LMFDB - Embedded newform 378.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("378.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{105}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.62348$ of defining polynomial
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} +7.00000 q^{7} -8.00000 q^{8} -39.7409 q^{10} +39.7409 q^{11} +88.6113 q^{13} -14.0000 q^{14} +16.0000 q^{16} -72.7409 q^{17} +38.0000 q^{19} +79.4817 q^{20} -79.4817 q^{22} -7.12957 q^{23} +269.834 q^{25} -177.223 q^{26} +28.0000 q^{28} -257.575 q^{29} -48.6113 q^{31} -32.0000 q^{32} +145.482 q^{34} +139.093 q^{35} -343.279 q^{37} -76.0000 q^{38} -158.963 q^{40} -217.093 q^{41} +447.445 q^{43} +158.963 q^{44} +14.2591 q^{46} -387.279 q^{47} +49.0000 q^{49} -539.668 q^{50} +354.445 q^{52} +322.093 q^{53} +789.668 q^{55} -56.0000 q^{56} +515.149 q^{58} +539.113 q^{59} +303.555 q^{61} +97.2226 q^{62} +64.0000 q^{64} +1760.74 q^{65} -365.834 q^{67} -290.963 q^{68} -278.186 q^{70} +615.389 q^{71} +123.555 q^{73} +686.558 q^{74} +152.000 q^{76} +278.186 q^{77} +399.166 q^{79} +317.927 q^{80} +434.186 q^{82} -633.133 q^{83} -1445.39 q^{85} -894.890 q^{86} -317.927 q^{88} -823.541 q^{89} +620.279 q^{91} -28.5183 q^{92} +774.558 q^{94} +755.076 q^{95} -607.448 q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} + 14 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 + 14 * q^7 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} + 14 q^{7} - 16 q^{8} - 18 q^{10} + 18 q^{11} + 85 q^{13} - 28 q^{14} + 32 q^{16} - 84 q^{17} + 76 q^{19} + 36 q^{20} - 36 q^{22} - 45 q^{23} + 263 q^{25} - 170 q^{26} + 56 q^{28} - 177 q^{29} - 5 q^{31} - 64 q^{32} + 168 q^{34} + 63 q^{35} - 41 q^{37} - 152 q^{38} - 72 q^{40} - 219 q^{41} + 526 q^{43} + 72 q^{44} + 90 q^{46} - 129 q^{47} + 98 q^{49} - 526 q^{50} + 340 q^{52} + 429 q^{53} + 1026 q^{55} - 112 q^{56} + 354 q^{58} + 156 q^{59} + 976 q^{61} + 10 q^{62} + 128 q^{64} + 1800 q^{65} - 455 q^{67} - 336 q^{68} - 126 q^{70} + 1323 q^{71} + 616 q^{73} + 82 q^{74} + 304 q^{76} + 126 q^{77} + 1075 q^{79} + 144 q^{80} + 438 q^{82} + 363 q^{83} - 1323 q^{85} - 1052 q^{86} - 144 q^{88} + 597 q^{89} + 595 q^{91} - 180 q^{92} + 258 q^{94} + 342 q^{95} + 814 q^{97} - 196 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 + 14 * q^7 - 16 * q^8 - 18 * q^10 + 18 * q^11 + 85 * q^13 - 28 * q^14 + 32 * q^16 - 84 * q^17 + 76 * q^19 + 36 * q^20 - 36 * q^22 - 45 * q^23 + 263 * q^25 - 170 * q^26 + 56 * q^28 - 177 * q^29 - 5 * q^31 - 64 * q^32 + 168 * q^34 + 63 * q^35 - 41 * q^37 - 152 * q^38 - 72 * q^40 - 219 * q^41 + 526 * q^43 + 72 * q^44 + 90 * q^46 - 129 * q^47 + 98 * q^49 - 526 * q^50 + 340 * q^52 + 429 * q^53 + 1026 * q^55 - 112 * q^56 + 354 * q^58 + 156 * q^59 + 976 * q^61 + 10 * q^62 + 128 * q^64 + 1800 * q^65 - 455 * q^67 - 336 * q^68 - 126 * q^70 + 1323 * q^71 + 616 * q^73 + 82 * q^74 + 304 * q^76 + 126 * q^77 + 1075 * q^79 + 144 * q^80 + 438 * q^82 + 363 * q^83 - 1323 * q^85 - 1052 * q^86 - 144 * q^88 + 597 * q^89 + 595 * q^91 - 180 * q^92 + 258 * q^94 + 342 * q^95 + 814 * q^97 - 196 * 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
$5$	19.8704	1.77726	0.888632	−	0.458620i	$-0.151656\pi$
$5$	19.8704	1.77726	0.888632	+	0.458620i	$0.151656\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−39.7409	−1.25672
$11$	39.7409	1.08930	0.544651	−	0.838663i	$-0.316662\pi$
$11$	39.7409	1.08930	0.544651	+	0.838663i	$0.316662\pi$
$12$	0	0
$13$	88.6113	1.89049	0.945244	−	0.326364i	$-0.105824\pi$
$13$	88.6113	1.89049	0.945244	+	0.326364i	$0.105824\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−72.7409	−1.03778	−0.518890	−	0.854841i	$-0.673654\pi$
$17$	−72.7409	−1.03778	−0.518890	+	0.854841i	$0.673654\pi$
$18$	0	0
$19$	38.0000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	38.0000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	79.4817	0.888632
$21$	0	0
$22$	−79.4817	−0.770253
$23$	−7.12957	−0.0646356	−0.0323178	−	0.999478i	$-0.510289\pi$
$23$	−7.12957	−0.0646356	−0.0323178	+	0.999478i	$0.510289\pi$
$24$	0	0
$25$	269.834	2.15867
$26$	−177.223	−1.33678
$27$	0	0
$28$	28.0000	0.188982
$29$	−257.575	−1.64932	−0.824662	−	0.565625i	$-0.808635\pi$
$29$	−257.575	−1.64932	−0.824662	+	0.565625i	$0.808635\pi$
$30$	0	0
$31$	−48.6113	−0.281640	−0.140820	−	0.990035i	$-0.544974\pi$
$31$	−48.6113	−0.281640	−0.140820	+	0.990035i	$0.544974\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	145.482	0.733821
$35$	139.093	0.671743
$36$	0	0
$37$	−343.279	−1.52526	−0.762631	−	0.646834i	$-0.776093\pi$
$37$	−343.279	−1.52526	−0.762631	+	0.646834i	$0.776093\pi$
$38$	−76.0000	−0.324443
$39$	0	0
$40$	−158.963	−0.628358
$41$	−217.093	−0.826932	−0.413466	−	0.910519i	$-0.635682\pi$
$41$	−217.093	−0.826932	−0.413466	+	0.910519i	$0.635682\pi$
$42$	0	0
$43$	447.445	1.58685	0.793427	−	0.608665i	$-0.208295\pi$
$43$	447.445	1.58685	0.793427	+	0.608665i	$0.208295\pi$
$44$	158.963	0.544651
$45$	0	0
$46$	14.2591	0.0457043
$47$	−387.279	−1.20192	−0.600962	−	0.799277i	$-0.705216\pi$
$47$	−387.279	−1.20192	−0.600962	+	0.799277i	$0.705216\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−539.668	−1.52641
$51$	0	0
$52$	354.445	0.945244
