LMFDB - Embedded newform 1710.4.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,4,Mod(1,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, 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("1710.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1710.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:	$100.893266110$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 410x^{2} + 4362x - 12540 $ x^4 - x^3 - 410x^2 + 4362x - 12540
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.58500$ of defining polynomial
Character	$\chi$	$=$	1710.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +36.1008 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +36.1008 q^{7} -8.00000 q^{8} +10.0000 q^{10} +0.930760 q^{11} +45.8117 q^{13} -72.2015 q^{14} +16.0000 q^{16} +103.133 q^{17} +19.0000 q^{19} -20.0000 q^{20} -1.86152 q^{22} -17.0195 q^{23} +25.0000 q^{25} -91.6235 q^{26} +144.403 q^{28} +307.261 q^{29} +38.1135 q^{31} -32.0000 q^{32} -206.267 q^{34} -180.504 q^{35} +59.7728 q^{37} -38.0000 q^{38} +40.0000 q^{40} +269.284 q^{41} -351.964 q^{43} +3.72304 q^{44} +34.0389 q^{46} -472.748 q^{47} +960.265 q^{49} -50.0000 q^{50} +183.247 q^{52} +685.438 q^{53} -4.65380 q^{55} -288.806 q^{56} -614.522 q^{58} -272.918 q^{59} +258.765 q^{61} -76.2269 q^{62} +64.0000 q^{64} -229.059 q^{65} -630.625 q^{67} +412.534 q^{68} +361.008 q^{70} -133.460 q^{71} +631.242 q^{73} -119.546 q^{74} +76.0000 q^{76} +33.6011 q^{77} +640.921 q^{79} -80.0000 q^{80} -538.568 q^{82} -192.203 q^{83} -515.667 q^{85} +703.929 q^{86} -7.44608 q^{88} -1574.22 q^{89} +1653.84 q^{91} -68.0778 q^{92} +945.497 q^{94} -95.0000 q^{95} -1406.62 q^{97} -1920.53 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 36 q^{7} - 32 q^{8}+O(q^{10}) $ 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 36 * q^7 - 32 * q^8	$ 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 36 q^{7} - 32 q^{8} + 40 q^{10} - 54 q^{11} + 46 q^{13} - 72 q^{14} + 64 q^{16} - 14 q^{17} + 76 q^{19} - 80 q^{20} + 108 q^{22} - 104 q^{23} + 100 q^{25} - 92 q^{26} + 144 q^{28} - 14 q^{29} + 30 q^{31} - 128 q^{32} + 28 q^{34} - 180 q^{35} + 30 q^{37} - 152 q^{38} + 160 q^{40} + 36 q^{41} + 102 q^{43} - 216 q^{44} + 208 q^{46} - 408 q^{47} + 480 q^{49} - 200 q^{50} + 184 q^{52} + 176 q^{53} + 270 q^{55} - 288 q^{56} + 28 q^{58} - 66 q^{59} + 60 q^{61} - 60 q^{62} + 256 q^{64} - 230 q^{65} - 152 q^{67} - 56 q^{68} + 360 q^{70} - 172 q^{71} + 284 q^{73} - 60 q^{74} + 304 q^{76} + 300 q^{77} + 554 q^{79} - 320 q^{80} - 72 q^{82} + 394 q^{83} + 70 q^{85} - 204 q^{86} + 432 q^{88} + 60 q^{89} + 32 q^{91} - 416 q^{92} + 816 q^{94} - 380 q^{95} - 922 q^{97} - 960 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 36 * q^7 - 32 * q^8 + 40 * q^10 - 54 * q^11 + 46 * q^13 - 72 * q^14 + 64 * q^16 - 14 * q^17 + 76 * q^19 - 80 * q^20 + 108 * q^22 - 104 * q^23 + 100 * q^25 - 92 * q^26 + 144 * q^28 - 14 * q^29 + 30 * q^31 - 128 * q^32 + 28 * q^34 - 180 * q^35 + 30 * q^37 - 152 * q^38 + 160 * q^40 + 36 * q^41 + 102 * q^43 - 216 * q^44 + 208 * q^46 - 408 * q^47 + 480 * q^49 - 200 * q^50 + 184 * q^52 + 176 * q^53 + 270 * q^55 - 288 * q^56 + 28 * q^58 - 66 * q^59 + 60 * q^61 - 60 * q^62 + 256 * q^64 - 230 * q^65 - 152 * q^67 - 56 * q^68 + 360 * q^70 - 172 * q^71 + 284 * q^73 - 60 * q^74 + 304 * q^76 + 300 * q^77 + 554 * q^79 - 320 * q^80 - 72 * q^82 + 394 * q^83 + 70 * q^85 - 204 * q^86 + 432 * q^88 + 60 * q^89 + 32 * q^91 - 416 * q^92 + 816 * q^94 - 380 * q^95 - 922 * q^97 - 960 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	36.1008	1.94926	0.974629	−	0.223826i	$-0.0718549\pi$
$7$	36.1008	1.94926	0.974629	+	0.223826i	$0.0718549\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	0.930760	0.0255122	0.0127561	−	0.999919i	$-0.495939\pi$
$11$	0.930760	0.0255122	0.0127561	+	0.999919i	$0.495939\pi$
$12$	0	0
$13$	45.8117	0.977376	0.488688	−	0.872459i	$-0.337476\pi$
$13$	45.8117	0.977376	0.488688	+	0.872459i	$0.337476\pi$
$14$	−72.2015	−1.37833
$15$	0	0
$16$	16.0000	0.250000
$17$	103.133	1.47138	0.735692	−	0.677316i	$-0.236857\pi$
$17$	103.133	1.47138	0.735692	+	0.677316i	$0.236857\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−1.86152	−0.0180399
$23$	−17.0195	−0.154296	−0.0771479	−	0.997020i	$-0.524581\pi$
$23$	−17.0195	−0.154296	−0.0771479	+	0.997020i	$0.524581\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−91.6235	−0.691109
$27$	0	0
$28$	144.403	0.974629
$29$	307.261	1.96748	0.983740	−	0.179601i	$-0.0574807\pi$
$29$	307.261	1.96748	0.983740	+	0.179601i	$0.0574807\pi$
$30$	0	0
$31$	38.1135	0.220819	0.110409	−	0.993886i	$-0.464784\pi$
$31$	38.1135	0.220819	0.110409	+	0.993886i	$0.464784\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−206.267	−1.04043
$35$	−180.504	−0.871735
$36$	0	0
$37$	59.7728	0.265584	0.132792	−	0.991144i	$-0.457606\pi$
$37$	59.7728	0.265584	0.132792	+	0.991144i	$0.457606\pi$
$38$	−38.0000	−0.162221
$39$	0	0
$40$	40.0000	0.158114
$41$	269.284	1.02573	0.512867	−	0.858468i	$-0.328583\pi$
$41$	269.284	1.02573	0.512867	+	0.858468i	$0.328583\pi$
$42$	0	0
$43$	−351.964	−1.24823	−0.624117	−	0.781331i	$-0.714541\pi$
$43$	−351.964	−1.24823	−0.624117	+	0.781331i	$0.714541\pi$
$44$	3.72304	0.0127561
$45$	0	0
$46$	34.0389	0.109104
