LMFDB - Embedded newform 495.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.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:	$29.2059454528$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 $ x^7 - 2x^6 - 41x^5 + 40x^4 + 424x^3 - 168x^2 - 1042x - 388
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.28302$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28302 q^{2} -2.78784 q^{4} +5.00000 q^{5} +13.7084 q^{7} +24.6288 q^{8} +O(q^{10})$	\(q-2.28302 q^{2} -2.78784 q^{4} +5.00000 q^{5} +13.7084 q^{7} +24.6288 q^{8} -11.4151 q^{10} +11.0000 q^{11} +20.3581 q^{13} -31.2964 q^{14} -33.9252 q^{16} +25.5281 q^{17} +77.9636 q^{19} -13.9392 q^{20} -25.1132 q^{22} -203.500 q^{23} +25.0000 q^{25} -46.4779 q^{26} -38.2167 q^{28} +46.1656 q^{29} +167.295 q^{31} -119.579 q^{32} -58.2810 q^{34} +68.5418 q^{35} -244.645 q^{37} -177.992 q^{38} +123.144 q^{40} -204.756 q^{41} +489.867 q^{43} -30.6663 q^{44} +464.593 q^{46} -324.309 q^{47} -155.081 q^{49} -57.0754 q^{50} -56.7552 q^{52} +402.012 q^{53} +55.0000 q^{55} +337.621 q^{56} -105.397 q^{58} +381.003 q^{59} +819.734 q^{61} -381.936 q^{62} +544.402 q^{64} +101.791 q^{65} +867.608 q^{67} -71.1683 q^{68} -156.482 q^{70} -687.144 q^{71} +719.128 q^{73} +558.528 q^{74} -217.350 q^{76} +150.792 q^{77} -1147.70 q^{79} -169.626 q^{80} +467.461 q^{82} +345.721 q^{83} +127.640 q^{85} -1118.37 q^{86} +270.917 q^{88} -273.431 q^{89} +279.076 q^{91} +567.325 q^{92} +740.402 q^{94} +389.818 q^{95} +583.204 q^{97} +354.052 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8}+O(q^{10}) $ 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8	$ 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8} + 25 q^{10} + 77 q^{11} + 38 q^{13} + 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} + 165 q^{20} + 55 q^{22} + 334 q^{23} + 175 q^{25} - 372 q^{26} + 812 q^{28} - 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} + 150 q^{35} + 466 q^{37} - 494 q^{38} + 225 q^{40} - 258 q^{41} + 308 q^{43} + 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} + 125 q^{50} + 512 q^{52} + 110 q^{53} + 385 q^{55} + 20 q^{56} + 1362 q^{58} - 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} + 190 q^{65} + 2268 q^{67} - 1186 q^{68} + 100 q^{70} - 166 q^{71} + 200 q^{73} - 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1545 q^{80} - 1006 q^{82} + 370 q^{83} + 60 q^{85} + 106 q^{86} + 495 q^{88} - 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} + 1130 q^{95} + 3698 q^{97} + 697 q^{98}+O(q^{100}) $ 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8 + 25 * q^10 + 77 * q^11 + 38 * q^13 + 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 + 165 * q^20 + 55 * q^22 + 334 * q^23 + 175 * q^25 - 372 * q^26 + 812 * q^28 - 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 + 150 * q^35 + 466 * q^37 - 494 * q^38 + 225 * q^40 - 258 * q^41 + 308 * q^43 + 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 + 125 * q^50 + 512 * q^52 + 110 * q^53 + 385 * q^55 + 20 * q^56 + 1362 * q^58 - 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 + 190 * q^65 + 2268 * q^67 - 1186 * q^68 + 100 * q^70 - 166 * q^71 + 200 * q^73 - 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1545 * q^80 - 1006 * q^82 + 370 * q^83 + 60 * q^85 + 106 * q^86 + 495 * q^88 - 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 + 1130 * q^95 + 3698 * q^97 + 697 * 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 $

$2$	−2.28302	−0.807168	−0.403584	−	0.914943i	$-0.632236\pi$
$2$	−2.28302	−0.807168	−0.403584	+	0.914943i	$0.632236\pi$
$3$	0	0
$3$	0	0
$4$	−2.78784	−0.348480
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	13.7084	0.740182	0.370091	−	0.928996i	$-0.379327\pi$
$7$	13.7084	0.740182	0.370091	+	0.928996i	$0.379327\pi$
$8$	24.6288	1.08845
$9$	0	0
$10$	−11.4151	−0.360976
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	20.3581	0.434333	0.217166	−	0.976135i	$-0.430319\pi$
$13$	20.3581	0.434333	0.217166	+	0.976135i	$0.430319\pi$
$14$	−31.2964	−0.597451
$15$	0	0
$16$	−33.9252	−0.530081
$17$	25.5281	0.364204	0.182102	−	0.983280i	$-0.441710\pi$
$17$	25.5281	0.364204	0.182102	+	0.983280i	$0.441710\pi$
$18$	0	0
$19$	77.9636	0.941372	0.470686	−	0.882301i	$-0.344006\pi$
$19$	77.9636	0.941372	0.470686	+	0.882301i	$0.344006\pi$
$20$	−13.9392	−0.155845
$21$	0	0
$22$	−25.1132	−0.243370
$23$	−203.500	−1.84490	−0.922449	−	0.386119i	$-0.873815\pi$
$23$	−203.500	−1.84490	−0.922449	+	0.386119i	$0.873815\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−46.4779	−0.350579
$27$	0	0
$28$	−38.2167	−0.257939
$29$	46.1656	0.295612	0.147806	−	0.989016i	$-0.452779\pi$
$29$	46.1656	0.295612	0.147806	+	0.989016i	$0.452779\pi$
$30$	0	0
$31$	167.295	0.969258	0.484629	−	0.874720i	$-0.338955\pi$
$31$	167.295	0.969258	0.484629	+	0.874720i	$0.338955\pi$
$32$	−119.579	−0.660585
$33$	0	0
$34$	−58.2810	−0.293974
$35$	68.5418	0.331019
$36$	0	0
$37$	−244.645	−1.08701	−0.543505	−	0.839406i	$-0.682903\pi$
$37$	−244.645	−1.08701	−0.543505	+	0.839406i	$0.682903\pi$
$38$	−177.992	−0.759845
$39$	0	0
$40$	123.144	0.486770
$41$	−204.756	−0.779939	−0.389969	−	0.920828i	$-0.627514\pi$
$41$	−204.756	−0.779939	−0.389969	+	0.920828i	$0.627514\pi$
$42$	0	0
$43$	489.867	1.73730	0.868651	−	0.495425i	$-0.164988\pi$
$43$	489.867	1.73730	0.868651	+	0.495425i	$0.164988\pi$
