LMFDB - Embedded newform 2025.4.a.bk.1.4

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 - 91x^14 + 3268x^12 - 59128x^10 + 571975x^8 - 2881141x^6 + 6555196x^4 - 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.08740$ 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)

$f(q)$	$=$	$q-3.08740 q^{2} +1.53204 q^{4} -31.3204 q^{7} +19.9692 q^{8} +O(q^{10})$	$q-3.08740 q^{2} +1.53204 q^{4} -31.3204 q^{7} +19.9692 q^{8} -19.1937 q^{11} -20.9909 q^{13} +96.6986 q^{14} -73.9092 q^{16} -6.19137 q^{17} -96.6654 q^{19} +59.2586 q^{22} +162.986 q^{23} +64.8074 q^{26} -47.9842 q^{28} +7.64922 q^{29} +225.025 q^{31} +68.4339 q^{32} +19.1153 q^{34} +155.911 q^{37} +298.445 q^{38} -315.221 q^{41} -192.759 q^{43} -29.4056 q^{44} -503.202 q^{46} +318.312 q^{47} +637.967 q^{49} -32.1590 q^{52} +277.459 q^{53} -625.442 q^{56} -23.6162 q^{58} -429.292 q^{59} -89.9348 q^{61} -694.741 q^{62} +379.991 q^{64} +583.304 q^{67} -9.48546 q^{68} +132.605 q^{71} +259.933 q^{73} -481.361 q^{74} -148.096 q^{76} +601.153 q^{77} +124.649 q^{79} +973.214 q^{82} +36.7198 q^{83} +595.123 q^{86} -383.282 q^{88} +333.424 q^{89} +657.443 q^{91} +249.701 q^{92} -982.757 q^{94} -1029.25 q^{97} -1969.66 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 54 q^{4}+O(q^{10}) $ 16 * q + 54 * q^4	$ 16 q + 54 q^{4} - 90 q^{11} - 102 q^{14} + 146 q^{16} + 4 q^{19} - 468 q^{26} - 516 q^{29} + 38 q^{31} + 212 q^{34} - 576 q^{41} - 1644 q^{44} - 290 q^{46} - 4 q^{49} - 2430 q^{56} - 2202 q^{59} + 20 q^{61} - 322 q^{64} - 2952 q^{71} - 4080 q^{74} - 396 q^{76} - 218 q^{79} - 6108 q^{86} - 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) $ 16 * q + 54 * q^4 - 90 * q^11 - 102 * q^14 + 146 * q^16 + 4 * q^19 - 468 * q^26 - 516 * q^29 + 38 * q^31 + 212 * q^34 - 576 * q^41 - 1644 * q^44 - 290 * q^46 - 4 * q^49 - 2430 * q^56 - 2202 * q^59 + 20 * q^61 - 322 * q^64 - 2952 * q^71 - 4080 * q^74 - 396 * q^76 - 218 * q^79 - 6108 * q^86 - 4074 * q^89 - 942 * q^91 - 1078 * q^94

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 $

$2$	−3.08740	−1.09156	−0.545781	−	0.837928i	$-0.683767\pi$
$2$	−3.08740	−1.09156	−0.545781	+	0.837928i	$0.683767\pi$
$3$	0	0
$3$	0	0
$4$	1.53204	0.191506
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−31.3204	−1.69114	−0.845571	−	0.533863i	$-0.820740\pi$
$7$	−31.3204	−1.69114	−0.845571	+	0.533863i	$0.820740\pi$
$8$	19.9692	0.882521
$9$	0	0
$10$	0	0
$11$	−19.1937	−0.526101	−0.263051	−	0.964782i	$-0.584729\pi$
$11$	−19.1937	−0.526101	−0.263051	+	0.964782i	$0.584729\pi$
$12$	0	0
$13$	−20.9909	−0.447833	−0.223917	−	0.974608i	$-0.571884\pi$
$13$	−20.9909	−0.447833	−0.223917	+	0.974608i	$0.571884\pi$
$14$	96.6986	1.84598
$15$	0	0
$16$	−73.9092	−1.15483
$17$	−6.19137	−0.0883311	−0.0441656	−	0.999024i	$-0.514063\pi$
$17$	−6.19137	−0.0883311	−0.0441656	+	0.999024i	$0.514063\pi$
$18$	0	0
$19$	−96.6654	−1.16719	−0.583594	−	0.812046i	$-0.698354\pi$
$19$	−96.6654	−1.16719	−0.583594	+	0.812046i	$0.698354\pi$
$20$	0	0
$21$	0	0
$22$	59.2586	0.574271
$23$	162.986	1.47760	0.738801	−	0.673923i	$-0.235392\pi$
$23$	162.986	1.47760	0.738801	+	0.673923i	$0.235392\pi$
$24$	0	0
$25$	0	0
$26$	64.8074	0.488837
$27$	0	0
$28$	−47.9842	−0.323863
$29$	7.64922	0.0489802	0.0244901	−	0.999700i	$-0.492204\pi$
$29$	7.64922	0.0489802	0.0244901	+	0.999700i	$0.492204\pi$
$30$	0	0
$31$	225.025	1.30373	0.651865	−	0.758335i	$-0.273987\pi$
$31$	225.025	1.30373	0.651865	+	0.758335i	$0.273987\pi$
$32$	68.4339	0.378048
$33$	0	0
$34$	19.1153	0.0964188
$35$	0	0
$36$	0	0
$37$	155.911	0.692748	0.346374	−	0.938097i	$-0.387413\pi$
$37$	155.911	0.692748	0.346374	+	0.938097i	$0.387413\pi$
$38$	298.445	1.27406
$39$	0	0
$40$	0	0
$41$	−315.221	−1.20071	−0.600357	−	0.799732i	$-0.704975\pi$
$41$	−315.221	−1.20071	−0.600357	+	0.799732i	$0.704975\pi$
$42$	0	0
$43$	−192.759	−0.683614	−0.341807	−	0.939770i	$-0.611039\pi$
$43$	−192.759	−0.683614	−0.341807	+	0.939770i	$0.611039\pi$
$44$	−29.4056	−0.100751
$45$	0	0
$46$	−503.202	−1.61289
$47$	318.312	0.987885	0.493942	−	0.869495i	$-0.335555\pi$
$47$	318.312	0.987885	0.493942	+	0.869495i	$0.335555\pi$
$48$	0	0
$49$	637.967	1.85996
$50$	0	0
$51$	0	0
$52$	−32.1590	−0.0857625
$53$	277.459	0.719093	0.359547	−	0.933127i	$-0.382931\pi$
$53$	277.459	0.719093	0.359547	+	0.933127i	$0.382931\pi$
$54$	0	0
$55$	0	0
$56$	−625.442	−1.49247
$57$	0	0
$58$	−23.6162	−0.0534648
$59$	−429.292	−0.947272	−0.473636	−	0.880721i	$-0.657059\pi$
$59$	−429.292	−0.947272	−0.473636	+	0.880721i	$0.657059\pi$
$60$	0	0
$61$	−89.9348	−0.188770	−0.0943850	−	0.995536i	$-0.530088\pi$
$61$	−89.9348	−0.188770	−0.0943850	+	0.995536i	$0.530088\pi$
$62$	−694.741	−1.42310
$63$	0	0
$64$	379.991	0.742169
$65$	0	0
$66$	0	0
$67$	583.304	1.06361	0.531805	−	0.846867i	$-0.321514\pi$
$67$	583.304	1.06361	0.531805	+	0.846867i	$0.321514\pi$
$68$	−9.48546	−0.0169159
$69$	0	0
$70$	0	0
