LMFDB - Embedded newform 1225.4.a.z.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 32x^{2} - 35x + 120 $ x^4 - 32x^2 - 35x + 120
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.50478$ 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)

$f(q)$	$=$	$q+0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} +2.15229 q^{6} -7.94792 q^{8} -8.82008 q^{9} +O(q^{10})$	\(q+0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} +2.15229 q^{6} -7.94792 q^{8} -8.82008 q^{9} +54.8800 q^{11} -33.0239 q^{12} +16.0073 q^{13} +57.9496 q^{16} -0.422056 q^{17} -4.45223 q^{18} -127.501 q^{19} +27.7025 q^{22} -51.1101 q^{23} -33.8883 q^{24} +8.08024 q^{26} -152.729 q^{27} +41.4750 q^{29} -192.354 q^{31} +92.8353 q^{32} +233.997 q^{33} -0.213047 q^{34} +68.3132 q^{36} +189.232 q^{37} -64.3605 q^{38} +68.2519 q^{39} +76.3187 q^{41} -294.499 q^{43} -425.056 q^{44} -25.7996 q^{46} +540.297 q^{47} +247.085 q^{48} -1.79956 q^{51} -123.980 q^{52} +661.316 q^{53} -77.0953 q^{54} -543.639 q^{57} +20.9359 q^{58} -410.312 q^{59} -46.0495 q^{61} -97.0974 q^{62} -416.735 q^{64} +118.118 q^{66} -10.4074 q^{67} +3.26890 q^{68} -217.923 q^{69} -491.117 q^{71} +70.1012 q^{72} -814.540 q^{73} +95.5215 q^{74} +987.522 q^{76} +34.4525 q^{78} -858.725 q^{79} -413.064 q^{81} +38.5244 q^{82} -1055.80 q^{83} -148.658 q^{86} +176.841 q^{87} -436.182 q^{88} -341.567 q^{89} +395.858 q^{92} -820.159 q^{93} +272.733 q^{94} +395.831 q^{96} -1417.21 q^{97} -484.046 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9	$ 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26} - 441 q^{27} + 129 q^{29} - 114 q^{31} + 310 q^{32} - 865 q^{33} - 203 q^{34} + 1101 q^{36} - 403 q^{37} + 555 q^{38} + 674 q^{39} - 671 q^{41} + 411 q^{43} + 438 q^{44} - 997 q^{46} - 8 q^{47} - 523 q^{48} - 885 q^{51} + 74 q^{52} - 90 q^{53} + 2777 q^{54} - 233 q^{57} - 673 q^{58} - 1018 q^{59} - 50 q^{61} - 1626 q^{62} - 2421 q^{64} + 3841 q^{66} - 424 q^{67} - 617 q^{68} - 1080 q^{69} + 215 q^{71} - 2940 q^{72} - 1207 q^{73} + 623 q^{74} + 3257 q^{76} - 278 q^{78} - 951 q^{79} + 28 q^{81} + 1695 q^{82} - 3035 q^{83} - 99 q^{86} + 2210 q^{87} - 163 q^{88} - 2819 q^{89} - 3073 q^{92} + 852 q^{93} + 3056 q^{94} + 1345 q^{96} - 1100 q^{97} + 2383 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9 + 100 * q^11 - 165 * q^12 + 44 * q^13 + 160 * q^16 - 53 * q^17 - 433 * q^18 + 29 * q^19 + 152 * q^22 - 295 * q^23 + 21 * q^24 - 700 * q^26 - 441 * q^27 + 129 * q^29 - 114 * q^31 + 310 * q^32 - 865 * q^33 - 203 * q^34 + 1101 * q^36 - 403 * q^37 + 555 * q^38 + 674 * q^39 - 671 * q^41 + 411 * q^43 + 438 * q^44 - 997 * q^46 - 8 * q^47 - 523 * q^48 - 885 * q^51 + 74 * q^52 - 90 * q^53 + 2777 * q^54 - 233 * q^57 - 673 * q^58 - 1018 * q^59 - 50 * q^61 - 1626 * q^62 - 2421 * q^64 + 3841 * q^66 - 424 * q^67 - 617 * q^68 - 1080 * q^69 + 215 * q^71 - 2940 * q^72 - 1207 * q^73 + 623 * q^74 + 3257 * q^76 - 278 * q^78 - 951 * q^79 + 28 * q^81 + 1695 * q^82 - 3035 * q^83 - 99 * q^86 + 2210 * q^87 - 163 * q^88 - 2819 * q^89 - 3073 * q^92 + 852 * q^93 + 3056 * q^94 + 1345 * q^96 - 1100 * q^97 + 2383 * 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 $

$2$	0.504784	0.178468	0.0892340	−	0.996011i	$-0.471558\pi$
$2$	0.504784	0.178468	0.0892340	+	0.996011i	$0.471558\pi$
$3$	4.26379	0.820567	0.410284	−	0.911958i	$-0.365430\pi$
$3$	4.26379	0.820567	0.410284	+	0.911958i	$0.365430\pi$
$4$	−7.74519	−0.968149
$5$	0	0
$5$	0	0
$6$	2.15229	0.146445
$7$	0	0
$7$	0	0
$8$	−7.94792	−0.351252
$9$	−8.82008	−0.326670
$10$	0	0
$11$	54.8800	1.50427	0.752134	−	0.659010i	$-0.229025\pi$
$11$	54.8800	1.50427	0.752134	+	0.659010i	$0.229025\pi$
$12$	−33.0239	−0.794431
$13$	16.0073	0.341510	0.170755	−	0.985313i	$-0.445379\pi$
$13$	16.0073	0.341510	0.170755	+	0.985313i	$0.445379\pi$
$14$	0	0
$15$	0	0
$16$	57.9496	0.905462
$17$	−0.422056	−0.00602139	−0.00301069	−	0.999995i	$-0.500958\pi$
$17$	−0.422056	−0.00602139	−0.00301069	+	0.999995i	$0.500958\pi$
$18$	−4.45223	−0.0583000
$19$	−127.501	−1.53952	−0.769758	−	0.638336i	$-0.779623\pi$
$19$	−127.501	−1.53952	−0.769758	+	0.638336i	$0.779623\pi$
$20$	0	0
$21$	0	0
$22$	27.7025	0.268464
$23$	−51.1101	−0.463357	−0.231678	−	0.972792i	$-0.574422\pi$
$23$	−51.1101	−0.463357	−0.231678	+	0.972792i	$0.574422\pi$
$24$	−33.8883	−0.288226
$25$	0	0
$26$	8.08024	0.0609487
$27$	−152.729	−1.08862
$28$	0	0
$29$	41.4750	0.265576	0.132788	−	0.991144i	$-0.457607\pi$
$29$	41.4750	0.265576	0.132788	+	0.991144i	$0.457607\pi$
$30$	0	0
$31$	−192.354	−1.11445	−0.557224	−	0.830362i	$-0.688134\pi$
$31$	−192.354	−1.11445	−0.557224	+	0.830362i	$0.688134\pi$
$32$	92.8353	0.512848
$33$	233.997	1.23435
$34$	−0.213047	−0.00107462
$35$	0	0
$36$	68.3132	0.316265
$37$	189.232	0.840801	0.420400	−	0.907339i	$-0.361890\pi$
$37$	189.232	0.840801	0.420400	+	0.907339i	$0.361890\pi$
$38$	−64.3605	−0.274754
$39$	68.2519	0.280232
$40$	0	0
$41$	76.3187	0.290707	0.145353	−	0.989380i	$-0.453568\pi$
$41$	76.3187	0.290707	0.145353	+	0.989380i	$0.453568\pi$
$42$	0	0
$43$	−294.499	−1.04443	−0.522216	−	0.852813i	$-0.674895\pi$
$43$	−294.499	−1.04443	−0.522216	+	0.852813i	$0.674895\pi$
$44$	−425.056	−1.45636
$45$	0	0
$46$	−25.7996	−0.0826943
$47$	540.297	1.67682	0.838408	−	0.545043i	$-0.183487\pi$
$47$	540.297	1.67682	0.838408	+	0.545043i	$0.183487\pi$
$48$	247.085	0.742992
$49$	0	0
