LMFDB - Embedded newform 2025.4.a.be.1.8

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + \cdots + 73008 $ x^12 - 4x^11 - 64x^10 + 241x^9 + 1452x^8 - 5130x^7 - 13834x^6 + 45247x^5 + 52669x^4 - 144610x^3 - 96816x^2 + 143136*x + 73008
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.33965$ 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)

$f(q)$	$=$	$q+1.33965 q^{2} -6.20534 q^{4} +22.3290 q^{7} -19.0302 q^{8} +O(q^{10})$	$q+1.33965 q^{2} -6.20534 q^{4} +22.3290 q^{7} -19.0302 q^{8} +28.6102 q^{11} -75.7274 q^{13} +29.9131 q^{14} +24.1490 q^{16} +126.236 q^{17} -72.6849 q^{19} +38.3277 q^{22} -124.018 q^{23} -101.448 q^{26} -138.559 q^{28} -83.6816 q^{29} +37.8376 q^{31} +184.593 q^{32} +169.112 q^{34} -256.266 q^{37} -97.3722 q^{38} +84.6503 q^{41} +32.9793 q^{43} -177.536 q^{44} -166.141 q^{46} -62.4545 q^{47} +155.585 q^{49} +469.914 q^{52} +310.844 q^{53} -424.925 q^{56} -112.104 q^{58} +200.503 q^{59} +493.888 q^{61} +50.6891 q^{62} +54.0976 q^{64} +187.741 q^{67} -783.336 q^{68} -1003.80 q^{71} +930.875 q^{73} -343.307 q^{74} +451.034 q^{76} +638.838 q^{77} -821.208 q^{79} +113.402 q^{82} +833.493 q^{83} +44.1807 q^{86} -544.457 q^{88} +537.343 q^{89} -1690.92 q^{91} +769.575 q^{92} -83.6672 q^{94} +106.339 q^{97} +208.430 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) $ 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8	$ 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} - 29 q^{11} - 24 q^{13} + 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} + 18 q^{22} - 318 q^{23} + 154 q^{26} - 96 q^{28} - 106 q^{29} + 60 q^{31} - 914 q^{32} - 108 q^{34} + 84 q^{37} - 640 q^{38} + 353 q^{41} + 426 q^{43} - 571 q^{44} + 270 q^{46} - 1210 q^{47} + 666 q^{49} + 75 q^{52} - 448 q^{53} + 570 q^{56} - 594 q^{58} - 482 q^{59} + 402 q^{61} - 2544 q^{62} + 975 q^{64} + 201 q^{67} - 3437 q^{68} + 944 q^{71} + 453 q^{73} - 10 q^{74} - 462 q^{76} - 2652 q^{77} + 258 q^{79} - 873 q^{82} - 3012 q^{83} - 1952 q^{86} + 216 q^{88} + 738 q^{89} - 618 q^{91} - 5232 q^{92} + 63 q^{94} + 318 q^{97} - 7511 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8 - 29 * q^11 - 24 * q^13 + 69 * q^14 + 192 * q^16 - 79 * q^17 - 75 * q^19 + 18 * q^22 - 318 * q^23 + 154 * q^26 - 96 * q^28 - 106 * q^29 + 60 * q^31 - 914 * q^32 - 108 * q^34 + 84 * q^37 - 640 * q^38 + 353 * q^41 + 426 * q^43 - 571 * q^44 + 270 * q^46 - 1210 * q^47 + 666 * q^49 + 75 * q^52 - 448 * q^53 + 570 * q^56 - 594 * q^58 - 482 * q^59 + 402 * q^61 - 2544 * q^62 + 975 * q^64 + 201 * q^67 - 3437 * q^68 + 944 * q^71 + 453 * q^73 - 10 * q^74 - 462 * q^76 - 2652 * q^77 + 258 * q^79 - 873 * q^82 - 3012 * q^83 - 1952 * q^86 + 216 * q^88 + 738 * q^89 - 618 * q^91 - 5232 * q^92 + 63 * q^94 + 318 * q^97 - 7511 * 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 $

$2$	1.33965	0.473638	0.236819	−	0.971554i	$-0.423895\pi$
$2$	1.33965	0.473638	0.236819	+	0.971554i	$0.423895\pi$
$3$	0	0
