LMFDB - Embedded newform 2025.4.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, 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("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2025.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:	$119.478867762$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 $ x^7 - 2x^6 - 44x^5 + 74x^4 + 479x^3 - 460x^2 - 1200x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19444$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19444 q^{2} -3.18442 q^{4} -2.76605 q^{7} +24.5436 q^{8} +O(q^{10})$	$q-2.19444 q^{2} -3.18442 q^{4} -2.76605 q^{7} +24.5436 q^{8} +52.6590 q^{11} -20.4535 q^{13} +6.06994 q^{14} -28.3842 q^{16} +3.66084 q^{17} -95.6705 q^{19} -115.557 q^{22} -89.8411 q^{23} +44.8840 q^{26} +8.80825 q^{28} +227.780 q^{29} +279.139 q^{31} -134.061 q^{32} -8.03351 q^{34} -273.725 q^{37} +209.944 q^{38} -64.8647 q^{41} -418.762 q^{43} -167.688 q^{44} +197.151 q^{46} +138.709 q^{47} -335.349 q^{49} +65.1323 q^{52} -197.063 q^{53} -67.8887 q^{56} -499.852 q^{58} -741.103 q^{59} +488.468 q^{61} -612.554 q^{62} +521.263 q^{64} +411.468 q^{67} -11.6576 q^{68} +310.343 q^{71} +51.0260 q^{73} +600.675 q^{74} +304.655 q^{76} -145.657 q^{77} +1208.00 q^{79} +142.342 q^{82} +905.221 q^{83} +918.949 q^{86} +1292.44 q^{88} -663.633 q^{89} +56.5752 q^{91} +286.092 q^{92} -304.389 q^{94} +725.336 q^{97} +735.905 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 36 q^{4} - 22 q^{7} - 18 q^{8}+O(q^{10}) $ 7 * q - 2 * q^2 + 36 * q^4 - 22 * q^7 - 18 * q^8	$ 7 q - 2 q^{2} + 36 q^{4} - 22 q^{7} - 18 q^{8} + 23 q^{11} - 96 q^{13} - 21 q^{14} + 324 q^{16} - 161 q^{17} + 279 q^{19} - 311 q^{22} - 96 q^{23} - 358 q^{26} - 337 q^{28} - 296 q^{29} + 244 q^{31} + 314 q^{32} + 125 q^{34} - 404 q^{37} - 305 q^{38} - 47 q^{41} - 525 q^{43} + 55 q^{44} + 717 q^{46} - 164 q^{47} + 1225 q^{49} - 1682 q^{52} - 506 q^{53} - 981 q^{56} - 1183 q^{58} - 85 q^{59} + 828 q^{61} + 786 q^{62} + 2236 q^{64} - 1093 q^{67} - 2473 q^{68} + 328 q^{71} - 2085 q^{73} - 1316 q^{74} + 2789 q^{76} - 24 q^{77} + 2110 q^{79} + 62 q^{82} - 1290 q^{83} - 2569 q^{86} - 2271 q^{88} - 3048 q^{89} + 3338 q^{91} - 2763 q^{92} - 517 q^{94} - 1787 q^{97} - 1279 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 36 * q^4 - 22 * q^7 - 18 * q^8 + 23 * q^11 - 96 * q^13 - 21 * q^14 + 324 * q^16 - 161 * q^17 + 279 * q^19 - 311 * q^22 - 96 * q^23 - 358 * q^26 - 337 * q^28 - 296 * q^29 + 244 * q^31 + 314 * q^32 + 125 * q^34 - 404 * q^37 - 305 * q^38 - 47 * q^41 - 525 * q^43 + 55 * q^44 + 717 * q^46 - 164 * q^47 + 1225 * q^49 - 1682 * q^52 - 506 * q^53 - 981 * q^56 - 1183 * q^58 - 85 * q^59 + 828 * q^61 + 786 * q^62 + 2236 * q^64 - 1093 * q^67 - 2473 * q^68 + 328 * q^71 - 2085 * q^73 - 1316 * q^74 + 2789 * q^76 - 24 * q^77 + 2110 * q^79 + 62 * q^82 - 1290 * q^83 - 2569 * q^86 - 2271 * q^88 - 3048 * q^89 + 3338 * q^91 - 2763 * q^92 - 517 * q^94 - 1787 * q^97 - 1279 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.19444	−0.775853	−0.387927	−	0.921690i	$-0.626809\pi$
$2$	−2.19444	−0.775853	−0.387927	+	0.921690i	$0.626809\pi$
$3$	0	0
$3$	0	0
$4$	−3.18442	−0.398052
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.76605	−0.149353	−0.0746763	−	0.997208i	$-0.523792\pi$
$7$	−2.76605	−0.149353	−0.0746763	+	0.997208i	$0.523792\pi$
$8$	24.5436	1.08468
$9$	0	0
$10$	0	0
$11$	52.6590	1.44339	0.721694	−	0.692212i	$-0.243364\pi$
$11$	52.6590	1.44339	0.721694	+	0.692212i	$0.243364\pi$
$12$	0	0
$13$	−20.4535	−0.436367	−0.218183	−	0.975908i	$-0.570013\pi$
$13$	−20.4535	−0.436367	−0.218183	+	0.975908i	$0.570013\pi$
$14$	6.06994	0.115876
$15$	0	0
$16$	−28.3842	−0.443503
$17$	3.66084	0.0522285	0.0261142	−	0.999659i	$-0.491687\pi$
$17$	3.66084	0.0522285	0.0261142	+	0.999659i	$0.491687\pi$
$18$	0	0
$19$	−95.6705	−1.15517	−0.577587	−	0.816329i	$-0.696006\pi$
$19$	−95.6705	−1.15517	−0.577587	+	0.816329i	$0.696006\pi$
$20$	0	0
$21$	0	0
$22$	−115.557	−1.11986
$23$	−89.8411	−0.814486	−0.407243	−	0.913320i	$-0.633510\pi$
$23$	−89.8411	−0.814486	−0.407243	+	0.913320i	$0.633510\pi$
$24$	0	0
$25$	0	0
$26$	44.8840	0.338556
$27$	0	0
$28$	8.80825	0.0594501
$29$	227.780	1.45854	0.729272	−	0.684224i	$-0.239859\pi$
$29$	227.780	1.45854	0.729272	+	0.684224i	$0.239859\pi$
$30$	0	0
$31$	279.139	1.61725	0.808625	−	0.588324i	$-0.200212\pi$
$31$	279.139	1.61725	0.808625	+	0.588324i	$0.200212\pi$
$32$	−134.061	−0.740590
$33$	0	0
$34$	−8.03351	−0.0405216
$35$	0	0
$36$	0	0
$37$	−273.725	−1.21622	−0.608110	−	0.793852i	$-0.708072\pi$
$37$	−273.725	−1.21622	−0.608110	+	0.793852i	$0.708072\pi$
$38$	209.944	0.896246
$39$	0	0
$40$	0	0
$41$	−64.8647	−0.247077	−0.123539	−	0.992340i	$-0.539424\pi$
$41$	−64.8647	−0.247077	−0.123539	+	0.992340i	$0.539424\pi$
$42$	0	0
$43$	−418.762	−1.48513	−0.742565	−	0.669774i	$-0.766391\pi$
$43$	−418.762	−1.48513	−0.742565	+	0.669774i	$0.766391\pi$
$44$	−167.688	−0.574544
$45$	0	0
$46$	197.151	0.631921
$47$	138.709	0.430484	0.215242	−	0.976561i	$-0.430946\pi$
