LMFDB - Embedded newform 2025.4.a.bb.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:	$0$
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			1.4
Root			$0.225250$ 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)

$f(q)$	$=$	$q+0.225250 q^{2} -7.94926 q^{4} +31.1940 q^{7} -3.59257 q^{8} +O(q^{10})$	$q+0.225250 q^{2} -7.94926 q^{4} +31.1940 q^{7} -3.59257 q^{8} +18.1285 q^{11} +50.1562 q^{13} +7.02645 q^{14} +62.7849 q^{16} +131.631 q^{17} +23.2428 q^{19} +4.08344 q^{22} +32.9856 q^{23} +11.2977 q^{26} -247.969 q^{28} +125.817 q^{29} +125.113 q^{31} +42.8828 q^{32} +29.6499 q^{34} -99.9894 q^{37} +5.23544 q^{38} +245.326 q^{41} -139.176 q^{43} -144.108 q^{44} +7.43001 q^{46} +472.961 q^{47} +630.067 q^{49} -398.705 q^{52} -421.529 q^{53} -112.067 q^{56} +28.3404 q^{58} -742.413 q^{59} +8.97736 q^{61} +28.1816 q^{62} -492.620 q^{64} -588.906 q^{67} -1046.37 q^{68} -48.5526 q^{71} -409.800 q^{73} -22.5226 q^{74} -184.763 q^{76} +565.500 q^{77} +530.527 q^{79} +55.2595 q^{82} +294.590 q^{83} -31.3495 q^{86} -65.1278 q^{88} +852.817 q^{89} +1564.57 q^{91} -262.211 q^{92} +106.534 q^{94} +388.091 q^{97} +141.922 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.225250	0.0796378	0.0398189	−	0.999207i	$-0.487322\pi$
$2$	0.225250	0.0796378	0.0398189	+	0.999207i	$0.487322\pi$
$3$	0	0
$3$	0	0
$4$	−7.94926	−0.993658
$5$	0	0
$5$	0	0
$6$	0	0
$7$	31.1940	1.68432	0.842159	−	0.539229i	$-0.181284\pi$
$7$	31.1940	1.68432	0.842159	+	0.539229i	$0.181284\pi$
$8$	−3.59257	−0.158771
$9$	0	0
$10$	0	0
$11$	18.1285	0.496904	0.248452	−	0.968644i	$-0.420078\pi$
$11$	18.1285	0.496904	0.248452	+	0.968644i	$0.420078\pi$
$12$	0	0
$13$	50.1562	1.07006	0.535032	−	0.844832i	$-0.320300\pi$
$13$	50.1562	1.07006	0.535032	+	0.844832i	$0.320300\pi$
$14$	7.02645	0.134136
$15$	0	0
$16$	62.7849	0.981014
$17$	131.631	1.87795	0.938977	−	0.343981i	$-0.111776\pi$
$17$	131.631	1.87795	0.938977	+	0.343981i	$0.111776\pi$
$18$	0	0
$19$	23.2428	0.280646	0.140323	−	0.990106i	$-0.455186\pi$
$19$	23.2428	0.280646	0.140323	+	0.990106i	$0.455186\pi$
$20$	0	0
$21$	0	0
$22$	4.08344	0.0395723
$23$	32.9856	0.299043	0.149521	−	0.988759i	$-0.452227\pi$
$23$	32.9856	0.299043	0.149521	+	0.988759i	$0.452227\pi$
$24$	0	0
$25$	0	0
$26$	11.2977	0.0852176
$27$	0	0
$28$	−247.969	−1.67364
$29$	125.817	0.805645	0.402823	−	0.915278i	$-0.368029\pi$
$29$	125.817	0.805645	0.402823	+	0.915278i	$0.368029\pi$
$30$	0	0
$31$	125.113	0.724868	0.362434	−	0.932009i	$-0.381946\pi$
$31$	125.113	0.724868	0.362434	+	0.932009i	$0.381946\pi$
$32$	42.8828	0.236896
$33$	0	0
$34$	29.6499	0.149556
$35$	0	0
$36$	0	0
$37$	−99.9894	−0.444274	−0.222137	−	0.975015i	$-0.571303\pi$
$37$	−99.9894	−0.444274	−0.222137	+	0.975015i	$0.571303\pi$
$38$	5.23544	0.0223500
$39$	0	0
$40$	0	0
$41$	245.326	0.934473	0.467237	−	0.884132i	$-0.345250\pi$
$41$	245.326	0.934473	0.467237	+	0.884132i	$0.345250\pi$
$42$	0	0
$43$	−139.176	−0.493586	−0.246793	−	0.969068i	$-0.579377\pi$
$43$	−139.176	−0.493586	−0.246793	+	0.969068i	$0.579377\pi$
$44$	−144.108	−0.493752
$45$	0	0
$46$	7.43001	0.0238151
$47$	472.961	1.46784	0.733919	−	0.679237i	$-0.237689\pi$
$47$	472.961	1.46784	0.733919	+	0.679237i	$0.237689\pi$
$48$	0	0
$49$	630.067	1.83693
$50$	0	0
$51$	0	0
$52$	−398.705	−1.06328
$53$	−421.529	−1.09248	−0.546240	−	0.837628i	$-0.683941\pi$
$53$	−421.529	−1.09248	−0.546240	+	0.837628i	$0.683941\pi$
$54$	0	0
$55$	0	0
$56$	−112.067	−0.267420
$57$	0	0
$58$	28.3404	0.0641598
$59$	−742.413	−1.63820	−0.819101	−	0.573649i	$-0.805527\pi$
$59$	−742.413	−1.63820	−0.819101	+	0.573649i	$0.805527\pi$
$60$	0	0
$61$	8.97736	0.0188432	0.00942158	−	0.999956i	$-0.497001\pi$
$61$	8.97736	0.0188432	0.00942158	+	0.999956i	$0.497001\pi$
$62$	28.1816	0.0577269
$63$	0	0
$64$	−492.620	−0.962148
$65$	0	0
$66$	0	0
$67$	−588.906	−1.07383	−0.536913	−	0.843638i	$-0.680410\pi$
$67$	−588.906	−1.07383	−0.536913	+	0.843638i	$0.680410\pi$
$68$	−1046.37	−1.86604
$69$	0	0
$70$	0	0
$71$	−48.5526	−0.0811568	−0.0405784	−	0.999176i	$-0.512920\pi$
$71$	−48.5526	−0.0811568	−0.0405784	+	0.999176i	$0.512920\pi$
$72$	0	0
$73$	−409.800	−0.657034	−0.328517	−	0.944498i	$-0.606549\pi$
$73$	−409.800	−0.657034	−0.328517	+	0.944498i	$0.606549\pi$
$74$	−22.5226	−0.0353811
$75$	0	0
$76$	−184.763	−0.278866
$77$	565.500	0.836944
$78$	0	0
$79$	530.527	0.755556	0.377778	−	0.925896i	$-0.376688\pi$
$79$	530.527	0.755556	0.377778	+	0.925896i	$0.376688\pi$
$80$	0	0
$81$	0	0
$82$	55.2595	0.0744195
$83$	294.590	0.389584	0.194792	−	0.980845i	$-0.437597\pi$
$83$	294.590	0.389584	0.194792	+	0.980845i	$0.437597\pi$
$84$	0	0
$85$	0	0
$86$	−31.3495	−0.0393081