$47$	−472.748	−1.46718	−0.733590	−	0.679593i	$-0.762157\pi$
$47$	−472.748	−1.46718	−0.733590	+	0.679593i	$0.762157\pi$
$48$	0	0
$49$	960.265	2.79961
$50$	−50.0000	−0.141421
$51$	0	0
$52$	183.247	0.488688
$53$	685.438	1.77646	0.888228	−	0.459404i	$-0.151937\pi$
$53$	685.438	1.77646	0.888228	+	0.459404i	$0.151937\pi$
$54$	0	0
$55$	−4.65380	−0.0114094
$56$	−288.806	−0.689167
$57$	0	0
$58$	−614.522	−1.39122
$59$	−272.918	−0.602218	−0.301109	−	0.953590i	$-0.597357\pi$
$59$	−272.918	−0.602218	−0.301109	+	0.953590i	$0.597357\pi$
$60$	0	0
$61$	258.765	0.543140	0.271570	−	0.962419i	$-0.412457\pi$
$61$	258.765	0.543140	0.271570	+	0.962419i	$0.412457\pi$
$62$	−76.2269	−0.156142
$63$	0	0
$64$	64.0000	0.125000
$65$	−229.059	−0.437096
$66$	0	0
$67$	−630.625	−1.14990	−0.574948	−	0.818190i	$-0.694978\pi$
$67$	−630.625	−1.14990	−0.574948	+	0.818190i	$0.694978\pi$
$68$	412.534	0.735692
$69$	0	0
$70$	361.008	0.616410
$71$	−133.460	−0.223081	−0.111540	−	0.993760i	$-0.535578\pi$
$71$	−133.460	−0.223081	−0.111540	+	0.993760i	$0.535578\pi$
$72$	0	0
$73$	631.242	1.01207	0.506036	−	0.862512i	$-0.331110\pi$
$73$	631.242	1.01207	0.506036	+	0.862512i	$0.331110\pi$
$74$	−119.546	−0.187796
$75$	0	0
$76$	76.0000	0.114708
$77$	33.6011	0.0497300
$78$	0	0
$79$	640.921	0.912776	0.456388	−	0.889781i	$-0.349143\pi$
$79$	640.921	0.912776	0.456388	+	0.889781i	$0.349143\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	−538.568	−0.725303
$83$	−192.203	−0.254181	−0.127091	−	0.991891i	$-0.540564\pi$
$83$	−192.203	−0.254181	−0.127091	+	0.991891i	$0.540564\pi$
$84$	0	0
$85$	−515.667	−0.658023
$86$	703.929	0.882635
$87$	0	0
$88$	−7.44608	−0.00901994
$89$	−1574.22	−1.87491	−0.937456	−	0.348103i	$-0.886826\pi$
$89$	−1574.22	−1.87491	−0.937456	+	0.348103i	$0.886826\pi$
$90$	0	0
$91$	1653.84	1.90516
$92$	−68.0778	−0.0771479
$93$	0	0
$94$	945.497	1.03745
$95$	−95.0000	−0.102598
$96$	0	0
$97$	−1406.62	−1.47238	−0.736188	−	0.676777i	$-0.763376\pi$
$97$	−1406.62	−1.47238	−0.736188	+	0.676777i	$0.763376\pi$
$98$	−1920.53	−1.97962
$99$	0	0
$100$	100.000	0.100000
$101$	993.051	0.978340	0.489170	−	0.872189i	$-0.337300\pi$
$101$	993.051	0.978340	0.489170	+	0.872189i	$0.337300\pi$
$102$	0	0
$103$	−597.901	−0.571970	−0.285985	−	0.958234i	$-0.592321\pi$
$103$	−597.901	−0.571970	−0.285985	+	0.958234i	$0.592321\pi$
$104$	−366.494	−0.345555
$105$	0	0
$106$	−1370.88	−1.25614
$107$	−2123.32	−1.91840	−0.959202	−	0.282721i	$-0.908763\pi$
$107$	−2123.32	−1.91840	−0.959202	+	0.282721i	$0.908763\pi$
$108$	0	0
$109$	−1282.57	−1.12704	−0.563522	−	0.826101i	$-0.690554\pi$
$109$	−1282.57	−1.12704	−0.563522	+	0.826101i	$0.690554\pi$
$110$	9.30760	0.00806768
$111$	0	0
$112$	577.612	0.487315
$113$	1435.84	1.19533	0.597664	−	0.801746i	$-0.296095\pi$
$113$	1435.84	1.19533	0.597664	+	0.801746i	$0.296095\pi$
$114$	0	0
$115$	85.0973	0.0690031
$116$	1229.04	0.983740
$117$	0	0
$118$	545.836	0.425832
$119$	3723.20	2.86811
$120$	0	0
$121$	−1330.13	−0.999349
$122$	−517.531	−0.384058
$123$	0	0
$124$	152.454	0.110409
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−103.018	−0.0719794	−0.0359897	−	0.999352i	$-0.511458\pi$
$127$	−103.018	−0.0719794	−0.0359897	+	0.999352i	$0.511458\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	458.117	0.309073
$131$	263.824	0.175957	0.0879787	−	0.996122i	$-0.471959\pi$
$131$	263.824	0.175957	0.0879787	+	0.996122i	$0.471959\pi$
$132$	0	0
$133$	685.915	0.447190
$134$	1261.25	0.813100
$135$	0	0
$136$	−825.068	−0.520213
$137$	834.524	0.520425	0.260213	−	0.965551i	$-0.416207\pi$
$137$	834.524	0.520425	0.260213	+	0.965551i	$0.416207\pi$
$138$	0	0
$139$	985.778	0.601529	0.300765	−	0.953698i	$-0.402758\pi$
$139$	985.778	0.601529	0.300765	+	0.953698i	$0.402758\pi$
$140$	−722.015	−0.435867
$141$	0	0
$142$	266.919	0.157742
$143$	42.6397	0.0249351
$144$	0	0
$145$	−1536.30	−0.879883
$146$	−1262.48	−0.715643
$147$	0	0
$148$	239.091	0.132792
$149$	−2555.14	−1.40487	−0.702433	−	0.711750i	$-0.747903\pi$
$149$	−2555.14	−1.40487	−0.702433	+	0.711750i	$0.747903\pi$
$150$	0	0
$151$	1527.25	0.823086	0.411543	−	0.911390i	$-0.364990\pi$
$151$	1527.25	0.823086	0.411543	+	0.911390i	$0.364990\pi$
$152$	−152.000	−0.0811107
$153$	0	0
$154$	−67.2023	−0.0351644
$155$	−190.567	−0.0987531
$156$	0	0
$157$	−1159.89	−0.589613	−0.294807	−	0.955557i	$-0.595255\pi$
$157$	−1159.89	−0.589613	−0.294807	+	0.955557i	$0.595255\pi$
$158$	−1281.84	−0.645430
$159$	0	0
$160$	160.000	0.0790569
$161$	−614.415	−0.300762
$162$	0	0
$163$	755.501	0.363039	0.181520	−	0.983387i	$-0.441898\pi$
$163$	755.501	0.363039	0.181520	+	0.983387i	$0.441898\pi$
$164$	1077.14	0.512867
$165$	0	0
$166$	384.406	0.179733
$167$	2545.46	1.17948	0.589742	−	0.807592i	$-0.299230\pi$
$167$	2545.46	1.17948	0.589742	+	0.807592i	$0.299230\pi$
$168$	0	0
$169$	−98.2849	−0.0447360
$170$	1031.33	0.465293
$171$	0	0
$172$	−1407.86	−0.624117
$173$	−249.551	−0.109670	−0.0548352	−	0.998495i	$-0.517463\pi$
$173$	−249.551	−0.109670	−0.0548352	+	0.998495i	$0.517463\pi$
$174$	0	0
$175$	902.519	0.389852
$176$	14.8922	0.00637806
$177$	0	0
$178$	3148.44	1.32576
$179$	−2176.94	−0.909008	−0.454504	−	0.890745i	$-0.650183\pi$
$179$	−2176.94	−0.909008	−0.454504	+	0.890745i	$0.650183\pi$
$180$	0	0
$181$	−1829.01	−0.751103	−0.375551	−	0.926802i	$-0.622547\pi$
