866760i	$-0.666198\pi$
$311$	−5470.56	−0.997450	−0.498725	+	0.866760i	$0.666198\pi$
$312$	322.790	0.0585718
$313$	−903.461	−0.163152	−0.0815761	−	0.996667i	$-0.525995\pi$
$313$	−903.461	−0.163152	−0.0815761	+	0.996667i	$0.525995\pi$
$314$	487.264	0.0875729
$315$	0	0
$316$	285.930	0.0509014
$317$	880.773	0.156054	0.0780270	−	0.996951i	$-0.475138\pi$
$317$	880.773	0.156054	0.0780270	+	0.996951i	$0.475138\pi$
$318$	−245.360	−0.0432676
$319$	−1187.35	−0.208398
$320$	0	0
$321$	1384.43	0.240721
$322$	6092.35	1.05439
$323$	313.373	0.0539832
$324$	372.002	0.0637864
$325$	0	0
$326$	3051.07	0.518353
$327$	−4548.19	−0.769160
$328$	3318.16	0.558582
$329$	−362.035	−0.0606677
$330$	0	0
$331$	609.140	0.101152	0.0505761	−	0.998720i	$-0.483894\pi$
$331$	609.140	0.101152	0.0505761	+	0.998720i	$0.483894\pi$
$332$	−751.338	−0.124202
$333$	−2009.54	−0.330698
$334$	3195.05	0.523430
$335$	0	0
$336$	5081.60	0.825071
$337$	9473.47	1.53131	0.765657	−	0.643250i	$-0.222414\pi$
$337$	9473.47	1.53131	0.765657	+	0.643250i	$0.222414\pi$
$338$	−7515.30	−1.20940
$339$	481.217	0.0770977
$340$	0	0
$341$	−3361.83	−0.533881
$342$	1302.37	0.205918
$343$	−4970.62	−0.782474
$344$	650.365	0.101934
$345$	0	0
$346$	1652.87	0.256817
$347$	−3436.86	−0.531702	−0.265851	−	0.964014i	$-0.585653\pi$
$347$	−3436.86	−0.531702	−0.265851	+	0.964014i	$0.585653\pi$
$348$	−399.558	−0.0615476
$349$	703.930	0.107967	0.0539835	−	0.998542i	$-0.482808\pi$
$349$	703.930	0.107967	0.0539835	+	0.998542i	$0.482808\pi$
$350$	0	0
$351$	−240.261	−0.0365362
$352$	7611.77	1.15258
$353$	−2982.17	−0.449646	−0.224823	−	0.974400i	$-0.572180\pi$
$353$	−2982.17	−0.449646	−0.224823	+	0.974400i	$0.572180\pi$
$354$	4253.79	0.638663
$355$	0	0
$356$	6300.78	0.938036
$357$	−490.286	−0.0726854
$358$	4059.27	0.599271
$359$	8585.52	1.26219	0.631095	−	0.775705i	$-0.282606\pi$
$359$	8585.52	1.26219	0.631095	+	0.775705i	$0.282606\pi$
$360$	0	0
$361$	−5196.10	−0.757559
$362$	−1226.46	−0.178070
$363$	−1036.02	−0.149798
$364$	869.122	0.125149
$365$	0	0
$366$	−4133.86	−0.590384
$367$	−6989.50	−0.994139	−0.497070	−	0.867711i	$-0.665591\pi$
$367$	−6989.50	−0.994139	−0.497070	+	0.867711i	$0.665591\pi$
$368$	−6429.93	−0.910824
$369$	−2469.80	−0.348435
$370$	0	0
$371$	490.144	0.0685904
$372$	−1131.30	−0.157675
$373$	1100.55	0.152773	0.0763865	−	0.997078i	$-0.475662\pi$
$373$	1100.55	0.152773	0.0763865	+	0.997078i	$0.475662\pi$
$374$	−1116.52	−0.154369
$375$	0	0
$376$	205.840	0.0282324
$377$	258.059	0.0352538
$378$	−2037.61	−0.277258
$379$	7950.59	1.07756	0.538779	−	0.842447i	$-0.318886\pi$
$379$	7950.59	1.07756	0.538779	+	0.842447i	$0.318886\pi$
$380$	0	0
$381$	7750.91	1.04223
$382$	10930.9	1.46407
$383$	−10160.2	−1.35551	−0.677755	−	0.735288i	$-0.737047\pi$
$383$	−10160.2	−1.35551	−0.677755	+	0.735288i	$0.737047\pi$
$384$	−4221.96	−0.561071
$385$	0	0
$386$	−16187.9	−2.13456
$387$	−484.085	−0.0635850
$388$	−5803.40	−0.759338
$389$	9770.47	1.27348	0.636738	−	0.771080i	$-0.280283\pi$
$389$	9770.47	1.27348	0.636738	+	0.771080i	$0.280283\pi$
$390$	0	0
$391$	620.377	0.0802399
$392$	−1321.25	−0.170238
$393$	453.631	0.0582256
$394$	−8946.52	−1.14396
$395$	0	0
$396$	−1692.33	−0.214754
$397$	3191.11	0.403418	0.201709	−	0.979445i	$-0.435350\pi$
$397$	3191.11	0.403418	0.201709	+	0.979445i	$0.435350\pi$
$398$	−10647.6	−1.34099
$399$	−2601.68	−0.326434