$53$	322.093	0.834771	0.417386	−	0.908729i	$-0.362946\pi$
$53$	322.093	0.834771	0.417386	+	0.908729i	$0.362946\pi$
$54$	0	0
$55$	789.668	1.93598
$56$	−56.0000	−0.133631
$57$	0	0
$58$	515.149	1.16625
$59$	539.113	1.18960	0.594801	−	0.803873i	$-0.297231\pi$
$59$	539.113	1.18960	0.594801	+	0.803873i	$0.297231\pi$
$60$	0	0
$61$	303.555	0.637151	0.318576	−	0.947897i	$-0.396796\pi$
$61$	303.555	0.637151	0.318576	+	0.947897i	$0.396796\pi$
$62$	97.2226	0.199150
$63$	0	0
$64$	64.0000	0.125000
$65$	1760.74	3.35990
$66$	0	0
$67$	−365.834	−0.667070	−0.333535	−	0.942738i	$-0.608242\pi$
$67$	−365.834	−0.667070	−0.333535	+	0.942738i	$0.608242\pi$
$68$	−290.963	−0.518890
$69$	0	0
$70$	−278.186	−0.474994
$71$	615.389	1.02864	0.514318	−	0.857599i	$-0.328045\pi$
$71$	615.389	1.02864	0.514318	+	0.857599i	$0.328045\pi$
$72$	0	0
$73$	123.555	0.198096	0.0990480	−	0.995083i	$-0.468420\pi$
$73$	123.555	0.198096	0.0990480	+	0.995083i	$0.468420\pi$
$74$	686.558	1.07852
$75$	0	0
$76$	152.000	0.229416
$77$	278.186	0.411717
$78$	0	0
$79$	399.166	0.568477	0.284239	−	0.958754i	$-0.408259\pi$
$79$	399.166	0.568477	0.284239	+	0.958754i	$0.408259\pi$
$80$	317.927	0.444316
$81$	0	0
$82$	434.186	0.584729
$83$	−633.133	−0.837293	−0.418647	−	0.908149i	$-0.637495\pi$
$83$	−633.133	−0.837293	−0.418647	+	0.908149i	$0.637495\pi$
$84$	0	0
$85$	−1445.39	−1.84441
$86$	−894.890	−1.12208
$87$	0	0
$88$	−317.927	−0.385126
$89$	−823.541	−0.980845	−0.490422	−	0.871485i	$-0.663157\pi$
$89$	−823.541	−0.980845	−0.490422	+	0.871485i	$0.663157\pi$
$90$	0	0
$91$	620.279	0.714537
$92$	−28.5183	−0.0323178
$93$	0	0
$94$	774.558	0.849889
$95$	755.076	0.815465
$96$	0	0
$97$	−607.448	−0.635845	−0.317923	−	0.948117i	$-0.602985\pi$
$97$	−607.448	−0.635845	−0.317923	+	0.948117i	$0.602985\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	1079.34	1.07934
$101$	−1064.82	−1.04904	−0.524521	−	0.851397i	$-0.675756\pi$
$101$	−1064.82	−1.04904	−0.524521	+	0.851397i	$0.675756\pi$
$102$	0	0
$103$	1494.50	1.42969	0.714844	−	0.699284i	$-0.246498\pi$
$103$	1494.50	1.42969	0.714844	+	0.699284i	$0.246498\pi$
$104$	−708.890	−0.668389
$105$	0	0
$106$	−644.186	−0.590272
$107$	1733.30	1.56602	0.783009	−	0.622010i	$-0.213684\pi$
$107$	1733.30	1.56602	0.783009	+	0.622010i	$0.213684\pi$
$108$	0	0
$109$	−33.0534	−0.0290453	−0.0145227	−	0.999895i	$-0.504623\pi$
$109$	−33.0534	−0.0290453	−0.0145227	+	0.999895i	$0.504623\pi$
$110$	−1579.34	−1.36894
$111$	0	0
$112$	112.000	0.0944911
$113$	242.963	0.202266	0.101133	−	0.994873i	$-0.467753\pi$
$113$	242.963	0.202266	0.101133	+	0.994873i	$0.467753\pi$
$114$	0	0
$115$	−141.668	−0.114875
$116$	−1030.30	−0.824662
$117$	0	0
$118$	−1078.23	−0.841176
$119$	−509.186	−0.392244
$120$	0	0
$121$	248.335	0.186578
$122$	−607.110	−0.450534
$123$	0	0
$124$	−194.445	−0.140820
$125$	2877.91	2.05926
$126$	0	0
$127$	1600.06	1.11797	0.558987	−	0.829177i	$-0.311190\pi$
$127$	1600.06	1.11797	0.558987	+	0.829177i	$0.311190\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−3521.49	−2.37581
$131$	−2470.58	−1.64775	−0.823875	−	0.566771i	$-0.808192\pi$
$131$	−2470.58	−1.64775	−0.823875	+	0.566771i	$0.808192\pi$
$132$	0	0
$133$	266.000	0.173422
$134$	731.668	0.471690
$135$	0	0
$136$	581.927	0.366910
$137$	241.966	0.150895	0.0754474	−	0.997150i	$-0.475962\pi$
$137$	241.966	0.150895	0.0754474	+	0.997150i	$0.475962\pi$
$138$	0	0
$139$	307.780	0.187810	0.0939050	−	0.995581i	$-0.470065\pi$
$139$	307.780	0.187810	0.0939050	+	0.995581i	$0.470065\pi$
$140$	556.372	0.335872
$141$	0	0
$142$	−1230.78	−0.727356
$143$	3521.49	2.05931
$144$	0	0
$145$	−5118.12	−2.93129
$146$	−247.110	−0.140075
$147$	0	0
$148$	−1373.12	−0.762631
$149$	−1298.46	−0.713922	−0.356961	−	0.934119i	$-0.616187\pi$
$149$	−1298.46	−0.713922	−0.356961	+	0.934119i	$0.616187\pi$
$150$	0	0
$151$	−1430.28	−0.770823	−0.385411	−	0.922745i	$-0.625940\pi$
$151$	−1430.28	−0.770823	−0.385411	+	0.922745i	$0.625940\pi$
$152$	−304.000	−0.162221
$153$	0	0
$154$	−556.372	−0.291128
$155$	−965.927	−0.500549
$156$	0	0
$157$	−3641.18	−1.85094	−0.925469	−	0.378822i	$-0.876329\pi$
$157$	−3641.18	−1.85094	−0.925469	+	0.378822i	$0.876329\pi$
$158$	−798.332	−0.401974
$159$	0	0
$160$	−635.854	−0.314179
$161$	−49.9070	−0.0244300
$162$	0	0
$163$	−3298.45	−1.58500	−0.792498	−	0.609874i	$-0.791220\pi$
$163$	−3298.45	−1.58500	−0.792498	+	0.609874i	$0.791220\pi$
$164$	−868.372	−0.413466
$165$	0	0
$166$	1266.27	0.592056
$167$	1402.69	0.649959	0.324980	−	0.945721i	$-0.394643\pi$
$167$	1402.69	0.649959	0.324980	+	0.945721i	$0.394643\pi$
$168$	0	0
$169$	5654.96	2.57395
$170$	2890.78	1.30419
$171$	0	0
$172$	1789.78	0.793427
$173$	1683.01	0.739635	0.369817	−	0.929104i	$-0.379420\pi$
$173$	1683.01	0.739635	0.369817	+	0.929104i	$0.379420\pi$
$174$	0	0
$175$	1888.84	0.815901
$176$	635.854	0.272325
$177$	0	0
$178$	1647.08	0.693562
$179$	−1492.08	−0.623035	−0.311517	−	0.950240i	$-0.600837\pi$
$179$	−1492.08	−0.623035	−0.311517	+	0.950240i	$0.600837\pi$
$180$	0	0
$181$	2945.73	1.20969	0.604846	−	0.796343i	$-0.293235\pi$
$181$	2945.73	1.20969	0.604846	+	0.796343i	$0.293235\pi$
$182$	−1240.56	−0.505254
$183$	0	0
$184$	57.0366	0.0228521
$185$	−6821.10	−2.71080
$186$	0	0
$187$	−2890.78	−1.13045
$188$	−1549.12	−0.600962
