LMFDB - Embedded newform 2254.4.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, 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("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 38x^{4} + 203x^{2} - 100 $ x^6 - 38x^4 + 203x^2 - 100
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-5.62980$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.92350 q^{3} +4.00000 q^{4} +13.4068 q^{5} +7.84699 q^{6} +8.00000 q^{8} -11.6062 q^{9} +26.8137 q^{10} -46.1770 q^{11} +15.6940 q^{12} -72.7465 q^{13} +52.6017 q^{15} +16.0000 q^{16} +51.6451 q^{17} -23.2123 q^{18} -115.382 q^{19} +53.6274 q^{20} -92.3541 q^{22} +23.0000 q^{23} +31.3880 q^{24} +54.7435 q^{25} -145.493 q^{26} -151.471 q^{27} -12.8001 q^{29} +105.203 q^{30} +41.0425 q^{31} +32.0000 q^{32} -181.175 q^{33} +103.290 q^{34} -46.4247 q^{36} +268.884 q^{37} -230.763 q^{38} -285.421 q^{39} +107.255 q^{40} +204.674 q^{41} +163.610 q^{43} -184.708 q^{44} -155.602 q^{45} +46.0000 q^{46} -259.589 q^{47} +62.7760 q^{48} +109.487 q^{50} +202.630 q^{51} -290.986 q^{52} -766.655 q^{53} -302.942 q^{54} -619.088 q^{55} -452.699 q^{57} -25.6003 q^{58} -584.803 q^{59} +210.407 q^{60} -713.386 q^{61} +82.0850 q^{62} +64.0000 q^{64} -975.301 q^{65} -362.351 q^{66} +1064.46 q^{67} +206.581 q^{68} +90.2404 q^{69} -659.926 q^{71} -92.8494 q^{72} +653.396 q^{73} +537.769 q^{74} +214.786 q^{75} -461.526 q^{76} -570.842 q^{78} -1291.12 q^{79} +214.509 q^{80} -280.930 q^{81} +409.348 q^{82} +91.8152 q^{83} +692.398 q^{85} +327.221 q^{86} -50.2214 q^{87} -369.416 q^{88} +532.663 q^{89} -311.204 q^{90} +92.0000 q^{92} +161.030 q^{93} -519.178 q^{94} -1546.90 q^{95} +125.552 q^{96} -480.841 q^{97} +535.939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 24 q^{4} + 48 q^{8} - 52 q^{9} - 84 q^{11} + 52 q^{15} + 96 q^{16} - 104 q^{18} - 168 q^{22} + 138 q^{23} - 286 q^{25} - 22 q^{29} + 104 q^{30} + 192 q^{32} - 208 q^{36} - 180 q^{37} - 414 q^{39}+ \cdots + 600 q^{99}+O(q^{100}) $ 6 * q + 12 * q^2 + 24 * q^4 + 48 * q^8 - 52 * q^9 - 84 * q^11 + 52 * q^15 + 96 * q^16 - 104 * q^18 - 168 * q^22 + 138 * q^23 - 286 * q^25 - 22 * q^29 + 104 * q^30 + 192 * q^32 - 208 * q^36 - 180 * q^37 - 414 * q^39 - 600 * q^43 - 336 * q^44 + 276 * q^46 - 572 * q^50 + 464 * q^51 - 876 * q^53 - 1452 * q^57 - 44 * q^58 + 208 * q^60 + 384 * q^64 - 1532 * q^65 - 92 * q^67 - 1434 * q^71 - 416 * q^72 - 360 * q^74 - 828 * q^78 - 2820 * q^79 - 2198 * q^81 - 184 * q^85 - 1200 * q^86 - 672 * q^88 + 552 * q^92 - 2742 * q^93 - 2888 * q^95 + 600 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.92350	0.755077	0.377539	−	0.925994i	$-0.376771\pi$
$3$	3.92350	0.755077	0.377539	+	0.925994i	$0.376771\pi$
$4$	4.00000	0.500000
$5$	13.4068	1.19914	0.599572	−	0.800321i	$-0.295337\pi$
$5$	13.4068	1.19914	0.599572	+	0.800321i	$0.295337\pi$
$6$	7.84699	0.533920
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−11.6062	−0.429858
$10$	26.8137	0.847923
$11$	−46.1770	−1.26572	−0.632859	−	0.774267i	$-0.718119\pi$
$11$	−46.1770	−1.26572	−0.632859	+	0.774267i	$0.718119\pi$
$12$	15.6940	0.377539
$13$	−72.7465	−1.55202	−0.776010	−	0.630721i	$-0.782759\pi$
$13$	−72.7465	−1.55202	−0.776010	+	0.630721i	$0.782759\pi$
$14$	0	0
$15$	52.6017	0.905447
$16$	16.0000	0.250000
$17$	51.6451	0.736811	0.368405	−	0.929665i	$-0.379904\pi$
$17$	51.6451	0.736811	0.368405	+	0.929665i	$0.379904\pi$
$18$	−23.2123	−0.303956
$19$	−115.382	−1.39318	−0.696588	−	0.717471i	$-0.745299\pi$
$19$	−115.382	−1.39318	−0.696588	+	0.717471i	$0.745299\pi$
$20$	53.6274	0.599572
$21$	0	0
$22$	−92.3541	−0.894998
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	31.3880	0.266960
$25$	54.7435	0.437948
$26$	−145.493	−1.09744
$27$	−151.471	−1.07965
$28$	0	0
$29$	−12.8001	−0.0819630	−0.0409815	−	0.999160i	$-0.513048\pi$
$29$	−12.8001	−0.0819630	−0.0409815	+	0.999160i	$0.513048\pi$
$30$	105.203	0.640248
$31$	41.0425	0.237789	0.118894	−	0.992907i	$-0.462065\pi$
$31$	41.0425	0.237789	0.118894	+	0.992907i	$0.462065\pi$
$32$	32.0000	0.176777
$33$	−181.175	−0.955715
$34$	103.290	0.521004
$35$	0	0
$36$	−46.4247	−0.214929
$37$	268.884	1.19471	0.597356	−	0.801977i	$-0.296218\pi$
$37$	268.884	1.19471	0.597356	+	0.801977i	$0.296218\pi$
$38$	−230.763	−0.985124
$39$	−285.421	−1.17189
$40$	107.255	0.423962
$41$	204.674	0.779627	0.389813	−	0.920894i	$-0.372540\pi$
$41$	204.674	0.779627	0.389813	+	0.920894i	$0.372540\pi$
$42$	0	0
$43$	163.610	0.580240	0.290120	−	0.956990i	$-0.406305\pi$
$43$	163.610	0.580240	0.290120	+	0.956990i	$0.406305\pi$
$44$	−184.708	−0.632859
$45$	−155.602	−0.515462
$46$	46.0000	0.147442
$47$	−259.589	−0.805637	−0.402818	−	0.915280i	$-0.631969\pi$
$47$	−259.589	−0.805637	−0.402818	+	0.915280i	$0.631969\pi$
$48$	62.7760	0.188769
$49$	0	0
$50$	109.487	0.309676
$51$	202.630	0.556349
$52$	−290.986	−0.776010
$53$	−766.655	−1.98695	−0.993473	−	0.114069i	$-0.963612\pi$
$53$	−766.655	−1.98695	−0.993473	+	0.114069i	$0.963612\pi$
$54$	−302.942	−0.763430
$55$	−619.088	−1.51778
$56$	0	0
$57$	−452.699	−1.05196
$58$	−25.6003	−0.0579566
$59$	−584.803	−1.29042	−0.645211	−	0.764004i	$-0.723231\pi$
$59$	−584.803	−1.29042	−0.645211	+	0.764004i	$0.723231\pi$
$60$	210.407	0.452723
$61$	−713.386	−1.49737	−0.748686	−	0.662925i	$-0.769315\pi$
$61$	−713.386	−1.49737	−0.748686	+	0.662925i	$0.769315\pi$
$62$	82.0850	0.168142
$63$	0	0
$64$	64.0000	0.125000
$65$	−975.301	−1.86110
$66$	−362.351	−0.675793
