LMFDB - Embedded newform 1225.4.a.bj.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1163891200.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 $ x^6 - 2x^5 - 23x^4 + 12x^3 + 154x^2 + 152*x + 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.04490$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.369315 q^{2} +9.74070 q^{3} -7.86361 q^{4} +3.59738 q^{6} -5.85867 q^{8} +67.8812 q^{9} +O(q^{10})$	\(q+0.369315 q^{2} +9.74070 q^{3} -7.86361 q^{4} +3.59738 q^{6} -5.85867 q^{8} +67.8812 q^{9} +30.6361 q^{11} -76.5970 q^{12} -36.4622 q^{13} +60.7452 q^{16} +79.7341 q^{17} +25.0695 q^{18} -152.418 q^{19} +11.3144 q^{22} -22.2207 q^{23} -57.0675 q^{24} -13.4660 q^{26} +398.211 q^{27} +101.285 q^{29} +249.956 q^{31} +69.3034 q^{32} +298.417 q^{33} +29.4470 q^{34} -533.791 q^{36} -7.55765 q^{37} -56.2904 q^{38} -355.168 q^{39} +142.280 q^{41} +237.530 q^{43} -240.910 q^{44} -8.20644 q^{46} +331.129 q^{47} +591.700 q^{48} +776.666 q^{51} +286.725 q^{52} +487.337 q^{53} +147.065 q^{54} -1484.66 q^{57} +37.4059 q^{58} -717.355 q^{59} +354.592 q^{61} +92.3126 q^{62} -460.367 q^{64} +110.210 q^{66} -57.5883 q^{67} -626.997 q^{68} -216.445 q^{69} -696.174 q^{71} -397.693 q^{72} +261.035 q^{73} -2.79115 q^{74} +1198.56 q^{76} -131.169 q^{78} +271.344 q^{79} +2046.06 q^{81} +52.5460 q^{82} -681.441 q^{83} +87.7234 q^{86} +986.584 q^{87} -179.487 q^{88} +160.668 q^{89} +174.735 q^{92} +2434.75 q^{93} +122.291 q^{94} +675.063 q^{96} +167.841 q^{97} +2079.61 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 16 q^{3} + 14 q^{4} - 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 + 16 * q^3 + 14 * q^4 - 24 * q^6 + 66 * q^8 + 70 * q^9	$ 6 q + 2 q^{2} + 16 q^{3} + 14 q^{4} - 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} + 160 q^{12} + 168 q^{13} + 298 q^{16} - 4 q^{17} - 354 q^{18} - 308 q^{19} + 236 q^{22} + 336 q^{23} + 92 q^{24} - 56 q^{26} + 964 q^{27} + 176 q^{29} - 392 q^{31} + 770 q^{32} + 188 q^{33} - 812 q^{34} + 230 q^{36} + 140 q^{37} + 20 q^{38} + 140 q^{39} - 656 q^{41} + 388 q^{43} - 160 q^{44} - 388 q^{46} + 628 q^{47} + 1396 q^{48} + 744 q^{51} + 1520 q^{52} + 676 q^{53} - 2284 q^{54} - 1468 q^{57} + 2012 q^{58} - 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 3620 q^{66} - 1768 q^{67} - 2940 q^{68} + 1048 q^{69} - 224 q^{71} - 2858 q^{72} + 2640 q^{73} + 928 q^{74} + 1340 q^{76} - 8 q^{78} + 1636 q^{79} + 4442 q^{81} - 1756 q^{82} + 140 q^{83} + 1180 q^{86} + 1940 q^{87} + 5652 q^{88} + 1904 q^{89} + 1952 q^{92} + 1592 q^{93} + 3332 q^{94} + 6460 q^{96} + 516 q^{97} - 2804 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + 16 * q^3 + 14 * q^4 - 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 + 160 * q^12 + 168 * q^13 + 298 * q^16 - 4 * q^17 - 354 * q^18 - 308 * q^19 + 236 * q^22 + 336 * q^23 + 92 * q^24 - 56 * q^26 + 964 * q^27 + 176 * q^29 - 392 * q^31 + 770 * q^32 + 188 * q^33 - 812 * q^34 + 230 * q^36 + 140 * q^37 + 20 * q^38 + 140 * q^39 - 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 + 628 * q^47 + 1396 * q^48 + 744 * q^51 + 1520 * q^52 + 676 * q^53 - 2284 * q^54 - 1468 * q^57 + 2012 * q^58 - 996 * q^59 - 740 * q^61 - 364 * q^62 + 1426 * q^64 + 3620 * q^66 - 1768 * q^67 - 2940 * q^68 + 1048 * q^69 - 224 * q^71 - 2858 * q^72 + 2640 * q^73 + 928 * q^74 + 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 - 1756 * q^82 + 140 * q^83 + 1180 * q^86 + 1940 * q^87 + 5652 * q^88 + 1904 * q^89 + 1952 * q^92 + 1592 * q^93 + 3332 * q^94 + 6460 * q^96 + 516 * q^97 - 2804 * q^99

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 $

$2$	0.369315	0.130573	0.0652863	−	0.997867i	$-0.479204\pi$
$2$	0.369315	0.130573	0.0652863	+	0.997867i	$0.479204\pi$
$3$	9.74070	1.87460	0.937299	−	0.348526i	$-0.113318\pi$
$3$	9.74070	1.87460	0.937299	+	0.348526i	$0.113318\pi$
$4$	−7.86361	−0.982951
$5$	0	0
$5$	0	0
$6$	3.59738	0.244771
$7$	0	0
$7$	0	0
$8$	−5.85867	−0.258919
$9$	67.8812	2.51412
$10$	0	0
$11$	30.6361	0.839739	0.419870	−	0.907584i	$-0.362076\pi$
$11$	30.6361	0.839739	0.419870	+	0.907584i	$0.362076\pi$
$12$	−76.5970	−1.84264
$13$	−36.4622	−0.777908	−0.388954	−	0.921257i	$-0.627164\pi$
$13$	−36.4622	−0.777908	−0.388954	+	0.921257i	$0.627164\pi$
$14$	0	0
$15$	0	0
$16$	60.7452	0.949143
$17$	79.7341	1.13755	0.568775	−	0.822493i	$-0.307417\pi$
$17$	79.7341	1.13755	0.568775	+	0.822493i	$0.307417\pi$
$18$	25.0695	0.328275
$19$	−152.418	−1.84038	−0.920189	−	0.391474i	$-0.871965\pi$
$19$	−152.418	−1.84038	−0.920189	+	0.391474i	$0.871965\pi$
$20$	0	0
$21$	0	0
$22$	11.3144	0.109647
$23$	−22.2207	−0.201450	−0.100725	−	0.994914i	$-0.532116\pi$
$23$	−22.2207	−0.201450	−0.100725	+	0.994914i	$0.532116\pi$
$24$	−57.0675	−0.485369
$25$	0	0
$26$	−13.4660	−0.101573
$27$	398.211	2.83836
$28$	0	0
$29$	101.285	0.648555	0.324278	−	0.945962i	$-0.394879\pi$
$29$	101.285	0.648555	0.324278	+	0.945962i	$0.394879\pi$
$30$	0	0
$31$	249.956	1.44818	0.724089	−	0.689707i	$-0.242261\pi$
$31$	249.956	1.44818	0.724089	+	0.689707i	$0.242261\pi$
$32$	69.3034	0.382851
$33$	298.417	1.57417
$34$	29.4470	0.148533
$35$	0	0
$36$	−533.791	−2.47125
$37$	−7.55765	−0.0335803	−0.0167901	−	0.999859i	$-0.505345\pi$
$37$	−7.55765	−0.0335803	−0.0167901	+	0.999859i	$0.505345\pi$
$38$	−56.2904	−0.240303
$39$	−355.168	−1.45826
$40$	0	0
$41$	142.280	0.541960	0.270980	−	0.962585i	$-0.412652\pi$
$41$	142.280	0.541960	0.270980	+	0.962585i	$0.412652\pi$
$42$	0	0
