LMFDB - Embedded newform 1620.4.d.b.649.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(649,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1620.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.d (of order $2$, degree $1$, 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:	$95.5830942093$
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} - 2 x^{9} + 74 x^{8} + 552 x^{7} - 11155 x^{6} + 179300 x^{5} - 1394375 x^{4} + \cdots + 30517578125 $ x^10 - 2x^9 + 74x^8 + 552x^7 - 11155x^6 + 179300x^5 - 1394375x^4 + 8625000x^3 + 144531250x^2 - 488281250*x + 30517578125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$0.265737 - 11.1772i$ of defining polynomial
Character	$\chi$	$=$	1620.649
Dual form			1620.4.d.b.649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.265737 + 11.1772i) q^{5} +28.9801i q^{7} +O(q^{10})$	$q+(0.265737 + 11.1772i) q^{5} +28.9801i q^{7} -65.0045 q^{11} +31.0957i q^{13} -115.990i q^{17} +97.1930 q^{19} +56.2280i q^{23} +(-124.859 + 5.94039i) q^{25} -153.303 q^{29} -249.496 q^{31} +(-323.916 + 7.70110i) q^{35} +264.796i q^{37} -70.6145 q^{41} +63.6228i q^{43} -45.8720i q^{47} -496.847 q^{49} +247.021i q^{53} +(-17.2741 - 726.567i) q^{55} +716.658 q^{59} -584.385 q^{61} +(-347.562 + 8.26329i) q^{65} -489.315i q^{67} +245.450 q^{71} -549.757i q^{73} -1883.84i q^{77} +1064.40 q^{79} -979.423i q^{83} +(1296.44 - 30.8229i) q^{85} +793.118 q^{89} -901.157 q^{91} +(25.8278 + 1086.34i) q^{95} +1423.98i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5}+O(q^{10}) $ 10 * q + 2 * q^5	$ 10 q + 2 q^{5} - 56 q^{11} - 42 q^{19} - 144 q^{25} - 96 q^{29} - 44 q^{31} - 110 q^{35} + 298 q^{41} - 636 q^{49} - 88 q^{55} + 370 q^{59} - 232 q^{61} - 420 q^{65} - 672 q^{71} + 168 q^{79} + 1010 q^{85} + 2036 q^{89} + 6 q^{91} + 12 q^{95}+O(q^{100}) $ 10 * q + 2 * q^5 - 56 * q^11 - 42 * q^19 - 144 * q^25 - 96 * q^29 - 44 * q^31 - 110 * q^35 + 298 * q^41 - 636 * q^49 - 88 * q^55 + 370 * q^59 - 232 * q^61 - 420 * q^65 - 672 * q^71 + 168 * q^79 + 1010 * q^85 + 2036 * q^89 + 6 * q^91 + 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.265737	+	11.1772i	0.0237683	+	0.999717i
$5$	0.265737	+	11.1772i	0.0237683	+	0.999717i
$6$		0			0
$7$			28.9801i			1.56478i	0.622789	+	0.782390i	$0.285999\pi$
$7$			28.9801i			1.56478i	−0.622789	+	0.782390i	$0.714001\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−65.0045			−1.78178			−0.890891	−	0.454218i	$-0.849919\pi$
$11$	−65.0045			−1.78178			−0.890891	+	0.454218i	$0.849919\pi$
$12$		0			0
$13$			31.0957i			0.663415i	0.943382	+	0.331708i	$0.107625\pi$
$13$			31.0957i			0.663415i	−0.943382	+	0.331708i	$0.892375\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	115.990i		−	1.65481i	−0.561606	−	0.827405i	$-0.689816\pi$
$17$		−	115.990i		−	1.65481i	0.561606	−	0.827405i	$-0.310184\pi$
$18$		0			0
$19$	97.1930			1.17356			0.586779	−	0.809747i	$-0.300396\pi$
$19$	97.1930			1.17356			0.586779	+	0.809747i	$0.300396\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			56.2280i			0.509754i	0.966973	+	0.254877i	$0.0820350\pi$
$23$			56.2280i			0.509754i	−0.966973	+	0.254877i	$0.917965\pi$
$24$		0			0
$25$	−124.859	+	5.94039i	−0.998870	+	0.0475231i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−153.303			−0.981641			−0.490820	−	0.871261i	$-0.663303\pi$
$29$	−153.303			−0.981641			−0.490820	+	0.871261i	$0.663303\pi$
$30$		0			0
$31$	−249.496			−1.44551			−0.722754	−	0.691106i	$-0.757124\pi$
$31$	−249.496			−1.44551			−0.722754	+	0.691106i	$0.757124\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−323.916	+	7.70110i	−1.56434	+	0.0371921i
$36$		0			0
$37$			264.796i			1.17654i	0.808663	+	0.588272i	$0.200192\pi$
$37$			264.796i			1.17654i	−0.808663	+	0.588272i	$0.799808\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−70.6145			−0.268979			−0.134489	−	0.990915i	$-0.542939\pi$
$41$	−70.6145			−0.268979			−0.134489	+	0.990915i	$0.542939\pi$
$42$		0			0
$43$			63.6228i			0.225637i	0.993616	+	0.112818i	$0.0359878\pi$
$43$			63.6228i			0.225637i	−0.993616	+	0.112818i	$0.964012\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	45.8720i		−	0.142364i	−0.997463	−	0.0711822i	$-0.977323\pi$
$47$		−	45.8720i		−	0.142364i	0.997463	−	0.0711822i	$-0.0226772\pi$
$48$		0			0
$49$	−496.847			−1.44853
$50$		0			0
$51$		0			0
$52$		0			0
$53$			247.021i			0.640206i	0.947383	+	0.320103i	$0.103717\pi$
$53$			247.021i			0.640206i	−0.947383	+	0.320103i	$0.896283\pi$
$54$		0			0
$55$	−17.2741	−	726.567i	−0.0423499	−	1.78128i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	716.658			1.58137			0.790686	−	0.612222i	$-0.209724\pi$
$59$	716.658			1.58137			0.790686	+	0.612222i	$0.209724\pi$
$60$		0			0
$61$	−584.385			−1.22660			−0.613302	−	0.789849i	$-0.710159\pi$
$61$	−584.385			−1.22660			−0.613302	+	0.789849i	$0.710159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−347.562	+	8.26329i	−0.663228	+	0.0157682i
$66$		0			0
$67$		−	489.315i		−	0.892229i	−0.894976	−	0.446115i	$-0.852807\pi$
$67$		−	489.315i		−	0.892229i	0.894976	−	0.446115i	$-0.147193\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	245.450			0.410276			0.205138	−	0.978733i	$-0.434236\pi$