$67$	1064.46	1.94096	0.970482	−	0.241173i	$-0.0775322\pi$
$67$	1064.46	1.94096	0.970482	+	0.241173i	$0.0775322\pi$
$68$	206.581	0.368405
$69$	90.2404	0.157445
$70$	0	0
$71$	−659.926	−1.10308	−0.551541	−	0.834148i	$-0.685960\pi$
$71$	−659.926	−1.10308	−0.551541	+	0.834148i	$0.685960\pi$
$72$	−92.8494	−0.151978
$73$	653.396	1.04759	0.523796	−	0.851844i	$-0.324515\pi$
$73$	653.396	1.04759	0.523796	+	0.851844i	$0.324515\pi$
$74$	537.769	0.844788
$75$	214.786	0.330684
$76$	−461.526	−0.696588
$77$	0	0
$78$	−570.842	−0.828655
$79$	−1291.12	−1.83876	−0.919381	−	0.393369i	$-0.871309\pi$
$79$	−1291.12	−1.83876	−0.919381	+	0.393369i	$0.871309\pi$
$80$	214.509	0.299786
$81$	−280.930	−0.385364
$82$	409.348	0.551279
$83$	91.8152	0.121422	0.0607110	−	0.998155i	$-0.480663\pi$
$83$	91.8152	0.121422	0.0607110	+	0.998155i	$0.480663\pi$
$84$	0	0
$85$	692.398	0.883543
$86$	327.221	0.410292
$87$	−50.2214	−0.0618884
$88$	−369.416	−0.447499
$89$	532.663	0.634406	0.317203	−	0.948358i	$-0.397256\pi$
$89$	532.663	0.634406	0.317203	+	0.948358i	$0.397256\pi$
$90$	−311.204	−0.364487
$91$	0	0
$92$	92.0000	0.104257
$93$	161.030	0.179549
$94$	−519.178	−0.569671
$95$	−1546.90	−1.67062
$96$	125.552	0.133480
$97$	−480.841	−0.503320	−0.251660	−	0.967816i	$-0.580976\pi$
$97$	−480.841	−0.503320	−0.251660	+	0.967816i	$0.580976\pi$
$98$	0	0
$99$	535.939	0.544079
$100$	218.974	0.218974
$101$	−43.4054	−0.0427624	−0.0213812	−	0.999771i	$-0.506806\pi$
$101$	−43.4054	−0.0427624	−0.0213812	+	0.999771i	$0.506806\pi$
$102$	405.259	0.393398
$103$	−1443.38	−1.38078	−0.690390	−	0.723437i	$-0.742561\pi$
$103$	−1443.38	−1.38078	−0.690390	+	0.723437i	$0.742561\pi$
$104$	−581.972	−0.548722
$105$	0	0
$106$	−1533.31	−1.40498
$107$	−1076.06	−0.972211	−0.486105	−	0.873900i	$-0.661583\pi$
$107$	−1076.06	−0.972211	−0.486105	+	0.873900i	$0.661583\pi$
$108$	−605.885	−0.539827
$109$	−536.625	−0.471553	−0.235777	−	0.971807i	$-0.575763\pi$
$109$	−536.625	−0.471553	−0.235777	+	0.971807i	$0.575763\pi$
$110$	−1238.18	−1.07323
$111$	1054.97	0.902099
$112$	0	0
$113$	−238.411	−0.198476	−0.0992382	−	0.995064i	$-0.531641\pi$
$113$	−238.411	−0.198476	−0.0992382	+	0.995064i	$0.531641\pi$
$114$	−905.398	−0.743845
$115$	308.357	0.250039
$116$	−51.2006	−0.0409815
$117$	844.308	0.667148
$118$	−1169.61	−0.912466
$119$	0	0
$120$	420.814	0.320124
$121$	801.319	0.602043
$122$	−1426.77	−1.05880
$123$	803.038	0.588678
$124$	164.170	0.118894
$125$	−941.919	−0.673982
$126$	0	0
$127$	−2643.06	−1.84673	−0.923363	−	0.383929i	$-0.874571\pi$
$127$	−2643.06	−1.84673	−0.923363	+	0.383929i	$0.874571\pi$
$128$	128.000	0.0883883
$129$	641.925	0.438126
$130$	−1950.60	−1.31599
$131$	22.5348	0.0150296	0.00751479	−	0.999972i	$-0.497608\pi$
$131$	22.5348	0.0150296	0.00751479	+	0.999972i	$0.497608\pi$
$132$	−724.702	−0.477858
$133$	0	0
$134$	2128.92	1.37247
$135$	−2030.75	−1.29466
$136$	413.161	0.260502
$137$	−2035.85	−1.26959	−0.634797	−	0.772679i	$-0.718916\pi$
$137$	−2035.85	−1.26959	−0.634797	+	0.772679i	$0.718916\pi$
$138$	180.481	0.111330
$139$	2152.00	1.31317	0.656585	−	0.754252i	$-0.272000\pi$
$139$	2152.00	1.31317	0.656585	+	0.754252i	$0.272000\pi$
$140$	0	0
$141$	−1018.50	−0.608318
$142$	−1319.85	−0.779997
$143$	3359.22	1.96442
$144$	−185.699	−0.107465
$145$	−171.610	−0.0982855
$146$	1306.79	0.740759
$147$	0	0
$148$	1075.54	0.597356
$149$	3410.39	1.87510	0.937551	−	0.347847i	$-0.113087\pi$
$149$	3410.39	1.87510	0.937551	+	0.347847i	$0.113087\pi$
$150$	429.572	0.233829
$151$	−1475.33	−0.795106	−0.397553	−	0.917579i	$-0.630140\pi$
$151$	−1475.33	−0.795106	−0.397553	+	0.917579i	$0.630140\pi$
$152$	−923.052	−0.492562
$153$	−599.402	−0.316724
$154$	0	0
$155$	550.250	0.285143
$156$	−1141.68	−0.585947
$157$	1811.96	0.921084	0.460542	−	0.887638i	$-0.347655\pi$
$157$	1811.96	0.921084	0.460542	+	0.887638i	$0.347655\pi$
$158$	−2582.24	−1.30020
$159$	−3007.97	−1.50030
$160$	429.019	0.211981
$161$	0	0
$162$	−561.861	−0.272493
$163$	636.545	0.305877	0.152939	−	0.988236i	$-0.451126\pi$
$163$	636.545	0.305877	0.152939	+	0.988236i	$0.451126\pi$
$164$	818.696	0.389813
$165$	−2428.99	−1.14604
$166$	183.630	0.0858583
$167$	2913.65	1.35009	0.675044	−	0.737777i	$-0.264125\pi$
$167$	2913.65	1.35009	0.675044	+	0.737777i	$0.264125\pi$
$168$	0	0
$169$	3095.06	1.40876
$170$	1384.80	0.624759
$171$	1339.14	0.598868
$172$	654.441	0.290120
$173$	1717.24	0.754680	0.377340	−	0.926075i	$-0.376839\pi$
$173$	1717.24	0.754680	0.377340	+	0.926075i	$0.376839\pi$
$174$	−100.443	−0.0437617
$175$	0	0
$176$	−738.833	−0.316430
$177$	−2294.47	−0.974369
$178$	1065.33	0.448593
$179$	342.553	0.143037	0.0715185	−	0.997439i	$-0.477215\pi$
$179$	342.553	0.143037	0.0715185	+	0.997439i	$0.477215\pi$
$180$	−622.408	−0.257731
$181$	−2475.32	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$181$	−2475.32	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$182$	0	0
$183$	−2798.97	−1.13063
$184$	184.000	0.0737210
$185$	3604.89	1.43263
$186$	322.060	0.126960
$187$	−2384.82	−0.932595
$188$	−1038.36	−0.402818
$189$	0	0
$190$	−3093.81	−1.18131
$191$	1716.05	0.650101	0.325050	−	0.945697i	$-0.394619\pi$
$191$	1716.05	0.650101	0.325050	+	0.945697i	$0.394619\pi$
$192$	251.104	0.0943847
$193$	297.315	0.110887	0.0554436	−	0.998462i	$-0.482343\pi$
$193$	297.315	0.110887	0.0554436	+	0.998462i	$0.482343\pi$
$194$	−961.682	−0.355901
$195$	−3826.59	−1.40527
$196$	0	0