$47$	138.709	0.430484	0.215242	+	0.976561i	$0.430946\pi$
$48$	0	0
$49$	−335.349	−0.977694
$50$	0	0
$51$	0	0
$52$	65.1323	0.173697
$53$	−197.063	−0.510730	−0.255365	−	0.966845i	$-0.582196\pi$
$53$	−197.063	−0.510730	−0.255365	+	0.966845i	$0.582196\pi$
$54$	0	0
$55$	0	0
$56$	−67.8887	−0.162000
$57$	0	0
$58$	−499.852	−1.13162
$59$	−741.103	−1.63531	−0.817656	−	0.575707i	$-0.804727\pi$
$59$	−741.103	−1.63531	−0.817656	+	0.575707i	$0.804727\pi$
$60$	0	0
$61$	488.468	1.02528	0.512639	−	0.858604i	$-0.328668\pi$
$61$	488.468	1.02528	0.512639	+	0.858604i	$0.328668\pi$
$62$	−612.554	−1.25475
$63$	0	0
$64$	521.263	1.01809
$65$	0	0
$66$	0	0
$67$	411.468	0.750280	0.375140	−	0.926968i	$-0.377595\pi$
$67$	411.468	0.750280	0.375140	+	0.926968i	$0.377595\pi$
$68$	−11.6576	−0.0207896
$69$	0	0
$70$	0	0
$71$	310.343	0.518746	0.259373	−	0.965777i	$-0.416484\pi$
$71$	310.343	0.518746	0.259373	+	0.965777i	$0.416484\pi$
$72$	0	0
$73$	51.0260	0.0818101	0.0409051	−	0.999163i	$-0.486976\pi$
$73$	51.0260	0.0818101	0.0409051	+	0.999163i	$0.486976\pi$
$74$	600.675	0.943609
$75$	0	0
$76$	304.655	0.459820
$77$	−145.657	−0.215574
$78$	0	0
$79$	1208.00	1.72039	0.860193	−	0.509969i	$-0.170343\pi$
$79$	1208.00	1.72039	0.860193	+	0.509969i	$0.170343\pi$
$80$	0	0
$81$	0	0
$82$	142.342	0.191695
$83$	905.221	1.19712	0.598560	−	0.801078i	$-0.295740\pi$
$83$	905.221	1.19712	0.598560	+	0.801078i	$0.295740\pi$
$84$	0	0
$85$	0	0
$86$	918.949	1.15224
$87$	0	0
$88$	1292.44	1.56562
$89$	−663.633	−0.790393	−0.395197	−	0.918597i	$-0.629324\pi$
$89$	−663.633	−0.790393	−0.395197	+	0.918597i	$0.629324\pi$
$90$	0	0
$91$	56.5752	0.0651725
$92$	286.092	0.324208
$93$	0	0
$94$	−304.389	−0.333992
$95$	0	0
$96$	0	0
$97$	725.336	0.759244	0.379622	−	0.925142i	$-0.376054\pi$
$97$	725.336	0.759244	0.379622	+	0.925142i	$0.376054\pi$
$98$	735.905	0.758547
$99$	0	0
$100$	0	0
$101$	−977.782	−0.963296	−0.481648	−	0.876365i	$-0.659962\pi$
$101$	−977.782	−0.963296	−0.481648	+	0.876365i	$0.659962\pi$
$102$	0	0
$103$	−1587.70	−1.51885	−0.759423	−	0.650597i	$-0.774519\pi$
$103$	−1587.70	−1.51885	−0.759423	+	0.650597i	$0.774519\pi$
$104$	−502.001	−0.473319
$105$	0	0
$106$	432.444	0.396252
$107$	897.731	0.811093	0.405546	−	0.914074i	$-0.367081\pi$
$107$	897.731	0.811093	0.405546	+	0.914074i	$0.367081\pi$
$108$	0	0
$109$	855.492	0.751754	0.375877	−	0.926669i	$-0.377341\pi$
$109$	855.492	0.751754	0.375877	+	0.926669i	$0.377341\pi$
$110$	0	0
$111$	0	0
$112$	78.5120	0.0662383
$113$	910.241	0.757772	0.378886	−	0.925443i	$-0.376307\pi$
$113$	910.241	0.757772	0.378886	+	0.925443i	$0.376307\pi$
$114$	0	0
$115$	0	0
$116$	−725.348	−0.580576
$117$	0	0
$118$	1626.31	1.26876
$119$	−10.1261	−0.00780046
$120$	0	0
$121$	1441.97	1.08337
$122$	−1071.92	−0.795465
$123$	0	0
$124$	−888.893	−0.643750
$125$	0	0
$126$	0	0
$127$	2038.25	1.42414	0.712068	−	0.702111i	$-0.247759\pi$
$127$	2038.25	1.42414	0.712068	+	0.702111i	$0.247759\pi$
$128$	−71.3936	−0.0492997
$129$	0	0
$130$	0	0
$131$	−235.645	−0.157163	−0.0785817	−	0.996908i	$-0.525039\pi$
$131$	−235.645	−0.157163	−0.0785817	+	0.996908i	$0.525039\pi$
$132$	0	0
$133$	264.629	0.172528
$134$	−902.942	−0.582107
$135$	0	0
$136$	89.8501	0.0566513
$137$	−2972.89	−1.85395	−0.926975	−	0.375123i	$-0.877600\pi$
$137$	−2972.89	−1.85395	−0.926975	+	0.375123i	$0.877600\pi$
$138$	0	0
$139$	−2094.63	−1.27816	−0.639080	−	0.769140i	$-0.720685\pi$
$139$	−2094.63	−1.27816	−0.639080	+	0.769140i	$0.720685\pi$
$140$	0	0
$141$	0	0
$142$	−681.031	−0.402471
$143$	−1077.06	−0.629847
$144$	0	0
$145$	0	0
$146$	−111.974	−0.0634726
$147$	0	0
$148$	871.655	0.484119
$149$	−273.937	−0.150616	−0.0753081	−	0.997160i	$-0.523994\pi$
$149$	−273.937	−0.150616	−0.0753081	+	0.997160i	$0.523994\pi$
$150$	0	0
$151$	−936.844	−0.504896	−0.252448	−	0.967610i	$-0.581236\pi$
$151$	−936.844	−0.504896	−0.252448	+	0.967610i	$0.581236\pi$
$152$	−2348.10	−1.25300
$153$	0	0
$154$	319.637	0.167254
$155$	0	0
$156$	0	0
$157$	−398.436	−0.202539	−0.101269	−	0.994859i	$-0.532290\pi$
$157$	−398.436	−0.202539	−0.101269	+	0.994859i	$0.532290\pi$
$158$	−2650.89	−1.33477
$159$	0	0
$160$	0	0
$161$	248.505	0.121646
$162$	0	0
$163$	−478.154	−0.229766	−0.114883	−	0.993379i	$-0.536649\pi$
$163$	−478.154	−0.229766	−0.114883	+	0.993379i	$0.536649\pi$
$164$	206.556	0.0983495
$165$	0	0
$166$	−1986.46	−0.928789
$167$	−340.362	−0.157713	−0.0788564	−	0.996886i	$-0.525127\pi$
$167$	−340.362	−0.157713	−0.0788564	+	0.996886i	$0.525127\pi$
$168$	0	0
$169$	−1778.66	−0.809584
$170$	0	0
$171$	0	0
$172$	1333.51	0.591159
$173$	3776.07	1.65948	0.829738	−	0.558154i	$-0.188490\pi$
$173$	3776.07	1.65948	0.829738	+	0.558154i	$0.188490\pi$
$174$	0	0
$175$	0	0
$176$	−1494.68	−0.640147
$177$	0	0
$178$	1456.31	0.613229
$179$	−186.652	−0.0779385	−0.0389693	−	0.999240i	$-0.512407\pi$
$179$	−186.652	−0.0779385	−0.0389693	+	0.999240i	$0.512407\pi$
$180$	0	0
$181$	1438.75	0.590837	0.295418	−	0.955368i	$-0.404541\pi$
$181$	1438.75	0.590837	0.295418	+	0.955368i	$0.404541\pi$
$182$	−124.151	−0.0505643
$183$	0	0
$184$	−2205.02	−0.883459
$185$	0	0