$71$	132.605	0.221653	0.110826	−	0.993840i	$-0.464650\pi$
$71$	132.605	0.221653	0.110826	+	0.993840i	$0.464650\pi$
$72$	0	0
$73$	259.933	0.416752	0.208376	−	0.978049i	$-0.433182\pi$
$73$	259.933	0.416752	0.208376	+	0.978049i	$0.433182\pi$
$74$	−481.361	−0.756176
$75$	0	0
$76$	−148.096	−0.223523
$77$	601.153	0.889712
$78$	0	0
$79$	124.649	0.177520	0.0887601	−	0.996053i	$-0.471710\pi$
$79$	124.649	0.177520	0.0887601	+	0.996053i	$0.471710\pi$
$80$	0	0
$81$	0	0
$82$	973.214	1.31065
$83$	36.7198	0.0485605	0.0242802	−	0.999705i	$-0.492271\pi$
$83$	36.7198	0.0485605	0.0242802	+	0.999705i	$0.492271\pi$
$84$	0	0
$85$	0	0
$86$	595.123	0.746207
$87$	0	0
$88$	−383.282	−0.464295
$89$	333.424	0.397111	0.198555	−	0.980090i	$-0.436375\pi$
$89$	333.424	0.397111	0.198555	+	0.980090i	$0.436375\pi$
$90$	0	0
$91$	657.443	0.757349
$92$	249.701	0.282969
$93$	0	0
$94$	−982.757	−1.07834
$95$	0	0
$96$	0	0
$97$	−1029.25	−1.07736	−0.538681	−	0.842510i	$-0.681077\pi$
$97$	−1029.25	−1.07736	−0.538681	+	0.842510i	$0.681077\pi$
$98$	−1969.66	−2.03026
$99$	0	0
$100$	0	0
$101$	926.962	0.913230	0.456615	−	0.889664i	$-0.349062\pi$
$101$	926.962	0.913230	0.456615	+	0.889664i	$0.349062\pi$
$102$	0	0
$103$	1687.60	1.61441	0.807205	−	0.590271i	$-0.200979\pi$
$103$	1687.60	1.61441	0.807205	+	0.590271i	$0.200979\pi$
$104$	−419.171	−0.395222
$105$	0	0
$106$	−856.627	−0.784934
$107$	−750.833	−0.678372	−0.339186	−	0.940719i	$-0.610152\pi$
$107$	−750.833	−0.678372	−0.339186	+	0.940719i	$0.610152\pi$
$108$	0	0
$109$	401.243	0.352589	0.176294	−	0.984338i	$-0.443589\pi$
$109$	401.243	0.352589	0.176294	+	0.984338i	$0.443589\pi$
$110$	0	0
$111$	0	0
$112$	2314.86	1.95298
$113$	−220.450	−0.183524	−0.0917620	−	0.995781i	$-0.529250\pi$
$113$	−220.450	−0.183524	−0.0917620	+	0.995781i	$0.529250\pi$
$114$	0	0
$115$	0	0
$116$	11.7189	0.00937997
$117$	0	0
$118$	1325.40	1.03401
$119$	193.916	0.149380
$120$	0	0
$121$	−962.603	−0.723218
$122$	277.665	0.206054
$123$	0	0
$124$	344.748	0.249671
$125$	0	0
$126$	0	0
$127$	1185.99	0.828659	0.414329	−	0.910127i	$-0.364016\pi$
$127$	1185.99	0.828659	0.414329	+	0.910127i	$0.364016\pi$
$128$	−1720.65	−1.18817
$129$	0	0
$130$	0	0
$131$	−2770.80	−1.84799	−0.923993	−	0.382408i	$-0.875095\pi$
$131$	−2770.80	−1.84799	−0.923993	+	0.382408i	$0.875095\pi$
$132$	0	0
$133$	3027.60	1.97388
$134$	−1800.89	−1.16100
$135$	0	0
$136$	−123.637	−0.0779541
$137$	1321.94	0.824384	0.412192	−	0.911097i	$-0.364763\pi$
$137$	1321.94	0.824384	0.412192	+	0.911097i	$0.364763\pi$
$138$	0	0
$139$	−1810.45	−1.10475	−0.552374	−	0.833596i	$-0.686278\pi$
$139$	−1810.45	−1.10475	−0.552374	+	0.833596i	$0.686278\pi$
$140$	0	0
$141$	0	0
$142$	−409.405	−0.241947
$143$	402.893	0.235606
$144$	0	0
$145$	0	0
$146$	−802.518	−0.454910
$147$	0	0
$148$	238.863	0.132665
$149$	−254.768	−0.140076	−0.0700382	−	0.997544i	$-0.522312\pi$
$149$	−254.768	−0.140076	−0.0700382	+	0.997544i	$0.522312\pi$
$150$	0	0
$151$	2844.74	1.53313	0.766563	−	0.642170i	$-0.221966\pi$
$151$	2844.74	1.53313	0.766563	+	0.642170i	$0.221966\pi$
$152$	−1930.33	−1.03007
$153$	0	0
$154$	−1856.00	−0.971175
$155$	0	0
$156$	0	0
$157$	−1745.14	−0.887115	−0.443558	−	0.896246i	$-0.646284\pi$
$157$	−1745.14	−0.887115	−0.443558	+	0.896246i	$0.646284\pi$
$158$	−384.841	−0.193774
$159$	0	0
$160$	0	0
$161$	−5104.77	−2.49884
$162$	0	0
$163$	2607.78	1.25311	0.626554	−	0.779378i	$-0.284465\pi$
$163$	2607.78	1.25311	0.626554	+	0.779378i	$0.284465\pi$
$164$	−482.933	−0.229943
$165$	0	0
$166$	−113.369	−0.0530067
$167$	−1223.32	−0.566847	−0.283423	−	0.958995i	$-0.591470\pi$
$167$	−1223.32	−0.566847	−0.283423	+	0.958995i	$0.591470\pi$
$168$	0	0
$169$	−1756.38	−0.799445
$170$	0	0
$171$	0	0
$172$	−295.315	−0.130916
$173$	2789.93	1.22610	0.613048	−	0.790045i	$-0.289943\pi$
$173$	2789.93	1.22610	0.613048	+	0.790045i	$0.289943\pi$
$174$	0	0
$175$	0	0
$176$	1418.59	0.607558
$177$	0	0
$178$	−1029.41	−0.433471
$179$	769.584	0.321349	0.160674	−	0.987007i	$-0.448633\pi$
$179$	769.584	0.321349	0.160674	+	0.987007i	$0.448633\pi$
$180$	0	0
$181$	2302.38	0.945495	0.472747	−	0.881198i	$-0.343262\pi$
$181$	2302.38	0.945495	0.472747	+	0.881198i	$0.343262\pi$
$182$	−2029.79	−0.826693
$183$	0	0
$184$	3254.69	1.30402
$185$	0	0
$186$	0	0
$187$	118.835	0.0464711
$188$	487.668	0.189185
$189$	0	0
$190$	0	0
$191$	−4228.96	−1.60208	−0.801039	−	0.598612i	$-0.795719\pi$
$191$	−4228.96	−1.60208	−0.801039	+	0.598612i	$0.795719\pi$
$192$	0	0
$193$	4364.09	1.62764	0.813818	−	0.581119i	$-0.197385\pi$
$193$	4364.09	1.62764	0.813818	+	0.581119i	$0.197385\pi$
$194$	3177.69	1.17601
$195$	0	0
$196$	977.393	0.356193
$197$	1031.50	0.373052	0.186526	−	0.982450i	$-0.440277\pi$
$197$	1031.50	0.373052	0.186526	+	0.982450i	$0.440277\pi$
$198$	0	0
$199$	1271.59	0.452966	0.226483	−	0.974015i	$-0.427277\pi$
$199$	1271.59	0.452966	0.226483	+	0.974015i	$0.427277\pi$
$200$	0	0
$201$	0	0