$43$	237.530	0.842396	0.421198	−	0.906969i	$-0.361610\pi$
$43$	237.530	0.842396	0.421198	+	0.906969i	$0.361610\pi$
$44$	−240.910	−0.825422
$45$	0	0
$46$	−8.20644	−0.0263038
$47$	331.129	1.02766	0.513830	−	0.857892i	$-0.328226\pi$
$47$	331.129	1.02766	0.513830	+	0.857892i	$0.328226\pi$
$48$	591.700	1.77926
$49$	0	0
$50$	0	0
$51$	776.666	2.13245
$52$	286.725	0.764645
$53$	487.337	1.26303	0.631517	−	0.775362i	$-0.282432\pi$
$53$	487.337	1.26303	0.631517	+	0.775362i	$0.282432\pi$
$54$	147.065	0.370612
$55$	0	0
$56$	0	0
$57$	−1484.66	−3.44997
$58$	37.4059	0.0846835
$59$	−717.355	−1.58291	−0.791454	−	0.611228i	$-0.790676\pi$
$59$	−717.355	−1.58291	−0.791454	+	0.611228i	$0.790676\pi$
$60$	0	0
$61$	354.592	0.744276	0.372138	−	0.928177i	$-0.378625\pi$
$61$	354.592	0.744276	0.372138	+	0.928177i	$0.378625\pi$
$62$	92.3126	0.189092
$63$	0	0
$64$	−460.367	−0.899153
$65$	0	0
$66$	110.210	0.205544
$67$	−57.5883	−0.105008	−0.0525040	−	0.998621i	$-0.516720\pi$
$67$	−57.5883	−0.105008	−0.0525040	+	0.998621i	$0.516720\pi$
$68$	−626.997	−1.11816
$69$	−216.445	−0.377637
$70$	0	0
$71$	−696.174	−1.16367	−0.581836	−	0.813306i	$-0.697665\pi$
$71$	−696.174	−1.16367	−0.581836	+	0.813306i	$0.697665\pi$
$72$	−397.693	−0.650952
$73$	261.035	0.418518	0.209259	−	0.977860i	$-0.432895\pi$
$73$	261.035	0.418518	0.209259	+	0.977860i	$0.432895\pi$
$74$	−2.79115	−0.00438466
$75$	0	0
$76$	1198.56	1.80900
$77$	0	0
$78$	−131.169	−0.190409
$79$	271.344	0.386438	0.193219	−	0.981156i	$-0.438107\pi$
$79$	271.344	0.386438	0.193219	+	0.981156i	$0.438107\pi$
$80$	0	0
$81$	2046.06	2.80667
$82$	52.5460	0.0707651
$83$	−681.441	−0.901179	−0.450590	−	0.892731i	$-0.648786\pi$
$83$	−681.441	−0.901179	−0.450590	+	0.892731i	$0.648786\pi$
$84$	0	0
$85$	0	0
$86$	87.7234	0.109994
$87$	986.584	1.21578
$88$	−179.487	−0.217424
$89$	160.668	0.191357	0.0956784	−	0.995412i	$-0.469498\pi$
$89$	160.668	0.191357	0.0956784	+	0.995412i	$0.469498\pi$
$90$	0	0
$91$	0	0
$92$	174.735	0.198015
$93$	2434.75	2.71475
$94$	122.291	0.134184
$95$	0	0
$96$	675.063	0.717691
$97$	167.841	0.175687	0.0878436	−	0.996134i	$-0.472002\pi$
$97$	167.841	0.175687	0.0878436	+	0.996134i	$0.472002\pi$
$98$	0	0
$99$	2079.61	2.11120
$100$	0	0
$101$	−413.394	−0.407270	−0.203635	−	0.979047i	$-0.565276\pi$
$101$	−413.394	−0.407270	−0.203635	+	0.979047i	$0.565276\pi$
$102$	286.834	0.278439
$103$	1451.11	1.38817	0.694086	−	0.719892i	$-0.255809\pi$
$103$	1451.11	1.38817	0.694086	+	0.719892i	$0.255809\pi$
$104$	213.620	0.201415
$105$	0	0
$106$	179.981	0.164918
$107$	1780.63	1.60878	0.804391	−	0.594100i	$-0.202492\pi$
$107$	1780.63	1.60878	0.804391	+	0.594100i	$0.202492\pi$
$108$	−3131.38	−2.78997
$109$	37.8920	0.0332972	0.0166486	−	0.999861i	$-0.494700\pi$
$109$	37.8920	0.0332972	0.0166486	+	0.999861i	$0.494700\pi$
$110$	0	0
$111$	−73.6168	−0.0629495
$112$	0	0
$113$	−457.320	−0.380717	−0.190359	−	0.981715i	$-0.560965\pi$
$113$	−457.320	−0.380717	−0.190359	+	0.981715i	$0.560965\pi$
$114$	−548.308	−0.450471
$115$	0	0
$116$	−796.463	−0.637498
$117$	−2475.10	−1.95575
$118$	−264.930	−0.206684
$119$	0	0
$120$	0	0
$121$	−392.430	−0.294838
$122$	130.956	0.0971820
$123$	1385.90	1.01596
$124$	−1965.56	−1.42349
$125$	0	0
$126$	0	0
$127$	2545.41	1.77849	0.889246	−	0.457429i	$-0.151229\pi$
$127$	2545.41	1.77849	0.889246	+	0.457429i	$0.151229\pi$
$128$	−724.447	−0.500256
$129$	2313.71	1.57915
$130$	0	0
$131$	970.873	0.647523	0.323762	−	0.946139i	$-0.395052\pi$
$131$	970.873	0.647523	0.323762	+	0.946139i	$0.395052\pi$
$132$	−2346.63	−1.54733
$133$	0	0
$134$	−21.2682	−0.0137112
$135$	0	0
$136$	−467.135	−0.294533
$137$	−183.635	−0.114518	−0.0572592	−	0.998359i	$-0.518236\pi$
$137$	−183.635	−0.114518	−0.0572592	+	0.998359i	$0.518236\pi$
$138$	−79.9365	−0.0493090
$139$	1078.90	0.658356	0.329178	−	0.944268i	$-0.393228\pi$
$139$	1078.90	0.658356	0.329178	+	0.944268i	$0.393228\pi$
$140$	0	0
$141$	3225.42	1.92645
$142$	−257.108	−0.151944
$143$	−1117.06	−0.653240
$144$	4123.45	2.38626
$145$	0	0
$146$	96.4040	0.0546469
$147$	0	0
$148$	59.4304	0.0330077
$149$	1211.24	0.665962	0.332981	−	0.942934i	$-0.391946\pi$
$149$	1211.24	0.665962	0.332981	+	0.942934i	$0.391946\pi$
$150$	0	0
$151$	−602.244	−0.324569	−0.162285	−	0.986744i	$-0.551886\pi$
$151$	−602.244	−0.324569	−0.162285	+	0.986744i	$0.551886\pi$
$152$	892.969	0.476509
$153$	5412.44	2.85994
$154$	0	0
$155$	0	0
$156$	2792.90	1.43340
$157$	1602.07	0.814389	0.407195	−	0.913341i	$-0.366507\pi$
$157$	1602.07	0.814389	0.407195	+	0.913341i	$0.366507\pi$
$158$	100.212	0.0504582
$159$	4747.00	2.36768
$160$	0	0
$161$	0	0
$162$	755.641	0.366474
$163$	3091.61	1.48561	0.742803	−	0.669510i	$-0.233496\pi$
$163$	3091.61	1.48561	0.742803	+	0.669510i	$0.233496\pi$
$164$	−1118.83	−0.532720
$165$	0	0
$166$	−251.666	−0.117669
$167$	−3251.19	−1.50649	−0.753247	−	0.657738i	$-0.771513\pi$
$167$	−3251.19	−1.50649	−0.753247	+	0.657738i	$0.771513\pi$
$168$	0	0
$169$	−867.505	−0.394859
$170$	0	0
$171$	−10346.3	−4.62693
$172$	−1867.84	−0.828034
$173$	2254.55	0.990809	0.495405	−	0.868662i	$-0.335020\pi$
$173$	2254.55	0.990809	0.495405	+	0.868662i	$0.335020\pi$
$174$	364.360	0.158747
$175$	0	0
$176$	1860.99	0.797033
$177$	−6987.53	−2.96732
$178$	59.3370	0.0249859
$179$	−2685.46	−1.12135	−0.560673	−	0.828038i	$-0.689457\pi$
$179$	−2685.46	−1.12135	−0.560673	+	0.828038i	$0.689457\pi$
$180$	0	0
$181$	−1293.25	−0.531087	−0.265543	−	0.964099i	$-0.585551\pi$