$50$	0	0
$51$	−1.79956	−0.00494095
$52$	−123.980	−0.330633
$53$	661.316	1.71394	0.856969	−	0.515368i	$-0.172345\pi$
$53$	661.316	1.71394	0.856969	+	0.515368i	$0.172345\pi$
$54$	−77.0953	−0.194284
$55$	0	0
$56$	0	0
$57$	−543.639	−1.26328
$58$	20.9359	0.0473969
$59$	−410.312	−0.905390	−0.452695	−	0.891665i	$-0.649537\pi$
$59$	−410.312	−0.905390	−0.452695	+	0.891665i	$0.649537\pi$
$60$	0	0
$61$	−46.0495	−0.0966563	−0.0483281	−	0.998832i	$-0.515389\pi$
$61$	−46.0495	−0.0966563	−0.0483281	+	0.998832i	$0.515389\pi$
$62$	−97.0974	−0.198893
$63$	0	0
$64$	−416.735	−0.813935
$65$	0	0
$66$	118.118	0.220292
$67$	−10.4074	−0.0189771	−0.00948854	−	0.999955i	$-0.503020\pi$
$67$	−10.4074	−0.0189771	−0.00948854	+	0.999955i	$0.503020\pi$
$68$	3.26890	0.00582960
$69$	−217.923	−0.380215
$70$	0	0
$71$	−491.117	−0.820913	−0.410456	−	0.911880i	$-0.634631\pi$
$71$	−491.117	−0.820913	−0.410456	+	0.911880i	$0.634631\pi$
$72$	70.1012	0.114743
$73$	−814.540	−1.30595	−0.652977	−	0.757378i	$-0.726480\pi$
$73$	−814.540	−1.30595	−0.652977	+	0.757378i	$0.726480\pi$
$74$	95.5215	0.150056
$75$	0	0
$76$	987.522	1.49048
$77$	0	0
$78$	34.4525	0.0500125
$79$	−858.725	−1.22296	−0.611482	−	0.791258i	$-0.709426\pi$
$79$	−858.725	−1.22296	−0.611482	+	0.791258i	$0.709426\pi$
$80$	0	0
$81$	−413.064	−0.566617
$82$	38.5244	0.0518818
$83$	−1055.80	−1.39626	−0.698129	−	0.715972i	$-0.745984\pi$
$83$	−1055.80	−1.39626	−0.698129	+	0.715972i	$0.745984\pi$
$84$	0	0
$85$	0	0
$86$	−148.658	−0.186398
$87$	176.841	0.217923
$88$	−436.182	−0.528376
$89$	−341.567	−0.406809	−0.203405	−	0.979095i	$-0.565201\pi$
$89$	−341.567	−0.406809	−0.203405	+	0.979095i	$0.565201\pi$
$90$	0	0
$91$	0	0
$92$	395.858	0.448598
$93$	−820.159	−0.914479
$94$	272.733	0.299258
$95$	0	0
$96$	395.831	0.420826
$97$	−1417.21	−1.48346	−0.741731	−	0.670697i	$-0.765995\pi$
$97$	−1417.21	−1.48346	−0.741731	+	0.670697i	$0.765995\pi$
$98$	0	0
$99$	−484.046	−0.491398
$100$	0	0
$101$	121.051	0.119258	0.0596289	−	0.998221i	$-0.481008\pi$
$101$	121.051	0.119258	0.0596289	+	0.998221i	$0.481008\pi$
$102$	−0.908388	−0.000881802 0
$103$	−655.594	−0.627161	−0.313581	−	0.949562i	$-0.601529\pi$
$103$	−655.594	−0.627161	−0.313581	+	0.949562i	$0.601529\pi$
$104$	−127.225	−0.119956
$105$	0	0
$106$	333.821	0.305883
$107$	−2069.81	−1.87006	−0.935031	−	0.354567i	$-0.884628\pi$
$107$	−2069.81	−1.87006	−0.935031	+	0.354567i	$0.884628\pi$
$108$	1182.92	1.05395
$109$	904.218	0.794573	0.397286	−	0.917695i	$-0.369952\pi$
$109$	904.218	0.794573	0.397286	+	0.917695i	$0.369952\pi$
$110$	0	0
$111$	806.848	0.689934
$112$	0	0
$113$	843.350	0.702086	0.351043	−	0.936359i	$-0.385827\pi$
$113$	843.350	0.702086	0.351043	+	0.936359i	$0.385827\pi$
$114$	−274.420	−0.225454
$115$	0	0
$116$	−321.232	−0.257117
$117$	−141.186	−0.111561
$118$	−207.119	−0.161583
$119$	0	0
$120$	0	0
$121$	1680.81	1.26282
$122$	−23.2450	−0.0172501
$123$	325.407	0.238544
$124$	1489.82	1.07895
$125$	0	0
$126$	0	0
$127$	−2030.50	−1.41872	−0.709362	−	0.704845i	$-0.751017\pi$
$127$	−2030.50	−1.41872	−0.709362	+	0.704845i	$0.751017\pi$
$128$	−953.043	−0.658109
$129$	−1255.68	−0.857027
$130$	0	0
$131$	−2872.47	−1.91579	−0.957897	−	0.287113i	$-0.907305\pi$
$131$	−2872.47	−1.91579	−0.957897	+	0.287113i	$0.907305\pi$
$132$	−1812.35	−1.19504
$133$	0	0
$134$	−5.25348	−0.00338680
$135$	0	0
$136$	3.35446	0.00211502
$137$	1621.62	1.01127	0.505636	−	0.862747i	$-0.331258\pi$
$137$	1621.62	1.01127	0.505636	+	0.862747i	$0.331258\pi$
$138$	−110.004	−0.0678562
$139$	−2617.72	−1.59735	−0.798677	−	0.601760i	$-0.794466\pi$
$139$	−2617.72	−1.59735	−0.798677	+	0.601760i	$0.794466\pi$
$140$	0	0
$141$	2303.71	1.37594
$142$	−247.908	−0.146507
$143$	878.482	0.513723
$144$	−511.120	−0.295787
$145$	0	0
$146$	−411.166	−0.233071
$147$	0	0
$148$	−1465.64	−0.814021
$149$	2252.63	1.23854	0.619270	−	0.785178i	$-0.287429\pi$
$149$	2252.63	1.23854	0.619270	+	0.785178i	$0.287429\pi$
$150$	0	0
$151$	−1849.61	−0.996813	−0.498407	−	0.866943i	$-0.666081\pi$
$151$	−1849.61	−0.996813	−0.498407	+	0.866943i	$0.666081\pi$
$152$	1013.37	0.540757
$153$	3.72257	0.00196700
$154$	0	0
$155$	0	0
$156$	−528.624	−0.271307
$157$	−3337.59	−1.69661	−0.848307	−	0.529505i	$-0.822378\pi$
$157$	−3337.59	−1.69661	−0.848307	+	0.529505i	$0.822378\pi$
$158$	−433.471	−0.218260
$159$	2819.71	1.40640
$160$	0	0
$161$	0	0
$162$	−208.508	−0.101123
$163$	471.099	0.226376	0.113188	−	0.993574i	$-0.463894\pi$
$163$	471.099	0.226376	0.113188	+	0.993574i	$0.463894\pi$
$164$	−591.103	−0.281447
$165$	0	0
$166$	−532.952	−0.249187
$167$	−3676.72	−1.70367	−0.851836	−	0.523809i	$-0.824511\pi$
$167$	−3676.72	−1.70367	−0.851836	+	0.523809i	$0.824511\pi$
$168$	0	0
$169$	−1940.77	−0.883371
$170$	0	0
$171$	1124.57	0.502913
$172$	2280.95	1.01117
$173$	1009.11	0.443475	0.221737	−	0.975106i	$-0.428827\pi$
$173$	1009.11	0.443475	0.221737	+	0.975106i	$0.428827\pi$
$174$	89.2663	0.0388923
$175$	0	0
$176$	3180.27	1.36206
$177$	−1749.48	−0.742934
$178$	−172.417	−0.0726024
$179$	3201.27	1.33673	0.668364	−	0.743835i	$-0.266995\pi$
$179$	3201.27	1.33673	0.668364	+	0.743835i	$0.266995\pi$
$180$	0	0
$181$	1970.14	0.809056	0.404528	−	0.914526i	$-0.367436\pi$
$181$	1970.14	0.809056	0.404528	+	0.914526i	$0.367436\pi$
$182$	0	0
$183$	−196.345	−0.0793130
$184$	406.219	0.162755
$185$	0	0
$186$	−414.003	−0.163205
$187$	−23.1624	−0.00905778