$400$	0	0
$401$	11101.4	1.38249	0.691247	−	0.722619i	$-0.257062\pi$
$401$	11101.4	1.38249	0.691247	+	0.722619i	$0.257062\pi$
$402$	4252.30	0.527576
$403$	730.660	0.0903146
$404$	682.154	0.0840060
$405$	0	0
$406$	2188.55	0.267526
$407$	9141.89	1.11338
$408$	278.759	0.0338250
$409$	−5263.84	−0.636382	−0.318191	−	0.948027i	$-0.603075\pi$
$409$	−5263.84	−0.636382	−0.318191	+	0.948027i	$0.603075\pi$
$410$	0	0
$411$	−772.221	−0.0926784
$412$	−6822.75	−0.815856
$413$	−8497.61	−1.01245
$414$	2578.27	0.306074
$415$	0	0
$416$	−1654.34	−0.194978
$417$	6883.27	0.808333
$418$	−5924.78	−0.693279
$419$	5365.09	0.625541	0.312771	−	0.949829i	$-0.398743\pi$
$419$	5365.09	0.625541	0.312771	+	0.949829i	$0.398743\pi$
$420$	0	0
$421$	4453.87	0.515602	0.257801	−	0.966198i	$-0.417002\pi$
$421$	4453.87	0.515602	0.257801	+	0.966198i	$0.417002\pi$
$422$	2992.90	0.345242
$423$	−153.212	−0.0176110
$424$	−278.678	−0.0319193
$425$	0	0
$426$	−982.486	−0.111741
$427$	8258.03	0.935911
$428$	−2119.39	−0.239356
$429$	1093.01	0.123009
$430$	0	0
$431$	−15164.6	−1.69478	−0.847392	−	0.530968i	$-0.821828\pi$
$431$	−15164.6	−1.69478	−0.847392	+	0.530968i	$0.821828\pi$
$432$	2150.52	0.239507
$433$	−5748.48	−0.638001	−0.319000	−	0.947755i	$-0.603347\pi$
$433$	−5748.48	−0.638001	−0.319000	+	0.947755i	$0.603347\pi$
$434$	6196.59	0.685359
$435$	0	0
$436$	6962.70	0.764800
$437$	3292.01	0.360362
$438$	10426.6	1.13745
$439$	−2413.19	−0.262358	−0.131179	−	0.991359i	$-0.541876\pi$
$439$	−2413.19	−0.262358	−0.131179	+	0.991359i	$0.541876\pi$
$440$	0	0
$441$	983.446	0.106192
$442$	242.665	0.0261140
$443$	5598.47	0.600432	0.300216	−	0.953871i	$-0.402941\pi$
$443$	5598.47	0.600432	0.300216	+	0.953871i	$0.402941\pi$
$444$	3076.36	0.328823
$445$	0	0
$446$	−3793.03	−0.402703
$447$	2507.23	0.265297
$448$	−479.224	−0.0505384
$449$	1387.50	0.145836	0.0729179	−	0.997338i	$-0.476769\pi$
$449$	1387.50	0.145836	0.0729179	+	0.997338i	$0.476769\pi$
$450$	0	0
$451$	11235.7	1.17310
$452$	−736.682	−0.0766607
$453$	10111.8	1.04877
$454$	−5993.26	−0.619554
$455$	0	0
$456$	1479.22	0.151910
$457$	1208.57	0.123707	0.0618537	−	0.998085i	$-0.480299\pi$
$457$	1208.57	0.123707	0.0618537	+	0.998085i	$0.480299\pi$
$458$	−10948.5	−1.11701
$459$	−207.488	−0.0210995
$460$	0	0
$461$	−1838.27	−0.185720	−0.0928599	−	0.995679i	$-0.529601\pi$
$461$	−1838.27	−0.185720	−0.0928599	+	0.995679i	$0.529601\pi$
$462$	9269.58	0.933463
$463$	13925.4	1.39777	0.698885	−	0.715234i	$-0.253680\pi$
$463$	13925.4	1.39777	0.698885	+	0.715234i	$0.253680\pi$
$464$	−2309.82	−0.231100
$465$	0	0
$466$	17070.3	1.69692
$467$	8692.90	0.861369	0.430685	−	0.902502i	$-0.358272\pi$
$467$	8692.90	0.861369	0.430685	+	0.902502i	$0.358272\pi$
$468$	367.810	0.0363291
$469$	−8494.63	−0.836344
$470$	0	0
$471$	−411.934	−0.0402992
$472$	4831.43	0.471154
$473$	2202.21	0.214076
$474$	−662.795	−0.0642261
$475$	0	0
$476$	750.566	0.0722734
$477$	207.428	0.0199108
$478$	−6141.45	−0.587664
$479$	9915.68	0.945843	0.472922	−	0.881105i	$-0.343199\pi$
$479$	9915.68	0.945843	0.472922	+	0.881105i	$0.343199\pi$
$480$	0	0
$481$	−1986.90	−0.188347
$482$	−5931.50	−0.560523
$483$	−5150.48	−0.485207
$484$	1586.01	0.148949
$485$	0	0
$486$	−862.312	−0.0804841
$487$	−16633.8	−1.54774	−0.773871	−	0.633344i	$-0.781682\pi$
$487$	−16633.8	−1.54774	−0.773871	+	0.633344i	$0.781682\pi$