$186$	0	0
$187$	192.776	0.0753860
$188$	−441.706	−0.171355
$189$	0	0
$190$	0	0
$191$	390.435	0.147910	0.0739552	−	0.997262i	$-0.476438\pi$
$191$	390.435	0.147910	0.0739552	+	0.997262i	$0.476438\pi$
$192$	0	0
$193$	3915.05	1.46016	0.730081	−	0.683361i	$-0.239482\pi$
$193$	3915.05	1.46016	0.730081	+	0.683361i	$0.239482\pi$
$194$	−1591.71	−0.589062
$195$	0	0
$196$	1067.89	0.389173
$197$	−892.680	−0.322847	−0.161423	−	0.986885i	$-0.551608\pi$
$197$	−892.680	−0.322847	−0.161423	+	0.986885i	$0.551608\pi$
$198$	0	0
$199$	−2770.50	−0.986913	−0.493457	−	0.869770i	$-0.664267\pi$
$199$	−2770.50	−0.986913	−0.493457	+	0.869770i	$0.664267\pi$
$200$	0	0
$201$	0	0
$202$	2145.69	0.747376
$203$	−630.052	−0.217837
$204$	0	0
$205$	0	0
$206$	3484.13	1.17840
$207$	0	0
$208$	580.554	0.193530
$209$	−5037.91	−1.66737
$210$	0	0
$211$	4582.51	1.49513	0.747566	−	0.664187i	$-0.231222\pi$
$211$	4582.51	1.49513	0.747566	+	0.664187i	$0.231222\pi$
$212$	627.531	0.203297
$213$	0	0
$214$	−1970.02	−0.629289
$215$	0	0
$216$	0	0
$217$	−772.111	−0.241541
$218$	−1877.33	−0.583251
$219$	0	0
$220$	0	0
$221$	−74.8768	−0.0227908
$222$	0	0
$223$	−4095.34	−1.22979	−0.614897	−	0.788608i	$-0.710802\pi$
$223$	−4095.34	−1.22979	−0.614897	+	0.788608i	$0.710802\pi$
$224$	370.820	0.110609
$225$	0	0
$226$	−1997.47	−0.587920
$227$	−3869.35	−1.13136	−0.565678	−	0.824626i	$-0.691386\pi$
$227$	−3869.35	−1.13136	−0.565678	+	0.824626i	$0.691386\pi$
$228$	0	0
$229$	1236.01	0.356672	0.178336	−	0.983970i	$-0.442929\pi$
$229$	1236.01	0.356672	0.178336	+	0.983970i	$0.442929\pi$
$230$	0	0
$231$	0	0
$232$	5590.55	1.58206
$233$	2207.05	0.620552	0.310276	−	0.950647i	$-0.399579\pi$
$233$	2207.05	0.620552	0.310276	+	0.950647i	$0.399579\pi$
$234$	0	0
$235$	0	0
$236$	2359.98	0.650939
$237$	0	0
$238$	22.2211	0.00605201
$239$	−876.648	−0.237262	−0.118631	−	0.992938i	$-0.537851\pi$
$239$	−876.648	−0.237262	−0.118631	+	0.992938i	$0.537851\pi$
$240$	0	0
$241$	−477.861	−0.127725	−0.0638626	−	0.997959i	$-0.520342\pi$
$241$	−477.861	−0.127725	−0.0638626	+	0.997959i	$0.520342\pi$
$242$	−3164.32	−0.840537
$243$	0	0
$244$	−1555.49	−0.408114
$245$	0	0
$246$	0	0
$247$	1956.79	0.504080
$248$	6851.06	1.75420
$249$	0	0
$250$	0	0
$251$	−6892.28	−1.73322	−0.866608	−	0.498990i	$-0.833704\pi$
$251$	−6892.28	−1.73322	−0.866608	+	0.498990i	$0.833704\pi$
$252$	0	0
$253$	−4730.94	−1.17562
$254$	−4472.82	−1.10492
$255$	0	0
$256$	−4013.43	−0.979842
$257$	−7256.71	−1.76133	−0.880664	−	0.473742i	$-0.842903\pi$
$257$	−7256.71	−1.76133	−0.880664	+	0.473742i	$0.842903\pi$
$258$	0	0
$259$	757.138	0.181646
$260$	0	0
$261$	0	0
$262$	517.110	0.121936
$263$	−6317.55	−1.48121	−0.740603	−	0.671943i	$-0.765460\pi$
$263$	−6317.55	−1.48121	−0.740603	+	0.671943i	$0.765460\pi$
$264$	0	0
$265$	0	0
$266$	−580.714	−0.133857
$267$	0	0
$268$	−1310.28	−0.298650
$269$	−5746.22	−1.30243	−0.651214	−	0.758894i	$-0.725740\pi$
$269$	−5746.22	−1.30243	−0.651214	+	0.758894i	$0.725740\pi$
$270$	0	0
$271$	4925.20	1.10400	0.552001	−	0.833844i	$-0.313865\pi$
$271$	4925.20	1.10400	0.552001	+	0.833844i	$0.313865\pi$
$272$	−103.910	−0.0231635
$273$	0	0
$274$	6523.84	1.43839
$275$	0	0
$276$	0	0
$277$	−2325.88	−0.504508	−0.252254	−	0.967661i	$-0.581172\pi$
$277$	−2325.88	−0.504508	−0.252254	+	0.967661i	$0.581172\pi$
$278$	4596.55	0.991665
$279$	0	0
$280$	0	0
$281$	−3283.42	−0.697055	−0.348527	−	0.937299i	$-0.613318\pi$
$281$	−3283.42	−0.697055	−0.348527	+	0.937299i	$0.613318\pi$
$282$	0	0
$283$	−2014.21	−0.423083	−0.211541	−	0.977369i	$-0.567848\pi$
$283$	−2014.21	−0.423083	−0.211541	+	0.977369i	$0.567848\pi$
$284$	−988.262	−0.206488
$285$	0	0
$286$	2363.54	0.488669
$287$	179.419	0.0369016
$288$	0	0
$289$	−4899.60	−0.997272
$290$	0	0
$291$	0	0
$292$	−162.488	−0.0325647
$293$	480.697	0.0958450	0.0479225	−	0.998851i	$-0.484740\pi$
$293$	480.697	0.0958450	0.0479225	+	0.998851i	$0.484740\pi$
$294$	0	0
$295$	0	0
$296$	−6718.20	−1.31921
$297$	0	0
$298$	601.140	0.116856
$299$	1837.56	0.355414
$300$	0	0
$301$	1158.32	0.221808
$302$	2055.85	0.391725
$303$	0	0
$304$	2715.53	0.512323
$305$	0	0
$306$	0	0
$307$	−3222.21	−0.599026	−0.299513	−	0.954092i	$-0.596824\pi$
$307$	−3222.21	−0.599026	−0.299513	+	0.954092i	$0.596824\pi$
$308$	463.833	0.0858096
$309$	0	0
$310$	0	0
$311$	−2414.01	−0.440147	−0.220074	−	0.975483i	$-0.570630\pi$
$311$	−2414.01	−0.440147	−0.220074	+	0.975483i	$0.570630\pi$
$312$	0	0
$313$	2506.94	0.452717	0.226359	−	0.974044i	$-0.427318\pi$
$313$	2506.94	0.452717	0.226359	+	0.974044i	$0.427318\pi$
$314$	874.344	0.157140
$315$	0	0
$316$	−3846.77	−0.684803
$317$	3707.12	0.656822	0.328411	−	0.944535i	$-0.393487\pi$
$317$	3707.12	0.656822	0.328411	+	0.944535i	$0.393487\pi$
$318$	0	0
$319$	11994.7	2.10525
$320$	0	0
$321$	0	0
$322$	−545.330	−0.0943791
$323$	−350.234	−0.0603330
$324$	0	0
$325$	0	0
$326$	1049.28	0.178265
$327$	0	0
$328$	−1592.01	−0.268000
$329$	−383.675	−0.0642939
$330$	0	0
$331$	2553.45	0.424018	0.212009	−	0.977268i	$-0.431999\pi$
$331$	2553.45	0.424018	0.212009	+	0.977268i	$0.431999\pi$