$181$	−1293.25	−0.531087	−0.265543	+	0.964099i	$0.585551\pi$
$182$	0	0
$183$	3453.97	1.39522
$184$	130.184	0.0521591
$185$	0	0
$186$	899.189	0.354472
$187$	2442.74	0.955245
$188$	−2603.86	−1.01014
$189$	0	0
$190$	0	0
$191$	−139.878	−0.0529905	−0.0264953	−	0.999649i	$-0.508435\pi$
$191$	−139.878	−0.0529905	−0.0264953	+	0.999649i	$0.508435\pi$
$192$	−4484.29	−1.68555
$193$	−2384.81	−0.889442	−0.444721	−	0.895669i	$-0.646697\pi$
$193$	−2384.81	−0.889442	−0.444721	+	0.895669i	$0.646697\pi$
$194$	61.9861	0.0229399
$195$	0	0
$196$	0	0
$197$	1008.67	0.364797	0.182399	−	0.983225i	$-0.441614\pi$
$197$	1008.67	0.364797	0.182399	+	0.983225i	$0.441614\pi$
$198$	768.032	0.275665
$199$	−995.036	−0.354453	−0.177227	−	0.984170i	$-0.556713\pi$
$199$	−995.036	−0.354453	−0.177227	+	0.984170i	$0.556713\pi$
$200$	0	0
$201$	−560.950	−0.196848
$202$	−152.672	−0.0531782
$203$	0	0
$204$	−6107.39	−2.09609
$205$	0	0
$206$	535.915	0.181257
$207$	−1508.37	−0.506468
$208$	−2214.90	−0.738346
$209$	−4669.51	−1.54544
$210$	0	0
$211$	2307.30	0.752802	0.376401	−	0.926457i	$-0.377161\pi$
$211$	2307.30	0.752802	0.376401	+	0.926457i	$0.377161\pi$
$212$	−3832.22	−1.24150
$213$	−6781.22	−2.18142
$214$	657.612	0.210063
$215$	0	0
$216$	−2332.99	−0.734905
$217$	0	0
$218$	13.9941	0.00434770
$219$	2542.66	0.784552
$220$	0	0
$221$	−2907.28	−0.884910
$222$	−27.1878	−0.00821947
$223$	1629.00	0.489173	0.244587	−	0.969627i	$-0.421348\pi$
$223$	1629.00	0.489173	0.244587	+	0.969627i	$0.421348\pi$
$224$	0	0
$225$	0	0
$226$	−168.895	−0.0497112
$227$	−2410.92	−0.704926	−0.352463	−	0.935826i	$-0.614656\pi$
$227$	−2410.92	−0.704926	−0.352463	+	0.935826i	$0.614656\pi$
$228$	11674.8	3.39115
$229$	−3666.59	−1.05806	−0.529028	−	0.848604i	$-0.677443\pi$
$229$	−3666.59	−1.05806	−0.529028	+	0.848604i	$0.677443\pi$
$230$	0	0
$231$	0	0
$232$	−593.393	−0.167923
$233$	3018.33	0.848658	0.424329	−	0.905508i	$-0.360510\pi$
$233$	3018.33	0.848658	0.424329	+	0.905508i	$0.360510\pi$
$234$	−914.091	−0.255367
$235$	0	0
$236$	5640.99	1.55592
$237$	2643.08	0.724417
$238$	0	0
$239$	−546.873	−0.148010	−0.0740048	−	0.997258i	$-0.523578\pi$
$239$	−546.873	−0.148010	−0.0740048	+	0.997258i	$0.523578\pi$
$240$	0	0
$241$	−3135.27	−0.838010	−0.419005	−	0.907984i	$-0.637621\pi$
$241$	−3135.27	−0.838010	−0.419005	+	0.907984i	$0.637621\pi$
$242$	−144.930	−0.0384978
$243$	9178.37	2.42302
$244$	−2788.37	−0.731587
$245$	0	0
$246$	511.835	0.132656
$247$	5557.52	1.43165
$248$	−1464.41	−0.374960
$249$	−6637.71	−1.68935
$250$	0	0
$251$	914.967	0.230088	0.115044	−	0.993360i	$-0.463299\pi$
$251$	914.967	0.230088	0.115044	+	0.993360i	$0.463299\pi$
$252$	0	0
$253$	−680.756	−0.169165
$254$	940.058	0.232222
$255$	0	0
$256$	3415.38	0.833834
$257$	−6334.18	−1.53741	−0.768707	−	0.639602i	$-0.779099\pi$
$257$	−6334.18	−1.53741	−0.768707	+	0.639602i	$0.779099\pi$
$258$	854.487	0.206194
$259$	0	0
$260$	0	0
$261$	6875.33	1.63054
$262$	358.558	0.0845487
$263$	−4126.75	−0.967553	−0.483777	−	0.875191i	$-0.660735\pi$
$263$	−4126.75	−0.967553	−0.483777	+	0.875191i	$0.660735\pi$
$264$	−1748.32	−0.407583
$265$	0	0
$266$	0	0
$267$	1565.02	0.358717
$268$	452.852	0.103218
$269$	5887.07	1.33435	0.667176	−	0.744900i	$-0.267503\pi$
$269$	5887.07	1.33435	0.667176	+	0.744900i	$0.267503\pi$
$270$	0	0
$271$	−7237.72	−1.62236	−0.811181	−	0.584795i	$-0.801175\pi$
$271$	−7237.72	−1.62236	−0.811181	+	0.584795i	$0.801175\pi$
$272$	4843.46	1.07970
$273$	0	0
$274$	−67.8192	−0.0149530
$275$	0	0
$276$	1702.04	0.371199
$277$	−5640.17	−1.22341	−0.611705	−	0.791086i	$-0.709516\pi$
$277$	−5640.17	−1.22341	−0.611705	+	0.791086i	$0.709516\pi$
$278$	398.456	0.0859632
$279$	16967.3	3.64089
$280$	0	0
$281$	−2593.05	−0.550492	−0.275246	−	0.961374i	$-0.588759\pi$
$281$	−2593.05	−0.550492	−0.275246	+	0.961374i	$0.588759\pi$
$282$	1191.20	0.251542
$283$	2533.26	0.532109	0.266054	−	0.963958i	$-0.414280\pi$
$283$	2533.26	0.532109	0.266054	+	0.963958i	$0.414280\pi$
$284$	5474.44	1.14383
$285$	0	0
$286$	−412.547	−0.0852952
$287$	0	0
$288$	4704.40	0.962532
$289$	1444.52	0.294021
$290$	0	0
$291$	1634.89	0.329343
$292$	−2052.67	−0.411382
$293$	−589.215	−0.117482	−0.0587411	−	0.998273i	$-0.518709\pi$
$293$	−589.215	−0.117482	−0.0587411	+	0.998273i	$0.518709\pi$
$294$	0	0
$295$	0	0
$296$	44.2777	0.00869456
$297$	12199.6	2.38348
$298$	447.327	0.0869563
$299$	810.217	0.156709
$300$	0	0
$301$	0	0
$302$	−222.418	−0.0423798
$303$	−4026.74	−0.763467
$304$	−9258.68	−1.74678
$305$	0	0
$306$	1998.90	0.373429
$307$	5273.75	0.980420	0.490210	−	0.871604i	$-0.336920\pi$
$307$	5273.75	0.980420	0.490210	+	0.871604i	$0.336920\pi$
$308$	0	0
$309$	14134.8	2.60226
$310$	0	0
$311$	3295.08	0.600795	0.300397	−	0.953814i	$-0.402881\pi$
$311$	3295.08	0.600795	0.300397	+	0.953814i	$0.402881\pi$
$312$	2080.81	0.377572
$313$	−1819.37	−0.328553	−0.164276	−	0.986414i	$-0.552529\pi$
$313$	−1819.37	−0.328553	−0.164276	+	0.986414i	$0.552529\pi$
$314$	591.668	0.106337
$315$	0	0
$316$	−2133.75	−0.379850
$317$	−1887.91	−0.334496	−0.167248	−	0.985915i	$-0.553488\pi$
$317$	−1887.91	−0.334496	−0.167248	+	0.985915i	$0.553488\pi$
$318$	1753.14	0.309154
$319$	3102.97	0.544617
$320$	0	0
$321$	17344.6	3.01582
$322$	0	0
$323$	−12152.9	−2.09352
$324$	−16089.4	−2.75882
$325$	0	0
$326$	1141.78	0.193979
$327$	369.094	0.0624188
$328$	−833.569	−0.140324
$329$	0	0
$330$	0	0
$331$	−456.387	−0.0757863	−0.0378932	−	0.999282i	$-0.512065\pi$
$331$	−456.387	−0.0757863	−0.0378932	+	0.999282i	$0.512065\pi$