$488$	−4695.21	−0.435538
$489$	−2579.38	−0.238535
$490$	0	0
$491$	−291.435	−0.0267867	−0.0133933	−	0.999910i	$-0.504263\pi$
$491$	−291.435	−0.0267867	−0.0133933	+	0.999910i	$0.504263\pi$
$492$	3780.95	0.346460
$493$	222.857	0.0203590
$494$	1287.69	0.117279
$495$	0	0
$496$	−6539.95	−0.592041
$497$	1962.67	0.177138
$498$	1741.62	0.156715
$499$	11845.7	1.06270	0.531350	−	0.847152i	$-0.321685\pi$
$499$	11845.7	1.06270	0.531350	+	0.847152i	$0.321685\pi$
$500$	0	0
$501$	−2701.10	−0.240871
$502$	−18893.3	−1.67978
$503$	16484.6	1.46126	0.730629	−	0.682775i	$-0.239227\pi$
$503$	16484.6	1.46126	0.730629	+	0.682775i	$0.239227\pi$
$504$	−2314.31	−0.204538
$505$	0	0
$506$	−11729.1	−1.03048
$507$	6353.45	0.556541
$508$	−11865.7	−1.03633
$509$	11521.1	1.00327	0.501636	−	0.865079i	$-0.332732\pi$
$509$	11521.1	1.00327	0.501636	+	0.865079i	$0.332732\pi$
$510$	0	0
$511$	−20828.7	−1.80315
$512$	7103.01	0.613109
$513$	−1101.03	−0.0947592
$514$	−7076.63	−0.607270
$515$	0	0
$516$	741.072	0.0632246
$517$	696.999	0.0592920
$518$	−16850.5	−1.42928
$519$	−1397.34	−0.118182
$520$	0	0
$521$	6623.78	0.556993	0.278496	−	0.960437i	$-0.410164\pi$
$521$	6623.78	0.556993	0.278496	+	0.960437i	$0.410164\pi$
$522$	926.187	0.0776592
$523$	13552.8	1.13312	0.566559	−	0.824021i	$-0.308274\pi$
$523$	13552.8	1.13312	0.566559	+	0.824021i	$0.308274\pi$
$524$	−694.452	−0.0578956
$525$	0	0
$526$	5929.75	0.491538
$527$	630.992	0.0521564
$528$	−9783.21	−0.806363
$529$	−5649.91	−0.464363
$530$	0	0
$531$	−3596.17	−0.293899
$532$	3982.85	0.324584
$533$	−2441.97	−0.198449
$534$	−14605.4	−1.18359
$535$	0	0
$536$	4829.73	0.389203
$537$	−3431.71	−0.275772
$538$	3160.20	0.253245
$539$	−4473.93	−0.357524
$540$	0	0
$541$	10971.0	0.871866	0.435933	−	0.899979i	$-0.356418\pi$
$541$	10971.0	0.871866	0.435933	+	0.899979i	$0.356418\pi$
$542$	23597.6	1.87012
$543$	1036.85	0.0819442
$544$	−1428.67	−0.112599
$545$	0	0
$546$	−2014.65	−0.157910
$547$	12428.0	0.971453	0.485726	−	0.874111i	$-0.338555\pi$
$547$	12428.0	0.971453	0.485726	+	0.874111i	$0.338555\pi$
$548$	1182.17	0.0921531
$549$	3494.78	0.271682
$550$	0	0
$551$	1182.58	0.0914333
$552$	2928.38	0.225797
$553$	1324.03	0.101815
$554$	−22120.5	−1.69641
$555$	0	0
$556$	−10537.4	−0.803751
$557$	−22737.5	−1.72966	−0.864828	−	0.502069i	$-0.832572\pi$
$557$	−22737.5	−1.72966	−0.864828	+	0.502069i	$0.832572\pi$
$558$	2622.38	0.198950
$559$	−478.629	−0.0362144
$560$	0	0
$561$	943.910	0.0710373
$562$	−10964.7	−0.822988
$563$	20051.1	1.50099	0.750493	−	0.660878i	$-0.229816\pi$
$563$	20051.1	1.50099	0.750493	+	0.660878i	$0.229816\pi$
$564$	234.549	0.0175111
$565$	0	0
$566$	−5833.96	−0.433250
$567$	1722.60	0.127588
$568$	−1115.90	−0.0824334
$569$	10035.7	0.739397	0.369699	−	0.929152i	$-0.379461\pi$
$569$	10035.7	0.739397	0.369699	+	0.929152i	$0.379461\pi$
$570$	0	0
$571$	22946.3	1.68174	0.840870	−	0.541237i	$-0.182044\pi$
$571$	22946.3	1.68174	0.840870	+	0.541237i	$0.182044\pi$
$572$	−1673.25	−0.122312
$573$	−9241.00	−0.673732
$574$	−20709.9	−1.50595
$575$	0	0
$576$	−202.806	−0.0146706
$577$	−14230.3	−1.02672	−0.513358	−	0.858174i	$-0.671599\pi$
$577$	−14230.3	−1.02672	−0.513358	+	0.858174i	$0.671599\pi$
$578$	−17224.7	−1.23954
$579$	13685.3	0.982281
$580$	0	0
$581$	−3479.16	−0.248434
$582$	13452.5	0.958114
$583$	−943.637	−0.0670351
$584$	11842.5	0.839117
$585$	0	0