$332$	−2882.60	−0.476516
$333$	0	0
$334$	746.906	0.122362
$335$	0	0
$336$	0	0
$337$	−3552.89	−0.574297	−0.287148	−	0.957886i	$-0.592707\pi$
$337$	−3552.89	−0.574297	−0.287148	+	0.957886i	$0.592707\pi$
$338$	3903.16	0.628118
$339$	0	0
$340$	0	0
$341$	14699.2	2.33432
$342$	0	0
$343$	1876.35	0.295374
$344$	−10277.9	−1.61089
$345$	0	0
$346$	−8286.37	−1.28751
$347$	−7768.02	−1.20176	−0.600878	−	0.799341i	$-0.705182\pi$
$347$	−7768.02	−1.20176	−0.600878	+	0.799341i	$0.705182\pi$
$348$	0	0
$349$	696.007	0.106752	0.0533759	−	0.998574i	$-0.483002\pi$
$349$	696.007	0.106752	0.0533759	+	0.998574i	$0.483002\pi$
$350$	0	0
$351$	0	0
$352$	−7059.52	−1.06896
$353$	−5451.87	−0.822022	−0.411011	−	0.911630i	$-0.634824\pi$
$353$	−5451.87	−0.822022	−0.411011	+	0.911630i	$0.634824\pi$
$354$	0	0
$355$	0	0
$356$	2113.28	0.314618
$357$	0	0
$358$	409.596	0.0604688
$359$	−4036.41	−0.593408	−0.296704	−	0.954969i	$-0.595887\pi$
$359$	−4036.41	−0.593408	−0.296704	+	0.954969i	$0.595887\pi$
$360$	0	0
$361$	2293.85	0.334429
$362$	−3157.26	−0.458403
$363$	0	0
$364$	−180.159	−0.0259420
$365$	0	0
$366$	0	0
$367$	−11239.9	−1.59869	−0.799345	−	0.600872i	$-0.794820\pi$
$367$	−11239.9	−1.59869	−0.799345	+	0.600872i	$0.794820\pi$
$368$	2550.07	0.361227
$369$	0	0
$370$	0	0
$371$	545.086	0.0762789
$372$	0	0
$373$	6320.72	0.877411	0.438706	−	0.898631i	$-0.355437\pi$
$373$	6320.72	0.877411	0.438706	+	0.898631i	$0.355437\pi$
$374$	−423.036	−0.0584885
$375$	0	0
$376$	3404.41	0.466939
$377$	−4658.90	−0.636460
$378$	0	0
$379$	9325.49	1.26390	0.631950	−	0.775009i	$-0.282255\pi$
$379$	9325.49	1.26390	0.631950	+	0.775009i	$0.282255\pi$
$380$	0	0
$381$	0	0
$382$	−856.788	−0.114757
$383$	11148.0	1.48730	0.743652	−	0.668567i	$-0.233092\pi$
$383$	11148.0	1.48730	0.743652	+	0.668567i	$0.233092\pi$
$384$	0	0
$385$	0	0
$386$	−8591.35	−1.13287
$387$	0	0
$388$	−2309.77	−0.302219
$389$	−6570.81	−0.856436	−0.428218	−	0.903676i	$-0.640858\pi$
$389$	−6570.81	−0.856436	−0.428218	+	0.903676i	$0.640858\pi$
$390$	0	0
$391$	−328.894	−0.0425394
$392$	−8230.66	−1.06049
$393$	0	0
$394$	1958.94	0.250482
$395$	0	0
$396$	0	0
$397$	−3969.33	−0.501800	−0.250900	−	0.968013i	$-0.580727\pi$
$397$	−3969.33	−0.501800	−0.250900	+	0.968013i	$0.580727\pi$
$398$	6079.71	0.765700
$399$	0	0
$400$	0	0
$401$	−4374.70	−0.544793	−0.272396	−	0.962185i	$-0.587816\pi$
$401$	−4374.70	−0.544793	−0.272396	+	0.962185i	$0.587816\pi$
$402$	0	0
$403$	−5709.35	−0.705714
$404$	3113.66	0.383442
$405$	0	0
$406$	1382.61	0.169010
$407$	−14414.1	−1.75548
$408$	0	0
$409$	−8872.33	−1.07264	−0.536318	−	0.844016i	$-0.680185\pi$
$409$	−8872.33	−1.07264	−0.536318	+	0.844016i	$0.680185\pi$
$410$	0	0
$411$	0	0
$412$	5055.91	0.604580
$413$	2049.93	0.244238
$414$	0	0
$415$	0	0
$416$	2742.01	0.323169
$417$	0	0
$418$	11055.4	1.29363
$419$	1014.17	0.118247	0.0591236	−	0.998251i	$-0.481169\pi$
$419$	1014.17	0.118247	0.0591236	+	0.998251i	$0.481169\pi$
$420$	0	0
$421$	−14893.1	−1.72410	−0.862051	−	0.506822i	$-0.830820\pi$
$421$	−14893.1	−1.72410	−0.862051	+	0.506822i	$0.830820\pi$
$422$	−10056.1	−1.16000
$423$	0	0
$424$	−4836.63	−0.553980
$425$	0	0
$426$	0	0
$427$	−1351.13	−0.153128
$428$	−2858.75	−0.322857
$429$	0	0
$430$	0	0
$431$	4363.90	0.487707	0.243853	−	0.969812i	$-0.421588\pi$
$431$	4363.90	0.487707	0.243853	+	0.969812i	$0.421588\pi$
$432$	0	0
$433$	−9301.59	−1.03235	−0.516173	−	0.856484i	$-0.672644\pi$
$433$	−9301.59	−1.03235	−0.516173	+	0.856484i	$0.672644\pi$
$434$	1694.35	0.187400
$435$	0	0
$436$	−2724.24	−0.299237
$437$	8595.15	0.940873
$438$	0	0
$439$	1520.15	0.165269	0.0826343	−	0.996580i	$-0.473667\pi$
$439$	1520.15	0.165269	0.0826343	+	0.996580i	$0.473667\pi$
$440$	0	0
$441$	0	0
$442$	164.313	0.0176823
$443$	4462.48	0.478598	0.239299	−	0.970946i	$-0.423082\pi$
$443$	4462.48	0.478598	0.239299	+	0.970946i	$0.423082\pi$
$444$	0	0
$445$	0	0
$446$	8986.98	0.954139
$447$	0	0
$448$	−1441.84	−0.152055
$449$	5371.66	0.564598	0.282299	−	0.959326i	$-0.408903\pi$
$449$	5371.66	0.564598	0.282299	+	0.959326i	$0.408903\pi$
$450$	0	0
$451$	−3415.71	−0.356628
$452$	−2898.59	−0.301633
$453$	0	0
$454$	8491.08	0.877767
$455$	0	0
$456$	0	0
$457$	−15525.4	−1.58917	−0.794583	−	0.607155i	$-0.792311\pi$
$457$	−15525.4	−1.58917	−0.794583	+	0.607155i	$0.792311\pi$
$458$	−2712.36	−0.276725
$459$	0	0
$460$	0	0
$461$	56.4179	0.00569988	0.00284994	−	0.999996i	$-0.499093\pi$
$461$	56.4179	0.00569988	0.00284994	+	0.999996i	$0.499093\pi$
$462$	0	0
$463$	−13373.8	−1.34240	−0.671201	−	0.741275i	$-0.734221\pi$
$463$	−13373.8	−1.34240	−0.671201	+	0.741275i	$0.734221\pi$
$464$	−6465.36	−0.646868
$465$	0	0
$466$	−4843.25	−0.481457
$467$	2677.46	0.265306	0.132653	−	0.991163i	$-0.457650\pi$
$467$	2677.46	0.265306	0.132653	+	0.991163i	$0.457650\pi$
$468$	0	0
$469$	−1138.14	−0.112056
$470$	0	0
$471$	0	0
$472$	−18189.3	−1.77380
$473$	−22051.6	−2.14362
$474$	0	0
$475$	0	0
$476$	32.2456	0.00310499
$477$	0	0
$478$	1923.75	0.184080
$479$	3469.93	0.330992	0.165496	−	0.986210i	$-0.447078\pi$
$479$	3469.93	0.330992	0.165496	+	0.986210i	$0.447078\pi$