$202$	−2861.90	−0.996846
$203$	−239.577	−0.0828324
$204$	0	0
$205$	0	0
$206$	−5210.30	−1.76223
$207$	0	0
$208$	1551.42	0.517172
$209$	1855.36	0.614059
$210$	0	0
$211$	346.295	0.112986	0.0564928	−	0.998403i	$-0.482008\pi$
$211$	346.295	0.112986	0.0564928	+	0.998403i	$0.482008\pi$
$212$	425.080	0.137710
$213$	0	0
$214$	2318.12	0.740484
$215$	0	0
$216$	0	0
$217$	−7047.86	−2.20479
$218$	−1238.80	−0.384872
$219$	0	0
$220$	0	0
$221$	129.963	0.0395576
$222$	0	0
$223$	4039.33	1.21297	0.606487	−	0.795093i	$-0.292578\pi$
$223$	4039.33	1.21297	0.606487	+	0.795093i	$0.292578\pi$
$224$	−2143.38	−0.639332
$225$	0	0
$226$	680.618	0.200328
$227$	−862.451	−0.252171	−0.126086	−	0.992019i	$-0.540241\pi$
$227$	−862.451	−0.252171	−0.126086	+	0.992019i	$0.540241\pi$
$228$	0	0
$229$	958.217	0.276510	0.138255	−	0.990397i	$-0.455851\pi$
$229$	958.217	0.276510	0.138255	+	0.990397i	$0.455851\pi$
$230$	0	0
$231$	0	0
$232$	152.749	0.0432260
$233$	1159.05	0.325887	0.162944	−	0.986635i	$-0.447901\pi$
$233$	1159.05	0.325887	0.162944	+	0.986635i	$0.447901\pi$
$234$	0	0
$235$	0	0
$236$	−657.694	−0.181408
$237$	0	0
$238$	−598.697	−0.163058
$239$	−4046.40	−1.09515	−0.547573	−	0.836758i	$-0.684448\pi$
$239$	−4046.40	−1.09515	−0.547573	+	0.836758i	$0.684448\pi$
$240$	0	0
$241$	−2425.49	−0.648296	−0.324148	−	0.946006i	$-0.605078\pi$
$241$	−2425.49	−0.648296	−0.324148	+	0.946006i	$0.605078\pi$
$242$	2971.94	0.789436
$243$	0	0
$244$	−137.784	−0.0361505
$245$	0	0
$246$	0	0
$247$	2029.09	0.522705
$248$	4493.55	1.15057
$249$	0	0
$250$	0	0
$251$	7272.75	1.82889	0.914445	−	0.404709i	$-0.132627\pi$
$251$	7272.75	1.82889	0.914445	+	0.404709i	$0.132627\pi$
$252$	0	0
$253$	−3128.29	−0.777368
$254$	−3661.63	−0.904532
$255$	0	0
$256$	2272.43	0.554792
$257$	3234.18	0.784990	0.392495	−	0.919754i	$-0.371612\pi$
$257$	3234.18	0.784990	0.392495	+	0.919754i	$0.371612\pi$
$258$	0	0
$259$	−4883.20	−1.17153
$260$	0	0
$261$	0	0
$262$	8554.58	2.01719
$263$	1056.62	0.247733	0.123866	−	0.992299i	$-0.460471\pi$
$263$	1056.62	0.247733	0.123866	+	0.992299i	$0.460471\pi$
$264$	0	0
$265$	0	0
$266$	−9347.41	−2.15461
$267$	0	0
$268$	893.647	0.203687
$269$	−1528.13	−0.346363	−0.173182	−	0.984890i	$-0.555405\pi$
$269$	−1528.13	−0.346363	−0.173182	+	0.984890i	$0.555405\pi$
$270$	0	0
$271$	−7368.77	−1.65174	−0.825869	−	0.563862i	$-0.809315\pi$
$271$	−7368.77	−1.65174	−0.825869	+	0.563862i	$0.809315\pi$
$272$	457.599	0.102008
$273$	0	0
$274$	−4081.35	−0.899865
$275$	0	0
$276$	0	0
$277$	1854.32	0.402221	0.201111	−	0.979569i	$-0.435545\pi$
$277$	1854.32	0.402221	0.201111	+	0.979569i	$0.435545\pi$
$278$	5589.57	1.20590
$279$	0	0
$280$	0	0
$281$	415.132	0.0881306	0.0440653	−	0.999029i	$-0.485969\pi$
$281$	415.132	0.0881306	0.0440653	+	0.999029i	$0.485969\pi$
$282$	0	0
$283$	−4116.22	−0.864607	−0.432303	−	0.901728i	$-0.642299\pi$
$283$	−4116.22	−0.864607	−0.432303	+	0.901728i	$0.642299\pi$
$284$	203.157	0.0424477
$285$	0	0
$286$	−1243.89	−0.257178
$287$	9872.84	2.03058
$288$	0	0
$289$	−4874.67	−0.992198
$290$	0	0
$291$	0	0
$292$	398.229	0.0798103
$293$	−4906.20	−0.978236	−0.489118	−	0.872218i	$-0.662681\pi$
$293$	−4906.20	−0.978236	−0.489118	+	0.872218i	$0.662681\pi$
$294$	0	0
$295$	0	0
$296$	3113.42	0.611364
$297$	0	0
$298$	786.570	0.152902
$299$	−3421.22	−0.661720
$300$	0	0
$301$	6037.28	1.15609
$302$	−8782.86	−1.67350
$303$	0	0
$304$	7144.46	1.34790
$305$	0	0
$306$	0	0
$307$	−6909.30	−1.28448	−0.642239	−	0.766504i	$-0.721994\pi$
$307$	−6909.30	−1.28448	−0.642239	+	0.766504i	$0.721994\pi$
$308$	920.994	0.170385
$309$	0	0
$310$	0	0
$311$	−5373.87	−0.979820	−0.489910	−	0.871773i	$-0.662970\pi$
$311$	−5373.87	−0.979820	−0.489910	+	0.871773i	$0.662970\pi$
$312$	0	0
$313$	−2392.60	−0.432070	−0.216035	−	0.976386i	$-0.569313\pi$
$313$	−2392.60	−0.432070	−0.216035	+	0.976386i	$0.569313\pi$
$314$	5387.94	0.968340
$315$	0	0
$316$	190.968	0.0339961
$317$	1183.68	0.209723	0.104861	−	0.994487i	$-0.466560\pi$
$317$	1183.68	0.209723	0.104861	+	0.994487i	$0.466560\pi$
$318$	0	0
$319$	−146.817	−0.0257685
$320$	0	0
$321$	0	0
$322$	15760.5	2.72763
$323$	598.492	0.103099
$324$	0	0
$325$	0	0
$326$	−8051.25	−1.36784
$327$	0	0
$328$	−6294.70	−1.05965
$329$	−9969.66	−1.67065
$330$	0	0
$331$	−9386.43	−1.55868	−0.779342	−	0.626598i	$-0.784447\pi$
$331$	−9386.43	−1.55868	−0.779342	+	0.626598i	$0.784447\pi$
$332$	56.2563	0.00929960
$333$	0	0
$334$	3776.88	0.618748
$335$	0	0
$336$	0	0
$337$	−12171.6	−1.96745	−0.983724	−	0.179685i	$-0.942492\pi$
$337$	−12171.6	−1.96745	−0.983724	+	0.179685i	$0.942492\pi$
$338$	5422.65	0.872644
$339$	0	0
$340$	0	0
$341$	−4319.05	−0.685893
$342$	0	0
$343$	−9238.47	−1.45432
$344$	−3849.23	−0.603304
$345$	0	0
$346$	−8613.64	−1.33836
$347$	−2785.57	−0.430943	−0.215471	−	0.976510i	$-0.569129\pi$
$347$	−2785.57	−0.430943	−0.215471	+	0.976510i	$0.569129\pi$
$348$	0	0