$44$	−30.6663	−0.105071
$45$	0	0
$46$	464.593	1.48914
$47$	−324.309	−1.00650	−0.503248	−	0.864142i	$-0.667862\pi$
$47$	−324.309	−1.00650	−0.503248	+	0.864142i	$0.667862\pi$
$48$	0	0
$49$	−155.081	−0.452131
$50$	−57.0754	−0.161434
$51$	0	0
$52$	−56.7552	−0.151356
$53$	402.012	1.04190	0.520950	−	0.853587i	$-0.325578\pi$
$53$	402.012	1.04190	0.520950	+	0.853587i	$0.325578\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	337.621	0.805651
$57$	0	0
$58$	−105.397	−0.238608
$59$	381.003	0.840719	0.420359	−	0.907358i	$-0.361904\pi$
$59$	381.003	0.840719	0.420359	+	0.907358i	$0.361904\pi$
$60$	0	0
$61$	819.734	1.72059	0.860296	−	0.509795i	$-0.170279\pi$
$61$	819.734	1.72059	0.860296	+	0.509795i	$0.170279\pi$
$62$	−381.936	−0.782354
$63$	0	0
$64$	544.402	1.06328
$65$	101.791	0.194240
$66$	0	0
$67$	867.608	1.58202	0.791009	−	0.611805i	$-0.209556\pi$
$67$	867.608	1.58202	0.791009	+	0.611805i	$0.209556\pi$
$68$	−71.1683	−0.126918
$69$	0	0
$70$	−156.482	−0.267188
$71$	−687.144	−1.14858	−0.574288	−	0.818653i	$-0.694721\pi$
$71$	−687.144	−1.14858	−0.574288	+	0.818653i	$0.694721\pi$
$72$	0	0
$73$	719.128	1.15298	0.576490	−	0.817104i	$-0.304422\pi$
$73$	719.128	1.15298	0.576490	+	0.817104i	$0.304422\pi$
$74$	558.528	0.877400
$75$	0	0
$76$	−217.350	−0.328050
$77$	150.792	0.223173
$78$	0	0
$79$	−1147.70	−1.63451	−0.817254	−	0.576277i	$-0.804505\pi$
$79$	−1147.70	−1.63451	−0.817254	+	0.576277i	$0.804505\pi$
$80$	−169.626	−0.237060
$81$	0	0
$82$	467.461	0.629541
$83$	345.721	0.457203	0.228602	−	0.973520i	$-0.426585\pi$
$83$	345.721	0.457203	0.228602	+	0.973520i	$0.426585\pi$
$84$	0	0
$85$	127.640	0.162877
$86$	−1118.37	−1.40229
$87$	0	0
$88$	270.917	0.328180
$89$	−273.431	−0.325658	−0.162829	−	0.986654i	$-0.552062\pi$
$89$	−273.431	−0.325658	−0.162829	+	0.986654i	$0.552062\pi$
$90$	0	0
$91$	279.076	0.321485
$92$	567.325	0.642910
$93$	0	0
$94$	740.402	0.812411
$95$	389.818	0.420994
$96$	0	0
$97$	583.204	0.610468	0.305234	−	0.952277i	$-0.401265\pi$
$97$	583.204	0.610468	0.305234	+	0.952277i	$0.401265\pi$
$98$	354.052	0.364945
$99$	0	0
$100$	−69.6960	−0.0696960
$101$	1255.37	1.23677	0.618386	−	0.785874i	$-0.287787\pi$
$101$	1255.37	1.23677	0.618386	+	0.785874i	$0.287787\pi$
$102$	0	0
$103$	616.207	0.589482	0.294741	−	0.955577i	$-0.404767\pi$
$103$	616.207	0.589482	0.294741	+	0.955577i	$0.404767\pi$
$104$	501.396	0.472749
$105$	0	0
$106$	−917.801	−0.840987
$107$	727.562	0.657346	0.328673	−	0.944444i	$-0.393399\pi$
$107$	727.562	0.657346	0.328673	+	0.944444i	$0.393399\pi$
$108$	0	0
$109$	−222.737	−0.195728	−0.0978640	−	0.995200i	$-0.531201\pi$
$109$	−222.737	−0.195728	−0.0978640	+	0.995200i	$0.531201\pi$
$110$	−125.566	−0.108838
$111$	0	0
$112$	−465.059	−0.392357
$113$	1415.43	1.17834	0.589170	−	0.808009i	$-0.299455\pi$
$113$	1415.43	1.17834	0.589170	+	0.808009i	$0.299455\pi$
$114$	0	0
$115$	−1017.50	−0.825063
$116$	−128.702	−0.103015
$117$	0	0
$118$	−869.836	−0.678601
$119$	349.948	0.269577
$120$	0	0
$121$	121.000	0.0909091
$122$	−1871.46	−1.38881
$123$	0	0
$124$	−466.391	−0.337767
$125$	125.000	0.0894427
$126$	0	0
$127$	832.759	0.581853	0.290927	−	0.956745i	$-0.406036\pi$
$127$	832.759	0.581853	0.290927	+	0.956745i	$0.406036\pi$
$128$	−286.248	−0.197664
$129$	0	0
$130$	−232.389	−0.156784
$131$	2460.23	1.64085	0.820424	−	0.571755i	$-0.193737\pi$
$131$	2460.23	1.64085	0.820424	+	0.571755i	$0.193737\pi$
$132$	0	0
$133$	1068.75	0.696787
$134$	−1980.76	−1.27695
$135$	0	0
$136$	628.726	0.396418
$137$	382.732	0.238679	0.119339	−	0.992854i	$-0.461922\pi$
$137$	382.732	0.238679	0.119339	+	0.992854i	$0.461922\pi$
$138$	0	0
$139$	525.932	0.320928	0.160464	−	0.987042i	$-0.448701\pi$
$139$	525.932	0.320928	0.160464	+	0.987042i	$0.448701\pi$
$140$	−191.084	−0.115354
$141$	0	0
$142$	1568.76	0.927094
$143$	223.939	0.130956
$144$	0	0
$145$	230.828	0.132202
$146$	−1641.78	−0.930648
$147$	0	0
$148$	682.031	0.378802
$149$	−428.728	−0.235723	−0.117862	−	0.993030i	$-0.537604\pi$
$149$	−428.728	−0.235723	−0.117862	+	0.993030i	$0.537604\pi$
$150$	0	0
$151$	−162.677	−0.0876720	−0.0438360	−	0.999039i	$-0.513958\pi$
$151$	−162.677	−0.0876720	−0.0438360	+	0.999039i	$0.513958\pi$
$152$	1920.15	1.02464
$153$	0	0
$154$	−344.260	−0.180138
$155$	836.473	0.433465
$156$	0	0
$157$	−2170.64	−1.10341	−0.551706	−	0.834039i	$-0.686023\pi$
$157$	−2170.64	−1.10341	−0.551706	+	0.834039i	$0.686023\pi$
$158$	2620.21	1.31932
$159$	0	0
$160$	−597.893	−0.295423
$161$	−2789.65	−1.36556
$162$	0	0
$163$	3044.84	1.46313	0.731565	−	0.681772i	$-0.238790\pi$
$163$	3044.84	1.46313	0.731565	+	0.681772i	$0.238790\pi$
$164$	570.827	0.271793
$165$	0	0
$166$	−789.287	−0.369040
$167$	−2736.19	−1.26786	−0.633931	−	0.773389i	$-0.718560\pi$
$167$	−2736.19	−1.26786	−0.633931	+	0.773389i	$0.718560\pi$
$168$	0	0
$169$	−1782.55	−0.811355
$170$	−291.405	−0.131469
$171$	0	0
$172$	−1365.67	−0.605415
$173$	1755.48	0.771482	0.385741	−	0.922607i	$-0.373946\pi$
$173$	1755.48	0.771482	0.385741	+	0.922607i	$0.373946\pi$
$174$	0	0
$175$	342.709	0.148036
$176$	−373.177	−0.159826
$177$	0	0
$178$	624.246	0.262861
$179$	−273.721	−0.114295	−0.0571477	−	0.998366i	$-0.518201\pi$
$179$	−273.721	−0.114295	−0.0571477	+	0.998366i	$0.518201\pi$
$180$	0	0
$181$	1451.05	0.595888	0.297944	−	0.954583i	$-0.403699\pi$