$480$	0	0
$481$	5598.63	0.530718
$482$	1048.64	0.0990960
$483$	0	0
$484$	−4591.82	−0.431238
$485$	0	0
$486$	0	0
$487$	14040.6	1.30645	0.653224	−	0.757165i	$-0.273416\pi$
$487$	14040.6	1.30645	0.653224	+	0.757165i	$0.273416\pi$
$488$	11988.8	1.11210
$489$	0	0
$490$	0	0
$491$	−9815.44	−0.902169	−0.451084	−	0.892481i	$-0.648963\pi$
$491$	−9815.44	−0.902169	−0.451084	+	0.892481i	$0.648963\pi$
$492$	0	0
$493$	833.868	0.0761775
$494$	−4294.07	−0.391092
$495$	0	0
$496$	−7923.12	−0.717255
$497$	−858.425	−0.0774761
$498$	0	0
$499$	−12052.0	−1.08121	−0.540603	−	0.841278i	$-0.681804\pi$
$499$	−12052.0	−1.08121	−0.540603	+	0.841278i	$0.681804\pi$
$500$	0	0
$501$	0	0
$502$	15124.7	1.34472
$503$	−4695.09	−0.416191	−0.208095	−	0.978109i	$-0.566726\pi$
$503$	−4695.09	−0.416191	−0.208095	+	0.978109i	$0.566726\pi$
$504$	0	0
$505$	0	0
$506$	10381.8	0.912108
$507$	0	0
$508$	−6490.62	−0.566880
$509$	819.813	0.0713902	0.0356951	−	0.999363i	$-0.488635\pi$
$509$	819.813	0.0713902	0.0356951	+	0.999363i	$0.488635\pi$
$510$	0	0
$511$	−141.140	−0.0122186
$512$	9378.41	0.809514
$513$	0	0
$514$	15924.4	1.36653
$515$	0	0
$516$	0	0
$517$	7304.26	0.621356
$518$	−1661.50	−0.140930
$519$	0	0
$520$	0	0
$521$	−3282.80	−0.276050	−0.138025	−	0.990429i	$-0.544075\pi$
$521$	−3282.80	−0.276050	−0.138025	+	0.990429i	$0.544075\pi$
$522$	0	0
$523$	10768.1	0.900300	0.450150	−	0.892953i	$-0.351371\pi$
$523$	10768.1	0.900300	0.450150	+	0.892953i	$0.351371\pi$
$524$	750.392	0.0625592
$525$	0	0
$526$	13863.5	1.14920
$527$	1021.88	0.0844665
$528$	0	0
$529$	−4095.57	−0.336613
$530$	0	0
$531$	0	0
$532$	−842.690	−0.0686752
$533$	1326.71	0.107816
$534$	0	0
$535$	0	0
$536$	10098.9	0.813816
$537$	0	0
$538$	12609.8	1.01049
$539$	−17659.1	−1.41119
$540$	0	0
$541$	−16037.9	−1.27453	−0.637266	−	0.770644i	$-0.719935\pi$
$541$	−16037.9	−1.27453	−0.637266	+	0.770644i	$0.719935\pi$
$542$	−10808.1	−0.856543
$543$	0	0
$544$	−490.776	−0.0386799
$545$	0	0
$546$	0	0
$547$	−2049.12	−0.160172	−0.0800862	−	0.996788i	$-0.525520\pi$
$547$	−2049.12	−0.160172	−0.0800862	+	0.996788i	$0.525520\pi$
$548$	9466.92	0.737968
$549$	0	0
$550$	0	0
$551$	−21791.9	−1.68487
$552$	0	0
$553$	−3341.38	−0.256944
$554$	5104.02	0.391424
$555$	0	0
$556$	6670.18	0.508774
$557$	3644.07	0.277207	0.138603	−	0.990348i	$-0.455739\pi$
$557$	3644.07	0.277207	0.138603	+	0.990348i	$0.455739\pi$
$558$	0	0
$559$	8565.12	0.648061
$560$	0	0
$561$	0	0
$562$	7205.28	0.540812
$563$	350.736	0.0262553	0.0131277	−	0.999914i	$-0.495821\pi$
$563$	350.736	0.0262553	0.0131277	+	0.999914i	$0.495821\pi$
$564$	0	0
$565$	0	0
$566$	4420.07	0.328250
$567$	0	0
$568$	7616.93	0.562675
$569$	19448.7	1.43292	0.716460	−	0.697628i	$-0.245761\pi$
$569$	19448.7	1.43292	0.716460	+	0.697628i	$0.245761\pi$
$570$	0	0
$571$	−15145.9	−1.11005	−0.555024	−	0.831834i	$-0.687291\pi$
$571$	−15145.9	−1.11005	−0.555024	+	0.831834i	$0.687291\pi$
$572$	3429.80	0.250712
$573$	0	0
$574$	−393.725	−0.0286302
$575$	0	0
$576$	0	0
$577$	6365.11	0.459243	0.229621	−	0.973280i	$-0.426251\pi$
$577$	6365.11	0.459243	0.229621	+	0.973280i	$0.426251\pi$
$578$	10751.9	0.773737
$579$	0	0
$580$	0	0
$581$	−2503.89	−0.178793
$582$	0	0
$583$	−10377.1	−0.737182
$584$	1252.36	0.0887381
$585$	0	0
$586$	−1054.86	−0.0743617
$587$	10284.4	0.723136	0.361568	−	0.932346i	$-0.382241\pi$
$587$	10284.4	0.723136	0.361568	+	0.932346i	$0.382241\pi$
$588$	0	0
$589$	−26705.3	−1.86821
$590$	0	0
$591$	0	0
$592$	7769.47	0.539397
$593$	666.566	0.0461595	0.0230798	−	0.999734i	$-0.492653\pi$
$593$	666.566	0.0461595	0.0230798	+	0.999734i	$0.492653\pi$
$594$	0	0
$595$	0	0
$596$	872.330	0.0599530
$597$	0	0
$598$	−4032.43	−0.275749
$599$	−25213.3	−1.71984	−0.859922	−	0.510426i	$-0.829488\pi$
$599$	−25213.3	−1.71984	−0.859922	+	0.510426i	$0.829488\pi$
$600$	0	0
$601$	−20618.7	−1.39942	−0.699712	−	0.714426i	$-0.746688\pi$
$601$	−20618.7	−1.39942	−0.699712	+	0.714426i	$0.746688\pi$
$602$	−2541.86	−0.172090
$603$	0	0
$604$	2983.30	0.200975
$605$	0	0
$606$	0	0
$607$	5083.52	0.339923	0.169962	−	0.985451i	$-0.445636\pi$
$607$	5083.52	0.339923	0.169962	+	0.985451i	$0.445636\pi$
$608$	12825.7	0.855511
$609$	0	0
$610$	0	0
$611$	−2837.07	−0.187849
$612$	0	0
$613$	−2625.18	−0.172969	−0.0864845	−	0.996253i	$-0.527563\pi$
$613$	−2625.18	−0.172969	−0.0864845	+	0.996253i	$0.527563\pi$
$614$	7070.95	0.464756
$615$	0	0
$616$	−3574.95	−0.233829
$617$	−1631.32	−0.106442	−0.0532208	−	0.998583i	$-0.516949\pi$
$617$	−1631.32	−0.106442	−0.0532208	+	0.998583i	$0.516949\pi$
$618$	0	0
$619$	3184.67	0.206789	0.103395	−	0.994640i	$-0.467030\pi$
$619$	3184.67	0.206789	0.103395	+	0.994640i	$0.467030\pi$
$620$	0	0
$621$	0	0
$622$	5297.40	0.341489
$623$	1835.64	0.118047
$624$	0	0
$625$	0	0
$626$	−5501.33	−0.351242
$627$	0	0
$628$	1268.78	0.0806210
$629$	−1002.06	−0.0635214
$630$	0	0
$631$	10436.7	0.658447	0.329223	−	0.944252i	$-0.393213\pi$
$631$	10436.7	0.658447	0.329223	+	0.944252i	$0.393213\pi$
$632$	29648.6	1.86607
$633$	0	0
$634$	−8135.07	−0.509598
$635$	0	0
$636$	0	0
