LMFDB - Embedded newform 2025.4.a.bb.1.2

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			1.2
Root			$-3.04174$ 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)

$f(q)$	$=$	$q-3.04174 q^{2} +1.25219 q^{4} +13.7122 q^{7} +20.5251 q^{8} +O(q^{10})$	$q-3.04174 q^{2} +1.25219 q^{4} +13.7122 q^{7} +20.5251 q^{8} +31.8068 q^{11} -58.2432 q^{13} -41.7090 q^{14} -72.4495 q^{16} -109.055 q^{17} +129.695 q^{19} -96.7481 q^{22} -79.6684 q^{23} +177.161 q^{26} +17.1703 q^{28} +9.03537 q^{29} -33.3809 q^{31} +56.1719 q^{32} +331.717 q^{34} +22.1645 q^{37} -394.500 q^{38} +121.740 q^{41} -10.1417 q^{43} +39.8281 q^{44} +242.331 q^{46} +441.492 q^{47} -154.975 q^{49} -72.9314 q^{52} +593.610 q^{53} +281.445 q^{56} -27.4833 q^{58} +442.460 q^{59} +144.576 q^{61} +101.536 q^{62} +408.736 q^{64} -862.721 q^{67} -136.557 q^{68} -818.541 q^{71} -495.052 q^{73} -67.4187 q^{74} +162.403 q^{76} +436.142 q^{77} +1170.53 q^{79} -370.300 q^{82} +424.043 q^{83} +30.8485 q^{86} +652.838 q^{88} +1031.37 q^{89} -798.643 q^{91} -99.7598 q^{92} -1342.90 q^{94} -1598.71 q^{97} +471.394 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.04174	−1.07542	−0.537709	−	0.843131i	$-0.680710\pi$
$2$	−3.04174	−1.07542	−0.537709	+	0.843131i	$0.680710\pi$
$3$	0	0
$3$	0	0
$4$	1.25219	0.156523
$5$	0	0
$5$	0	0
$6$	0	0
$7$	13.7122	0.740390	0.370195	−	0.928954i	$-0.379291\pi$
$7$	13.7122	0.740390	0.370195	+	0.928954i	$0.379291\pi$
$8$	20.5251	0.907090
$9$	0	0
$10$	0	0
$11$	31.8068	0.871829	0.435914	−	0.899988i	$-0.356425\pi$
$11$	31.8068	0.871829	0.435914	+	0.899988i	$0.356425\pi$
$12$	0	0
$13$	−58.2432	−1.24260	−0.621298	−	0.783574i	$-0.713394\pi$
$13$	−58.2432	−1.24260	−0.621298	+	0.783574i	$0.713394\pi$
$14$	−41.7090	−0.796229
$15$	0	0
$16$	−72.4495	−1.13202
$17$	−109.055	−1.55586	−0.777932	−	0.628348i	$-0.783731\pi$
$17$	−109.055	−1.55586	−0.777932	+	0.628348i	$0.783731\pi$
$18$	0	0
$19$	129.695	1.56601	0.783004	−	0.622017i	$-0.213687\pi$
$19$	129.695	1.56601	0.783004	+	0.622017i	$0.213687\pi$
$20$	0	0
$21$	0	0
$22$	−96.7481	−0.937580
$23$	−79.6684	−0.722261	−0.361131	−	0.932515i	$-0.617609\pi$
$23$	−79.6684	−0.722261	−0.361131	+	0.932515i	$0.617609\pi$
$24$	0	0
$25$	0	0
$26$	177.161	1.33631
$27$	0	0
$28$	17.1703	0.115888
$29$	9.03537	0.0578561	0.0289280	−	0.999581i	$-0.490791\pi$
$29$	9.03537	0.0578561	0.0289280	+	0.999581i	$0.490791\pi$
$30$	0	0
$31$	−33.3809	−0.193400	−0.0966998	−	0.995314i	$-0.530829\pi$
$31$	−33.3809	−0.193400	−0.0966998	+	0.995314i	$0.530829\pi$
$32$	56.1719	0.310309
$33$	0	0
$34$	331.717	1.67320
$35$	0	0
$36$	0	0
$37$	22.1645	0.0984817	0.0492408	−	0.998787i	$-0.484320\pi$
$37$	22.1645	0.0984817	0.0492408	+	0.998787i	$0.484320\pi$
$38$	−394.500	−1.68411
$39$	0	0
$40$	0	0
$41$	121.740	0.463720	0.231860	−	0.972749i	$-0.425519\pi$
$41$	121.740	0.463720	0.231860	+	0.972749i	$0.425519\pi$
$42$	0	0
$43$	−10.1417	−0.0359674	−0.0179837	−	0.999838i	$-0.505725\pi$
$43$	−10.1417	−0.0359674	−0.0179837	+	0.999838i	$0.505725\pi$
$44$	39.8281	0.136462
$45$	0	0
$46$	242.331	0.776733
$47$	441.492	1.37018	0.685088	−	0.728460i	$-0.259764\pi$
$47$	441.492	1.37018	0.685088	+	0.728460i	$0.259764\pi$
$48$	0	0
$49$	−154.975	−0.451822
$50$	0	0
$51$	0	0
$52$	−72.9314	−0.194495
$53$	593.610	1.53846	0.769232	−	0.638969i	$-0.220639\pi$
$53$	593.610	1.53846	0.769232	+	0.638969i	$0.220639\pi$
$54$	0	0
$55$	0	0
$56$	281.445	0.671601
$57$	0	0
$58$	−27.4833	−0.0622195
$59$	442.460	0.976328	0.488164	−	0.872752i	$-0.337667\pi$
$59$	442.460	0.976328	0.488164	+	0.872752i	$0.337667\pi$
$60$	0	0
$61$	144.576	0.303460	0.151730	−	0.988422i	$-0.451516\pi$
$61$	144.576	0.303460	0.151730	+	0.988422i	$0.451516\pi$
$62$	101.536	0.207985
$63$	0	0
$64$	408.736	0.798312
$65$	0	0
$66$	0	0
$67$	−862.721	−1.57311	−0.786553	−	0.617523i	$-0.788136\pi$
$67$	−862.721	−1.57311	−0.786553	+	0.617523i	$0.788136\pi$
$68$	−136.557	−0.243529
$69$	0	0
$70$	0	0
$71$	−818.541	−1.36821	−0.684105	−	0.729384i	$-0.739807\pi$
$71$	−818.541	−1.36821	−0.684105	+	0.729384i	$0.739807\pi$
$72$	0	0
$73$	−495.052	−0.793719	−0.396859	−	0.917879i	$-0.629900\pi$
$73$	−495.052	−0.793719	−0.396859	+	0.917879i	$0.629900\pi$
$74$	−67.4187	−0.105909
$75$	0	0
$76$	162.403	0.245117
$77$	436.142	0.645494
$78$	0	0
$79$	1170.53	1.66702	0.833510	−	0.552505i	$-0.186328\pi$
$79$	1170.53	1.66702	0.833510	+	0.552505i	$0.186328\pi$
$80$	0	0
$81$	0	0
$82$	−370.300	−0.498693
$83$	424.043	0.560781	0.280390	−	0.959886i	$-0.409536\pi$
$83$	424.043	0.560781	0.280390	+	0.959886i	$0.409536\pi$
$84$	0	0
$85$	0	0
$86$	30.8485	0.0386799
$87$	0	0
$88$	652.838	0.790827