$71$	245.450			0.410276			0.205138	+	0.978733i	$0.434236\pi$
$72$		0			0
$73$		−	549.757i		−	0.881427i	−0.897648	−	0.440713i	$-0.854726\pi$
$73$		−	549.757i		−	0.881427i	0.897648	−	0.440713i	$-0.145274\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1883.84i		−	2.78809i
$78$		0			0
$79$	1064.40			1.51588			0.757939	−	0.652326i	$-0.226207\pi$
$79$	1064.40			1.51588			0.757939	+	0.652326i	$0.226207\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	979.423i		−	1.29525i	−0.761960	−	0.647624i	$-0.775763\pi$
$83$		−	979.423i		−	1.29525i	0.761960	−	0.647624i	$-0.224237\pi$
$84$		0			0
$85$	1296.44	−	30.8229i	1.65434	−	0.0393320i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	793.118			0.944610			0.472305	−	0.881435i	$-0.343422\pi$
$89$	793.118			0.944610			0.472305	+	0.881435i	$0.343422\pi$
$90$		0			0
$91$	−901.157			−1.03810
$92$		0			0
$93$		0			0
$94$		0			0
$95$	25.8278	+	1086.34i	0.0278935	+	1.17323i
$96$		0			0
$97$			1423.98i			1.49055i	0.666758	+	0.745275i	$0.267682\pi$
$97$			1423.98i			1.49055i	−0.666758	+	0.745275i	$0.732318\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−382.034			−0.376374			−0.188187	−	0.982133i	$-0.560261\pi$
$101$	−382.034			−0.376374			−0.188187	+	0.982133i	$0.560261\pi$
$102$		0			0
$103$			234.765i			0.224584i	0.993675	+	0.112292i	$0.0358192\pi$
$103$			234.765i			0.224584i	−0.993675	+	0.112292i	$0.964181\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	858.137i		−	0.775320i	−0.921802	−	0.387660i	$-0.873284\pi$
$107$		−	858.137i		−	0.775320i	0.921802	−	0.387660i	$-0.126716\pi$
$108$		0			0
$109$	−420.545			−0.369550			−0.184775	−	0.982781i	$-0.559156\pi$
$109$	−420.545			−0.369550			−0.184775	+	0.982781i	$0.559156\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			690.723i			0.575024i	0.957777	+	0.287512i	$0.0928282\pi$
$113$			690.723i			0.575024i	−0.957777	+	0.287512i	$0.907172\pi$
$114$		0			0
$115$	−628.471	+	14.9419i	−0.509610	+	0.0121160i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3361.41			2.58941
$120$		0			0
$121$	2894.58			2.17474
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−99.5765	−	1393.99i	−0.0712511	−	0.997458i
$126$		0			0
$127$			416.132i			0.290754i	0.989376	+	0.145377i	$0.0464395\pi$
$127$			416.132i			0.290754i	−0.989376	+	0.145377i	$0.953560\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1197.91			−0.798945			−0.399473	−	0.916745i	$-0.630807\pi$
$131$	−1197.91			−0.798945			−0.399473	+	0.916745i	$0.630807\pi$
$132$		0			0
$133$			2816.66i			1.83636i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1864.92i			1.16300i	0.813547	+	0.581500i	$0.197534\pi$
$137$			1864.92i			1.16300i	−0.813547	+	0.581500i	$0.802466\pi$
$138$		0			0
$139$	−2669.89			−1.62919			−0.814595	−	0.580030i	$-0.803041\pi$
$139$	−2669.89			−1.62919			−0.814595	+	0.580030i	$0.803041\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2021.36i		−	1.18206i
$144$		0			0
$145$	−40.7383	−	1713.49i	−0.0233319	−	0.981363i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−524.732			−0.288508			−0.144254	−	0.989541i	$-0.546078\pi$
$149$	−524.732			−0.288508			−0.144254	+	0.989541i	$0.546078\pi$
$150$		0			0
$151$	743.912			0.400919			0.200459	−	0.979702i	$-0.435757\pi$
$151$	743.912			0.400919			0.200459	+	0.979702i	$0.435757\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−66.3003	−	2788.66i	−0.0343572	−	1.44510i
$156$		0			0
$157$			3097.45i			1.57455i	0.616605	+	0.787273i	$0.288508\pi$
$157$			3097.45i			1.57455i	−0.616605	+	0.787273i	$0.711492\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1629.49			−0.797653
$162$		0			0
$163$		−	3408.38i		−	1.63782i	−0.573921	−	0.818911i	$-0.694578\pi$
$163$		−	3408.38i		−	1.63782i	0.573921	−	0.818911i	$-0.305422\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4097.41i		−	1.89860i	−0.314365	−	0.949302i	$-0.601792\pi$
$167$		−	4097.41i		−	1.89860i	0.314365	−	0.949302i	$-0.398208\pi$
$168$		0			0
$169$	1230.06			0.559880
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	3949.30i		−	1.73560i	−0.496910	−	0.867802i	$-0.665532\pi$
$173$		−	3949.30i		−	1.73560i	0.496910	−	0.867802i	$-0.334468\pi$
$174$		0			0
$175$	−172.153	−	3618.42i	−0.0743632	−	1.56301i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2132.87			0.890604			0.445302	−	0.895380i	$-0.353096\pi$
$179$	2132.87			0.890604			0.445302	+	0.895380i	$0.353096\pi$
$180$		0			0
$181$	4641.65			1.90614			0.953069	−	0.302753i	$-0.0979057\pi$
$181$	4641.65			1.90614			0.953069	+	0.302753i	$0.0979057\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2959.67	+	70.3661i	−1.17621	+	0.0279644i
$186$		0			0
$187$			7539.89i			2.94851i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	860.781			0.326094			0.163047	−	0.986618i	$-0.447868\pi$
$191$	860.781			0.326094			0.163047	+	0.986618i	$0.447868\pi$
$192$		0			0
$193$		−	4191.94i		−	1.56343i	−0.623633	−	0.781717i	$-0.714344\pi$
$193$		−	4191.94i		−	1.56343i	0.623633	−	0.781717i	$-0.285656\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			704.069i			0.254634i	0.991862	+	0.127317i	$0.0406365\pi$