$189$	0	0
$190$	−1510.15	−0.576621
$191$	1913.44	0.724879	0.362439	−	0.932007i	$-0.381944\pi$
$191$	1913.44	0.724879	0.362439	+	0.932007i	$0.381944\pi$
$192$	0	0
$193$	−59.3263	−0.0221264	−0.0110632	−	0.999939i	$-0.503522\pi$
$193$	−59.3263	−0.0221264	−0.0110632	+	0.999939i	$0.503522\pi$
$194$	1214.90	0.449611
$195$	0	0
$196$	196.000	0.0714286
$197$	−287.780	−0.104079	−0.0520394	−	0.998645i	$-0.516572\pi$
$197$	−287.780	−0.104079	−0.0520394	+	0.998645i	$0.516572\pi$
$198$	0	0
$199$	−4014.51	−1.43006	−0.715028	−	0.699096i	$-0.753586\pi$
$199$	−4014.51	−1.43006	−0.715028	+	0.699096i	$0.753586\pi$
$200$	−2158.67	−0.763205
$201$	0	0
$202$	2129.63	0.741785
$203$	−1803.02	−0.623386
$204$	0	0
$205$	−4313.73	−1.46968
$206$	−2989.01	−1.01094
$207$	0	0
$208$	1417.78	0.472622
$209$	1510.15	0.499806
$210$	0	0
$211$	−4810.30	−1.56945	−0.784726	−	0.619842i	$-0.787197\pi$
$211$	−4810.30	−1.56945	−0.784726	+	0.619842i	$0.787197\pi$
$212$	1288.37	0.417386
$213$	0	0
$214$	−3466.59	−1.10734
$215$	8890.93	2.82026
$216$	0	0
$217$	−340.279	−0.106450
$218$	66.1068	0.0205381
$219$	0	0
$220$	3158.67	0.967989
$221$	−6445.66	−1.96191
$222$	0	0
$223$	3453.12	1.03694	0.518470	−	0.855096i	$-0.326502\pi$
$223$	3453.12	1.03694	0.518470	+	0.855096i	$0.326502\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−485.927	−0.143024
$227$	158.127	0.0462345	0.0231172	−	0.999733i	$-0.492641\pi$
$227$	158.127	0.0462345	0.0231172	+	0.999733i	$0.492641\pi$
$228$	0	0
$229$	−612.332	−0.176699	−0.0883495	−	0.996090i	$-0.528159\pi$
$229$	−612.332	−0.176699	−0.0883495	+	0.996090i	$0.528159\pi$
$230$	283.335	0.0812286
$231$	0	0
$232$	2060.60	0.583124
$233$	−5206.39	−1.46387	−0.731936	−	0.681374i	$-0.761383\pi$
$233$	−5206.39	−1.46387	−0.731936	+	0.681374i	$0.761383\pi$
$234$	0	0
$235$	−7695.40	−2.13614
$236$	2156.45	0.594801
$237$	0	0
$238$	1018.37	0.277358
$239$	647.854	0.175340	0.0876698	−	0.996150i	$-0.472058\pi$
$239$	647.854	0.175340	0.0876698	+	0.996150i	$0.472058\pi$
$240$	0	0
$241$	6141.67	1.64158	0.820788	−	0.571232i	$-0.193535\pi$
$241$	6141.67	1.64158	0.820788	+	0.571232i	$0.193535\pi$
$242$	−496.671	−0.131931
$243$	0	0
$244$	1214.22	0.318576
$245$	973.651	0.253895
$246$	0	0
$247$	3367.23	0.867415
$248$	388.890	0.0995748
$249$	0	0
$250$	−5755.82	−1.45612
$251$	4779.84	1.20199	0.600997	−	0.799251i	$-0.294770\pi$
$251$	4779.84	1.20199	0.600997	+	0.799251i	$0.294770\pi$
$252$	0	0
$253$	−283.335	−0.0704077
$254$	−3200.12	−0.790526
$255$	0	0
$256$	256.000	0.0625000
$257$	−1207.85	−0.293165	−0.146583	−	0.989198i	$-0.546827\pi$
$257$	−1207.85	−0.293165	−0.146583	+	0.989198i	$0.546827\pi$
$258$	0	0
$259$	−2402.95	−0.576495
$260$	7042.98	1.67995
$261$	0	0
$262$	4941.16	1.16514
$263$	−2869.66	−0.672817	−0.336408	−	0.941716i	$-0.609212\pi$
$263$	−2869.66	−0.672817	−0.336408	+	0.941716i	$0.609212\pi$
$264$	0	0
$265$	6400.12	1.48361
$266$	−532.000	−0.122628
$267$	0	0
$268$	−1463.34	−0.333535
$269$	−5871.69	−1.33087	−0.665434	−	0.746457i	$-0.731753\pi$
$269$	−5871.69	−1.33087	−0.665434	+	0.746457i	$0.731753\pi$
$270$	0	0
$271$	1081.96	0.242525	0.121263	−	0.992620i	$-0.461306\pi$
$271$	1081.96	0.242525	0.121263	+	0.992620i	$0.461306\pi$
$272$	−1163.85	−0.259445
$273$	0	0
$274$	−483.933	−0.106699
$275$	10723.4	2.35144
$276$	0	0
$277$	931.511	0.202054	0.101027	−	0.994884i	$-0.467787\pi$
$277$	931.511	0.202054	0.101027	+	0.994884i	$0.467787\pi$
$278$	−615.561	−0.132802
$279$	0	0
$280$	−1112.74	−0.237497
$281$	1915.50	0.406652	0.203326	−	0.979111i	$-0.434825\pi$
$281$	1915.50	0.406652	0.203326	+	0.979111i	$0.434825\pi$
$282$	0	0
$283$	126.985	0.0266730	0.0133365	−	0.999911i	$-0.495755\pi$
$283$	126.985	0.0266730	0.0133365	+	0.999911i	$0.495755\pi$
$284$	2461.55	0.514318
$285$	0	0
$286$	−7042.98	−1.45615
$287$	−1519.65	−0.312551
$288$	0	0
$289$	378.232	0.0769859
$290$	10236.2	2.07273
$291$	0	0
$292$	494.220	0.0990480
$293$	4129.14	0.823300	0.411650	−	0.911342i	$-0.364953\pi$
$293$	4129.14	0.823300	0.411650	+	0.911342i	$0.364953\pi$
$294$	0	0
$295$	10712.4	2.11424
$296$	2746.23	0.539262
$297$	0	0
$298$	2596.93	0.504819
$299$	−631.761	−0.122193
$300$	0	0
$301$	3132.12	0.599775
$302$	2860.55	0.545054
$303$	0	0
$304$	608.000	0.114708
$305$	6031.76	1.13239
$306$	0	0
$307$	1528.43	0.284144	0.142072	−	0.989856i	$-0.454624\pi$
$307$	1528.43	0.284144	0.142072	+	0.989856i	$0.454624\pi$
$308$	1112.74	0.205859
$309$	0	0
$310$	1931.85	0.353942
$311$	−8985.18	−1.63827	−0.819136	−	0.573599i	$-0.805547\pi$
$311$	−8985.18	−1.63827	−0.819136	+	0.573599i	$0.805547\pi$
$312$	0	0
$313$	−9218.80	−1.66478	−0.832392	−	0.554187i	$-0.813029\pi$
$313$	−9218.80	−1.66478	−0.832392	+	0.554187i	$0.813029\pi$
$314$	7282.35	1.30881
$315$	0	0
$316$	1596.66	0.284239
$317$	−802.226	−0.142137	−0.0710686	−	0.997471i	$-0.522641\pi$
$317$	−802.226	−0.142137	−0.0710686	+	0.997471i	$0.522641\pi$
$318$	0	0
$319$	−10236.2	−1.79661
$320$	1271.71	0.222158
$321$	0	0
$322$	99.8140	0.0172746
$323$	−2764.15	−0.476166
$324$	0	0
$325$	23910.3	4.08094
$326$	6596.90	1.12076
$327$	0	0
$328$	1736.74	0.292365
$329$	−2710.95	−0.454285
$330$	0	0
$331$	−953.564	−0.158346	−0.0791731	−	0.996861i	$-0.525228\pi$
$331$	−953.564	−0.158346	−0.0791731	+	0.996861i	$0.525228\pi$