$332$	5358.58	0.885815
$333$	−513.022	−0.0844247
$334$	−1200.71	−0.196707
$335$	0	0
$336$	0	0
$337$	−8174.42	−1.32133	−0.660666	−	0.750680i	$-0.729726\pi$
$337$	−8174.42	−1.32133	−0.660666	+	0.750680i	$0.729726\pi$
$338$	−320.383	−0.0515577
$339$	−4454.61	−0.713691
$340$	0	0
$341$	7657.69	1.21609
$342$	−3821.06	−0.604150
$343$	0	0
$344$	−1391.61	−0.218112
$345$	0	0
$346$	832.637	0.129372
$347$	7935.70	1.22770	0.613849	−	0.789424i	$-0.289621\pi$
$347$	7935.70	1.22770	0.613849	+	0.789424i	$0.289621\pi$
$348$	−7758.11	−1.19505
$349$	−8649.40	−1.32662	−0.663312	−	0.748343i	$-0.730850\pi$
$349$	−8649.40	−1.32662	−0.663312	+	0.748343i	$0.730850\pi$
$350$	0	0
$351$	−14519.7	−2.20798
$352$	2123.19	0.321495
$353$	−4931.06	−0.743495	−0.371747	−	0.928334i	$-0.621241\pi$
$353$	−4931.06	−0.743495	−0.371747	+	0.928334i	$0.621241\pi$
$354$	−2580.60	−0.387450
$355$	0	0
$356$	−1263.43	−0.188094
$357$	0	0
$358$	−991.781	−0.146417
$359$	8243.64	1.21193	0.605965	−	0.795491i	$-0.292787\pi$
$359$	8243.64	1.21193	0.605965	+	0.795491i	$0.292787\pi$
$360$	0	0
$361$	16372.4	2.38699
$362$	−477.617	−0.0693453
$363$	−3822.54	−0.552703
$364$	0	0
$365$	0	0
$366$	1275.60	0.182177
$367$	−8179.11	−1.16334	−0.581670	−	0.813425i	$-0.697601\pi$
$367$	−8179.11	−1.16334	−0.581670	+	0.813425i	$0.697601\pi$
$368$	−1349.80	−0.191204
$369$	9658.12	1.36255
$370$	0	0
$371$	0	0
$372$	−19145.9	−2.66847
$373$	12795.5	1.77620	0.888102	−	0.459646i	$-0.152024\pi$
$373$	12795.5	1.77620	0.888102	+	0.459646i	$0.152024\pi$
$374$	902.140	0.124729
$375$	0	0
$376$	−1939.97	−0.266081
$377$	−3693.07	−0.504516
$378$	0	0
$379$	−5735.40	−0.777329	−0.388664	−	0.921379i	$-0.627063\pi$
$379$	−5735.40	−0.777329	−0.388664	+	0.921379i	$0.627063\pi$
$380$	0	0
$381$	24794.1	3.33396
$382$	−51.6589	−0.00691910
$383$	737.841	0.0984384	0.0492192	−	0.998788i	$-0.484327\pi$
$383$	737.841	0.0984384	0.0492192	+	0.998788i	$0.484327\pi$
$384$	−7056.62	−0.937778
$385$	0	0
$386$	−880.745	−0.116137
$387$	16123.8	2.11788
$388$	−1319.83	−0.172692
$389$	−13034.0	−1.69884	−0.849419	−	0.527718i	$-0.823048\pi$
$389$	−13034.0	−1.69884	−0.849419	+	0.527718i	$0.823048\pi$
$390$	0	0
$391$	−1771.75	−0.229159
$392$	0	0
$393$	9456.98	1.21385
$394$	372.518	0.0476325
$395$	0	0
$396$	−16353.3	−2.07521
$397$	8313.64	1.05101	0.525503	−	0.850792i	$-0.323877\pi$
$397$	8313.64	1.05101	0.525503	+	0.850792i	$0.323877\pi$
$398$	−367.481	−0.0462819
$399$	0	0
$400$	0	0
$401$	−14341.4	−1.78598	−0.892988	−	0.450080i	$-0.851396\pi$
$401$	−14341.4	−1.78598	−0.892988	+	0.450080i	$0.851396\pi$
$402$	−207.167	−0.0257029
$403$	−9113.97	−1.12655
$404$	3250.77	0.400326
$405$	0	0
$406$	0	0
$407$	−231.537	−0.0281987
$408$	−4550.22	−0.552131
$409$	6282.35	0.759517	0.379758	−	0.925086i	$-0.376007\pi$
$409$	6282.35	0.759517	0.379758	+	0.925086i	$0.376007\pi$
$410$	0	0
$411$	−1788.74	−0.214676
$412$	−11410.9	−1.36450
$413$	0	0
$414$	−557.063	−0.0661308
$415$	0	0
$416$	−2526.96	−0.297823
$417$	10509.3	1.23415
$418$	−1724.52	−0.201792
$419$	−15226.1	−1.77528	−0.887640	−	0.460538i	$-0.847656\pi$
$419$	−15226.1	−1.77528	−0.887640	+	0.460538i	$0.847656\pi$
$420$	0	0
$421$	−2026.55	−0.234603	−0.117302	−	0.993096i	$-0.537424\pi$
$421$	−2026.55	−0.234603	−0.117302	+	0.993096i	$0.537424\pi$
$422$	852.121	0.0982953
$423$	22477.4	2.58366
$424$	−2855.14	−0.327024
$425$	0	0
$426$	−2504.41	−0.284833
$427$	0	0
$428$	−14002.2	−1.58135
$429$	−10880.9	−1.22456
$430$	0	0
$431$	8983.56	1.00400	0.501999	−	0.864868i	$-0.332598\pi$
$431$	8983.56	1.00400	0.501999	+	0.864868i	$0.332598\pi$
$432$	24189.4	2.69401
$433$	−6836.59	−0.758766	−0.379383	−	0.925240i	$-0.623864\pi$
$433$	−6836.59	−0.758766	−0.379383	+	0.925240i	$0.623864\pi$
$434$	0	0
$435$	0	0
$436$	−297.968	−0.0327295
$437$	3386.85	0.370743
$438$	939.042	0.102441
$439$	−7802.96	−0.848325	−0.424163	−	0.905586i	$-0.639432\pi$
$439$	−7802.96	−0.848325	−0.424163	+	0.905586i	$0.639432\pi$
$440$	0	0
$441$	0	0
$442$	−1073.70	−0.115545
$443$	−5184.37	−0.556020	−0.278010	−	0.960578i	$-0.589675\pi$
$443$	−5184.37	−0.556020	−0.278010	+	0.960578i	$0.589675\pi$
$444$	578.893	0.0618762
$445$	0	0
$446$	601.612	0.0638726
$447$	11798.3	1.24841
$448$	0	0
$449$	−772.951	−0.0812424	−0.0406212	−	0.999175i	$-0.512934\pi$
$449$	−772.951	−0.0812424	−0.0406212	+	0.999175i	$0.512934\pi$
$450$	0	0
$451$	4358.90	0.455105
$452$	3596.18	0.374226
$453$	−5866.28	−0.608437
$454$	−890.387	−0.0920439
$455$	0	0
$456$	8698.14	0.893262
$457$	−11532.7	−1.18048	−0.590239	−	0.807228i	$-0.700967\pi$
$457$	−11532.7	−1.18048	−0.590239	+	0.807228i	$0.700967\pi$
$458$	−1354.13	−0.138153
$459$	31751.0	3.22878
$460$	0	0
$461$	−10400.7	−1.05078	−0.525391	−	0.850861i	$-0.676081\pi$
$461$	−10400.7	−1.05078	−0.525391	+	0.850861i	$0.676081\pi$
$462$	0	0
$463$	13855.4	1.39075	0.695374	−	0.718648i	$-0.255239\pi$
$463$	13855.4	1.39075	0.695374	+	0.718648i	$0.255239\pi$
$464$	6152.56	0.615572
$465$	0	0
$466$	1114.71	0.110811
$467$	−14734.5	−1.46002	−0.730011	−	0.683435i	$-0.760485\pi$
$467$	−14734.5	−1.46002	−0.730011	+	0.683435i	$0.760485\pi$
$468$	19463.2	1.92241
$469$	0	0
$470$	0	0
$471$	15605.3	1.52665
$472$	4202.74	0.409845
$473$	7277.00	0.707393
$474$	976.130	0.0945889
$475$	0	0
$476$	0	0
$477$	33081.0	3.17542
$478$	−201.968	−0.0193260
$479$	1471.23	0.140338	0.0701691	−	0.997535i	$-0.477646\pi$
$479$	1471.23	0.140338	0.0701691	+	0.997535i	$0.477646\pi$
$480$	0	0
$481$	275.569	0.0261224