$349$	−3359.76	−0.515311	−0.257656	−	0.966237i	$-0.582950\pi$
$349$	−3359.76	−0.515311	−0.257656	+	0.966237i	$0.582950\pi$
$350$	0	0
$351$	0	0
$352$	−1313.50	−0.198891
$353$	−5427.68	−0.818375	−0.409188	−	0.912450i	$-0.634188\pi$
$353$	−5427.68	−0.818375	−0.409188	+	0.912450i	$0.634188\pi$
$354$	0	0
$355$	0	0
$356$	510.820	0.0760489
$357$	0	0
$358$	−2376.01	−0.350772
$359$	−3920.74	−0.576404	−0.288202	−	0.957570i	$-0.593057\pi$
$359$	−3920.74	−0.576404	−0.288202	+	0.957570i	$0.593057\pi$
$360$	0	0
$361$	2485.20	0.362327
$362$	−7108.37	−1.03207
$363$	0	0
$364$	1007.23	0.145037
$365$	0	0
$366$	0	0
$367$	−1677.21	−0.238554	−0.119277	−	0.992861i	$-0.538058\pi$
$367$	−1677.21	−0.238554	−0.119277	+	0.992861i	$0.538058\pi$
$368$	−12046.1	−1.70638
$369$	0	0
$370$	0	0
$371$	−8690.13	−1.21609
$372$	0	0
$373$	−956.336	−0.132754	−0.0663769	−	0.997795i	$-0.521144\pi$
$373$	−956.336	−0.132754	−0.0663769	+	0.997795i	$0.521144\pi$
$374$	−366.892	−0.0507260
$375$	0	0
$376$	6356.43	0.871829
$377$	−160.564	−0.0219349
$378$	0	0
$379$	10019.0	1.35789	0.678943	−	0.734191i	$-0.262438\pi$
$379$	10019.0	1.35789	0.678943	+	0.734191i	$0.262438\pi$
$380$	0	0
$381$	0	0
$382$	13056.5	1.74877
$383$	−12886.2	−1.71920	−0.859600	−	0.510967i	$-0.829287\pi$
$383$	−12886.2	−1.71920	−0.859600	+	0.510967i	$0.829287\pi$
$384$	0	0
$385$	0	0
$386$	−13473.7	−1.77667
$387$	0	0
$388$	−1576.85	−0.206321
$389$	−688.486	−0.0897369	−0.0448684	−	0.998993i	$-0.514287\pi$
$389$	−688.486	−0.0897369	−0.0448684	+	0.998993i	$0.514287\pi$
$390$	0	0
$391$	−1009.11	−0.130518
$392$	12739.7	1.64145
$393$	0	0
$394$	−3184.65	−0.407209
$395$	0	0
$396$	0	0
$397$	7893.06	0.997837	0.498918	−	0.866649i	$-0.333731\pi$
$397$	7893.06	0.997837	0.498918	+	0.866649i	$0.333731\pi$
$398$	−3925.90	−0.494441
$399$	0	0
$400$	0	0
$401$	4192.82	0.522144	0.261072	−	0.965319i	$-0.415924\pi$
$401$	4192.82	0.522144	0.261072	+	0.965319i	$0.415924\pi$
$402$	0	0
$403$	−4723.47	−0.583853
$404$	1420.15	0.174889
$405$	0	0
$406$	739.669	0.0904166
$407$	−2992.51	−0.364455
$408$	0	0
$409$	9010.03	1.08928	0.544642	−	0.838668i	$-0.316665\pi$
$409$	9010.03	1.08928	0.544642	+	0.838668i	$0.316665\pi$
$410$	0	0
$411$	0	0
$412$	2585.48	0.309169
$413$	13445.6	1.60197
$414$	0	0
$415$	0	0
$416$	−1436.49	−0.169302
$417$	0	0
$418$	−5728.25	−0.670283
$419$	6859.80	0.799817	0.399908	−	0.916555i	$-0.369042\pi$
$419$	6859.80	0.799817	0.399908	+	0.916555i	$0.369042\pi$
$420$	0	0
$421$	12594.6	1.45801	0.729007	−	0.684507i	$-0.239982\pi$
$421$	12594.6	1.45801	0.729007	+	0.684507i	$0.239982\pi$
$422$	−1069.15	−0.123331
$423$	0	0
$424$	5540.63	0.634615
$425$	0	0
$426$	0	0
$427$	2816.79	0.319237
$428$	−1150.31	−0.129912
$429$	0	0
$430$	0	0
$431$	−8470.15	−0.946619	−0.473310	−	0.880896i	$-0.656941\pi$
$431$	−8470.15	−0.946619	−0.473310	+	0.880896i	$0.656941\pi$
$432$	0	0
$433$	−8182.86	−0.908183	−0.454092	−	0.890955i	$-0.650036\pi$
$433$	−8182.86	−0.908183	−0.454092	+	0.890955i	$0.650036\pi$
$434$	21759.6	2.40666
$435$	0	0
$436$	614.723	0.0675227
$437$	−15755.1	−1.72464
$438$	0	0
$439$	8353.58	0.908189	0.454094	−	0.890954i	$-0.349963\pi$
$439$	8353.58	0.908189	0.454094	+	0.890954i	$0.349963\pi$
$440$	0	0
$441$	0	0
$442$	−401.247	−0.0431795
$443$	2222.59	0.238371	0.119186	−	0.992872i	$-0.461972\pi$
$443$	2222.59	0.238371	0.119186	+	0.992872i	$0.461972\pi$
$444$	0	0
$445$	0	0
$446$	−12471.0	−1.32404
$447$	0	0
$448$	−11901.5	−1.25511
$449$	−5455.02	−0.573360	−0.286680	−	0.958026i	$-0.592552\pi$
$449$	−5455.02	−0.573360	−0.286680	+	0.958026i	$0.592552\pi$
$450$	0	0
$451$	6050.25	0.631697
$452$	−337.739	−0.0351459
$453$	0	0
$454$	2662.73	0.275260
$455$	0	0
$456$	0	0
$457$	−6974.13	−0.713865	−0.356932	−	0.934130i	$-0.616177\pi$
$457$	−6974.13	−0.713865	−0.356932	+	0.934130i	$0.616177\pi$
$458$	−2958.40	−0.301827
$459$	0	0
$460$	0	0
$461$	5623.10	0.568099	0.284050	−	0.958810i	$-0.408322\pi$
$461$	5623.10	0.568099	0.284050	+	0.958810i	$0.408322\pi$
$462$	0	0
$463$	−2881.60	−0.289242	−0.144621	−	0.989487i	$-0.546196\pi$
$463$	−2881.60	−0.289242	−0.144621	+	0.989487i	$0.546196\pi$
$464$	−565.348	−0.0565638
$465$	0	0
$466$	−3578.45	−0.355726
$467$	−16877.2	−1.67234	−0.836172	−	0.548468i	$-0.815211\pi$
$467$	−16877.2	−1.67234	−0.836172	+	0.548468i	$0.815211\pi$
$468$	0	0
$469$	−18269.3	−1.79872
$470$	0	0
$471$	0	0
$472$	−8572.61	−0.835988
$473$	3699.75	0.359650
$474$	0	0
$475$	0	0
$476$	297.088	0.0286072
$477$	0	0
$478$	12492.9	1.19542
$479$	−19088.2	−1.82079	−0.910397	−	0.413736i	$-0.864224\pi$
$479$	−19088.2	−1.82079	−0.910397	+	0.413736i	$0.864224\pi$
$480$	0	0
$481$	−3272.72	−0.310235
$482$	7488.45	0.707655
$483$	0	0
$484$	−1474.75	−0.138500
$485$	0	0
$486$	0	0
$487$	−12553.6	−1.16809	−0.584044	−	0.811722i	$-0.698530\pi$
$487$	−12553.6	−1.16809	−0.584044	+	0.811722i	$0.698530\pi$
$488$	−1795.92	−0.166593
$489$	0	0
$490$	0	0
$491$	1569.32	0.144241	0.0721205	−	0.997396i	$-0.477023\pi$