$586$	235.324	0.0165890
$587$	−12173.9	−0.855999	−0.428000	−	0.903779i	$-0.640782\pi$
$587$	−12173.9	−0.855999	−0.428000	+	0.903779i	$0.640782\pi$
$588$	−1505.53	−0.105590
$589$	3348.33	0.234237
$590$	0	0
$591$	7563.40	0.526424
$592$	17784.2	1.23467
$593$	−21548.0	−1.49219	−0.746097	−	0.665838i	$-0.768074\pi$
$593$	−21548.0	−1.49219	−0.746097	+	0.665838i	$0.768074\pi$
$594$	3922.86	0.270971
$595$	0	0
$596$	−3838.25	−0.263793
$597$	9001.47	0.617095
$598$	2549.21	0.174323
$599$	22162.4	1.51174	0.755870	−	0.654722i	$-0.227214\pi$
$599$	22162.4	1.51174	0.755870	+	0.654722i	$0.227214\pi$
$600$	0	0
$601$	2344.90	0.159152	0.0795761	−	0.996829i	$-0.474643\pi$
$601$	2344.90	0.159152	0.0795761	+	0.996829i	$0.474643\pi$
$602$	−4059.16	−0.274816
$603$	−3594.90	−0.242779
$604$	−15479.9	−1.04283
$605$	0	0
$606$	−1581.25	−0.105997
$607$	−4273.53	−0.285761	−0.142881	−	0.989740i	$-0.545637\pi$
$607$	−4273.53	−0.285761	−0.142881	+	0.989740i	$0.545637\pi$
$608$	−7581.20	−0.505688
$609$	−1850.20	−0.123110
$610$	0	0
$611$	−151.486	−0.0100302
$612$	317.637	0.0209799
$613$	28945.8	1.90719	0.953597	−	0.301085i	$-0.0973487\pi$
$613$	28945.8	1.90719	0.953597	+	0.301085i	$0.0973487\pi$
$614$	23905.3	1.57124
$615$	0	0
$616$	10528.3	0.688633
$617$	−885.511	−0.0577785	−0.0288892	−	0.999583i	$-0.509197\pi$
$617$	−885.511	−0.0577785	−0.0288892	+	0.999583i	$0.509197\pi$
$618$	15815.3	1.02943
$619$	−6120.02	−0.397390	−0.198695	−	0.980061i	$-0.563670\pi$
$619$	−6120.02	−0.397390	−0.198695	+	0.980061i	$0.563670\pi$
$620$	0	0
$621$	−2179.67	−0.140849
$622$	−19412.9	−1.25142
$623$	29176.5	1.87630
$624$	2126.28	0.136409
$625$	0	0
$626$	−3206.03	−0.204694
$627$	5008.82	0.319032
$628$	630.618	0.0400707
$629$	−1715.87	−0.108770
$630$	0	0
$631$	9436.47	0.595341	0.297670	−	0.954669i	$-0.403790\pi$
$631$	9436.47	0.595341	0.297670	+	0.954669i	$0.403790\pi$
$632$	−752.798	−0.0473808
$633$	−2530.20	−0.158873
$634$	3125.52	0.195789
$635$	0	0
$636$	−317.546	−0.0197980
$637$	972.363	0.0604810
$638$	−4213.44	−0.261460
$639$	830.595	0.0514207
$640$	0	0
$641$	−12119.8	−0.746807	−0.373403	−	0.927669i	$-0.621809\pi$
$641$	−12119.8	−0.746807	−0.373403	+	0.927669i	$0.621809\pi$
$642$	4912.80	0.302013
$643$	7693.03	0.471825	0.235912	−	0.971774i	$-0.424192\pi$
$643$	7693.03	0.471825	0.235912	+	0.971774i	$0.424192\pi$
$644$	7884.74	0.482457
$645$	0	0
$646$	1112.04	0.0677285
$647$	−9921.81	−0.602885	−0.301443	−	0.953484i	$-0.597468\pi$
$647$	−9921.81	−0.602885	−0.301443	+	0.953484i	$0.597468\pi$
$648$	−979.408	−0.0593747
$649$	16359.8	0.989489
$650$	0	0
$651$	−5238.61	−0.315388
$652$	3948.70	0.237183
$653$	−9973.03	−0.597665	−0.298832	−	0.954306i	$-0.596597\pi$
$653$	−9973.03	−0.597665	−0.298832	+	0.954306i	$0.596597\pi$
$654$	−16139.7	−0.965006
$655$	0	0
$656$	21857.4	1.30090
$657$	−8814.67	−0.523429
$658$	−1284.72	−0.0761150
$659$	−24918.9	−1.47299	−0.736495	−	0.676443i	$-0.763521\pi$
$659$	−24918.9	−1.47299	−0.736495	+	0.676443i	$0.763521\pi$
$660$	0	0
$661$	14562.7	0.856922	0.428461	−	0.903560i	$-0.359056\pi$
$661$	14562.7	0.856922	0.428461	+	0.903560i	$0.359056\pi$
$662$	2161.60	0.126908
$663$	−205.149	−0.0120171
$664$	1978.12	0.115612
$665$	0	0
$666$	−7131.08	−0.414901
$667$	2341.13	0.135905
$668$	4135.05	0.239506
$669$	3206.64	0.185315
$670$	0	0
$671$	−15898.6	−0.914690
$672$	11861.1	0.680882