$89$	1031.37	1.22837	0.614183	−	0.789163i	$-0.289486\pi$
$89$	1031.37	1.22837	0.614183	+	0.789163i	$0.289486\pi$
$90$	0	0
$91$	−798.643	−0.920006
$92$	−99.7598	−0.113051
$93$	0	0
$94$	−1342.90	−1.47351
$95$	0	0
$96$	0	0
$97$	−1598.71	−1.67345	−0.836725	−	0.547623i	$-0.815533\pi$
$97$	−1598.71	−1.67345	−0.836725	+	0.547623i	$0.815533\pi$
$98$	471.394	0.485897
$99$	0	0
$100$	0	0
$101$	−214.000	−0.210830	−0.105415	−	0.994428i	$-0.533617\pi$
$101$	−214.000	−0.210830	−0.105415	+	0.994428i	$0.533617\pi$
$102$	0	0
$103$	−1672.32	−1.59979	−0.799897	−	0.600138i	$-0.795112\pi$
$103$	−1672.32	−1.59979	−0.799897	+	0.600138i	$0.795112\pi$
$104$	−1195.45	−1.12715
$105$	0	0
$106$	−1805.61	−1.65449
$107$	600.699	0.542727	0.271363	−	0.962477i	$-0.412525\pi$
$107$	600.699	0.542727	0.271363	+	0.962477i	$0.412525\pi$
$108$	0	0
$109$	−771.570	−0.678009	−0.339005	−	0.940785i	$-0.610090\pi$
$109$	−771.570	−0.678009	−0.339005	+	0.940785i	$0.610090\pi$
$110$	0	0
$111$	0	0
$112$	−993.444	−0.838140
$113$	1166.68	0.971254	0.485627	−	0.874166i	$-0.338591\pi$
$113$	1166.68	0.971254	0.485627	+	0.874166i	$0.338591\pi$
$114$	0	0
$115$	0	0
$116$	11.3140	0.00905583
$117$	0	0
$118$	−1345.85	−1.04996
$119$	−1495.38	−1.15195
$120$	0	0
$121$	−319.327	−0.239915
$122$	−439.763	−0.326347
$123$	0	0
$124$	−41.7991	−0.0302716
$125$	0	0
$126$	0	0
$127$	−1630.10	−1.13896	−0.569479	−	0.822006i	$-0.692855\pi$
$127$	−1630.10	−1.13896	−0.569479	+	0.822006i	$0.692855\pi$
$128$	−1692.64	−1.16883
$129$	0	0
$130$	0	0
$131$	258.825	0.172623	0.0863115	−	0.996268i	$-0.472492\pi$
$131$	258.825	0.172623	0.0863115	+	0.996268i	$0.472492\pi$
$132$	0	0
$133$	1778.41	1.15946
$134$	2624.17	1.69175
$135$	0	0
$136$	−2238.36	−1.41131
$137$	−1052.77	−0.656527	−0.328264	−	0.944586i	$-0.606463\pi$
$137$	−1052.77	−0.656527	−0.328264	+	0.944586i	$0.606463\pi$
$138$	0	0
$139$	2384.50	1.45504	0.727519	−	0.686088i	$-0.240673\pi$
$139$	2384.50	1.45504	0.727519	+	0.686088i	$0.240673\pi$
$140$	0	0
$141$	0	0
$142$	2489.79	1.47140
$143$	−1852.53	−1.08333
$144$	0	0
$145$	0	0
$146$	1505.82	0.853579
$147$	0	0
$148$	27.7541	0.0154147
$149$	1146.29	0.630253	0.315126	−	0.949050i	$-0.397953\pi$
$149$	1146.29	0.630253	0.315126	+	0.949050i	$0.397953\pi$
$150$	0	0
$151$	−1166.12	−0.628458	−0.314229	−	0.949347i	$-0.601746\pi$
$151$	−1166.12	−0.628458	−0.314229	+	0.949347i	$0.601746\pi$
$152$	2662.01	1.42051
$153$	0	0
$154$	−1326.63	−0.694175
$155$	0	0
$156$	0	0
$157$	3475.62	1.76678	0.883390	−	0.468638i	$-0.155255\pi$
$157$	3475.62	1.76678	0.883390	+	0.468638i	$0.155255\pi$
$158$	−3560.44	−1.79274
$159$	0	0
$160$	0	0
$161$	−1092.43	−0.534755
$162$	0	0
$163$	2223.32	1.06837	0.534183	−	0.845369i	$-0.320619\pi$
$163$	2223.32	1.06837	0.534183	+	0.845369i	$0.320619\pi$
$164$	152.441	0.0725831
$165$	0	0
$166$	−1289.83	−0.603074
$167$	1587.98	0.735819	0.367909	−	0.929862i	$-0.380074\pi$
$167$	1587.98	0.735819	0.367909	+	0.929862i	$0.380074\pi$
$168$	0	0
$169$	1195.26	0.544044
$170$	0	0
$171$	0	0
$172$	−12.6993	−0.00562974
$173$	−1507.80	−0.662634	−0.331317	−	0.943520i	$-0.607493\pi$
$173$	−1507.80	−0.662634	−0.331317	+	0.943520i	$0.607493\pi$
$174$	0	0
$175$	0	0
$176$	−2304.39	−0.986931
$177$	0	0
$178$	−3137.15	−1.32101
$179$	1572.66	0.656683	0.328341	−	0.944559i	$-0.393510\pi$
$179$	1572.66	0.656683	0.328341	+	0.944559i	$0.393510\pi$
$180$	0	0
$181$	1984.41	0.814918	0.407459	−	0.913223i	$-0.366415\pi$
$181$	1984.41	0.814918	0.407459	+	0.913223i	$0.366415\pi$
$182$	2429.27	0.989391
$183$	0	0
$184$	−1635.20	−0.655156
$185$	0	0
$186$	0	0
$187$	−3468.69	−1.35645
$188$	552.831	0.214465
$189$	0	0
$190$	0	0
$191$	−2348.35	−0.889639	−0.444819	−	0.895620i	$-0.646732\pi$
$191$	−2348.35	−0.889639	−0.444819	+	0.895620i	$0.646732\pi$
$192$	0	0
$193$	234.169	0.0873362	0.0436681	−	0.999046i	$-0.486096\pi$
$193$	234.169	0.0873362	0.0436681	+	0.999046i	$0.486096\pi$
$194$	4862.87	1.79966
$195$	0	0
$196$	−194.058	−0.0707207
$197$	717.846	0.259616	0.129808	−	0.991539i	$-0.458564\pi$
$197$	717.846	0.259616	0.129808	+	0.991539i	$0.458564\pi$
$198$	0	0
$199$	1701.45	0.606094	0.303047	−	0.952976i	$-0.401996\pi$
$199$	1701.45	0.606094	0.303047	+	0.952976i	$0.401996\pi$
$200$	0	0
$201$	0	0
$202$	650.933	0.226730
$203$	123.895	0.0428361
$204$	0	0
$205$	0	0
$206$	5086.77	1.72045
$207$	0	0
$208$	4219.69	1.40665
$209$	4125.19	1.36529
$210$	0	0
$211$	−4938.66	−1.61133	−0.805667	−	0.592368i	$-0.798193\pi$
$211$	−4938.66	−1.61133	−0.805667	+	0.592368i	$0.798193\pi$
$212$	743.311	0.240806
$213$	0	0
$214$	−1827.17	−0.583658
$215$	0	0
$216$	0	0
$217$	−457.726	−0.143191
$218$	2346.92	0.729143