$332$	−2532.53	−0.418647
$333$	0	0
$334$	−2805.37	−0.459591
$335$	−7269.27	−1.18556
$336$	0	0
$337$	9659.00	1.56130	0.780652	−	0.624966i	$-0.214887\pi$
$337$	9659.00	1.56130	0.780652	+	0.624966i	$0.214887\pi$
$338$	−11309.9	−1.82005
$339$	0	0
$340$	−5781.57	−0.922204
$341$	−1931.85	−0.306791
$342$	0	0
$343$	343.000	0.0539949
$344$	−3579.56	−0.561038
$345$	0	0
$346$	−3366.02	−0.523001
$347$	−1854.34	−0.286876	−0.143438	−	0.989659i	$-0.545816\pi$
$347$	−1854.34	−0.286876	−0.143438	+	0.989659i	$0.545816\pi$
$348$	0	0
$349$	−2643.08	−0.405389	−0.202695	−	0.979242i	$-0.564970\pi$
$349$	−2643.08	−0.405389	−0.202695	+	0.979242i	$0.564970\pi$
$350$	−3777.67	−0.576929
$351$	0	0
$352$	−1271.71	−0.192563
$353$	3096.64	0.466905	0.233453	−	0.972368i	$-0.424998\pi$
$353$	3096.64	0.466905	0.233453	+	0.972368i	$0.424998\pi$
$354$	0	0
$355$	12228.0	1.82816
$356$	−3294.16	−0.490422
$357$	0	0
$358$	2984.16	0.440552
$359$	−5575.10	−0.819617	−0.409809	−	0.912172i	$-0.634405\pi$
$359$	−5575.10	−0.819617	−0.409809	+	0.912172i	$0.634405\pi$
$360$	0	0
$361$	−5415.00	−0.789474
$362$	−5891.45	−0.855381
$363$	0	0
$364$	2481.12	0.357269
$365$	2455.09	0.352069
$366$	0	0
$367$	8400.17	1.19478	0.597391	−	0.801950i	$-0.296204\pi$
$367$	8400.17	1.19478	0.597391	+	0.801950i	$0.296204\pi$
$368$	−114.073	−0.0161589
$369$	0	0
$370$	13642.2	1.91682
$371$	2254.65	0.315514
$372$	0	0
$373$	4612.62	0.640301	0.320151	−	0.947367i	$-0.396266\pi$
$373$	4612.62	0.640301	0.320151	+	0.947367i	$0.396266\pi$
$374$	5781.57	0.799352
$375$	0	0
$376$	3098.23	0.424944
$377$	−22824.0	−3.11803
$378$	0	0
$379$	669.166	0.0906933	0.0453466	−	0.998971i	$-0.485561\pi$
$379$	669.166	0.0906933	0.0453466	+	0.998971i	$0.485561\pi$
$380$	3020.30	0.407733
$381$	0	0
$382$	−3826.88	−0.512567
$383$	2190.08	0.292187	0.146094	−	0.989271i	$-0.453330\pi$
$383$	2190.08	0.292187	0.146094	+	0.989271i	$0.453330\pi$
$384$	0	0
$385$	5527.67	0.731731
$386$	118.653	0.0156457
$387$	0	0
$388$	−2429.79	−0.317923
$389$	−12281.9	−1.60082	−0.800410	−	0.599452i	$-0.795385\pi$
$389$	−12281.9	−1.60082	−0.800410	+	0.599452i	$0.795385\pi$
$390$	0	0
$391$	518.611	0.0670775
$392$	−392.000	−0.0505076
$393$	0	0
$394$	575.561	0.0735948
$395$	7931.60	1.01033
$396$	0	0
$397$	−11016.9	−1.39275	−0.696376	−	0.717677i	$-0.745206\pi$
$397$	−11016.9	−1.39275	−0.696376	+	0.717677i	$0.745206\pi$
$398$	8029.02	1.01120
$399$	0	0
$400$	4317.34	0.539668
$401$	2428.73	0.302456	0.151228	−	0.988499i	$-0.451677\pi$
$401$	2428.73	0.302456	0.151228	+	0.988499i	$0.451677\pi$
$402$	0	0
$403$	−4307.51	−0.532437
$404$	−4259.27	−0.524521
$405$	0	0
$406$	3606.05	0.440801
$407$	−13642.2	−1.66147
$408$	0	0
$409$	−8077.01	−0.976485	−0.488243	−	0.872708i	$-0.662362\pi$
$409$	−8077.01	−0.976485	−0.488243	+	0.872708i	$0.662362\pi$
$410$	8627.46	1.03922
$411$	0	0
$412$	5978.02	0.714844
$413$	3773.79	0.449627
$414$	0	0
$415$	−12580.6	−1.48809
$416$	−2835.56	−0.334194
$417$	0	0
$418$	−3020.30	−0.353416
$419$	10911.5	1.27223	0.636114	−	0.771595i	$-0.280541\pi$
$419$	10911.5	1.27223	0.636114	+	0.771595i	$0.280541\pi$
$420$	0	0
$421$	−3452.68	−0.399699	−0.199850	−	0.979827i	$-0.564045\pi$
$421$	−3452.68	−0.399699	−0.199850	+	0.979827i	$0.564045\pi$
$422$	9620.59	1.10977
$423$	0	0
$424$	−2576.74	−0.295136
$425$	−19627.9	−2.24022
$426$	0	0
$427$	2124.88	0.240821
$428$	6933.18	0.783009
$429$	0	0
$430$	−17781.9	−1.99423
$431$	−4418.73	−0.493834	−0.246917	−	0.969037i	$-0.579418\pi$
$431$	−4418.73	−0.493834	−0.246917	+	0.969037i	$0.579418\pi$
$432$	0	0
$433$	−517.365	−0.0574203	−0.0287102	−	0.999588i	$-0.509140\pi$
$433$	−517.365	−0.0574203	−0.0287102	+	0.999588i	$0.509140\pi$
$434$	680.558	0.0752715
$435$	0	0
$436$	−132.214	−0.0145227
$437$	−270.924	−0.0296568
$438$	0	0
$439$	12145.5	1.32044	0.660222	−	0.751071i	$-0.270462\pi$
$439$	12145.5	1.32044	0.660222	+	0.751071i	$0.270462\pi$
$440$	−6317.34	−0.684471
$441$	0	0
$442$	12891.3	1.38728
$443$	11814.7	1.26712	0.633558	−	0.773695i	$-0.281594\pi$
$443$	11814.7	1.26712	0.633558	+	0.773695i	$0.281594\pi$
$444$	0	0
$445$	−16364.1	−1.74322
$446$	−6906.23	−0.733228
$447$	0	0
$448$	448.000	0.0472456
$449$	−15969.1	−1.67846	−0.839231	−	0.543775i	$-0.816995\pi$
$449$	−15969.1	−1.67846	−0.839231	+	0.543775i	$0.816995\pi$
$450$	0	0
$451$	−8627.46	−0.900779
$452$	971.854	0.101133
$453$	0	0
$454$	−316.253	−0.0326927
$455$	12325.2	1.26992
$456$	0	0
$457$	12880.2	1.31840	0.659201	−	0.751967i	$-0.270895\pi$
$457$	12880.2	1.31840	0.659201	+	0.751967i	$0.270895\pi$
$458$	1224.66	0.124945
$459$	0	0
$460$	−566.671	−0.0574373
$461$	6764.57	0.683421	0.341711	−	0.939805i	$-0.388994\pi$
$461$	6764.57	0.683421	0.341711	+	0.939805i	$0.388994\pi$
$462$	0	0
$463$	1121.06	0.112527	0.0562636	−	0.998416i	$-0.482081\pi$
$463$	1121.06	0.112527	0.0562636	+	0.998416i	$0.482081\pi$
$464$	−4121.20	−0.412331
$465$	0	0
$466$	10412.8	1.03511
$467$	15671.4	1.55286	0.776431	−	0.630202i	$-0.217028\pi$
$467$	15671.4	1.55286	0.776431	+	0.630202i	$0.217028\pi$
$468$	0	0
$469$	−2560.84	−0.252129
$470$	15390.8	1.51048
$471$	0	0
$472$	−4312.90	−0.420588
$473$	17781.9	1.72856
$474$	0	0
$475$	10253.7	0.990466
$476$	−2036.74	−0.196122
$477$	0	0