$181$	−1829.01	−0.751103	−0.375551	+	0.926802i	$0.622547\pi$
$182$	−3307.68	−1.34715
$183$	0	0
$184$	136.156	0.0545518
$185$	−298.864	−0.118773
$186$	0	0
$187$	95.9925	0.0375383
$188$	−1890.99	−0.733590
$189$	0	0
$190$	190.000	0.0725476
$191$	−1315.52	−0.498366	−0.249183	−	0.968456i	$-0.580162\pi$
$191$	−1315.52	−0.498366	−0.249183	+	0.968456i	$0.580162\pi$
$192$	0	0
$193$	2585.22	0.964189	0.482095	−	0.876119i	$-0.339876\pi$
$193$	2585.22	0.964189	0.482095	+	0.876119i	$0.339876\pi$
$194$	2813.24	1.04113
$195$	0	0
$196$	3841.06	1.39980
$197$	5487.57	1.98464	0.992318	−	0.123713i	$-0.0394803\pi$
$197$	5487.57	1.98464	0.992318	+	0.123713i	$0.0394803\pi$
$198$	0	0
$199$	−5391.36	−1.92052	−0.960259	−	0.279111i	$-0.909960\pi$
$199$	−5391.36	−1.92052	−0.960259	+	0.279111i	$0.909960\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−1986.10	−0.691791
$203$	11092.3	3.83512
$204$	0	0
$205$	−1346.42	−0.458722
$206$	1195.80	0.404444
$207$	0	0
$208$	732.988	0.244344
$209$	17.6844	0.00585291
$210$	0	0
$211$	−3385.70	−1.10465	−0.552324	−	0.833629i	$-0.686259\pi$
$211$	−3385.70	−1.10465	−0.552324	+	0.833629i	$0.686259\pi$
$212$	2741.75	0.888228
$213$	0	0
$214$	4246.64	1.35652
$215$	1759.82	0.558227
$216$	0	0
$217$	1375.92	0.430433
$218$	2565.14	0.796941
$219$	0	0
$220$	−18.6152	−0.00570471
$221$	4724.72	1.43810
$222$	0	0
$223$	2915.74	0.875572	0.437786	−	0.899079i	$-0.355763\pi$
$223$	2915.74	0.875572	0.437786	+	0.899079i	$0.355763\pi$
$224$	−1155.22	−0.344583
$225$	0	0
$226$	−2871.67	−0.845225
$227$	88.2474	0.0258026	0.0129013	−	0.999917i	$-0.495893\pi$
$227$	88.2474	0.0258026	0.0129013	+	0.999917i	$0.495893\pi$
$228$	0	0
$229$	3130.32	0.903307	0.451653	−	0.892194i	$-0.350834\pi$
$229$	3130.32	0.903307	0.451653	+	0.892194i	$0.350834\pi$
$230$	−170.195	−0.0487926
$231$	0	0
$232$	−2458.09	−0.695609
$233$	−1590.28	−0.447136	−0.223568	−	0.974688i	$-0.571771\pi$
$233$	−1590.28	−0.447136	−0.223568	+	0.974688i	$0.571771\pi$
$234$	0	0
$235$	2363.74	0.656143
$236$	−1091.67	−0.301109
$237$	0	0
$238$	−7446.40	−2.02806
$239$	3604.37	0.975511	0.487755	−	0.872980i	$-0.337816\pi$
$239$	3604.37	0.975511	0.487755	+	0.872980i	$0.337816\pi$
$240$	0	0
$241$	6388.57	1.70757	0.853784	−	0.520628i	$-0.174302\pi$
$241$	6388.57	1.70757	0.853784	+	0.520628i	$0.174302\pi$
$242$	2660.27	0.706647
$243$	0	0
$244$	1035.06	0.271570
$245$	−4801.33	−1.25202
$246$	0	0
$247$	870.423	0.224225
$248$	−304.908	−0.0780712
$249$	0	0
$250$	250.000	0.0632456
$251$	−1386.54	−0.348677	−0.174338	−	0.984686i	$-0.555779\pi$
$251$	−1386.54	−0.348677	−0.174338	+	0.984686i	$0.555779\pi$
$252$	0	0
$253$	−15.8410	−0.00393643
$254$	206.036	0.0508971
$255$	0	0
$256$	256.000	0.0625000
$257$	4775.87	1.15919	0.579593	−	0.814906i	$-0.303212\pi$
$257$	4775.87	1.15919	0.579593	+	0.814906i	$0.303212\pi$
$258$	0	0
$259$	2157.84	0.517691
$260$	−916.235	−0.218548
$261$	0	0
$262$	−527.648	−0.124421
$263$	−4723.86	−1.10755	−0.553775	−	0.832666i	$-0.686813\pi$
$263$	−4723.86	−1.10755	−0.553775	+	0.832666i	$0.686813\pi$
$264$	0	0
$265$	−3427.19	−0.794455
$266$	−1371.83	−0.316211
$267$	0	0
$268$	−2522.50	−0.574948
$269$	4038.70	0.915405	0.457703	−	0.889105i	$-0.348672\pi$
$269$	4038.70	0.915405	0.457703	+	0.889105i	$0.348672\pi$
$270$	0	0
$271$	−7923.94	−1.77618	−0.888090	−	0.459669i	$-0.847968\pi$
$271$	−7923.94	−1.77618	−0.888090	+	0.459669i	$0.847968\pi$
$272$	1650.14	0.367846
$273$	0	0
$274$	−1669.05	−0.367996
$275$	23.2690	0.00510245
$276$	0	0
$277$	−3593.69	−0.779510	−0.389755	−	0.920919i	$-0.627440\pi$
$277$	−3593.69	−0.779510	−0.389755	+	0.920919i	$0.627440\pi$
$278$	−1971.56	−0.425345
$279$	0	0
$280$	1444.03	0.308205
$281$	3973.82	0.843623	0.421812	−	0.906684i	$-0.361394\pi$
$281$	3973.82	0.843623	0.421812	+	0.906684i	$0.361394\pi$
$282$	0	0
$283$	3155.29	0.662765	0.331383	−	0.943496i	$-0.392485\pi$
$283$	3155.29	0.662765	0.331383	+	0.943496i	$0.392485\pi$
$284$	−533.838	−0.111540
$285$	0	0
$286$	−85.2795	−0.0176317
$287$	9721.36	1.99942
$288$	0	0
$289$	5723.52	1.16497
$290$	3072.61	0.622172
$291$	0	0
$292$	2524.97	0.506036
$293$	−25.5662	−0.00509760	−0.00254880	−	0.999997i	$-0.500811\pi$
$293$	−25.5662	−0.00509760	−0.00254880	+	0.999997i	$0.500811\pi$
$294$	0	0
$295$	1364.59	0.269320
$296$	−478.183	−0.0938980
$297$	0	0
$298$	5110.27	0.993390
$299$	−779.691	−0.150805
$300$	0	0
$301$	−12706.2	−2.43313
$302$	−3054.50	−0.582009
$303$	0	0
$304$	304.000	0.0573539
$305$	−1293.83	−0.242899
$306$	0	0
$307$	−272.811	−0.0507171	−0.0253585	−	0.999678i	$-0.508073\pi$
$307$	−272.811	−0.0507171	−0.0253585	+	0.999678i	$0.508073\pi$
$308$	134.405	0.0248650
$309$	0	0
$310$	381.135	0.0698290
$311$	3768.46	0.687105	0.343553	−	0.939133i	$-0.388370\pi$
$311$	3768.46	0.687105	0.343553	+	0.939133i	$0.388370\pi$
$312$	0	0
$313$	−4788.92	−0.864811	−0.432406	−	0.901679i	$-0.642335\pi$
$313$	−4788.92	−0.864811	−0.432406	+	0.901679i	$0.642335\pi$
$314$	2319.78	0.416920
$315$	0	0
$316$	2563.69	0.456388
$317$	4040.14	0.715825	0.357913	−	0.933755i	$-0.383489\pi$
$317$	4040.14	0.715825	0.357913	+	0.933755i	$0.383489\pi$
$318$	0	0
$319$	285.986	0.0501948
$320$	−320.000	−0.0559017
$321$	0	0
$322$	1228.83	0.212671
$323$	1959.54	0.337559
$324$	0	0
$325$	1145.29	0.195475
$326$	−1511.00	−0.256708
$327$	0	0
$328$	−2154.27	−0.362652
$329$	−17066.6	−2.85991