$197$	2586.52	0.935442	0.467721	−	0.883876i	$-0.345075\pi$
$197$	2586.52	0.935442	0.467721	+	0.883876i	$0.345075\pi$
$198$	1071.88	0.384722
$199$	−652.033	−0.232268	−0.116134	−	0.993234i	$-0.537050\pi$
$199$	−652.033	−0.232268	−0.116134	+	0.993234i	$0.537050\pi$
$200$	437.948	0.154838
$201$	4176.41	1.46558
$202$	−86.8109	−0.0302376
$203$	0	0
$204$	810.518	0.278175
$205$	2744.03	0.934885
$206$	−2886.76	−0.976359
$207$	−266.942	−0.0896316
$208$	−1163.94	−0.388005
$209$	5327.98	1.76337
$210$	0	0
$211$	−376.981	−0.122997	−0.0614987	−	0.998107i	$-0.519588\pi$
$211$	−376.981	−0.122997	−0.0614987	+	0.998107i	$0.519588\pi$
$212$	−3066.62	−0.993473
$213$	−2589.22	−0.832912
$214$	−2152.12	−0.687457
$215$	2193.50	0.695792
$216$	−1211.77	−0.381715
$217$	0	0
$218$	−1073.25	−0.333439
$219$	2563.60	0.791013
$220$	−2476.35	−0.758890
$221$	−3757.00	−1.14354
$222$	2109.93	0.637881
$223$	−1944.66	−0.583963	−0.291982	−	0.956424i	$-0.594315\pi$
$223$	−1944.66	−0.583963	−0.291982	+	0.956424i	$0.594315\pi$
$224$	0	0
$225$	−635.362	−0.188255
$226$	−476.822	−0.140344
$227$	3226.07	0.943269	0.471634	−	0.881794i	$-0.343664\pi$
$227$	3226.07	0.943269	0.471634	+	0.881794i	$0.343664\pi$
$228$	−1810.80	−0.525978
$229$	6496.30	1.87462	0.937310	−	0.348498i	$-0.113308\pi$
$229$	6496.30	1.87462	0.937310	+	0.348498i	$0.113308\pi$
$230$	616.715	0.176804
$231$	0	0
$232$	−102.401	−0.0289783
$233$	−2936.37	−0.825613	−0.412806	−	0.910819i	$-0.635451\pi$
$233$	−2936.37	−0.825613	−0.412806	+	0.910819i	$0.635451\pi$
$234$	1688.62	0.471745
$235$	−3480.27	−0.966075
$236$	−2339.21	−0.645211
$237$	−5065.70	−1.38841
$238$	0	0
$239$	5555.07	1.50346	0.751731	−	0.659470i	$-0.229219\pi$
$239$	5555.07	1.50346	0.751731	+	0.659470i	$0.229219\pi$
$240$	841.627	0.226362
$241$	−4364.87	−1.16666	−0.583332	−	0.812234i	$-0.698251\pi$
$241$	−4364.87	−1.16666	−0.583332	+	0.812234i	$0.698251\pi$
$242$	1602.64	0.425708
$243$	2987.49	0.788674
$244$	−2853.54	−0.748686
$245$	0	0
$246$	1606.08	0.416259
$247$	8393.61	2.16224
$248$	328.340	0.0840710
$249$	360.237	0.0916830
$250$	−1883.84	−0.476577
$251$	2635.60	0.662779	0.331389	−	0.943494i	$-0.392483\pi$
$251$	2635.60	0.662779	0.331389	+	0.943494i	$0.392483\pi$
$252$	0	0
$253$	−1062.07	−0.263920
$254$	−5286.13	−1.30583
$255$	2716.62	0.667143
$256$	256.000	0.0625000
$257$	5809.66	1.41010	0.705052	−	0.709156i	$-0.250924\pi$
$257$	5809.66	1.41010	0.705052	+	0.709156i	$0.250924\pi$
$258$	1283.85	0.309802
$259$	0	0
$260$	−3901.20	−0.930548
$261$	148.561	0.0352325
$262$	45.0696	0.0106275
$263$	−1047.20	−0.245526	−0.122763	−	0.992436i	$-0.539175\pi$
$263$	−1047.20	−0.245526	−0.122763	+	0.992436i	$0.539175\pi$
$264$	−1449.40	−0.337896
$265$	−10278.4	−2.38264
$266$	0	0
$267$	2089.90	0.479026
$268$	4257.84	0.970482
$269$	−2832.87	−0.642094	−0.321047	−	0.947063i	$-0.604035\pi$
$269$	−2832.87	−0.642094	−0.321047	+	0.947063i	$0.604035\pi$
$270$	−4061.50	−0.915463
$271$	2953.80	0.662105	0.331053	−	0.943612i	$-0.392596\pi$
$271$	2953.80	0.662105	0.331053	+	0.943612i	$0.392596\pi$
$272$	826.322	0.184203
$273$	0	0
$274$	−4071.70	−0.897738
$275$	−2527.89	−0.554318
$276$	360.962	0.0787223
$277$	−8522.36	−1.84859	−0.924294	−	0.381680i	$-0.875346\pi$
$277$	−8522.36	−1.84859	−0.924294	+	0.381680i	$0.875346\pi$
$278$	4304.01	0.928551
$279$	−476.346	−0.102215
$280$	0	0
$281$	−4979.31	−1.05708	−0.528542	−	0.848907i	$-0.677261\pi$
$281$	−4979.31	−1.05708	−0.528542	+	0.848907i	$0.677261\pi$
$282$	−2036.99	−0.430146
$283$	−866.435	−0.181994	−0.0909969	−	0.995851i	$-0.529005\pi$
$283$	−866.435	−0.181994	−0.0909969	+	0.995851i	$0.529005\pi$
$284$	−2639.70	−0.551541
$285$	−6069.27	−1.26145
$286$	6718.44	1.38905
$287$	0	0
$288$	−371.397	−0.0759889
$289$	−2245.78	−0.457110
$290$	−343.219	−0.0694984
$291$	−1886.58	−0.380045
$292$	2613.58	0.523796
$293$	4521.27	0.901487	0.450743	−	0.892654i	$-0.351159\pi$
$293$	4521.27	0.901487	0.450743	+	0.892654i	$0.351159\pi$
$294$	0	0
$295$	−7840.37	−1.54740
$296$	2151.07	0.422394
$297$	6994.49	1.36654
$298$	6820.79	1.32590
$299$	−1673.17	−0.323618
$300$	859.143	0.165342
$301$	0	0
$302$	−2950.67	−0.562225
$303$	−170.301	−0.0322889
$304$	−1846.10	−0.348294
$305$	−9564.25	−1.79557
$306$	−1198.80	−0.223958
$307$	−2434.53	−0.452593	−0.226296	−	0.974058i	$-0.572662\pi$
$307$	−2434.53	−0.452593	−0.226296	+	0.974058i	$0.572662\pi$
$308$	0	0
$309$	−5663.09	−1.04260
$310$	1100.50	0.201627
$311$	−7161.06	−1.30568	−0.652840	−	0.757496i	$-0.726423\pi$
$311$	−7161.06	−1.30568	−0.652840	+	0.757496i	$0.726423\pi$
$312$	−2283.37	−0.414327
$313$	1586.38	0.286477	0.143238	−	0.989688i	$-0.454248\pi$
$313$	1586.38	0.286477	0.143238	+	0.989688i	$0.454248\pi$
$314$	3623.92	0.651304
$315$	0	0
$316$	−5164.47	−0.919381
$317$	4746.43	0.840966	0.420483	−	0.907300i	$-0.361861\pi$
$317$	4746.43	0.840966	0.420483	+	0.907300i	$0.361861\pi$
$318$	−6015.94	−1.06087
$319$	591.073	0.103742
$320$	858.038	0.149893
$321$	−4221.92	−0.734095
$322$	0	0
$323$	−5958.90	−1.02651
$324$	−1123.72	−0.192682
$325$	−3982.40	−0.679703
$326$	1273.09	0.216288
$327$	−2105.45	−0.356059
$328$	1637.39	0.275640
$329$	0	0
$330$	−4857.98	−0.810373
$331$	4126.46	0.685229	0.342614	−	0.939476i	$-0.388688\pi$
$331$	4126.46	0.685229	0.342614	+	0.939476i	$0.388688\pi$
$332$	367.261	0.0607110
$333$	−3120.72	−0.513556
$334$	5827.29	0.954657
$335$	14271.1	2.32750
$336$	0	0
$337$	2842.26	0.459429	0.229715	−	0.973258i	$-0.426221\pi$