$181$	1451.05	0.595888	0.297944	+	0.954583i	$0.403699\pi$
$182$	−637.136	−0.259493
$183$	0	0
$184$	−5011.96	−2.00808
$185$	−1223.22	−0.486126
$186$	0	0
$187$	280.809	0.109812
$188$	904.122	0.350744
$189$	0	0
$190$	−889.960	−0.339813
$191$	−1539.26	−0.583124	−0.291562	−	0.956552i	$-0.594175\pi$
$191$	−1539.26	−0.583124	−0.291562	+	0.956552i	$0.594175\pi$
$192$	0	0
$193$	−3445.06	−1.28487	−0.642437	−	0.766338i	$-0.722077\pi$
$193$	−3445.06	−1.28487	−0.642437	+	0.766338i	$0.722077\pi$
$194$	−1331.46	−0.492750
$195$	0	0
$196$	432.341	0.157559
$197$	−4763.53	−1.72278	−0.861389	−	0.507945i	$-0.830405\pi$
$197$	−4763.53	−1.72278	−0.861389	+	0.507945i	$0.830405\pi$
$198$	0	0
$199$	−583.888	−0.207994	−0.103997	−	0.994578i	$-0.533163\pi$
$199$	−583.888	−0.207994	−0.103997	+	0.994578i	$0.533163\pi$
$200$	615.720	0.217690
$201$	0	0
$202$	−2866.03	−0.998283
$203$	632.855	0.218806
$204$	0	0
$205$	−1023.78	−0.348799
$206$	−1406.81	−0.475811
$207$	0	0
$208$	−690.653	−0.230232
$209$	857.599	0.283834
$210$	0	0
$211$	−4673.83	−1.52493	−0.762464	−	0.647031i	$-0.776010\pi$
$211$	−4673.83	−1.52493	−0.762464	+	0.647031i	$0.776010\pi$
$212$	−1120.75	−0.363081
$213$	0	0
$214$	−1661.03	−0.530589
$215$	2449.33	0.776945
$216$	0	0
$217$	2293.34	0.717427
$218$	508.512	0.157985
$219$	0	0
$220$	−153.331	−0.0469891
$221$	519.704	0.158186
$222$	0	0
$223$	3522.86	1.05788	0.528942	−	0.848658i	$-0.322589\pi$
$223$	3522.86	1.05788	0.528942	+	0.848658i	$0.322589\pi$
$224$	−1639.23	−0.488953
$225$	0	0
$226$	−3231.45	−0.951118
$227$	3352.04	0.980099	0.490050	−	0.871695i	$-0.336979\pi$
$227$	3352.04	0.980099	0.490050	+	0.871695i	$0.336979\pi$
$228$	0	0
$229$	−50.1794	−0.0144801	−0.00724006	−	0.999974i	$-0.502305\pi$
$229$	−50.1794	−0.0144801	−0.00724006	+	0.999974i	$0.502305\pi$
$230$	2322.97	0.665964
$231$	0	0
$232$	1137.00	0.321758
$233$	601.386	0.169091	0.0845454	−	0.996420i	$-0.473056\pi$
$233$	601.386	0.169091	0.0845454	+	0.996420i	$0.473056\pi$
$234$	0	0
$235$	−1621.54	−0.450119
$236$	−1062.18	−0.292974
$237$	0	0
$238$	−798.937	−0.217594
$239$	−3739.76	−1.01216	−0.506078	−	0.862488i	$-0.668905\pi$
$239$	−3739.76	−1.01216	−0.506078	+	0.862488i	$0.668905\pi$
$240$	0	0
$241$	6669.24	1.78259	0.891294	−	0.453426i	$-0.149798\pi$
$241$	6669.24	1.78259	0.891294	+	0.453426i	$0.149798\pi$
$242$	−276.245	−0.0733789
$243$	0	0
$244$	−2285.29	−0.599592
$245$	−775.404	−0.202199
$246$	0	0
$247$	1587.19	0.408869
$248$	4120.27	1.05499
$249$	0	0
$250$	−285.377	−0.0721953
$251$	−7659.23	−1.92608	−0.963040	−	0.269359i	$-0.913188\pi$
$251$	−7659.23	−1.92608	−0.963040	+	0.269359i	$0.913188\pi$
$252$	0	0
$253$	−2238.50	−0.556258
$254$	−1901.20	−0.469653
$255$	0	0
$256$	−3701.71	−0.903737
$257$	3546.35	0.860759	0.430380	−	0.902648i	$-0.358380\pi$
$257$	3546.35	0.860759	0.430380	+	0.902648i	$0.358380\pi$
$258$	0	0
$259$	−3353.68	−0.804585
$260$	−283.776	−0.0676886
$261$	0	0
$262$	−5616.74	−1.32444
$263$	2594.98	0.608415	0.304207	−	0.952606i	$-0.401608\pi$
$263$	2594.98	0.608415	0.304207	+	0.952606i	$0.401608\pi$
$264$	0	0
$265$	2010.06	0.465952
$266$	−2439.98	−0.562424
$267$	0	0
$268$	−2418.75	−0.551302
$269$	3194.48	0.724056	0.362028	−	0.932167i	$-0.382084\pi$
$269$	3194.48	0.724056	0.362028	+	0.932167i	$0.382084\pi$
$270$	0	0
$271$	1304.12	0.292323	0.146162	−	0.989261i	$-0.453308\pi$
$271$	1304.12	0.292323	0.146162	+	0.989261i	$0.453308\pi$
$272$	−866.046	−0.193058
$273$	0	0
$274$	−873.783	−0.192654
$275$	275.000	0.0603023
$276$	0	0
$277$	−2790.61	−0.605313	−0.302656	−	0.953100i	$-0.597873\pi$
$277$	−2790.61	−0.605313	−0.302656	+	0.953100i	$0.597873\pi$
$278$	−1200.71	−0.259043
$279$	0	0
$280$	1688.10	0.360298
$281$	4004.66	0.850170	0.425085	−	0.905153i	$-0.360244\pi$
$281$	4004.66	0.850170	0.425085	+	0.905153i	$0.360244\pi$
$282$	0	0
$283$	1975.26	0.414902	0.207451	−	0.978245i	$-0.433483\pi$
$283$	1975.26	0.414902	0.207451	+	0.978245i	$0.433483\pi$
$284$	1915.65	0.400256
$285$	0	0
$286$	−511.257	−0.105704
$287$	−2806.87	−0.577297
$288$	0	0
$289$	−4261.32	−0.867355
$290$	−526.984	−0.106709
$291$	0	0
$292$	−2004.81	−0.401791
$293$	459.970	0.0917124	0.0458562	−	0.998948i	$-0.485398\pi$
$293$	459.970	0.0917124	0.0458562	+	0.998948i	$0.485398\pi$
$294$	0	0
$295$	1905.02	0.375981
$296$	−6025.31	−1.18316
$297$	0	0
$298$	978.793	0.190268
$299$	−4142.87	−0.801300
$300$	0	0
$301$	6715.27	1.28592
$302$	371.394	0.0707660
$303$	0	0
$304$	−2644.93	−0.499004
$305$	4098.67	0.769472
$306$	0	0
$307$	−5442.76	−1.01184	−0.505919	−	0.862581i	$-0.668847\pi$
$307$	−5442.76	−1.01184	−0.505919	+	0.862581i	$0.668847\pi$
$308$	−420.384	−0.0777715
$309$	0	0
$310$	−1909.68	−0.349879
$311$	−7012.79	−1.27865	−0.639323	−	0.768939i	$-0.720785\pi$
$311$	−7012.79	−1.27865	−0.639323	+	0.768939i	$0.720785\pi$
$312$	0	0
$313$	5185.18	0.936369	0.468185	−	0.883631i	$-0.344908\pi$
$313$	5185.18	0.936369	0.468185	+	0.883631i	$0.344908\pi$
$314$	4955.60	0.890639
$315$	0	0
$316$	3199.60	0.569594
$317$	−1238.29	−0.219399	−0.109700	−	0.993965i	$-0.534989\pi$
$317$	−1238.29	−0.219399	−0.109700	+	0.993965i	$0.534989\pi$
$318$	0	0
$319$	507.822	0.0891302
$320$	2722.01	0.475515
$321$	0	0
$322$	6368.81	1.10224
$323$	1990.26	0.342852
$324$	0	0
$325$	508.953	0.0868666
$326$	−6951.41	−1.18099
$327$	0	0
$328$	−5042.89	−0.848924
$329$	−4445.74	−0.744990