$197$			704.069i			0.254634i	−0.991862	+	0.127317i	$0.959363\pi$
$198$		0			0
$199$	2820.73			1.00481			0.502403	−	0.864633i	$-0.332449\pi$
$199$	2820.73			1.00481			0.502403	+	0.864633i	$0.332449\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	4442.73i		−	1.53605i
$204$		0			0
$205$	−18.7649	−	789.271i	−0.00639317	−	0.268903i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6317.98			−2.09102
$210$		0			0
$211$	−3793.29			−1.23763			−0.618817	−	0.785535i	$-0.712388\pi$
$211$	−3793.29			−1.23763			−0.618817	+	0.785535i	$0.712388\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−711.123	+	16.9070i	−0.225573	+	0.00536300i
$216$		0			0
$217$		−	7230.41i		−	2.26190i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3606.80			1.09783
$222$		0			0
$223$			315.421i			0.0947181i	0.998878	+	0.0473591i	$0.0150805\pi$
$223$			315.421i			0.0947181i	−0.998878	+	0.0473591i	$0.984920\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1757.97i			0.514011i	0.966410	+	0.257005i	$0.0827358\pi$
$227$			1757.97i			0.514011i	−0.966410	+	0.257005i	$0.917264\pi$
$228$		0			0
$229$	1479.27			0.426870			0.213435	−	0.976957i	$-0.431535\pi$
$229$	1479.27			0.426870			0.213435	+	0.976957i	$0.431535\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2493.45i		−	0.701078i	−0.936548	−	0.350539i	$-0.885998\pi$
$233$		−	2493.45i		−	0.701078i	0.936548	−	0.350539i	$-0.114002\pi$
$234$		0			0
$235$	512.720	−	12.1899i	0.142324	−	0.00338376i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3658.38			−0.990129			−0.495064	−	0.868856i	$-0.664856\pi$
$239$	−3658.38			−0.990129			−0.495064	+	0.868856i	$0.664856\pi$
$240$		0			0
$241$	−111.150			−0.0297086			−0.0148543	−	0.999890i	$-0.504728\pi$
$241$	−111.150			−0.0297086			−0.0148543	+	0.999890i	$0.504728\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−132.031	−	5553.35i	−0.0344292	−	1.44813i
$246$		0			0
$247$			3022.28i			0.778556i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4256.57			−1.07041			−0.535203	−	0.844723i	$-0.679765\pi$
$251$	−4256.57			−1.07041			−0.535203	+	0.844723i	$0.679765\pi$
$252$		0			0
$253$		−	3655.07i		−	0.908271i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	4088.34i		−	0.992311i	−0.868234	−	0.496155i	$-0.834745\pi$
$257$		−	4088.34i		−	0.992311i	0.868234	−	0.496155i	$-0.165255\pi$
$258$		0			0
$259$	−7673.81			−1.84103
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1129.90i		−	0.264914i	−0.991189	−	0.132457i	$-0.957713\pi$
$263$		−	1129.90i		−	0.264914i	0.991189	−	0.132457i	$-0.0422867\pi$
$264$		0			0
$265$	−2760.99	+	65.6426i	−0.640025	+	0.0152166i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−742.505			−0.168295			−0.0841475	−	0.996453i	$-0.526817\pi$
$269$	−742.505			−0.168295			−0.0841475	+	0.996453i	$0.526817\pi$
$270$		0			0
$271$	−6391.72			−1.43273			−0.716364	−	0.697727i	$-0.754195\pi$
$271$	−6391.72			−1.43273			−0.716364	+	0.697727i	$0.754195\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8116.38	−	386.152i	1.77977	−	0.0846758i
$276$		0			0
$277$			5204.47i			1.12890i	0.825466	+	0.564452i	$0.190912\pi$
$277$			5204.47i			1.12890i	−0.825466	+	0.564452i	$0.809088\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1931.04			0.409951			0.204975	−	0.978767i	$-0.434289\pi$
$281$	1931.04			0.409951			0.204975	+	0.978767i	$0.434289\pi$
$282$		0			0
$283$		−	1385.85i		−	0.291096i	−0.989351	−	0.145548i	$-0.953506\pi$
$283$		−	1385.85i		−	0.291096i	0.989351	−	0.145548i	$-0.0464945\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	2046.42i		−	0.420893i
$288$		0			0
$289$	−8540.73			−1.73839
$290$		0			0
$291$		0			0
$292$		0			0
$293$			4151.26i			0.827711i	0.910343	+	0.413855i	$0.135818\pi$
$293$			4151.26i			0.827711i	−0.910343	+	0.413855i	$0.864182\pi$
$294$		0			0
$295$	190.443	+	8010.21i	0.0375865	+	1.58092i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1748.45			−0.338179
$300$		0			0
$301$	−1843.80			−0.353072
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−155.293	−	6531.78i	−0.0291543	−	1.22626i
$306$		0			0
$307$			1489.18i			0.276847i	0.990373	+	0.138423i	$0.0442035\pi$
$307$			1489.18i			0.276847i	−0.990373	+	0.138423i	$0.955797\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4103.95			0.748276			0.374138	−	0.927373i	$-0.377939\pi$
$311$	4103.95			0.748276			0.374138	+	0.927373i	$0.377939\pi$
$312$		0			0
$313$			9323.15i			1.68363i	0.539767	+	0.841814i	$0.318512\pi$
$313$			9323.15i			1.68363i	−0.539767	+	0.841814i	$0.681488\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	5556.40i		−	0.984475i	−0.870461	−	0.492237i	$-0.836179\pi$
$317$		−	5556.40i		−	0.984475i	0.870461	−	0.492237i	$-0.163821\pi$
$318$		0			0
$319$	9965.36			1.74907
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	11273.4i		−	1.94201i
$324$		0			0
$325$	−184.721	−	3882.57i	−0.0315276	−	0.662666i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1329.38			0.222769
$330$		0			0
$331$	5808.89			0.964608			0.482304	−	0.876004i	$-0.339800\pi$
$331$	5808.89			0.964608			0.482304	+	0.876004i	$0.339800\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5469.16	−	130.029i	0.891977	−	0.0212068i