$337$	2842.26	0.459429	0.229715	+	0.973258i	$0.426221\pi$
$338$	6190.11	0.996147
$339$	−935.406	−0.149865
$340$	2769.59	0.441771
$341$	−1895.22	−0.300974
$342$	2678.28	0.423464
$343$	0	0
$344$	1308.88	0.205146
$345$	1209.84	0.188799
$346$	3434.49	0.533640
$347$	−7996.17	−1.23705	−0.618526	−	0.785764i	$-0.712270\pi$
$347$	−7996.17	−1.23705	−0.618526	+	0.785764i	$0.712270\pi$
$348$	−200.885	−0.0309442
$349$	9584.09	1.46998	0.734992	−	0.678076i	$-0.237186\pi$
$349$	9584.09	1.46998	0.734992	+	0.678076i	$0.237186\pi$
$350$	0	0
$351$	11019.0	1.67564
$352$	−1477.67	−0.223749
$353$	11083.1	1.67109	0.835543	−	0.549426i	$-0.185153\pi$
$353$	11083.1	1.67109	0.835543	+	0.549426i	$0.185153\pi$
$354$	−4588.95	−0.688983
$355$	−8847.52	−1.32275
$356$	2130.65	0.317203
$357$	0	0
$358$	685.107	0.101142
$359$	−3024.56	−0.444653	−0.222326	−	0.974972i	$-0.571365\pi$
$359$	−3024.56	−0.444653	−0.222326	+	0.974972i	$0.571365\pi$
$360$	−1244.82	−0.182243
$361$	6453.90	0.940940
$362$	−4950.63	−0.718783
$363$	3143.97	0.454589
$364$	0	0
$365$	8759.98	1.25621
$366$	−5597.93	−0.799477
$367$	−1663.24	−0.236568	−0.118284	−	0.992980i	$-0.537739\pi$
$367$	−1663.24	−0.236568	−0.118284	+	0.992980i	$0.537739\pi$
$368$	368.000	0.0521286
$369$	−2375.48	−0.335129
$370$	7209.78	1.01302
$371$	0	0
$372$	644.121	0.0897745
$373$	1236.97	0.171710	0.0858548	−	0.996308i	$-0.472638\pi$
$373$	1236.97	0.171710	0.0858548	+	0.996308i	$0.472638\pi$
$374$	−4769.64	−0.659444
$375$	−3695.61	−0.508909
$376$	−2076.71	−0.284836
$377$	931.166	0.127208
$378$	0	0
$379$	−8450.68	−1.14534	−0.572668	−	0.819788i	$-0.694091\pi$
$379$	−8450.68	−1.14534	−0.572668	+	0.819788i	$0.694091\pi$
$380$	−6187.61	−0.835310
$381$	−10370.1	−1.39442
$382$	3432.11	0.459691
$383$	5337.77	0.712134	0.356067	−	0.934460i	$-0.384118\pi$
$383$	5337.77	0.712134	0.356067	+	0.934460i	$0.384118\pi$
$384$	502.208	0.0667400
$385$	0	0
$386$	594.631	0.0784091
$387$	−1898.89	−0.249421
$388$	−1923.36	−0.251660
$389$	−14331.3	−1.86793	−0.933966	−	0.357361i	$-0.883677\pi$
$389$	−14331.3	−1.86793	−0.933966	+	0.357361i	$0.883677\pi$
$390$	−7653.18	−0.993677
$391$	1187.84	0.153636
$392$	0	0
$393$	88.4153	0.0113485
$394$	5173.05	0.661458
$395$	−17309.8	−2.20494
$396$	2143.75	0.272040
$397$	3622.07	0.457901	0.228950	−	0.973438i	$-0.426471\pi$
$397$	3622.07	0.457901	0.228950	+	0.973438i	$0.426471\pi$
$398$	−1304.07	−0.164238
$399$	0	0
$400$	875.895	0.109487
$401$	13011.2	1.62032	0.810158	−	0.586212i	$-0.199381\pi$
$401$	13011.2	1.62032	0.810158	+	0.586212i	$0.199381\pi$
$402$	8352.82	1.03632
$403$	−2985.70	−0.369053
$404$	−173.622	−0.0213812
$405$	−3766.39	−0.462107
$406$	0	0
$407$	−12416.3	−1.51217
$408$	1621.04	0.196699
$409$	−4224.19	−0.510691	−0.255346	−	0.966850i	$-0.582189\pi$
$409$	−4224.19	−0.510691	−0.255346	+	0.966850i	$0.582189\pi$
$410$	5488.06	0.661064
$411$	−7987.65	−0.958642
$412$	−5773.52	−0.690390
$413$	0	0
$414$	−533.884	−0.0633791
$415$	1230.95	0.145603
$416$	−2327.89	−0.274361
$417$	8443.38	0.991545
$418$	10656.0	1.24689
$419$	7385.94	0.861162	0.430581	−	0.902552i	$-0.358309\pi$
$419$	7385.94	0.861162	0.430581	+	0.902552i	$0.358309\pi$
$420$	0	0
$421$	4982.36	0.576782	0.288391	−	0.957513i	$-0.406880\pi$
$421$	4982.36	0.576782	0.288391	+	0.957513i	$0.406880\pi$
$422$	−753.962	−0.0869723
$423$	3012.83	0.346310
$424$	−6133.24	−0.702491
$425$	2827.23	0.322685
$426$	−5178.44	−0.588958
$427$	0	0
$428$	−4304.24	−0.486105
$429$	13179.9	1.48329
$430$	4386.99	0.491999
$431$	12505.4	1.39760	0.698799	−	0.715318i	$-0.253718\pi$
$431$	12505.4	1.39760	0.698799	+	0.715318i	$0.253718\pi$
$432$	−2423.54	−0.269913
$433$	−17336.3	−1.92409	−0.962044	−	0.272893i	$-0.912020\pi$
$433$	−17336.3	−1.92409	−0.962044	+	0.272893i	$0.912020\pi$
$434$	0	0
$435$	−673.310	−0.0742132
$436$	−2146.50	−0.235777
$437$	−2653.78	−0.290497
$438$	5127.19	0.559331
$439$	14336.8	1.55867	0.779337	−	0.626604i	$-0.215556\pi$
$439$	14336.8	1.55867	0.779337	+	0.626604i	$0.215556\pi$
$440$	−4952.71	−0.536616
$441$	0	0
$442$	−7514.01	−0.808608
$443$	−5855.98	−0.628050	−0.314025	−	0.949415i	$-0.601678\pi$
$443$	−5855.98	−0.628050	−0.314025	+	0.949415i	$0.601678\pi$
$444$	4219.87	0.451050
$445$	7141.33	0.760745
$446$	−3889.31	−0.412924
$447$	13380.7	1.41585
$448$	0	0
$449$	−464.947	−0.0488690	−0.0244345	−	0.999701i	$-0.507779\pi$
$449$	−464.947	−0.0488690	−0.0244345	+	0.999701i	$0.507779\pi$
$450$	−1270.72	−0.133117
$451$	−9451.23	−0.986788
$452$	−953.645	−0.0992382
$453$	−5788.47	−0.600366
$454$	6452.15	0.666992
$455$	0	0
$456$	−3621.59	−0.371923
$457$	10204.3	1.04450	0.522252	−	0.852791i	$-0.325092\pi$
$457$	10204.3	1.04450	0.522252	+	0.852791i	$0.325092\pi$
$458$	12992.6	1.32556
$459$	−7822.75	−0.795500
$460$	1233.43	0.125019
$461$	−6067.39	−0.612986	−0.306493	−	0.951873i	$-0.599156\pi$
$461$	−6067.39	−0.612986	−0.306493	+	0.951873i	$0.599156\pi$
$462$	0	0
$463$	−917.651	−0.0921098	−0.0460549	−	0.998939i	$-0.514665\pi$
$463$	−917.651	−0.0921098	−0.0460549	+	0.998939i	$0.514665\pi$
$464$	−204.802	−0.0204908
$465$	2158.91	0.215305
$466$	−5872.73	−0.583796
$467$	355.721	0.0352480	0.0176240	−	0.999845i	$-0.494390\pi$
$467$	355.721	0.0352480	0.0176240	+	0.999845i	$0.494390\pi$
$468$	3377.23	0.333574
$469$	0	0
$470$	−6960.54	−0.683118
$471$	7109.22	0.695489
$472$	−4678.43	−0.456233
$473$	−7555.04	−0.734421
$474$	−10131.4	−0.981752
$475$	−6316.38	−0.610138
$476$	0	0
$477$	8897.92	0.854105