$330$	0	0
$331$	10600.3	1.76026	0.880128	−	0.474737i	$-0.157457\pi$
$331$	10600.3	1.76026	0.880128	+	0.474737i	$0.157457\pi$
$332$	−963.816	−0.159326
$333$	0	0
$334$	6246.77	1.02338
$335$	4338.04	0.707500
$336$	0	0
$337$	−2515.05	−0.406538	−0.203269	−	0.979123i	$-0.565157\pi$
$337$	−2515.05	−0.406538	−0.203269	+	0.979123i	$0.565157\pi$
$338$	4069.58	0.654900
$339$	0	0
$340$	−355.841	−0.0567594
$341$	1840.24	0.292242
$342$	0	0
$343$	−6827.87	−1.07484
$344$	12064.8	1.89096
$345$	0	0
$346$	−4007.78	−0.622715
$347$	2637.98	0.408110	0.204055	−	0.978959i	$-0.434588\pi$
$347$	2637.98	0.408110	0.204055	+	0.978959i	$0.434588\pi$
$348$	0	0
$349$	70.3163	0.0107849	0.00539247	−	0.999985i	$-0.498284\pi$
$349$	70.3163	0.0107849	0.00539247	+	0.999985i	$0.498284\pi$
$350$	−782.410	−0.119490
$351$	0	0
$352$	−1315.37	−0.199174
$353$	6628.94	0.999499	0.499749	−	0.866170i	$-0.333425\pi$
$353$	6628.94	0.999499	0.499749	+	0.866170i	$0.333425\pi$
$354$	0	0
$355$	−3435.72	−0.513659
$356$	762.281	0.113485
$357$	0	0
$358$	624.909	0.0922556
$359$	−1294.62	−0.190328	−0.0951638	−	0.995462i	$-0.530337\pi$
$359$	−1294.62	−0.190328	−0.0951638	+	0.995462i	$0.530337\pi$
$360$	0	0
$361$	−780.682	−0.113819
$362$	−3312.77	−0.480982
$363$	0	0
$364$	−778.021	−0.112031
$365$	3595.64	0.515628
$366$	0	0
$367$	3891.15	0.553451	0.276725	−	0.960949i	$-0.410751\pi$
$367$	3891.15	0.553451	0.276725	+	0.960949i	$0.410751\pi$
$368$	6903.77	0.977946
$369$	0	0
$370$	2792.64	0.392385
$371$	5510.93	0.771195
$372$	0	0
$373$	9682.18	1.34403	0.672016	−	0.740536i	$-0.265429\pi$
$373$	9682.18	1.34403	0.672016	+	0.740536i	$0.265429\pi$
$374$	−641.091	−0.0886365
$375$	0	0
$376$	−7987.34	−1.09552
$377$	939.845	0.128394
$378$	0	0
$379$	10277.1	1.39288	0.696438	−	0.717617i	$-0.254767\pi$
$379$	10277.1	1.39288	0.696438	+	0.717617i	$0.254767\pi$
$380$	−1086.75	−0.146708
$381$	0	0
$382$	3514.15	0.470679
$383$	6308.68	0.841667	0.420833	−	0.907138i	$-0.361738\pi$
$383$	6308.68	0.841667	0.420833	+	0.907138i	$0.361738\pi$
$384$	0	0
$385$	753.960	0.0998061
$386$	7865.12	1.03711
$387$	0	0
$388$	−1625.88	−0.212736
$389$	6137.49	0.799956	0.399978	−	0.916525i	$-0.369018\pi$
$389$	6137.49	0.799956	0.399978	+	0.916525i	$0.369018\pi$
$390$	0	0
$391$	−5194.96	−0.671919
$392$	−3819.46	−0.492122
$393$	0	0
$394$	10875.2	1.39057
$395$	−5738.49	−0.730974
$396$	0	0
$397$	13295.5	1.68081	0.840406	−	0.541958i	$-0.182317\pi$
$397$	13295.5	1.68081	0.840406	+	0.541958i	$0.182317\pi$
$398$	1333.03	0.167886
$399$	0	0
$400$	−848.130	−0.106016
$401$	−1513.58	−0.188491	−0.0942453	−	0.995549i	$-0.530044\pi$
$401$	−1513.58	−0.188491	−0.0942453	+	0.995549i	$0.530044\pi$
$402$	0	0
$403$	3405.80	0.420981
$404$	−3499.77	−0.430991
$405$	0	0
$406$	−1444.82	−0.176613
$407$	−2691.09	−0.327746
$408$	0	0
$409$	2200.38	0.266020	0.133010	−	0.991115i	$-0.457536\pi$
$409$	2200.38	0.266020	0.133010	+	0.991115i	$0.457536\pi$
$410$	2337.30	0.281540
$411$	0	0
$412$	−1717.89	−0.205423
$413$	5222.93	0.622285
$414$	0	0
$415$	1728.61	0.204467
$416$	−2434.40	−0.286914
$417$	0	0
$418$	−1957.91	−0.229102
$419$	−228.547	−0.0266474	−0.0133237	−	0.999911i	$-0.504241\pi$
$419$	−228.547	−0.0266474	−0.0133237	+	0.999911i	$0.504241\pi$
$420$	0	0
$421$	843.076	0.0975987	0.0487993	−	0.998809i	$-0.484461\pi$
$421$	843.076	0.0975987	0.0487993	+	0.998809i	$0.484461\pi$
$422$	10670.4	1.23087
$423$	0	0
$424$	9901.09	1.13405
$425$	638.202	0.0728408
$426$	0	0
$427$	11237.2	1.27355
$428$	−2028.33	−0.229072
$429$	0	0
$430$	−5591.86	−0.627125
$431$	−13288.6	−1.48512	−0.742562	−	0.669777i	$-0.766390\pi$
$431$	−13288.6	−1.48512	−0.742562	+	0.669777i	$0.766390\pi$
$432$	0	0
$433$	3278.01	0.363813	0.181907	−	0.983316i	$-0.441773\pi$
$433$	3278.01	0.363813	0.181907	+	0.983316i	$0.441773\pi$
$434$	−5235.72	−0.579084
$435$	0	0
$436$	620.956	0.0682073
$437$	−15865.6	−1.73674
$438$	0	0
$439$	4817.02	0.523698	0.261849	−	0.965109i	$-0.415668\pi$
$439$	4817.02	0.523698	0.261849	+	0.965109i	$0.415668\pi$
$440$	1354.58	0.146767
$441$	0	0
$442$	−1186.49	−0.127682
$443$	10168.9	1.09060	0.545302	−	0.838240i	$-0.316415\pi$
$443$	10168.9	1.09060	0.545302	+	0.838240i	$0.316415\pi$
$444$	0	0
$445$	−1367.15	−0.145639
$446$	−8042.74	−0.853890
$447$	0	0
$448$	7462.85	0.787024
$449$	−15651.3	−1.64505	−0.822527	−	0.568727i	$-0.807436\pi$
$449$	−15651.3	−1.64505	−0.822527	+	0.568727i	$0.807436\pi$
$450$	0	0
$451$	−2252.31	−0.235160
$452$	−3946.00	−0.410628
$453$	0	0
$454$	−7652.75	−0.791105
$455$	1395.38	0.143773
$456$	0	0
$457$	−3898.42	−0.399038	−0.199519	−	0.979894i	$-0.563938\pi$
$457$	−3898.42	−0.399038	−0.199519	+	0.979894i	$0.563938\pi$
$458$	114.560	0.0116879
$459$	0	0
$460$	2836.63	0.287518
$461$	−8058.83	−0.814180	−0.407090	−	0.913388i	$-0.633456\pi$
$461$	−8058.83	−0.814180	−0.407090	+	0.913388i	$0.633456\pi$
$462$	0	0
$463$	−4841.72	−0.485991	−0.242996	−	0.970027i	$-0.578130\pi$
$463$	−4841.72	−0.485991	−0.242996	+	0.970027i	$0.578130\pi$
$464$	−1566.18	−0.156698
$465$	0	0
$466$	−1372.97	−0.136485
$467$	1776.39	0.176020	0.0880102	−	0.996120i	$-0.471949\pi$
$467$	1776.39	0.176020	0.0880102	+	0.996120i	$0.471949\pi$
$468$	0	0
$469$	11893.5	1.17098
$470$	3702.01	0.363321
$471$	0	0
$472$	9383.66	0.915080
$473$	5388.53	0.523816
$474$	0	0
$475$	1949.09	0.188274
$476$	−975.600	−0.0939424
$477$	0	0