$87$	0	0
$88$	−65.1278	−0.0788937
$89$	852.817	1.01571	0.507856	−	0.861442i	$-0.330438\pi$
$89$	852.817	1.01571	0.507856	+	0.861442i	$0.330438\pi$
$90$	0	0
$91$	1564.57	1.80233
$92$	−262.211	−0.297146
$93$	0	0
$94$	106.534	0.116896
$95$	0	0
$96$	0	0
$97$	388.091	0.406233	0.203117	−	0.979155i	$-0.434893\pi$
$97$	388.091	0.406233	0.203117	+	0.979155i	$0.434893\pi$
$98$	141.922	0.146289
$99$	0	0
$100$	0	0
$101$	−1079.99	−1.06399	−0.531997	−	0.846746i	$-0.678558\pi$
$101$	−1079.99	−1.06399	−0.531997	+	0.846746i	$0.678558\pi$
$102$	0	0
$103$	594.997	0.569192	0.284596	−	0.958648i	$-0.408141\pi$
$103$	594.997	0.569192	0.284596	+	0.958648i	$0.408141\pi$
$104$	−180.190	−0.169895
$105$	0	0
$106$	−94.9494	−0.0870028
$107$	498.693	0.450565	0.225282	−	0.974293i	$-0.427670\pi$
$107$	498.693	0.450565	0.225282	+	0.974293i	$0.427670\pi$
$108$	0	0
$109$	−959.301	−0.842976	−0.421488	−	0.906834i	$-0.638492\pi$
$109$	−959.301	−0.842976	−0.421488	+	0.906834i	$0.638492\pi$
$110$	0	0
$111$	0	0
$112$	1958.51	1.65234
$113$	−1728.79	−1.43921	−0.719607	−	0.694382i	$-0.755678\pi$
$113$	−1728.79	−1.43921	−0.719607	+	0.694382i	$0.755678\pi$
$114$	0	0
$115$	0	0
$116$	−1000.16	−0.800536
$117$	0	0
$118$	−167.228	−0.130463
$119$	4106.10	3.16307
$120$	0	0
$121$	−1002.36	−0.753087
$122$	2.02215	0.00150063
$123$	0	0
$124$	−994.554	−0.720271
$125$	0	0
$126$	0	0
$127$	−1019.95	−0.712647	−0.356324	−	0.934363i	$-0.615970\pi$
$127$	−1019.95	−0.712647	−0.356324	+	0.934363i	$0.615970\pi$
$128$	−454.025	−0.313520
$129$	0	0
$130$	0	0
$131$	−1702.39	−1.13541	−0.567705	−	0.823232i	$-0.692169\pi$
$131$	−1702.39	−1.13541	−0.567705	+	0.823232i	$0.692169\pi$
$132$	0	0
$133$	725.037	0.472697
$134$	−132.651	−0.0855172
$135$	0	0
$136$	−472.893	−0.298164
$137$	1049.68	0.654599	0.327300	−	0.944921i	$-0.393861\pi$
$137$	1049.68	0.654599	0.327300	+	0.944921i	$0.393861\pi$
$138$	0	0
$139$	1434.82	0.875540	0.437770	−	0.899087i	$-0.355768\pi$
$139$	1434.82	0.875540	0.437770	+	0.899087i	$0.355768\pi$
$140$	0	0
$141$	0	0
$142$	−10.9365	−0.00646316
$143$	909.256	0.531719
$144$	0	0
$145$	0	0
$146$	−92.3075	−0.0523248
$147$	0	0
$148$	794.842	0.441457
$149$	1218.46	0.669935	0.334968	−	0.942230i	$-0.391275\pi$
$149$	1218.46	0.669935	0.334968	+	0.942230i	$0.391275\pi$
$150$	0	0
$151$	1609.50	0.867414	0.433707	−	0.901054i	$-0.357205\pi$
$151$	1609.50	0.867414	0.433707	+	0.901054i	$0.357205\pi$
$152$	−83.5014	−0.0445583
$153$	0	0
$154$	127.379	0.0666524
$155$	0	0
$156$	0	0
$157$	−1286.34	−0.653890	−0.326945	−	0.945043i	$-0.606019\pi$
$157$	−1286.34	−0.653890	−0.326945	+	0.945043i	$0.606019\pi$
$158$	119.501	0.0601708
$159$	0	0
$160$	0	0
$161$	1028.95	0.503683
$162$	0	0
$163$	−1416.84	−0.680830	−0.340415	−	0.940275i	$-0.610568\pi$
$163$	−1416.84	−0.680830	−0.340415	+	0.940275i	$0.610568\pi$
$164$	−1950.16	−0.928547
$165$	0	0
$166$	66.3563	0.0310256
$167$	−894.975	−0.414702	−0.207351	−	0.978267i	$-0.566484\pi$
$167$	−894.975	−0.414702	−0.207351	+	0.978267i	$0.566484\pi$
$168$	0	0
$169$	318.645	0.145037
$170$	0	0
$171$	0	0
$172$	1106.35	0.490456
$173$	1735.10	0.762526	0.381263	−	0.924467i	$-0.375489\pi$
$173$	1735.10	0.762526	0.381263	+	0.924467i	$0.375489\pi$
$174$	0	0
$175$	0	0
$176$	1138.19	0.487469
$177$	0	0
$178$	192.097	0.0808892
$179$	2133.37	0.890815	0.445408	−	0.895328i	$-0.353059\pi$
$179$	2133.37	0.890815	0.445408	+	0.895328i	$0.353059\pi$
$180$	0	0
$181$	−3611.98	−1.48330	−0.741648	−	0.670789i	$-0.765955\pi$
$181$	−3611.98	−1.48330	−0.741648	+	0.670789i	$0.765955\pi$
$182$	352.420	0.143534
$183$	0	0
$184$	−118.503	−0.0474792
$185$	0	0
$186$	0	0
$187$	2386.27	0.933162
$188$	−3759.69	−1.45853
$189$	0	0
$190$	0	0
$191$	−596.990	−0.226160	−0.113080	−	0.993586i	$-0.536072\pi$
$191$	−596.990	−0.226160	−0.113080	+	0.993586i	$0.536072\pi$
$192$	0	0
$193$	1207.74	0.450442	0.225221	−	0.974308i	$-0.427690\pi$
$193$	1207.74	0.450442	0.225221	+	0.974308i	$0.427690\pi$
$194$	87.4173	0.0323515
$195$	0	0
$196$	−5008.57	−1.82528
$197$	3268.56	1.18211	0.591054	−	0.806632i	$-0.298712\pi$
$197$	3268.56	1.18211	0.591054	+	0.806632i	$0.298712\pi$
$198$	0	0
$199$	−2109.88	−0.751585	−0.375793	−	0.926704i	$-0.622629\pi$
$199$	−2109.88	−0.751585	−0.375793	+	0.926704i	$0.622629\pi$
$200$	0	0
$201$	0	0
$202$	−243.268	−0.0847342
$203$	3924.75	1.35696
$204$	0	0
$205$	0	0
$206$	134.023	0.0453292
$207$	0	0
$208$	3149.05	1.04975
$209$	421.357	0.139454
$210$	0	0
$211$	658.642	0.214895	0.107447	−	0.994211i	$-0.465732\pi$
$211$	658.642	0.214895	0.107447	+	0.994211i	$0.465732\pi$
$212$	3350.85	1.08555
$213$	0	0
$214$	112.330	0.0358820
$215$	0	0
$216$	0	0
$217$	3902.77	1.22091