$219$	0	0
$220$	0	0
$221$	6351.70	1.93331
$222$	0	0
$223$	3053.15	0.916833	0.458416	−	0.888737i	$-0.348417\pi$
$223$	3053.15	0.916833	0.458416	+	0.888737i	$0.348417\pi$
$224$	770.242	0.229750
$225$	0	0
$226$	−3548.73	−1.04450
$227$	839.953	0.245593	0.122797	−	0.992432i	$-0.460814\pi$
$227$	839.953	0.245593	0.122797	+	0.992432i	$0.460814\pi$
$228$	0	0
$229$	1401.51	0.404429	0.202214	−	0.979341i	$-0.435186\pi$
$229$	1401.51	0.404429	0.202214	+	0.979341i	$0.435186\pi$
$230$	0	0
$231$	0	0
$232$	185.452	0.0524807
$233$	2856.99	0.803295	0.401647	−	0.915794i	$-0.368438\pi$
$233$	2856.99	0.803295	0.401647	+	0.915794i	$0.368438\pi$
$234$	0	0
$235$	0	0
$236$	554.043	0.152818
$237$	0	0
$238$	4548.57	1.23882
$239$	−1709.83	−0.462761	−0.231381	−	0.972863i	$-0.574324\pi$
$239$	−1709.83	−0.462761	−0.231381	+	0.972863i	$0.574324\pi$
$240$	0	0
$241$	−4332.45	−1.15800	−0.579000	−	0.815328i	$-0.696557\pi$
$241$	−4332.45	−1.15800	−0.579000	+	0.815328i	$0.696557\pi$
$242$	971.309	0.258009
$243$	0	0
$244$	181.036	0.0474987
$245$	0	0
$246$	0	0
$247$	−7553.86	−1.94591
$248$	−685.146	−0.175431
$249$	0	0
$250$	0	0
$251$	−1724.73	−0.433722	−0.216861	−	0.976202i	$-0.569582\pi$
$251$	−1724.73	−0.433722	−0.216861	+	0.976202i	$0.569582\pi$
$252$	0	0
$253$	−2534.00	−0.629688
$254$	4958.33	1.22486
$255$	0	0
$256$	1878.70	0.458666
$257$	5853.25	1.42068	0.710342	−	0.703856i	$-0.248540\pi$
$257$	5853.25	1.42068	0.710342	+	0.703856i	$0.248540\pi$
$258$	0	0
$259$	303.925	0.0729149
$260$	0	0
$261$	0	0
$262$	−787.277	−0.185642
$263$	−392.360	−0.0919923	−0.0459962	−	0.998942i	$-0.514646\pi$
$263$	−392.360	−0.0919923	−0.0459962	+	0.998942i	$0.514646\pi$
$264$	0	0
$265$	0	0
$266$	−5409.47	−1.24690
$267$	0	0
$268$	−1080.29	−0.246228
$269$	−4610.33	−1.04497	−0.522485	−	0.852648i	$-0.674995\pi$
$269$	−4610.33	−1.04497	−0.522485	+	0.852648i	$0.674995\pi$
$270$	0	0
$271$	−1155.72	−0.259058	−0.129529	−	0.991576i	$-0.541347\pi$
$271$	−1155.72	−0.259058	−0.129529	+	0.991576i	$0.541347\pi$
$272$	7900.97	1.76128
$273$	0	0
$274$	3202.25	0.706041
$275$	0	0
$276$	0	0
$277$	608.410	0.131970	0.0659852	−	0.997821i	$-0.478981\pi$
$277$	608.410	0.131970	0.0659852	+	0.997821i	$0.478981\pi$
$278$	−7253.02	−1.56477
$279$	0	0
$280$	0	0
$281$	2749.24	0.583650	0.291825	−	0.956472i	$-0.405737\pi$
$281$	2749.24	0.583650	0.291825	+	0.956472i	$0.405737\pi$
$282$	0	0
$283$	−1021.56	−0.214578	−0.107289	−	0.994228i	$-0.534217\pi$
$283$	−1021.56	−0.214578	−0.107289	+	0.994228i	$0.534217\pi$
$284$	−1024.97	−0.214157
$285$	0	0
$286$	5634.91	1.16503
$287$	1669.32	0.343334
$288$	0	0
$289$	6979.96	1.42071
$290$	0	0
$291$	0	0
$292$	−619.898	−0.124236
$293$	5232.50	1.04330	0.521648	−	0.853161i	$-0.325317\pi$
$293$	5232.50	1.04330	0.521648	+	0.853161i	$0.325317\pi$
$294$	0	0
$295$	0	0
$296$	454.929	0.0893317
$297$	0	0
$298$	−3486.71	−0.677785
$299$	4640.14	0.897479
$300$	0	0
$301$	−139.065	−0.0266299
$302$	3547.02	0.675855
$303$	0	0
$304$	−9396.36	−1.77276
$305$	0	0
$306$	0	0
$307$	4912.68	0.913296	0.456648	−	0.889648i	$-0.349050\pi$
$307$	4912.68	0.913296	0.456648	+	0.889648i	$0.349050\pi$
$308$	546.132	0.101035
$309$	0	0
$310$	0	0
$311$	−828.860	−0.151126	−0.0755632	−	0.997141i	$-0.524075\pi$
$311$	−828.860	−0.151126	−0.0755632	+	0.997141i	$0.524075\pi$
$312$	0	0
$313$	−9237.21	−1.66811	−0.834054	−	0.551683i	$-0.813986\pi$
$313$	−9237.21	−1.66811	−0.834054	+	0.551683i	$0.813986\pi$
$314$	−10571.9	−1.90003
$315$	0	0
$316$	1465.72	0.260928
$317$	9098.41	1.61204	0.806021	−	0.591887i	$-0.201617\pi$
$317$	9098.41	1.61204	0.806021	+	0.591887i	$0.201617\pi$
$318$	0	0
$319$	287.386	0.0504406
$320$	0	0
$321$	0	0
$322$	3322.89	0.575086
$323$	−14143.9	−2.43649
$324$	0	0
$325$	0	0
$326$	−6762.76	−1.14894
$327$	0	0
$328$	2498.72	0.420636
$329$	6053.84	1.01447
$330$	0	0
$331$	5963.45	0.990275	0.495137	−	0.868815i	$-0.335118\pi$
$331$	5963.45	0.990275	0.495137	+	0.868815i	$0.335118\pi$
$332$	530.982	0.0877754
$333$	0	0
$334$	−4830.23	−0.791313
$335$	0	0
$336$	0	0
$337$	−596.854	−0.0964768	−0.0482384	−	0.998836i	$-0.515361\pi$
$337$	−596.854	−0.0964768	−0.0482384	+	0.998836i	$0.515361\pi$
$338$	−3635.69	−0.585075
$339$	0	0
$340$	0	0
$341$	−1061.74	−0.168611
$342$	0	0
$343$	−6828.34	−1.07492
$344$	−208.160	−0.0326256
$345$	0	0
$346$	4586.32	0.712608
$347$	4562.10	0.705782	0.352891	−	0.935664i	$-0.385199\pi$
$347$	4562.10	0.705782	0.352891	+	0.935664i	$0.385199\pi$
$348$	0	0
$349$	675.267	0.103571	0.0517854	−	0.998658i	$-0.483509\pi$
$349$	675.267	0.103571	0.0517854	+	0.998658i	$0.483509\pi$
$350$	0	0
$351$	0	0
$352$	1786.65	0.270536