$482$	−1157.90	−0.109421
$483$	0	0
$484$	3085.91	0.289812
$485$	0	0
$486$	3389.71	0.316379
$487$	−5610.00	−0.521998	−0.260999	−	0.965339i	$-0.584052\pi$
$487$	−5610.00	−0.521998	−0.260999	+	0.965339i	$0.584052\pi$
$488$	−2077.44	−0.192707
$489$	30114.5	2.78491
$490$	0	0
$491$	−3193.98	−0.293569	−0.146784	−	0.989169i	$-0.546892\pi$
$491$	−3193.98	−0.293569	−0.146784	+	0.989169i	$0.546892\pi$
$492$	−10898.2	−0.998636
$493$	8075.85	0.737764
$494$	2052.47	0.186933
$495$	0	0
$496$	15183.6	1.37453
$497$	0	0
$498$	−2451.41	−0.220583
$499$	8724.32	0.782673	0.391337	−	0.920248i	$-0.372013\pi$
$499$	8724.32	0.782673	0.391337	+	0.920248i	$0.372013\pi$
$500$	0	0
$501$	−31668.8	−2.82407
$502$	337.911	0.0300432
$503$	−14636.7	−1.29745	−0.648726	−	0.761022i	$-0.724698\pi$
$503$	−14636.7	−1.29745	−0.648726	+	0.761022i	$0.724698\pi$
$504$	0	0
$505$	0	0
$506$	−251.413	−0.0220883
$507$	−8450.11	−0.740202
$508$	−20016.1	−1.74817
$509$	−8530.56	−0.742850	−0.371425	−	0.928463i	$-0.621131\pi$
$509$	−8530.56	−0.742850	−0.371425	+	0.928463i	$0.621131\pi$
$510$	0	0
$511$	0	0
$512$	7056.93	0.609131
$513$	−60694.7	−5.22366
$514$	−2339.31	−0.200744
$515$	0	0
$516$	−18194.1	−1.55223
$517$	10144.5	0.862967
$518$	0	0
$519$	21960.9	1.85737
$520$	0	0
$521$	−13743.0	−1.15564	−0.577822	−	0.816163i	$-0.696097\pi$
$521$	−13743.0	−1.15564	−0.577822	+	0.816163i	$0.696097\pi$
$522$	2539.16	0.212904
$523$	−2236.17	−0.186961	−0.0934806	−	0.995621i	$-0.529799\pi$
$523$	−2236.17	−0.186961	−0.0934806	+	0.995621i	$0.529799\pi$
$524$	−7634.56	−0.636483
$525$	0	0
$526$	−1524.07	−0.126336
$527$	19930.0	1.64737
$528$	18127.4	1.49412
$529$	−11673.2	−0.959418
$530$	0	0
$531$	−48694.9	−3.97962
$532$	0	0
$533$	−5187.84	−0.421595
$534$	577.984	0.0468386
$535$	0	0
$536$	337.391	0.0271885
$537$	−26158.3	−2.10207
$538$	2174.18	0.174230
$539$	0	0
$540$	0	0
$541$	−19487.9	−1.54870	−0.774352	−	0.632755i	$-0.781924\pi$
$541$	−19487.9	−1.54870	−0.774352	+	0.632755i	$0.781924\pi$
$542$	−2673.00	−0.211836
$543$	−12597.2	−0.995574
$544$	5525.84	0.435512
$545$	0	0
$546$	0	0
$547$	15949.3	1.24670	0.623349	−	0.781944i	$-0.285772\pi$
$547$	15949.3	1.24670	0.623349	+	0.781944i	$0.285772\pi$
$548$	1444.04	0.112566
$549$	24070.1	1.87120
$550$	0	0
$551$	−15437.7	−1.19359
$552$	1268.08	0.0977773
$553$	0	0
$554$	−2083.00	−0.159744
$555$	0	0
$556$	−8484.08	−0.647132
$557$	6349.74	0.483029	0.241514	−	0.970397i	$-0.422356\pi$
$557$	6349.74	0.483029	0.241514	+	0.970397i	$0.422356\pi$
$558$	6266.29	0.475400
$559$	−8660.88	−0.655306
$560$	0	0
$561$	23794.0	1.79070
$562$	−957.651	−0.0718791
$563$	−14805.0	−1.10827	−0.554135	−	0.832427i	$-0.686951\pi$
$563$	−14805.0	−1.10827	−0.554135	+	0.832427i	$0.686951\pi$
$564$	−25363.5	−1.89361
$565$	0	0
$566$	935.571	0.0694788
$567$	0	0
$568$	4078.65	0.301297
$569$	24578.7	1.81088	0.905442	−	0.424469i	$-0.139539\pi$
$569$	24578.7	1.81088	0.905442	+	0.424469i	$0.139539\pi$
$570$	0	0
$571$	−15609.9	−1.14405	−0.572027	−	0.820235i	$-0.693843\pi$
$571$	−15609.9	−1.14405	−0.572027	+	0.820235i	$0.693843\pi$
$572$	8784.12	0.642103
$573$	−1362.51	−0.0993359
$574$	0	0
$575$	0	0
$576$	−31250.2	−2.26058
$577$	−15111.3	−1.09028	−0.545141	−	0.838344i	$-0.683524\pi$
$577$	−15111.3	−1.09028	−0.545141	+	0.838344i	$0.683524\pi$
$578$	533.484	0.0383910
$579$	−23229.7	−1.66735
$580$	0	0
$581$	0	0
$582$	603.788	0.0430031
$583$	14930.1	1.06062
$584$	−1529.31	−0.108362
$585$	0	0
$586$	−217.606	−0.0153399
$587$	−14983.2	−1.05353	−0.526766	−	0.850010i	$-0.676596\pi$
$587$	−14983.2	−1.05353	−0.526766	+	0.850010i	$0.676596\pi$
$588$	0	0
$589$	−38098.0	−2.66519
$590$	0	0
$591$	9825.18	0.683848
$592$	−459.090	−0.0318725
$593$	−6931.01	−0.479971	−0.239985	−	0.970777i	$-0.577143\pi$
$593$	−6931.01	−0.479971	−0.239985	+	0.970777i	$0.577143\pi$
$594$	4505.51	0.311217
$595$	0	0
$596$	−9524.68	−0.654608
$597$	−9692.34	−0.664457
$598$	299.225	0.0204619
$599$	12309.2	0.839632	0.419816	−	0.907609i	$-0.362095\pi$
$599$	12309.2	0.839632	0.419816	+	0.907609i	$0.362095\pi$
$600$	0	0
$601$	15293.6	1.03800	0.519001	−	0.854774i	$-0.326304\pi$
$601$	15293.6	1.03800	0.519001	+	0.854774i	$0.326304\pi$
$602$	0	0
$603$	−3909.16	−0.264002
$604$	4735.81	0.319036
$605$	0	0
$606$	−1487.14	−0.0996878
$607$	5338.80	0.356994	0.178497	−	0.983940i	$-0.442877\pi$
$607$	5338.80	0.356994	0.178497	+	0.983940i	$0.442877\pi$
$608$	−10563.1	−0.704590
$609$	0	0
$610$	0	0
$611$	−12073.7	−0.799426
$612$	−42561.3	−2.81118
$613$	28848.6	1.90079	0.950396	−	0.311043i	$-0.100678\pi$
$613$	28848.6	1.90079	0.950396	+	0.311043i	$0.100678\pi$
$614$	1947.68	0.128016
$615$	0	0
$616$	0	0
$617$	12346.6	0.805602	0.402801	−	0.915287i	$-0.368037\pi$
$617$	12346.6	0.805602	0.402801	+	0.915287i	$0.368037\pi$
$618$	5220.18	0.339784
$619$	−6383.16	−0.414477	−0.207238	−	0.978290i	$-0.566448\pi$
$619$	−6383.16	−0.414477	−0.207238	+	0.978290i	$0.566448\pi$
$620$	0	0
$621$	−8848.54	−0.571787
$622$	1216.92	0.0784473
$623$	0	0
$624$	−21574.7	−1.38410
$625$	0	0
$626$	−671.921	−0.0429000
$627$	−45484.2	−2.89707
$628$	−12598.1	−0.800505
$629$	−602.602	−0.0381992
$630$	0	0
$631$	25708.6	1.62194	0.810969	−	0.585090i	$-0.198941\pi$
$631$	25708.6	1.62194	0.810969	+	0.585090i	$0.198941\pi$
$632$	−1589.72	−0.100056
$633$	22474.7	1.41120
$634$	−697.232	−0.0436760
$635$	0	0
$636$	−37328.5	−2.32732
$637$	0	0
$638$	1145.97	0.0711120
$639$	−47257.1	−2.92561
$640$	0	0
$641$	−18617.5	−1.14719	−0.573594	−	0.819139i	$-0.694451\pi$