$478$	11110.1	1.06311
$479$	−8646.13	−0.824743	−0.412371	−	0.911016i	$-0.635299\pi$
$479$	−8646.13	−0.824743	−0.412371	+	0.911016i	$0.635299\pi$
$480$	1683.25	0.160062
$481$	−19560.4	−1.85421
$482$	−8729.74	−0.824956
$483$	0	0
$484$	3205.28	0.301021
$485$	−6446.56	−0.603553
$486$	5974.99	0.557677
$487$	−11184.7	−1.04071	−0.520357	−	0.853949i	$-0.674201\pi$
$487$	−11184.7	−1.04071	−0.520357	+	0.853949i	$0.674201\pi$
$488$	−5707.09	−0.529401
$489$	2497.48	0.230961
$490$	0	0
$491$	−568.402	−0.0522436	−0.0261218	−	0.999659i	$-0.508316\pi$
$491$	−568.402	−0.0522436	−0.0261218	+	0.999659i	$0.508316\pi$
$492$	3212.15	0.294339
$493$	−661.065	−0.0603913
$494$	16787.2	1.52893
$495$	7185.24	0.652430
$496$	656.680	0.0594472
$497$	0	0
$498$	720.473	0.0648297
$499$	6290.35	0.564318	0.282159	−	0.959368i	$-0.408949\pi$
$499$	6290.35	0.564318	0.282159	+	0.959368i	$0.408949\pi$
$500$	−3767.67	−0.336991
$501$	11431.7	1.01942
$502$	5271.20	0.468655
$503$	−20085.3	−1.78043	−0.890217	−	0.455537i	$-0.849447\pi$
$503$	−20085.3	−1.78043	−0.890217	+	0.455537i	$0.849447\pi$
$504$	0	0
$505$	−581.930	−0.0512783
$506$	−2124.14	−0.186620
$507$	12143.4	1.06373
$508$	−10572.3	−0.923363
$509$	5646.93	0.491740	0.245870	−	0.969303i	$-0.420926\pi$
$509$	5646.93	0.491740	0.245870	+	0.969303i	$0.420926\pi$
$510$	5433.24	0.471741
$511$	0	0
$512$	512.000	0.0441942
$513$	17477.0	1.50415
$514$	11619.3	0.997093
$515$	−19351.2	−1.65576
$516$	2567.70	0.219063
$517$	11987.0	1.01971
$518$	0	0
$519$	6737.61	0.569842
$520$	−7802.41	−0.657997
$521$	5409.83	0.454911	0.227456	−	0.973788i	$-0.426959\pi$
$521$	5409.83	0.454911	0.227456	+	0.973788i	$0.426959\pi$
$522$	297.121	0.0249131
$523$	10925.2	0.913433	0.456716	−	0.889612i	$-0.349025\pi$
$523$	10925.2	0.913433	0.456716	+	0.889612i	$0.349025\pi$
$524$	90.1393	0.00751479
$525$	0	0
$526$	−2094.40	−0.173613
$527$	2119.65	0.175205
$528$	−2898.81	−0.238929
$529$	529.000	0.0434783
$530$	−20556.8	−1.68478
$531$	6787.33	0.554699
$532$	0	0
$533$	−14889.3	−1.21000
$534$	4179.80	0.338722
$535$	−14426.6	−1.16582
$536$	8515.69	0.686234
$537$	1344.01	0.108004
$538$	−5665.74	−0.454029
$539$	0	0
$540$	−8123.00	−0.647330
$541$	−20304.8	−1.61363	−0.806813	−	0.590806i	$-0.798810\pi$
$541$	−20304.8	−1.61363	−0.806813	+	0.590806i	$0.798810\pi$
$542$	5907.60	0.468179
$543$	−9711.90	−0.767546
$544$	1652.64	0.130251
$545$	−7194.44	−0.565461
$546$	0	0
$547$	−21070.9	−1.64703	−0.823517	−	0.567292i	$-0.807991\pi$
$547$	−21070.9	−1.64703	−0.823517	+	0.567292i	$0.807991\pi$
$548$	−8143.40	−0.634797
$549$	8279.67	0.643657
$550$	−5055.78	−0.391962
$551$	1476.90	0.114189
$552$	721.924	0.0556650
$553$	0	0
$554$	−17044.7	−1.30715
$555$	14143.8	1.08175
$556$	8608.02	0.656585
$557$	23294.5	1.77203	0.886014	−	0.463658i	$-0.153463\pi$
$557$	23294.5	1.77203	0.886014	+	0.463658i	$0.153463\pi$
$558$	−952.693	−0.0722772
$559$	−11902.1	−0.900544
$560$	0	0
$561$	−9356.83	−0.704181
$562$	−9958.62	−0.747472
$563$	−7847.67	−0.587460	−0.293730	−	0.955888i	$-0.594897\pi$
$563$	−7847.67	−0.587460	−0.293730	+	0.955888i	$0.594897\pi$
$564$	−4073.99	−0.304159
$565$	−3196.34	−0.238002
$566$	−1732.87	−0.128689
$567$	0	0
$568$	−5279.41	−0.389998
$569$	483.831	0.0356472	0.0178236	−	0.999841i	$-0.494326\pi$
$569$	483.831	0.0356472	0.0178236	+	0.999841i	$0.494326\pi$
$570$	−12138.5	−0.891978
$571$	−8263.38	−0.605625	−0.302812	−	0.953050i	$-0.597926\pi$
$571$	−8263.38	−0.605625	−0.302812	+	0.953050i	$0.597926\pi$
$572$	13436.9	0.982210
$573$	6732.93	0.490877
$574$	0	0
$575$	1259.10	0.0913184
$576$	−742.795	−0.0537323
$577$	14891.3	1.07441	0.537203	−	0.843453i	$-0.319481\pi$
$577$	14891.3	1.07441	0.537203	+	0.843453i	$0.319481\pi$
$578$	−4491.56	−0.323225
$579$	1166.52	0.0837284
$580$	−686.438	−0.0491428
$581$	0	0
$582$	−3773.16	−0.268733
$583$	35401.8	2.51491
$584$	5227.17	0.370380
$585$	11319.5	0.800007
$586$	9042.54	0.637447
$587$	−25245.9	−1.77514	−0.887572	−	0.460669i	$-0.847610\pi$
$587$	−25245.9	−1.77514	−0.887572	+	0.460669i	$0.847610\pi$
$588$	0	0
$589$	−4735.55	−0.331282
$590$	−15680.7	−1.09418
$591$	10148.2	0.706331
$592$	4302.15	0.298678
$593$	−8469.56	−0.586515	−0.293257	−	0.956034i	$-0.594739\pi$
$593$	−8469.56	−0.586515	−0.293257	+	0.956034i	$0.594739\pi$
$594$	13989.0	0.966288
$595$	0	0
$596$	13641.6	0.937551
$597$	−2558.25	−0.175381
$598$	−3346.34	−0.228833
$599$	23546.6	1.60616	0.803079	−	0.595872i	$-0.203193\pi$
$599$	23546.6	1.60616	0.803079	+	0.595872i	$0.203193\pi$
$600$	1718.29	0.116915
$601$	−1709.76	−0.116045	−0.0580223	−	0.998315i	$-0.518479\pi$
$601$	−1709.76	−0.116045	−0.0580223	+	0.998315i	$0.518479\pi$
$602$	0	0
$603$	−12354.3	−0.834339
$604$	−5901.34	−0.397553
$605$	10743.2	0.721936
$606$	−340.602	−0.0228317
$607$	14877.4	0.994822	0.497411	−	0.867515i	$-0.334284\pi$
$607$	14877.4	0.994822	0.497411	+	0.867515i	$0.334284\pi$
$608$	−3692.21	−0.246281
$609$	0	0
$610$	−19128.5	−1.26966
$611$	18884.2	1.25036
$612$	−2397.61	−0.158362
$613$	−4408.81	−0.290489	−0.145245	−	0.989396i	$-0.546397\pi$
$613$	−4408.81	−0.290489	−0.145245	+	0.989396i	$0.546397\pi$
$614$	−4869.06	−0.320032
$615$	10766.2	0.705911
$616$	0	0
$617$	3601.95	0.235023	0.117511	−	0.993072i	$-0.462508\pi$
$617$	3601.95	0.235023	0.117511	+	0.993072i	$0.462508\pi$
$618$	−11326.2	−0.737227
$619$	8847.54	0.574496	0.287248	−	0.957856i	$-0.407260\pi$
$619$	8847.54	0.574496	0.287248	+	0.957856i	$0.407260\pi$