$491$	1569.32	0.144241	0.0721205	+	0.997396i	$0.477023\pi$
$492$	0	0
$493$	−47.3592	−0.00432647
$494$	−6264.63	−0.570565
$495$	0	0
$496$	−16631.4	−1.50559
$497$	−4153.25	−0.374846
$498$	0	0
$499$	−10714.7	−0.961232	−0.480616	−	0.876931i	$-0.659587\pi$
$499$	−10714.7	−0.961232	−0.480616	+	0.876931i	$0.659587\pi$
$500$	0	0
$501$	0	0
$502$	−22453.9	−1.99635
$503$	−19016.8	−1.68572	−0.842862	−	0.538130i	$-0.819131\pi$
$503$	−19016.8	−1.68572	−0.842862	+	0.538130i	$0.819131\pi$
$504$	0	0
$505$	0	0
$506$	9658.30	0.848545
$507$	0	0
$508$	1816.99	0.158693
$509$	16411.5	1.42913	0.714563	−	0.699571i	$-0.246625\pi$
$509$	16411.5	1.42913	0.714563	+	0.699571i	$0.246625\pi$
$510$	0	0
$511$	−8141.21	−0.704786
$512$	6749.35	0.582582
$513$	0	0
$514$	−9985.20	−0.856865
$515$	0	0
$516$	0	0
$517$	−6109.58	−0.519727
$518$	15076.4	1.27880
$519$	0	0
$520$	0	0
$521$	−9319.21	−0.783651	−0.391826	−	0.920040i	$-0.628156\pi$
$521$	−9319.21	−0.783651	−0.391826	+	0.920040i	$0.628156\pi$
$522$	0	0
$523$	−19719.6	−1.64871	−0.824357	−	0.566071i	$-0.808463\pi$
$523$	−19719.6	−1.64871	−0.824357	+	0.566071i	$0.808463\pi$
$524$	−4244.99	−0.353900
$525$	0	0
$526$	−3262.19	−0.270415
$527$	−1393.21	−0.115160
$528$	0	0
$529$	14397.3	1.18331
$530$	0	0
$531$	0	0
$532$	4638.41	0.378009
$533$	6616.78	0.537719
$534$	0	0
$535$	0	0
$536$	11648.1	0.938658
$537$	0	0
$538$	4717.95	0.378077
$539$	−12244.9	−0.978527
$540$	0	0
$541$	16592.1	1.31857	0.659287	−	0.751891i	$-0.270858\pi$
$541$	16592.1	1.31857	0.659287	+	0.751891i	$0.270858\pi$
$542$	22750.4	1.80297
$543$	0	0
$544$	−423.700	−0.0333934
$545$	0	0
$546$	0	0
$547$	−12264.5	−0.958667	−0.479333	−	0.877633i	$-0.659122\pi$
$547$	−12264.5	−0.958667	−0.479333	+	0.877633i	$0.659122\pi$
$548$	2025.26	0.157874
$549$	0	0
$550$	0	0
$551$	−739.415	−0.0571690
$552$	0	0
$553$	−3904.05	−0.300212
$554$	−5725.03	−0.439049
$555$	0	0
$556$	−2773.68	−0.211565
$557$	−1623.54	−0.123504	−0.0617518	−	0.998092i	$-0.519669\pi$
$557$	−1623.54	−0.123504	−0.0617518	+	0.998092i	$0.519669\pi$
$558$	0	0
$559$	4046.18	0.306145
$560$	0	0
$561$	0	0
$562$	−1281.68	−0.0961999
$563$	−6842.29	−0.512199	−0.256100	−	0.966650i	$-0.582438\pi$
$563$	−6842.29	−0.512199	−0.256100	+	0.966650i	$0.582438\pi$
$564$	0	0
$565$	0	0
$566$	12708.4	0.943771
$567$	0	0
$568$	2648.02	0.195613
$569$	−9076.58	−0.668735	−0.334367	−	0.942443i	$-0.608523\pi$
$569$	−9076.58	−0.668735	−0.334367	+	0.942443i	$0.608523\pi$
$570$	0	0
$571$	−5501.59	−0.403213	−0.201606	−	0.979467i	$-0.564616\pi$
$571$	−5501.59	−0.403213	−0.201606	+	0.979467i	$0.564616\pi$
$572$	617.250	0.0451198
$573$	0	0
$574$	−30481.4	−2.21650
$575$	0	0
$576$	0	0
$577$	−13600.1	−0.981245	−0.490622	−	0.871372i	$-0.663231\pi$
$577$	−13600.1	−0.981245	−0.490622	+	0.871372i	$0.663231\pi$
$578$	15050.1	1.08304
$579$	0	0
$580$	0	0
$581$	−1150.08	−0.0821226
$582$	0	0
$583$	−5325.46	−0.378316
$584$	5190.65	0.367792
$585$	0	0
$586$	15147.4	1.06780
$587$	21216.1	1.49179	0.745896	−	0.666062i	$-0.232021\pi$
$587$	21216.1	1.49179	0.745896	+	0.666062i	$0.232021\pi$
$588$	0	0
$589$	−21752.1	−1.52170
$590$	0	0
$591$	0	0
$592$	−11523.3	−0.800007
$593$	14973.6	1.03692	0.518458	−	0.855103i	$-0.326506\pi$
$593$	14973.6	1.03692	0.518458	+	0.855103i	$0.326506\pi$
$594$	0	0
$595$	0	0
$596$	−390.315	−0.0268254
$597$	0	0
$598$	10562.7	0.722307
$599$	−17904.8	−1.22132	−0.610659	−	0.791894i	$-0.709095\pi$
$599$	−17904.8	−1.22132	−0.610659	+	0.791894i	$0.709095\pi$
$600$	0	0
$601$	16455.0	1.11683	0.558414	−	0.829562i	$-0.311410\pi$
$601$	16455.0	1.11683	0.558414	+	0.829562i	$0.311410\pi$
$602$	−18639.5	−1.26194
$603$	0	0
$604$	4358.27	0.293602
$605$	0	0
$606$	0	0
$607$	5457.49	0.364930	0.182465	−	0.983212i	$-0.441592\pi$
$607$	5457.49	0.364930	0.182465	+	0.983212i	$0.441592\pi$
$608$	−6615.19	−0.441253
$609$	0	0
$610$	0	0
$611$	−6681.66	−0.442408
$612$	0	0
$613$	10945.3	0.721167	0.360584	−	0.932727i	$-0.382578\pi$
$613$	10945.3	0.721167	0.360584	+	0.932727i	$0.382578\pi$
$614$	21331.8	1.40209
$615$	0	0
$616$	12004.5	0.785189
$617$	25534.0	1.66606	0.833032	−	0.553225i	$-0.186603\pi$
$617$	25534.0	1.66606	0.833032	+	0.553225i	$0.186603\pi$
$618$	0	0
$619$	−7895.81	−0.512697	−0.256349	−	0.966584i	$-0.582520\pi$
$619$	−7895.81	−0.512697	−0.256349	+	0.966584i	$0.582520\pi$
$620$	0	0
$621$	0	0
$622$	16591.3	1.06953
$623$	−10443.0	−0.671570
$624$	0	0
$625$	0	0
$626$	7386.93	0.471631
$627$	0	0
$628$	−2673.63	−0.169887
$629$	−965.305	−0.0611912
$630$	0	0
$631$	−5659.61	−0.357061	−0.178531	−	0.983934i	$-0.557134\pi$
$631$	−5659.61	−0.357061	−0.178531	+	0.983934i	$0.557134\pi$
$632$	2489.14	0.156665
$633$	0	0
$634$	−3654.50	−0.228925
$635$	0	0
$636$	0	0
$637$	−13391.5	−0.832952
$638$	453.282	0.0281279
$639$	0	0
$640$	0	0
$641$	−16082.9	−0.991010	−0.495505	−	0.868605i	$-0.665017\pi$
$641$	−16082.9	−0.991010	−0.495505	+	0.868605i	$0.665017\pi$