$353$	3361.83	0.506890	0.253445	−	0.967350i	$-0.418436\pi$
$353$	3361.83	0.506890	0.253445	+	0.967350i	$0.418436\pi$
$354$	0	0
$355$	0	0
$356$	1291.46	0.192268
$357$	0	0
$358$	−4783.63	−0.706208
$359$	7701.61	1.13224	0.566122	−	0.824322i	$-0.308443\pi$
$359$	7701.61	1.13224	0.566122	+	0.824322i	$0.308443\pi$
$360$	0	0
$361$	9961.87	1.45238
$362$	−6036.07	−0.876378
$363$	0	0
$364$	−1000.05	−0.144003
$365$	0	0
$366$	0	0
$367$	−5909.50	−0.840526	−0.420263	−	0.907402i	$-0.638062\pi$
$367$	−5909.50	−0.840526	−0.420263	+	0.907402i	$0.638062\pi$
$368$	5771.94	0.817617
$369$	0	0
$370$	0	0
$371$	8139.72	1.13906
$372$	0	0
$373$	8424.38	1.16943	0.584716	−	0.811238i	$-0.301206\pi$
$373$	8424.38	1.16943	0.584716	+	0.811238i	$0.301206\pi$
$374$	10550.8	1.45875
$375$	0	0
$376$	9061.67	1.24287
$377$	−526.249	−0.0718917
$378$	0	0
$379$	−3510.24	−0.475749	−0.237875	−	0.971296i	$-0.576451\pi$
$379$	−3510.24	−0.475749	−0.237875	+	0.971296i	$0.576451\pi$
$380$	0	0
$381$	0	0
$382$	7143.09	0.956733
$383$	−5732.03	−0.764733	−0.382367	−	0.924011i	$-0.624891\pi$
$383$	−5732.03	−0.764733	−0.382367	+	0.924011i	$0.624891\pi$
$384$	0	0
$385$	0	0
$386$	−712.283	−0.0939229
$387$	0	0
$388$	−2001.89	−0.261934
$389$	14367.2	1.87261	0.936306	−	0.351184i	$-0.114221\pi$
$389$	14367.2	1.87261	0.936306	+	0.351184i	$0.114221\pi$
$390$	0	0
$391$	8688.23	1.12374
$392$	−3180.88	−0.409843
$393$	0	0
$394$	−2183.50	−0.279196
$395$	0	0
$396$	0	0
$397$	9014.42	1.13960	0.569800	−	0.821784i	$-0.307021\pi$
$397$	9014.42	1.13960	0.569800	+	0.821784i	$0.307021\pi$
$398$	−5175.37	−0.651804
$399$	0	0
$400$	0	0
$401$	10511.1	1.30897	0.654487	−	0.756073i	$-0.272885\pi$
$401$	10511.1	1.30897	0.654487	+	0.756073i	$0.272885\pi$
$402$	0	0
$403$	1944.21	0.240317
$404$	−267.969	−0.0329998
$405$	0	0
$406$	−376.857	−0.0460667
$407$	704.982	0.0858591
$408$	0	0
$409$	3318.20	0.401160	0.200580	−	0.979677i	$-0.435717\pi$
$409$	3318.20	0.401160	0.200580	+	0.979677i	$0.435717\pi$
$410$	0	0
$411$	0	0
$412$	−2094.06	−0.250405
$413$	6067.11	0.722864
$414$	0	0
$415$	0	0
$416$	−3271.63	−0.385589
$417$	0	0
$418$	−12547.8	−1.46826
$419$	−486.138	−0.0566812	−0.0283406	−	0.999598i	$-0.509022\pi$
$419$	−486.138	−0.0566812	−0.0283406	+	0.999598i	$0.509022\pi$
$420$	0	0
$421$	5051.93	0.584836	0.292418	−	0.956291i	$-0.405540\pi$
$421$	5051.93	0.584836	0.292418	+	0.956291i	$0.405540\pi$
$422$	15022.1	1.73286
$423$	0	0
$424$	12183.9	1.39553
$425$	0	0
$426$	0	0
$427$	1982.46	0.224679
$428$	752.188	0.0849495
$429$	0	0
$430$	0	0
$431$	6944.24	0.776084	0.388042	−	0.921642i	$-0.373152\pi$
$431$	6944.24	0.776084	0.388042	+	0.921642i	$0.373152\pi$
$432$	0	0
$433$	13738.3	1.52476	0.762378	−	0.647131i	$-0.224032\pi$
$433$	13738.3	1.52476	0.762378	+	0.647131i	$0.224032\pi$
$434$	1392.28	0.153990
$435$	0	0
$436$	−966.150	−0.106124
$437$	−10332.6	−1.13107
$438$	0	0
$439$	8481.83	0.922132	0.461066	−	0.887366i	$-0.347467\pi$
$439$	8481.83	0.922132	0.461066	+	0.887366i	$0.347467\pi$
$440$	0	0
$441$	0	0
$442$	−19320.2	−2.07912
$443$	558.195	0.0598660	0.0299330	−	0.999552i	$-0.490471\pi$
$443$	558.195	0.0598660	0.0299330	+	0.999552i	$0.490471\pi$
$444$	0	0
$445$	0	0
$446$	−9286.88	−0.985979
$447$	0	0
$448$	5604.68	0.591063
$449$	14775.0	1.55295	0.776476	−	0.630147i	$-0.217005\pi$
$449$	14775.0	1.55295	0.776476	+	0.630147i	$0.217005\pi$
$450$	0	0
$451$	3872.15	0.404285
$452$	1460.90	0.152024
$453$	0	0
$454$	−2554.92	−0.264115
$455$	0	0
$456$	0	0
$457$	14207.9	1.45431	0.727154	−	0.686474i	$-0.240842\pi$
$457$	14207.9	1.45431	0.727154	+	0.686474i	$0.240842\pi$
$458$	−4263.02	−0.434930
$459$	0	0
$460$	0	0
$461$	−7702.51	−0.778182	−0.389091	−	0.921199i	$-0.627211\pi$
$461$	−7702.51	−0.778182	−0.389091	+	0.921199i	$0.627211\pi$
$462$	0	0
$463$	−7909.07	−0.793878	−0.396939	−	0.917845i	$-0.629928\pi$
$463$	−7909.07	−0.793878	−0.396939	+	0.917845i	$0.629928\pi$
$464$	−654.608	−0.0654945
$465$	0	0
$466$	−8690.22	−0.863877
$467$	−11639.8	−1.15337	−0.576685	−	0.816967i	$-0.695654\pi$
$467$	−11639.8	−1.15337	−0.576685	+	0.816967i	$0.695654\pi$
$468$	0	0
$469$	−11829.8	−1.16471
$470$	0	0
$471$	0	0
$472$	9081.53	0.885618
$473$	−322.576	−0.0313574
$474$	0	0
$475$	0	0
$476$	−1872.50	−0.180307
$477$	0	0
$478$	5200.87	0.497662
$479$	−18695.9	−1.78337	−0.891687	−	0.452653i	$-0.850478\pi$
$479$	−18695.9	−1.78337	−0.891687	+	0.452653i	$0.850478\pi$
$480$	0	0
$481$	−1290.93	−0.122373
$482$	13178.2	1.24533
$483$	0	0
$484$	−399.857	−0.0375523
$485$	0	0
$486$	0	0
$487$	2249.54	0.209315	0.104658	−	0.994508i	$-0.466625\pi$
$487$	2249.54	0.209315	0.104658	+	0.994508i	$0.466625\pi$
$488$	2967.44	0.275266