$218$	−216.082	−0.0671328
$219$	0	0
$220$	0	0
$221$	6602.11	2.00953
$222$	0	0
$223$	−1218.79	−0.365993	−0.182997	−	0.983114i	$-0.558580\pi$
$223$	−1218.79	−0.365993	−0.182997	+	0.983114i	$0.558580\pi$
$224$	1337.69	0.399009
$225$	0	0
$226$	−389.410	−0.114616
$227$	1525.57	0.446060	0.223030	−	0.974812i	$-0.428405\pi$
$227$	1525.57	0.446060	0.223030	+	0.974812i	$0.428405\pi$
$228$	0	0
$229$	−6182.65	−1.78411	−0.892055	−	0.451927i	$-0.850737\pi$
$229$	−6182.65	−1.78411	−0.892055	+	0.451927i	$0.850737\pi$
$230$	0	0
$231$	0	0
$232$	−452.008	−0.127913
$233$	2013.82	0.566222	0.283111	−	0.959087i	$-0.408633\pi$
$233$	2013.82	0.566222	0.283111	+	0.959087i	$0.408633\pi$
$234$	0	0
$235$	0	0
$236$	5901.64	1.62781
$237$	0	0
$238$	924.898	0.251900
$239$	5086.31	1.37659	0.688297	−	0.725429i	$-0.258359\pi$
$239$	5086.31	1.37659	0.688297	+	0.725429i	$0.258359\pi$
$240$	0	0
$241$	3286.83	0.878519	0.439259	−	0.898360i	$-0.355241\pi$
$241$	3286.83	0.878519	0.439259	+	0.898360i	$0.355241\pi$
$242$	−225.781	−0.0599742
$243$	0	0
$244$	−71.3634	−0.0187237
$245$	0	0
$246$	0	0
$247$	1165.77	0.300309
$248$	−449.476	−0.115088
$249$	0	0
$250$	0	0
$251$	3480.55	0.875260	0.437630	−	0.899155i	$-0.355818\pi$
$251$	3480.55	0.875260	0.437630	+	0.899155i	$0.355818\pi$
$252$	0	0
$253$	597.979	0.148595
$254$	−229.744	−0.0567537
$255$	0	0
$256$	3838.69	0.937180
$257$	6794.98	1.64926	0.824628	−	0.565675i	$-0.191384\pi$
$257$	6794.98	1.64926	0.824628	+	0.565675i	$0.191384\pi$
$258$	0	0
$259$	−3119.07	−0.748300
$260$	0	0
$261$	0	0
$262$	−383.464	−0.0904216
$263$	−5364.24	−1.25769	−0.628846	−	0.777530i	$-0.716472\pi$
$263$	−5364.24	−1.25769	−0.628846	+	0.777530i	$0.716472\pi$
$264$	0	0
$265$	0	0
$266$	163.314	0.0376445
$267$	0	0
$268$	4681.37	1.06702
$269$	48.4985	0.0109926	0.00549629	−	0.999985i	$-0.498250\pi$
$269$	48.4985	0.0109926	0.00549629	+	0.999985i	$0.498250\pi$
$270$	0	0
$271$	7643.16	1.71324	0.856622	−	0.515945i	$-0.172559\pi$
$271$	7643.16	1.71324	0.856622	+	0.515945i	$0.172559\pi$
$272$	8264.43	1.84230
$273$	0	0
$274$	236.440	0.0521309
$275$	0	0
$276$	0	0
$277$	−5146.02	−1.11623	−0.558113	−	0.829765i	$-0.688474\pi$
$277$	−5146.02	−1.11623	−0.558113	+	0.829765i	$0.688474\pi$
$278$	323.193	0.0697261
$279$	0	0
$280$	0	0
$281$	5854.74	1.24293	0.621467	−	0.783440i	$-0.286537\pi$
$281$	5854.74	1.24293	0.621467	+	0.783440i	$0.286537\pi$
$282$	0	0
$283$	−9393.84	−1.97317	−0.986583	−	0.163260i	$-0.947799\pi$
$283$	−9393.84	−1.97317	−0.986583	+	0.163260i	$0.947799\pi$
$284$	385.957	0.0806421
$285$	0	0
$286$	204.810	0.0423449
$287$	7652.69	1.57395
$288$	0	0
$289$	12413.7	2.52671
$290$	0	0
$291$	0	0
$292$	3257.61	0.652867
$293$	1780.58	0.355026	0.177513	−	0.984118i	$-0.443195\pi$
$293$	1780.58	0.355026	0.177513	+	0.984118i	$0.443195\pi$
$294$	0	0
$295$	0	0
$296$	359.219	0.0705377
$297$	0	0
$298$	274.458	0.0533522
$299$	1654.43	0.319995
$300$	0	0
$301$	−4341.47	−0.831356
$302$	362.540	0.0690790
$303$	0	0
$304$	1459.30	0.275317
$305$	0	0
$306$	0	0
$307$	−8480.18	−1.57651	−0.788256	−	0.615347i	$-0.789016\pi$
$307$	−8480.18	−1.57651	−0.788256	+	0.615347i	$0.789016\pi$
$308$	−4495.31	−0.831636
$309$	0	0
$310$	0	0
$311$	−6810.79	−1.24182	−0.620908	−	0.783884i	$-0.713236\pi$
$311$	−6810.79	−1.24182	−0.620908	+	0.783884i	$0.713236\pi$
$312$	0	0
$313$	−8046.81	−1.45314	−0.726570	−	0.687093i	$-0.758887\pi$
$313$	−8046.81	−1.45314	−0.726570	+	0.687093i	$0.758887\pi$
$314$	−289.747	−0.0520744
$315$	0	0
$316$	−4217.30	−0.750764
$317$	−1566.22	−0.277501	−0.138751	−	0.990327i	$-0.544309\pi$
$317$	−1566.22	−0.277501	−0.138751	+	0.990327i	$0.544309\pi$
$318$	0	0
$319$	2280.88	0.400328
$320$	0	0
$321$	0	0
$322$	231.772	0.0401122
$323$	3059.47	0.527039
$324$	0	0
$325$	0	0
$326$	−319.142	−0.0542198
$327$	0	0
$328$	−881.349	−0.148367
$329$	14753.5	2.47231
$330$	0	0
$331$	2442.56	0.405605	0.202802	−	0.979220i	$-0.434995\pi$
$331$	2442.56	0.405605	0.202802	+	0.979220i	$0.434995\pi$
$332$	−2341.77	−0.387113
$333$	0	0
$334$	−201.593	−0.0330260
$335$	0	0
$336$	0	0
$337$	−9472.98	−1.53123	−0.765617	−	0.643296i	$-0.777566\pi$
$337$	−9472.98	−1.53123	−0.765617	+	0.643296i	$0.777566\pi$
$338$	71.7748	0.0115504
$339$	0	0
$340$	0	0
$341$	2268.10	0.360190
$342$	0	0
$343$	8954.76	1.40966
$344$	500.001	0.0783670
$345$	0	0
$346$	390.830	0.0607259
$347$	3312.47	0.512457	0.256229	−	0.966616i	$-0.417520\pi$
$347$	3312.47	0.512457	0.256229	+	0.966616i	$0.417520\pi$
$348$	0	0
$349$	5668.95	0.869490	0.434745	−	0.900554i	$-0.356838\pi$
$349$	5668.95	0.869490	0.434745	+	0.900554i	$0.356838\pi$
$350$	0	0
$351$	0	0
$352$	777.400	0.117715