$642$	0	0
$643$	66.7491	0.00409382	0.00204691	−	0.999998i	$-0.499348\pi$
$643$	66.7491	0.00409382	0.00204691	+	0.999998i	$0.499348\pi$
$644$	−7820.74	−0.478541
$645$	0	0
$646$	−1847.78	−0.112539
$647$	−1263.42	−0.0767697	−0.0383849	−	0.999263i	$-0.512221\pi$
$647$	−1263.42	−0.0767697	−0.0383849	+	0.999263i	$0.512221\pi$
$648$	0	0
$649$	8239.69	0.498361
$650$	0	0
$651$	0	0
$652$	3995.23	0.239977
$653$	21867.4	1.31047	0.655235	−	0.755425i	$-0.272570\pi$
$653$	21867.4	1.31047	0.655235	+	0.755425i	$0.272570\pi$
$654$	0	0
$655$	0	0
$656$	23297.7	1.38662
$657$	0	0
$658$	30780.3	1.82362
$659$	362.493	0.0214275	0.0107138	−	0.999943i	$-0.496590\pi$
$659$	362.493	0.0214275	0.0107138	+	0.999943i	$0.496590\pi$
$660$	0	0
$661$	−398.741	−0.0234633	−0.0117316	−	0.999931i	$-0.503734\pi$
$661$	−398.741	−0.0234633	−0.0117316	+	0.999931i	$0.503734\pi$
$662$	28979.7	1.70140
$663$	0	0
$664$	733.263	0.0428556
$665$	0	0
$666$	0	0
$667$	1246.71	0.0723732
$668$	−1874.18	−0.108554
$669$	0	0
$670$	0	0
$671$	1726.18	0.0993121
$672$	0	0
$673$	−3104.18	−0.177797	−0.0888986	−	0.996041i	$-0.528335\pi$
$673$	−3104.18	−0.177797	−0.0888986	+	0.996041i	$0.528335\pi$
$674$	37578.7	2.14759
$675$	0	0
$676$	−2690.85	−0.153098
$677$	−24565.4	−1.39457	−0.697285	−	0.716794i	$-0.745609\pi$
$677$	−24565.4	−1.39457	−0.697285	+	0.716794i	$0.745609\pi$
$678$	0	0
$679$	32236.4	1.82197
$680$	0	0
$681$	0	0
$682$	13334.6	0.748694
$683$	−8921.93	−0.499836	−0.249918	−	0.968267i	$-0.580404\pi$
$683$	−8921.93	−0.499836	−0.249918	+	0.968267i	$0.580404\pi$
$684$	0	0
$685$	0	0
$686$	28522.9	1.58747
$687$	0	0
$688$	14246.6	0.789459
$689$	−5824.12	−0.322034
$690$	0	0
$691$	−6688.21	−0.368207	−0.184104	−	0.982907i	$-0.558938\pi$
$691$	−6688.21	−0.368207	−0.184104	+	0.982907i	$0.558938\pi$
$692$	4274.30	0.234804
$693$	0	0
$694$	8600.17	0.470400
$695$	0	0
$696$	0	0
$697$	1951.65	0.106060
$698$	10372.9	0.562494
$699$	0	0
$700$	0	0
$701$	33422.0	1.80076	0.900380	−	0.435105i	$-0.143289\pi$
$701$	33422.0	1.80076	0.900380	+	0.435105i	$0.143289\pi$
$702$	0	0
$703$	−15071.2	−0.808567
$704$	−7293.42	−0.390456
$705$	0	0
$706$	16757.4	0.893306
$707$	−29032.8	−1.54440
$708$	0	0
$709$	29293.5	1.55168	0.775840	−	0.630929i	$-0.217326\pi$
$709$	29293.5	1.55168	0.775840	+	0.630929i	$0.217326\pi$
$710$	0	0
$711$	0	0
$712$	6658.20	0.350459
$713$	36675.8	1.92639
$714$	0	0
$715$	0	0
$716$	1179.04	0.0615401
$717$	0	0
$718$	12104.9	0.629180
$719$	29311.2	1.52034	0.760169	−	0.649725i	$-0.225116\pi$
$719$	29311.2	1.52034	0.760169	+	0.649725i	$0.225116\pi$
$720$	0	0
$721$	−52856.3	−2.73020
$722$	−7672.81	−0.395502
$723$	0	0
$724$	3527.35	0.181067
$725$	0	0
$726$	0	0
$727$	4931.68	0.251590	0.125795	−	0.992056i	$-0.459852\pi$
$727$	4931.68	0.251590	0.125795	+	0.992056i	$0.459852\pi$
$728$	13128.6	0.668377
$729$	0	0
$730$	0	0
$731$	1193.44	0.0603844
$732$	0	0
$733$	8225.55	0.414485	0.207243	−	0.978290i	$-0.433551\pi$
$733$	8225.55	0.414485	0.207243	+	0.978290i	$0.433551\pi$
$734$	5178.21	0.260397
$735$	0	0
$736$	11153.8	0.558604
$737$	−11195.7	−0.559566
$738$	0	0
$739$	17847.1	0.888382	0.444191	−	0.895932i	$-0.353491\pi$
$739$	17847.1	0.888382	0.444191	+	0.895932i	$0.353491\pi$
$740$	0	0
$741$	0	0
$742$	26829.9	1.32743
$743$	−23810.6	−1.17568	−0.587838	−	0.808979i	$-0.700021\pi$
$743$	−23810.6	−1.17568	−0.587838	+	0.808979i	$0.700021\pi$
$744$	0	0
$745$	0	0
$746$	2952.59	0.144909
$747$	0	0
$748$	182.061	0.00889947
$749$	23516.4	1.14722
$750$	0	0
$751$	21295.8	1.03475	0.517374	−	0.855759i	$-0.326909\pi$
$751$	21295.8	1.03475	0.517374	+	0.855759i	$0.326909\pi$
$752$	−23526.2	−1.14084
$753$	0	0
$754$	495.726	0.0239433
$755$	0	0
$756$	0	0
$757$	13423.0	0.644472	0.322236	−	0.946659i	$-0.395565\pi$
$757$	13423.0	0.644472	0.322236	+	0.946659i	$0.395565\pi$
$758$	−30932.5	−1.48222
$759$	0	0
$760$	0	0
$761$	−24100.2	−1.14801	−0.574003	−	0.818853i	$-0.694610\pi$
$761$	−24100.2	−1.14801	−0.574003	+	0.818853i	$0.694610\pi$
$762$	0	0
$763$	−12567.1	−0.596277
$764$	−6478.96	−0.306807
$765$	0	0
$766$	39784.8	1.87661
$767$	9011.23	0.424220
$768$	0	0
$769$	−986.759	−0.0462724	−0.0231362	−	0.999732i	$-0.507365\pi$
$769$	−986.759	−0.0462724	−0.0231362	+	0.999732i	$0.507365\pi$
$770$	0	0
$771$	0	0
$772$	6685.98	0.311701
$773$	−7277.05	−0.338599	−0.169300	−	0.985565i	$-0.554151\pi$
$773$	−7277.05	−0.338599	−0.169300	+	0.985565i	$0.554151\pi$
$774$	0	0
$775$	0	0
$776$	−20553.2	−0.950794
$777$	0	0
$778$	2125.63	0.0979533
$779$	30471.0	1.40146
$780$	0	0
$781$	−2545.18	−0.116612
$782$	3115.51	0.142469
$783$	0	0
$784$	−47151.6	−2.14794
$785$	0	0
$786$	0	0
$787$	4939.44	0.223726	0.111863	−	0.993724i	$-0.464318\pi$
$787$	4939.44	0.223726	0.111863	+	0.993724i	$0.464318\pi$
$788$	1580.30	0.0714415
$789$	0	0
$790$	0	0
$791$	6904.59	0.310365
$792$	0	0
$793$	1887.81	0.0845375
$794$	−24369.0	−1.08920
$795$	0	0
$796$	1948.13	0.0867456