$478$	8537.94	0.816980
$479$	11919.1	1.13695	0.568475	−	0.822700i	$-0.307534\pi$
$479$	11919.1	1.13695	0.568475	+	0.822700i	$0.307534\pi$
$480$	0	0
$481$	−4980.51	−0.472124
$482$	−15226.0	−1.43885
$483$	0	0
$484$	−337.329	−0.0316800
$485$	2916.02	0.273009
$486$	0	0
$487$	−11711.4	−1.08972	−0.544862	−	0.838526i	$-0.683418\pi$
$487$	−11711.4	−1.08972	−0.544862	+	0.838526i	$0.683418\pi$
$488$	20189.1	1.87278
$489$	0	0
$490$	1770.26	0.163209
$491$	−15200.4	−1.39712	−0.698558	−	0.715554i	$-0.746174\pi$
$491$	−15200.4	−1.39712	−0.698558	+	0.715554i	$0.746174\pi$
$492$	0	0
$493$	1178.52	0.107663
$494$	−3623.58	−0.330026
$495$	0	0
$496$	−5675.51	−0.513786
$497$	−9419.61	−0.850156
$498$	0	0
$499$	−14456.5	−1.29692	−0.648460	−	0.761249i	$-0.724587\pi$
$499$	−14456.5	−1.29692	−0.648460	+	0.761249i	$0.724587\pi$
$500$	−348.480	−0.0311690
$501$	0	0
$502$	17486.1	1.55467
$503$	−7283.74	−0.645658	−0.322829	−	0.946457i	$-0.604634\pi$
$503$	−7283.74	−0.645658	−0.322829	+	0.946457i	$0.604634\pi$
$504$	0	0
$505$	6276.85	0.553102
$506$	5110.52	0.448993
$507$	0	0
$508$	−2321.60	−0.202764
$509$	−14538.6	−1.26603	−0.633016	−	0.774139i	$-0.718183\pi$
$509$	−14538.6	−1.26603	−0.633016	+	0.774139i	$0.718183\pi$
$510$	0	0
$511$	9858.06	0.853415
$512$	10741.0	0.927131
$513$	0	0
$514$	−8096.37	−0.694777
$515$	3081.04	0.263625
$516$	0	0
$517$	−3567.40	−0.303470
$518$	7656.50	0.649435
$519$	0	0
$520$	2506.98	0.211420
$521$	−17598.3	−1.47984	−0.739918	−	0.672697i	$-0.765136\pi$
$521$	−17598.3	−1.47984	−0.739918	+	0.672697i	$0.765136\pi$
$522$	0	0
$523$	−17967.1	−1.50219	−0.751095	−	0.660194i	$-0.770474\pi$
$523$	−17967.1	−1.50219	−0.751095	+	0.660194i	$0.770474\pi$
$524$	−6858.73	−0.571803
$525$	0	0
$526$	−5924.37	−0.491093
$527$	4270.71	0.353008
$528$	0	0
$529$	29245.2	2.40365
$530$	−4589.00	−0.376101
$531$	0	0
$532$	−2979.51	−0.242816
$533$	−4168.44	−0.338753
$534$	0	0
$535$	3637.81	0.293974
$536$	21368.1	1.72195
$537$	0	0
$538$	−7293.05	−0.584435
$539$	−1705.89	−0.136323
$540$	0	0
$541$	18207.7	1.44697	0.723486	−	0.690339i	$-0.242539\pi$
$541$	18207.7	1.44697	0.723486	+	0.690339i	$0.242539\pi$
$542$	−2977.32	−0.235954
$543$	0	0
$544$	−3052.62	−0.240588
$545$	−1113.69	−0.0875322
$546$	0	0
$547$	−12019.9	−0.939551	−0.469776	−	0.882786i	$-0.655665\pi$
$547$	−12019.9	−0.939551	−0.469776	+	0.882786i	$0.655665\pi$
$548$	−1067.00	−0.0831749
$549$	0	0
$550$	−627.829	−0.0486740
$551$	3599.23	0.278280
$552$	0	0
$553$	−15733.1	−1.20983
$554$	6371.01	0.488589
$555$	0	0
$556$	−1466.22	−0.111837
$557$	−1145.33	−0.0871258	−0.0435629	−	0.999051i	$-0.513871\pi$
$557$	−1145.33	−0.0871258	−0.0435629	+	0.999051i	$0.513871\pi$
$558$	0	0
$559$	9972.76	0.754567
$560$	−2325.29	−0.175467
$561$	0	0
$562$	−9142.69	−0.686230
$563$	−20321.6	−1.52123	−0.760614	−	0.649204i	$-0.775102\pi$
$563$	−20321.6	−1.52123	−0.760614	+	0.649204i	$0.775102\pi$
$564$	0	0
$565$	7077.15	0.526970
$566$	−4509.56	−0.334895
$567$	0	0
$568$	−16923.5	−1.25017
$569$	−9639.75	−0.710227	−0.355114	−	0.934823i	$-0.615558\pi$
$569$	−9639.75	−0.710227	−0.355114	+	0.934823i	$0.615558\pi$
$570$	0	0
$571$	−8819.44	−0.646378	−0.323189	−	0.946334i	$-0.604755\pi$
$571$	−8819.44	−0.646378	−0.323189	+	0.946334i	$0.604755\pi$
$572$	−624.307	−0.0456357
$573$	0	0
$574$	6408.12	0.465975
$575$	−5087.50	−0.368980
$576$	0	0
$577$	5152.39	0.371745	0.185873	−	0.982574i	$-0.440489\pi$
$577$	5152.39	0.371745	0.185873	+	0.982574i	$0.440489\pi$
$578$	9728.65	0.700101
$579$	0	0
$580$	−643.512	−0.0460696
$581$	4739.27	0.338413
$582$	0	0
$583$	4422.14	0.314144
$584$	17711.3	1.25496
$585$	0	0
$586$	−1050.12	−0.0740273
$587$	24056.3	1.69150	0.845750	−	0.533579i	$-0.179153\pi$
$587$	24056.3	1.69150	0.845750	+	0.533579i	$0.179153\pi$
$588$	0	0
$589$	13042.9	0.912433
$590$	−4349.18	−0.303480
$591$	0	0
$592$	8299.63	0.576204
$593$	19436.8	1.34599	0.672996	−	0.739646i	$-0.265007\pi$
$593$	19436.8	1.34599	0.672996	+	0.739646i	$0.265007\pi$
$594$	0	0
$595$	1749.74	0.120559
$596$	1195.23	0.0821449
$597$	0	0
$598$	9458.24	0.646783
$599$	8842.06	0.603133	0.301567	−	0.953445i	$-0.402490\pi$
$599$	8842.06	0.603133	0.301567	+	0.953445i	$0.402490\pi$
$600$	0	0
$601$	3193.52	0.216749	0.108375	−	0.994110i	$-0.465435\pi$
$601$	3193.52	0.216749	0.108375	+	0.994110i	$0.465435\pi$
$602$	−15331.1	−1.03795
$603$	0	0
$604$	453.518	0.0305520
$605$	605.000	0.0406558
$606$	0	0
$607$	−25592.2	−1.71130	−0.855648	−	0.517558i	$-0.826841\pi$
$607$	−25592.2	−1.71130	−0.855648	+	0.517558i	$0.826841\pi$
$608$	−9322.78	−0.621856
$609$	0	0
$610$	−9357.32	−0.621093
$611$	−6602.32	−0.437154
$612$	0	0
$613$	−812.439	−0.0535303	−0.0267652	−	0.999642i	$-0.508521\pi$
$613$	−812.439	−0.0535303	−0.0267652	+	0.999642i	$0.508521\pi$
$614$	12425.9	0.816724
$615$	0	0
$616$	3713.83	0.242913
$617$	10378.5	0.677183	0.338592	−	0.940933i	$-0.390050\pi$
$617$	10378.5	0.677183	0.338592	+	0.940933i	$0.390050\pi$
$618$	0	0
$619$	−27141.1	−1.76234	−0.881172	−	0.472795i	$-0.843245\pi$
$619$	−27141.1	−1.76234	−0.881172	+	0.472795i	$0.843245\pi$
$620$	−2331.96	−0.151054
$621$	0	0
$622$	16010.3	1.03208
$623$	−3748.28	−0.241046
$624$	0	0
$625$	625.000	0.0400000
$626$	−11837.8	−0.755807
$627$	0	0
$628$	6051.39	0.384517
$629$	−6245.32	−0.395894
$630$	0	0
$631$	−12566.9	−0.792840	−0.396420	−	0.918069i	$-0.629748\pi$