$330$	0	0
$331$	−9554.49	−1.58659	−0.793297	−	0.608835i	$-0.791637\pi$
$331$	−9554.49	−1.58659	−0.793297	+	0.608835i	$0.791637\pi$
$332$	−768.812	−0.127091
$333$	0	0
$334$	−5090.92	−0.834021
$335$	3153.12	0.514249
$336$	0	0
$337$	1784.39	0.288433	0.144217	−	0.989546i	$-0.453934\pi$
$337$	1784.39	0.288433	0.144217	+	0.989546i	$0.453934\pi$
$338$	196.570	0.0316331
$339$	0	0
$340$	−2062.67	−0.329012
$341$	35.4745	0.00563358
$342$	0	0
$343$	22283.7	3.50790
$344$	2815.71	0.441317
$345$	0	0
$346$	499.101	0.0775487
$347$	−364.399	−0.0563745	−0.0281873	−	0.999603i	$-0.508973\pi$
$347$	−364.399	−0.0563745	−0.0281873	+	0.999603i	$0.508973\pi$
$348$	0	0
$349$	8408.07	1.28961	0.644805	−	0.764348i	$-0.276939\pi$
$349$	8408.07	1.28961	0.644805	+	0.764348i	$0.276939\pi$
$350$	−1805.04	−0.275667
$351$	0	0
$352$	−29.7843	−0.00450997
$353$	−8910.59	−1.34352	−0.671760	−	0.740769i	$-0.734462\pi$
$353$	−8910.59	−1.34352	−0.671760	+	0.740769i	$0.734462\pi$
$354$	0	0
$355$	667.298	0.0997648
$356$	−6296.89	−0.937456
$357$	0	0
$358$	4353.89	0.642766
$359$	5941.53	0.873488	0.436744	−	0.899586i	$-0.356132\pi$
$359$	5941.53	0.873488	0.436744	+	0.899586i	$0.356132\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	3658.03	0.531110
$363$	0	0
$364$	6615.35	0.952579
$365$	−3156.21	−0.452612
$366$	0	0
$367$	8723.39	1.24076	0.620378	−	0.784303i	$-0.286979\pi$
$367$	8723.39	1.24076	0.620378	+	0.784303i	$0.286979\pi$
$368$	−272.311	−0.0385739
$369$	0	0
$370$	597.728	0.0839849
$371$	24744.8	3.46277
$372$	0	0
$373$	4892.51	0.679155	0.339577	−	0.940578i	$-0.389716\pi$
$373$	4892.51	0.679155	0.339577	+	0.940578i	$0.389716\pi$
$374$	−191.985	−0.0265436
$375$	0	0
$376$	3781.99	0.518726
$377$	14076.1	1.92297
$378$	0	0
$379$	1237.51	0.167722	0.0838609	−	0.996477i	$-0.473275\pi$
$379$	1237.51	0.167722	0.0838609	+	0.996477i	$0.473275\pi$
$380$	−380.000	−0.0512989
$381$	0	0
$382$	2631.05	0.352398
$383$	10955.6	1.46164	0.730818	−	0.682573i	$-0.239139\pi$
$383$	10955.6	1.46164	0.730818	+	0.682573i	$0.239139\pi$
$384$	0	0
$385$	−168.006	−0.0222399
$386$	−5170.45	−0.681785
$387$	0	0
$388$	−5626.47	−0.736188
$389$	−5211.34	−0.679242	−0.339621	−	0.940562i	$-0.610299\pi$
$389$	−5211.34	−0.679242	−0.339621	+	0.940562i	$0.610299\pi$
$390$	0	0
$391$	−1755.28	−0.227028
$392$	−7682.12	−0.989811
$393$	0	0
$394$	−10975.1	−1.40335
$395$	−3204.61	−0.408206
$396$	0	0
$397$	1136.18	0.143635	0.0718174	−	0.997418i	$-0.477120\pi$
$397$	1136.18	0.143635	0.0718174	+	0.997418i	$0.477120\pi$
$398$	10782.7	1.35801
$399$	0	0
$400$	400.000	0.0500000
$401$	3873.02	0.482318	0.241159	−	0.970486i	$-0.422472\pi$
$401$	3873.02	0.482318	0.241159	+	0.970486i	$0.422472\pi$
$402$	0	0
$403$	1746.04	0.215823
$404$	3972.21	0.489170
$405$	0	0
$406$	−22184.7	−2.71184
$407$	55.6342	0.00677563
$408$	0	0
$409$	5436.39	0.657243	0.328621	−	0.944462i	$-0.393416\pi$
$409$	5436.39	0.657243	0.328621	+	0.944462i	$0.393416\pi$
$410$	2692.84	0.324366
$411$	0	0
$412$	−2391.60	−0.285985
$413$	−9852.54	−1.17388
$414$	0	0
$415$	961.015	0.113673
$416$	−1465.98	−0.172777
$417$	0	0
$418$	−35.3689	−0.00413863
$419$	−4339.11	−0.505917	−0.252959	−	0.967477i	$-0.581404\pi$
$419$	−4339.11	−0.505917	−0.252959	+	0.967477i	$0.581404\pi$
$420$	0	0
$421$	10397.6	1.20367	0.601836	−	0.798619i	$-0.294436\pi$
$421$	10397.6	1.20367	0.601836	+	0.798619i	$0.294436\pi$
$422$	6771.39	0.781105
$423$	0	0
$424$	−5483.50	−0.628072
$425$	2578.34	0.294277
$426$	0	0
$427$	9341.63	1.05872
$428$	−8493.29	−0.959202
$429$	0	0
$430$	−3519.64	−0.394726
$431$	9819.17	1.09738	0.548692	−	0.836024i	$-0.315126\pi$
$431$	9819.17	1.09738	0.548692	+	0.836024i	$0.315126\pi$
$432$	0	0
$433$	8198.36	0.909903	0.454952	−	0.890516i	$-0.349657\pi$
$433$	8198.36	0.909903	0.454952	+	0.890516i	$0.349657\pi$
$434$	−2751.85	−0.304362
$435$	0	0
$436$	−5130.28	−0.563522
$437$	−323.370	−0.0353979
$438$	0	0
$439$	−849.949	−0.0924052	−0.0462026	−	0.998932i	$-0.514712\pi$
$439$	−849.949	−0.0924052	−0.0462026	+	0.998932i	$0.514712\pi$
$440$	37.2304	0.00403384
$441$	0	0
$442$	−9449.45	−1.01689
$443$	4275.18	0.458511	0.229255	−	0.973366i	$-0.426371\pi$
$443$	4275.18	0.458511	0.229255	+	0.973366i	$0.426371\pi$
$444$	0	0
$445$	7871.11	0.838486
$446$	−5831.48	−0.619123
$447$	0	0
$448$	2310.45	0.243657
$449$	−15175.3	−1.59503	−0.797515	−	0.603299i	$-0.793853\pi$
$449$	−15175.3	−1.59503	−0.797515	+	0.603299i	$0.793853\pi$
$450$	0	0
$451$	250.639	0.0261688
$452$	5743.35	0.597664
$453$	0	0
$454$	−176.495	−0.0182452
$455$	−8269.19	−0.852013
$456$	0	0
$457$	5141.61	0.526290	0.263145	−	0.964756i	$-0.415240\pi$
$457$	5141.61	0.526290	0.263145	+	0.964756i	$0.415240\pi$
$458$	−6260.64	−0.638734
$459$	0	0
$460$	340.389	0.0345016
$461$	1294.65	0.130798	0.0653992	−	0.997859i	$-0.479168\pi$
$461$	1294.65	0.130798	0.0653992	+	0.997859i	$0.479168\pi$
$462$	0	0
$463$	−12898.4	−1.29469	−0.647344	−	0.762198i	$-0.724120\pi$
$463$	−12898.4	−1.29469	−0.647344	+	0.762198i	$0.724120\pi$
$464$	4916.17	0.491870
$465$	0	0
$466$	3180.56	0.316173
$467$	−10324.8	−1.02307	−0.511535	−	0.859262i	$-0.670923\pi$
$467$	−10324.8	−1.02307	−0.511535	+	0.859262i	$0.670923\pi$
$468$	0	0
$469$	−22766.0	−2.24145
$470$	−4727.48	−0.463963
$471$	0	0
$472$	2183.34	0.212916
$473$	−327.594	−0.0318453
$474$	0	0