$188$	−4184.70	−1.62341
$189$	0	0
$190$	0	0
$191$	3953.39	1.49768	0.748842	−	0.662749i	$-0.230610\pi$
$191$	3953.39	1.49768	0.748842	+	0.662749i	$0.230610\pi$
$192$	−1776.87	−0.667888
$193$	−2425.45	−0.904598	−0.452299	−	0.891866i	$-0.649396\pi$
$193$	−2425.45	−0.904598	−0.452299	+	0.891866i	$0.649396\pi$
$194$	−715.385	−0.264751
$195$	0	0
$196$	0	0
$197$	−3126.50	−1.13073	−0.565366	−	0.824840i	$-0.691265\pi$
$197$	−3126.50	−1.13073	−0.565366	+	0.824840i	$0.691265\pi$
$198$	−244.338	−0.0876989
$199$	3566.13	1.27033	0.635166	−	0.772376i	$-0.280932\pi$
$199$	3566.13	1.27033	0.635166	+	0.772376i	$0.280932\pi$
$200$	0	0
$201$	−44.3749	−0.0155720
$202$	61.1046	0.0212837
$203$	0	0
$204$	13.9379	0.00478358
$205$	0	0
$206$	−330.933	−0.111928
$207$	450.795	0.151364
$208$	927.618	0.309225
$209$	−6997.27	−2.31584
$210$	0	0
$211$	−3699.18	−1.20693	−0.603465	−	0.797389i	$-0.706214\pi$
$211$	−3699.18	−1.20693	−0.603465	+	0.797389i	$0.706214\pi$
$212$	−5122.02	−1.65935
$213$	−2094.02	−0.673614
$214$	−1044.81	−0.333746
$215$	0	0
$216$	1213.88	0.382380
$217$	0	0
$218$	456.435	0.141806
$219$	−3473.03	−1.07162
$220$	0	0
$221$	−6.75599	−0.00205637
$222$	407.284	0.123131
$223$	−102.041	−0.0306419	−0.0153210	−	0.999883i	$-0.504877\pi$
$223$	−102.041	−0.0306419	−0.0153210	+	0.999883i	$0.504877\pi$
$224$	0	0
$225$	0	0
$226$	425.709	0.125300
$227$	5739.91	1.67829	0.839144	−	0.543909i	$-0.183057\pi$
$227$	5739.91	1.67829	0.839144	+	0.543909i	$0.183057\pi$
$228$	4210.59	1.22304
$229$	843.125	0.243298	0.121649	−	0.992573i	$-0.461182\pi$
$229$	843.125	0.243298	0.121649	+	0.992573i	$0.461182\pi$
$230$	0	0
$231$	0	0
$232$	−329.640	−0.0932841
$233$	2017.65	0.567300	0.283650	−	0.958928i	$-0.408455\pi$
$233$	2017.65	0.567300	0.283650	+	0.958928i	$0.408455\pi$
$234$	−71.2683	−0.0199101
$235$	0	0
$236$	3177.94	0.876553
$237$	−3661.43	−1.00352
$238$	0	0
$239$	243.685	0.0659526	0.0329763	−	0.999456i	$-0.489501\pi$
$239$	243.685	0.0659526	0.0329763	+	0.999456i	$0.489501\pi$
$240$	0	0
$241$	3060.58	0.818048	0.409024	−	0.912524i	$-0.365869\pi$
$241$	3060.58	0.818048	0.409024	+	0.912524i	$0.365869\pi$
$242$	848.448	0.225373
$243$	2362.47	0.623674
$244$	356.662	0.0935777
$245$	0	0
$246$	164.260	0.0425725
$247$	−2040.95	−0.525760
$248$	1528.82	0.391452
$249$	−4501.72	−1.14572
$250$	0	0
$251$	6594.14	1.65824	0.829120	−	0.559070i	$-0.188842\pi$
$251$	6594.14	1.65824	0.829120	+	0.559070i	$0.188842\pi$
$252$	0	0
$253$	−2804.92	−0.697012
$254$	−1024.96	−0.253197
$255$	0	0
$256$	2852.80	0.696484
$257$	512.646	0.124428	0.0622139	−	0.998063i	$-0.480184\pi$
$257$	512.646	0.124428	0.0622139	+	0.998063i	$0.480184\pi$
$258$	−633.847	−0.152952
$259$	0	0
$260$	0	0
$261$	−365.813	−0.0867557
$262$	−1449.98	−0.341908
$263$	3560.04	0.834681	0.417341	−	0.908750i	$-0.362962\pi$
$263$	3560.04	0.834681	0.417341	+	0.908750i	$0.362962\pi$
$264$	−1859.79	−0.433568
$265$	0	0
$266$	0	0
$267$	−1456.37	−0.333814
$268$	80.6072	0.0183726
$269$	549.744	0.124604	0.0623020	−	0.998057i	$-0.480156\pi$
$269$	549.744	0.124604	0.0623020	+	0.998057i	$0.480156\pi$
$270$	0	0
$271$	3944.91	0.884267	0.442134	−	0.896949i	$-0.354222\pi$
$271$	3944.91	0.884267	0.442134	+	0.896949i	$0.354222\pi$
$272$	−24.4580	−0.00545214
$273$	0	0
$274$	818.567	0.180480
$275$	0	0
$276$	1687.86	0.368105
$277$	1254.67	0.272150	0.136075	−	0.990699i	$-0.456551\pi$
$277$	1254.67	0.272150	0.136075	+	0.990699i	$0.456551\pi$
$278$	−1321.38	−0.285077
$279$	1696.58	0.364056
$280$	0	0
$281$	−681.496	−0.144679	−0.0723393	−	0.997380i	$-0.523046\pi$
$281$	−681.496	−0.144679	−0.0723393	+	0.997380i	$0.523046\pi$
$282$	1162.88	0.245561
$283$	−5946.88	−1.24914	−0.624568	−	0.780970i	$-0.714725\pi$
$283$	−5946.88	−1.24914	−0.624568	+	0.780970i	$0.714725\pi$
$284$	3803.79	0.794766
$285$	0	0
$286$	443.444	0.0916831
$287$	0	0
$288$	−818.815	−0.167532
$289$	−4912.82	−0.999964
$290$	0	0
$291$	−6042.69	−1.21728
$292$	6308.77	1.26436
$293$	2171.03	0.432877	0.216438	−	0.976296i	$-0.430556\pi$
$293$	2171.03	0.432877	0.216438	+	0.976296i	$0.430556\pi$
$294$	0	0
$295$	0	0
$296$	−1504.00	−0.295333
$297$	−8381.79	−1.63758
$298$	1137.09	0.221040
$299$	−818.137	−0.158241
$300$	0	0
$301$	0	0
$302$	−933.651	−0.177899
$303$	516.137	0.0978590
$304$	−7388.64	−1.39397
$305$	0	0
$306$	1.87909	0.000351047 0
$307$	−3644.59	−0.677549	−0.338775	−	0.940868i	$-0.610012\pi$
$307$	−3644.59	−0.677549	−0.338775	+	0.940868i	$0.610012\pi$
$308$	0	0
$309$	−2795.32	−0.514628
$310$	0	0
$311$	−2584.19	−0.471177	−0.235589	−	0.971853i	$-0.575702\pi$
$311$	−2584.19	−0.471177	−0.235589	+	0.971853i	$0.575702\pi$
$312$	−542.461	−0.0984320
$313$	6693.63	1.20878	0.604388	−	0.796690i	$-0.293418\pi$
$313$	6693.63	1.20878	0.604388	+	0.796690i	$0.293418\pi$
$314$	−1684.76	−0.302791
$315$	0	0
$316$	6650.99	1.18401
$317$	3843.36	0.680962	0.340481	−	0.940251i	$-0.389410\pi$
$317$	3843.36	0.680962	0.340481	+	0.940251i	$0.389410\pi$
$318$	1423.35	0.250998
$319$	2276.15	0.399498
$320$	0	0
$321$	−8825.26	−1.53451
$322$	0	0
$323$	53.8126	0.00927002
$324$	3199.26	0.548570
$325$	0	0
$326$	237.803	0.0404009
$327$	3855.40	0.652000
$328$	−606.575	−0.102111
$329$	0	0
$330$	0	0
$331$	−821.922	−0.136486	−0.0682431	−	0.997669i	$-0.521739\pi$
$331$	−821.922	−0.136486	−0.0682431	+	0.997669i	$0.521739\pi$
$332$	8177.39	1.35179
$333$	−1669.05	−0.274664
$334$	−1855.95	−0.304051