$489$	0	0
$490$	0	0
$491$	1988.35	0.182756	0.0913778	−	0.995816i	$-0.470873\pi$
$491$	1988.35	0.182756	0.0913778	+	0.995816i	$0.470873\pi$
$492$	0	0
$493$	−985.351	−0.0900162
$494$	22976.9	2.09267
$495$	0	0
$496$	2418.43	0.218933
$497$	−11224.0	−1.01301
$498$	0	0
$499$	2413.90	0.216556	0.108278	−	0.994121i	$-0.465466\pi$
$499$	2413.90	0.216556	0.108278	+	0.994121i	$0.465466\pi$
$500$	0	0
$501$	0	0
$502$	5246.19	0.466432
$503$	8758.56	0.776391	0.388196	−	0.921577i	$-0.373098\pi$
$503$	8758.56	0.776391	0.388196	+	0.921577i	$0.373098\pi$
$504$	0	0
$505$	0	0
$506$	7707.77	0.677178
$507$	0	0
$508$	−2041.19	−0.178274
$509$	−11848.0	−1.03174	−0.515869	−	0.856667i	$-0.672531\pi$
$509$	−11848.0	−1.03174	−0.515869	+	0.856667i	$0.672531\pi$
$510$	0	0
$511$	−6788.27	−0.587662
$512$	7826.64	0.675570
$513$	0	0
$514$	−17804.1	−1.52783
$515$	0	0
$516$	0	0
$517$	14042.5	1.19456
$518$	−924.460	−0.0784140
$519$	0	0
$520$	0	0
$521$	3816.55	0.320933	0.160466	−	0.987041i	$-0.448700\pi$
$521$	3816.55	0.320933	0.160466	+	0.987041i	$0.448700\pi$
$522$	0	0
$523$	12158.9	1.01658	0.508288	−	0.861187i	$-0.330278\pi$
$523$	12158.9	1.01658	0.508288	+	0.861187i	$0.330278\pi$
$524$	324.097	0.0270195
$525$	0	0
$526$	1193.46	0.0989302
$527$	3640.35	0.300903
$528$	0	0
$529$	−5819.94	−0.478338
$530$	0	0
$531$	0	0
$532$	2226.90	0.181482
$533$	−7090.50	−0.576217
$534$	0	0
$535$	0	0
$536$	−17707.4	−1.42695
$537$	0	0
$538$	14023.4	1.12378
$539$	−4929.26	−0.393911
$540$	0	0
$541$	14919.8	1.18568	0.592840	−	0.805321i	$-0.298007\pi$
$541$	14919.8	1.18568	0.592840	+	0.805321i	$0.298007\pi$
$542$	3515.39	0.278596
$543$	0	0
$544$	−6125.82	−0.482799
$545$	0	0
$546$	0	0
$547$	1027.16	0.0802889	0.0401444	−	0.999194i	$-0.487218\pi$
$547$	1027.16	0.0802889	0.0401444	+	0.999194i	$0.487218\pi$
$548$	−1318.27	−0.102762
$549$	0	0
$550$	0	0
$551$	1171.85	0.0906031
$552$	0	0
$553$	16050.5	1.23425
$554$	−1850.63	−0.141923
$555$	0	0
$556$	2985.84	0.227748
$557$	10590.8	0.805648	0.402824	−	0.915277i	$-0.368029\pi$
$557$	10590.8	0.805648	0.402824	+	0.915277i	$0.368029\pi$
$558$	0	0
$559$	590.685	0.0446929
$560$	0	0
$561$	0	0
$562$	−8362.46	−0.627668
$563$	2192.07	0.164093	0.0820467	−	0.996628i	$-0.473854\pi$
$563$	2192.07	0.164093	0.0820467	+	0.996628i	$0.473854\pi$
$564$	0	0
$565$	0	0
$566$	3107.32	0.230761
$567$	0	0
$568$	−16800.6	−1.24109
$569$	−16569.8	−1.22081	−0.610405	−	0.792089i	$-0.708993\pi$
$569$	−16569.8	−1.22081	−0.610405	+	0.792089i	$0.708993\pi$
$570$	0	0
$571$	7279.33	0.533503	0.266752	−	0.963765i	$-0.414050\pi$
$571$	7279.33	0.533503	0.266752	+	0.963765i	$0.414050\pi$
$572$	−2319.71	−0.169567
$573$	0	0
$574$	−5077.64	−0.369227
$575$	0	0
$576$	0	0
$577$	17938.0	1.29422	0.647112	−	0.762395i	$-0.275976\pi$
$577$	17938.0	1.29422	0.647112	+	0.762395i	$0.275976\pi$
$578$	−21231.2	−1.52786
$579$	0	0
$580$	0	0
$581$	5814.58	0.415197
$582$	0	0
$583$	18880.8	1.34128
$584$	−10161.0	−0.719974
$585$	0	0
$586$	−15915.9	−1.12198
$587$	−11509.7	−0.809293	−0.404646	−	0.914473i	$-0.632605\pi$
$587$	−11509.7	−0.809293	−0.404646	+	0.914473i	$0.632605\pi$
$588$	0	0
$589$	−4329.35	−0.302865
$590$	0	0
$591$	0	0
$592$	−1605.81	−0.111484
$593$	9612.80	0.665684	0.332842	−	0.942983i	$-0.391992\pi$
$593$	9612.80	0.665684	0.332842	+	0.942983i	$0.391992\pi$
$594$	0	0
$595$	0	0
$596$	1435.37	0.0986493
$597$	0	0
$598$	−14114.1	−0.965165
$599$	7371.99	0.502857	0.251429	−	0.967876i	$-0.419100\pi$
$599$	7371.99	0.502857	0.251429	+	0.967876i	$0.419100\pi$
$600$	0	0
$601$	−24190.6	−1.64185	−0.820927	−	0.571033i	$-0.806543\pi$
$601$	−24190.6	−1.64185	−0.820927	+	0.571033i	$0.806543\pi$
$602$	423.001	0.0286383
$603$	0	0
$604$	−1460.20	−0.0983684
$605$	0	0
$606$	0	0
$607$	1317.73	0.0881138	0.0440569	−	0.999029i	$-0.485972\pi$
$607$	1317.73	0.0881138	0.0440569	+	0.999029i	$0.485972\pi$
$608$	7285.23	0.485946
$609$	0	0
$610$	0	0
$611$	−25713.9	−1.70257
$612$	0	0
$613$	4137.52	0.272615	0.136307	−	0.990667i	$-0.456477\pi$
$613$	4137.52	0.272615	0.136307	+	0.990667i	$0.456477\pi$
$614$	−14943.1	−0.982174
$615$	0	0
$616$	8951.86	0.585521
$617$	7791.61	0.508393	0.254197	−	0.967153i	$-0.418189\pi$
$617$	7791.61	0.508393	0.254197	+	0.967153i	$0.418189\pi$
$618$	0	0
$619$	21956.4	1.42569	0.712844	−	0.701322i	$-0.247407\pi$
$619$	21956.4	1.42569	0.712844	+	0.701322i	$0.247407\pi$
$620$	0	0
$621$	0	0
$622$	2521.18	0.162524
$623$	14142.3	0.909471
$624$	0	0
$625$	0	0
$626$	28097.2	1.79391
$627$	0	0
$628$	4352.13	0.276543
$629$	−2417.15	−0.153224
$630$	0	0
$631$	−11152.7	−0.703618	−0.351809	−	0.936072i	$-0.614433\pi$
$631$	−11152.7	−0.703618	−0.351809	+	0.936072i	$0.614433\pi$