$336$		0			0
$337$		−	5174.73i		−	0.836455i	−0.908342	−	0.418228i	$-0.862651\pi$
$337$		−	5174.73i		−	0.836455i	0.908342	−	0.418228i	$-0.137349\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16218.3			2.57558
$342$		0			0
$343$		−	4458.52i		−	0.701858i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	1378.72i		−	0.213296i	−0.994297	−	0.106648i	$-0.965988\pi$
$347$		−	1378.72i		−	0.213296i	0.994297	−	0.106648i	$-0.0340117\pi$
$348$		0			0
$349$	−6907.98			−1.05953			−0.529765	−	0.848145i	$-0.677720\pi$
$349$	−6907.98			−1.05953			−0.529765	+	0.848145i	$0.677720\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3589.29i			0.541185i	0.962694	+	0.270593i	$0.0872197\pi$
$353$			3589.29i			0.541185i	−0.962694	+	0.270593i	$0.912780\pi$
$354$		0			0
$355$	65.2253	+	2743.44i	0.00975155	+	0.410160i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1844.44			−0.271159			−0.135579	−	0.990767i	$-0.543290\pi$
$359$	−1844.44			−0.271159			−0.135579	+	0.990767i	$0.543290\pi$
$360$		0			0
$361$	2587.48			0.377239
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6144.73	−	146.091i	0.881178	−	0.0209500i
$366$		0			0
$367$			1794.81i			0.255281i	0.991821	+	0.127640i	$0.0407403\pi$
$367$			1794.81i			0.255281i	−0.991821	+	0.127640i	$0.959260\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7158.69			−1.00178
$372$		0			0
$373$			6603.38i			0.916649i	0.888785	+	0.458325i	$0.151550\pi$
$373$			6603.38i			0.916649i	−0.888785	+	0.458325i	$0.848450\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4767.05i		−	0.651235i
$378$		0			0
$379$	−7149.56			−0.968993			−0.484497	−	0.874793i	$-0.660997\pi$
$379$	−7149.56			−0.968993			−0.484497	+	0.874793i	$0.660997\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	9225.52i		−	1.23081i	−0.788209	−	0.615407i	$-0.788992\pi$
$383$		−	9225.52i		−	1.23081i	0.788209	−	0.615407i	$-0.211008\pi$
$384$		0			0
$385$	21056.0	−	500.606i	2.78731	−	0.0662682i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3948.50			0.514645			0.257322	−	0.966326i	$-0.417160\pi$
$389$	3948.50			0.514645			0.257322	+	0.966326i	$0.417160\pi$
$390$		0			0
$391$	6521.90			0.843546
$392$		0			0
$393$		0			0
$394$		0			0
$395$	282.851	+	11897.0i	0.0360298	+	1.51545i
$396$		0			0
$397$		−	5580.53i		−	0.705488i	−0.935720	−	0.352744i	$-0.885249\pi$
$397$		−	5580.53i		−	0.705488i	0.935720	−	0.352744i	$-0.114751\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3158.78			−0.393371			−0.196686	−	0.980467i	$-0.563018\pi$
$401$	−3158.78			−0.393371			−0.196686	+	0.980467i	$0.563018\pi$
$402$		0			0
$403$		−	7758.24i		−	0.958972i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	17212.9i		−	2.09634i
$408$		0			0
$409$	−6981.61			−0.844055			−0.422028	−	0.906583i	$-0.638681\pi$
$409$	−6981.61			−0.844055			−0.422028	+	0.906583i	$0.638681\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			20768.8i			2.47450i
$414$		0			0
$415$	10947.2	−	260.269i	1.29488	−	0.0307858i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16283.5			−1.89857			−0.949287	−	0.314410i	$-0.898193\pi$
$419$	−16283.5			−1.89857			−0.949287	+	0.314410i	$0.898193\pi$
$420$		0			0
$421$	−9861.57			−1.14162			−0.570811	−	0.821081i	$-0.693371\pi$
$421$	−9861.57			−1.14162			−0.570811	+	0.821081i	$0.693371\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	689.027	+	14482.4i	0.0786417	+	1.65294i
$426$		0			0
$427$		−	16935.6i		−	1.91936i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11997.0			−1.34078			−0.670390	−	0.742009i	$-0.733873\pi$
$431$	−11997.0			−1.34078			−0.670390	+	0.742009i	$0.733873\pi$
$432$		0			0
$433$		−	695.887i		−	0.0772337i	−0.999254	−	0.0386168i	$-0.987705\pi$
$433$		−	695.887i		−	0.0772337i	0.999254	−	0.0386168i	$-0.0122952\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5464.97i			0.598226i
$438$		0			0
$439$	−5206.64			−0.566058			−0.283029	−	0.959111i	$-0.591339\pi$
$439$	−5206.64			−0.566058			−0.283029	+	0.959111i	$0.591339\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			369.418i			0.0396198i	0.999804	+	0.0198099i	$0.00630610\pi$
$443$			369.418i			0.0396198i	−0.999804	+	0.0198099i	$0.993694\pi$
$444$		0			0
$445$	210.761	+	8864.82i	0.0224518	+	0.944344i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3637.82			0.382360			0.191180	−	0.981555i	$-0.438769\pi$
$449$	3637.82			0.382360			0.191180	+	0.981555i	$0.438769\pi$
$450$		0			0
$451$	4590.26			0.479261
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−239.471	−	10072.4i	−0.0246738	−	1.03781i
$456$		0			0
$457$		−	16601.6i		−	1.69932i	−0.527331	−	0.849660i	$-0.676807\pi$
$457$		−	16601.6i		−	1.69932i	0.527331	−	0.849660i	$-0.323193\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17833.4			1.80170			0.900850	−	0.434131i	$-0.142944\pi$
$461$	17833.4			1.80170			0.900850	+	0.434131i	$0.142944\pi$
$462$		0			0
$463$		−	7961.67i		−	0.799159i	−0.916699	−	0.399579i	$-0.869156\pi$
$463$		−	7961.67i		−	0.799159i	0.916699	−	0.399579i	$-0.130844\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	9295.41i		−	0.921071i	−0.887641	−	0.460535i	$-0.847657\pi$