$3$	0	0
$4$	−6.20534	−0.775667
$5$	0	0
$5$	0	0
$6$	0	0
$7$	22.3290	1.20565	0.602827	−	0.797872i	$-0.294041\pi$
$7$	22.3290	1.20565	0.602827	+	0.797872i	$0.294041\pi$
$8$	−19.0302	−0.841023
$9$	0	0
$10$	0	0
$11$	28.6102	0.784209	0.392105	−	0.919921i	$-0.371747\pi$
$11$	28.6102	0.784209	0.392105	+	0.919921i	$0.371747\pi$
$12$	0	0
$13$	−75.7274	−1.61562	−0.807808	−	0.589446i	$-0.799346\pi$
$13$	−75.7274	−1.61562	−0.807808	+	0.589446i	$0.799346\pi$
$14$	29.9131	0.571043
$15$	0	0
$16$	24.1490	0.377327
$17$	126.236	1.80098	0.900490	−	0.434876i	$-0.143208\pi$
$17$	126.236	1.80098	0.900490	+	0.434876i	$0.143208\pi$
$18$	0	0
$19$	−72.6849	−0.877634	−0.438817	−	0.898576i	$-0.644602\pi$
$19$	−72.6849	−0.877634	−0.438817	+	0.898576i	$0.644602\pi$
$20$	0	0
$21$	0	0
$22$	38.3277	0.371431
$23$	−124.018	−1.12433	−0.562165	−	0.827025i	$-0.690031\pi$
$23$	−124.018	−1.12433	−0.562165	+	0.827025i	$0.690031\pi$
$24$	0	0
$25$	0	0
$26$	−101.448	−0.765217
$27$	0	0
$28$	−138.559	−0.935187
$29$	−83.6816	−0.535837	−0.267919	−	0.963442i	$-0.586336\pi$
$29$	−83.6816	−0.535837	−0.267919	+	0.963442i	$0.586336\pi$
$30$	0	0
$31$	37.8376	0.219220	0.109610	−	0.993975i	$-0.465040\pi$
$31$	37.8376	0.219220	0.109610	+	0.993975i	$0.465040\pi$
$32$	184.593	1.01974
$33$	0	0
$34$	169.112	0.853012
$35$	0	0
$36$	0	0
$37$	−256.266	−1.13865	−0.569323	−	0.822114i	$-0.692795\pi$
$37$	−256.266	−1.13865	−0.569323	+	0.822114i	$0.692795\pi$
$38$	−97.3722	−0.415681
$39$	0	0
$40$	0	0
$41$	84.6503	0.322443	0.161221	−	0.986918i	$-0.448457\pi$
$41$	84.6503	0.322443	0.161221	+	0.986918i	$0.448457\pi$
$42$	0	0
$43$	32.9793	0.116960	0.0584802	−	0.998289i	$-0.481375\pi$
$43$	32.9793	0.116960	0.0584802	+	0.998289i	$0.481375\pi$
$44$	−177.536	−0.608286
$45$	0	0
$46$	−166.141	−0.532525
$47$	−62.4545	−0.193828	−0.0969141	−	0.995293i	$-0.530897\pi$
$47$	−62.4545	−0.193828	−0.0969141	+	0.995293i	$0.530897\pi$
$48$	0	0
$49$	155.585	0.453602
$50$	0	0
$51$	0	0
$52$	469.914	1.25318
$53$	310.844	0.805618	0.402809	−	0.915284i	$-0.368034\pi$
$53$	310.844	0.805618	0.402809	+	0.915284i	$0.368034\pi$
$54$	0	0
$55$	0	0
$56$	−424.925	−1.01398
$57$	0	0
$58$	−112.104	−0.253793
$59$	200.503	0.442428	0.221214	−	0.975225i	$-0.428998\pi$
$59$	200.503	0.442428	0.221214	+	0.975225i	$0.428998\pi$
$60$	0	0
$61$	493.888	1.03665	0.518327	−	0.855183i	$-0.326555\pi$
$61$	493.888	1.03665	0.518327	+	0.855183i	$0.326555\pi$
$62$	50.6891	0.103831
$63$	0	0
$64$	54.0976	0.105659
$65$	0	0
$66$	0	0
$67$	187.741	0.342331	0.171165	−	0.985242i	$-0.445247\pi$
$67$	187.741	0.342331	0.171165	+	0.985242i	$0.445247\pi$
$68$	−783.336	−1.39696
$69$	0	0
$70$	0	0
$71$	−1003.80	−1.67787	−0.838935	−	0.544232i	$-0.816821\pi$
$71$	−1003.80	−1.67787	−0.838935	+	0.544232i	$0.816821\pi$
$72$	0	0
$73$	930.875	1.49248	0.746238	−	0.665679i	$-0.231858\pi$
$73$	930.875	1.49248	0.746238	+	0.665679i	$0.231858\pi$
$74$	−343.307	−0.539305
$75$	0	0
$76$	451.034	0.680752