$641$	−18617.5	−1.14719	−0.573594	+	0.819139i	$0.694451\pi$
$642$	6405.60	0.393783
$643$	−21168.8	−1.29832	−0.649158	−	0.760654i	$-0.724879\pi$
$643$	−21168.8	−1.29832	−0.649158	+	0.760654i	$0.724879\pi$
$644$	0	0
$645$	0	0
$646$	−4488.26	−0.273357
$647$	−5952.47	−0.361693	−0.180847	−	0.983511i	$-0.557884\pi$
$647$	−5952.47	−0.361693	−0.180847	+	0.983511i	$0.557884\pi$
$648$	−11987.2	−0.726700
$649$	−21976.9	−1.32923
$650$	0	0
$651$	0	0
$652$	−24311.2	−1.46028
$653$	3040.63	0.182219	0.0911096	−	0.995841i	$-0.470959\pi$
$653$	3040.63	0.182219	0.0911096	+	0.995841i	$0.470959\pi$
$654$	136.312	0.00815018
$655$	0	0
$656$	8642.81	0.514398
$657$	17719.3	1.05220
$658$	0	0
$659$	−3335.95	−0.197193	−0.0985966	−	0.995127i	$-0.531435\pi$
$659$	−3335.95	−0.197193	−0.0985966	+	0.995127i	$0.531435\pi$
$660$	0	0
$661$	1547.90	0.0910838	0.0455419	−	0.998962i	$-0.485499\pi$
$661$	1547.90	0.0910838	0.0455419	+	0.998962i	$0.485499\pi$
$662$	−168.550	−0.00989561
$663$	−28319.0	−1.65885
$664$	3992.34	0.233332
$665$	0	0
$666$	−189.467	−0.0110235
$667$	−2250.62	−0.130651
$668$	25566.0	1.48081
$669$	15867.5	0.917003
$670$	0	0
$671$	10863.3	0.624998
$672$	0	0
$673$	−19632.4	−1.12448	−0.562239	−	0.826975i	$-0.690060\pi$
$673$	−19632.4	−1.12448	−0.562239	+	0.826975i	$0.690060\pi$
$674$	−3018.94	−0.172530
$675$	0	0
$676$	6821.72	0.388127
$677$	30318.2	1.72116	0.860579	−	0.509317i	$-0.170102\pi$
$677$	30318.2	1.72116	0.860579	+	0.509317i	$0.170102\pi$
$678$	−1645.15	−0.0931885
$679$	0	0
$680$	0	0
$681$	−23484.0	−1.32145
$682$	2828.10	0.158788
$683$	−28970.8	−1.62304	−0.811520	−	0.584325i	$-0.801359\pi$
$683$	−28970.8	−1.62304	−0.811520	+	0.584325i	$0.801359\pi$
$684$	81359.6	4.54804
$685$	0	0
$686$	0	0
$687$	−35715.1	−1.98343
$688$	14428.8	0.799554
$689$	−17769.4	−0.982525
$690$	0	0
$691$	3434.91	0.189103	0.0945514	−	0.995520i	$-0.469858\pi$
$691$	3434.91	0.189103	0.0945514	+	0.995520i	$0.469858\pi$
$692$	−17728.9	−0.973917
$693$	0	0
$694$	2930.77	0.160304
$695$	0	0
$696$	−5780.06	−0.314788
$697$	11344.5	0.616507
$698$	−3194.35	−0.173221
$699$	29400.6	1.59089
$700$	0	0
$701$	16304.4	0.878471	0.439235	−	0.898372i	$-0.355249\pi$
$701$	16304.4	0.878471	0.439235	+	0.898372i	$0.355249\pi$
$702$	−5362.33	−0.288302
$703$	1151.92	0.0618004
$704$	−14103.8	−0.755054
$705$	0	0
$706$	−1821.11	−0.0970800
$707$	0	0
$708$	54947.2	2.91673
$709$	16094.4	0.852519	0.426260	−	0.904601i	$-0.359831\pi$
$709$	16094.4	0.852519	0.426260	+	0.904601i	$0.359831\pi$
$710$	0	0
$711$	18419.2	0.971552
$712$	−941.299	−0.0495459
$713$	−5554.21	−0.291735
$714$	0	0
$715$	0	0
$716$	21117.4	1.10223
$717$	−5326.93	−0.277458
$718$	3044.50	0.158245
$719$	−8651.93	−0.448766	−0.224383	−	0.974501i	$-0.572037\pi$
$719$	−8651.93	−0.448766	−0.224383	+	0.974501i	$0.572037\pi$
$720$	0	0
$721$	0	0
$722$	6046.56	0.311676
$723$	−30539.7	−1.57093
$724$	10169.6	0.522032
$725$	0	0
$726$	−1411.72	−0.0721679
$727$	−6999.43	−0.357076	−0.178538	−	0.983933i	$-0.557137\pi$
$727$	−6999.43	−0.357076	−0.178538	+	0.983933i	$0.557137\pi$
$728$	0	0
$729$	34160.1	1.73551
$730$	0	0
$731$	18939.3	0.958268
$732$	−27160.7	−1.37143
$733$	5167.74	0.260402	0.130201	−	0.991488i	$-0.458438\pi$
$733$	5167.74	0.260402	0.130201	+	0.991488i	$0.458438\pi$
$734$	−3020.67	−0.151900
$735$	0	0
$736$	−1539.97	−0.0771251
$737$	−1764.28	−0.0881793
$738$	3566.89	0.177912
$739$	−12319.1	−0.613214	−0.306607	−	0.951836i	$-0.599194\pi$
$739$	−12319.1	−0.613214	−0.306607	+	0.951836i	$0.599194\pi$
$740$	0	0
$741$	54134.1	2.68376
$742$	0	0
$743$	−16942.4	−0.836548	−0.418274	−	0.908321i	$-0.637365\pi$
$743$	−16942.4	−0.836548	−0.418274	+	0.908321i	$0.637365\pi$
$744$	−14264.4	−0.702900
$745$	0	0
$746$	4725.56	0.231923
$747$	−46257.0	−2.26567
$748$	−19208.8	−0.938959
$749$	0	0
$750$	0	0
$751$	32917.9	1.59946	0.799728	−	0.600363i	$-0.204977\pi$
$751$	32917.9	1.59946	0.799728	+	0.600363i	$0.204977\pi$
$752$	20114.5	0.975397
$753$	8912.41	0.431323
$754$	−1363.90	−0.0658760
$755$	0	0
$756$	0	0
$757$	9433.45	0.452926	0.226463	−	0.974020i	$-0.427284\pi$
$757$	9433.45	0.452926	0.226463	+	0.974020i	$0.427284\pi$
$758$	−2118.17	−0.101498
$759$	−6631.04	−0.317117
$760$	0	0
$761$	34551.0	1.64582	0.822912	−	0.568170i	$-0.192348\pi$
$761$	34551.0	1.64582	0.822912	+	0.568170i	$0.192348\pi$
$762$	9156.82	0.435323
$763$	0	0
$764$	1099.94	0.0520871
$765$	0	0
$766$	272.495	0.0128533
$767$	26156.4	1.23136
$768$	33268.2	1.56310
$769$	348.011	0.0163194	0.00815970	−	0.999967i	$-0.497403\pi$
$769$	348.011	0.0163194	0.00815970	+	0.999967i	$0.497403\pi$
$770$	0	0
$771$	−61699.3	−2.88203
$772$	18753.2	0.874277
$773$	−30083.9	−1.39979	−0.699897	−	0.714243i	$-0.746771\pi$
$773$	−30083.9	−1.39979	−0.699897	+	0.714243i	$0.746771\pi$
$774$	5954.77	0.276537
$775$	0	0
$776$	−983.323	−0.0454887
$777$	0	0
$778$	−4813.64	−0.221822
$779$	−21686.1	−0.997412
$780$	0	0
$781$	−21328.1	−0.977181
$782$	−654.333	−0.0299219
$783$	40332.7	1.84083
$784$	0	0
$785$	0	0
$786$	3492.60	0.158495
$787$	31958.4	1.44751	0.723757	−	0.690055i	$-0.242413\pi$
$787$	31958.4	1.44751	0.723757	+	0.690055i	$0.242413\pi$
$788$	−7931.81	−0.358578
$789$	−40197.5	−1.81377
$790$	0	0
$791$	0	0
$792$	−12183.8	−0.546630
$793$	−12929.2	−0.578979
$794$	3070.35	0.137233
$795$	0	0
$796$	7824.57	0.348410
$797$	−364.165	−0.0161849	−0.00809247	−	0.999967i	$-0.502576\pi$
$797$	−364.165	−0.0161849	−0.00809247	+	0.999967i	$0.502576\pi$
$798$	0	0