$620$	2201.00	0.142572
$621$	−3483.84	−0.225123
$622$	−14322.1	−0.923255
$623$	0	0
$624$	−4566.73	−0.292974
$625$	−19471.1	−1.24615
$626$	3172.75	0.202570
$627$	20904.3	1.33148
$628$	7247.84	0.460542
$629$	13886.6	0.880276
$630$	0	0
$631$	17087.4	1.07803	0.539017	−	0.842295i	$-0.318796\pi$
$631$	17087.4	1.07803	0.539017	+	0.842295i	$0.318796\pi$
$632$	−10328.9	−0.650100
$633$	−1479.08	−0.0928725
$634$	9492.87	0.594653
$635$	−35435.2	−2.21449
$636$	−12031.9	−0.750149
$637$	0	0
$638$	1182.15	0.0733568
$639$	7659.21	0.474169
$640$	1716.08	0.105990
$641$	−28613.4	−1.76312	−0.881560	−	0.472071i	$-0.843507\pi$
$641$	−28613.4	−1.76312	−0.881560	+	0.472071i	$0.843507\pi$
$642$	−8443.83	−0.519083
$643$	26152.7	1.60398	0.801991	−	0.597336i	$-0.203774\pi$
$643$	26152.7	1.60398	0.801991	+	0.597336i	$0.203774\pi$
$644$	0	0
$645$	8606.18	0.525377
$646$	−11917.8	−0.725850
$647$	−3572.62	−0.217085	−0.108543	−	0.994092i	$-0.534618\pi$
$647$	−3572.62	−0.217085	−0.108543	+	0.994092i	$0.534618\pi$
$648$	−2247.44	−0.136247
$649$	27004.5	1.63331
$650$	−7964.79	−0.480623
$651$	0	0
$652$	2546.18	0.152939
$653$	4225.89	0.253249	0.126625	−	0.991951i	$-0.459586\pi$
$653$	4225.89	0.253249	0.126625	+	0.991951i	$0.459586\pi$
$654$	−4210.89	−0.251772
$655$	302.121	0.0180226
$656$	3274.78	0.194907
$657$	−7583.42	−0.450316
$658$	0	0
$659$	−480.617	−0.0284100	−0.0142050	−	0.999899i	$-0.504522\pi$
$659$	−480.617	−0.0284100	−0.0142050	+	0.999899i	$0.504522\pi$
$660$	−9715.97	−0.573020
$661$	−8118.31	−0.477709	−0.238855	−	0.971055i	$-0.576772\pi$
$661$	−8118.31	−0.477709	−0.238855	+	0.971055i	$0.576772\pi$
$662$	8252.92	0.484530
$663$	−14740.6	−0.863465
$664$	734.521	0.0429292
$665$	0	0
$666$	−6241.43	−0.363139
$667$	−294.403	−0.0170905
$668$	11654.6	0.675044
$669$	−7629.85	−0.440937
$670$	28542.1	1.64579
$671$	32942.0	1.89525
$672$	0	0
$673$	12717.0	0.728387	0.364193	−	0.931323i	$-0.381345\pi$
$673$	12717.0	0.728387	0.364193	+	0.931323i	$0.381345\pi$
$674$	5684.52	0.324866
$675$	−8292.06	−0.472832
$676$	12380.2	0.704382
$677$	−9158.89	−0.519948	−0.259974	−	0.965616i	$-0.583714\pi$
$677$	−9158.89	−0.519948	−0.259974	+	0.965616i	$0.583714\pi$
$678$	−1870.81	−0.105971
$679$	0	0
$680$	5539.19	0.312379
$681$	12657.5	0.712241
$682$	−3790.44	−0.212820
$683$	−26020.5	−1.45775	−0.728877	−	0.684645i	$-0.759957\pi$
$683$	−26020.5	−1.45775	−0.728877	+	0.684645i	$0.759957\pi$
$684$	5356.55	0.299434
$685$	−27294.3	−1.52243
$686$	0	0
$687$	25488.2	1.41548
$688$	2617.76	0.145060
$689$	55771.5	3.08378
$690$	2419.68	0.133501
$691$	−19424.2	−1.06936	−0.534682	−	0.845053i	$-0.679569\pi$
$691$	−19424.2	−1.06936	−0.534682	+	0.845053i	$0.679569\pi$
$692$	6868.98	0.377340
$693$	0	0
$694$	−15992.3	−0.874728
$695$	28851.6	1.57468
$696$	−401.771	−0.0218809
$697$	10570.4	0.574437
$698$	19168.2	1.03944
$699$	−11520.8	−0.623401
$700$	0	0
$701$	−4667.12	−0.251462	−0.125731	−	0.992064i	$-0.540128\pi$
$701$	−4667.12	−0.251462	−0.125731	+	0.992064i	$0.540128\pi$
$702$	22038.0	1.18486
$703$	−31024.3	−1.66444
$704$	−2955.33	−0.158215
$705$	−13654.8	−0.729461
$706$	22166.2	1.18164
$707$	0	0
$708$	−9177.90	−0.487184
$709$	7897.34	0.418323	0.209161	−	0.977881i	$-0.432927\pi$
$709$	7897.34	0.418323	0.209161	+	0.977881i	$0.432927\pi$
$710$	−17695.0	−0.935329
$711$	14984.9	0.790406
$712$	4261.30	0.224296
$713$	943.978	0.0495824
$714$	0	0
$715$	45036.5	2.35562
$716$	1370.21	0.0715185
$717$	21795.3	1.13523
$718$	−6049.13	−0.314417
$719$	26374.3	1.36801	0.684003	−	0.729479i	$-0.260237\pi$
$719$	26374.3	1.36801	0.684003	+	0.729479i	$0.260237\pi$
$720$	−2489.63	−0.128866
$721$	0	0
$722$	12907.8	0.665345
$723$	−17125.6	−0.880921
$724$	−9901.27	−0.508257
$725$	−700.724	−0.0358955
$726$	6287.94	0.321443
$727$	−22247.4	−1.13495	−0.567475	−	0.823390i	$-0.692080\pi$
$727$	−22247.4	−1.13495	−0.567475	+	0.823390i	$0.692080\pi$
$728$	0	0
$729$	19306.5	0.980874
$730$	17520.0	0.888277
$731$	8449.67	0.427527
$732$	−11195.9	−0.565316
$733$	−15509.2	−0.781506	−0.390753	−	0.920496i	$-0.627785\pi$
$733$	−15509.2	−0.781506	−0.390753	+	0.920496i	$0.627785\pi$
$734$	−3326.48	−0.167279
$735$	0	0
$736$	736.000	0.0368605
$737$	−49153.7	−2.45671
$738$	−4750.96	−0.236972
$739$	−10116.7	−0.503583	−0.251792	−	0.967781i	$-0.581020\pi$
$739$	−10116.7	−0.503583	−0.251792	+	0.967781i	$0.581020\pi$
$740$	14419.6	0.716316
$741$	32932.3	1.63266
$742$	0	0
$743$	14506.2	0.716261	0.358130	−	0.933672i	$-0.383414\pi$
$743$	14506.2	0.716261	0.358130	+	0.933672i	$0.383414\pi$
$744$	1288.24	0.0634801
$745$	45722.6	2.24852
$746$	2473.93	0.121417
$747$	−1065.62	−0.0521942
$748$	−9539.28	−0.466297
$749$	0	0
$750$	−7391.23	−0.359853
$751$	−22064.7	−1.07211	−0.536054	−	0.844184i	$-0.680085\pi$
$751$	−22064.7	−1.07211	−0.536054	+	0.844184i	$0.680085\pi$
$752$	−4153.42	−0.201409
$753$	10340.8	0.500449
$754$	1862.33	0.0899498
$755$	−19779.6	−0.953447
$756$	0	0
$757$	11951.9	0.573844	0.286922	−	0.957954i	$-0.407368\pi$
$757$	11951.9	0.573844	0.286922	+	0.957954i	$0.407368\pi$
$758$	−16901.4	−0.809874
$759$	−4167.04	−0.199280
$760$	−12375.2	−0.590653
$761$	20108.0	0.957839	0.478920	−	0.877859i	$-0.341029\pi$
$761$	20108.0	0.957839	0.478920	+	0.877859i	$0.341029\pi$
$762$	−20740.1	−0.986004
$763$	0	0
$764$	6864.21	0.325050
$765$	−8036.09	−0.379798
$766$	10675.5	0.503555
$767$	42542.4	2.00276
$768$	1004.42	0.0471923
$769$	−10292.2	−0.482635	−0.241317	−	0.970446i	$-0.577580\pi$