$631$	−12566.9	−0.792840	−0.396420	+	0.918069i	$0.629748\pi$
$632$	−28266.4	−1.77908
$633$	0	0
$634$	2827.04	0.177092
$635$	4163.79	0.260213
$636$	0	0
$637$	−3157.15	−0.196375
$638$	−1159.36	−0.0719431
$639$	0	0
$640$	−1431.24	−0.0883979
$641$	−15387.1	−0.948136	−0.474068	−	0.880488i	$-0.657215\pi$
$641$	−15387.1	−0.948136	−0.474068	+	0.880488i	$0.657215\pi$
$642$	0	0
$643$	−11582.3	−0.710358	−0.355179	−	0.934798i	$-0.615580\pi$
$643$	−11582.3	−0.710358	−0.355179	+	0.934798i	$0.615580\pi$
$644$	7777.10	0.475871
$645$	0	0
$646$	−4543.80	−0.276739
$647$	−3656.07	−0.222156	−0.111078	−	0.993812i	$-0.535430\pi$
$647$	−3656.07	−0.222156	−0.111078	+	0.993812i	$0.535430\pi$
$648$	0	0
$649$	4191.04	0.253486
$650$	−1161.95	−0.0701159
$651$	0	0
$652$	−8488.53	−0.509872
$653$	30701.2	1.83987	0.919933	−	0.392076i	$-0.128243\pi$
$653$	30701.2	1.83987	0.919933	+	0.392076i	$0.128243\pi$
$654$	0	0
$655$	12301.1	0.733810
$656$	6946.38	0.413431
$657$	0	0
$658$	10149.7	0.601332
$659$	−26615.9	−1.57331	−0.786653	−	0.617395i	$-0.788188\pi$
$659$	−26615.9	−1.57331	−0.786653	+	0.617395i	$0.788188\pi$
$660$	0	0
$661$	−828.195	−0.0487338	−0.0243669	−	0.999703i	$-0.507757\pi$
$661$	−828.195	−0.0487338	−0.0243669	+	0.999703i	$0.507757\pi$
$662$	−24200.6	−1.42082
$663$	0	0
$664$	8514.70	0.497642
$665$	5343.76	0.311612
$666$	0	0
$667$	−9394.69	−0.545373
$668$	7628.08	0.441825
$669$	0	0
$670$	−9903.81	−0.571071
$671$	9017.07	0.518778
$672$	0	0
$673$	−8920.70	−0.510948	−0.255474	−	0.966816i	$-0.582231\pi$
$673$	−8920.70	−0.510948	−0.255474	+	0.966816i	$0.582231\pi$
$674$	5741.89	0.328144
$675$	0	0
$676$	4969.46	0.282741
$677$	12683.1	0.720014	0.360007	−	0.932950i	$-0.382774\pi$
$677$	12683.1	0.720014	0.360007	+	0.932950i	$0.382774\pi$
$678$	0	0
$679$	7994.77	0.451857
$680$	3143.63	0.177283
$681$	0	0
$682$	−4201.30	−0.235889
$683$	−13061.6	−0.731756	−0.365878	−	0.930663i	$-0.619231\pi$
$683$	−13061.6	−0.731756	−0.365878	+	0.930663i	$0.619231\pi$
$684$	0	0
$685$	1913.66	0.106740
$686$	15588.1	0.867577
$687$	0	0
$688$	−16618.8	−0.920911
$689$	8184.22	0.452531
$690$	0	0
$691$	−9533.73	−0.524863	−0.262432	−	0.964951i	$-0.584524\pi$
$691$	−9533.73	−0.524863	−0.262432	+	0.964951i	$0.584524\pi$
$692$	−4893.99	−0.268846
$693$	0	0
$694$	−6022.55	−0.329413
$695$	2629.66	0.143523
$696$	0	0
$697$	−5227.03	−0.284057
$698$	−160.533	−0.00870525
$699$	0	0
$700$	−955.418	−0.0515877
$701$	23152.1	1.24742	0.623711	−	0.781655i	$-0.285624\pi$
$701$	23152.1	1.24742	0.623711	+	0.781655i	$0.285624\pi$
$702$	0	0
$703$	−19073.4	−1.02328
$704$	5988.42	0.320592
$705$	0	0
$706$	−15134.0	−0.806763
$707$	17209.1	0.915437
$708$	0	0
$709$	−23980.8	−1.27026	−0.635132	−	0.772404i	$-0.719054\pi$
$709$	−23980.8	−1.27026	−0.635132	+	0.772404i	$0.719054\pi$
$710$	7843.80	0.414609
$711$	0	0
$712$	−6734.27	−0.354463
$713$	−34044.4	−1.78818
$714$	0	0
$715$	1119.70	0.0585654
$716$	763.091	0.0398297
$717$	0	0
$718$	2955.64	0.153626
$719$	36893.7	1.91363	0.956816	−	0.290693i	$-0.0938858\pi$
$719$	36893.7	1.91363	0.956816	+	0.290693i	$0.0938858\pi$
$720$	0	0
$721$	8447.19	0.436324
$722$	1782.31	0.0918708
$723$	0	0
$724$	−4045.30	−0.207655
$725$	1154.14	0.0591223
$726$	0	0
$727$	−8831.11	−0.450520	−0.225260	−	0.974299i	$-0.572323\pi$
$727$	−8831.11	−0.450520	−0.225260	+	0.974299i	$0.572323\pi$
$728$	6873.32	0.349921
$729$	0	0
$730$	−8208.90	−0.416199
$731$	12505.4	0.632732
$732$	0	0
$733$	−12954.4	−0.652770	−0.326385	−	0.945237i	$-0.605831\pi$
$733$	−12954.4	−0.652770	−0.326385	+	0.945237i	$0.605831\pi$
$734$	−8883.56	−0.446728
$735$	0	0
$736$	24334.2	1.21871
$737$	9543.69	0.476996
$738$	0	0
$739$	−13003.4	−0.647276	−0.323638	−	0.946181i	$-0.604906\pi$
$739$	−13003.4	−0.647276	−0.323638	+	0.946181i	$0.604906\pi$
$740$	3410.16	0.169405
$741$	0	0
$742$	−12581.5	−0.622484
$743$	−98.2053	−0.00484899	−0.00242450	−	0.999997i	$-0.500772\pi$
$743$	−98.2053	−0.00484899	−0.00242450	+	0.999997i	$0.500772\pi$
$744$	0	0
$745$	−2143.64	−0.105419
$746$	−22104.6	−1.08486
$747$	0	0
$748$	−782.851	−0.0382672
$749$	9973.68	0.486556
$750$	0	0
$751$	−8247.02	−0.400716	−0.200358	−	0.979723i	$-0.564211\pi$
$751$	−8247.02	−0.400716	−0.200358	+	0.979723i	$0.564211\pi$
$752$	11002.2	0.533525
$753$	0	0
$754$	−2145.68	−0.103635
$755$	−813.385	−0.0392081
$756$	0	0
$757$	−33058.3	−1.58722	−0.793610	−	0.608426i	$-0.791801\pi$
$757$	−33058.3	−1.58722	−0.793610	+	0.608426i	$0.791801\pi$
$758$	−23462.8	−1.12428
$759$	0	0
$760$	9600.75	0.458231
$761$	−18789.4	−0.895029	−0.447514	−	0.894277i	$-0.647691\pi$
$761$	−18789.4	−0.895029	−0.447514	+	0.894277i	$0.647691\pi$
$762$	0	0
$763$	−3053.36	−0.144874
$764$	4291.20	0.203207
$765$	0	0
$766$	−14402.8	−0.679366
$767$	7756.51	0.365152
$768$	0	0
$769$	6813.72	0.319518	0.159759	−	0.987156i	$-0.448928\pi$
$769$	6813.72	0.319518	0.159759	+	0.987156i	$0.448928\pi$
$770$	−1721.30	−0.0805603
$771$	0	0
$772$	9604.28	0.447753
$773$	−6837.32	−0.318139	−0.159069	−	0.987267i	$-0.550849\pi$
$773$	−6837.32	−0.318139	−0.159069	+	0.987267i	$0.550849\pi$
$774$	0	0
$775$	4182.37	0.193852
$776$	14363.6	0.664463
$777$	0	0
$778$	−14012.0	−0.645699
$779$	−15963.5	−0.734213
$780$	0	0
$781$	−7558.58	−0.346309
$782$	11860.2	0.542352
$783$	0	0
$784$	5261.15	0.239666
$785$	−10853.2	−0.493461
$786$	0	0