$799$	26402.2	1.16902
$800$	0	0
$801$	10906.3	0.481093
$802$	−5296.50	−0.233199
$803$	7997.08	0.351446
$804$	4411.09	0.193492
$805$	0	0
$806$	−3365.92	−0.147096
$807$	57344.2	2.50138
$808$	2421.94	0.105450
$809$	30714.8	1.33482	0.667412	−	0.744688i	$-0.267402\pi$
$809$	30714.8	1.33482	0.667412	+	0.744688i	$0.267402\pi$
$810$	0	0
$811$	−18967.6	−0.821260	−0.410630	−	0.911802i	$-0.634691\pi$
$811$	−18967.6	−0.821260	−0.410630	+	0.911802i	$0.634691\pi$
$812$	0	0
$813$	−70500.4	−3.04128
$814$	−85.5100	−0.00368197
$815$	0	0
$816$	47178.7	2.02400
$817$	−36204.0	−1.55033
$818$	2320.17	0.0991720
$819$	0	0
$820$	0	0
$821$	−14817.0	−0.629860	−0.314930	−	0.949115i	$-0.601981\pi$
$821$	−14817.0	−0.629860	−0.314930	+	0.949115i	$0.601981\pi$
$822$	−660.606	−0.0280308
$823$	26588.1	1.12613	0.563063	−	0.826414i	$-0.309623\pi$
$823$	26588.1	1.12613	0.563063	+	0.826414i	$0.309623\pi$
$824$	−8501.54	−0.359424
$825$	0	0
$826$	0	0
$827$	11287.0	0.474592	0.237296	−	0.971437i	$-0.423739\pi$
$827$	11287.0	0.474592	0.237296	+	0.971437i	$0.423739\pi$
$828$	11861.2	0.497833
$829$	−14448.6	−0.605332	−0.302666	−	0.953097i	$-0.597877\pi$
$829$	−14448.6	−0.605332	−0.302666	+	0.953097i	$0.597877\pi$
$830$	0	0
$831$	−54939.2	−2.29340
$832$	16786.0	0.699459
$833$	0	0
$834$	3881.23	0.161146
$835$	0	0
$836$	36719.2	1.51909
$837$	99535.4	4.11045
$838$	−5623.21	−0.231803
$839$	−40032.0	−1.64727	−0.823635	−	0.567120i	$-0.808057\pi$
$839$	−40032.0	−1.64727	−0.823635	+	0.567120i	$0.808057\pi$
$840$	0	0
$841$	−14130.4	−0.579376
$842$	−748.434	−0.0306327
$843$	−25258.1	−1.03195
$844$	−18143.7	−0.739967
$845$	0	0
$846$	8301.23	0.337355
$847$	0	0
$848$	29603.3	1.19880
$849$	24675.7	0.997490
$850$	0	0
$851$	167.936	0.00676473
$852$	53324.9	2.14423
$853$	10864.4	0.436098	0.218049	−	0.975938i	$-0.430031\pi$
$853$	10864.4	0.436098	0.218049	+	0.975938i	$0.430031\pi$
$854$	0	0
$855$	0	0
$856$	−10432.1	−0.416544
$857$	42013.7	1.67463	0.837316	−	0.546719i	$-0.184123\pi$
$857$	42013.7	1.67463	0.837316	+	0.546719i	$0.184123\pi$
$858$	−4018.50	−0.159894
$859$	−16135.6	−0.640908	−0.320454	−	0.947264i	$-0.603835\pi$
$859$	−16135.6	−0.640908	−0.320454	+	0.947264i	$0.603835\pi$
$860$	0	0
$861$	0	0
$862$	3317.76	0.131094
$863$	46246.8	1.82417	0.912086	−	0.409998i	$-0.134471\pi$
$863$	46246.8	1.82417	0.912086	+	0.409998i	$0.134471\pi$
$864$	27597.4	1.08667
$865$	0	0
$866$	−2524.85	−0.0990739
$867$	14070.7	0.551171
$868$	0	0
$869$	8312.93	0.324507
$870$	0	0
$871$	2099.80	0.0816865
$872$	−221.996	−0.00862127
$873$	11393.2	0.441698
$874$	1250.81	0.0484089
$875$	0	0
$876$	−19994.5	−0.771177
$877$	25094.1	0.966212	0.483106	−	0.875562i	$-0.339509\pi$
$877$	25094.1	0.966212	0.483106	+	0.875562i	$0.339509\pi$
$878$	−2881.75	−0.110768
$879$	−5739.36	−0.220232
$880$	0	0
$881$	27546.6	1.05342	0.526712	−	0.850044i	$-0.323424\pi$
$881$	27546.6	1.05342	0.526712	+	0.850044i	$0.323424\pi$
$882$	0	0
$883$	−5825.31	−0.222013	−0.111006	−	0.993820i	$-0.535407\pi$
$883$	−5825.31	−0.222013	−0.111006	+	0.993820i	$0.535407\pi$
$884$	22861.7	0.869823
$885$	0	0
$886$	−1914.67	−0.0726010
$887$	6214.29	0.235237	0.117619	−	0.993059i	$-0.462474\pi$
$887$	6214.29	0.235237	0.117619	+	0.993059i	$0.462474\pi$
$888$	431.296	0.0162988
$889$	0	0
$890$	0	0
$891$	62683.4	2.35687
$892$	−12809.8	−0.480833
$893$	−50470.1	−1.89128
$894$	4357.28	0.163008
$895$	0	0
$896$	0	0
$897$	7892.08	0.293767
$898$	−285.462	−0.0106080
$899$	25316.8	0.939223
$900$	0	0
$901$	38857.3	1.43677
$902$	1609.80	0.0594242
$903$	0	0
$904$	2679.28	0.0985748
$905$	0	0
$906$	−2166.50	−0.0794451
$907$	−24415.9	−0.893845	−0.446923	−	0.894573i	$-0.647480\pi$
$907$	−24415.9	−0.893845	−0.446923	+	0.894573i	$0.647480\pi$
$908$	18958.5	0.692907
$909$	−28061.7	−1.02392
$910$	0	0
$911$	−3493.35	−0.127047	−0.0635236	−	0.997980i	$-0.520234\pi$
$911$	−3493.35	−0.127047	−0.0635236	+	0.997980i	$0.520234\pi$
$912$	−90186.0	−3.27451
$913$	−20876.7	−0.756755
$914$	−4259.21	−0.154138
$915$	0	0
$916$	28832.6	1.04002
$917$	0	0
$918$	11726.1	0.421590
$919$	15656.7	0.561989	0.280995	−	0.959709i	$-0.409336\pi$
$919$	15656.7	0.561989	0.280995	+	0.959709i	$0.409336\pi$
$920$	0	0
$921$	51370.0	1.83789
$922$	−3841.14	−0.137203
$923$	25384.1	0.905230
$924$	0	0
$925$	0	0
$926$	5117.02	0.181594
$927$	98502.8	3.49003
$928$	7019.38	0.248300
$929$	30504.5	1.07731	0.538654	−	0.842527i	$-0.318933\pi$
$929$	30504.5	1.07731	0.538654	+	0.842527i	$0.318933\pi$
$930$	0	0
$931$	0	0
$932$	−23735.0	−0.834189
$933$	32096.4	1.12625
$934$	−5441.66	−0.190639
$935$	0	0
$936$	14500.8	0.506381
$937$	50290.7	1.75339	0.876694	−	0.481049i	$-0.159744\pi$
$937$	50290.7	1.75339	0.876694	+	0.481049i	$0.159744\pi$
$938$	0	0
$939$	−17722.0	−0.615904
$940$	0	0
$941$	−18681.7	−0.647189	−0.323595	−	0.946196i	$-0.604891\pi$
$941$	−18681.7	−0.647189	−0.323595	+	0.946196i	$0.604891\pi$
$942$	5763.26	0.199339
$943$	−3161.56	−0.109178
$944$	−43575.8	−1.50241
$945$	0	0
$946$	2687.50	0.0923660
$947$	−16332.6	−0.560440	−0.280220	−	0.959936i	$-0.590407\pi$
$947$	−16332.6	−0.560440	−0.280220	+	0.959936i	$0.590407\pi$
$948$	−20784.2	−0.712066
$949$	−9517.91	−0.325568
$950$	0	0
$951$	−18389.5	−0.627046
$952$	0	0
$953$	34233.6	1.16362	0.581812	−	0.813323i	$-0.302344\pi$
$953$	34233.6	1.16362	0.581812	+	0.813323i	$0.302344\pi$
$954$	12217.3	0.414622
$955$	0	0
$956$	4300.40	0.145486
$957$	30225.1	1.02094
$958$	543.345	0.0183243
$959$	0	0
$960$	0	0
$961$	32687.2	1.09722