$475$	475.000	0.0458831
$476$	14892.8	1.43405
$477$	0	0
$478$	−7208.73	−0.689790
$479$	−5406.05	−0.515676	−0.257838	−	0.966188i	$-0.583010\pi$
$479$	−5406.05	−0.515676	−0.257838	+	0.966188i	$0.583010\pi$
$480$	0	0
$481$	2738.30	0.259575
$482$	−12777.1	−1.20743
$483$	0	0
$484$	−5320.53	−0.499675
$485$	7033.09	0.658466
$486$	0	0
$487$	15664.5	1.45755	0.728773	−	0.684756i	$-0.240091\pi$
$487$	15664.5	1.45755	0.728773	+	0.684756i	$0.240091\pi$
$488$	−2070.12	−0.192029
$489$	0	0
$490$	9602.65	0.885313
$491$	9832.59	0.903745	0.451872	−	0.892083i	$-0.350756\pi$
$491$	9832.59	0.903745	0.451872	+	0.892083i	$0.350756\pi$
$492$	0	0
$493$	31688.9	2.89492
$494$	−1740.85	−0.158551
$495$	0	0
$496$	609.815	0.0552047
$497$	−4817.99	−0.434842
$498$	0	0
$499$	12193.9	1.09393	0.546966	−	0.837155i	$-0.315783\pi$
$499$	12193.9	1.09393	0.546966	+	0.837155i	$0.315783\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	2773.09	0.246552
$503$	−20823.3	−1.84585	−0.922926	−	0.384977i	$-0.874209\pi$
$503$	−20823.3	−1.84585	−0.922926	+	0.384977i	$0.874209\pi$
$504$	0	0
$505$	−4965.26	−0.437527
$506$	31.6821	0.00278348
$507$	0	0
$508$	−412.073	−0.0359897
$509$	−262.903	−0.0228939	−0.0114469	−	0.999934i	$-0.503644\pi$
$509$	−262.903	−0.0228939	−0.0114469	+	0.999934i	$0.503644\pi$
$510$	0	0
$511$	22788.3	1.97279
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−9551.75	−0.819668
$515$	2989.51	0.255793
$516$	0	0
$517$	−440.015	−0.0374310
$518$	−4315.69	−0.366063
$519$	0	0
$520$	1832.47	0.154537
$521$	13595.2	1.14321	0.571607	−	0.820527i	$-0.306320\pi$
$521$	13595.2	1.14321	0.571607	+	0.820527i	$0.306320\pi$
$522$	0	0
$523$	16842.1	1.40813	0.704066	−	0.710134i	$-0.251366\pi$
$523$	16842.1	1.40813	0.704066	+	0.710134i	$0.251366\pi$
$524$	1055.30	0.0879787
$525$	0	0
$526$	9447.72	0.783156
$527$	3930.77	0.324909
$528$	0	0
$529$	−11877.3	−0.976193
$530$	6854.38	0.561764
$531$	0	0
$532$	2743.66	0.223595
$533$	12336.4	1.00253
$534$	0	0
$535$	10616.6	0.857937
$536$	5045.00	0.406550
$537$	0	0
$538$	−8077.41	−0.647289
$539$	893.776	0.0714243
$540$	0	0
$541$	11469.7	0.911500	0.455750	−	0.890108i	$-0.349371\pi$
$541$	11469.7	0.911500	0.455750	+	0.890108i	$0.349371\pi$
$542$	15847.9	1.25595
$543$	0	0
$544$	−3300.27	−0.260107
$545$	6412.85	0.504030
$546$	0	0
$547$	7568.23	0.591580	0.295790	−	0.955253i	$-0.404417\pi$
$547$	7568.23	0.591580	0.295790	+	0.955253i	$0.404417\pi$
$548$	3338.10	0.260213
$549$	0	0
$550$	−46.5380	−0.00360798
$551$	5837.95	0.451371
$552$	0	0
$553$	23137.7	1.77924
$554$	7187.39	0.551197
$555$	0	0
$556$	3943.11	0.300765
$557$	−10701.2	−0.814051	−0.407025	−	0.913417i	$-0.633434\pi$
$557$	−10701.2	−0.814051	−0.407025	+	0.913417i	$0.633434\pi$
$558$	0	0
$559$	−16124.1	−1.21999
$560$	−2888.06	−0.217934
$561$	0	0
$562$	−7947.64	−0.596532
$563$	−12327.0	−0.922772	−0.461386	−	0.887200i	$-0.652648\pi$
$563$	−12327.0	−0.922772	−0.461386	+	0.887200i	$0.652648\pi$
$564$	0	0
$565$	−7179.18	−0.534567
$566$	−6310.58	−0.468646
$567$	0	0
$568$	1067.68	0.0788710
$569$	−18887.8	−1.39160	−0.695799	−	0.718236i	$-0.744950\pi$
$569$	−18887.8	−1.39160	−0.695799	+	0.718236i	$0.744950\pi$
$570$	0	0
$571$	−15534.3	−1.13851	−0.569256	−	0.822161i	$-0.692769\pi$
$571$	−15534.3	−1.13851	−0.569256	+	0.822161i	$0.692769\pi$
$572$	170.559	0.0124675
$573$	0	0
$574$	−19442.7	−1.41380
$575$	−425.486	−0.0308591
$576$	0	0
$577$	5670.77	0.409146	0.204573	−	0.978851i	$-0.434419\pi$
$577$	5670.77	0.409146	0.204573	+	0.978851i	$0.434419\pi$
$578$	−11447.0	−0.823761
$579$	0	0
$580$	−6145.22	−0.439942
$581$	−6938.68	−0.495464
$582$	0	0
$583$	637.978	0.0453214
$584$	−5049.93	−0.357821
$585$	0	0
$586$	51.1325	0.00360455
$587$	8951.67	0.629429	0.314715	−	0.949186i	$-0.398091\pi$
$587$	8951.67	0.629429	0.314715	+	0.949186i	$0.398091\pi$
$588$	0	0
$589$	724.156	0.0506593
$590$	−2729.18	−0.190438
$591$	0	0
$592$	956.365	0.0663959
$593$	−16888.6	−1.16953	−0.584765	−	0.811203i	$-0.698813\pi$
$593$	−16888.6	−1.16953	−0.584765	+	0.811203i	$0.698813\pi$
$594$	0	0
$595$	−18616.0	−1.28266
$596$	−10220.5	−0.702433
$597$	0	0
$598$	1559.38	0.106635
$599$	4473.77	0.305164	0.152582	−	0.988291i	$-0.451241\pi$
$599$	4473.77	0.305164	0.152582	+	0.988291i	$0.451241\pi$
$600$	0	0
$601$	−19271.9	−1.30802	−0.654009	−	0.756487i	$-0.726914\pi$
$601$	−19271.9	−1.30802	−0.654009	+	0.756487i	$0.726914\pi$
$602$	25412.4	1.72048
$603$	0	0
$604$	6109.00	0.411543
$605$	6650.67	0.446923
$606$	0	0
$607$	9000.09	0.601816	0.300908	−	0.953653i	$-0.402710\pi$
$607$	9000.09	0.601816	0.300908	+	0.953653i	$0.402710\pi$
$608$	−608.000	−0.0405554
$609$	0	0
$610$	2587.65	0.171756
$611$	−21657.4	−1.43399
$612$	0	0
$613$	−11357.5	−0.748328	−0.374164	−	0.927363i	$-0.622070\pi$
$613$	−11357.5	−0.748328	−0.374164	+	0.927363i	$0.622070\pi$
$614$	545.622	0.0358624
$615$	0	0
$616$	−268.809	−0.0175822
$617$	−7071.80	−0.461426	−0.230713	−	0.973022i	$-0.574106\pi$
$617$	−7071.80	−0.461426	−0.230713	+	0.973022i	$0.574106\pi$
$618$	0	0
$619$	−19272.4	−1.25141	−0.625706	−	0.780059i	$-0.715189\pi$
$619$	−19272.4	−1.25141	−0.625706	+	0.780059i	$0.715189\pi$
$620$	−762.269	−0.0493766
$621$	0	0
$622$	−7536.92	−0.485857
$623$	−56830.6	−3.65469
$624$	0	0
$625$	625.000	0.0400000
$626$	9577.84	0.611514
$627$	0	0
$628$	−4639.56	−0.294807
$629$	6164.58	0.390776