$637$	6859.04	0.426633
$638$	−26321.7	−1.63336
$639$	0	0
$640$	0	0
$641$	−9083.45	−0.559711	−0.279856	−	0.960042i	$-0.590287\pi$
$641$	−9083.45	−0.559711	−0.279856	+	0.960042i	$0.590287\pi$
$642$	0	0
$643$	−17187.6	−1.05414	−0.527071	−	0.849821i	$-0.676710\pi$
$643$	−17187.6	−1.05414	−0.527071	+	0.849821i	$0.676710\pi$
$644$	−791.343	−0.0484213
$645$	0	0
$646$	768.570	0.0468096
$647$	2359.17	0.143352	0.0716758	−	0.997428i	$-0.477165\pi$
$647$	2359.17	0.143352	0.0716758	+	0.997428i	$0.477165\pi$
$648$	0	0
$649$	−39025.7	−2.36039
$650$	0	0
$651$	0	0
$652$	1522.64	0.0914589
$653$	5436.11	0.325776	0.162888	−	0.986645i	$-0.447919\pi$
$653$	5436.11	0.325776	0.162888	+	0.986645i	$0.447919\pi$
$654$	0	0
$655$	0	0
$656$	1841.13	0.109579
$657$	0	0
$658$	841.954	0.0498826
$659$	−3088.91	−0.182590	−0.0912950	−	0.995824i	$-0.529101\pi$
$659$	−3088.91	−0.182590	−0.0912950	+	0.995824i	$0.529101\pi$
$660$	0	0
$661$	−19053.9	−1.12120	−0.560598	−	0.828088i	$-0.689429\pi$
$661$	−19053.9	−1.12120	−0.560598	+	0.828088i	$0.689429\pi$
$662$	−5603.39	−0.328976
$663$	0	0
$664$	22217.4	1.29850
$665$	0	0
$666$	0	0
$667$	−20464.1	−1.18796
$668$	1083.86	0.0627779
$669$	0	0
$670$	0	0
$671$	25722.2	1.47987
$672$	0	0
$673$	29237.5	1.67462	0.837312	−	0.546725i	$-0.184126\pi$
$673$	29237.5	1.67462	0.837312	+	0.546725i	$0.184126\pi$
$674$	7796.61	0.445570
$675$	0	0
$676$	5663.98	0.322257
$677$	14471.2	0.821529	0.410764	−	0.911742i	$-0.365262\pi$
$677$	14471.2	0.821529	0.410764	+	0.911742i	$0.365262\pi$
$678$	0	0
$679$	−2006.31	−0.113395
$680$	0	0
$681$	0	0
$682$	−32256.5	−1.81109
$683$	5782.94	0.323980	0.161990	−	0.986792i	$-0.448209\pi$
$683$	5782.94	0.323980	0.161990	+	0.986792i	$0.448209\pi$
$684$	0	0
$685$	0	0
$686$	−4117.54	−0.229167
$687$	0	0
$688$	11886.2	0.658659
$689$	4030.62	0.222866
$690$	0	0
$691$	11055.1	0.608620	0.304310	−	0.952573i	$-0.401574\pi$
$691$	11055.1	0.608620	0.304310	+	0.952573i	$0.401574\pi$
$692$	−12024.6	−0.660557
$693$	0	0
$694$	17046.5	0.932386
$695$	0	0
$696$	0	0
$697$	−237.459	−0.0129045
$698$	−1527.35	−0.0828238
$699$	0	0
$700$	0	0
$701$	13554.2	0.730294	0.365147	−	0.930950i	$-0.381019\pi$
$701$	13554.2	0.730294	0.365147	+	0.930950i	$0.381019\pi$
$702$	0	0
$703$	26187.4	1.40495
$704$	27449.2	1.46950
$705$	0	0
$706$	11963.8	0.637768
$707$	2704.59	0.143871
$708$	0	0
$709$	−9389.17	−0.497345	−0.248672	−	0.968588i	$-0.579994\pi$
$709$	−9389.17	−0.497345	−0.248672	+	0.968588i	$0.579994\pi$
$710$	0	0
$711$	0	0
$712$	−16287.9	−0.857326
$713$	−25078.1	−1.31723
$714$	0	0
$715$	0	0
$716$	594.376	0.0310236
$717$	0	0
$718$	8857.67	0.460398
$719$	26301.8	1.36424	0.682122	−	0.731238i	$-0.261057\pi$
$719$	26301.8	1.36424	0.682122	+	0.731238i	$0.261057\pi$
$720$	0	0
$721$	4391.67	0.226844
$722$	−5033.72	−0.259467
$723$	0	0
$724$	−4581.58	−0.235184
$725$	0	0
$726$	0	0
$727$	25538.6	1.30285	0.651427	−	0.758711i	$-0.274171\pi$
$727$	25538.6	1.30285	0.651427	+	0.758711i	$0.274171\pi$
$728$	1388.56	0.0706915
$729$	0	0
$730$	0	0
$731$	−1533.02	−0.0775660
$732$	0	0
$733$	−6982.12	−0.351829	−0.175914	−	0.984405i	$-0.556288\pi$
$733$	−6982.12	−0.351829	−0.175914	+	0.984405i	$0.556288\pi$
$734$	24665.4	1.24035
$735$	0	0
$736$	12044.2	0.603200
$737$	21667.5	1.08295
$738$	0	0
$739$	8863.91	0.441224	0.220612	−	0.975362i	$-0.429195\pi$
$739$	8863.91	0.441224	0.220612	+	0.975362i	$0.429195\pi$
$740$	0	0
$741$	0	0
$742$	−1196.16	−0.0591812
$743$	−38944.9	−1.92295	−0.961473	−	0.274899i	$-0.911356\pi$
$743$	−38944.9	−1.92295	−0.961473	+	0.274899i	$0.911356\pi$
$744$	0	0
$745$	0	0
$746$	−13870.5	−0.680742
$747$	0	0
$748$	−613.879	−0.0300075
$749$	−2483.17	−0.121139
$750$	0	0
$751$	17590.3	0.854700	0.427350	−	0.904086i	$-0.359447\pi$
$751$	17590.3	0.854700	0.427350	+	0.904086i	$0.359447\pi$
$752$	−3937.13	−0.190921
$753$	0	0
$754$	10223.7	0.493799
$755$	0	0
$756$	0	0
$757$	4075.85	0.195693	0.0978463	−	0.995202i	$-0.468805\pi$
$757$	4075.85	0.195693	0.0978463	+	0.995202i	$0.468805\pi$
$758$	−20464.3	−0.980601
$759$	0	0
$760$	0	0
$761$	−40537.0	−1.93097	−0.965483	−	0.260466i	$-0.916124\pi$
$761$	−40537.0	−1.93097	−0.965483	+	0.260466i	$0.916124\pi$
$762$	0	0
$763$	−2366.33	−0.112276
$764$	−1243.31	−0.0588761
$765$	0	0
$766$	−24463.7	−1.15393
$767$	15158.1	0.713596
$768$	0	0
$769$	24032.7	1.12697	0.563485	−	0.826126i	$-0.309460\pi$
$769$	24032.7	1.12697	0.563485	+	0.826126i	$0.309460\pi$
$770$	0	0
$771$	0	0
$772$	−12467.1	−0.581220
$773$	−20881.5	−0.971611	−0.485805	−	0.874067i	$-0.661474\pi$
$773$	−20881.5	−0.971611	−0.485805	+	0.874067i	$0.661474\pi$
$774$	0	0
$775$	0	0
$776$	17802.3	0.823539
$777$	0	0
$778$	14419.3	0.664468
$779$	6205.64	0.285417
$780$	0	0
$781$	16342.4	0.748752
$782$	721.740	0.0330043
$783$	0	0
$784$	9518.60	0.433610
$785$	0	0
$786$	0	0
$787$	5136.39	0.232646	0.116323	−	0.993211i	$-0.462889\pi$
$787$	5136.39	0.232646	0.116323	+	0.993211i	$0.462889\pi$
$788$	2842.67	0.128510
$789$	0	0
$790$	0	0
$791$	−2517.77	−0.113175
$792$	0	0
$793$	−9990.86	−0.447397
$794$	8710.46	0.389323
$795$	0	0