$478$	−1295.71	−0.123984
$479$	11211.7	1.06947	0.534734	−	0.845020i	$-0.320412\pi$
$479$	11211.7	1.06947	0.534734	+	0.845020i	$0.320412\pi$
$480$	0	0
$481$	−30418.4	−2.88349
$482$	−12283.3	−1.16077
$483$	0	0
$484$	993.341	0.0932890
$485$	−12070.3	−1.13007
$486$	0	0
$487$	−5261.00	−0.489525	−0.244762	−	0.969583i	$-0.578710\pi$
$487$	−5261.00	−0.489525	−0.244762	+	0.969583i	$0.578710\pi$
$488$	−2428.44	−0.225267
$489$	0	0
$490$	−1947.30	−0.179531
$491$	−8754.40	−0.804645	−0.402322	−	0.915498i	$-0.631797\pi$
$491$	−8754.40	−0.804645	−0.402322	+	0.915498i	$0.631797\pi$
$492$	0	0
$493$	18736.2	1.71164
$494$	−6734.46	−0.613355
$495$	0	0
$496$	−777.780	−0.0704100
$497$	4307.72	0.388788
$498$	0	0
$499$	−18969.8	−1.70181	−0.850905	−	0.525319i	$-0.823946\pi$
$499$	−18969.8	−1.70181	−0.850905	+	0.525319i	$0.823946\pi$
$500$	11511.6	1.02963
$501$	0	0
$502$	−9559.67	−0.849938
$503$	3563.27	0.315862	0.157931	−	0.987450i	$-0.449518\pi$
$503$	3563.27	0.315862	0.157931	+	0.987450i	$0.449518\pi$
$504$	0	0
$505$	−21158.4	−1.86443
$506$	566.671	0.0497857
$507$	0	0
$508$	6400.25	0.558987
$509$	−722.492	−0.0629153	−0.0314577	−	0.999505i	$-0.510015\pi$
$509$	−722.492	−0.0629153	−0.0314577	+	0.999505i	$0.510015\pi$
$510$	0	0
$511$	864.884	0.0748732
$512$	−512.000	−0.0441942
$513$	0	0
$514$	2415.70	0.207299
$515$	29696.4	2.54094
$516$	0	0
$517$	−15390.8	−1.30926
$518$	4805.91	0.407644
$519$	0	0
$520$	−14086.0	−1.18790
$521$	−10687.6	−0.898721	−0.449361	−	0.893350i	$-0.648348\pi$
$521$	−10687.6	−0.898721	−0.449361	+	0.893350i	$0.648348\pi$
$522$	0	0
$523$	−4810.42	−0.402189	−0.201095	−	0.979572i	$-0.564450\pi$
$523$	−4810.42	−0.402189	−0.201095	+	0.979572i	$0.564450\pi$
$524$	−9882.31	−0.823875
$525$	0	0
$526$	5739.32	0.475753
$527$	3536.03	0.292280
$528$	0	0
$529$	−12116.2	−0.995822
$530$	−12800.2	−1.04907
$531$	0	0
$532$	1064.00	0.0867110
$533$	−19236.9	−1.56331
$534$	0	0
$535$	34441.3	2.78323
$536$	2926.67	0.235845
$537$	0	0
$538$	11743.4	0.941065
$539$	1947.30	0.155615
$540$	0	0
$541$	−8090.06	−0.642918	−0.321459	−	0.946923i	$-0.604173\pi$
$541$	−8090.06	−0.642918	−0.321459	+	0.946923i	$0.604173\pi$
$542$	−2163.92	−0.171491
$543$	0	0
$544$	2327.71	0.183455
$545$	−656.785	−0.0516212
$546$	0	0
$547$	−11831.9	−0.924852	−0.462426	−	0.886658i	$-0.653021\pi$
$547$	−11831.9	−0.924852	−0.462426	+	0.886658i	$0.653021\pi$
$548$	967.866	0.0754474
$549$	0	0
$550$	−21446.9	−1.66272
$551$	−9787.84	−0.756762
$552$	0	0
$553$	2794.16	0.214864
$554$	−1863.02	−0.142874
$555$	0	0
$556$	1231.12	0.0939050
$557$	−15783.2	−1.20064	−0.600321	−	0.799759i	$-0.704960\pi$
$557$	−15783.2	−1.20064	−0.600321	+	0.799759i	$0.704960\pi$
$558$	0	0
$559$	39648.7	2.99993
$560$	2225.49	0.167936
$561$	0	0
$562$	−3831.00	−0.287546
$563$	4353.81	0.325917	0.162958	−	0.986633i	$-0.447896\pi$
$563$	4353.81	0.325917	0.162958	+	0.986633i	$0.447896\pi$
$564$	0	0
$565$	4827.79	0.359481
$566$	−253.970	−0.0188607
$567$	0	0
$568$	−4923.11	−0.363678
$569$	16526.4	1.21761	0.608806	−	0.793319i	$-0.291649\pi$
$569$	16526.4	1.21761	0.608806	+	0.793319i	$0.291649\pi$
$570$	0	0
$571$	−24307.8	−1.78152	−0.890761	−	0.454471i	$-0.849828\pi$
$571$	−24307.8	−1.78152	−0.890761	+	0.454471i	$0.849828\pi$
$572$	14086.0	1.02966
$573$	0	0
$574$	3039.30	0.221007
$575$	−1923.80	−0.139527
$576$	0	0
$577$	−5474.80	−0.395007	−0.197503	−	0.980302i	$-0.563283\pi$
$577$	−5474.80	−0.395007	−0.197503	+	0.980302i	$0.563283\pi$
$578$	−756.463	−0.0544372
$579$	0	0
$580$	−20472.5	−1.46564
$581$	−4431.93	−0.316467
$582$	0	0
$583$	12800.2	0.909318
$584$	−988.439	−0.0700375
$585$	0	0
$586$	−8258.28	−0.582161
$587$	12868.2	0.904818	0.452409	−	0.891811i	$-0.350565\pi$
$587$	12868.2	0.904818	0.452409	+	0.891811i	$0.350565\pi$
$588$	0	0
$589$	−1847.23	−0.129225
$590$	−21424.8	−1.49499
$591$	0	0
$592$	−5492.46	−0.381316
$593$	16734.6	1.15886	0.579432	−	0.815020i	$-0.303274\pi$
$593$	16734.6	1.15886	0.579432	+	0.815020i	$0.303274\pi$
$594$	0	0
$595$	−10117.7	−0.697121
$596$	−5193.86	−0.356961
$597$	0	0
$598$	1263.52	0.0864034
$599$	7640.15	0.521149	0.260574	−	0.965454i	$-0.416088\pi$
$599$	7640.15	0.521149	0.260574	+	0.965454i	$0.416088\pi$
$600$	0	0
$601$	−6068.66	−0.411890	−0.205945	−	0.978564i	$-0.566027\pi$
$601$	−6068.66	−0.411890	−0.205945	+	0.978564i	$0.566027\pi$
$602$	−6264.23	−0.424105
$603$	0	0
$604$	−5721.10	−0.385411
$605$	4934.53	0.331599
$606$	0	0
$607$	11318.1	0.756814	0.378407	−	0.925639i	$-0.376472\pi$
$607$	11318.1	0.756814	0.378407	+	0.925639i	$0.376472\pi$
$608$	−1216.00	−0.0811107
$609$	0	0
$610$	−12063.5	−0.800718
$611$	−34317.3	−2.27222
$612$	0	0
$613$	16973.1	1.11833	0.559167	−	0.829055i	$-0.311121\pi$
$613$	16973.1	1.11833	0.559167	+	0.829055i	$0.311121\pi$
$614$	−3056.87	−0.200920
$615$	0	0
$616$	−2225.49	−0.145564
$617$	−13351.7	−0.871182	−0.435591	−	0.900145i	$-0.643461\pi$
$617$	−13351.7	−0.871182	−0.435591	+	0.900145i	$0.643461\pi$
$618$	0	0
$619$	−7258.04	−0.471284	−0.235642	−	0.971840i	$-0.575719\pi$
$619$	−7258.04	−0.471284	−0.235642	+	0.971840i	$0.575719\pi$
$620$	−3863.71	−0.250275
$621$	0	0
$622$	17970.4	1.15843
$623$	−5764.79	−0.370724
$624$	0	0
$625$	23456.1	1.50119
$626$	18437.6	1.17718
$627$	0	0
$628$	−14564.7	−0.925469