$962$	101.772	0.00341086
$963$	120871.	4.04467
$964$	24654.5	0.823723
$965$	0	0
$966$	0	0
$967$	−53792.0	−1.78887	−0.894434	−	0.447200i	$-0.852421\pi$
$967$	−53792.0	−1.78887	−0.894434	+	0.447200i	$0.852421\pi$
$968$	2299.11	0.0763392
$969$	−118378.	−3.92451
$970$	0	0
$971$	−826.331	−0.0273102	−0.0136551	−	0.999907i	$-0.504347\pi$
$971$	−826.331	−0.0273102	−0.0136551	+	0.999907i	$0.504347\pi$
$972$	−72175.1	−2.38171
$973$	0	0
$974$	−2071.85	−0.0681586
$975$	0	0
$976$	21539.8	0.706425
$977$	−8703.52	−0.285005	−0.142503	−	0.989794i	$-0.545515\pi$
$977$	−8703.52	−0.285005	−0.142503	+	0.989794i	$0.545515\pi$
$978$	11121.7	0.363633
$979$	4922.23	0.160690
$980$	0	0
$981$	2572.15	0.0837130
$982$	−1179.58	−0.0383320
$983$	41000.9	1.33034	0.665170	−	0.746692i	$-0.268359\pi$
$983$	41000.9	1.33034	0.665170	+	0.746692i	$0.268359\pi$
$984$	−8119.55	−0.263051
$985$	0	0
$986$	2982.53	0.0963317
$987$	0	0
$988$	−43702.1	−1.40724
$989$	−5278.09	−0.169700
$990$	0	0
$991$	−18341.0	−0.587912	−0.293956	−	0.955819i	$-0.594972\pi$
$991$	−18341.0	−0.587912	−0.293956	+	0.955819i	$0.594972\pi$
$992$	17322.8	0.554436
$993$	−4445.52	−0.142069
$994$	0	0
$995$	0	0
$996$	52196.3	1.66055
$997$	−57335.4	−1.82129	−0.910646	−	0.413187i	$-0.864416\pi$
$997$	−57335.4	−1.82129	−0.910646	+	0.413187i	$0.864416\pi$
$998$	3222.02	0.102196
$999$	−3009.54	−0.0953129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.bj.1.4		6
5.4	even	2		245.4.a.o.1.3	✓	6
7.6	odd	2		1225.4.a.bi.1.4		6
15.14	odd	2		2205.4.a.bz.1.4		6
35.4	even	6		245.4.e.q.226.4		12
35.9	even	6		245.4.e.q.116.4		12
35.19	odd	6		245.4.e.p.116.4		12
35.24	odd	6		245.4.e.p.226.4		12
35.34	odd	2		245.4.a.p.1.3	yes	6
105.104	even	2		2205.4.a.ca.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.o.1.3	✓	6	5.4	even	2
245.4.a.p.1.3	yes	6	35.34	odd	2
245.4.e.p.116.4		12	35.19	odd	6
245.4.e.p.226.4		12	35.24	odd	6
245.4.e.q.116.4		12	35.9	even	6
245.4.e.q.226.4		12	35.4	even	6
1225.4.a.bi.1.4		6	7.6	odd	2
1225.4.a.bj.1.4		6	1.1	even	1	trivial
2205.4.a.bz.1.4		6	15.14	odd	2
2205.4.a.ca.1.4		6	105.104	even	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.369315 q^{2} +9.74070 q^{3} -7.86361 q^{4} +3.59738 q^{6} -5.85867 q^{8} +67.8812 q^{9} +O(q^{10})\)	\(q+0.369315 q^{2} +9.74070 q^{3} -7.86361 q^{4} +3.59738 q^{6} -5.85867 q^{8} +67.8812 q^{9} +30.6361 q^{11} -76.5970 q^{12} -36.4622 q^{13} +60.7452 q^{16} +79.7341 q^{17} +25.0695 q^{18} -152.418 q^{19} +11.3144 q^{22} -22.2207 q^{23} -57.0675 q^{24} -13.4660 q^{26} +398.211 q^{27} +101.285 q^{29} +249.956 q^{31} +69.3034 q^{32} +298.417 q^{33} +29.4470 q^{34} -533.791 q^{36} -7.55765 q^{37} -56.2904 q^{38} -355.168 q^{39} +142.280 q^{41} +237.530 q^{43} -240.910 q^{44} -8.20644 q^{46} +331.129 q^{47} +591.700 q^{48} +776.666 q^{51} +286.725 q^{52} +487.337 q^{53} +147.065 q^{54} -1484.66 q^{57} +37.4059 q^{58} -717.355 q^{59} +354.592 q^{61} +92.3126 q^{62} -460.367 q^{64} +110.210 q^{66} -57.5883 q^{67} -626.997 q^{68} -216.445 q^{69} -696.174 q^{71} -397.693 q^{72} +261.035 q^{73} -2.79115 q^{74} +1198.56 q^{76} -131.169 q^{78} +271.344 q^{79} +2046.06 q^{81} +52.5460 q^{82} -681.441 q^{83} +87.7234 q^{86} +986.584 q^{87} -179.487 q^{88} +160.668 q^{89} +174.735 q^{92} +2434.75 q^{93} +122.291 q^{94} +675.063 q^{96} +167.841 q^{97} +2079.61 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 16 q^{3} + 14 q^{4} - 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + 16 * q^3 + 14 * q^4 - 24 * q^6 + 66 * q^8 + 70 * q^9	\( 6 q + 2 q^{2} + 16 q^{3} + 14 q^{4} - 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} + 160 q^{12} + 168 q^{13} + 298 q^{16} - 4 q^{17} - 354 q^{18} - 308 q^{19} + 236 q^{22} + 336 q^{23} + 92 q^{24} - 56 q^{26} + 964 q^{27} + 176 q^{29} - 392 q^{31} + 770 q^{32} + 188 q^{33} - 812 q^{34} + 230 q^{36} + 140 q^{37} + 20 q^{38} + 140 q^{39} - 656 q^{41} + 388 q^{43} - 160 q^{44} - 388 q^{46} + 628 q^{47} + 1396 q^{48} + 744 q^{51} + 1520 q^{52} + 676 q^{53} - 2284 q^{54} - 1468 q^{57} + 2012 q^{58} - 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 3620 q^{66} - 1768 q^{67} - 2940 q^{68} + 1048 q^{69} - 224 q^{71} - 2858 q^{72} + 2640 q^{73} + 928 q^{74} + 1340 q^{76} - 8 q^{78} + 1636 q^{79} + 4442 q^{81} - 1756 q^{82} + 140 q^{83} + 1180 q^{86} + 1940 q^{87} + 5652 q^{88} + 1904 q^{89} + 1952 q^{92} + 1592 q^{93} + 3332 q^{94} + 6460 q^{96} + 516 q^{97} - 2804 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 16 * q^3 + 14 * q^4 - 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 + 160 * q^12 + 168 * q^13 + 298 * q^16 - 4 * q^17 - 354 * q^18 - 308 * q^19 + 236 * q^22 + 336 * q^23 + 92 * q^24 - 56 * q^26 + 964 * q^27 + 176 * q^29 - 392 * q^31 + 770 * q^32 + 188 * q^33 - 812 * q^34 + 230 * q^36 + 140 * q^37 + 20 * q^38 + 140 * q^39 - 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 + 628 * q^47 + 1396 * q^48 + 744 * q^51 + 1520 * q^52 + 676 * q^53 - 2284 * q^54 - 1468 * q^57 + 2012 * q^58 - 996 * q^59 - 740 * q^61 - 364 * q^62 + 1426 * q^64 + 3620 * q^66 - 1768 * q^67 - 2940 * q^68 + 1048 * q^69 - 224 * q^71 - 2858 * q^72 + 2640 * q^73 + 928 * q^74 + 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 - 1756 * q^82 + 140 * q^83 + 1180 * q^86 + 1940 * q^87 + 5652 * q^88 + 1904 * q^89 + 1952 * q^92 + 1592 * q^93 + 3332 * q^94 + 6460 * q^96 + 516 * q^97 - 2804 * q^99

Introduction

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