$467$		−	9295.41i		−	0.921071i	0.887641	−	0.460535i	$-0.152343\pi$
$468$		0			0
$469$	14180.4			1.39614
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4135.77i		−	0.402036i
$474$		0			0
$475$	−12135.4	+	577.365i	−1.17223	+	0.0557712i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6204.04			−0.591795			−0.295898	−	0.955220i	$-0.595619\pi$
$479$	−6204.04			−0.591795			−0.295898	+	0.955220i	$0.595619\pi$
$480$		0			0
$481$	−8234.01			−0.780537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15916.1	+	378.405i	−1.49013	+	0.0354278i
$486$		0			0
$487$			13876.9i			1.29121i	0.763670	+	0.645607i	$0.223396\pi$
$487$			13876.9i			1.29121i	−0.763670	+	0.645607i	$0.776604\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15913.2			−1.46264			−0.731318	−	0.682037i	$-0.761094\pi$
$491$	−15913.2			−1.46264			−0.731318	+	0.682037i	$0.761094\pi$
$492$		0			0
$493$			17781.6i			1.62443i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			7113.18i			0.641991i
$498$		0			0
$499$	−4373.68			−0.392370			−0.196185	−	0.980567i	$-0.562855\pi$
$499$	−4373.68			−0.392370			−0.196185	+	0.980567i	$0.562855\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9752.94i		−	0.864537i	−0.901745	−	0.432268i	$-0.857713\pi$
$503$		−	9752.94i		−	0.864537i	0.901745	−	0.432268i	$-0.142287\pi$
$504$		0			0
$505$	−101.521	−	4270.06i	−0.00894576	−	0.376268i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16447.4			−1.43226			−0.716129	−	0.697968i	$-0.754088\pi$
$509$	−16447.4			−1.43226			−0.716129	+	0.697968i	$0.754088\pi$
$510$		0			0
$511$	15932.0			1.37924
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2624.02	+	62.3860i	−0.224520	+	0.00533797i
$516$		0			0
$517$			2981.89i			0.253662i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18516.3			−1.55703			−0.778517	−	0.627623i	$-0.784028\pi$
$521$	−18516.3			−1.55703			−0.778517	+	0.627623i	$0.784028\pi$
$522$		0			0
$523$			9786.74i			0.818249i	0.912478	+	0.409125i	$0.134166\pi$
$523$			9786.74i			0.818249i	−0.912478	+	0.409125i	$0.865834\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			28939.0i			2.39204i
$528$		0			0
$529$	9005.41			0.740150
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2195.81i		−	0.178445i
$534$		0			0
$535$	9591.55	−	228.039i	0.775101	−	0.0184280i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	32297.3			2.58097
$540$		0			0
$541$	11416.2			0.907247			0.453624	−	0.891193i	$-0.350131\pi$
$541$	11416.2			0.907247			0.453624	+	0.891193i	$0.350131\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−111.755	−	4700.51i	−0.00878356	−	0.369445i
$546$		0			0
$547$		−	13334.6i		−	1.04231i	−0.853461	−	0.521156i	$-0.825501\pi$
$547$		−	13334.6i		−	1.04231i	0.853461	−	0.521156i	$-0.174499\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−14899.9			−1.15201
$552$		0			0
$553$			30846.4i			2.37201i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			6761.43i			0.514346i	0.966365	+	0.257173i	$0.0827911\pi$
$557$			6761.43i			0.514346i	−0.966365	+	0.257173i	$0.917209\pi$
$558$		0			0
$559$	−1978.40			−0.149691
$560$		0			0
$561$		0			0
$562$		0			0
$563$			9913.12i			0.742075i	0.928618	+	0.371038i	$0.120998\pi$
$563$			9913.12i			0.742075i	−0.928618	+	0.371038i	$0.879002\pi$
$564$		0			0
$565$	−7720.34	+	183.551i	−0.574862	+	0.0136673i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2178.12			−0.160477			−0.0802385	−	0.996776i	$-0.525568\pi$
$569$	−2178.12			−0.160477			−0.0802385	+	0.996776i	$0.525568\pi$
$570$		0			0
$571$	9003.12			0.659840			0.329920	−	0.944009i	$-0.392978\pi$
$571$	9003.12			0.659840			0.329920	+	0.944009i	$0.392978\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−334.016	−	7020.56i	−0.0242251	−	0.509178i
$576$		0			0
$577$		−	14524.4i		−	1.04794i	−0.851738	−	0.523969i	$-0.824451\pi$
$577$		−	14524.4i		−	1.04794i	0.851738	−	0.523969i	$-0.175549\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	28383.8			2.02678
$582$		0			0
$583$		−	16057.5i		−	1.14071i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	12449.9i		−	0.875405i	−0.899120	−	0.437703i	$-0.855792\pi$
$587$		−	12449.9i		−	0.875405i	0.899120	−	0.437703i	$-0.144208\pi$
$588$		0			0
$589$	−24249.2			−1.69639
$590$		0			0
$591$		0			0
$592$		0			0
$593$			10201.8i			0.706470i	0.935535	+	0.353235i	$0.114918\pi$
$593$			10201.8i			0.706470i	−0.935535	+	0.353235i	$0.885082\pi$
$594$		0			0
$595$	893.253	+	37571.1i	0.0615459	+	2.58868i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15209.7			1.03748			0.518741	−	0.854932i	$-0.326401\pi$
$599$	15209.7			1.03748			0.518741	+	0.854932i	$0.326401\pi$
$600$		0			0
$601$	−864.886			−0.0587012			−0.0293506	−	0.999569i	$-0.509344\pi$
$601$	−864.886			−0.0587012			−0.0293506	+	0.999569i	$0.509344\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	769.200	+	32353.3i	0.0516899	+	2.17413i
$606$		0			0
$607$		−	211.775i		−	0.0141609i	−0.999975	−	0.00708046i	$-0.997746\pi$
$607$		−	211.775i		−	0.0141609i	0.999975	−	0.00708046i	$-0.00225380\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1426.42			0.0944467
$612$		0			0