$353$	1739.08	0.262215	0.131108	−	0.991368i	$-0.458147\pi$
$353$	1739.08	0.262215	0.131108	+	0.991368i	$0.458147\pi$
$354$	0	0
$355$	0	0
$356$	−6779.27	−1.00927
$357$	0	0
$358$	480.542	0.0709426
$359$	8624.58	1.26793	0.633966	−	0.773361i	$-0.281426\pi$
$359$	8624.58	1.26793	0.633966	+	0.773361i	$0.281426\pi$
$360$	0	0
$361$	−6318.77	−0.921238
$362$	−813.598	−0.118126
$363$	0	0
$364$	−12437.2	−1.79090
$365$	0	0
$366$	0	0
$367$	6109.17	0.868927	0.434463	−	0.900689i	$-0.356938\pi$
$367$	6109.17	0.868927	0.434463	+	0.900689i	$0.356938\pi$
$368$	2071.00	0.293365
$369$	0	0
$370$	0	0
$371$	−13149.2	−1.84009
$372$	0	0
$373$	−6555.33	−0.909979	−0.454990	−	0.890497i	$-0.650357\pi$
$373$	−6555.33	−0.909979	−0.454990	+	0.890497i	$0.650357\pi$
$374$	537.507	0.0743150
$375$	0	0
$376$	−1699.14	−0.233050
$377$	6310.52	0.862092
$378$	0	0
$379$	5032.40	0.682050	0.341025	−	0.940054i	$-0.389226\pi$
$379$	5032.40	0.682050	0.341025	+	0.940054i	$0.389226\pi$
$380$	0	0
$381$	0	0
$382$	−134.472	−0.0180109
$383$	−3662.99	−0.488695	−0.244347	−	0.969688i	$-0.578574\pi$
$383$	−3662.99	−0.488695	−0.244347	+	0.969688i	$0.578574\pi$
$384$	0	0
$385$	0	0
$386$	272.044	0.0358722
$387$	0	0
$388$	−3085.03	−0.403657
$389$	−4870.96	−0.634878	−0.317439	−	0.948279i	$-0.602823\pi$
$389$	−4870.96	−0.634878	−0.317439	+	0.948279i	$0.602823\pi$
$390$	0	0
$391$	4341.93	0.561588
$392$	−2263.56	−0.291650
$393$	0	0
$394$	736.243	0.0941406
$395$	0	0
$396$	0	0
$397$	3744.62	0.473393	0.236696	−	0.971584i	$-0.423935\pi$
$397$	3744.62	0.473393	0.236696	+	0.971584i	$0.423935\pi$
$398$	−475.250	−0.0598546
$399$	0	0
$400$	0	0
$401$	−1800.87	−0.224267	−0.112134	−	0.993693i	$-0.535768\pi$
$401$	−1800.87	−0.224267	−0.112134	+	0.993693i	$0.535768\pi$
$402$	0	0
$403$	6275.18	0.775655
$404$	8585.16	1.05725
$405$	0	0
$406$	884.049	0.108066
$407$	−1812.66	−0.220762
$408$	0	0
$409$	15932.0	1.92613	0.963064	−	0.269274i	$-0.0867837\pi$
$409$	15932.0	1.92613	0.963064	+	0.269274i	$0.0867837\pi$
$410$	0	0
$411$	0	0
$412$	−4729.79	−0.565582
$413$	−23158.8	−2.75926
$414$	0	0
$415$	0	0
$416$	2150.84	0.253494
$417$	0	0
$418$	94.9106	0.0111058
$419$	−3447.75	−0.401989	−0.200995	−	0.979592i	$-0.564417\pi$
$419$	−3447.75	−0.401989	−0.200995	+	0.979592i	$0.564417\pi$
$420$	0	0
$421$	5615.98	0.650133	0.325066	−	0.945691i	$-0.394613\pi$
$421$	5615.98	0.650133	0.325066	+	0.945691i	$0.394613\pi$
$422$	148.359	0.0171137
$423$	0	0
$424$	1514.37	0.173454
$425$	0	0
$426$	0	0
$427$	280.040	0.0317379
$428$	−3964.24	−0.447707
$429$	0	0
$430$	0	0
$431$	−3534.04	−0.394962	−0.197481	−	0.980307i	$-0.563276\pi$
$431$	−3534.04	−0.394962	−0.197481	+	0.980307i	$0.563276\pi$
$432$	0	0
$433$	−6674.80	−0.740809	−0.370405	−	0.928871i	$-0.620781\pi$
$433$	−6674.80	−0.740809	−0.370405	+	0.928871i	$0.620781\pi$
$434$	879.098	0.0972305
$435$	0	0
$436$	7625.74	0.837630
$437$	766.679	0.0839250
$438$	0	0
$439$	−6543.52	−0.711401	−0.355700	−	0.934600i	$-0.615758\pi$
$439$	−6543.52	−0.711401	−0.355700	+	0.934600i	$0.615758\pi$
$440$	0	0
$441$	0	0
$442$	1487.12	0.160035
$443$	−14309.7	−1.53471	−0.767353	−	0.641225i	$-0.778427\pi$
$443$	−14309.7	−1.53471	−0.767353	+	0.641225i	$0.778427\pi$
$444$	0	0
$445$	0	0
$446$	−274.533	−0.0291469
$447$	0	0
$448$	−15366.8	−1.62056
$449$	−14587.4	−1.53324	−0.766618	−	0.642104i	$-0.778062\pi$
$449$	−14587.4	−1.53324	−0.766618	+	0.642104i	$0.778062\pi$
$450$	0	0
$451$	4447.38	0.464343
$452$	13742.6	1.43009
$453$	0	0
$454$	343.634	0.0355232
$455$	0	0
$456$	0	0
$457$	−2025.77	−0.207356	−0.103678	−	0.994611i	$-0.533061\pi$
$457$	−2025.77	−0.207356	−0.103678	+	0.994611i	$0.533061\pi$
$458$	−1392.64	−0.142083
$459$	0	0
$460$	0	0
$461$	−3556.19	−0.359280	−0.179640	−	0.983732i	$-0.557493\pi$
$461$	−3556.19	−0.359280	−0.179640	+	0.983732i	$0.557493\pi$
$462$	0	0
$463$	10282.7	1.03213	0.516066	−	0.856549i	$-0.327396\pi$
$463$	10282.7	1.03213	0.516066	+	0.856549i	$0.327396\pi$
$464$	7899.43	0.790349
$465$	0	0
$466$	453.613	0.0450927
$467$	−8217.44	−0.814256	−0.407128	−	0.913371i	$-0.633470\pi$
$467$	−8217.44	−0.814256	−0.407128	+	0.913371i	$0.633470\pi$
$468$	0	0
$469$	−18370.3	−1.80866
$470$	0	0
$471$	0	0
$472$	2667.17	0.260098
$473$	−2523.06	−0.245265
$474$	0	0
$475$	0	0
$476$	−32640.5	−3.14301
$477$	0	0
$478$	1145.69	0.109629
$479$	13685.2	1.30541	0.652705	−	0.757612i	$-0.273634\pi$
$479$	13685.2	1.30541	0.652705	+	0.757612i	$0.273634\pi$
$480$	0	0
$481$	−5015.09	−0.475402
$482$	740.357	0.0699633
$483$	0	0
$484$	7968.01	0.748310
$485$	0	0
$486$	0	0
$487$	3239.59	0.301437	0.150718	−	0.988577i	$-0.451841\pi$
$487$	3239.59	0.301437	0.150718	+	0.988577i	$0.451841\pi$
$488$	−32.2518	−0.00299174