$673$	−10744.1	−0.615388	−0.307694	−	0.951485i	$-0.599557\pi$
$673$	−10744.1	−0.615388	−0.307694	+	0.951485i	$0.599557\pi$
$674$	33617.6	1.92122
$675$	0	0
$676$	−9726.32	−0.553387
$677$	−842.278	−0.0478159	−0.0239080	−	0.999714i	$-0.507611\pi$
$677$	−842.278	−0.0478159	−0.0239080	+	0.999714i	$0.507611\pi$
$678$	1707.65	0.0967285
$679$	−26873.4	−1.51886
$680$	0	0
$681$	5066.71	0.285105
$682$	−11929.8	−0.669819
$683$	8313.36	0.465742	0.232871	−	0.972508i	$-0.425188\pi$
$683$	8313.36	0.465742	0.232871	+	0.972508i	$0.425188\pi$
$684$	1685.53	0.0942220
$685$	0	0
$686$	−17638.8	−0.981709
$687$	9255.91	0.514025
$688$	4284.08	0.237397
$689$	205.090	0.0113401
$690$	0	0
$691$	16891.1	0.929909	0.464954	−	0.885335i	$-0.346071\pi$
$691$	16891.1	0.929909	0.464954	+	0.885335i	$0.346071\pi$
$692$	2139.15	0.117512
$693$	−7836.52	−0.429559
$694$	−12196.1	−0.667085
$695$	0	0
$696$	1051.96	0.0572907
$697$	−2108.86	−0.114604
$698$	2497.97	0.135458
$699$	−14431.2	−0.780887
$700$	0	0
$701$	26296.2	1.41682	0.708412	−	0.705799i	$-0.249412\pi$
$701$	26296.2	1.41682	0.708412	+	0.705799i	$0.249412\pi$
$702$	−852.594	−0.0458391
$703$	−9105.18	−0.488490
$704$	922.613	0.0493925
$705$	0	0
$706$	−10582.6	−0.564136
$707$	3158.80	0.168032
$708$	5505.27	0.292233
$709$	26644.7	1.41137	0.705686	−	0.708525i	$-0.250639\pi$
$709$	26644.7	1.41137	0.705686	+	0.708525i	$0.250639\pi$
$710$	0	0
$711$	560.328	0.0295555
$712$	−16588.7	−0.873158
$713$	6628.61	0.348167
$714$	−1739.83	−0.0911927
$715$	0	0
$716$	5253.52	0.274208
$717$	5191.99	0.270430
$718$	30466.7	1.58357
$719$	5825.10	0.302141	0.151071	−	0.988523i	$-0.451728\pi$
$719$	5825.10	0.302141	0.151071	+	0.988523i	$0.451728\pi$
$720$	0	0
$721$	−31593.6	−1.63191
$722$	−18438.9	−0.950450
$723$	5014.50	0.257941
$724$	−1587.29	−0.0814797
$725$	0	0
$726$	−3676.42	−0.187940
$727$	−6715.29	−0.342581	−0.171290	−	0.985221i	$-0.554794\pi$
$727$	−6715.29	−0.342581	−0.171290	+	0.985221i	$0.554794\pi$
$728$	−2288.22	−0.116493
$729$	729.000	0.0370370
$730$	0	0
$731$	−413.340	−0.0209137
$732$	−5350.06	−0.270142
$733$	−14206.6	−0.715870	−0.357935	−	0.933747i	$-0.616519\pi$
$733$	−14206.6	−0.715870	−0.357935	+	0.933747i	$0.616519\pi$
$734$	−24803.0	−1.24727
$735$	0	0
$736$	−15008.3	−0.751649
$737$	16354.1	0.817381
$738$	−8764.36	−0.437155
$739$	−30931.2	−1.53968	−0.769840	−	0.638237i	$-0.779664\pi$
$739$	−30931.2	−1.53968	−0.769840	+	0.638237i	$0.779664\pi$
$740$	0	0
$741$	−1088.62	−0.0539694
$742$	1739.33	0.0860550
$743$	32851.1	1.62206	0.811030	−	0.585005i	$-0.198907\pi$
$743$	32851.1	1.62206	0.811030	+	0.585005i	$0.198907\pi$
$744$	2978.48	0.146769
$745$	0	0
$746$	3905.42	0.191672
$747$	−1472.37	−0.0721168
$748$	−1445.01	−0.0706346
$749$	−9814.07	−0.478769
$750$	0	0
$751$	36055.8	1.75192	0.875961	−	0.482382i	$-0.160228\pi$
$751$	36055.8	1.75192	0.875961	+	0.482382i	$0.160228\pi$
$752$	1355.91	0.0657512
$753$	15972.4	0.772998
$754$	915.749	0.0442302
$755$	0	0
$756$	−2637.08	−0.126865
$757$	7094.16	0.340610	0.170305	−	0.985391i	$-0.445525\pi$
$757$	7094.16	0.340610	0.170305	+	0.985391i	$0.445525\pi$
$758$	28213.5	1.35193
$759$	9915.83	0.474205
$760$	0	0
$761$	23664.3	1.12724	0.563620	−	0.826034i	$-0.309408\pi$
$761$	23664.3	1.12724	0.563620	+	0.826034i	$0.309408\pi$
$762$	27505.0	1.30761
$763$	32241.6	1.52978
$764$	14146.8	0.669913
$765$	0	0