$613$			26556.6i			1.74978i	0.484326	+	0.874888i	$0.339065\pi$
$613$			26556.6i			1.74978i	−0.484326	+	0.874888i	$0.660935\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			22094.4i			1.44163i	0.693127	+	0.720816i	$0.256233\pi$
$617$			22094.4i			1.44163i	−0.693127	+	0.720816i	$0.743767\pi$
$618$		0			0
$619$	3473.03			0.225514			0.112757	−	0.993623i	$-0.464032\pi$
$619$	3473.03			0.225514			0.112757	+	0.993623i	$0.464032\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			22984.7i			1.47811i
$624$		0			0
$625$	15554.4	−	1483.42i	0.995483	−	0.0949389i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	30713.7			1.94696
$630$		0			0
$631$	−7702.65			−0.485955			−0.242978	−	0.970032i	$-0.578124\pi$
$631$	−7702.65			−0.485955			−0.242978	+	0.970032i	$0.578124\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4651.19	+	110.582i	−0.290672	+	0.00691072i
$636$		0			0
$637$		−	15449.8i		−	0.960980i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4122.37			0.254015			0.127008	−	0.991902i	$-0.459463\pi$
$641$	4122.37			0.254015			0.127008	+	0.991902i	$0.459463\pi$
$642$		0			0
$643$		−	20162.1i		−	1.23657i	−0.785953	−	0.618286i	$-0.787827\pi$
$643$		−	20162.1i		−	1.23657i	0.785953	−	0.618286i	$-0.212173\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			17801.3i			1.08167i	0.841128	+	0.540836i	$0.181892\pi$
$647$			17801.3i			1.08167i	−0.841128	+	0.540836i	$0.818108\pi$
$648$		0			0
$649$	−46586.0			−2.81766
$650$		0			0
$651$		0			0
$652$		0			0
$653$			9785.37i			0.586418i	0.956048	+	0.293209i	$0.0947232\pi$
$653$			9785.37i			0.586418i	−0.956048	+	0.293209i	$0.905277\pi$
$654$		0			0
$655$	−318.330	−	13389.3i	−0.0189896	−	0.798720i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2546.92			0.150552			0.0752760	−	0.997163i	$-0.476016\pi$
$659$	2546.92			0.150552			0.0752760	+	0.997163i	$0.476016\pi$
$660$		0			0
$661$	27359.8			1.60995			0.804973	−	0.593312i	$-0.202180\pi$
$661$	27359.8			1.60995			0.804973	+	0.593312i	$0.202180\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−31482.4	+	748.493i	−1.83584	+	0.0436471i
$666$		0			0
$667$		−	8619.90i		−	0.500396i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	37987.7			2.18554
$672$		0			0
$673$		−	28498.0i		−	1.63227i	−0.577860	−	0.816136i	$-0.696112\pi$
$673$		−	28498.0i		−	1.63227i	0.577860	−	0.816136i	$-0.303888\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			21448.4i			1.21762i	0.793315	+	0.608811i	$0.208353\pi$
$677$			21448.4i			1.21762i	−0.793315	+	0.608811i	$0.791647\pi$
$678$		0			0
$679$	−41267.1			−2.33238
$680$		0			0
$681$		0			0
$682$		0			0
$683$			17257.8i			0.966839i	0.875389	+	0.483419i	$0.160605\pi$
$683$			17257.8i			0.966839i	−0.875389	+	0.483419i	$0.839395\pi$
$684$		0			0
$685$	−20844.5	+	495.579i	−1.16267	+	0.0276425i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7681.28			−0.424722
$690$		0			0
$691$	300.818			0.0165610			0.00828051	−	0.999966i	$-0.497364\pi$
$691$	300.818			0.0165610			0.00828051	+	0.999966i	$0.497364\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−709.491	−	29841.9i	−0.0387230	−	1.62873i
$696$		0			0
$697$			8190.59i			0.445109i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12714.8			−0.685065			−0.342532	−	0.939506i	$-0.611285\pi$
$701$	−12714.8			−0.685065			−0.342532	+	0.939506i	$0.611285\pi$
$702$		0			0
$703$			25736.3i			1.38074i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	11071.4i		−	0.588942i
$708$		0			0
$709$	−20519.8			−1.08694			−0.543469	−	0.839430i	$-0.682889\pi$
$709$	−20519.8			−1.08694			−0.543469	+	0.839430i	$0.682889\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	14028.6i		−	0.736854i
$714$		0			0
$715$	22593.1	−	537.151i	1.18173	−	0.0280956i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14933.0			0.774556			0.387278	−	0.921963i	$-0.373415\pi$
$719$	14933.0			0.774556			0.387278	+	0.921963i	$0.373415\pi$
$720$		0			0
$721$	−6803.53			−0.351424
$722$		0			0
$723$		0			0
$724$		0			0
$725$	19141.2	−	910.678i	0.980532	−	0.0466506i
$726$		0			0
$727$		−	30061.1i		−	1.53357i	−0.641906	−	0.766784i	$-0.721856\pi$
$727$		−	30061.1i		−	1.53357i	0.641906	−	0.766784i	$-0.278144\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7379.62			0.373386
$732$		0			0
$733$			5364.37i			0.270310i	0.990824	+	0.135155i	$0.0431533\pi$
$733$			5364.37i			0.270310i	−0.990824	+	0.135155i	$0.956847\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			31807.7i			1.58976i
$738$		0			0
$739$	−9318.57			−0.463855			−0.231928	−	0.972733i	$-0.574503\pi$
$739$	−9318.57			−0.463855			−0.231928	+	0.972733i	$0.574503\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24494.4i		−	1.20944i	−0.796438	−	0.604720i	$-0.793285\pi$
$743$		−	24494.4i		−	1.20944i	0.796438	−	0.604720i	$-0.206715\pi$
$744$		0			0
$745$	−139.441	−	5865.03i	−0.00685735	−	0.288427i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24868.9			1.21320
$750$		0			0
$751$	5587.58			0.271496			0.135748	−	0.990743i	$-0.456656\pi$
$751$	5587.58			0.271496			0.135748	+	0.990743i	$0.456656\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	197.685	+	8314.84i	0.00952915	+	0.400805i