$489$	0	0
$490$	0	0
$491$	11128.1	1.02282	0.511409	−	0.859338i	$-0.329124\pi$
$491$	11128.1	1.02282	0.511409	+	0.859338i	$0.329124\pi$
$492$	0	0
$493$	16561.5	1.51296
$494$	262.590	0.0239159
$495$	0	0
$496$	7855.18	0.711105
$497$	−1514.55	−0.136694
$498$	0	0
$499$	−5427.22	−0.486885	−0.243443	−	0.969915i	$-0.578277\pi$
$499$	−5427.22	−0.486885	−0.243443	+	0.969915i	$0.578277\pi$
$500$	0	0
$501$	0	0
$502$	783.993	0.0697038
$503$	9600.22	0.850999	0.425500	−	0.904959i	$-0.360098\pi$
$503$	9600.22	0.850999	0.425500	+	0.904959i	$0.360098\pi$
$504$	0	0
$505$	0	0
$506$	134.695	0.0118338
$507$	0	0
$508$	8107.87	0.708127
$509$	−18939.9	−1.64930	−0.824652	−	0.565641i	$-0.808629\pi$
$509$	−18939.9	−1.64930	−0.824652	+	0.565641i	$0.808629\pi$
$510$	0	0
$511$	−12783.3	−1.10666
$512$	4496.87	0.388155
$513$	0	0
$514$	1530.57	0.131343
$515$	0	0
$516$	0	0
$517$	8574.06	0.729375
$518$	−702.570	−0.0595930
$519$	0	0
$520$	0	0
$521$	19292.6	1.62231	0.811155	−	0.584831i	$-0.198839\pi$
$521$	19292.6	1.62231	0.811155	+	0.584831i	$0.198839\pi$
$522$	0	0
$523$	−17967.5	−1.50223	−0.751114	−	0.660172i	$-0.770483\pi$
$523$	−17967.5	−1.50223	−0.751114	+	0.660172i	$0.770483\pi$
$524$	13532.8	1.12821
$525$	0	0
$526$	−1208.29	−0.100160
$527$	16468.7	1.36127
$528$	0	0
$529$	−11078.9	−0.910574
$530$	0	0
$531$	0	0
$532$	−5763.51	−0.469699
$533$	12304.6	0.999946
$534$	0	0
$535$	0	0
$536$	2115.69	0.170492
$537$	0	0
$538$	10.9243	0.000875426 0
$539$	11422.1	0.912777
$540$	0	0
$541$	−8299.36	−0.659552	−0.329776	−	0.944059i	$-0.606973\pi$
$541$	−8299.36	−0.659552	−0.329776	+	0.944059i	$0.606973\pi$
$542$	1721.62	0.136439
$543$	0	0
$544$	5644.71	0.444880
$545$	0	0
$546$	0	0
$547$	−84.7210	−0.00662232	−0.00331116	−	0.999995i	$-0.501054\pi$
$547$	−84.7210	−0.00662232	−0.00331116	+	0.999995i	$0.501054\pi$
$548$	−8344.17	−0.650448
$549$	0	0
$550$	0	0
$551$	2924.35	0.226101
$552$	0	0
$553$	16549.3	1.27260
$554$	−1159.14	−0.0888938
$555$	0	0
$556$	−11405.8	−0.869987
$557$	20914.0	1.59094	0.795469	−	0.605994i	$-0.207224\pi$
$557$	20914.0	1.59094	0.795469	+	0.605994i	$0.207224\pi$
$558$	0	0
$559$	−6980.56	−0.528169
$560$	0	0
$561$	0	0
$562$	1318.78	0.0989846
$563$	−11041.0	−0.826503	−0.413252	−	0.910617i	$-0.635607\pi$
$563$	−11041.0	−0.826503	−0.413252	+	0.910617i	$0.635607\pi$
$564$	0	0
$565$	0	0
$566$	−2115.96	−0.157139
$567$	0	0
$568$	174.429	0.0128853
$569$	−6014.62	−0.443139	−0.221569	−	0.975145i	$-0.571118\pi$
$569$	−6014.62	−0.443139	−0.221569	+	0.975145i	$0.571118\pi$
$570$	0	0
$571$	−21937.3	−1.60779	−0.803896	−	0.594770i	$-0.797243\pi$
$571$	−21937.3	−1.60779	−0.803896	+	0.594770i	$0.797243\pi$
$572$	−7227.91	−0.528346
$573$	0	0
$574$	1723.77	0.125346
$575$	0	0
$576$	0	0
$577$	−473.507	−0.0341635	−0.0170818	−	0.999854i	$-0.505438\pi$
$577$	−473.507	−0.0341635	−0.0170818	+	0.999854i	$0.505438\pi$
$578$	2796.19	0.201222
$579$	0	0
$580$	0	0
$581$	9189.44	0.656183
$582$	0	0
$583$	−7641.68	−0.542858
$584$	1472.24	0.104318
$585$	0	0
$586$	401.076	0.0282735
$587$	−13020.3	−0.915514	−0.457757	−	0.889077i	$-0.651347\pi$
$587$	−13020.3	−0.915514	−0.457757	+	0.889077i	$0.651347\pi$
$588$	0	0
$589$	2907.97	0.203431
$590$	0	0
$591$	0	0
$592$	−6277.82	−0.435839
$593$	12887.3	0.892441	0.446220	−	0.894923i	$-0.352770\pi$
$593$	12887.3	0.892441	0.446220	+	0.894923i	$0.352770\pi$
$594$	0	0
$595$	0	0
$596$	−9685.88	−0.665686
$597$	0	0
$598$	372.661	0.0254837
$599$	−8674.48	−0.591702	−0.295851	−	0.955234i	$-0.595603\pi$
$599$	−8674.48	−0.591702	−0.295851	+	0.955234i	$0.595603\pi$
$600$	0	0
$601$	10935.7	0.742225	0.371112	−	0.928588i	$-0.378976\pi$
$601$	10935.7	0.742225	0.371112	+	0.928588i	$0.378976\pi$
$602$	−977.916	−0.0662074
$603$	0	0
$604$	−12794.4	−0.861913
$605$	0	0
$606$	0	0
$607$	−21670.4	−1.44905	−0.724527	−	0.689246i	$-0.757942\pi$
$607$	−21670.4	−1.44905	−0.724527	+	0.689246i	$0.757942\pi$
$608$	996.718	0.0664840
$609$	0	0
$610$	0	0
$611$	23721.9	1.57068
$612$	0	0
$613$	15571.2	1.02596	0.512982	−	0.858399i	$-0.328541\pi$
$613$	15571.2	1.02596	0.512982	+	0.858399i	$0.328541\pi$
$614$	−1910.16	−0.125550
$615$	0	0
$616$	−2031.60	−0.132882
$617$	−9726.54	−0.634645	−0.317322	−	0.948318i	$-0.602784\pi$
$617$	−9726.54	−0.634645	−0.317322	+	0.948318i	$0.602784\pi$
$618$	0	0
$619$	6297.59	0.408920	0.204460	−	0.978875i	$-0.434456\pi$
$619$	6297.59	0.408920	0.204460	+	0.978875i	$0.434456\pi$
$620$	0	0
$621$	0	0
$622$	−1534.13	−0.0988955
$623$	26602.8	1.71078
$624$	0	0
$625$	0	0
$626$	−1812.54	−0.115725
$627$	0	0
$628$	10225.4	0.649743
$629$	−13161.7	−0.834327
$630$	0	0
$631$	−5670.98	−0.357778	−0.178889	−	0.983869i	$-0.557250\pi$