$787$	−11939.6	−0.540787	−0.270394	−	0.962750i	$-0.587154\pi$
$787$	−11939.6	−0.540787	−0.270394	+	0.962750i	$0.587154\pi$
$788$	13280.0	0.600354
$789$	0	0
$790$	13101.1	0.590019
$791$	19403.2	0.872186
$792$	0	0
$793$	16688.2	0.747310
$794$	−30353.8	−1.35670
$795$	0	0
$796$	1627.79	0.0724817
$797$	27176.0	1.20781	0.603904	−	0.797057i	$-0.293611\pi$
$797$	27176.0	1.20781	0.603904	+	0.797057i	$0.293611\pi$
$798$	0	0
$799$	−8278.99	−0.366570
$800$	−2989.47	−0.132117
$801$	0	0
$802$	3455.53	0.152144
$803$	7910.41	0.347637
$804$	0	0
$805$	−13948.2	−0.610697
$806$	−7775.50	−0.339802
$807$	0	0
$808$	30918.3	1.34617
$809$	8062.09	0.350369	0.175184	−	0.984536i	$-0.443948\pi$
$809$	8062.09	0.350369	0.175184	+	0.984536i	$0.443948\pi$
$810$	0	0
$811$	6087.93	0.263596	0.131798	−	0.991277i	$-0.457925\pi$
$811$	6087.93	0.263596	0.131798	+	0.991277i	$0.457925\pi$
$812$	−1764.30	−0.0762497
$813$	0	0
$814$	6143.81	0.264546
$815$	15224.2	0.654331
$816$	0	0
$817$	38191.7	1.63545
$818$	−5023.51	−0.214722
$819$	0	0
$820$	2854.13	0.121550
$821$	−19313.2	−0.820994	−0.410497	−	0.911862i	$-0.634645\pi$
$821$	−19313.2	−0.820994	−0.410497	+	0.911862i	$0.634645\pi$
$822$	0	0
$823$	−29674.5	−1.25685	−0.628425	−	0.777870i	$-0.716300\pi$
$823$	−29674.5	−1.25685	−0.628425	+	0.777870i	$0.716300\pi$
$824$	15176.4	0.641622
$825$	0	0
$826$	−11924.0	−0.502288
$827$	−4272.71	−0.179658	−0.0898288	−	0.995957i	$-0.528632\pi$
$827$	−4272.71	−0.179658	−0.0898288	+	0.995957i	$0.528632\pi$
$828$	0	0
$829$	−32533.5	−1.36301	−0.681506	−	0.731813i	$-0.738674\pi$
$829$	−32533.5	−1.36301	−0.681506	+	0.731813i	$0.738674\pi$
$830$	−3946.44	−0.165039
$831$	0	0
$832$	11083.0	0.461819
$833$	−3958.92	−0.164668
$834$	0	0
$835$	−13681.0	−0.567005
$836$	−2390.85	−0.0989107
$837$	0	0
$838$	521.776	0.0215089
$839$	7348.64	0.302388	0.151194	−	0.988504i	$-0.451688\pi$
$839$	7348.64	0.302388	0.151194	+	0.988504i	$0.451688\pi$
$840$	0	0
$841$	−22257.7	−0.912614
$842$	−1924.76	−0.0787785
$843$	0	0
$844$	13029.9	0.531407
$845$	−8912.74	−0.362849
$846$	0	0
$847$	1658.71	0.0672893
$848$	−13638.4	−0.552291
$849$	0	0
$850$	−1457.03	−0.0587948
$851$	49785.2	2.00542
$852$	0	0
$853$	−1748.08	−0.0701676	−0.0350838	−	0.999384i	$-0.511170\pi$
$853$	−1748.08	−0.0701676	−0.0350838	+	0.999384i	$0.511170\pi$
$854$	−25654.7	−1.02797
$855$	0	0
$856$	17919.0	0.715488
$857$	−258.968	−0.0103223	−0.00516113	−	0.999987i	$-0.501643\pi$
$857$	−258.968	−0.0103223	−0.00516113	+	0.999987i	$0.501643\pi$
$858$	0	0
$859$	−40297.0	−1.60060	−0.800301	−	0.599599i	$-0.795327\pi$
$859$	−40297.0	−1.60060	−0.800301	+	0.599599i	$0.795327\pi$
$860$	−6828.35	−0.270750
$861$	0	0
$862$	30338.0	1.19874
$863$	−32833.3	−1.29508	−0.647542	−	0.762030i	$-0.724203\pi$
$863$	−32833.3	−1.29508	−0.647542	+	0.762030i	$0.724203\pi$
$864$	0	0
$865$	8777.38	0.345017
$866$	−7483.74	−0.293658
$867$	0	0
$868$	−6393.46	−0.250009
$869$	−12624.7	−0.492823
$870$	0	0
$871$	17662.9	0.687122
$872$	−5485.75	−0.213040
$873$	0	0
$874$	36221.3	1.40184
$875$	1713.54	0.0662039
$876$	0	0
$877$	19847.7	0.764205	0.382102	−	0.924120i	$-0.375200\pi$
$877$	19847.7	0.764205	0.382102	+	0.924120i	$0.375200\pi$
$878$	−10997.3	−0.422712
$879$	0	0
$880$	−1865.89	−0.0714762
$881$	11363.8	0.434569	0.217285	−	0.976108i	$-0.430280\pi$
$881$	11363.8	0.434569	0.217285	+	0.976108i	$0.430280\pi$
$882$	0	0
$883$	38240.6	1.45742	0.728708	−	0.684824i	$-0.240121\pi$
$883$	38240.6	1.45742	0.728708	+	0.684824i	$0.240121\pi$
$884$	−1448.85	−0.0551246
$885$	0	0
$886$	−23215.7	−0.880301
$887$	−17616.9	−0.666875	−0.333438	−	0.942772i	$-0.608209\pi$
$887$	−17616.9	−0.666875	−0.333438	+	0.942772i	$0.608209\pi$
$888$	0	0
$889$	11415.8	0.430677
$890$	3121.23	0.117555
$891$	0	0
$892$	−9821.18	−0.368652
$893$	−25284.3	−0.947487
$894$	0	0
$895$	−1368.61	−0.0511145
$896$	−3923.98	−0.146307
$897$	0	0
$898$	35732.1	1.32783
$899$	7723.26	0.286524
$900$	0	0
$901$	10262.6	0.379464
$902$	5142.07	0.189814
$903$	0	0
$904$	34860.4	1.28256
$905$	7255.25	0.266489
$906$	0	0
$907$	−35946.4	−1.31597	−0.657983	−	0.753033i	$-0.728590\pi$
$907$	−35946.4	−1.31597	−0.657983	+	0.753033i	$0.728590\pi$
$908$	−9344.95	−0.341545
$909$	0	0
$910$	−3185.68	−0.116049
$911$	−26741.0	−0.972523	−0.486261	−	0.873813i	$-0.661640\pi$
$911$	−26741.0	−0.972523	−0.486261	+	0.873813i	$0.661640\pi$
$912$	0	0
$913$	3802.93	0.137852
$914$	8900.15	0.322091
$915$	0	0
$916$	139.892	0.00504603
$917$	33725.7	1.21453
$918$	0	0
$919$	−12650.0	−0.454064	−0.227032	−	0.973887i	$-0.572902\pi$
$919$	−12650.0	−0.454064	−0.227032	+	0.973887i	$0.572902\pi$
$920$	−25059.8	−0.898040
$921$	0	0
$922$	18398.4	0.657180
$923$	−13989.0	−0.498864
$924$	0	0
$925$	−6116.12	−0.217402
$926$	11053.7	0.392276
$927$	0	0
$928$	−5520.42	−0.195277
$929$	27323.5	0.964968	0.482484	−	0.875905i	$-0.339735\pi$
$929$	27323.5	0.964968	0.482484	+	0.875905i	$0.339735\pi$
$930$	0	0
$931$	−12090.7	−0.425623
$932$	−1676.57	−0.0589248
$933$	0	0
$934$	−4055.53	−0.142078
$935$	1404.04	0.0491093
$936$	0	0
$937$	26089.8	0.909623	0.454812	−	0.890588i	$-0.349707\pi$
$937$	26089.8	0.909623	0.454812	+	0.890588i	$0.349707\pi$
$938$	−27153.0	−0.945178
$939$	0	0
$940$	4520.61	0.156857
$941$	54789.7	1.89808	0.949041	−	0.315154i	$-0.102056\pi$
$941$	54789.7	1.89808	0.949041	+	0.315154i	$0.102056\pi$
$942$	0	0
$943$	41667.8	1.43891