$335$	0	0
$336$	0	0
$337$	−1032.55	−0.166904	−0.0834520	−	0.996512i	$-0.526595\pi$
$337$	−1032.55	−0.166904	−0.0834520	+	0.996512i	$0.526595\pi$
$338$	−979.667	−0.157653
$339$	3595.87	0.576108
$340$	0	0
$341$	−10556.4	−1.67643
$342$	567.665	0.0897538
$343$	0	0
$344$	2340.65	0.366859
$345$	0	0
$346$	509.382	0.0791461
$347$	2587.70	0.400332	0.200166	−	0.979762i	$-0.435852\pi$
$347$	2587.70	0.400332	0.200166	+	0.979762i	$0.435852\pi$
$348$	−1369.67	−0.210982
$349$	8454.24	1.29669	0.648345	−	0.761346i	$-0.275461\pi$
$349$	8454.24	1.29669	0.648345	+	0.761346i	$0.275461\pi$
$350$	0	0
$351$	−2444.79	−0.371776
$352$	5094.80	0.771460
$353$	−4129.63	−0.622658	−0.311329	−	0.950302i	$-0.600774\pi$
$353$	−4129.63	−0.622658	−0.311329	+	0.950302i	$0.600774\pi$
$354$	−883.111	−0.132590
$355$	0	0
$356$	2645.50	0.393852
$357$	0	0
$358$	1615.95	0.238563
$359$	−4536.69	−0.666957	−0.333478	−	0.942758i	$-0.608222\pi$
$359$	−4536.69	−0.666957	−0.333478	+	0.942758i	$0.608222\pi$
$360$	0	0
$361$	9397.56	1.37011
$362$	994.492	0.144390
$363$	7166.65	1.03623
$364$	0	0
$365$	0	0
$366$	−99.1120	−0.0141548
$367$	1643.00	0.233689	0.116844	−	0.993150i	$-0.462722\pi$
$367$	1643.00	0.233689	0.116844	+	0.993150i	$0.462722\pi$
$368$	−2961.81	−0.419552
$369$	−673.137	−0.0949650
$370$	0	0
$371$	0	0
$372$	6352.29	0.885352
$373$	−9019.53	−1.25205	−0.626024	−	0.779804i	$-0.715319\pi$
$373$	−9019.53	−1.25205	−0.626024	+	0.779804i	$0.715319\pi$
$374$	−11.6920	−0.00161652
$375$	0	0
$376$	−4294.23	−0.588984
$377$	663.904	0.0906971
$378$	0	0
$379$	−2463.63	−0.333900	−0.166950	−	0.985965i	$-0.553392\pi$
$379$	−2463.63	−0.333900	−0.166950	+	0.985965i	$0.553392\pi$
$380$	0	0
$381$	−8657.63	−1.16416
$382$	1995.61	0.267289
$383$	−11007.2	−1.46852	−0.734261	−	0.678867i	$-0.762471\pi$
$383$	−11007.2	−1.46852	−0.734261	+	0.678867i	$0.762471\pi$
$384$	−4063.58	−0.540023
$385$	0	0
$386$	−1224.33	−0.161442
$387$	2597.50	0.341184
$388$	10976.6	1.43621
$389$	6162.79	0.803254	0.401627	−	0.915803i	$-0.368445\pi$
$389$	6162.79	0.803254	0.401627	+	0.915803i	$0.368445\pi$
$390$	0	0
$391$	21.5713	0.00279005
$392$	0	0
$393$	−12247.6	−1.57204
$394$	−1578.21	−0.201799
$395$	0	0
$396$	3749.03	0.475747
$397$	−15137.9	−1.91373	−0.956864	−	0.290537i	$-0.906166\pi$
$397$	−15137.9	−1.91373	−0.956864	+	0.290537i	$0.906166\pi$
$398$	1800.12	0.226714
$399$	0	0
$400$	0	0
$401$	3302.29	0.411243	0.205622	−	0.978632i	$-0.434078\pi$
$401$	3302.29	0.411243	0.205622	+	0.978632i	$0.434078\pi$
$402$	−22.3997	−0.00277910
$403$	−3079.08	−0.380596
$404$	−937.564	−0.115459
$405$	0	0
$406$	0	0
$407$	10385.1	1.26479
$408$	14.3027	0.00173552
$409$	−4576.40	−0.553273	−0.276636	−	0.960975i	$-0.589220\pi$
$409$	−4576.40	−0.553273	−0.276636	+	0.960975i	$0.589220\pi$
$410$	0	0
$411$	6914.25	0.829817
$412$	5077.70	0.607186
$413$	0	0
$414$	227.554	0.0270137
$415$	0	0
$416$	1486.05	0.175143
$417$	−11161.4	−1.31074
$418$	−3532.11	−0.413304
$419$	6778.05	0.790285	0.395142	−	0.918620i	$-0.370695\pi$
$419$	6778.05	0.790285	0.395142	+	0.918620i	$0.370695\pi$
$420$	0	0
$421$	−3952.68	−0.457582	−0.228791	−	0.973476i	$-0.573477\pi$
$421$	−3952.68	−0.457582	−0.228791	+	0.973476i	$0.573477\pi$
$422$	−1867.29	−0.215398
$423$	−4765.46	−0.547765
$424$	−5256.08	−0.602024
$425$	0	0
$426$	−1057.03	−0.120219
$427$	0	0
$428$	16031.1	1.81050
$429$	3745.67	0.421544
$430$	0	0
$431$	−3435.10	−0.383905	−0.191952	−	0.981404i	$-0.561482\pi$
$431$	−3435.10	−0.383905	−0.191952	+	0.981404i	$0.561482\pi$
$432$	−8850.60	−0.985705
$433$	−5457.82	−0.605742	−0.302871	−	0.953032i	$-0.597945\pi$
$433$	−5457.82	−0.605742	−0.302871	+	0.953032i	$0.597945\pi$
$434$	0	0
$435$	0	0
$436$	−7003.35	−0.769265
$437$	6516.61	0.713344
$438$	−1753.13	−0.191250
$439$	−4188.10	−0.455323	−0.227662	−	0.973740i	$-0.573108\pi$
$439$	−4188.10	−0.455323	−0.227662	+	0.973740i	$0.573108\pi$
$440$	0	0
$441$	0	0
$442$	−3.41031	−0.000366996 0
$443$	−2097.50	−0.224955	−0.112478	−	0.993654i	$-0.535879\pi$
$443$	−2097.50	−0.224955	−0.112478	+	0.993654i	$0.535879\pi$
$444$	−6249.19	−0.667959
$445$	0	0
$446$	−51.5085	−0.00546860
$447$	9604.74	1.01631
$448$	0	0
$449$	11318.2	1.18962	0.594809	−	0.803867i	$-0.297228\pi$
$449$	11318.2	1.18962	0.594809	+	0.803867i	$0.297228\pi$
$450$	0	0
$451$	4188.37	0.437301
$452$	−6531.91	−0.679724
$453$	−7886.34	−0.817952
$454$	2897.41	0.299521
$455$	0	0
$456$	4320.79	0.443728
$457$	11696.4	1.19723	0.598617	−	0.801036i	$-0.295717\pi$
$457$	11696.4	1.19723	0.598617	+	0.801036i	$0.295717\pi$
$458$	425.596	0.0434209
$459$	64.4603	0.00655501
$460$	0	0
$461$	−10685.0	−1.07951	−0.539753	−	0.841823i	$-0.681482\pi$
$461$	−10685.0	−1.07951	−0.539753	+	0.841823i	$0.681482\pi$
$462$	0	0
$463$	10407.2	1.04463	0.522315	−	0.852752i	$-0.325068\pi$
$463$	10407.2	1.04463	0.522315	+	0.852752i	$0.325068\pi$
$464$	2403.46	0.240469
$465$	0	0
$466$	1018.48	0.101245
$467$	−2978.24	−0.295111	−0.147555	−	0.989054i	$-0.547140\pi$
$467$	−2978.24	−0.295111	−0.147555	+	0.989054i	$0.547140\pi$
$468$	1093.51	0.108008
$469$	0	0
$470$	0	0
$471$	−14230.8	−1.39219
$472$	3261.12	0.318020
$473$	−16162.1	−1.57111
$474$	−1848.23	−0.179097
$475$	0	0
$476$	0	0
$477$	−5832.86	−0.559892
$478$	123.008	0.0117704
$479$	14214.3	1.35588	0.677940	−	0.735117i	$-0.262873\pi$
$479$	14214.3	1.35588	0.677940	+	0.735117i	$0.262873\pi$
$480$	0	0