$629$	24970.4	1.58289
$630$	0	0
$631$	−9332.14	−0.588758	−0.294379	−	0.955689i	$-0.595113\pi$
$631$	−9332.14	−0.588758	−0.294379	+	0.955689i	$0.595113\pi$
$632$	−3193.33	−0.200987
$633$	0	0
$634$	1604.45	0.100506
$635$	31793.9	1.98693
$636$	0	0
$637$	4341.95	0.270070
$638$	20472.5	1.27040
$639$	0	0
$640$	−2543.41	−0.157090
$641$	9908.84	0.610571	0.305285	−	0.952261i	$-0.401248\pi$
$641$	9908.84	0.610571	0.305285	+	0.952261i	$0.401248\pi$
$642$	0	0
$643$	−7468.14	−0.458032	−0.229016	−	0.973423i	$-0.573551\pi$
$643$	−7468.14	−0.458032	−0.229016	+	0.973423i	$0.573551\pi$
$644$	−199.628	−0.0122150
$645$	0	0
$646$	5528.30	0.336700
$647$	−4863.05	−0.295496	−0.147748	−	0.989025i	$-0.547202\pi$
$647$	−4863.05	−0.295496	−0.147748	+	0.989025i	$0.547202\pi$
$648$	0	0
$649$	21424.8	1.29584
$650$	−47820.6	−2.88566
$651$	0	0
$652$	−13193.8	−0.792498
$653$	−1144.48	−0.0685863	−0.0342931	−	0.999412i	$-0.510918\pi$
$653$	−1144.48	−0.0685863	−0.0342931	+	0.999412i	$0.510918\pi$
$654$	0	0
$655$	−49091.4	−2.92849
$656$	−3473.49	−0.206733
$657$	0	0
$658$	5421.91	0.321228
$659$	20783.3	1.22853	0.614267	−	0.789098i	$-0.289452\pi$
$659$	20783.3	1.22853	0.614267	+	0.789098i	$0.289452\pi$
$660$	0	0
$661$	6071.70	0.357280	0.178640	−	0.983915i	$-0.442830\pi$
$661$	6071.70	0.357280	0.178640	+	0.983915i	$0.442830\pi$
$662$	1907.13	0.111968
$663$	0	0
$664$	5065.06	0.296028
$665$	5285.53	0.308217
$666$	0	0
$667$	1836.40	0.106605
$668$	5610.75	0.324980
$669$	0	0
$670$	14538.5	0.838318
$671$	12063.5	0.694050
$672$	0	0
$673$	−25493.0	−1.46015	−0.730077	−	0.683365i	$-0.760516\pi$
$673$	−25493.0	−1.46015	−0.730077	+	0.683365i	$0.760516\pi$
$674$	−19318.0	−1.10401
$675$	0	0
$676$	22619.8	1.28697
$677$	14911.3	0.846511	0.423256	−	0.906010i	$-0.360887\pi$
$677$	14911.3	0.846511	0.423256	+	0.906010i	$0.360887\pi$
$678$	0	0
$679$	−4252.14	−0.240327
$680$	11563.1	0.652097
$681$	0	0
$682$	3863.71	0.216934
$683$	−7005.90	−0.392494	−0.196247	−	0.980555i	$-0.562875\pi$
$683$	−7005.90	−0.392494	−0.196247	+	0.980555i	$0.562875\pi$
$684$	0	0
$685$	4807.98	0.268180
$686$	−686.000	−0.0381802
$687$	0	0
$688$	7159.12	0.396714
$689$	28541.1	1.57813
$690$	0	0
$691$	2772.02	0.152609	0.0763045	−	0.997085i	$-0.475688\pi$
$691$	2772.02	0.152609	0.0763045	+	0.997085i	$0.475688\pi$
$692$	6732.04	0.369817
$693$	0	0
$694$	3708.68	0.202852
$695$	6115.73	0.333788
$696$	0	0
$697$	15791.5	0.858173
$698$	5286.16	0.286654
$699$	0	0
$700$	7555.35	0.407950
$701$	9933.72	0.535223	0.267612	−	0.963527i	$-0.413766\pi$
$701$	9933.72	0.535223	0.267612	+	0.963527i	$0.413766\pi$
$702$	0	0
$703$	−13044.6	−0.699838
$704$	2543.41	0.136163
$705$	0	0
$706$	−6193.28	−0.330152
$707$	−7453.72	−0.396501
$708$	0	0
$709$	−13314.4	−0.705266	−0.352633	−	0.935762i	$-0.614714\pi$
$709$	−13314.4	−0.705266	−0.352633	+	0.935762i	$0.614714\pi$
$710$	−24456.1	−1.29270
$711$	0	0
$712$	6588.33	0.346781
$713$	346.578	0.0182040
$714$	0	0
$715$	69973.5	3.65994
$716$	−5968.32	−0.311517
$717$	0	0
$718$	11150.2	0.579557
$719$	17436.0	0.904386	0.452193	−	0.891920i	$-0.350642\pi$
$719$	17436.0	0.904386	0.452193	+	0.891920i	$0.350642\pi$
$720$	0	0
$721$	10461.5	0.540371
$722$	10830.0	0.558242
$723$	0	0
$724$	11782.9	0.604846
$725$	−69502.4	−3.56035
$726$	0	0
$727$	5380.39	0.274481	0.137240	−	0.990538i	$-0.456177\pi$
$727$	5380.39	0.274481	0.137240	+	0.990538i	$0.456177\pi$
$728$	−4962.23	−0.252627
$729$	0	0
$730$	−4910.18	−0.248950
$731$	−32547.5	−1.64680
$732$	0	0
$733$	−37200.7	−1.87454	−0.937271	−	0.348600i	$-0.886657\pi$
$733$	−37200.7	−1.87454	−0.937271	+	0.348600i	$0.886657\pi$
$734$	−16800.3	−0.844839
$735$	0	0
$736$	228.146	0.0114261
$737$	−14538.5	−0.726641
$738$	0	0
$739$	17236.5	0.857988	0.428994	−	0.903307i	$-0.358868\pi$
$739$	17236.5	0.857988	0.428994	+	0.903307i	$0.358868\pi$
$740$	−27284.4	−1.35540
$741$	0	0
$742$	−4509.30	−0.223102
$743$	16450.2	0.812246	0.406123	−	0.913818i	$-0.366880\pi$
$743$	16450.2	0.812246	0.406123	+	0.913818i	$0.366880\pi$
$744$	0	0
$745$	−25801.1	−1.26883
$746$	−9225.24	−0.452761
$747$	0	0
$748$	−11563.1	−0.565227
$749$	12133.1	0.591900
$750$	0	0
$751$	−29439.6	−1.43045	−0.715223	−	0.698897i	$-0.753675\pi$
$751$	−29439.6	−1.43045	−0.715223	+	0.698897i	$0.753675\pi$
$752$	−6196.46	−0.300481
$753$	0	0
$754$	45648.0	2.20478
$755$	−28420.2	−1.36996
$756$	0	0
$757$	12924.0	0.620518	0.310259	−	0.950652i	$-0.399584\pi$
$757$	12924.0	0.620518	0.310259	+	0.950652i	$0.399584\pi$
$758$	−1338.33	−0.0641298
$759$	0	0
$760$	−6040.61	−0.288310
$761$	12513.7	0.596087	0.298043	−	0.954552i	$-0.403666\pi$
$761$	12513.7	0.596087	0.298043	+	0.954552i	$0.403666\pi$
$762$	0	0
$763$	−231.374	−0.0109781
$764$	7653.77	0.362439
$765$	0	0
$766$	−4380.16	−0.206608
$767$	47771.5	2.24893
$768$	0	0
$769$	−12823.8	−0.601351	−0.300675	−	0.953727i	$-0.597212\pi$
$769$	−12823.8	−0.601351	−0.300675	+	0.953727i	$0.597212\pi$
$770$	−11055.3	−0.517412
$771$	0	0
$772$	−237.305	−0.0110632
$773$	8348.45	0.388451	0.194226	−	0.980957i	$-0.437781\pi$
$773$	8348.45	0.388451	0.194226	+	0.980957i	$0.437781\pi$
$774$	0	0
$775$	−13117.0	−0.607968
$776$	4859.59	0.224805
$777$	0	0
$778$	24563.9	1.13195
$779$	−8249.53	−0.379423
$780$	0	0
$781$	24456.1	1.12050