$630$	0	0
$631$	30363.6	1.91562	0.957810	−	0.287403i	$-0.0927918\pi$
$631$	30363.6	1.91562	0.957810	+	0.287403i	$0.0927918\pi$
$632$	−5127.37	−0.322715
$633$	0	0
$634$	−8080.27	−0.506165
$635$	515.091	0.0321902
$636$	0	0
$637$	43991.4	2.73627
$638$	−571.972	−0.0354931
$639$	0	0
$640$	640.000	0.0395285
$641$	−4344.27	−0.267688	−0.133844	−	0.991002i	$-0.542732\pi$
$641$	−4344.27	−0.267688	−0.133844	+	0.991002i	$0.542732\pi$
$642$	0	0
$643$	−19953.3	−1.22376	−0.611882	−	0.790949i	$-0.709587\pi$
$643$	−19953.3	−1.22376	−0.611882	+	0.790949i	$0.709587\pi$
$644$	−2457.66	−0.150381
$645$	0	0
$646$	−3919.07	−0.238690
$647$	−2474.77	−0.150376	−0.0751880	−	0.997169i	$-0.523956\pi$
$647$	−2474.77	−0.150376	−0.0751880	+	0.997169i	$0.523956\pi$
$648$	0	0
$649$	−254.021	−0.0153639
$650$	−2290.59	−0.138222
$651$	0	0
$652$	3022.00	0.181520
$653$	−30781.5	−1.84468	−0.922339	−	0.386382i	$-0.873725\pi$
$653$	−30781.5	−1.84468	−0.922339	+	0.386382i	$0.873725\pi$
$654$	0	0
$655$	−1319.12	−0.0786905
$656$	4308.54	0.256433
$657$	0	0
$658$	34133.2	2.02226
$659$	−13752.5	−0.812930	−0.406465	−	0.913666i	$-0.633239\pi$
$659$	−13752.5	−0.812930	−0.406465	+	0.913666i	$0.633239\pi$
$660$	0	0
$661$	2059.46	0.121186	0.0605928	−	0.998163i	$-0.480701\pi$
$661$	2059.46	0.121186	0.0605928	+	0.998163i	$0.480701\pi$
$662$	19109.0	1.12189
$663$	0	0
$664$	1537.62	0.0898666
$665$	−3429.57	−0.199990
$666$	0	0
$667$	−5229.41	−0.303574
$668$	10181.8	0.589742
$669$	0	0
$670$	−6306.25	−0.363629
$671$	240.848	0.0138567
$672$	0	0
$673$	−22647.6	−1.29718	−0.648588	−	0.761139i	$-0.724640\pi$
$673$	−22647.6	−1.29718	−0.648588	+	0.761139i	$0.724640\pi$
$674$	−3568.78	−0.203953
$675$	0	0
$676$	−393.140	−0.0223680
$677$	−19995.2	−1.13512	−0.567562	−	0.823331i	$-0.692113\pi$
$677$	−19995.2	−1.13512	−0.567562	+	0.823331i	$0.692113\pi$
$678$	0	0
$679$	−50780.0	−2.87004
$680$	4125.34	0.232646
$681$	0	0
$682$	−70.9490	−0.00398354
$683$	16395.8	0.918550	0.459275	−	0.888294i	$-0.348109\pi$
$683$	16395.8	0.918550	0.459275	+	0.888294i	$0.348109\pi$
$684$	0	0
$685$	−4172.62	−0.232741
$686$	−44567.5	−2.48046
$687$	0	0
$688$	−5631.43	−0.312058
$689$	31401.1	1.73626
$690$	0	0
$691$	−8258.78	−0.454673	−0.227336	−	0.973816i	$-0.573002\pi$
$691$	−8258.78	−0.454673	−0.227336	+	0.973816i	$0.573002\pi$
$692$	−998.202	−0.0548352
$693$	0	0
$694$	728.798	0.0398628
$695$	−4928.89	−0.269012
$696$	0	0
$697$	27772.2	1.50925
$698$	−16816.1	−0.911891
$699$	0	0
$700$	3610.08	0.194926
$701$	12578.1	0.677701	0.338851	−	0.940840i	$-0.389962\pi$
$701$	12578.1	0.677701	0.338851	+	0.940840i	$0.389962\pi$
$702$	0	0
$703$	1135.68	0.0609290
$704$	59.5686	0.00318903
$705$	0	0
$706$	17821.2	0.950013
$707$	35849.9	1.90704
$708$	0	0
$709$	−18242.9	−0.966329	−0.483164	−	0.875530i	$-0.660513\pi$
$709$	−18242.9	−0.966329	−0.483164	+	0.875530i	$0.660513\pi$
$710$	−1334.60	−0.0705444
$711$	0	0
$712$	12593.8	0.662882
$713$	−648.670	−0.0340714
$714$	0	0
$715$	−213.199	−0.0111513
$716$	−8707.77	−0.454504
$717$	0	0
$718$	−11883.1	−0.617649
$719$	2709.59	0.140543	0.0702717	−	0.997528i	$-0.477613\pi$
$719$	2709.59	0.140543	0.0702717	+	0.997528i	$0.477613\pi$
$720$	0	0
$721$	−21584.7	−1.11492
$722$	−722.000	−0.0372161
$723$	0	0
$724$	−7316.06	−0.375551
$725$	7681.52	0.393496
$726$	0	0
$727$	−4649.35	−0.237187	−0.118594	−	0.992943i	$-0.537839\pi$
$727$	−4649.35	−0.237187	−0.118594	+	0.992943i	$0.537839\pi$
$728$	−13230.7	−0.673575
$729$	0	0
$730$	6312.42	0.320045
$731$	−36299.3	−1.83663
$732$	0	0
$733$	−7179.88	−0.361794	−0.180897	−	0.983502i	$-0.557900\pi$
$733$	−7179.88	−0.361794	−0.180897	+	0.983502i	$0.557900\pi$
$734$	−17446.8	−0.877347
$735$	0	0
$736$	544.623	0.0272759
$737$	−586.960	−0.0293364
$738$	0	0
$739$	−31181.9	−1.55216	−0.776079	−	0.630635i	$-0.782794\pi$
$739$	−31181.9	−1.55216	−0.776079	+	0.630635i	$0.782794\pi$
$740$	−1195.46	−0.0593863
$741$	0	0
$742$	−49489.7	−2.44855
$743$	−2085.25	−0.102961	−0.0514807	−	0.998674i	$-0.516394\pi$
$743$	−2085.25	−0.102961	−0.0514807	+	0.998674i	$0.516394\pi$
$744$	0	0
$745$	12775.7	0.628275
$746$	−9785.02	−0.480235
$747$	0	0
$748$	383.970	0.0187692
$749$	−76653.5	−3.73947
$750$	0	0
$751$	−34782.8	−1.69007	−0.845034	−	0.534713i	$-0.820420\pi$
$751$	−34782.8	−1.69007	−0.845034	+	0.534713i	$0.820420\pi$
$752$	−7563.97	−0.366795
$753$	0	0
$754$	−28152.3	−1.35974
$755$	−7636.26	−0.368095
$756$	0	0
$757$	17185.7	0.825133	0.412566	−	0.910928i	$-0.364632\pi$
$757$	17185.7	0.825133	0.412566	+	0.910928i	$0.364632\pi$
$758$	−2475.02	−0.118597
$759$	0	0
$760$	760.000	0.0362738
$761$	−20792.1	−0.990425	−0.495213	−	0.868772i	$-0.664910\pi$
$761$	−20792.1	−0.990425	−0.495213	+	0.868772i	$0.664910\pi$
$762$	0	0
$763$	−46301.7	−2.19690
$764$	−5262.09	−0.249183
$765$	0	0
$766$	−21911.2	−1.03353
$767$	−12502.8	−0.588594
$768$	0	0
$769$	19857.3	0.931174	0.465587	−	0.885002i	$-0.345843\pi$
$769$	19857.3	0.931174	0.465587	+	0.885002i	$0.345843\pi$
$770$	336.011	0.0157260
$771$	0	0
$772$	10340.9	0.482095
$773$	21528.0	1.00169	0.500847	−	0.865536i	$-0.333022\pi$
$773$	21528.0	1.00169	0.500847	+	0.865536i	$0.333022\pi$
$774$	0	0
$775$	952.836	0.0441637
$776$	11252.9	0.520563
$777$	0	0
$778$	10422.7	0.480297
$779$	5116.40	0.235320
$780$	0	0
$781$	−124.219	−0.00569129
$782$	3510.55	0.160533