$632$	24025.2	1.51214
$633$	0	0
$634$	−27675.0	−1.73362
$635$	0	0
$636$	0	0
$637$	9026.23	0.561432
$638$	−874.155	−0.0542447
$639$	0	0
$640$	0	0
$641$	4809.44	0.296352	0.148176	−	0.988961i	$-0.452660\pi$
$641$	4809.44	0.296352	0.148176	+	0.988961i	$0.452660\pi$
$642$	0	0
$643$	2553.03	0.156581	0.0782904	−	0.996931i	$-0.475054\pi$
$643$	2553.03	0.156581	0.0782904	+	0.996931i	$0.475054\pi$
$644$	−1367.93	−0.0837018
$645$	0	0
$646$	43022.1	2.62025
$647$	−8446.00	−0.513210	−0.256605	−	0.966516i	$-0.582604\pi$
$647$	−8446.00	−0.513210	−0.256605	+	0.966516i	$0.582604\pi$
$648$	0	0
$649$	14073.2	0.851191
$650$	0	0
$651$	0	0
$652$	2784.01	0.167224
$653$	−8094.87	−0.485110	−0.242555	−	0.970138i	$-0.577985\pi$
$653$	−8094.87	−0.485110	−0.242555	+	0.970138i	$0.577985\pi$
$654$	0	0
$655$	0	0
$656$	−8819.97	−0.524942
$657$	0	0
$658$	−18414.2	−1.09097
$659$	2175.33	0.128587	0.0642936	−	0.997931i	$-0.479521\pi$
$659$	2175.33	0.128587	0.0642936	+	0.997931i	$0.479521\pi$
$660$	0	0
$661$	−8876.95	−0.522350	−0.261175	−	0.965291i	$-0.584110\pi$
$661$	−8876.95	−0.522350	−0.261175	+	0.965291i	$0.584110\pi$
$662$	−18139.3	−1.06496
$663$	0	0
$664$	8703.53	0.508679
$665$	0	0
$666$	0	0
$667$	−719.834	−0.0417872
$668$	1988.45	0.115173
$669$	0	0
$670$	0	0
$671$	4598.51	0.264565
$672$	0	0
$673$	9167.92	0.525108	0.262554	−	0.964917i	$-0.415435\pi$
$673$	9167.92	0.525108	0.262554	+	0.964917i	$0.415435\pi$
$674$	1815.47	0.103753
$675$	0	0
$676$	1496.70	0.0851557
$677$	18877.9	1.07169	0.535846	−	0.844316i	$-0.319993\pi$
$677$	18877.9	1.07169	0.535846	+	0.844316i	$0.319993\pi$
$678$	0	0
$679$	−21921.9	−1.23901
$680$	0	0
$681$	0	0
$682$	3229.54	0.181328
$683$	27207.5	1.52426	0.762128	−	0.647426i	$-0.224155\pi$
$683$	27207.5	1.52426	0.762128	+	0.647426i	$0.224155\pi$
$684$	0	0
$685$	0	0
$686$	20770.1	1.15598
$687$	0	0
$688$	734.762	0.0407159
$689$	−34573.7	−1.91169
$690$	0	0
$691$	−17407.2	−0.958325	−0.479162	−	0.877726i	$-0.659060\pi$
$691$	−17407.2	−0.958325	−0.479162	+	0.877726i	$0.659060\pi$
$692$	−1888.04	−0.103718
$693$	0	0
$694$	−13876.7	−0.759011
$695$	0	0
$696$	0	0
$697$	−13276.3	−0.721486
$698$	−2053.99	−0.111382
$699$	0	0
$700$	0	0
$701$	18543.0	0.999086	0.499543	−	0.866289i	$-0.333501\pi$
$701$	18543.0	0.999086	0.499543	+	0.866289i	$0.333501\pi$
$702$	0	0
$703$	2874.63	0.154223
$704$	13000.6	0.695991
$705$	0	0
$706$	−10225.8	−0.545118
$707$	−2934.42	−0.156096
$708$	0	0
$709$	26233.8	1.38961	0.694803	−	0.719200i	$-0.255492\pi$
$709$	26233.8	1.38961	0.694803	+	0.719200i	$0.255492\pi$
$710$	0	0
$711$	0	0
$712$	21168.9	1.11424
$713$	2659.40	0.139685
$714$	0	0
$715$	0	0
$716$	1969.27	0.102786
$717$	0	0
$718$	−23426.3	−1.21763
$719$	−3043.06	−0.157840	−0.0789199	−	0.996881i	$-0.525147\pi$
$719$	−3043.06	−0.157840	−0.0789199	+	0.996881i	$0.525147\pi$
$720$	0	0
$721$	−22931.2	−1.18447
$722$	−30301.4	−1.56192
$723$	0	0
$724$	2484.86	0.127554
$725$	0	0
$726$	0	0
$727$	−21942.5	−1.11940	−0.559698	−	0.828696i	$-0.689083\pi$
$727$	−21942.5	−1.11940	−0.559698	+	0.828696i	$0.689083\pi$
$728$	−16392.2	−0.834528
$729$	0	0
$730$	0	0
$731$	1106.00	0.0559603
$732$	0	0
$733$	26576.9	1.33921	0.669604	−	0.742718i	$-0.266464\pi$
$733$	26576.9	1.33921	0.669604	+	0.742718i	$0.266464\pi$
$734$	17975.2	0.903917
$735$	0	0
$736$	−4475.13	−0.224124
$737$	−27440.4	−1.37148
$738$	0	0
$739$	28787.7	1.43298	0.716489	−	0.697598i	$-0.245748\pi$
$739$	28787.7	1.43298	0.716489	+	0.697598i	$0.245748\pi$
$740$	0	0
$741$	0	0
$742$	−24758.9	−1.22497
$743$	4503.02	0.222341	0.111171	−	0.993801i	$-0.464540\pi$
$743$	4503.02	0.222341	0.111171	+	0.993801i	$0.464540\pi$
$744$	0	0
$745$	0	0
$746$	−25624.8	−1.25763
$747$	0	0
$748$	−4343.45	−0.212316
$749$	8236.92	0.401830
$750$	0	0
$751$	5705.65	0.277233	0.138617	−	0.990346i	$-0.455734\pi$
$751$	5705.65	0.277233	0.138617	+	0.990346i	$0.455734\pi$
$752$	−31985.9	−1.55107
$753$	0	0
$754$	1600.71	0.0773136
$755$	0	0
$756$	0	0
$757$	−17397.6	−0.835305	−0.417652	−	0.908607i	$-0.637147\pi$
$757$	−17397.6	−0.835305	−0.417652	+	0.908607i	$0.637147\pi$
$758$	10677.2	0.511629
$759$	0	0
$760$	0	0
$761$	−35514.4	−1.69171	−0.845857	−	0.533410i	$-0.820910\pi$
$761$	−35514.4	−1.69171	−0.845857	+	0.533410i	$0.820910\pi$
$762$	0	0
$763$	−10579.9	−0.501992
$764$	−2940.58	−0.139249
$765$	0	0
$766$	17435.3	0.822408
$767$	−25770.3	−1.21318
$768$	0	0
$769$	31263.7	1.46605	0.733027	−	0.680199i	$-0.238107\pi$
$769$	31263.7	1.46605	0.733027	+	0.680199i	$0.238107\pi$
$770$	0	0
$771$	0	0
$772$	293.224	0.0136702
$773$	6676.47	0.310654	0.155327	−	0.987863i	$-0.450357\pi$