$782$	−1037.22	−0.0474310
$783$	0	0
$784$	784.000	0.0357143
$785$	−72351.7	−3.28961
$786$	0	0
$787$	−23593.9	−1.06865	−0.534327	−	0.845278i	$-0.679435\pi$
$787$	−23593.9	−1.06865	−0.534327	+	0.845278i	$0.679435\pi$
$788$	−1151.12	−0.0520394
$789$	0	0
$790$	−15863.2	−0.714415
$791$	1700.74	0.0764494
$792$	0	0
$793$	26898.4	1.20453
$794$	22033.8	0.984825
$795$	0	0
$796$	−16058.0	−0.715028
$797$	−25753.3	−1.14458	−0.572289	−	0.820052i	$-0.693945\pi$
$797$	−25753.3	−1.14458	−0.572289	+	0.820052i	$0.693945\pi$
$798$	0	0
$799$	28171.0	1.24733
$800$	−8634.68	−0.381603
$801$	0	0
$802$	−4857.46	−0.213869
$803$	4910.18	0.215786
$804$	0	0
$805$	−991.674	−0.0434185
$806$	8615.02	0.376490
$807$	0	0
$808$	8518.54	0.370892
$809$	−15768.2	−0.685266	−0.342633	−	0.939469i	$-0.611319\pi$
$809$	−15768.2	−0.685266	−0.342633	+	0.939469i	$0.611319\pi$
$810$	0	0
$811$	−18577.0	−0.804351	−0.402175	−	0.915563i	$-0.631746\pi$
$811$	−18577.0	−0.804351	−0.402175	+	0.915563i	$0.631746\pi$
$812$	−7212.09	−0.311693
$813$	0	0
$814$	27284.4	1.17484
$815$	−65541.6	−2.81696
$816$	0	0
$817$	17002.9	0.728099
$818$	16154.0	0.690479
$819$	0	0
$820$	−17254.9	−0.734839
$821$	−25406.7	−1.08002	−0.540012	−	0.841657i	$-0.681580\pi$
$821$	−25406.7	−1.08002	−0.540012	+	0.841657i	$0.681580\pi$
$822$	0	0
$823$	43664.6	1.84940	0.924698	−	0.380702i	$-0.124318\pi$
$823$	43664.6	1.84940	0.924698	+	0.380702i	$0.124318\pi$
$824$	−11956.0	−0.505471
$825$	0	0
$826$	−7547.58	−0.317934
$827$	23206.4	0.975773	0.487887	−	0.872907i	$-0.337768\pi$
$827$	23206.4	0.975773	0.487887	+	0.872907i	$0.337768\pi$
$828$	0	0
$829$	10247.3	0.429316	0.214658	−	0.976689i	$-0.431136\pi$
$829$	10247.3	0.429316	0.214658	+	0.976689i	$0.431136\pi$
$830$	25161.2	1.05224
$831$	0	0
$832$	5671.12	0.236311
$833$	−3564.30	−0.148254
$834$	0	0
$835$	27872.0	1.15515
$836$	6040.61	0.249903
$837$	0	0
$838$	−21823.1	−0.899600
$839$	−12322.0	−0.507035	−0.253518	−	0.967331i	$-0.581588\pi$
$839$	−12322.0	−0.507035	−0.253518	+	0.967331i	$0.581588\pi$
$840$	0	0
$841$	41955.7	1.72027
$842$	6905.37	0.282630
$843$	0	0
$844$	−19241.2	−0.784726
$845$	112366.	4.57458
$846$	0	0
$847$	1738.35	0.0705199
$848$	5153.49	0.208693
$849$	0	0
$850$	39255.9	1.58408
$851$	2447.43	0.0985863
$852$	0	0
$853$	38665.8	1.55204	0.776020	−	0.630708i	$-0.217235\pi$
$853$	38665.8	1.55204	0.776020	+	0.630708i	$0.217235\pi$
$854$	−4249.77	−0.170286
$855$	0	0
$856$	−13866.4	−0.553671
$857$	6516.94	0.259760	0.129880	−	0.991530i	$-0.458541\pi$
$857$	6516.94	0.259760	0.129880	+	0.991530i	$0.458541\pi$
$858$	0	0
$859$	45946.7	1.82501	0.912504	−	0.409068i	$-0.134146\pi$
$859$	45946.7	1.82501	0.912504	+	0.409068i	$0.134146\pi$
$860$	35563.7	1.41013
$861$	0	0
$862$	8837.45	0.349193
$863$	25825.2	1.01866	0.509328	−	0.860573i	$-0.329894\pi$
$863$	25825.2	1.01866	0.509328	+	0.860573i	$0.329894\pi$
$864$	0	0
$865$	33442.1	1.31453
$866$	1034.73	0.0406023
$867$	0	0
$868$	−1361.12	−0.0532250
$869$	15863.2	0.619243
$870$	0	0
$871$	−32417.0	−1.26109
$872$	264.427	0.0102691
$873$	0	0
$874$	541.848	0.0209706
$875$	20145.4	0.778329
$876$	0	0
$877$	36784.9	1.41635	0.708174	−	0.706038i	$-0.249519\pi$
$877$	36784.9	1.41635	0.708174	+	0.706038i	$0.249519\pi$
$878$	−24291.1	−0.933695
$879$	0	0
$880$	12634.7	0.483994
$881$	41594.5	1.59064	0.795321	−	0.606189i	$-0.207302\pi$
$881$	41594.5	1.59064	0.795321	+	0.606189i	$0.207302\pi$
$882$	0	0
$883$	−22923.9	−0.873670	−0.436835	−	0.899542i	$-0.643901\pi$
$883$	−22923.9	−0.873670	−0.436835	+	0.899542i	$0.643901\pi$
$884$	−25782.6	−0.980955
$885$	0	0
$886$	−23629.4	−0.895986
$887$	−2603.19	−0.0985418	−0.0492709	−	0.998785i	$-0.515690\pi$
$887$	−2603.19	−0.0985418	−0.0492709	+	0.998785i	$0.515690\pi$
$888$	0	0
$889$	11200.4	0.422554
$890$	32728.2	1.23264
$891$	0	0
$892$	13812.5	0.518470
$893$	−14716.6	−0.551481
$894$	0	0
$895$	−29648.2	−1.10730
$896$	−896.000	−0.0334077
$897$	0	0
$898$	31938.3	1.18685
$899$	12521.0	0.464516
$900$	0	0
$901$	−23429.3	−0.866308
$902$	17254.9	0.636947
$903$	0	0
$904$	−1943.71	−0.0715119
$905$	58532.9	2.14994
$906$	0	0
$907$	29167.9	1.06781	0.533905	−	0.845544i	$-0.320724\pi$
$907$	29167.9	1.06781	0.533905	+	0.845544i	$0.320724\pi$
$908$	632.506	0.0231172
$909$	0	0
$910$	−24650.4	−0.897971
$911$	−9798.14	−0.356341	−0.178171	−	0.984000i	$-0.557018\pi$
$911$	−9798.14	−0.356341	−0.178171	+	0.984000i	$0.557018\pi$
$912$	0	0
$913$	−25161.2	−0.912065
$914$	−25760.4	−0.932251
$915$	0	0
$916$	−2449.33	−0.0883495
$917$	−17294.0	−0.622791
$918$	0	0
$919$	−49986.6	−1.79424	−0.897120	−	0.441786i	$-0.854345\pi$
$919$	−49986.6	−1.79424	−0.897120	+	0.441786i	$0.854345\pi$
$920$	1133.34	0.0406143
$921$	0	0
$922$	−13529.1	−0.483252
$923$	54530.4	1.94463
$924$	0	0
$925$	−92628.3	−3.29254
$926$	−2242.12	−0.0795687
$927$	0	0
$928$	8242.39	0.291562
$929$	−50651.4	−1.78883	−0.894413	−	0.447242i	$-0.852406\pi$
$929$	−50651.4	−1.78883	−0.894413	+	0.447242i	$0.852406\pi$
$930$	0	0
$931$	1862.00	0.0655474
$932$	−20825.6	−0.731936
$933$	0	0
$934$	−31342.8	−1.09804
$935$	−57441.1	−2.00912
$936$	0	0
$937$	29457.1	1.02702	0.513511	−	0.858083i	$-0.328344\pi$
$937$	29457.1	1.02702	0.513511	+	0.858083i	$0.328344\pi$
$938$	5121.67	0.178282
$939$	0	0