$783$	0	0
$784$	15364.2	0.699902
$785$	5799.45	0.263683
$786$	0	0
$787$	−39904.4	−1.80742	−0.903710	−	0.428145i	$-0.859167\pi$
$787$	−39904.4	−1.80742	−0.903710	+	0.428145i	$0.859167\pi$
$788$	21950.3	0.992318
$789$	0	0
$790$	6409.21	0.288645
$791$	51834.8	2.33000
$792$	0	0
$793$	11854.5	0.530852
$794$	−2272.35	−0.101565
$795$	0	0
$796$	−21565.4	−0.960259
$797$	−30219.8	−1.34309	−0.671543	−	0.740966i	$-0.734368\pi$
$797$	−30219.8	−1.34309	−0.671543	+	0.740966i	$0.734368\pi$
$798$	0	0
$799$	−48756.2	−2.15879
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−7746.04	−0.341050
$803$	587.535	0.0258202
$804$	0	0
$805$	3072.08	0.134505
$806$	−3492.09	−0.152610
$807$	0	0
$808$	−7944.41	−0.345895
$809$	−34900.3	−1.51673	−0.758363	−	0.651833i	$-0.774000\pi$
$809$	−34900.3	−1.51673	−0.758363	+	0.651833i	$0.774000\pi$
$810$	0	0
$811$	−36071.1	−1.56181	−0.780905	−	0.624650i	$-0.785242\pi$
$811$	−36071.1	−1.56181	−0.780905	+	0.624650i	$0.785242\pi$
$812$	44369.4	1.91756
$813$	0	0
$814$	−111.268	−0.00479110
$815$	−3777.51	−0.162356
$816$	0	0
$817$	−6687.32	−0.286365
$818$	−10872.8	−0.464741
$819$	0	0
$820$	−5385.68	−0.229361
$821$	−3695.95	−0.157113	−0.0785563	−	0.996910i	$-0.525031\pi$
$821$	−3695.95	−0.157113	−0.0785563	+	0.996910i	$0.525031\pi$
$822$	0	0
$823$	33632.8	1.42450	0.712250	−	0.701925i	$-0.247676\pi$
$823$	33632.8	1.42450	0.712250	+	0.701925i	$0.247676\pi$
$824$	4783.21	0.202222
$825$	0	0
$826$	19705.1	0.830057
$827$	9896.52	0.416125	0.208063	−	0.978116i	$-0.433284\pi$
$827$	9896.52	0.416125	0.208063	+	0.978116i	$0.433284\pi$
$828$	0	0
$829$	2514.76	0.105357	0.0526787	−	0.998612i	$-0.483224\pi$
$829$	2514.76	0.105357	0.0526787	+	0.998612i	$0.483224\pi$
$830$	−1922.03	−0.0803791
$831$	0	0
$832$	2931.95	0.122172
$833$	99035.5	4.11930
$834$	0	0
$835$	−12727.3	−0.527481
$836$	70.7378	0.00292646
$837$	0	0
$838$	8678.22	0.357738
$839$	17494.6	0.719880	0.359940	−	0.932975i	$-0.382797\pi$
$839$	17494.6	0.719880	0.359940	+	0.932975i	$0.382797\pi$
$840$	0	0
$841$	70020.2	2.87097
$842$	−20795.1	−0.851125
$843$	0	0
$844$	−13542.8	−0.552324
$845$	491.425	0.0200065
$846$	0	0
$847$	−48018.8	−1.94799
$848$	10967.0	0.444114
$849$	0	0
$850$	−5156.67	−0.208085
$851$	−1017.30	−0.0409784
$852$	0	0
$853$	−17651.5	−0.708531	−0.354266	−	0.935145i	$-0.615269\pi$
$853$	−17651.5	−0.708531	−0.354266	+	0.935145i	$0.615269\pi$
$854$	−18683.3	−0.748627
$855$	0	0
$856$	16986.6	0.678258
$857$	10617.1	0.423188	0.211594	−	0.977358i	$-0.432135\pi$
$857$	10617.1	0.423188	0.211594	+	0.977358i	$0.432135\pi$
$858$	0	0
$859$	−940.655	−0.0373629	−0.0186814	−	0.999825i	$-0.505947\pi$
$859$	−940.655	−0.0373629	−0.0186814	+	0.999825i	$0.505947\pi$
$860$	7039.29	0.279114
$861$	0	0
$862$	−19638.3	−0.775968
$863$	34694.3	1.36849	0.684246	−	0.729251i	$-0.260131\pi$
$863$	34694.3	1.36849	0.684246	+	0.729251i	$0.260131\pi$
$864$	0	0
$865$	1247.75	0.0490461
$866$	−16396.7	−0.643399
$867$	0	0
$868$	5503.70	0.215216
$869$	596.544	0.0232870
$870$	0	0
$871$	−28890.0	−1.12388
$872$	10260.6	0.398470
$873$	0	0
$874$	646.739	0.0250301
$875$	−4512.60	−0.174347
$876$	0	0
$877$	−30130.8	−1.16014	−0.580072	−	0.814565i	$-0.696976\pi$
$877$	−30130.8	−1.16014	−0.580072	+	0.814565i	$0.696976\pi$
$878$	1699.90	0.0653403
$879$	0	0
$880$	−74.4608	−0.00285236
$881$	5044.75	0.192919	0.0964597	−	0.995337i	$-0.469248\pi$
$881$	5044.75	0.192919	0.0964597	+	0.995337i	$0.469248\pi$
$882$	0	0
$883$	15728.5	0.599440	0.299720	−	0.954027i	$-0.403107\pi$
$883$	15728.5	0.599440	0.299720	+	0.954027i	$0.403107\pi$
$884$	18898.9	0.719048
$885$	0	0
$886$	−8550.37	−0.324216
$887$	16440.5	0.622342	0.311171	−	0.950354i	$-0.399279\pi$
$887$	16440.5	0.622342	0.311171	+	0.950354i	$0.399279\pi$
$888$	0	0
$889$	−3719.03	−0.140306
$890$	−15742.2	−0.592899
$891$	0	0
$892$	11663.0	0.437786
$893$	−8982.22	−0.336594
$894$	0	0
$895$	10884.7	0.406521
$896$	−4620.90	−0.172292
$897$	0	0
$898$	30350.7	1.12786
$899$	11710.8	0.434456
$900$	0	0
$901$	70691.6	2.61385
$902$	−501.278	−0.0185041
$903$	0	0
$904$	−11486.7	−0.422613
$905$	9145.07	0.335903
$906$	0	0
$907$	−27363.4	−1.00175	−0.500874	−	0.865520i	$-0.666988\pi$
$907$	−27363.4	−1.00175	−0.500874	+	0.865520i	$0.666988\pi$
$908$	352.990	0.0129013
$909$	0	0
$910$	16538.4	0.602464
$911$	15939.3	0.579684	0.289842	−	0.957075i	$-0.406397\pi$
$911$	15939.3	0.579684	0.289842	+	0.957075i	$0.406397\pi$
$912$	0	0
$913$	−178.895	−0.00648473
$914$	−10283.2	−0.372143
$915$	0	0
$916$	12521.3	0.451653
$917$	9524.25	0.342986
$918$	0	0
$919$	26217.5	0.941061	0.470530	−	0.882384i	$-0.344063\pi$
$919$	26217.5	0.941061	0.470530	+	0.882384i	$0.344063\pi$
$920$	−680.778	−0.0243963
$921$	0	0
$922$	−2589.31	−0.0924884
$923$	−6114.02	−0.218034
$924$	0	0
$925$	1494.32	0.0531167
$926$	25796.8	0.915482
$927$	0	0
$928$	−9832.34	−0.347804
$929$	−20817.0	−0.735181	−0.367591	−	0.929988i	$-0.619817\pi$
$929$	−20817.0	−0.735181	−0.367591	+	0.929988i	$0.619817\pi$
$930$	0	0
$931$	18245.0	0.642274
$932$	−6361.12	−0.223568
$933$	0	0
$934$	20649.6	0.723420
$935$	−479.963	−0.0167877
$936$	0	0
$937$	−37517.5	−1.30805	−0.654025	−	0.756473i	$-0.726921\pi$
$937$	−37517.5	−1.30805	−0.654025	+	0.756473i	$0.726921\pi$
$938$	45532.1	1.58494
$939$	0	0
$940$	9454.97	0.328071
$941$	26725.6	0.925856	0.462928	−	0.886396i	$-0.346799\pi$