$797$	12447.6	0.553219	0.276610	−	0.960982i	$-0.410789\pi$
$797$	12447.6	0.553219	0.276610	+	0.960982i	$0.410789\pi$
$798$	0	0
$799$	−1970.79	−0.0872610
$800$	0	0
$801$	0	0
$802$	−12944.9	−0.569952
$803$	−4989.07	−0.219254
$804$	0	0
$805$	0	0
$806$	14583.2	0.637311
$807$	0	0
$808$	18510.7	0.805944
$809$	−37250.9	−1.61888	−0.809438	−	0.587205i	$-0.800228\pi$
$809$	−37250.9	−1.61888	−0.809438	+	0.587205i	$0.800228\pi$
$810$	0	0
$811$	7201.80	0.311824	0.155912	−	0.987771i	$-0.450168\pi$
$811$	7201.80	0.311824	0.155912	+	0.987771i	$0.450168\pi$
$812$	−367.042	−0.0158629
$813$	0	0
$814$	9239.08	0.397825
$815$	0	0
$816$	0	0
$817$	18633.1	0.797906
$818$	−27817.6	−1.18902
$819$	0	0
$820$	0	0
$821$	10824.6	0.460147	0.230074	−	0.973173i	$-0.426103\pi$
$821$	10824.6	0.460147	0.230074	+	0.973173i	$0.426103\pi$
$822$	0	0
$823$	−18031.3	−0.763710	−0.381855	−	0.924222i	$-0.624715\pi$
$823$	−18031.3	−0.763710	−0.381855	+	0.924222i	$0.624715\pi$
$824$	33700.0	1.42475
$825$	0	0
$826$	−41511.9	−1.74865
$827$	−13172.5	−0.553873	−0.276937	−	0.960888i	$-0.589319\pi$
$827$	−13172.5	−0.553873	−0.276937	+	0.960888i	$0.589319\pi$
$828$	0	0
$829$	−11929.1	−0.499776	−0.249888	−	0.968275i	$-0.580394\pi$
$829$	−11929.1	−0.499776	−0.249888	+	0.968275i	$0.580394\pi$
$830$	0	0
$831$	0	0
$832$	−7976.35	−0.332368
$833$	−3949.89	−0.164292
$834$	0	0
$835$	0	0
$836$	2842.50	0.117596
$837$	0	0
$838$	−21179.0	−0.873049
$839$	25393.4	1.04491	0.522453	−	0.852668i	$-0.325017\pi$
$839$	25393.4	1.04491	0.522453	+	0.852668i	$0.325017\pi$
$840$	0	0
$841$	−24330.5	−0.997601
$842$	−38884.6	−1.59151
$843$	0	0
$844$	530.540	0.0216373
$845$	0	0
$846$	0	0
$847$	30149.1	1.22306
$848$	−20506.8	−0.830431
$849$	0	0
$850$	0	0
$851$	25411.3	1.02361
$852$	0	0
$853$	39373.6	1.58045	0.790225	−	0.612816i	$-0.209963\pi$
$853$	39373.6	1.58045	0.790225	+	0.612816i	$0.209963\pi$
$854$	−8696.57	−0.348466
$855$	0	0
$856$	−14993.5	−0.598677
$857$	5240.83	0.208895	0.104448	−	0.994530i	$-0.466693\pi$
$857$	5240.83	0.208895	0.104448	+	0.994530i	$0.466693\pi$
$858$	0	0
$859$	−8308.93	−0.330032	−0.165016	−	0.986291i	$-0.552768\pi$
$859$	−8308.93	−0.330032	−0.165016	+	0.986291i	$0.552768\pi$
$860$	0	0
$861$	0	0
$862$	26150.8	1.03329
$863$	5719.30	0.225593	0.112797	−	0.993618i	$-0.464019\pi$
$863$	5719.30	0.225593	0.112797	+	0.993618i	$0.464019\pi$
$864$	0	0
$865$	0	0
$866$	25263.8	0.991338
$867$	0	0
$868$	−10797.6	−0.422230
$869$	−2392.47	−0.0933936
$870$	0	0
$871$	−12244.1	−0.476320
$872$	8012.50	0.311167
$873$	0	0
$874$	48642.2	1.88255
$875$	0	0
$876$	0	0
$877$	−12317.6	−0.474272	−0.237136	−	0.971476i	$-0.576209\pi$
$877$	−12317.6	−0.474272	−0.237136	+	0.971476i	$0.576209\pi$
$878$	−25790.9	−0.991343
$879$	0	0
$880$	0	0
$881$	−4540.33	−0.173630	−0.0868148	−	0.996224i	$-0.527669\pi$
$881$	−4540.33	−0.173630	−0.0868148	+	0.996224i	$0.527669\pi$
$882$	0	0
$883$	19437.5	0.740796	0.370398	−	0.928873i	$-0.379221\pi$
$883$	19437.5	0.740796	0.370398	+	0.928873i	$0.379221\pi$
$884$	199.108	0.00757550
$885$	0	0
$886$	−6862.03	−0.260197
$887$	2625.66	0.0993925	0.0496962	−	0.998764i	$-0.484175\pi$
$887$	2625.66	0.0993925	0.0496962	+	0.998764i	$0.484175\pi$
$888$	0	0
$889$	−37145.7	−1.40138
$890$	0	0
$891$	0	0
$892$	6188.43	0.232291
$893$	−30769.8	−1.15305
$894$	0	0
$895$	0	0
$896$	53891.6	2.00936
$897$	0	0
$898$	16841.8	0.625857
$899$	1721.26	0.0638569
$900$	0	0
$901$	−1717.85	−0.0635183
$902$	−18679.5	−0.689535
$903$	0	0
$904$	−4402.21	−0.161964
$905$	0	0
$906$	0	0
$907$	44880.6	1.64304	0.821519	−	0.570182i	$-0.193127\pi$
$907$	44880.6	1.64304	0.821519	+	0.570182i	$0.193127\pi$
$908$	−1321.31	−0.0482922
$909$	0	0
$910$	0	0
$911$	52766.0	1.91901	0.959503	−	0.281698i	$-0.0908976\pi$
$911$	52766.0	1.91901	0.959503	+	0.281698i	$0.0908976\pi$
$912$	0	0
$913$	−704.788	−0.0255477
$914$	21531.9	0.779227
$915$	0	0
$916$	1468.03	0.0529531
$917$	86782.7	3.12521
$918$	0	0
$919$	−33853.3	−1.21514	−0.607571	−	0.794265i	$-0.707856\pi$
$919$	−33853.3	−1.21514	−0.607571	+	0.794265i	$0.707856\pi$
$920$	0	0
$921$	0	0
$922$	−17360.8	−0.620115
$923$	−2783.50	−0.0992634
$924$	0	0
$925$	0	0
$926$	8896.65	0.315726
$927$	0	0
$928$	523.466	0.0185168
$929$	−5402.22	−0.190787	−0.0953935	−	0.995440i	$-0.530411\pi$
$929$	−5402.22	−0.190787	−0.0953935	+	0.995440i	$0.530411\pi$
$930$	0	0
$931$	−61669.3	−2.17092
$932$	1775.71	0.0624093
$933$	0	0
$934$	52106.7	1.82546
$935$	0	0
$936$	0	0
$937$	−37756.4	−1.31638	−0.658190	−	0.752852i	$-0.728677\pi$
$937$	−37756.4	−1.31638	−0.658190	+	0.752852i	$0.728677\pi$
$938$	56404.6	1.96341
$939$	0	0
$940$	0	0
$941$	3170.18	0.109825	0.0549123	−	0.998491i	$-0.482512\pi$
$941$	3170.18	0.109825	0.0549123	+	0.998491i	$0.482512\pi$
$942$	0	0
$943$	−51376.5	−1.77418
$944$	31728.6	1.09394
$945$	0	0
$946$	−11422.6	−0.392580
$947$	−41521.2	−1.42477	−0.712385	−	0.701789i	$-0.752385\pi$