$940$	−30781.6	−1.06807
$941$	12285.0	0.425590	0.212795	−	0.977097i	$-0.431743\pi$
$941$	12285.0	0.425590	0.212795	+	0.977097i	$0.431743\pi$
$942$	0	0
$943$	1547.78	0.0534493
$944$	8625.80	0.297400
$945$	0	0
$946$	−35563.7	−1.22228
$947$	10735.0	0.368363	0.184182	−	0.982892i	$-0.441037\pi$
$947$	10735.0	0.368363	0.184182	+	0.982892i	$0.441037\pi$
$948$	0	0
$949$	10948.4	0.374498
$950$	−20507.4	−0.700365
$951$	0	0
$952$	4073.49	0.138679
$953$	−101.551	−0.00345178	−0.00172589	−	0.999999i	$-0.500549\pi$
$953$	−101.551	−0.00345178	−0.00172589	+	0.999999i	$0.500549\pi$
$954$	0	0
$955$	38020.9	1.28830
$956$	2591.41	0.0876698
$957$	0	0
$958$	−22423.4	−0.756228
$959$	1693.76	0.0570329
$960$	0	0
$961$	−27427.9	−0.920679
$962$	60836.8	2.03894
$963$	0	0
$964$	24566.7	0.820788
$965$	−1178.84	−0.0393245
$966$	0	0
$967$	5751.74	0.191276	0.0956378	−	0.995416i	$-0.469511\pi$
$967$	5751.74	0.191276	0.0956378	+	0.995416i	$0.469511\pi$
$968$	−1986.68	−0.0659653
$969$	0	0
$970$	24140.5	0.799077
$971$	−1143.79	−0.0378024	−0.0189012	−	0.999821i	$-0.506017\pi$
$971$	−1143.79	−0.0378024	−0.0189012	+	0.999821i	$0.506017\pi$
$972$	0	0
$973$	2154.46	0.0709855
$974$	10522.0	0.346146
$975$	0	0
$976$	4856.88	0.159288
$977$	16323.6	0.534532	0.267266	−	0.963623i	$-0.413880\pi$
$977$	16323.6	0.534532	0.267266	+	0.963623i	$0.413880\pi$
$978$	0	0
$979$	−32728.2	−1.06844
$980$	3894.60	0.126947
$981$	0	0
$982$	17508.8	0.568970
$983$	19016.4	0.617017	0.308509	−	0.951222i	$-0.400170\pi$
$983$	19016.4	0.617017	0.308509	+	0.951222i	$0.400170\pi$
$984$	0	0
$985$	−5718.32	−0.184975
$986$	−37472.4	−1.21031
$987$	0	0
$988$	13468.9	0.433708
$989$	−3190.09	−0.102567
$990$	0	0
$991$	−53795.4	−1.72439	−0.862193	−	0.506580i	$-0.830909\pi$
$991$	−53795.4	−1.72439	−0.862193	+	0.506580i	$0.830909\pi$
$992$	1555.56	0.0497874
$993$	0	0
$994$	−8615.44	−0.274915
$995$	−79770.0	−2.54159
$996$	0	0
$997$	29029.7	0.922145	0.461073	−	0.887362i	$-0.347465\pi$
$997$	29029.7	0.922145	0.461073	+	0.887362i	$0.347465\pi$
$998$	37939.5	1.20336
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.a.o.1.2	✓	2
3.2	odd	2		378.4.a.p.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.a.o.1.2	✓	2	1.1	even	1	trivial
378.4.a.p.1.1	yes	2	3.2	odd	2

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} +7.00000 q^{7} -8.00000 q^{8} -39.7409 q^{10} +39.7409 q^{11} +88.6113 q^{13} -14.0000 q^{14} +16.0000 q^{16} -72.7409 q^{17} +38.0000 q^{19} +79.4817 q^{20} -79.4817 q^{22} -7.12957 q^{23} +269.834 q^{25} -177.223 q^{26} +28.0000 q^{28} -257.575 q^{29} -48.6113 q^{31} -32.0000 q^{32} +145.482 q^{34} +139.093 q^{35} -343.279 q^{37} -76.0000 q^{38} -158.963 q^{40} -217.093 q^{41} +447.445 q^{43} +158.963 q^{44} +14.2591 q^{46} -387.279 q^{47} +49.0000 q^{49} -539.668 q^{50} +354.445 q^{52} +322.093 q^{53} +789.668 q^{55} -56.0000 q^{56} +515.149 q^{58} +539.113 q^{59} +303.555 q^{61} +97.2226 q^{62} +64.0000 q^{64} +1760.74 q^{65} -365.834 q^{67} -290.963 q^{68} -278.186 q^{70} +615.389 q^{71} +123.555 q^{73} +686.558 q^{74} +152.000 q^{76} +278.186 q^{77} +399.166 q^{79} +317.927 q^{80} +434.186 q^{82} -633.133 q^{83} -1445.39 q^{85} -894.890 q^{86} -317.927 q^{88} -823.541 q^{89} +620.279 q^{91} -28.5183 q^{92} +774.558 q^{94} +755.076 q^{95} -607.448 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} + 14 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 + 14 * q^7 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + 9 q^{5} + 14 q^{7} - 16 q^{8} - 18 q^{10} + 18 q^{11} + 85 q^{13} - 28 q^{14} + 32 q^{16} - 84 q^{17} + 76 q^{19} + 36 q^{20} - 36 q^{22} - 45 q^{23} + 263 q^{25} - 170 q^{26} + 56 q^{28} - 177 q^{29} - 5 q^{31} - 64 q^{32} + 168 q^{34} + 63 q^{35} - 41 q^{37} - 152 q^{38} - 72 q^{40} - 219 q^{41} + 526 q^{43} + 72 q^{44} + 90 q^{46} - 129 q^{47} + 98 q^{49} - 526 q^{50} + 340 q^{52} + 429 q^{53} + 1026 q^{55} - 112 q^{56} + 354 q^{58} + 156 q^{59} + 976 q^{61} + 10 q^{62} + 128 q^{64} + 1800 q^{65} - 455 q^{67} - 336 q^{68} - 126 q^{70} + 1323 q^{71} + 616 q^{73} + 82 q^{74} + 304 q^{76} + 126 q^{77} + 1075 q^{79} + 144 q^{80} + 438 q^{82} + 363 q^{83} - 1323 q^{85} - 1052 q^{86} - 144 q^{88} + 597 q^{89} + 595 q^{91} - 180 q^{92} + 258 q^{94} + 342 q^{95} + 814 q^{97} - 196 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 9 * q^5 + 14 * q^7 - 16 * q^8 - 18 * q^10 + 18 * q^11 + 85 * q^13 - 28 * q^14 + 32 * q^16 - 84 * q^17 + 76 * q^19 + 36 * q^20 - 36 * q^22 - 45 * q^23 + 263 * q^25 - 170 * q^26 + 56 * q^28 - 177 * q^29 - 5 * q^31 - 64 * q^32 + 168 * q^34 + 63 * q^35 - 41 * q^37 - 152 * q^38 - 72 * q^40 - 219 * q^41 + 526 * q^43 + 72 * q^44 + 90 * q^46 - 129 * q^47 + 98 * q^49 - 526 * q^50 + 340 * q^52 + 429 * q^53 + 1026 * q^55 - 112 * q^56 + 354 * q^58 + 156 * q^59 + 976 * q^61 + 10 * q^62 + 128 * q^64 + 1800 * q^65 - 455 * q^67 - 336 * q^68 - 126 * q^70 + 1323 * q^71 + 616 * q^73 + 82 * q^74 + 304 * q^76 + 126 * q^77 + 1075 * q^79 + 144 * q^80 + 438 * q^82 + 363 * q^83 - 1323 * q^85 - 1052 * q^86 - 144 * q^88 + 597 * q^89 + 595 * q^91 - 180 * q^92 + 258 * q^94 + 342 * q^95 + 814 * q^97 - 196 * q^98

Introduction

L-functions

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