$769$	−10292.2	−0.482635	−0.241317	+	0.970446i	$0.577580\pi$
$770$	0	0
$771$	22794.2	1.06474
$772$	1189.26	0.0554436
$773$	16385.0	0.762391	0.381195	−	0.924495i	$-0.375513\pi$
$773$	16385.0	0.762391	0.381195	+	0.924495i	$0.375513\pi$
$774$	−3797.78	−0.176367
$775$	2246.81	0.104139
$776$	−3846.73	−0.177950
$777$	0	0
$778$	−28662.6	−1.32083
$779$	−23615.6	−1.08616
$780$	−15306.4	−0.702636
$781$	30473.4	1.39619
$782$	2375.68	0.108637
$783$	1938.85	0.0884917
$784$	0	0
$785$	24292.7	1.10451
$786$	176.831	0.00802460
$787$	−19333.1	−0.875669	−0.437835	−	0.899056i	$-0.644254\pi$
$787$	−19333.1	−0.875669	−0.437835	+	0.899056i	$0.644254\pi$
$788$	10346.1	0.467721
$789$	−4108.70	−0.185391
$790$	−34619.6	−1.55913
$791$	0	0
$792$	4287.51	0.192361
$793$	51896.3	2.32395
$794$	7244.14	0.323785
$795$	−40327.4	−1.79907
$796$	−2608.13	−0.116134
$797$	−39244.1	−1.74416	−0.872082	−	0.489360i	$-0.837230\pi$
$797$	−39244.1	−1.74416	−0.872082	+	0.489360i	$0.837230\pi$
$798$	0	0
$799$	−13406.5	−0.593602
$800$	1751.79	0.0774189
$801$	−6182.17	−0.272705
$802$	26022.3	1.14574
$803$	−30171.9	−1.32596
$804$	16705.6	0.732789
$805$	0	0
$806$	−5971.40	−0.260960
$807$	−11114.8	−0.484830
$808$	−347.244	−0.0151188
$809$	−2695.64	−0.117149	−0.0585747	−	0.998283i	$-0.518656\pi$
$809$	−2695.64	−0.117149	−0.0585747	+	0.998283i	$0.518656\pi$
$810$	−7532.78	−0.326759
$811$	41226.8	1.78504	0.892522	−	0.451005i	$-0.148934\pi$
$811$	41226.8	1.78504	0.892522	+	0.451005i	$0.148934\pi$
$812$	0	0
$813$	11589.2	0.499941
$814$	−24832.6	−1.06926
$815$	8534.05	0.366791
$816$	3242.07	0.139087
$817$	−18877.6	−0.808377
$818$	−8448.38	−0.361113
$819$	0	0
$820$	10976.1	0.467443
$821$	15286.8	0.649834	0.324917	−	0.945742i	$-0.394664\pi$
$821$	15286.8	0.649834	0.324917	+	0.945742i	$0.394664\pi$
$822$	−15975.3	−0.677862
$823$	−38784.6	−1.64270	−0.821352	−	0.570422i	$-0.806780\pi$
$823$	−38784.6	−1.64270	−0.821352	+	0.570422i	$0.806780\pi$
$824$	−11547.0	−0.488180
$825$	−9918.17	−0.418553
$826$	0	0
$827$	17195.8	0.723045	0.361523	−	0.932363i	$-0.382257\pi$
$827$	17195.8	0.723045	0.361523	+	0.932363i	$0.382257\pi$
$828$	−1067.77	−0.0448158
$829$	−40600.0	−1.70096	−0.850481	−	0.526007i	$-0.823689\pi$
$829$	−40600.0	−1.70096	−0.850481	+	0.526007i	$0.823689\pi$
$830$	2461.90	0.102957
$831$	−33437.5	−1.39583
$832$	−4655.78	−0.194002
$833$	0	0
$834$	16886.8	0.701128
$835$	39062.8	1.61895
$836$	21311.9	0.881684
$837$	−6216.76	−0.256729
$838$	14771.9	0.608933
$839$	1563.80	0.0643483	0.0321741	−	0.999482i	$-0.489757\pi$
$839$	1563.80	0.0643483	0.0321741	+	0.999482i	$0.489757\pi$
$840$	0	0
$841$	−24225.2	−0.993282
$842$	9964.72	0.407847
$843$	−19536.3	−0.798181
$844$	−1507.92	−0.0614987
$845$	41494.9	1.68931
$846$	6025.67	0.244878
$847$	0	0
$848$	−12266.5	−0.496736
$849$	−3399.46	−0.137419
$850$	5654.47	0.228172
$851$	6184.34	0.249114
$852$	−10356.9	−0.416456
$853$	−41988.4	−1.68541	−0.842705	−	0.538375i	$-0.819038\pi$
$853$	−41988.4	−1.68541	−0.842705	+	0.538375i	$0.819038\pi$
$854$	0	0
$855$	17953.6	0.718129
$856$	−8608.47	−0.343728
$857$	−16539.9	−0.659269	−0.329634	−	0.944109i	$-0.606925\pi$
$857$	−16539.9	−0.659269	−0.329634	+	0.944109i	$0.606925\pi$
$858$	26359.8	1.04884
$859$	−14022.0	−0.556955	−0.278477	−	0.960443i	$-0.589830\pi$
$859$	−14022.0	−0.556955	−0.278477	+	0.960443i	$0.589830\pi$
$860$	8773.99	0.347896
$861$	0	0
$862$	25010.8	0.988252
$863$	4713.59	0.185924	0.0929621	−	0.995670i	$-0.470366\pi$
$863$	4713.59	0.185924	0.0929621	+	0.995670i	$0.470366\pi$
$864$	−4847.08	−0.190858
$865$	23022.8	0.904971
$866$	−34672.6	−1.36054
$867$	−8811.31	−0.345153
$868$	0	0
$869$	59620.0	2.32735
$870$	−1346.62	−0.0524766
$871$	−77435.8	−3.01241
$872$	−4293.00	−0.166719
$873$	5580.72	0.216356
$874$	−5307.55	−0.205413
$875$	0	0
$876$	10254.4	0.395506
$877$	−15422.3	−0.593812	−0.296906	−	0.954907i	$-0.595955\pi$
$877$	−15422.3	−0.593812	−0.296906	+	0.954907i	$0.595955\pi$
$878$	28673.6	1.10215
$879$	17739.2	0.680692
$880$	−9905.41	−0.379445
$881$	−13646.3	−0.521855	−0.260928	−	0.965358i	$-0.584028\pi$
$881$	−13646.3	−0.521855	−0.260928	+	0.965358i	$0.584028\pi$
$882$	0	0
$883$	−7608.14	−0.289959	−0.144980	−	0.989435i	$-0.546312\pi$
$883$	−7608.14	−0.289959	−0.144980	+	0.989435i	$0.546312\pi$
$884$	−15028.0	−0.571772
$885$	−30761.7	−1.16841
$886$	−11712.0	−0.444098
$887$	4133.06	0.156454	0.0782270	−	0.996936i	$-0.475074\pi$
$887$	4133.06	0.156454	0.0782270	+	0.996936i	$0.475074\pi$
$888$	8439.73	0.318940
$889$	0	0
$890$	14282.7	0.537928
$891$	12972.5	0.487762
$892$	−7778.62	−0.291982
$893$	29951.8	1.12239
$894$	26761.3	1.00116
$895$	4592.56	0.171522
$896$	0	0
$897$	−6564.68	−0.244357
$898$	−929.893	−0.0345556
$899$	−525.350	−0.0194899
$900$	−2541.45	−0.0941277
$901$	−39594.0	−1.46400
$902$	−18902.5	−0.697764
$903$	0	0
$904$	−1907.29	−0.0701720
$905$	−33186.2	−1.21895
$906$	−11576.9	−0.424523
$907$	−44460.3	−1.62765	−0.813825	−	0.581110i	$-0.802619\pi$
$907$	−44460.3	−1.62765	−0.813825	+	0.581110i	$0.802619\pi$
$908$	12904.3	0.471634
$909$	503.771	0.0183818
$910$	0	0
$911$	6678.95	0.242902	0.121451	−	0.992597i	$-0.461245\pi$
$911$	6678.95	0.242902	0.121451	+	0.992597i	$0.461245\pi$
$912$	−7243.19	−0.262989
$913$	−4239.75	−0.153686
$914$	20408.7	0.738576
$915$	−37525.3	−1.35579
$916$	25985.2	0.937310
$917$	0	0
$918$	−15645.5	−0.562504
$919$	−37718.6	−1.35389	−0.676944	−	0.736035i	$-0.736696\pi$