$481$	3029.11	0.287142
$482$	1544.93	0.145995
$483$	0	0
$484$	−13018.2	−1.22260
$485$	0	0
$486$	1192.54	0.111306
$487$	−18097.0	−1.68389	−0.841946	−	0.539562i	$-0.818590\pi$
$487$	−18097.0	−1.68389	−0.841946	+	0.539562i	$0.818590\pi$
$488$	365.998	0.0339507
$489$	2008.67	0.185757
$490$	0	0
$491$	−13750.8	−1.26388	−0.631940	−	0.775017i	$-0.717741\pi$
$491$	−13750.8	−1.26388	−0.631940	+	0.775017i	$0.717741\pi$
$492$	−2520.34	−0.230947
$493$	−17.5048	−0.00159914
$494$	−1030.24	−0.0938314
$495$	0	0
$496$	−11146.9	−1.00909
$497$	0	0
$498$	−2272.40	−0.204475
$499$	15684.2	1.40706	0.703528	−	0.710667i	$-0.251607\pi$
$499$	15684.2	1.40706	0.703528	+	0.710667i	$0.251607\pi$
$500$	0	0
$501$	−15676.8	−1.39798
$502$	3328.61	0.295943
$503$	977.937	0.0866880	0.0433440	−	0.999060i	$-0.486199\pi$
$503$	977.937	0.0866880	0.0433440	+	0.999060i	$0.486199\pi$
$504$	0	0
$505$	0	0
$506$	−1415.88	−0.124394
$507$	−8275.02	−0.724865
$508$	15726.6	1.37354
$509$	19664.6	1.71242	0.856208	−	0.516632i	$-0.172814\pi$
$509$	19664.6	1.71242	0.856208	+	0.516632i	$0.172814\pi$
$510$	0	0
$511$	0	0
$512$	9064.39	0.782409
$513$	19473.2	1.67595
$514$	258.775	0.0222064
$515$	0	0
$516$	9725.49	0.829730
$517$	29651.5	2.52238
$518$	0	0
$519$	4302.63	0.363901
$520$	0	0
$521$	−12522.8	−1.05304	−0.526519	−	0.850163i	$-0.676503\pi$
$521$	−12522.8	−1.05304	−0.526519	+	0.850163i	$0.676503\pi$
$522$	−184.656	−0.0154831
$523$	1104.42	0.0923382	0.0461691	−	0.998934i	$-0.485299\pi$
$523$	1104.42	0.0923382	0.0461691	+	0.998934i	$0.485299\pi$
$524$	22247.8	1.85477
$525$	0	0
$526$	1797.05	0.148964
$527$	81.1843	0.00671052
$528$	13560.0	1.11766
$529$	−9554.75	−0.785301
$530$	0	0
$531$	3618.98	0.295763
$532$	0	0
$533$	1221.66	0.0992794
$534$	−735.152	−0.0595751
$535$	0	0
$536$	82.7170	0.00666573
$537$	13649.5	1.09687
$538$	277.502	0.0222378
$539$	0	0
$540$	0	0
$541$	−2126.45	−0.168989	−0.0844945	−	0.996424i	$-0.526928\pi$
$541$	−2126.45	−0.168989	−0.0844945	+	0.996424i	$0.526928\pi$
$542$	1991.33	0.157813
$543$	8400.25	0.663884
$544$	−39.1817	−0.00308805
$545$	0	0
$546$	0	0
$547$	2137.04	0.167044	0.0835220	−	0.996506i	$-0.473383\pi$
$547$	2137.04	0.167044	0.0835220	+	0.996506i	$0.473383\pi$
$548$	−12559.8	−0.979063
$549$	406.160	0.0315747
$550$	0	0
$551$	−5288.11	−0.408859
$552$	1732.03	0.133551
$553$	0	0
$554$	633.335	0.0485701
$555$	0	0
$556$	20274.8	1.54648
$557$	19435.1	1.47844	0.739219	−	0.673465i	$-0.235195\pi$
$557$	19435.1	1.47844	0.739219	+	0.673465i	$0.235195\pi$
$558$	856.407	0.0649724
$559$	−4714.14	−0.356685
$560$	0	0
$561$	−98.7598	−0.00743252
$562$	−344.008	−0.0258205
$563$	1467.52	0.109855	0.0549276	−	0.998490i	$-0.482507\pi$
$563$	1467.52	0.109855	0.0549276	+	0.998490i	$0.482507\pi$
$564$	−17842.7	−1.33212
$565$	0	0
$566$	−3001.89	−0.222931
$567$	0	0
$568$	3903.35	0.288347
$569$	15080.3	1.11107	0.555536	−	0.831493i	$-0.312513\pi$
$569$	15080.3	1.11107	0.555536	+	0.831493i	$0.312513\pi$
$570$	0	0
$571$	15800.8	1.15804	0.579022	−	0.815312i	$-0.303435\pi$
$571$	15800.8	1.15804	0.579022	+	0.815312i	$0.303435\pi$
$572$	−6804.02	−0.497361
$573$	16856.5	1.22895
$574$	0	0
$575$	0	0
$576$	3675.63	0.265888
$577$	6165.64	0.444851	0.222425	−	0.974950i	$-0.428603\pi$
$577$	6165.64	0.444851	0.222425	+	0.974950i	$0.428603\pi$
$578$	−2479.91	−0.178461
$579$	−10341.6	−0.742283
$580$	0	0
$581$	0	0
$582$	−3050.25	−0.217246
$583$	36293.0	2.57822
$584$	6473.89	0.458719
$585$	0	0
$586$	1095.90	0.0772547
$587$	23104.0	1.62454	0.812268	−	0.583285i	$-0.198233\pi$
$587$	23104.0	1.62454	0.812268	+	0.583285i	$0.198233\pi$
$588$	0	0
$589$	24525.4	1.71571
$590$	0	0
$591$	−13330.8	−0.927841
$592$	10965.9	0.761313
$593$	6544.43	0.453200	0.226600	−	0.973988i	$-0.427239\pi$
$593$	6544.43	0.453200	0.226600	+	0.973988i	$0.427239\pi$
$594$	−4230.99	−0.292255
$595$	0	0
$596$	−17447.0	−1.19909
$597$	15205.2	1.04239
$598$	−412.982	−0.0282410
$599$	3215.71	0.219349	0.109675	−	0.993968i	$-0.465019\pi$
$599$	3215.71	0.219349	0.109675	+	0.993968i	$0.465019\pi$
$600$	0	0
$601$	234.653	0.0159263	0.00796313	−	0.999968i	$-0.497465\pi$
$601$	234.653	0.0159263	0.00796313	+	0.999968i	$0.497465\pi$
$602$	0	0
$603$	91.7939	0.00619923
$604$	14325.6	0.965064
$605$	0	0
$606$	260.537	0.0174647
$607$	−14528.7	−0.971501	−0.485750	−	0.874098i	$-0.661454\pi$
$607$	−14528.7	−0.971501	−0.485750	+	0.874098i	$0.661454\pi$
$608$	−11836.6	−0.789537
$609$	0	0
$610$	0	0
$611$	8648.71	0.572650
$612$	−28.8320	−0.00190435
$613$	−11332.7	−0.746696	−0.373348	−	0.927691i	$-0.621790\pi$
$613$	−11332.7	−0.746696	−0.373348	+	0.927691i	$0.621790\pi$
$614$	−1839.73	−0.120921
$615$	0	0
$616$	0	0
$617$	17400.2	1.13534	0.567672	−	0.823255i	$-0.307844\pi$
$617$	17400.2	1.13534	0.567672	+	0.823255i	$0.307844\pi$
$618$	−1411.03	−0.0918446
$619$	11987.8	0.778402	0.389201	−	0.921153i	$-0.372751\pi$
$619$	11987.8	0.778402	0.389201	+	0.921153i	$0.372751\pi$
$620$	0	0
$621$	7806.02	0.504420
$622$	−1304.46	−0.0840900
$623$	0	0
$624$	3955.17	0.253740
$625$	0	0
$626$	3378.84	0.215728
$627$	−29834.9	−1.90030
$628$	25850.3	1.64258
$629$	−79.8667	−0.00506279
$630$	0	0
$631$	−3800.38	−0.239764	−0.119882	−	0.992788i	$-0.538252\pi$
$631$	−3800.38	−0.239764	−0.119882	+	0.992788i	$0.538252\pi$
$632$	6825.08	0.429568
$633$	−15772.6	−0.990368
$634$	1940.07	0.121530
$635$	0	0
$636$	−21839.2	−1.36161
$637$	0	0
$638$	1148.96	0.0712976