$766$	−36054.5	−1.70065
$767$	−3555.64	−0.167388
$768$	−15522.9	−0.729342
$769$	16743.8	0.785172	0.392586	−	0.919715i	$-0.371581\pi$
$769$	16743.8	0.785172	0.392586	+	0.919715i	$0.371581\pi$
$770$	0	0
$771$	5982.59	0.279452
$772$	−20950.4	−0.976713
$773$	2198.53	0.102297	0.0511485	−	0.998691i	$-0.483712\pi$
$773$	2198.53	0.102297	0.0511485	+	0.998691i	$0.483712\pi$
$774$	−1717.83	−0.0797752
$775$	0	0
$776$	15279.2	0.706819
$777$	14245.4	0.657725
$778$	34671.6	1.59773
$779$	−11190.6	−0.514691
$780$	0	0
$781$	−3778.57	−0.173122
$782$	2201.47	0.100671
$783$	−783.000	−0.0357371
$784$	−8703.37	−0.396473
$785$	0	0
$786$	1609.76	0.0730512
$787$	10092.7	0.457134	0.228567	−	0.973528i	$-0.426596\pi$
$787$	10092.7	0.457134	0.228567	+	0.973528i	$0.426596\pi$
$788$	−11578.6	−0.523440
$789$	−5013.02	−0.226195
$790$	0	0
$791$	−3411.30	−0.153340
$792$	4455.56	0.199901
$793$	3455.39	0.154735
$794$	11324.0	0.506138
$795$	0	0
$796$	−13780.1	−0.613597
$797$	35534.7	1.57930	0.789652	−	0.613555i	$-0.210261\pi$
$797$	35534.7	1.57930	0.789652	+	0.613555i	$0.210261\pi$
$798$	−9232.36	−0.409551
$799$	−130.822	−0.00579241
$800$	0	0
$801$	12347.4	0.544663
$802$	39394.7	1.73451
$803$	40100.0	1.76226
$804$	5503.34	0.241403
$805$	0	0
$806$	2592.83	0.113311
$807$	−2671.64	−0.116538
$808$	−1795.98	−0.0781958
$809$	27966.3	1.21538	0.607691	−	0.794173i	$-0.292096\pi$
$809$	27966.3	1.21538	0.607691	+	0.794173i	$0.292096\pi$
$810$	0	0
$811$	−19134.4	−0.828481	−0.414241	−	0.910167i	$-0.635953\pi$
$811$	−19134.4	−0.828481	−0.414241	+	0.910167i	$0.635953\pi$
$812$	2832.42	0.122412
$813$	−19949.5	−0.860589
$814$	32441.0	1.39687
$815$	0	0
$816$	1836.24	0.0787760
$817$	−2193.37	−0.0939246
$818$	−18679.3	−0.798420
$819$	1703.19	0.0726669
$820$	0	0
$821$	−21420.6	−0.910577	−0.455288	−	0.890344i	$-0.650464\pi$
$821$	−21420.6	−0.910577	−0.455288	+	0.890344i	$0.650464\pi$
$822$	−2740.31	−0.116276
$823$	16122.4	0.682855	0.341428	−	0.939908i	$-0.389090\pi$
$823$	16122.4	0.682855	0.341428	+	0.939908i	$0.389090\pi$
$824$	17962.9	0.759428
$825$	0	0
$826$	−30154.7	−1.27024
$827$	−22131.9	−0.930596	−0.465298	−	0.885154i	$-0.654053\pi$
$827$	−22131.9	−0.930596	−0.465298	+	0.885154i	$0.654053\pi$
$828$	3336.80	0.140050
$829$	135.660	0.00568355	0.00284178	−	0.999996i	$-0.499095\pi$
$829$	135.660	0.00568355	0.00284178	+	0.999996i	$0.499095\pi$
$830$	0	0
$831$	18700.7	0.780652
$832$	−200.521	−0.00835553
$833$	839.724	0.0349276
$834$	24426.0	1.01415
$835$	0	0
$836$	−7667.88	−0.317224
$837$	−2216.97	−0.0915526
$838$	19038.6	0.784818
$839$	−32154.4	−1.32311	−0.661557	−	0.749895i	$-0.730104\pi$
$839$	−32154.4	−1.32311	−0.661557	+	0.749895i	$0.730104\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	15805.0	0.646886
$843$	9269.60	0.378721
$844$	3873.42	0.157972
$845$	0	0
$846$	−543.691	−0.0220951
$847$	7344.21	0.297934
$848$	−1835.71	−0.0743378
$849$	4932.04	0.199372
$850$	0	0
$851$	−18025.3	−0.726085
$852$	−1271.54	−0.0511292
$853$	−2797.50	−0.112291	−0.0561457	−	0.998423i	$-0.517881\pi$
$853$	−2797.50	−0.112291	−0.0561457	+	0.998423i	$0.517881\pi$
$854$	29304.5	1.17422
$855$	0	0
$856$	5579.92	0.222801
$857$	−9053.14	−0.360851	−0.180425	−	0.983589i	$-0.557747\pi$
$857$	−9053.14	−0.360851	−0.180425	+	0.983589i	$0.557747\pi$
$858$	3878.65	0.154330
$859$	23178.7	0.920660	0.460330	−	0.887748i	$-0.347731\pi$