$756$		0			0
$757$		−	2805.57i		−	0.134703i	−0.997729	−	0.0673516i	$-0.978545\pi$
$757$		−	2805.57i		−	0.134703i	0.997729	−	0.0673516i	$-0.0214549\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−33191.1			−1.58105			−0.790523	−	0.612433i	$-0.790191\pi$
$761$	−33191.1			−1.58105			−0.790523	+	0.612433i	$0.790191\pi$
$762$		0			0
$763$		−	12187.4i		−	0.578264i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			22285.0i			1.04911i
$768$		0			0
$769$	−4081.45			−0.191393			−0.0956963	−	0.995411i	$-0.530508\pi$
$769$	−4081.45			−0.191393			−0.0956963	+	0.995411i	$0.530508\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			17542.3i			0.816238i	0.912929	+	0.408119i	$0.133815\pi$
$773$			17542.3i			0.816238i	−0.912929	+	0.408119i	$0.866185\pi$
$774$		0			0
$775$	31151.7	−	1482.10i	1.44387	−	0.0686951i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6863.24			−0.315662
$780$		0			0
$781$	−15955.4			−0.731022
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−34620.8	+	823.109i	−1.57410	+	0.0374242i
$786$		0			0
$787$			25436.7i			1.15212i	0.817407	+	0.576061i	$0.195411\pi$
$787$			25436.7i			1.15212i	−0.817407	+	0.576061i	$0.804589\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−20017.2			−0.899787
$792$		0			0
$793$		−	18171.9i		−	0.813748i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			32245.7i			1.43313i	0.697522	+	0.716564i	$0.254286\pi$
$797$			32245.7i			1.43313i	−0.697522	+	0.716564i	$0.745714\pi$
$798$		0			0
$799$	−5320.71			−0.235586
$800$		0			0
$801$		0			0
$802$		0			0
$803$			35736.7i			1.57051i
$804$		0			0
$805$	−433.018	−	18213.2i	−0.0189588	−	0.797428i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18379.4			0.798744			0.399372	−	0.916789i	$-0.369228\pi$
$809$	18379.4			0.798744			0.399372	+	0.916789i	$0.369228\pi$
$810$		0			0
$811$	4360.73			0.188811			0.0944057	−	0.995534i	$-0.469905\pi$
$811$	4360.73			0.188811			0.0944057	+	0.995534i	$0.469905\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	38096.1	−	905.734i	1.63736	−	0.0389282i
$816$		0			0
$817$			6183.69i			0.264798i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−44001.7			−1.87049			−0.935244	−	0.354003i	$-0.884820\pi$
$821$	−44001.7			−1.87049			−0.935244	+	0.354003i	$0.884820\pi$
$822$		0			0
$823$		−	25590.2i		−	1.08386i	−0.840422	−	0.541932i	$-0.817693\pi$
$823$		−	25590.2i		−	1.08386i	0.840422	−	0.541932i	$-0.182307\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	5226.69i		−	0.219770i	−0.993944	−	0.109885i	$-0.964952\pi$
$827$		−	5226.69i		−	0.219770i	0.993944	−	0.109885i	$-0.0350483\pi$
$828$		0			0
$829$	13745.4			0.575871			0.287936	−	0.957650i	$-0.407031\pi$
$829$	13745.4			0.575871			0.287936	+	0.957650i	$0.407031\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			57629.4i			2.39705i
$834$		0			0
$835$	45797.5	−	1088.83i	1.89807	−	0.0451266i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19554.5			−0.804643			−0.402322	−	0.915498i	$-0.631797\pi$
$839$	−19554.5			−0.804643			−0.402322	+	0.915498i	$0.631797\pi$
$840$		0			0
$841$	−887.306			−0.0363814
$842$		0			0
$843$		0			0
$844$		0			0
$845$	326.872	+	13748.6i	0.0133074	+	0.559722i
$846$		0			0
$847$			83885.4i			3.40299i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14888.9			−0.599748
$852$		0			0
$853$		−	40556.2i		−	1.62792i	−0.580920	−	0.813961i	$-0.697307\pi$
$853$		−	40556.2i		−	1.62792i	0.580920	−	0.813961i	$-0.302693\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			22482.7i			0.896142i	0.893998	+	0.448071i	$0.147889\pi$
$857$			22482.7i			0.896142i	−0.893998	+	0.448071i	$0.852111\pi$
$858$		0			0
$859$	−21844.4			−0.867661			−0.433831	−	0.900994i	$-0.642838\pi$
$859$	−21844.4			−0.867661			−0.433831	+	0.900994i	$0.642838\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	48697.1i		−	1.92082i	−0.278588	−	0.960411i	$-0.589866\pi$
$863$		−	48697.1i		−	1.92082i	0.278588	−	0.960411i	$-0.410134\pi$
$864$		0			0
$865$	44142.0	−	1049.48i	1.73511	−	0.0412523i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−69190.8			−2.70096
$870$		0			0
$871$	15215.6			0.591918
$872$		0			0
$873$		0			0
$874$		0			0
$875$	40398.0	−	2885.74i	1.56080	−	0.111492i
$876$		0			0
$877$			10545.0i			0.406022i	0.979177	+	0.203011i	$0.0650726\pi$
$877$			10545.0i			0.406022i	−0.979177	+	0.203011i	$0.934927\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7971.56			−0.304845			−0.152423	−	0.988315i	$-0.548707\pi$
$881$	−7971.56			−0.304845			−0.152423	+	0.988315i	$0.548707\pi$
$882$		0			0
$883$			46532.0i			1.77342i	0.462328	+	0.886709i	$0.347014\pi$
$883$			46532.0i			1.77342i	−0.462328	+	0.886709i	$0.652986\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	27733.8i		−	1.04984i	−0.851152	−	0.524920i	$-0.824095\pi$
$887$		−	27733.8i		−	1.04984i	0.851152	−	0.524920i	$-0.175905\pi$
$888$		0			0
$889$	−12059.6			−0.454966
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	4458.44i		−	0.167073i
$894$		0			0
$895$	566.783	+	23839.5i	0.0211681	+	0.890352i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	38248.3			1.41897
$900$		0			0