$639$	4331.69	0.268167
$640$	0	0
$641$	12294.2	0.757552	0.378776	−	0.925488i	$-0.376345\pi$
$641$	12294.2	0.757552	0.378776	+	0.925488i	$0.376345\pi$
$642$	−4454.85	−0.273861
$643$	14615.4	0.896382	0.448191	−	0.893938i	$-0.352068\pi$
$643$	14615.4	0.896382	0.448191	+	0.893938i	$0.352068\pi$
$644$	0	0
$645$	0	0
$646$	27.1637	0.00165440
$647$	−16047.0	−0.975071	−0.487536	−	0.873103i	$-0.662104\pi$
$647$	−16047.0	−0.975071	−0.487536	+	0.873103i	$0.662104\pi$
$648$	3283.00	0.199025
$649$	−22517.9	−1.36195
$650$	0	0
$651$	0	0
$652$	−3648.75	−0.219166
$653$	−623.425	−0.0373607	−0.0186803	−	0.999826i	$-0.505946\pi$
$653$	−623.425	−0.0373607	−0.0186803	+	0.999826i	$0.505946\pi$
$654$	1946.14	0.116361
$655$	0	0
$656$	4422.63	0.263224
$657$	7184.30	0.426616
$658$	0	0
$659$	963.459	0.0569515	0.0284758	−	0.999594i	$-0.490935\pi$
$659$	963.459	0.0569515	0.0284758	+	0.999594i	$0.490935\pi$
$660$	0	0
$661$	−24183.6	−1.42304	−0.711522	−	0.702664i	$-0.751994\pi$
$661$	−24183.6	−1.42304	−0.711522	+	0.702664i	$0.751994\pi$
$662$	−414.893	−0.0243584
$663$	−28.8061	−0.00168739
$664$	8391.43	0.490438
$665$	0	0
$666$	−842.507	−0.0490187
$667$	−2119.79	−0.123057
$668$	28476.9	1.64941
$669$	−435.080	−0.0251438
$670$	0	0
$671$	−2527.20	−0.145397
$672$	0	0
$673$	−14167.0	−0.811437	−0.405718	−	0.913998i	$-0.632979\pi$
$673$	−14167.0	−0.811437	−0.405718	+	0.913998i	$0.632979\pi$
$674$	−521.215	−0.0297870
$675$	0	0
$676$	15031.6	0.855235
$677$	−19397.1	−1.10117	−0.550584	−	0.834780i	$-0.685595\pi$
$677$	−19397.1	−1.10117	−0.550584	+	0.834780i	$0.685595\pi$
$678$	1815.14	0.102817
$679$	0	0
$680$	0	0
$681$	24473.8	1.37715
$682$	−5328.71	−0.299189
$683$	−20984.8	−1.17564	−0.587820	−	0.808992i	$-0.700014\pi$
$683$	−20984.8	−1.17564	−0.587820	+	0.808992i	$0.700014\pi$
$684$	−8710.02	−0.486895
$685$	0	0
$686$	0	0
$687$	3594.91	0.199643
$688$	−17066.1	−0.945694
$689$	10585.9	0.585328
$690$	0	0
$691$	18079.2	0.995320	0.497660	−	0.867372i	$-0.334193\pi$
$691$	18079.2	0.995320	0.497660	+	0.867372i	$0.334193\pi$
$692$	−7815.75	−0.429350
$693$	0	0
$694$	1306.23	0.0714465
$695$	0	0
$696$	−1405.52	−0.0765459
$697$	−32.2107	−0.00175046
$698$	4267.56	0.231418
$699$	8602.86	0.465508
$700$	0	0
$701$	18660.6	1.00542	0.502711	−	0.864455i	$-0.332336\pi$
$701$	18660.6	1.00542	0.502711	+	0.864455i	$0.332336\pi$
$702$	−1234.09	−0.0663500
$703$	−24127.4	−1.29443
$704$	−22870.4	−1.22438
$705$	0	0
$706$	−2084.57	−0.111125
$707$	0	0
$708$	13550.1	0.719271
$709$	−24545.4	−1.30017	−0.650085	−	0.759862i	$-0.725267\pi$
$709$	−24545.4	−1.30017	−0.650085	+	0.759862i	$0.725267\pi$
$710$	0	0
$711$	7574.02	0.399505
$712$	2714.74	0.142892
$713$	9831.26	0.516387
$714$	0	0
$715$	0	0
$716$	−24794.5	−1.29415
$717$	1039.02	0.0541185
$718$	−2290.05	−0.119030
$719$	7300.32	0.378659	0.189330	−	0.981914i	$-0.439369\pi$
$719$	7300.32	0.378659	0.189330	+	0.981914i	$0.439369\pi$
$720$	0	0
$721$	0	0
$722$	4743.74	0.244520
$723$	13049.7	0.671263
$724$	−15259.1	−0.783286
$725$	0	0
$726$	3617.61	0.184934
$727$	15466.0	0.788998	0.394499	−	0.918896i	$-0.370918\pi$
$727$	15466.0	0.788998	0.394499	+	0.918896i	$0.370918\pi$
$728$	0	0
$729$	21225.8	1.07838
$730$	0	0
$731$	124.295	0.00628893
$732$	1520.73	0.0767868
$733$	−13830.5	−0.696917	−0.348459	−	0.937324i	$-0.613295\pi$
$733$	−13830.5	−0.696917	−0.348459	+	0.937324i	$0.613295\pi$
$734$	829.358	0.0417060
$735$	0	0
$736$	−4744.83	−0.237631
$737$	−571.157	−0.0285466
$738$	−339.788	−0.0169482
$739$	11714.5	0.583118	0.291559	−	0.956553i	$-0.405826\pi$
$739$	11714.5	0.583118	0.291559	+	0.956553i	$0.405826\pi$
$740$	0	0
$741$	−8702.20	−0.431422
$742$	0	0
$743$	−28963.4	−1.43010	−0.715050	−	0.699074i	$-0.753596\pi$
$743$	−28963.4	−1.43010	−0.715050	+	0.699074i	$0.753596\pi$
$744$	6518.56	0.321212
$745$	0	0
$746$	−4552.91	−0.223450
$747$	9312.26	0.456115
$748$	179.397	0.00876928
$749$	0	0
$750$	0	0
$751$	6331.99	0.307667	0.153833	−	0.988097i	$-0.450838\pi$
$751$	6331.99	0.307667	0.153833	+	0.988097i	$0.450838\pi$
$752$	31310.0	1.51829
$753$	28116.0	1.36070
$754$	335.128	0.0161865
$755$	0	0
$756$	0	0
$757$	6256.80	0.300406	0.150203	−	0.988655i	$-0.452007\pi$
$757$	6256.80	0.300406	0.150203	+	0.988655i	$0.452007\pi$
$758$	−1243.60	−0.0595905
$759$	−11959.6	−0.571945
$760$	0	0
$761$	−13953.7	−0.664679	−0.332340	−	0.943160i	$-0.607838\pi$
$761$	−13953.7	−0.664679	−0.332340	+	0.943160i	$0.607838\pi$
$762$	−4370.23	−0.207765
$763$	0	0
$764$	−30619.8	−1.44998
$765$	0	0
$766$	−5556.28	−0.262084
$767$	−6568.00	−0.309200
$768$	12163.7	0.571512
$769$	−34278.1	−1.60741	−0.803707	−	0.595025i	$-0.797142\pi$
$769$	−34278.1	−1.60741	−0.803707	+	0.595025i	$0.797142\pi$
$770$	0	0
$771$	2185.82	0.102101
$772$	18785.5	0.875786
$773$	8769.31	0.408034	0.204017	−	0.978967i	$-0.434600\pi$
$773$	8769.31	0.408034	0.204017	+	0.978967i	$0.434600\pi$
$774$	1311.18	0.0608905
$775$	0	0
$776$	11263.9	0.521069
$777$	0	0
$778$	3110.88	0.143355
$779$	−9730.73	−0.447547
$780$	0	0
$781$	−26952.5	−1.23487
$782$	10.8889	0.000497934 0
$783$	−6334.45	−0.289112
$784$	0	0
$785$	0	0
$786$	−6182.40	−0.280558
$787$	−3546.15	−0.160618	−0.0803092	−	0.996770i	$-0.525591\pi$
$787$	−3546.15	−0.160618	−0.0803092	+	0.996770i	$0.525591\pi$
$788$	24215.4	1.09472
$789$	15179.3	0.684912
$790$	0	0
$791$	0	0
$792$	3847.16	0.172604
$793$	−737.130	−0.0330091