$773$	6676.47	0.310654	0.155327	+	0.987863i	$0.450357\pi$
$774$	0	0
$775$	0	0
$776$	−32813.7	−1.51797
$777$	0	0
$778$	−43701.3	−2.01384
$779$	15789.1	0.726189
$780$	0	0
$781$	−26035.2	−1.19284
$782$	−26427.3	−1.20849
$783$	0	0
$784$	11227.9	0.511473
$785$	0	0
$786$	0	0
$787$	−9433.83	−0.427293	−0.213646	−	0.976911i	$-0.568534\pi$
$787$	−9433.83	−0.427293	−0.213646	+	0.976911i	$0.568534\pi$
$788$	898.878	0.0406361
$789$	0	0
$790$	0	0
$791$	15997.7	0.719107
$792$	0	0
$793$	−8420.57	−0.377078
$794$	−27419.5	−1.22555
$795$	0	0
$796$	2130.54	0.0948679
$797$	23476.3	1.04338	0.521689	−	0.853136i	$-0.325302\pi$
$797$	23476.3	1.04338	0.521689	+	0.853136i	$0.325302\pi$
$798$	0	0
$799$	−48146.9	−2.13181
$800$	0	0
$801$	0	0
$802$	−31972.0	−1.40769
$803$	−15746.0	−0.691987
$804$	0	0
$805$	0	0
$806$	−5913.78	−0.258442
$807$	0	0
$808$	−4392.38	−0.191242
$809$	−33269.8	−1.44586	−0.722932	−	0.690919i	$-0.757206\pi$
$809$	−33269.8	−1.44586	−0.722932	+	0.690919i	$0.757206\pi$
$810$	0	0
$811$	27892.8	1.20771	0.603853	−	0.797096i	$-0.293631\pi$
$811$	27892.8	1.20771	0.603853	+	0.797096i	$0.293631\pi$
$812$	155.140	0.00670485
$813$	0	0
$814$	−2144.37	−0.0923345
$815$	0	0
$816$	0	0
$817$	−1315.33	−0.0563252
$818$	−10093.1	−0.431415
$819$	0	0
$820$	0	0
$821$	−3697.32	−0.157171	−0.0785856	−	0.996907i	$-0.525040\pi$
$821$	−3697.32	−0.157171	−0.0785856	+	0.996907i	$0.525040\pi$
$822$	0	0
$823$	−7144.51	−0.302602	−0.151301	−	0.988488i	$-0.548346\pi$
$823$	−7144.51	−0.302602	−0.151301	+	0.988488i	$0.548346\pi$
$824$	−34324.6	−1.45116
$825$	0	0
$826$	−18454.6	−0.777381
$827$	−19866.9	−0.835357	−0.417679	−	0.908595i	$-0.637156\pi$
$827$	−19866.9	−0.835357	−0.417679	+	0.908595i	$0.637156\pi$
$828$	0	0
$829$	−36736.5	−1.53910	−0.769548	−	0.638589i	$-0.779518\pi$
$829$	−36736.5	−1.53910	−0.769548	+	0.638589i	$0.779518\pi$
$830$	0	0
$831$	0	0
$832$	−23806.1	−0.991979
$833$	16900.8	0.702974
$834$	0	0
$835$	0	0
$836$	5165.52	0.213700
$837$	0	0
$838$	1478.71	0.0609560
$839$	30012.3	1.23497	0.617485	−	0.786583i	$-0.288152\pi$
$839$	30012.3	1.23497	0.617485	+	0.786583i	$0.288152\pi$
$840$	0	0
$841$	−24307.4	−0.996653
$842$	−15366.6	−0.628943
$843$	0	0
$844$	−6184.13	−0.252212
$845$	0	0
$846$	0	0
$847$	−4378.68	−0.177631
$848$	−43006.8	−1.74158
$849$	0	0
$850$	0	0
$851$	−1765.81	−0.0711295
$852$	0	0
$853$	5381.02	0.215994	0.107997	−	0.994151i	$-0.465556\pi$
$853$	5381.02	0.215994	0.107997	+	0.994151i	$0.465556\pi$
$854$	−6030.13	−0.241624
$855$	0	0
$856$	12329.4	0.492302
$857$	−14639.8	−0.583530	−0.291765	−	0.956490i	$-0.594242\pi$
$857$	−14639.8	−0.583530	−0.291765	+	0.956490i	$0.594242\pi$
$858$	0	0
$859$	29540.0	1.17333	0.586667	−	0.809828i	$-0.300440\pi$
$859$	29540.0	1.17333	0.586667	+	0.809828i	$0.300440\pi$
$860$	0	0
$861$	0	0
$862$	−21122.6	−0.834615
$863$	−3387.63	−0.133622	−0.0668112	−	0.997766i	$-0.521283\pi$
$863$	−3387.63	−0.133622	−0.0668112	+	0.997766i	$0.521283\pi$
$864$	0	0
$865$	0	0
$866$	−41788.3	−1.63975
$867$	0	0
$868$	−573.159	−0.0224128
$869$	37230.7	1.45336
$870$	0	0
$871$	50247.6	1.95474
$872$	−15836.5	−0.615015
$873$	0	0
$874$	31429.2	1.21637
$875$	0	0
$876$	0	0
$877$	244.104	0.00939888	0.00469944	−	0.999989i	$-0.498504\pi$
$877$	244.104	0.00939888	0.00469944	+	0.999989i	$0.498504\pi$
$878$	−25799.5	−0.991677
$879$	0	0
$880$	0	0
$881$	13910.2	0.531948	0.265974	−	0.963980i	$-0.414307\pi$
$881$	13910.2	0.531948	0.265974	+	0.963980i	$0.414307\pi$
$882$	0	0
$883$	8805.87	0.335607	0.167803	−	0.985820i	$-0.446333\pi$
$883$	8805.87	0.335607	0.167803	+	0.985820i	$0.446333\pi$
$884$	7953.52	0.302608
$885$	0	0
$886$	−1697.89	−0.0643810
$887$	−13280.8	−0.502735	−0.251368	−	0.967892i	$-0.580880\pi$
$887$	−13280.8	−0.502735	−0.251368	+	0.967892i	$0.580880\pi$
$888$	0	0
$889$	−22352.2	−0.843273
$890$	0	0
$891$	0	0
$892$	3823.11	0.143506
$893$	57259.5	2.14571
$894$	0	0
$895$	0	0
$896$	−23209.9	−0.865389
$897$	0	0
$898$	−44941.7	−1.67007
$899$	−301.609	−0.0111893
$900$	0	0
$901$	−64736.1	−2.39364
$902$	−11778.1	−0.434775
$903$	0	0
$904$	23946.2	0.881015
$905$	0	0
$906$	0	0
$907$	1084.24	0.0396932	0.0198466	−	0.999803i	$-0.493682\pi$
$907$	1084.24	0.0396932	0.0198466	+	0.999803i	$0.493682\pi$
$908$	1051.78	0.0384411
$909$	0	0
$910$	0	0
$911$	50188.7	1.82528	0.912638	−	0.408769i	$-0.134042\pi$
$911$	50188.7	1.82528	0.912638	+	0.408769i	$0.134042\pi$
$912$	0	0
$913$	13487.5	0.488905
$914$	−43216.8	−1.56399
$915$	0	0
$916$	1754.95	0.0633026
$917$	3549.06	0.127808
$918$	0	0
$919$	30376.1	1.09033	0.545166	−	0.838328i	$-0.316467\pi$
$919$	30376.1	1.09033	0.545166	+	0.838328i	$0.316467\pi$
$920$	0	0
$921$	0	0