$796$	8822.43	0.392843
$797$	21996.0	0.977587	0.488794	−	0.872400i	$-0.337437\pi$
$797$	21996.0	0.977587	0.488794	+	0.872400i	$0.337437\pi$
$798$	0	0
$799$	507.791	0.0224835
$800$	0	0
$801$	0	0
$802$	9600.03	0.422679
$803$	2686.98	0.118084
$804$	0	0
$805$	0	0
$806$	12528.8	0.547531
$807$	0	0
$808$	−23998.3	−1.04487
$809$	−31094.2	−1.35132	−0.675658	−	0.737215i	$-0.736140\pi$
$809$	−31094.2	−1.35132	−0.675658	+	0.737215i	$0.736140\pi$
$810$	0	0
$811$	19130.6	0.828320	0.414160	−	0.910204i	$-0.364075\pi$
$811$	19130.6	0.828320	0.414160	+	0.910204i	$0.364075\pi$
$812$	2006.35	0.0867106
$813$	0	0
$814$	31630.9	1.36199
$815$	0	0
$816$	0	0
$817$	40063.1	1.71558
$818$	19469.8	0.832208
$819$	0	0
$820$	0	0
$821$	3557.62	0.151232	0.0756162	−	0.997137i	$-0.475908\pi$
$821$	3557.62	0.151232	0.0756162	+	0.997137i	$0.475908\pi$
$822$	0	0
$823$	−6149.38	−0.260454	−0.130227	−	0.991484i	$-0.541571\pi$
$823$	−6149.38	−0.260454	−0.130227	+	0.991484i	$0.541571\pi$
$824$	−38967.9	−1.64747
$825$	0	0
$826$	−4498.45	−0.189493
$827$	21152.8	0.889425	0.444713	−	0.895673i	$-0.353306\pi$
$827$	21152.8	0.889425	0.444713	+	0.895673i	$0.353306\pi$
$828$	0	0
$829$	−17402.4	−0.729083	−0.364541	−	0.931187i	$-0.618774\pi$
$829$	−17402.4	−0.729083	−0.364541	+	0.931187i	$0.618774\pi$
$830$	0	0
$831$	0	0
$832$	−10661.6	−0.444261
$833$	−1227.66	−0.0510635
$834$	0	0
$835$	0	0
$836$	16042.8	0.663698
$837$	0	0
$838$	−2225.55	−0.0917425
$839$	18074.2	0.743733	0.371867	−	0.928286i	$-0.378718\pi$
$839$	18074.2	0.743733	0.371867	+	0.928286i	$0.378718\pi$
$840$	0	0
$841$	27495.0	1.12735
$842$	32682.1	1.33765
$843$	0	0
$844$	−14592.6	−0.595140
$845$	0	0
$846$	0	0
$847$	−3988.55	−0.161804
$848$	5593.47	0.226510
$849$	0	0
$850$	0	0
$851$	24591.8	0.990595
$852$	0	0
$853$	−40792.3	−1.63740	−0.818699	−	0.574222i	$-0.805304\pi$
$853$	−40792.3	−1.63740	−0.818699	+	0.574222i	$0.805304\pi$
$854$	2964.97	0.118805
$855$	0	0
$856$	22033.5	0.879778
$857$	−11057.9	−0.440758	−0.220379	−	0.975414i	$-0.570729\pi$
$857$	−11057.9	−0.440758	−0.220379	+	0.975414i	$0.570729\pi$
$858$	0	0
$859$	−1753.63	−0.0696543	−0.0348272	−	0.999393i	$-0.511088\pi$
$859$	−1753.63	−0.0696543	−0.0348272	+	0.999393i	$0.511088\pi$
$860$	0	0
$861$	0	0
$862$	−9576.33	−0.378389
$863$	−19186.5	−0.756796	−0.378398	−	0.925643i	$-0.623525\pi$
$863$	−19186.5	−0.756796	−0.378398	+	0.925643i	$0.623525\pi$
$864$	0	0
$865$	0	0
$866$	20411.8	0.800949
$867$	0	0
$868$	2458.72	0.0961457
$869$	63612.0	2.48319
$870$	0	0
$871$	−8415.93	−0.327397
$872$	20996.8	0.815415
$873$	0	0
$874$	−18861.6	−0.729980
$875$	0	0
$876$	0	0
$877$	8514.60	0.327842	0.163921	−	0.986473i	$-0.447586\pi$
$877$	8514.60	0.327842	0.163921	+	0.986473i	$0.447586\pi$
$878$	−3335.89	−0.128224
$879$	0	0
$880$	0	0
$881$	−41177.0	−1.57467	−0.787337	−	0.616522i	$-0.788541\pi$
$881$	−41177.0	−1.57467	−0.787337	+	0.616522i	$0.788541\pi$
$882$	0	0
$883$	−32540.4	−1.24017	−0.620086	−	0.784533i	$-0.712902\pi$
$883$	−32540.4	−1.24017	−0.620086	+	0.784533i	$0.712902\pi$
$884$	238.439	0.00907191
$885$	0	0
$886$	−9792.67	−0.371322
$887$	2166.10	0.0819960	0.0409980	−	0.999159i	$-0.486946\pi$
$887$	2166.10	0.0819960	0.0409980	+	0.999159i	$0.486946\pi$
$888$	0	0
$889$	−5637.89	−0.212698
$890$	0	0
$891$	0	0
$892$	13041.2	0.489522
$893$	−13270.3	−0.497284
$894$	0	0
$895$	0	0
$896$	197.478	0.00736304
$897$	0	0
$898$	−11787.8	−0.438045
$899$	63582.3	2.35883
$900$	0	0
$901$	−721.416	−0.0266747
$902$	7495.58	0.276691
$903$	0	0
$904$	22340.6	0.821943
$905$	0	0
$906$	0	0
$907$	706.062	0.0258483	0.0129241	−	0.999916i	$-0.495886\pi$
$907$	706.062	0.0258483	0.0129241	+	0.999916i	$0.495886\pi$
$908$	12321.6	0.450339
$909$	0	0
$910$	0	0
$911$	495.485	0.0180199	0.00900997	−	0.999959i	$-0.497132\pi$
$911$	495.485	0.0180199	0.00900997	+	0.999959i	$0.497132\pi$
$912$	0	0
$913$	47668.0	1.72791
$914$	34069.7	1.23296
$915$	0	0
$916$	−3935.98	−0.141974
$917$	651.806	0.0234728
$918$	0	0
$919$	20473.6	0.734889	0.367445	−	0.930045i	$-0.380233\pi$
$919$	20473.6	0.734889	0.367445	+	0.930045i	$0.380233\pi$
$920$	0	0
$921$	0	0
$922$	−123.806	−0.00442227
$923$	−6347.59	−0.226363
$924$	0	0
$925$	0	0
$926$	29348.0	1.04151
$927$	0	0
$928$	−30536.5	−1.08018
$929$	−49715.3	−1.75577	−0.877883	−	0.478875i	$-0.841045\pi$
$929$	−49715.3	−1.75577	−0.877883	+	0.478875i	$0.841045\pi$
$930$	0	0
$931$	32083.0	1.12941
$932$	−7028.16	−0.247012
$933$	0	0
$934$	−5875.53	−0.205839
$935$	0	0
$936$	0	0
$937$	31524.9	1.09912	0.549560	−	0.835454i	$-0.314796\pi$
$937$	31524.9	1.09912	0.549560	+	0.835454i	$0.314796\pi$
$938$	2497.58	0.0869392
$939$	0	0
$940$	0	0
$941$	−37528.5	−1.30010	−0.650050	−	0.759892i	$-0.725252\pi$
$941$	−37528.5	−1.30010	−0.650050	+	0.759892i	$0.725252\pi$
$942$	0	0
$943$	5827.52	0.201241
$944$	21035.6	0.725265
$945$	0	0
$946$	48390.9	1.66313
$947$	25863.4	0.887483	0.443741	−	0.896155i	$-0.353651\pi$
$947$	25863.4	0.887483	0.443741	+	0.896155i	$0.353651\pi$
$948$	0	0
$949$	−1043.66	−0.0356992
$950$	0	0
$951$	0	0
$952$	−248.530	−0.00846102