$794$	−7641.37	−0.341539
$795$	0	0
$796$	−27620.3	−1.22987
$797$	−18169.4	−0.807520	−0.403760	−	0.914865i	$-0.632297\pi$
$797$	−18169.4	−0.807520	−0.403760	+	0.914865i	$0.632297\pi$
$798$	0	0
$799$	−228.035	−0.0100968
$800$	0	0
$801$	3012.65	0.132892
$802$	1666.94	0.0733938
$803$	−44702.0	−1.96451
$804$	343.692	0.0150760
$805$	0	0
$806$	−1554.27	−0.0679241
$807$	2343.99	0.102246
$808$	−962.104	−0.0418895
$809$	11853.0	0.515115	0.257557	−	0.966263i	$-0.417082\pi$
$809$	11853.0	0.515115	0.257557	+	0.966263i	$0.417082\pi$
$810$	0	0
$811$	11921.1	0.516161	0.258080	−	0.966123i	$-0.416910\pi$
$811$	11921.1	0.516161	0.258080	+	0.966123i	$0.416910\pi$
$812$	0	0
$813$	16820.3	0.725601
$814$	5242.22	0.225724
$815$	0	0
$816$	−104.284	−0.00447384
$817$	37548.9	1.60792
$818$	−2310.09	−0.0987415
$819$	0	0
$820$	0	0
$821$	−42094.1	−1.78940	−0.894698	−	0.446671i	$-0.852610\pi$
$821$	−42094.1	−1.78940	−0.894698	+	0.446671i	$0.852610\pi$
$822$	3490.20	0.148096
$823$	−1622.17	−0.0687064	−0.0343532	−	0.999410i	$-0.510937\pi$
$823$	−1622.17	−0.0687064	−0.0343532	+	0.999410i	$0.510937\pi$
$824$	5210.61	0.220291
$825$	0	0
$826$	0	0
$827$	−28171.7	−1.18455	−0.592277	−	0.805735i	$-0.701771\pi$
$827$	−28171.7	−1.18455	−0.592277	+	0.805735i	$0.701771\pi$
$828$	−3491.50	−0.146543
$829$	9244.23	0.387293	0.193646	−	0.981071i	$-0.437969\pi$
$829$	9244.23	0.387293	0.193646	+	0.981071i	$0.437969\pi$
$830$	0	0
$831$	5349.63	0.223317
$832$	−6670.81	−0.277967
$833$	0	0
$834$	−5634.10	−0.233924
$835$	0	0
$836$	54195.2	2.24208
$837$	29378.2	1.21321
$838$	3421.45	0.141041
$839$	43012.5	1.76991	0.884956	−	0.465674i	$-0.154188\pi$
$839$	43012.5	1.76991	0.884956	+	0.465674i	$0.154188\pi$
$840$	0	0
$841$	−22668.8	−0.929469
$842$	−1995.25	−0.0816637
$843$	−2905.76	−0.118718
$844$	28650.9	1.16849
$845$	0	0
$846$	−2405.53	−0.0977585
$847$	0	0
$848$	38323.0	1.55191
$849$	−25356.3	−1.02500
$850$	0	0
$851$	−9671.70	−0.389591
$852$	16218.6	0.652159
$853$	−19756.9	−0.793041	−0.396521	−	0.918026i	$-0.629783\pi$
$853$	−19756.9	−0.793041	−0.396521	+	0.918026i	$0.629783\pi$
$854$	0	0
$855$	0	0
$856$	16450.7	0.656862
$857$	325.658	0.0129805	0.00649024	−	0.999979i	$-0.497934\pi$
$857$	325.658	0.0129805	0.00649024	+	0.999979i	$0.497934\pi$
$858$	1890.75	0.0752321
$859$	−3275.22	−0.130092	−0.0650461	−	0.997882i	$-0.520719\pi$
$859$	−3275.22	−0.130092	−0.0650461	+	0.997882i	$0.520719\pi$
$860$	0	0
$861$	0	0
$862$	−1733.98	−0.0685147
$863$	−44585.0	−1.75862	−0.879311	−	0.476247i	$-0.841997\pi$
$863$	−44585.0	−1.75862	−0.879311	+	0.476247i	$0.841997\pi$
$864$	−14178.7	−0.558297
$865$	0	0
$866$	−2755.02	−0.108106
$867$	−20947.3	−0.820537
$868$	0	0
$869$	−47126.9	−1.83967
$870$	0	0
$871$	−166.594	−0.00648087
$872$	−7186.65	−0.279095
$873$	12499.9	0.484602
$874$	3289.48	0.127309
$875$	0	0
$876$	26899.3	1.03749
$877$	−16433.8	−0.632761	−0.316380	−	0.948632i	$-0.602468\pi$
$877$	−16433.8	−0.632761	−0.316380	+	0.948632i	$0.602468\pi$
$878$	−2114.08	−0.0812607
$879$	9256.82	0.355205
$880$	0	0
$881$	39080.2	1.49449	0.747245	−	0.664549i	$-0.231376\pi$
$881$	39080.2	1.49449	0.747245	+	0.664549i	$0.231376\pi$
$882$	0	0
$883$	−48548.2	−1.85026	−0.925129	−	0.379652i	$-0.876044\pi$
$883$	−48548.2	−1.85026	−0.925129	+	0.379652i	$0.876044\pi$
$884$	52.3264	0.00199087
$885$	0	0
$886$	−1058.78	−0.0401473
$887$	−13447.1	−0.509029	−0.254514	−	0.967069i	$-0.581916\pi$
$887$	−13447.1	−0.509029	−0.254514	+	0.967069i	$0.581916\pi$
$888$	−6412.76	−0.242340
$889$	0	0
$890$	0	0
$891$	−22669.0	−0.852344
$892$	790.325	0.0296660
$893$	−68888.5	−2.58148
$894$	4848.32	0.181378
$895$	0	0
$896$	0	0
$897$	−3488.37	−0.129847
$898$	5713.24	0.212309
$899$	−7977.90	−0.295971
$900$	0	0
$901$	−279.112	−0.0103203
$902$	2114.22	0.0780442
$903$	0	0
$904$	−6702.87	−0.246609
$905$	0	0
$906$	−3980.89	−0.145978
$907$	−15750.3	−0.576604	−0.288302	−	0.957539i	$-0.593091\pi$
$907$	−15750.3	−0.576604	−0.288302	+	0.957539i	$0.593091\pi$
$908$	−44456.7	−1.62483
$909$	−1067.68	−0.0389579
$910$	0	0
$911$	31449.7	1.14377	0.571886	−	0.820333i	$-0.306212\pi$
$911$	31449.7	1.14377	0.571886	+	0.820333i	$0.306212\pi$
$912$	−31503.6	−1.14385
$913$	−57942.4	−2.10034
$914$	5904.16	0.213668
$915$	0	0
$916$	−6530.17	−0.235549
$917$	0	0
$918$	32.5385	0.00116986
$919$	−30355.6	−1.08960	−0.544798	−	0.838567i	$-0.683394\pi$
$919$	−30355.6	−1.08960	−0.544798	+	0.838567i	$0.683394\pi$
$920$	0	0
$921$	−15539.8	−0.555975
$922$	−5393.64	−0.192657
$923$	−7861.46	−0.280350
$924$	0	0
$925$	0	0
$926$	5253.39	0.186433
$927$	5782.39	0.204874
$928$	3850.34	0.136200
$929$	28533.8	1.00771	0.503856	−	0.863788i	$-0.331914\pi$
$929$	28533.8	1.00771	0.503856	+	0.863788i	$0.331914\pi$
$930$	0	0
$931$	0	0
$932$	−15627.1	−0.549231
$933$	−11018.5	−0.386632
$934$	−1503.37	−0.0526678
$935$	0	0
$936$	1122.13	0.0391860
$937$	−19269.1	−0.671817	−0.335908	−	0.941895i	$-0.609043\pi$
$937$	−19269.1	−0.671817	−0.335908	+	0.941895i	$0.609043\pi$
$938$	0	0
$939$	28540.3	0.991881
$940$	0	0
$941$	2097.84	0.0726756	0.0363378	−	0.999340i	$-0.488431\pi$
$941$	2097.84	0.0726756	0.0363378	+	0.999340i	$0.488431\pi$
$942$	−7183.46	−0.248461
$943$	−3900.66	−0.134701
$944$	−23777.4	−0.819797
$945$	0	0
$946$	−8158.36	−0.280392
$947$	−5072.14	−0.174047	−0.0870235	−	0.996206i	$-0.527736\pi$
$947$	−5072.14	−0.174047	−0.0870235	+	0.996206i	$0.527736\pi$