$859$	23178.7	0.920660	0.460330	+	0.887748i	$0.347731\pi$
$860$	0	0
$861$	17508.2	0.693004
$862$	−53813.1	−2.12631
$863$	7117.61	0.280749	0.140374	−	0.990098i	$-0.455169\pi$
$863$	7117.61	0.280749	0.140374	+	0.990098i	$0.455169\pi$
$864$	5019.59	0.197650
$865$	0	0
$866$	−20399.1	−0.800450
$867$	14561.8	0.570410
$868$	8019.65	0.313600
$869$	−2549.06	−0.0995064
$870$	0	0
$871$	−3554.39	−0.138273
$872$	−18331.4	−0.711904
$873$	−11372.7	−0.440903
$874$	11682.0	0.452118
$875$	0	0
$876$	13494.1	0.520462
$877$	−44014.6	−1.69472	−0.847359	−	0.531020i	$-0.821809\pi$
$877$	−44014.6	−1.69472	−0.847359	+	0.531020i	$0.821809\pi$
$878$	−8563.47	−0.329161
$879$	−198.943	−0.00763388
$880$	0	0
$881$	−31970.4	−1.22260	−0.611299	−	0.791399i	$-0.709353\pi$
$881$	−31970.4	−1.22260	−0.611299	+	0.791399i	$0.709353\pi$
$882$	3489.86	0.133231
$883$	−13620.9	−0.519116	−0.259558	−	0.965727i	$-0.583577\pi$
$883$	−13620.9	−0.519116	−0.259558	+	0.965727i	$0.583577\pi$
$884$	314.058	0.0119490
$885$	0	0
$886$	19866.8	0.753315
$887$	−23494.2	−0.889357	−0.444678	−	0.895690i	$-0.646682\pi$
$887$	−23494.2	−0.889357	−0.444678	+	0.895690i	$0.646682\pi$
$888$	−8099.44	−0.306080
$889$	−54945.4	−2.07290
$890$	0	0
$891$	−3316.39	−0.124695
$892$	−4908.96	−0.184265
$893$	−694.200	−0.0260140
$894$	8897.17	0.332848
$895$	0	0
$896$	29929.1	1.11591
$897$	−2155.11	−0.0802195
$898$	4923.70	0.182969
$899$	2381.19	0.0883392
$900$	0	0
$901$	177.114	0.00654886
$902$	39871.1	1.47180
$903$	3431.62	0.126464
$904$	1939.54	0.0713585
$905$	0	0
$906$	35882.8	1.31581
$907$	−11695.6	−0.428165	−0.214083	−	0.976816i	$-0.568676\pi$
$907$	−11695.6	−0.428165	−0.214083	+	0.976816i	$0.568676\pi$
$908$	−7756.49	−0.283489
$909$	1336.79	0.0487774
$910$	0	0
$911$	−30991.2	−1.12710	−0.563549	−	0.826083i	$-0.690564\pi$
$911$	−30991.2	−1.12710	−0.563549	+	0.826083i	$0.690564\pi$
$912$	9743.93	0.353787
$913$	6698.16	0.242800
$914$	4288.72	0.155206
$915$	0	0
$916$	−14169.6	−0.511111
$917$	−3215.74	−0.115805
$918$	−736.292	−0.0264720
$919$	−16612.6	−0.596300	−0.298150	−	0.954519i	$-0.596370\pi$
$919$	−16612.6	−0.596300	−0.298150	+	0.954519i	$0.596370\pi$
$920$	0	0
$921$	−20209.6	−0.723049
$922$	−6523.30	−0.233008
$923$	821.235	0.0292863
$924$	11996.7	0.427125
$925$	0	0
$926$	49415.8	1.75367
$927$	−13370.3	−0.473720
$928$	−5391.41	−0.190713
$929$	53475.3	1.88856	0.944278	−	0.329149i	$-0.106762\pi$
$929$	53475.3	1.88856	0.944278	+	0.329149i	$0.106762\pi$
$930$	0	0
$931$	4455.96	0.156862
$932$	22092.4	0.776460
$933$	16411.7	0.575878
$934$	30847.7	1.08069
$935$	0	0
$936$	−968.370	−0.0338164
$937$	−21279.8	−0.741923	−0.370961	−	0.928648i	$-0.620972\pi$
$937$	−21279.8	−0.741923	−0.370961	+	0.928648i	$0.620972\pi$
$938$	−30144.1	−1.04930
$939$	2710.38	0.0941960
$940$	0	0
$941$	13268.0	0.459645	0.229822	−	0.973233i	$-0.426186\pi$
$941$	13268.0	0.459645	0.229822	+	0.973233i	$0.426186\pi$
$942$	−1461.79	−0.0505602
$943$	−22153.7	−0.765031
$944$	31825.6	1.09728
$945$	0	0
$946$	7814.80	0.268585
$947$	−21016.6	−0.721170	−0.360585	−	0.932726i	$-0.617423\pi$
$947$	−21016.6	−0.721170	−0.360585	+	0.932726i	$0.617423\pi$
$948$	−857.791	−0.0293879
$949$	−8715.33	−0.298115
$950$	0	0
$951$	−2642.32	−0.0900979
$952$	−1976.09	−0.0672746
$953$	−42402.9	−1.44131	−0.720654	−	0.693295i	$-0.756158\pi$
$953$	−42402.9	−1.44131	−0.720654	+	0.693295i	$0.756158\pi$