$631$	−5670.98	−0.357778	−0.178889	+	0.983869i	$0.557250\pi$
$632$	−1905.95	−0.119960
$633$	0	0
$634$	−352.792	−0.0220996
$635$	0	0
$636$	0	0
$637$	31601.8	1.96563
$638$	513.767	0.0318813
$639$	0	0
$640$	0	0
$641$	21664.5	1.33494	0.667470	−	0.744637i	$-0.267377\pi$
$641$	21664.5	1.33494	0.667470	+	0.744637i	$0.267377\pi$
$642$	0	0
$643$	−6223.72	−0.381710	−0.190855	−	0.981618i	$-0.561126\pi$
$643$	−6223.72	−0.381710	−0.190855	+	0.981618i	$0.561126\pi$
$644$	−8179.43	−0.500488
$645$	0	0
$646$	689.146	0.0419723
$647$	−20451.9	−1.24273	−0.621365	−	0.783521i	$-0.713422\pi$
$647$	−20451.9	−1.24273	−0.621365	+	0.783521i	$0.713422\pi$
$648$	0	0
$649$	−13458.8	−0.814029
$650$	0	0
$651$	0	0
$652$	11262.8	0.676512
$653$	−7448.44	−0.446371	−0.223185	−	0.974776i	$-0.571646\pi$
$653$	−7448.44	−0.446371	−0.223185	+	0.974776i	$0.571646\pi$
$654$	0	0
$655$	0	0
$656$	15402.7	0.916731
$657$	0	0
$658$	3323.23	0.196889
$659$	826.370	0.0488480	0.0244240	−	0.999702i	$-0.492225\pi$
$659$	826.370	0.0488480	0.0244240	+	0.999702i	$0.492225\pi$
$660$	0	0
$661$	−6258.22	−0.368255	−0.184127	−	0.982902i	$-0.558946\pi$
$661$	−6258.22	−0.368255	−0.184127	+	0.982902i	$0.558946\pi$
$662$	550.186	0.0323015
$663$	0	0
$664$	−1058.33	−0.0618544
$665$	0	0
$666$	0	0
$667$	4150.17	0.240922
$668$	7114.39	0.412072
$669$	0	0
$670$	0	0
$671$	162.746	0.00936324
$672$	0	0
$673$	10079.8	0.577337	0.288668	−	0.957429i	$-0.406787\pi$
$673$	10079.8	0.577337	0.288668	+	0.957429i	$0.406787\pi$
$674$	−2133.79	−0.121944
$675$	0	0
$676$	−2532.99	−0.144117
$677$	25710.4	1.45958	0.729788	−	0.683674i	$-0.239619\pi$
$677$	25710.4	1.45958	0.729788	+	0.683674i	$0.239619\pi$
$678$	0	0
$679$	12106.1	0.684226
$680$	0	0
$681$	0	0
$682$	510.890	0.0286847
$683$	32091.0	1.79784	0.898922	−	0.438109i	$-0.144352\pi$
$683$	32091.0	1.79784	0.898922	+	0.438109i	$0.144352\pi$
$684$	0	0
$685$	0	0
$686$	2017.06	0.112262
$687$	0	0
$688$	−8738.17	−0.484215
$689$	−21142.3	−1.16902
$690$	0	0
$691$	10565.7	0.581679	0.290839	−	0.956772i	$-0.406066\pi$
$691$	10565.7	0.581679	0.290839	+	0.956772i	$0.406066\pi$
$692$	−13792.7	−0.757690
$693$	0	0
$694$	746.133	0.0408110
$695$	0	0
$696$	0	0
$697$	32292.4	1.75490
$698$	1276.93	0.0692443
$699$	0	0
$700$	0	0
$701$	13081.8	0.704837	0.352419	−	0.935842i	$-0.385359\pi$
$701$	13081.8	0.704837	0.352419	+	0.935842i	$0.385359\pi$
$702$	0	0
$703$	−2324.04	−0.124684
$704$	−8930.44	−0.478095
$705$	0	0
$706$	391.728	0.0208823
$707$	−33689.4	−1.79211
$708$	0	0
$709$	−28221.2	−1.49488	−0.747440	−	0.664329i	$-0.768717\pi$
$709$	−28221.2	−1.49488	−0.747440	+	0.664329i	$0.768717\pi$
$710$	0	0
$711$	0	0
$712$	−3063.80	−0.161265
$713$	4126.92	0.216766
$714$	0	0
$715$	0	0
$716$	−16958.8	−0.885165
$717$	0	0
$718$	1942.68	0.100975
$719$	−11471.4	−0.595010	−0.297505	−	0.954720i	$-0.596154\pi$
$719$	−11471.4	−0.595010	−0.297505	+	0.954720i	$0.596154\pi$
$720$	0	0
$721$	18560.3	0.958701
$722$	−1423.30	−0.0733654
$723$	0	0
$724$	28712.6	1.47389
$725$	0	0
$726$	0	0
$727$	−21659.3	−1.10495	−0.552475	−	0.833530i	$-0.686316\pi$
$727$	−21659.3	−1.10495	−0.552475	+	0.833530i	$0.686316\pi$
$728$	−5620.84	−0.286157
$729$	0	0
$730$	0	0
$731$	−18319.9	−0.926932
$732$	0	0
$733$	−5681.72	−0.286302	−0.143151	−	0.989701i	$-0.545723\pi$
$733$	−5681.72	−0.286302	−0.143151	+	0.989701i	$0.545723\pi$
$734$	1376.09	0.0691995
$735$	0	0
$736$	1414.52	0.0708421
$737$	−10676.0	−0.533588
$738$	0	0
$739$	261.324	0.0130080	0.00650402	−	0.999979i	$-0.497930\pi$
$739$	261.324	0.0130080	0.00650402	+	0.999979i	$0.497930\pi$
$740$	0	0
$741$	0	0
$742$	−2961.85	−0.146540
$743$	16405.9	0.810061	0.405031	−	0.914303i	$-0.367261\pi$
$743$	16405.9	0.810061	0.405031	+	0.914303i	$0.367261\pi$
$744$	0	0
$745$	0	0
$746$	−1476.59	−0.0724688
$747$	0	0
$748$	−18969.1	−0.927244
$749$	15556.2	0.758895
$750$	0	0
$751$	21474.6	1.04343	0.521716	−	0.853119i	$-0.325292\pi$
$751$	21474.6	1.04343	0.521716	+	0.853119i	$0.325292\pi$
$752$	29694.8	1.43997
$753$	0	0
$754$	1421.44	0.0686551
$755$	0	0
$756$	0	0
$757$	13643.2	0.655046	0.327523	−	0.944843i	$-0.393786\pi$
$757$	13643.2	0.655046	0.327523	+	0.944843i	$0.393786\pi$
$758$	1133.55	0.0543170
$759$	0	0
$760$	0	0
$761$	26937.9	1.28318	0.641589	−	0.767048i	$-0.278275\pi$
$761$	26937.9	1.28318	0.641589	+	0.767048i	$0.278275\pi$
$762$	0	0
$763$	−29924.4	−1.41984
$764$	4745.63	0.224726
$765$	0	0
$766$	−825.088	−0.0389186
$767$	−37236.6	−1.75298
$768$	0	0
$769$	−28885.4	−1.35453	−0.677265	−	0.735739i	$-0.736835\pi$
$769$	−28885.4	−1.35453	−0.677265	+	0.735739i	$0.736835\pi$
$770$	0	0
$771$	0	0
$772$	−9600.67	−0.447585