$948$	28358.5	0.971561
$949$	−13038.6	−0.445997
$950$	0	0
$951$	16387.3	0.558775
$952$	0	0
$953$	23464.1	0.797564	0.398782	−	0.917046i	$-0.369433\pi$
$953$	23464.1	0.797564	0.398782	+	0.917046i	$0.369433\pi$
$954$	−2944.33	−0.0999227
$955$	0	0
$956$	−1887.39	−0.0638519
$957$	9705.02	0.327815
$958$	7175.13	0.241981
$959$	0	0
$960$	0	0
$961$	7209.24	0.241994
$962$	1529.04	0.0512457
$963$	18255.9	0.610892
$964$	−23704.8	−0.791992
$965$	0	0
$966$	0	0
$967$	6994.27	0.232596	0.116298	−	0.993214i	$-0.462897\pi$
$967$	6994.27	0.232596	0.116298	+	0.993214i	$0.462897\pi$
$968$	−13359.0	−0.443568
$969$	229.446	0.00760667
$970$	0	0
$971$	−37752.5	−1.24772	−0.623860	−	0.781536i	$-0.714436\pi$
$971$	−37752.5	−1.24772	−0.623860	+	0.781536i	$0.714436\pi$
$972$	−18297.8	−0.603809
$973$	0	0
$974$	−9135.09	−0.300521
$975$	0	0
$976$	−2668.55	−0.0875186
$977$	6380.21	0.208926	0.104463	−	0.994529i	$-0.466688\pi$
$977$	6380.21	0.208926	0.104463	+	0.994529i	$0.466688\pi$
$978$	1013.94	0.0331517
$979$	−18745.2	−0.611950
$980$	0	0
$981$	−7975.28	−0.259563
$982$	−6941.18	−0.225562
$983$	17330.0	0.562299	0.281150	−	0.959664i	$-0.409284\pi$
$983$	17330.0	0.562299	0.281150	+	0.959664i	$0.409284\pi$
$984$	−2586.31	−0.0837891
$985$	0	0
$986$	−8.83612	−0.000285395 0
$987$	0	0
$988$	15807.6	0.509014
$989$	15051.9	0.483945
$990$	0	0
$991$	32191.8	1.03189	0.515947	−	0.856621i	$-0.327440\pi$
$991$	32191.8	1.03189	0.515947	+	0.856621i	$0.327440\pi$
$992$	−17857.3	−0.571542
$993$	−3504.50	−0.111996
$994$	0	0
$995$	0	0
$996$	34866.7	1.10923
$997$	−2678.20	−0.0850748	−0.0425374	−	0.999095i	$-0.513544\pi$
$997$	−2678.20	−0.0850748	−0.0425374	+	0.999095i	$0.513544\pi$
$998$	7917.13	0.251114
$999$	−28901.4	−0.915314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.z.1.3		4
5.4	even	2		1225.4.a.bd.1.2		4
7.6	odd	2		175.4.a.g.1.3	✓	4
21.20	even	2		1575.4.a.bl.1.2		4
35.13	even	4		175.4.b.f.99.4		8
35.27	even	4		175.4.b.f.99.5		8
35.34	odd	2		175.4.a.h.1.2	yes	4
105.104	even	2		1575.4.a.bg.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.g.1.3	✓	4	7.6	odd	2
175.4.a.h.1.2	yes	4	35.34	odd	2
175.4.b.f.99.4		8	35.13	even	4
175.4.b.f.99.5		8	35.27	even	4
1225.4.a.z.1.3		4	1.1	even	1	trivial
1225.4.a.bd.1.2		4	5.4	even	2
1575.4.a.bg.1.3		4	105.104	even	2
1575.4.a.bl.1.2		4	21.20	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)

\(f(q)\)	\(=\)	\(q+0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} +2.15229 q^{6} -7.94792 q^{8} -8.82008 q^{9} +O(q^{10})\)	\(q+0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} +2.15229 q^{6} -7.94792 q^{8} -8.82008 q^{9} +54.8800 q^{11} -33.0239 q^{12} +16.0073 q^{13} +57.9496 q^{16} -0.422056 q^{17} -4.45223 q^{18} -127.501 q^{19} +27.7025 q^{22} -51.1101 q^{23} -33.8883 q^{24} +8.08024 q^{26} -152.729 q^{27} +41.4750 q^{29} -192.354 q^{31} +92.8353 q^{32} +233.997 q^{33} -0.213047 q^{34} +68.3132 q^{36} +189.232 q^{37} -64.3605 q^{38} +68.2519 q^{39} +76.3187 q^{41} -294.499 q^{43} -425.056 q^{44} -25.7996 q^{46} +540.297 q^{47} +247.085 q^{48} -1.79956 q^{51} -123.980 q^{52} +661.316 q^{53} -77.0953 q^{54} -543.639 q^{57} +20.9359 q^{58} -410.312 q^{59} -46.0495 q^{61} -97.0974 q^{62} -416.735 q^{64} +118.118 q^{66} -10.4074 q^{67} +3.26890 q^{68} -217.923 q^{69} -491.117 q^{71} +70.1012 q^{72} -814.540 q^{73} +95.5215 q^{74} +987.522 q^{76} +34.4525 q^{78} -858.725 q^{79} -413.064 q^{81} +38.5244 q^{82} -1055.80 q^{83} -148.658 q^{86} +176.841 q^{87} -436.182 q^{88} -341.567 q^{89} +395.858 q^{92} -820.159 q^{93} +272.733 q^{94} +395.831 q^{96} -1417.21 q^{97} -484.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9	\( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26} - 441 q^{27} + 129 q^{29} - 114 q^{31} + 310 q^{32} - 865 q^{33} - 203 q^{34} + 1101 q^{36} - 403 q^{37} + 555 q^{38} + 674 q^{39} - 671 q^{41} + 411 q^{43} + 438 q^{44} - 997 q^{46} - 8 q^{47} - 523 q^{48} - 885 q^{51} + 74 q^{52} - 90 q^{53} + 2777 q^{54} - 233 q^{57} - 673 q^{58} - 1018 q^{59} - 50 q^{61} - 1626 q^{62} - 2421 q^{64} + 3841 q^{66} - 424 q^{67} - 617 q^{68} - 1080 q^{69} + 215 q^{71} - 2940 q^{72} - 1207 q^{73} + 623 q^{74} + 3257 q^{76} - 278 q^{78} - 951 q^{79} + 28 q^{81} + 1695 q^{82} - 3035 q^{83} - 99 q^{86} + 2210 q^{87} - 163 q^{88} - 2819 q^{89} - 3073 q^{92} + 852 q^{93} + 3056 q^{94} + 1345 q^{96} - 1100 q^{97} + 2383 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 3 * q^3 + 36 * q^4 - q^6 - 27 * q^8 + 61 * q^9 + 100 * q^11 - 165 * q^12 + 44 * q^13 + 160 * q^16 - 53 * q^17 - 433 * q^18 + 29 * q^19 + 152 * q^22 - 295 * q^23 + 21 * q^24 - 700 * q^26 - 441 * q^27 + 129 * q^29 - 114 * q^31 + 310 * q^32 - 865 * q^33 - 203 * q^34 + 1101 * q^36 - 403 * q^37 + 555 * q^38 + 674 * q^39 - 671 * q^41 + 411 * q^43 + 438 * q^44 - 997 * q^46 - 8 * q^47 - 523 * q^48 - 885 * q^51 + 74 * q^52 - 90 * q^53 + 2777 * q^54 - 233 * q^57 - 673 * q^58 - 1018 * q^59 - 50 * q^61 - 1626 * q^62 - 2421 * q^64 + 3841 * q^66 - 424 * q^67 - 617 * q^68 - 1080 * q^69 + 215 * q^71 - 2940 * q^72 - 1207 * q^73 + 623 * q^74 + 3257 * q^76 - 278 * q^78 - 951 * q^79 + 28 * q^81 + 1695 * q^82 - 3035 * q^83 - 99 * q^86 + 2210 * q^87 - 163 * q^88 - 2819 * q^89 - 3073 * q^92 + 852 * q^93 + 3056 * q^94 + 1345 * q^96 - 1100 * q^97 + 2383 * q^99

Introduction

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