$919$	−37718.6	−1.35389	−0.676944	+	0.736035i	$0.736696\pi$
$920$	2466.86	0.0884021
$921$	−9551.88	−0.341743
$922$	−12134.8	−0.433446
$923$	48007.3	1.71200
$924$	0	0
$925$	14719.7	0.523221
$926$	−1835.30	−0.0651315
$927$	16752.1	0.593540
$928$	−409.605	−0.0144892
$929$	20995.3	0.741478	0.370739	−	0.928737i	$-0.379104\pi$
$929$	20995.3	0.741478	0.370739	+	0.928737i	$0.379104\pi$
$930$	4317.81	0.152244
$931$	0	0
$932$	−11745.5	−0.412806
$933$	−28096.4	−0.985889
$934$	711.442	0.0249241
$935$	−31972.9	−1.11832
$936$	6754.47	0.235872
$937$	49138.2	1.71321	0.856603	−	0.515976i	$-0.172571\pi$
$937$	49138.2	1.71321	0.856603	+	0.515976i	$0.172571\pi$
$938$	0	0
$939$	6224.14	0.216312
$940$	−13921.1	−0.483038
$941$	1382.17	0.0478826	0.0239413	−	0.999713i	$-0.492379\pi$
$941$	1382.17	0.0478826	0.0239413	+	0.999713i	$0.492379\pi$
$942$	14218.4	0.491785
$943$	4707.50	0.162563
$944$	−9356.85	−0.322606
$945$	0	0
$946$	−15110.1	−0.519314
$947$	−27905.3	−0.957550	−0.478775	−	0.877938i	$-0.658919\pi$
$947$	−27905.3	−0.957550	−0.478775	+	0.877938i	$0.658919\pi$
$948$	−20262.8	−0.694203
$949$	−47532.3	−1.62588
$950$	−12632.8	−0.431433
$951$	18622.6	0.634995
$952$	0	0
$953$	7385.18	0.251028	0.125514	−	0.992092i	$-0.459942\pi$
$953$	7385.18	0.251028	0.125514	+	0.992092i	$0.459942\pi$
$954$	17795.8	0.603943
$955$	23006.9	0.779565
$956$	22220.3	0.751731
$957$	2319.07	0.0783333
$958$	−17292.3	−0.583181
$959$	0	0
$960$	3366.51	0.113181
$961$	−28106.5	−0.943457
$962$	−39120.8	−1.31113
$963$	12488.9	0.417913
$964$	−17459.5	−0.583332
$965$	3986.06	0.132970
$966$	0	0
$967$	24368.4	0.810377	0.405188	−	0.914233i	$-0.367206\pi$
$967$	24368.4	0.810377	0.405188	+	0.914233i	$0.367206\pi$
$968$	6410.55	0.212854
$969$	−23379.7	−0.775092
$970$	−12893.1	−0.426777
$971$	33043.9	1.09210	0.546050	−	0.837752i	$-0.316131\pi$
$971$	33043.9	1.09210	0.546050	+	0.837752i	$0.316131\pi$
$972$	11950.0	0.394337
$973$	0	0
$974$	−22369.4	−0.735896
$975$	−15624.9	−0.513229
$976$	−11414.2	−0.374343
$977$	59118.4	1.93589	0.967946	−	0.251158i	$-0.0808113\pi$
$977$	59118.4	1.93589	0.967946	+	0.251158i	$0.0808113\pi$
$978$	4994.96	0.163314
$979$	−24596.8	−0.802979
$980$	0	0
$981$	6228.16	0.202701
$982$	−1136.80	−0.0369418
$983$	16999.5	0.551576	0.275788	−	0.961218i	$-0.411061\pi$
$983$	16999.5	0.551576	0.275788	+	0.961218i	$0.411061\pi$
$984$	6424.30	0.208129
$985$	34677.1	1.12173
$986$	−1322.13	−0.0427031
$987$	0	0
$988$	33574.4	1.08112
$989$	3763.04	0.120988
$990$	14370.5	0.461337
$991$	30213.2	0.968471	0.484235	−	0.874938i	$-0.339098\pi$
$991$	30213.2	0.968471	0.484235	+	0.874938i	$0.339098\pi$
$992$	1313.36	0.0420355
$993$	16190.2	0.517401
$994$	0	0
$995$	−8741.70	−0.278523
$996$	1440.95	0.0458415
$997$	31255.0	0.992835	0.496417	−	0.868084i	$-0.334649\pi$
$997$	31255.0	0.992835	0.496417	+	0.868084i	$0.334649\pi$
$998$	12580.7	0.399033
$999$	−40728.2	−1.28987

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.n.1.5	yes	6
7.6	odd	2	inner	2254.4.a.n.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.n.1.2	✓	6	7.6	odd	2	inner
2254.4.a.n.1.5	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.92350 q^{3} +4.00000 q^{4} +13.4068 q^{5} +7.84699 q^{6} +8.00000 q^{8} -11.6062 q^{9} +26.8137 q^{10} -46.1770 q^{11} +15.6940 q^{12} -72.7465 q^{13} +52.6017 q^{15} +16.0000 q^{16} +51.6451 q^{17} -23.2123 q^{18} -115.382 q^{19} +53.6274 q^{20} -92.3541 q^{22} +23.0000 q^{23} +31.3880 q^{24} +54.7435 q^{25} -145.493 q^{26} -151.471 q^{27} -12.8001 q^{29} +105.203 q^{30} +41.0425 q^{31} +32.0000 q^{32} -181.175 q^{33} +103.290 q^{34} -46.4247 q^{36} +268.884 q^{37} -230.763 q^{38} -285.421 q^{39} +107.255 q^{40} +204.674 q^{41} +163.610 q^{43} -184.708 q^{44} -155.602 q^{45} +46.0000 q^{46} -259.589 q^{47} +62.7760 q^{48} +109.487 q^{50} +202.630 q^{51} -290.986 q^{52} -766.655 q^{53} -302.942 q^{54} -619.088 q^{55} -452.699 q^{57} -25.6003 q^{58} -584.803 q^{59} +210.407 q^{60} -713.386 q^{61} +82.0850 q^{62} +64.0000 q^{64} -975.301 q^{65} -362.351 q^{66} +1064.46 q^{67} +206.581 q^{68} +90.2404 q^{69} -659.926 q^{71} -92.8494 q^{72} +653.396 q^{73} +537.769 q^{74} +214.786 q^{75} -461.526 q^{76} -570.842 q^{78} -1291.12 q^{79} +214.509 q^{80} -280.930 q^{81} +409.348 q^{82} +91.8152 q^{83} +692.398 q^{85} +327.221 q^{86} -50.2214 q^{87} -369.416 q^{88} +532.663 q^{89} -311.204 q^{90} +92.0000 q^{92} +161.030 q^{93} -519.178 q^{94} -1546.90 q^{95} +125.552 q^{96} -480.841 q^{97} +535.939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 24 q^{4} + 48 q^{8} - 52 q^{9} - 84 q^{11} + 52 q^{15} + 96 q^{16} - 104 q^{18} - 168 q^{22} + 138 q^{23} - 286 q^{25} - 22 q^{29} + 104 q^{30} + 192 q^{32} - 208 q^{36} - 180 q^{37} - 414 q^{39}+ \cdots + 600 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 24 * q^4 + 48 * q^8 - 52 * q^9 - 84 * q^11 + 52 * q^15 + 96 * q^16 - 104 * q^18 - 168 * q^22 + 138 * q^23 - 286 * q^25 - 22 * q^29 + 104 * q^30 + 192 * q^32 - 208 * q^36 - 180 * q^37 - 414 * q^39 - 600 * q^43 - 336 * q^44 + 276 * q^46 - 572 * q^50 + 464 * q^51 - 876 * q^53 - 1452 * q^57 - 44 * q^58 + 208 * q^60 + 384 * q^64 - 1532 * q^65 - 92 * q^67 - 1434 * q^71 - 416 * q^72 - 360 * q^74 - 828 * q^78 - 2820 * q^79 - 2198 * q^81 - 184 * q^85 - 1200 * q^86 - 672 * q^88 + 552 * q^92 - 2742 * q^93 - 2888 * q^95 + 600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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