$773$	−3031.34	−0.141048	−0.0705238	−	0.997510i	$-0.522467\pi$
$773$	−3031.34	−0.141048	−0.0705238	+	0.997510i	$0.522467\pi$
$774$	0	0
$775$	0	0
$776$	−1394.24	−0.0644979
$777$	0	0
$778$	−1097.18	−0.0505603
$779$	5702.06	0.262256
$780$	0	0
$781$	−880.185	−0.0403271
$782$	978.019	0.0447236
$783$	0	0
$784$	39558.7	1.80205
$785$	0	0
$786$	0	0
$787$	−15154.8	−0.686416	−0.343208	−	0.939259i	$-0.611514\pi$
$787$	−15154.8	−0.686416	−0.343208	+	0.939259i	$0.611514\pi$
$788$	−25982.7	−1.17461
$789$	0	0
$790$	0	0
$791$	−53928.0	−2.42409
$792$	0	0
$793$	450.270	0.0201634
$794$	843.474	0.0377000
$795$	0	0
$796$	16772.0	0.746819
$797$	−28380.6	−1.26135	−0.630673	−	0.776048i	$-0.717221\pi$
$797$	−28380.6	−1.26135	−0.630673	+	0.776048i	$0.717221\pi$
$798$	0	0
$799$	62256.3	2.75653
$800$	0	0
$801$	0	0
$802$	−405.646	−0.0178602
$803$	−7429.06	−0.326483
$804$	0	0
$805$	0	0
$806$	1413.48	0.0617715
$807$	0	0
$808$	3879.95	0.168931
$809$	−14569.6	−0.633175	−0.316588	−	0.948563i	$-0.602537\pi$
$809$	−14569.6	−0.633175	−0.316588	+	0.948563i	$0.602537\pi$
$810$	0	0
$811$	27927.7	1.20921	0.604607	−	0.796524i	$-0.293330\pi$
$811$	27927.7	1.20921	0.604607	+	0.796524i	$0.293330\pi$
$812$	−31198.9	−1.34836
$813$	0	0
$814$	−408.300	−0.0175810
$815$	0	0
$816$	0	0
$817$	−3234.85	−0.138523
$818$	3588.68	0.153393
$819$	0	0
$820$	0	0
$821$	18044.1	0.767043	0.383521	−	0.923532i	$-0.374711\pi$
$821$	18044.1	0.767043	0.383521	+	0.923532i	$0.374711\pi$
$822$	0	0
$823$	32252.8	1.36606	0.683028	−	0.730393i	$-0.260663\pi$
$823$	32252.8	1.36606	0.683028	+	0.730393i	$0.260663\pi$
$824$	−2137.57	−0.0903710
$825$	0	0
$826$	−5216.53	−0.219741
$827$	−13569.5	−0.570566	−0.285283	−	0.958443i	$-0.592088\pi$
$827$	−13569.5	−0.570566	−0.285283	+	0.958443i	$0.592088\pi$
$828$	0	0
$829$	742.559	0.0311099	0.0155550	−	0.999879i	$-0.495049\pi$
$829$	742.559	0.0311099	0.0155550	+	0.999879i	$0.495049\pi$
$830$	0	0
$831$	0	0
$832$	−24707.9	−1.02956
$833$	82936.3	3.44967
$834$	0	0
$835$	0	0
$836$	−3349.48	−0.138569
$837$	0	0
$838$	−776.605	−0.0320136
$839$	−35504.7	−1.46097	−0.730487	−	0.682927i	$-0.760707\pi$
$839$	−35504.7	−1.46097	−0.730487	+	0.682927i	$0.760707\pi$
$840$	0	0
$841$	−8558.98	−0.350936
$842$	1265.00	0.0517752
$843$	0	0
$844$	−5235.72	−0.213532
$845$	0	0
$846$	0	0
$847$	−31267.6	−1.26844
$848$	−26465.7	−1.07174
$849$	0	0
$850$	0	0
$851$	−3298.21	−0.132857
$852$	0	0
$853$	−375.048	−0.0150544	−0.00752719	−	0.999972i	$-0.502396\pi$
$853$	−375.048	−0.0150544	−0.00752719	+	0.999972i	$0.502396\pi$
$854$	63.0790	0.00252754
$855$	0	0
$856$	−1791.59	−0.0715365
$857$	−32194.8	−1.28326	−0.641629	−	0.767015i	$-0.721741\pi$
$857$	−32194.8	−1.28326	−0.641629	+	0.767015i	$0.721741\pi$
$858$	0	0
$859$	−28137.5	−1.11763	−0.558813	−	0.829294i	$-0.688743\pi$
$859$	−28137.5	−1.11763	−0.558813	+	0.829294i	$0.688743\pi$
$860$	0	0
$861$	0	0
$862$	−796.042	−0.0314539
$863$	−15724.6	−0.620244	−0.310122	−	0.950697i	$-0.600370\pi$
$863$	−15724.6	−0.620244	−0.310122	+	0.950697i	$0.600370\pi$
$864$	0	0
$865$	0	0
$866$	−1503.50	−0.0589964
$867$	0	0
$868$	−31024.1	−1.21317
$869$	9617.64	0.375438
$870$	0	0
$871$	−29537.3	−1.14906
$872$	3446.35	0.133840
$873$	0	0
$874$	172.694	0.00668361
$875$	0	0
$876$	0	0
$877$	−15280.5	−0.588352	−0.294176	−	0.955751i	$-0.595045\pi$
$877$	−15280.5	−0.588352	−0.294176	+	0.955751i	$0.595045\pi$
$878$	−1473.93	−0.0566544
$879$	0	0
$880$	0	0
$881$	−24687.6	−0.944092	−0.472046	−	0.881574i	$-0.656484\pi$
$881$	−24687.6	−0.944092	−0.472046	+	0.881574i	$0.656484\pi$
$882$	0	0
$883$	−6562.59	−0.250112	−0.125056	−	0.992150i	$-0.539911\pi$
$883$	−6562.59	−0.250112	−0.125056	+	0.992150i	$0.539911\pi$
$884$	−52481.9	−1.99678
$885$	0	0
$886$	−3223.26	−0.122221
$887$	−50511.1	−1.91206	−0.956030	−	0.293270i	$-0.905256\pi$
$887$	−50511.1	−1.91206	−0.956030	+	0.293270i	$0.905256\pi$
$888$	0	0
$889$	−31816.4	−1.20032
$890$	0	0
$891$	0	0
$892$	9688.51	0.363672
$893$	10992.9	0.411943
$894$	0	0
$895$	0	0
$896$	−14162.9	−0.528067
$897$	0	0
$898$	−3285.81	−0.122104
$899$	15741.4	0.583986
$900$	0	0
$901$	−55486.3	−2.05163
$902$	1001.77	0.0369793
$903$	0	0
$904$	6210.81	0.228505
$905$	0	0
$906$	0	0
$907$	7017.44	0.256902	0.128451	−	0.991716i	$-0.458999\pi$
$907$	7017.44	0.256902	0.128451	+	0.991716i	$0.458999\pi$
$908$	−12127.2	−0.443231
$909$	0	0
$910$	0	0
$911$	−6124.59	−0.222741	−0.111370	−	0.993779i	$-0.535524\pi$
$911$	−6124.59	−0.222741	−0.111370	+	0.993779i	$0.535524\pi$
$912$	0	0
$913$	5340.47	0.193586
$914$	−456.305	−0.0165134
$915$	0	0
$916$	49147.5	1.77279
$917$	−53104.4	−1.91239
$918$	0	0
$919$	23780.7	0.853594	0.426797	−	0.904347i	$-0.359642\pi$