$941$	26725.6	0.925856	0.462928	+	0.886396i	$0.346799\pi$
$942$	0	0
$943$	−4583.07	−0.158266
$944$	−4366.68	−0.150555
$945$	0	0
$946$	655.189	0.0225180
$947$	−25719.8	−0.882557	−0.441279	−	0.897370i	$-0.645475\pi$
$947$	−25719.8	−0.882557	−0.441279	+	0.897370i	$0.645475\pi$
$948$	0	0
$949$	28918.3	0.989175
$950$	−950.000	−0.0324443
$951$	0	0
$952$	−29785.6	−1.01403
$953$	4559.63	0.154985	0.0774926	−	0.996993i	$-0.475309\pi$
$953$	4559.63	0.154985	0.0774926	+	0.996993i	$0.475309\pi$
$954$	0	0
$955$	6577.62	0.222876
$956$	14417.5	0.487755
$957$	0	0
$958$	10812.1	0.364638
$959$	30127.0	1.01444
$960$	0	0
$961$	−28338.4	−0.951239
$962$	−5476.59	−0.183547
$963$	0	0
$964$	25554.3	0.853784
$965$	−12926.1	−0.431199
$966$	0	0
$967$	53457.4	1.77774	0.888869	−	0.458161i	$-0.151492\pi$
$967$	53457.4	1.77774	0.888869	+	0.458161i	$0.151492\pi$
$968$	10641.1	0.353323
$969$	0	0
$970$	−14066.2	−0.465606
$971$	−12095.4	−0.399753	−0.199876	−	0.979821i	$-0.564054\pi$
$971$	−12095.4	−0.399753	−0.199876	+	0.979821i	$0.564054\pi$
$972$	0	0
$973$	35587.3	1.17254
$974$	−31328.9	−1.03064
$975$	0	0
$976$	4140.25	0.135785
$977$	24895.5	0.815227	0.407614	−	0.913155i	$-0.366361\pi$
$977$	24895.5	0.815227	0.407614	+	0.913155i	$0.366361\pi$
$978$	0	0
$979$	−1465.22	−0.0478332
$980$	−19205.3	−0.626011
$981$	0	0
$982$	−19665.2	−0.639044
$983$	−10329.2	−0.335146	−0.167573	−	0.985860i	$-0.553593\pi$
$983$	−10329.2	−0.335146	−0.167573	+	0.985860i	$0.553593\pi$
$984$	0	0
$985$	−27437.9	−0.887556
$986$	−63377.7	−2.04702
$987$	0	0
$988$	3481.69	0.112113
$989$	5990.24	0.192597
$990$	0	0
$991$	−44011.4	−1.41076	−0.705382	−	0.708827i	$-0.749225\pi$
$991$	−44011.4	−1.41076	−0.705382	+	0.708827i	$0.749225\pi$
$992$	−1219.63	−0.0390356
$993$	0	0
$994$	9635.99	0.307480
$995$	26956.8	0.858882
$996$	0	0
$997$	−11358.7	−0.360815	−0.180407	−	0.983592i	$-0.557742\pi$
$997$	−11358.7	−0.360815	−0.180407	+	0.983592i	$0.557742\pi$
$998$	−24387.7	−0.773527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.a.bc.1.4		4
3.2	odd	2		570.4.a.s.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.4.a.s.1.4	✓	4	3.2	odd	2
1710.4.a.bc.1.4		4	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:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +36.1008 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +36.1008 q^{7} -8.00000 q^{8} +10.0000 q^{10} +0.930760 q^{11} +45.8117 q^{13} -72.2015 q^{14} +16.0000 q^{16} +103.133 q^{17} +19.0000 q^{19} -20.0000 q^{20} -1.86152 q^{22} -17.0195 q^{23} +25.0000 q^{25} -91.6235 q^{26} +144.403 q^{28} +307.261 q^{29} +38.1135 q^{31} -32.0000 q^{32} -206.267 q^{34} -180.504 q^{35} +59.7728 q^{37} -38.0000 q^{38} +40.0000 q^{40} +269.284 q^{41} -351.964 q^{43} +3.72304 q^{44} +34.0389 q^{46} -472.748 q^{47} +960.265 q^{49} -50.0000 q^{50} +183.247 q^{52} +685.438 q^{53} -4.65380 q^{55} -288.806 q^{56} -614.522 q^{58} -272.918 q^{59} +258.765 q^{61} -76.2269 q^{62} +64.0000 q^{64} -229.059 q^{65} -630.625 q^{67} +412.534 q^{68} +361.008 q^{70} -133.460 q^{71} +631.242 q^{73} -119.546 q^{74} +76.0000 q^{76} +33.6011 q^{77} +640.921 q^{79} -80.0000 q^{80} -538.568 q^{82} -192.203 q^{83} -515.667 q^{85} +703.929 q^{86} -7.44608 q^{88} -1574.22 q^{89} +1653.84 q^{91} -68.0778 q^{92} +945.497 q^{94} -95.0000 q^{95} -1406.62 q^{97} -1920.53 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 36 q^{7} - 32 q^{8}+O(q^{10}) \) 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 36 * q^7 - 32 * q^8	\( 4 q - 8 q^{2} + 16 q^{4} - 20 q^{5} + 36 q^{7} - 32 q^{8} + 40 q^{10} - 54 q^{11} + 46 q^{13} - 72 q^{14} + 64 q^{16} - 14 q^{17} + 76 q^{19} - 80 q^{20} + 108 q^{22} - 104 q^{23} + 100 q^{25} - 92 q^{26} + 144 q^{28} - 14 q^{29} + 30 q^{31} - 128 q^{32} + 28 q^{34} - 180 q^{35} + 30 q^{37} - 152 q^{38} + 160 q^{40} + 36 q^{41} + 102 q^{43} - 216 q^{44} + 208 q^{46} - 408 q^{47} + 480 q^{49} - 200 q^{50} + 184 q^{52} + 176 q^{53} + 270 q^{55} - 288 q^{56} + 28 q^{58} - 66 q^{59} + 60 q^{61} - 60 q^{62} + 256 q^{64} - 230 q^{65} - 152 q^{67} - 56 q^{68} + 360 q^{70} - 172 q^{71} + 284 q^{73} - 60 q^{74} + 304 q^{76} + 300 q^{77} + 554 q^{79} - 320 q^{80} - 72 q^{82} + 394 q^{83} + 70 q^{85} - 204 q^{86} + 432 q^{88} + 60 q^{89} + 32 q^{91} - 416 q^{92} + 816 q^{94} - 380 q^{95} - 922 q^{97} - 960 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 - 20 * q^5 + 36 * q^7 - 32 * q^8 + 40 * q^10 - 54 * q^11 + 46 * q^13 - 72 * q^14 + 64 * q^16 - 14 * q^17 + 76 * q^19 - 80 * q^20 + 108 * q^22 - 104 * q^23 + 100 * q^25 - 92 * q^26 + 144 * q^28 - 14 * q^29 + 30 * q^31 - 128 * q^32 + 28 * q^34 - 180 * q^35 + 30 * q^37 - 152 * q^38 + 160 * q^40 + 36 * q^41 + 102 * q^43 - 216 * q^44 + 208 * q^46 - 408 * q^47 + 480 * q^49 - 200 * q^50 + 184 * q^52 + 176 * q^53 + 270 * q^55 - 288 * q^56 + 28 * q^58 - 66 * q^59 + 60 * q^61 - 60 * q^62 + 256 * q^64 - 230 * q^65 - 152 * q^67 - 56 * q^68 + 360 * q^70 - 172 * q^71 + 284 * q^73 - 60 * q^74 + 304 * q^76 + 300 * q^77 + 554 * q^79 - 320 * q^80 - 72 * q^82 + 394 * q^83 + 70 * q^85 - 204 * q^86 + 432 * q^88 + 60 * q^89 + 32 * q^91 - 416 * q^92 + 816 * q^94 - 380 * q^95 - 922 * q^97 - 960 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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