$901$	28652.0			1.05942
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1233.46	+	51880.5i	0.0453056	+	1.90560i
$906$		0			0
$907$			28760.4i			1.05289i	0.850208	+	0.526446i	$0.176476\pi$
$907$			28760.4i			1.05289i	−0.850208	+	0.526446i	$0.823524\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	32523.8			1.18283			0.591417	−	0.806366i	$-0.298569\pi$
$911$	32523.8			1.18283			0.591417	+	0.806366i	$0.298569\pi$
$912$		0			0
$913$			63666.9i			2.30785i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	34715.6i		−	1.25017i
$918$		0			0
$919$	4986.86			0.179000			0.0895002	−	0.995987i	$-0.471473\pi$
$919$	4986.86			0.179000			0.0895002	+	0.995987i	$0.471473\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			7632.45i			0.272183i
$924$		0			0
$925$	−1572.99	−	33062.1i	−0.0559131	−	1.17521i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−36837.1			−1.30095			−0.650477	−	0.759526i	$-0.725431\pi$
$929$	−36837.1			−1.30095			−0.650477	+	0.759526i	$0.725431\pi$
$930$		0			0
$931$	−48290.1			−1.69994
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−84274.7	+	2003.63i	−2.94768	+	0.0700810i
$936$		0			0
$937$			18905.9i			0.659155i	0.944129	+	0.329577i	$0.106906\pi$
$937$			18905.9i			0.659155i	−0.944129	+	0.329577i	$0.893094\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13353.1			0.462592			0.231296	−	0.972883i	$-0.425703\pi$
$941$	13353.1			0.462592			0.231296	+	0.972883i	$0.425703\pi$
$942$		0			0
$943$		−	3970.51i		−	0.137113i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	25564.5i		−	0.877229i	−0.898675	−	0.438615i	$-0.855469\pi$
$947$		−	25564.5i		−	0.877229i	0.898675	−	0.438615i	$-0.144531\pi$
$948$		0			0
$949$	17095.1			0.584752
$950$		0			0
$951$		0			0
$952$		0			0
$953$			41308.7i			1.40411i	0.712121	+	0.702057i	$0.247735\pi$
$953$			41308.7i			1.40411i	−0.712121	+	0.702057i	$0.752265\pi$
$954$		0			0
$955$	228.742	+	9621.10i	0.00775069	+	0.326002i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−54045.6			−1.81984
$960$		0			0
$961$	32457.1			1.08949
$962$		0			0
$963$		0			0
$964$		0			0
$965$	46854.1	−	1113.96i	1.56299	−	0.0371601i
$966$		0			0
$967$		−	17406.0i		−	0.578840i	−0.957202	−	0.289420i	$-0.906538\pi$
$967$		−	17406.0i		−	0.578840i	0.957202	−	0.289420i	$-0.0934624\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33608.1			1.11075			0.555373	−	0.831601i	$-0.312575\pi$
$971$	33608.1			1.11075			0.555373	+	0.831601i	$0.312575\pi$
$972$		0			0
$973$		−	77373.8i		−	2.54932i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	57195.5i		−	1.87292i	−0.350770	−	0.936461i	$-0.614080\pi$
$977$		−	57195.5i		−	1.87292i	0.350770	−	0.936461i	$-0.385920\pi$
$978$		0			0
$979$	−51556.2			−1.68309
$980$		0			0
$981$		0			0
$982$		0			0
$983$			16739.1i			0.543127i	0.962421	+	0.271563i	$0.0875406\pi$
$983$			16739.1i			0.543127i	−0.962421	+	0.271563i	$0.912459\pi$
$984$		0			0
$985$	−7869.51	+	187.098i	−0.254562	+	0.00605221i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3577.38			−0.115019
$990$		0			0
$991$	18256.4			0.585201			0.292600	−	0.956235i	$-0.405479\pi$
$991$	18256.4			0.585201			0.292600	+	0.956235i	$0.405479\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	749.575	+	31527.9i	0.0238825	+	1.00452i
$996$		0			0
$997$			13907.3i			0.441774i	0.975299	+	0.220887i	$0.0708951\pi$
$997$			13907.3i			0.441774i	−0.975299	+	0.220887i	$0.929105\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.d.b.649.6	yes	10
3.2	odd	2		1620.4.d.a.649.5	✓	10
5.4	even	2	inner	1620.4.d.b.649.5	yes	10
15.14	odd	2		1620.4.d.a.649.6	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.d.a.649.5	✓	10	3.2	odd	2
1620.4.d.a.649.6	yes	10	15.14	odd	2
1620.4.d.b.649.5	yes	10	5.4	even	2	inner
1620.4.d.b.649.6	yes	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.265737 + 11.1772i) q^{5} +28.9801i q^{7} +O(q^{10})\)	\(q+(0.265737 + 11.1772i) q^{5} +28.9801i q^{7} -65.0045 q^{11} +31.0957i q^{13} -115.990i q^{17} +97.1930 q^{19} +56.2280i q^{23} +(-124.859 + 5.94039i) q^{25} -153.303 q^{29} -249.496 q^{31} +(-323.916 + 7.70110i) q^{35} +264.796i q^{37} -70.6145 q^{41} +63.6228i q^{43} -45.8720i q^{47} -496.847 q^{49} +247.021i q^{53} +(-17.2741 - 726.567i) q^{55} +716.658 q^{59} -584.385 q^{61} +(-347.562 + 8.26329i) q^{65} -489.315i q^{67} +245.450 q^{71} -549.757i q^{73} -1883.84i q^{77} +1064.40 q^{79} -979.423i q^{83} +(1296.44 - 30.8229i) q^{85} +793.118 q^{89} -901.157 q^{91} +(25.8278 + 1086.34i) q^{95} +1423.98i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5}+O(q^{10}) \) 10 * q + 2 * q^5	\( 10 q + 2 q^{5} - 56 q^{11} - 42 q^{19} - 144 q^{25} - 96 q^{29} - 44 q^{31} - 110 q^{35} + 298 q^{41} - 636 q^{49} - 88 q^{55} + 370 q^{59} - 232 q^{61} - 420 q^{65} - 672 q^{71} + 168 q^{79} + 1010 q^{85} + 2036 q^{89} + 6 q^{91} + 12 q^{95}+O(q^{100}) \) 10 * q + 2 * q^5 - 56 * q^11 - 42 * q^19 - 144 * q^25 - 96 * q^29 - 44 * q^31 - 110 * q^35 + 298 * q^41 - 636 * q^49 - 88 * q^55 + 370 * q^59 - 232 * q^61 - 420 * q^65 - 672 * q^71 + 168 * q^79 + 1010 * q^85 + 2036 * q^89 + 6 * q^91 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more