$919$	23780.7	0.853594	0.426797	+	0.904347i	$0.359642\pi$
$920$	0	0
$921$	0	0
$922$	−801.031	−0.0286123
$923$	−2435.22	−0.0868430
$924$	0	0
$925$	0	0
$926$	2316.17	0.0821967
$927$	0	0
$928$	5395.41	0.190854
$929$	10189.1	0.359842	0.179921	−	0.983681i	$-0.442416\pi$
$929$	10189.1	0.359842	0.179921	+	0.983681i	$0.442416\pi$
$930$	0	0
$931$	14644.5	0.515526
$932$	−16008.4	−0.562631
$933$	0	0
$934$	−1850.98	−0.0648456
$935$	0	0
$936$	0	0
$937$	3880.22	0.135284	0.0676421	−	0.997710i	$-0.478452\pi$
$937$	3880.22	0.135284	0.0676421	+	0.997710i	$0.478452\pi$
$938$	−4137.92	−0.144038
$939$	0	0
$940$	0	0
$941$	−38673.9	−1.33978	−0.669890	−	0.742461i	$-0.733659\pi$
$941$	−38673.9	−1.33978	−0.669890	+	0.742461i	$0.733659\pi$
$942$	0	0
$943$	8092.22	0.279447
$944$	−46612.3	−1.60710
$945$	0	0
$946$	−568.318	−0.0195324
$947$	7754.88	0.266103	0.133052	−	0.991109i	$-0.457522\pi$
$947$	7754.88	0.266103	0.133052	+	0.991109i	$0.457522\pi$
$948$	0	0
$949$	−20554.0	−0.703069
$950$	0	0
$951$	0	0
$952$	−14751.4	−0.502203
$953$	22527.9	0.765739	0.382870	−	0.923802i	$-0.374936\pi$
$953$	22527.9	0.765739	0.382870	+	0.923802i	$0.374936\pi$
$954$	0	0
$955$	0	0
$956$	−40432.4	−1.36786
$957$	0	0
$958$	3082.58	0.103960
$959$	32743.7	1.10255
$960$	0	0
$961$	−14137.8	−0.474567
$962$	−1129.65	−0.0378600
$963$	0	0
$964$	−26127.8	−0.872947
$965$	0	0
$966$	0	0
$967$	5652.41	0.187972	0.0939861	−	0.995574i	$-0.470039\pi$
$967$	5652.41	0.187972	0.0939861	+	0.995574i	$0.470039\pi$
$968$	3601.04	0.119568
$969$	0	0
$970$	0	0
$971$	19612.4	0.648188	0.324094	−	0.946025i	$-0.394941\pi$
$971$	19612.4	0.648188	0.324094	+	0.946025i	$0.394941\pi$
$972$	0	0
$973$	44757.9	1.47469
$974$	729.716	0.0240058
$975$	0	0
$976$	563.642	0.0184854
$977$	27683.6	0.906528	0.453264	−	0.891376i	$-0.350259\pi$
$977$	27683.6	0.906528	0.453264	+	0.891376i	$0.350259\pi$
$978$	0	0
$979$	15460.3	0.504712
$980$	0	0
$981$	0	0
$982$	2506.60	0.0814549
$983$	30616.4	0.993400	0.496700	−	0.867922i	$-0.334545\pi$
$983$	30616.4	0.993400	0.496700	+	0.867922i	$0.334545\pi$
$984$	0	0
$985$	0	0
$986$	3730.47	0.120489
$987$	0	0
$988$	−9267.02	−0.298404
$989$	−4590.82	−0.147603
$990$	0	0
$991$	−14462.4	−0.463587	−0.231793	−	0.972765i	$-0.574459\pi$
$991$	−14462.4	−0.463587	−0.231793	+	0.972765i	$0.574459\pi$
$992$	5365.19	0.171719
$993$	0	0
$994$	−341.152	−0.0108860
$995$	0	0
$996$	0	0
$997$	6897.26	0.219096	0.109548	−	0.993982i	$-0.465060\pi$
$997$	6897.26	0.219096	0.109548	+	0.993982i	$0.465060\pi$
$998$	−1222.48	−0.0387745
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bb.1.4		7
3.2	odd	2		2025.4.a.ba.1.4		7
5.4	even	2		405.4.a.m.1.4		7
9.4	even	3		225.4.e.d.151.4		14
9.7	even	3		225.4.e.d.76.4		14
15.14	odd	2		405.4.a.n.1.4		7
45.4	even	6		45.4.e.c.16.4	✓	14
45.7	odd	12		225.4.k.d.49.8		28
45.13	odd	12		225.4.k.d.124.8		28
45.14	odd	6		135.4.e.c.46.4		14
45.22	odd	12		225.4.k.d.124.7		28
45.29	odd	6		135.4.e.c.91.4		14
45.34	even	6		45.4.e.c.31.4	yes	14
45.43	odd	12		225.4.k.d.49.7		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.4	✓	14	45.4	even	6
45.4.e.c.31.4	yes	14	45.34	even	6
135.4.e.c.46.4		14	45.14	odd	6
135.4.e.c.91.4		14	45.29	odd	6
225.4.e.d.76.4		14	9.7	even	3
225.4.e.d.151.4		14	9.4	even	3
225.4.k.d.49.7		28	45.43	odd	12
225.4.k.d.49.8		28	45.7	odd	12
225.4.k.d.124.7		28	45.22	odd	12
225.4.k.d.124.8		28	45.13	odd	12
405.4.a.m.1.4		7	5.4	even	2
405.4.a.n.1.4		7	15.14	odd	2
2025.4.a.ba.1.4		7	3.2	odd	2
2025.4.a.bb.1.4		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.225250 q^{2} -7.94926 q^{4} +31.1940 q^{7} -3.59257 q^{8} +O(q^{10})\)	\(q+0.225250 q^{2} -7.94926 q^{4} +31.1940 q^{7} -3.59257 q^{8} +18.1285 q^{11} +50.1562 q^{13} +7.02645 q^{14} +62.7849 q^{16} +131.631 q^{17} +23.2428 q^{19} +4.08344 q^{22} +32.9856 q^{23} +11.2977 q^{26} -247.969 q^{28} +125.817 q^{29} +125.113 q^{31} +42.8828 q^{32} +29.6499 q^{34} -99.9894 q^{37} +5.23544 q^{38} +245.326 q^{41} -139.176 q^{43} -144.108 q^{44} +7.43001 q^{46} +472.961 q^{47} +630.067 q^{49} -398.705 q^{52} -421.529 q^{53} -112.067 q^{56} +28.3404 q^{58} -742.413 q^{59} +8.97736 q^{61} +28.1816 q^{62} -492.620 q^{64} -588.906 q^{67} -1046.37 q^{68} -48.5526 q^{71} -409.800 q^{73} -22.5226 q^{74} -184.763 q^{76} +565.500 q^{77} +530.527 q^{79} +55.2595 q^{82} +294.590 q^{83} -31.3495 q^{86} -65.1278 q^{88} +852.817 q^{89} +1564.57 q^{91} -262.211 q^{92} +106.534 q^{94} +388.091 q^{97} +141.922 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 36 q^{4} - 22 q^{7} + 18 q^{8}+O(q^{10}) \) 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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