$953$	1060.03	0.0360312	0.0180156	−	0.999838i	$-0.494265\pi$
$953$	1060.03	0.0360312	0.0180156	+	0.999838i	$0.494265\pi$
$954$	0	0
$955$	0	0
$956$	2791.61	0.0944426
$957$	0	0
$958$	−7614.56	−0.256801
$959$	8223.16	0.276892
$960$	0	0
$961$	48127.3	1.61550
$962$	−12285.9	−0.411759
$963$	0	0
$964$	1521.71	0.0508412
$965$	0	0
$966$	0	0
$967$	5185.01	0.172429	0.0862145	−	0.996277i	$-0.472523\pi$
$967$	5185.01	0.172429	0.0862145	+	0.996277i	$0.472523\pi$
$968$	35391.0	1.17511
$969$	0	0
$970$	0	0
$971$	−28314.9	−0.935805	−0.467903	−	0.883780i	$-0.654990\pi$
$971$	−28314.9	−0.935805	−0.467903	+	0.883780i	$0.654990\pi$
$972$	0	0
$973$	5793.85	0.190897
$974$	−30811.3	−1.01361
$975$	0	0
$976$	−13864.8	−0.454713
$977$	44896.7	1.47019	0.735093	−	0.677967i	$-0.237139\pi$
$977$	44896.7	1.47019	0.735093	+	0.677967i	$0.237139\pi$
$978$	0	0
$979$	−34946.3	−1.14084
$980$	0	0
$981$	0	0
$982$	21539.4	0.699950
$983$	−14572.8	−0.472838	−0.236419	−	0.971651i	$-0.575974\pi$
$983$	−14572.8	−0.472838	−0.236419	+	0.971651i	$0.575974\pi$
$984$	0	0
$985$	0	0
$986$	−1829.88	−0.0591026
$987$	0	0
$988$	−6231.24	−0.200650
$989$	37622.0	1.20962
$990$	0	0
$991$	10602.1	0.339844	0.169922	−	0.985457i	$-0.445648\pi$
$991$	10602.1	0.339844	0.169922	+	0.985457i	$0.445648\pi$
$992$	−37421.6	−1.19772
$993$	0	0
$994$	1883.76	0.0601100
$995$	0	0
$996$	0	0
$997$	−27897.0	−0.886164	−0.443082	−	0.896481i	$-0.646115\pi$
$997$	−27897.0	−0.886164	−0.443082	+	0.896481i	$0.646115\pi$
$998$	26447.4	0.838857
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.ba.1.3		7
3.2	odd	2		2025.4.a.bb.1.5		7
5.4	even	2		405.4.a.n.1.5		7
9.2	odd	6		225.4.e.d.76.3		14
9.5	odd	6		225.4.e.d.151.3		14
15.14	odd	2		405.4.a.m.1.3		7
45.2	even	12		225.4.k.d.49.10		28
45.4	even	6		135.4.e.c.46.3		14
45.14	odd	6		45.4.e.c.16.5	✓	14
45.23	even	12		225.4.k.d.124.10		28
45.29	odd	6		45.4.e.c.31.5	yes	14
45.32	even	12		225.4.k.d.124.5		28
45.34	even	6		135.4.e.c.91.3		14
45.38	even	12		225.4.k.d.49.5		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.5	✓	14	45.14	odd	6
45.4.e.c.31.5	yes	14	45.29	odd	6
135.4.e.c.46.3		14	45.4	even	6
135.4.e.c.91.3		14	45.34	even	6
225.4.e.d.76.3		14	9.2	odd	6
225.4.e.d.151.3		14	9.5	odd	6
225.4.k.d.49.5		28	45.38	even	12
225.4.k.d.49.10		28	45.2	even	12
225.4.k.d.124.5		28	45.32	even	12
225.4.k.d.124.10		28	45.23	even	12
405.4.a.m.1.3		7	15.14	odd	2
405.4.a.n.1.5		7	5.4	even	2
2025.4.a.ba.1.3		7	1.1	even	1	trivial
2025.4.a.bb.1.5		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.19444 q^{2} -3.18442 q^{4} -2.76605 q^{7} +24.5436 q^{8} +O(q^{10})\)	\(q-2.19444 q^{2} -3.18442 q^{4} -2.76605 q^{7} +24.5436 q^{8} +52.6590 q^{11} -20.4535 q^{13} +6.06994 q^{14} -28.3842 q^{16} +3.66084 q^{17} -95.6705 q^{19} -115.557 q^{22} -89.8411 q^{23} +44.8840 q^{26} +8.80825 q^{28} +227.780 q^{29} +279.139 q^{31} -134.061 q^{32} -8.03351 q^{34} -273.725 q^{37} +209.944 q^{38} -64.8647 q^{41} -418.762 q^{43} -167.688 q^{44} +197.151 q^{46} +138.709 q^{47} -335.349 q^{49} +65.1323 q^{52} -197.063 q^{53} -67.8887 q^{56} -499.852 q^{58} -741.103 q^{59} +488.468 q^{61} -612.554 q^{62} +521.263 q^{64} +411.468 q^{67} -11.6576 q^{68} +310.343 q^{71} +51.0260 q^{73} +600.675 q^{74} +304.655 q^{76} -145.657 q^{77} +1208.00 q^{79} +142.342 q^{82} +905.221 q^{83} +918.949 q^{86} +1292.44 q^{88} -663.633 q^{89} +56.5752 q^{91} +286.092 q^{92} -304.389 q^{94} +725.336 q^{97} +735.905 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 36 q^{4} - 22 q^{7} - 18 q^{8}+O(q^{10}) \) 7 * q - 2 * q^2 + 36 * q^4 - 22 * q^7 - 18 * q^8	\( 7 q - 2 q^{2} + 36 q^{4} - 22 q^{7} - 18 q^{8} + 23 q^{11} - 96 q^{13} - 21 q^{14} + 324 q^{16} - 161 q^{17} + 279 q^{19} - 311 q^{22} - 96 q^{23} - 358 q^{26} - 337 q^{28} - 296 q^{29} + 244 q^{31} + 314 q^{32} + 125 q^{34} - 404 q^{37} - 305 q^{38} - 47 q^{41} - 525 q^{43} + 55 q^{44} + 717 q^{46} - 164 q^{47} + 1225 q^{49} - 1682 q^{52} - 506 q^{53} - 981 q^{56} - 1183 q^{58} - 85 q^{59} + 828 q^{61} + 786 q^{62} + 2236 q^{64} - 1093 q^{67} - 2473 q^{68} + 328 q^{71} - 2085 q^{73} - 1316 q^{74} + 2789 q^{76} - 24 q^{77} + 2110 q^{79} + 62 q^{82} - 1290 q^{83} - 2569 q^{86} - 2271 q^{88} - 3048 q^{89} + 3338 q^{91} - 2763 q^{92} - 517 q^{94} - 1787 q^{97} - 1279 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 36 * q^4 - 22 * q^7 - 18 * q^8 + 23 * q^11 - 96 * q^13 - 21 * q^14 + 324 * q^16 - 161 * q^17 + 279 * q^19 - 311 * q^22 - 96 * q^23 - 358 * q^26 - 337 * q^28 - 296 * q^29 + 244 * q^31 + 314 * q^32 + 125 * q^34 - 404 * q^37 - 305 * q^38 - 47 * q^41 - 525 * q^43 + 55 * q^44 + 717 * q^46 - 164 * q^47 + 1225 * q^49 - 1682 * q^52 - 506 * q^53 - 981 * q^56 - 1183 * q^58 - 85 * q^59 + 828 * q^61 + 786 * q^62 + 2236 * q^64 - 1093 * q^67 - 2473 * q^68 + 328 * q^71 - 2085 * q^73 - 1316 * q^74 + 2789 * q^76 - 24 * q^77 + 2110 * q^79 + 62 * q^82 - 1290 * q^83 - 2569 * q^86 - 2271 * q^88 - 3048 * q^89 + 3338 * q^91 - 2763 * q^92 - 517 * q^94 - 1787 * q^97 - 1279 * q^98

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