$947$	−41521.2	−1.42477	−0.712385	+	0.701789i	$0.752385\pi$
$948$	0	0
$949$	−5456.23	−0.186635
$950$	0	0
$951$	0	0
$952$	3872.35	0.131831
$953$	18886.5	0.641965	0.320982	−	0.947085i	$-0.395987\pi$
$953$	18886.5	0.641965	0.320982	+	0.947085i	$0.395987\pi$
$954$	0	0
$955$	0	0
$956$	−6199.27	−0.209727
$957$	0	0
$958$	58932.8	1.98751
$959$	−41403.5	−1.39415
$960$	0	0
$961$	20845.1	0.699710
$962$	10104.2	0.338641
$963$	0	0
$964$	−3715.96	−0.124152
$965$	0	0
$966$	0	0
$967$	39799.9	1.32355	0.661777	−	0.749700i	$-0.269802\pi$
$967$	39799.9	1.32355	0.661777	+	0.749700i	$0.269802\pi$
$968$	−19222.4	−0.638255
$969$	0	0
$970$	0	0
$971$	−39063.6	−1.29105	−0.645525	−	0.763739i	$-0.723361\pi$
$971$	−39063.6	−1.29105	−0.645525	+	0.763739i	$0.723361\pi$
$972$	0	0
$973$	56703.8	1.86829
$974$	38758.1	1.27504
$975$	0	0
$976$	6647.01	0.217997
$977$	5087.15	0.166584	0.0832918	−	0.996525i	$-0.473457\pi$
$977$	5087.15	0.166584	0.0832918	+	0.996525i	$0.473457\pi$
$978$	0	0
$979$	−6399.63	−0.208920
$980$	0	0
$981$	0	0
$982$	−4845.12	−0.157448
$983$	−32012.1	−1.03869	−0.519343	−	0.854566i	$-0.673823\pi$
$983$	−32012.1	−1.03869	−0.519343	+	0.854566i	$0.673823\pi$
$984$	0	0
$985$	0	0
$986$	146.217	0.00472261
$987$	0	0
$988$	3108.66	0.100101
$989$	−31416.9	−1.01011
$990$	0	0
$991$	3717.85	0.119174	0.0595870	−	0.998223i	$-0.481022\pi$
$991$	3717.85	0.119174	0.0595870	+	0.998223i	$0.481022\pi$
$992$	15399.3	0.492872
$993$	0	0
$994$	12822.7	0.409168
$995$	0	0
$996$	0	0
$997$	−5517.11	−0.175254	−0.0876272	−	0.996153i	$-0.527928\pi$
$997$	−5517.11	−0.175254	−0.0876272	+	0.996153i	$0.527928\pi$
$998$	33080.5	1.04924
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bk.1.4		16
3.2	odd	2		2025.4.a.bl.1.13		16
5.2	odd	4		405.4.b.e.244.4		16
5.3	odd	4		405.4.b.e.244.13		16
5.4	even	2	inner	2025.4.a.bk.1.13		16
9.4	even	3		225.4.e.g.151.13		32
9.7	even	3		225.4.e.g.76.13		32
15.2	even	4		405.4.b.f.244.13		16
15.8	even	4		405.4.b.f.244.4		16
15.14	odd	2		2025.4.a.bl.1.4		16
45.2	even	12		135.4.j.a.64.13		32
45.4	even	6		225.4.e.g.151.4		32
45.7	odd	12		45.4.j.a.4.4	✓	32
45.13	odd	12		45.4.j.a.34.4	yes	32
45.22	odd	12		45.4.j.a.34.13	yes	32
45.23	even	12		135.4.j.a.19.13		32
45.32	even	12		135.4.j.a.19.4		32
45.34	even	6		225.4.e.g.76.4		32
45.38	even	12		135.4.j.a.64.4		32
45.43	odd	12		45.4.j.a.4.13	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.4	✓	32	45.7	odd	12
45.4.j.a.4.13	yes	32	45.43	odd	12
45.4.j.a.34.4	yes	32	45.13	odd	12
45.4.j.a.34.13	yes	32	45.22	odd	12
135.4.j.a.19.4		32	45.32	even	12
135.4.j.a.19.13		32	45.23	even	12
135.4.j.a.64.4		32	45.38	even	12
135.4.j.a.64.13		32	45.2	even	12
225.4.e.g.76.4		32	45.34	even	6
225.4.e.g.76.13		32	9.7	even	3
225.4.e.g.151.4		32	45.4	even	6
225.4.e.g.151.13		32	9.4	even	3
405.4.b.e.244.4		16	5.2	odd	4
405.4.b.e.244.13		16	5.3	odd	4
405.4.b.f.244.4		16	15.8	even	4
405.4.b.f.244.13		16	15.2	even	4
2025.4.a.bk.1.4		16	1.1	even	1	trivial
2025.4.a.bk.1.13		16	5.4	even	2	inner
2025.4.a.bl.1.4		16	15.14	odd	2
2025.4.a.bl.1.13		16	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)

\(f(q)\)	\(=\)	\(q-3.08740 q^{2} +1.53204 q^{4} -31.3204 q^{7} +19.9692 q^{8} +O(q^{10})\)	\(q-3.08740 q^{2} +1.53204 q^{4} -31.3204 q^{7} +19.9692 q^{8} -19.1937 q^{11} -20.9909 q^{13} +96.6986 q^{14} -73.9092 q^{16} -6.19137 q^{17} -96.6654 q^{19} +59.2586 q^{22} +162.986 q^{23} +64.8074 q^{26} -47.9842 q^{28} +7.64922 q^{29} +225.025 q^{31} +68.4339 q^{32} +19.1153 q^{34} +155.911 q^{37} +298.445 q^{38} -315.221 q^{41} -192.759 q^{43} -29.4056 q^{44} -503.202 q^{46} +318.312 q^{47} +637.967 q^{49} -32.1590 q^{52} +277.459 q^{53} -625.442 q^{56} -23.6162 q^{58} -429.292 q^{59} -89.9348 q^{61} -694.741 q^{62} +379.991 q^{64} +583.304 q^{67} -9.48546 q^{68} +132.605 q^{71} +259.933 q^{73} -481.361 q^{74} -148.096 q^{76} +601.153 q^{77} +124.649 q^{79} +973.214 q^{82} +36.7198 q^{83} +595.123 q^{86} -383.282 q^{88} +333.424 q^{89} +657.443 q^{91} +249.701 q^{92} -982.757 q^{94} -1029.25 q^{97} -1969.66 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 54 q^{4}+O(q^{10}) \) 16 * q + 54 * q^4	\( 16 q + 54 q^{4} - 90 q^{11} - 102 q^{14} + 146 q^{16} + 4 q^{19} - 468 q^{26} - 516 q^{29} + 38 q^{31} + 212 q^{34} - 576 q^{41} - 1644 q^{44} - 290 q^{46} - 4 q^{49} - 2430 q^{56} - 2202 q^{59} + 20 q^{61} - 322 q^{64} - 2952 q^{71} - 4080 q^{74} - 396 q^{76} - 218 q^{79} - 6108 q^{86} - 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) \) 16 * q + 54 * q^4 - 90 * q^11 - 102 * q^14 + 146 * q^16 + 4 * q^19 - 468 * q^26 - 516 * q^29 + 38 * q^31 + 212 * q^34 - 576 * q^41 - 1644 * q^44 - 290 * q^46 - 4 * q^49 - 2430 * q^56 - 2202 * q^59 + 20 * q^61 - 322 * q^64 - 2952 * q^71 - 4080 * q^74 - 396 * q^76 - 218 * q^79 - 6108 * q^86 - 4074 * q^89 - 942 * q^91 - 1078 * q^94

Introduction

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