$944$	−12925.6	−0.445649
$945$	0	0
$946$	−12302.1	−0.422807
$947$	−32935.2	−1.13015	−0.565074	−	0.825040i	$-0.691152\pi$
$947$	−32935.2	−1.13015	−0.565074	+	0.825040i	$0.691152\pi$
$948$	0	0
$949$	14640.1	0.500777
$950$	−4449.80	−0.151969
$951$	0	0
$952$	8618.81	0.293421
$953$	−29576.5	−1.00533	−0.502664	−	0.864482i	$-0.667647\pi$
$953$	−29576.5	−1.00533	−0.502664	+	0.864482i	$0.667647\pi$
$954$	0	0
$955$	−7696.28	−0.260781
$956$	10425.9	0.352716
$957$	0	0
$958$	−27211.6	−0.917710
$959$	5246.63	0.176666
$960$	0	0
$961$	−1803.50	−0.0605384
$962$	11370.6	0.381083
$963$	0	0
$964$	−18592.8	−0.621197
$965$	−17225.3	−0.574613
$966$	0	0
$967$	29429.9	0.978697	0.489349	−	0.872088i	$-0.337235\pi$
$967$	29429.9	0.978697	0.489349	+	0.872088i	$0.337235\pi$
$968$	2980.09	0.0989500
$969$	0	0
$970$	−6657.31	−0.220364
$971$	12648.5	0.418033	0.209017	−	0.977912i	$-0.432974\pi$
$971$	12648.5	0.418033	0.209017	+	0.977912i	$0.432974\pi$
$972$	0	0
$973$	7209.67	0.237545
$974$	26737.4	0.879590
$975$	0	0
$976$	−27809.6	−0.912054
$977$	39737.9	1.30126	0.650629	−	0.759396i	$-0.274505\pi$
$977$	39737.9	1.30126	0.650629	+	0.759396i	$0.274505\pi$
$978$	0	0
$979$	−3007.74	−0.0981897
$980$	2161.70	0.0704624
$981$	0	0
$982$	34702.7	1.12771
$983$	18848.0	0.611555	0.305778	−	0.952103i	$-0.401084\pi$
$983$	18848.0	0.611555	0.305778	+	0.952103i	$0.401084\pi$
$984$	0	0
$985$	−23817.6	−0.770450
$986$	−2690.58	−0.0869021
$987$	0	0
$988$	−4424.84	−0.142483
$989$	−99687.8	−3.20514
$990$	0	0
$991$	44198.4	1.41676	0.708380	−	0.705832i	$-0.249427\pi$
$991$	44198.4	1.41676	0.708380	+	0.705832i	$0.249427\pi$
$992$	−20004.9	−0.640278
$993$	0	0
$994$	21505.1	0.686218
$995$	−2919.44	−0.0930176
$996$	0	0
$997$	−2068.77	−0.0657158	−0.0328579	−	0.999460i	$-0.510461\pi$
$997$	−2068.77	−0.0657158	−0.0328579	+	0.999460i	$0.510461\pi$
$998$	33004.5	1.04683
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.p.1.2	yes	7
3.2	odd	2		495.4.a.o.1.6	✓	7
5.4	even	2		2475.4.a.bp.1.6		7
15.14	odd	2		2475.4.a.bt.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.6	✓	7	3.2	odd	2
495.4.a.p.1.2	yes	7	1.1	even	1	trivial
2475.4.a.bp.1.6		7	5.4	even	2
2475.4.a.bt.1.2		7	15.14	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-2.28302 q^{2} -2.78784 q^{4} +5.00000 q^{5} +13.7084 q^{7} +24.6288 q^{8} +O(q^{10})\)	\(q-2.28302 q^{2} -2.78784 q^{4} +5.00000 q^{5} +13.7084 q^{7} +24.6288 q^{8} -11.4151 q^{10} +11.0000 q^{11} +20.3581 q^{13} -31.2964 q^{14} -33.9252 q^{16} +25.5281 q^{17} +77.9636 q^{19} -13.9392 q^{20} -25.1132 q^{22} -203.500 q^{23} +25.0000 q^{25} -46.4779 q^{26} -38.2167 q^{28} +46.1656 q^{29} +167.295 q^{31} -119.579 q^{32} -58.2810 q^{34} +68.5418 q^{35} -244.645 q^{37} -177.992 q^{38} +123.144 q^{40} -204.756 q^{41} +489.867 q^{43} -30.6663 q^{44} +464.593 q^{46} -324.309 q^{47} -155.081 q^{49} -57.0754 q^{50} -56.7552 q^{52} +402.012 q^{53} +55.0000 q^{55} +337.621 q^{56} -105.397 q^{58} +381.003 q^{59} +819.734 q^{61} -381.936 q^{62} +544.402 q^{64} +101.791 q^{65} +867.608 q^{67} -71.1683 q^{68} -156.482 q^{70} -687.144 q^{71} +719.128 q^{73} +558.528 q^{74} -217.350 q^{76} +150.792 q^{77} -1147.70 q^{79} -169.626 q^{80} +467.461 q^{82} +345.721 q^{83} +127.640 q^{85} -1118.37 q^{86} +270.917 q^{88} -273.431 q^{89} +279.076 q^{91} +567.325 q^{92} +740.402 q^{94} +389.818 q^{95} +583.204 q^{97} +354.052 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8}+O(q^{10}) \) 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8	\( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8} + 25 q^{10} + 77 q^{11} + 38 q^{13} + 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} + 165 q^{20} + 55 q^{22} + 334 q^{23} + 175 q^{25} - 372 q^{26} + 812 q^{28} - 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} + 150 q^{35} + 466 q^{37} - 494 q^{38} + 225 q^{40} - 258 q^{41} + 308 q^{43} + 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} + 125 q^{50} + 512 q^{52} + 110 q^{53} + 385 q^{55} + 20 q^{56} + 1362 q^{58} - 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} + 190 q^{65} + 2268 q^{67} - 1186 q^{68} + 100 q^{70} - 166 q^{71} + 200 q^{73} - 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1545 q^{80} - 1006 q^{82} + 370 q^{83} + 60 q^{85} + 106 q^{86} + 495 q^{88} - 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} + 1130 q^{95} + 3698 q^{97} + 697 q^{98}+O(q^{100}) \) 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8 + 25 * q^10 + 77 * q^11 + 38 * q^13 + 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 + 165 * q^20 + 55 * q^22 + 334 * q^23 + 175 * q^25 - 372 * q^26 + 812 * q^28 - 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 + 150 * q^35 + 466 * q^37 - 494 * q^38 + 225 * q^40 - 258 * q^41 + 308 * q^43 + 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 + 125 * q^50 + 512 * q^52 + 110 * q^53 + 385 * q^55 + 20 * q^56 + 1362 * q^58 - 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 + 190 * q^65 + 2268 * q^67 - 1186 * q^68 + 100 * q^70 - 166 * q^71 + 200 * q^73 - 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1545 * q^80 - 1006 * q^82 + 370 * q^83 + 60 * q^85 + 106 * q^86 + 495 * q^88 - 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 + 1130 * q^95 + 3698 * q^97 + 697 * q^98

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