$954$	736.080	0.0249806
$955$	0	0
$956$	−7948.28	−0.268897
$957$	3562.05	0.120318
$958$	35186.9	1.18668
$959$	5474.19	0.184328
$960$	0	0
$961$	−23049.0	−0.773689
$962$	−7050.72	−0.236304
$963$	−4153.29	−0.138980
$964$	−7676.56	−0.256479
$965$	0	0
$966$	−18277.0	−0.608752
$967$	−31613.7	−1.05132	−0.525661	−	0.850694i	$-0.676182\pi$
$967$	−31613.7	−1.05132	−0.525661	+	0.850694i	$0.676182\pi$
$968$	−4175.65	−0.138647
$969$	−940.120	−0.0311672
$970$	0	0
$971$	−42637.4	−1.40916	−0.704582	−	0.709622i	$-0.748866\pi$
$971$	−42637.4	−1.40916	−0.704582	+	0.709622i	$0.748866\pi$
$972$	−1116.01	−0.0368271
$973$	−48794.8	−1.60770
$974$	−59026.9	−1.94183
$975$	0	0
$976$	−30928.3	−1.01433
$977$	7766.81	0.254332	0.127166	−	0.991881i	$-0.459412\pi$
$977$	7766.81	0.254332	0.127166	+	0.991881i	$0.459412\pi$
$978$	−9153.21	−0.299271
$979$	−56171.4	−1.83375
$980$	0	0
$981$	13644.6	0.444075
$982$	−1034.19	−0.0336072
$983$	13603.8	0.441397	0.220698	−	0.975342i	$-0.429166\pi$
$983$	13603.8	0.441397	0.220698	+	0.975342i	$0.429166\pi$
$984$	−9954.49	−0.322498
$985$	0	0
$986$	790.832	0.0255428
$987$	1086.11	0.0350265
$988$	1666.54	0.0536635
$989$	−4342.16	−0.139608
$990$	0	0
$991$	−13619.6	−0.436569	−0.218284	−	0.975885i	$-0.570046\pi$
$991$	−13619.6	−0.436569	−0.218284	+	0.975885i	$0.570046\pi$
$992$	−15265.1	−0.488576
$993$	−1827.42	−0.0584002
$994$	6964.74	0.222241
$995$	0	0
$996$	2254.01	0.0717080
$997$	−27262.0	−0.865993	−0.432996	−	0.901396i	$-0.642544\pi$
$997$	−27262.0	−0.865993	−0.432996	+	0.901396i	$0.642544\pi$
$998$	42035.8	1.33329
$999$	6028.63	0.190928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.k.1.5		6
5.4	even	2		435.4.a.h.1.2	✓	6
15.14	odd	2		1305.4.a.h.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.2	✓	6	5.4	even	2
1305.4.a.h.1.5		6	15.14	odd	2
2175.4.a.k.1.5		6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q+3.54861 q^{2} -3.00000 q^{3} +4.59262 q^{4} -10.6458 q^{6} +21.2667 q^{7} -12.0915 q^{8} +9.00000 q^{9} -40.9431 q^{11} -13.7779 q^{12} +8.89857 q^{13} +75.4671 q^{14} -79.6488 q^{16} +7.68473 q^{17} +31.9375 q^{18} +40.7787 q^{19} -63.8000 q^{21} -145.291 q^{22} +80.7285 q^{23} +36.2744 q^{24} +31.5775 q^{26} -27.0000 q^{27} +97.6698 q^{28} +29.0000 q^{29} +82.1098 q^{31} -185.911 q^{32} +122.829 q^{33} +27.2701 q^{34} +41.3336 q^{36} -223.283 q^{37} +144.708 q^{38} -26.6957 q^{39} -274.422 q^{41} -226.401 q^{42} -53.7872 q^{43} -188.036 q^{44} +286.474 q^{46} -17.0236 q^{47} +238.946 q^{48} +109.272 q^{49} -23.0542 q^{51} +40.8678 q^{52} +23.0475 q^{53} -95.8124 q^{54} -257.145 q^{56} -122.336 q^{57} +102.910 q^{58} -399.574 q^{59} +388.308 q^{61} +291.376 q^{62} +191.400 q^{63} -22.5340 q^{64} +435.873 q^{66} -399.434 q^{67} +35.2930 q^{68} -242.186 q^{69} +92.2884 q^{71} -108.823 q^{72} -979.407 q^{73} -792.343 q^{74} +187.281 q^{76} -870.724 q^{77} -94.7326 q^{78} +62.2586 q^{79} +81.0000 q^{81} -973.817 q^{82} -163.597 q^{83} -293.009 q^{84} -190.870 q^{86} -87.0000 q^{87} +495.062 q^{88} +1371.94 q^{89} +189.243 q^{91} +370.755 q^{92} -246.330 q^{93} -60.4101 q^{94} +557.732 q^{96} -1263.64 q^{97} +387.763 q^{98} -368.488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) \) 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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