$922$	23429.1	0.836871
$923$	47674.4	1.70013
$924$	0	0
$925$	0	0
$926$	24057.3	0.853750
$927$	0	0
$928$	507.534	0.0179533
$929$	−46197.6	−1.63153	−0.815766	−	0.578382i	$-0.803684\pi$
$929$	−46197.6	−1.63153	−0.815766	+	0.578382i	$0.803684\pi$
$930$	0	0
$931$	−20099.5	−0.707557
$932$	3577.49	0.125734
$933$	0	0
$934$	35405.1	1.24035
$935$	0	0
$936$	0	0
$937$	−37004.7	−1.29017	−0.645085	−	0.764111i	$-0.723178\pi$
$937$	−37004.7	−1.29017	−0.645085	+	0.764111i	$0.723178\pi$
$938$	35983.2	1.25255
$939$	0	0
$940$	0	0
$941$	43814.4	1.51786	0.758932	−	0.651170i	$-0.225722\pi$
$941$	43814.4	1.51786	0.758932	+	0.651170i	$0.225722\pi$
$942$	0	0
$943$	−9698.80	−0.334927
$944$	−32056.0	−1.10523
$945$	0	0
$946$	981.191	0.0337223
$947$	−4609.73	−0.158180	−0.0790898	−	0.996867i	$-0.525201\pi$
$947$	−4609.73	−0.158180	−0.0790898	+	0.996867i	$0.525201\pi$
$948$	0	0
$949$	28833.4	0.986272
$950$	0	0
$951$	0	0
$952$	−30692.9	−1.04492
$953$	−4281.80	−0.145542	−0.0727708	−	0.997349i	$-0.523184\pi$
$953$	−4281.80	−0.145542	−0.0727708	+	0.997349i	$0.523184\pi$
$954$	0	0
$955$	0	0
$956$	−2141.03	−0.0724330
$957$	0	0
$958$	56868.0	1.91787
$959$	−14435.8	−0.486086
$960$	0	0
$961$	−28676.7	−0.962597
$962$	3926.68	0.131602
$963$	0	0
$964$	−5425.05	−0.181254
$965$	0	0
$966$	0	0
$967$	36229.8	1.20483	0.602416	−	0.798182i	$-0.294205\pi$
$967$	36229.8	1.20483	0.602416	+	0.798182i	$0.294205\pi$
$968$	−6554.21	−0.217624
$969$	0	0
$970$	0	0
$971$	844.928	0.0279249	0.0139624	−	0.999903i	$-0.495555\pi$
$971$	844.928	0.0279249	0.0139624	+	0.999903i	$0.495555\pi$
$972$	0	0
$973$	32696.7	1.07730
$974$	−6842.53	−0.225101
$975$	0	0
$976$	−10474.5	−0.343524
$977$	−45844.9	−1.50124	−0.750619	−	0.660735i	$-0.770245\pi$
$977$	−45844.9	−1.50124	−0.750619	+	0.660735i	$0.770245\pi$
$978$	0	0
$979$	32804.5	1.07093
$980$	0	0
$981$	0	0
$982$	−6048.05	−0.196539
$983$	40202.0	1.30442	0.652210	−	0.758039i	$-0.273842\pi$
$983$	40202.0	1.30442	0.652210	+	0.758039i	$0.273842\pi$
$984$	0	0
$985$	0	0
$986$	2997.18	0.0968050
$987$	0	0
$988$	−9458.86	−0.304581
$989$	807.974	0.0259778
$990$	0	0
$991$	1797.91	0.0576313	0.0288157	−	0.999585i	$-0.490826\pi$
$991$	1797.91	0.0576313	0.0288157	+	0.999585i	$0.490826\pi$
$992$	−1875.07	−0.0600136
$993$	0	0
$994$	34140.5	1.08941
$995$	0	0
$996$	0	0
$997$	−9132.07	−0.290086	−0.145043	−	0.989425i	$-0.546332\pi$
$997$	−9132.07	−0.290086	−0.145043	+	0.989425i	$0.546332\pi$
$998$	−7342.47	−0.232888
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bb.1.2		7
3.2	odd	2		2025.4.a.ba.1.6		7
5.4	even	2		405.4.a.m.1.6		7
9.4	even	3		225.4.e.d.151.6		14
9.7	even	3		225.4.e.d.76.6		14
15.14	odd	2		405.4.a.n.1.2		7
45.4	even	6		45.4.e.c.16.2	✓	14
45.7	odd	12		225.4.k.d.49.4		28
45.13	odd	12		225.4.k.d.124.4		28
45.14	odd	6		135.4.e.c.46.6		14
45.22	odd	12		225.4.k.d.124.11		28
45.29	odd	6		135.4.e.c.91.6		14
45.34	even	6		45.4.e.c.31.2	yes	14
45.43	odd	12		225.4.k.d.49.11		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.2	✓	14	45.4	even	6
45.4.e.c.31.2	yes	14	45.34	even	6
135.4.e.c.46.6		14	45.14	odd	6
135.4.e.c.91.6		14	45.29	odd	6
225.4.e.d.76.6		14	9.7	even	3
225.4.e.d.151.6		14	9.4	even	3
225.4.k.d.49.4		28	45.7	odd	12
225.4.k.d.49.11		28	45.43	odd	12
225.4.k.d.124.4		28	45.13	odd	12
225.4.k.d.124.11		28	45.22	odd	12
405.4.a.m.1.6		7	5.4	even	2
405.4.a.n.1.2		7	15.14	odd	2
2025.4.a.ba.1.6		7	3.2	odd	2
2025.4.a.bb.1.2		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)

\(f(q)\)	\(=\)	\(q-3.04174 q^{2} +1.25219 q^{4} +13.7122 q^{7} +20.5251 q^{8} +O(q^{10})\)	\(q-3.04174 q^{2} +1.25219 q^{4} +13.7122 q^{7} +20.5251 q^{8} +31.8068 q^{11} -58.2432 q^{13} -41.7090 q^{14} -72.4495 q^{16} -109.055 q^{17} +129.695 q^{19} -96.7481 q^{22} -79.6684 q^{23} +177.161 q^{26} +17.1703 q^{28} +9.03537 q^{29} -33.3809 q^{31} +56.1719 q^{32} +331.717 q^{34} +22.1645 q^{37} -394.500 q^{38} +121.740 q^{41} -10.1417 q^{43} +39.8281 q^{44} +242.331 q^{46} +441.492 q^{47} -154.975 q^{49} -72.9314 q^{52} +593.610 q^{53} +281.445 q^{56} -27.4833 q^{58} +442.460 q^{59} +144.576 q^{61} +101.536 q^{62} +408.736 q^{64} -862.721 q^{67} -136.557 q^{68} -818.541 q^{71} -495.052 q^{73} -67.4187 q^{74} +162.403 q^{76} +436.142 q^{77} +1170.53 q^{79} -370.300 q^{82} +424.043 q^{83} +30.8485 q^{86} +652.838 q^{88} +1031.37 q^{89} -798.643 q^{91} -99.7598 q^{92} -1342.90 q^{94} -1598.71 q^{97} +471.394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 36 q^{4} - 22 q^{7} + 18 q^{8}+O(q^{10}) \) 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

Introduction

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