$77$	638.838	0.945485
$78$	0	0
$79$	−821.208	−1.16953	−0.584766	−	0.811202i	$-0.698814\pi$
$79$	−821.208	−1.16953	−0.584766	+	0.811202i	$0.698814\pi$
$80$	0	0
$81$	0	0
$82$	113.402	0.152721
$83$	833.493	1.10226	0.551131	−	0.834419i	$-0.314197\pi$
$83$	833.493	1.10226	0.551131	+	0.834419i	$0.314197\pi$
$84$	0	0
$85$	0	0
$86$	44.1807	0.0553968
$87$	0	0
$88$	−544.457	−0.659538
$89$	537.343	0.639980	0.319990	−	0.947421i	$-0.396320\pi$
$89$	537.343	0.639980	0.319990	+	0.947421i	$0.396320\pi$
$90$	0	0
$91$	−1690.92	−1.94787
$92$	769.575	0.872106
$93$	0	0
$94$	−83.6672	−0.0918044
$95$	0	0
$96$	0	0
$97$	106.339	0.111310	0.0556549	−	0.998450i	$-0.482275\pi$
$97$	106.339	0.111310	0.0556549	+	0.998450i	$0.482275\pi$
$98$	208.430	0.214843
$99$	0	0
$100$	0	0
$101$	−1679.96	−1.65507	−0.827534	−	0.561416i	$-0.810257\pi$
$101$	−1679.96	−1.65507	−0.827534	+	0.561416i	$0.810257\pi$
$102$	0	0
$103$	−805.811	−0.770863	−0.385432	−	0.922736i	$-0.625947\pi$
$103$	−805.811	−0.770863	−0.385432	+	0.922736i	$0.625947\pi$
$104$	1441.11	1.35877
$105$	0	0
$106$	416.423	0.381571
$107$	−2001.69	−1.80851	−0.904256	−	0.426990i	$-0.859574\pi$
$107$	−2001.69	−1.80851	−0.904256	+	0.426990i	$0.859574\pi$
$108$	0	0
$109$	−1951.48	−1.71484	−0.857420	−	0.514617i	$-0.827934\pi$
$109$	−1951.48	−1.71484	−0.857420	+	0.514617i	$0.827934\pi$
$110$	0	0
$111$	0	0
$112$	539.223	0.454926
$113$	−341.434	−0.284243	−0.142121	−	0.989849i	$-0.545392\pi$
$113$	−341.434	−0.284243	−0.142121	+	0.989849i	$0.545392\pi$
$114$	0	0
$115$	0	0
$116$	519.273	0.415632
$117$	0	0
$118$	268.603	0.209550
$119$	2818.72	2.17136
$120$	0	0
$121$	−512.456	−0.385015
$122$	661.637	0.490998
$123$	0	0
$124$	−234.795	−0.170042
$125$	0	0
$126$	0	0
$127$	−808.728	−0.565063	−0.282531	−	0.959258i	$-0.591174\pi$
$127$	−808.728	−0.565063	−0.282531	+	0.959258i	$0.591174\pi$
$128$	−1404.27	−0.969695
$129$	0	0
$130$	0	0
$131$	2669.32	1.78031	0.890153	−	0.455662i	$-0.150598\pi$
$131$	2669.32	1.78031	0.890153	+	0.455662i	$0.150598\pi$
$132$	0	0
$133$	−1622.98	−1.05812
$134$	251.506	0.162141
$135$	0	0
$136$	−2402.29	−1.51467
$137$	−1949.48	−1.21573	−0.607866	−	0.794039i	$-0.707974\pi$
$137$	−1949.48	−1.21573	−0.607866	+	0.794039i	$0.707974\pi$
$138$	0	0
$139$	−2098.10	−1.28028	−0.640138	−	0.768260i	$-0.721123\pi$
$139$	−2098.10	−1.28028	−0.640138	+	0.768260i	$0.721123\pi$
$140$	0	0
$141$	0	0
$142$	−1344.74	−0.794702
$143$	−2166.58	−1.26698
$144$	0	0
$145$	0	0
$146$	1247.05	0.706893
$147$	0	0
$148$	1590.22	0.883210
$149$	1107.44	0.608893	0.304446	−	0.952529i	$-0.401529\pi$
$149$	1107.44	0.608893	0.304446	+	0.952529i	$0.401529\pi$
$150$	0	0
$151$	−501.047	−0.270031	−0.135015	−	0.990844i	$-0.543108\pi$
$151$	−501.047	−0.270031	−0.135015	+	0.990844i	$0.543108\pi$
$152$	1383.21	0.738110
$153$	0	0
$154$	855.819	0.447817
$155$	0	0
$156$	0	0
$157$	929.377	0.472435	0.236218	−	0.971700i	$-0.424092\pi$
$157$	929.377	0.472435	0.236218	+	0.971700i	$0.424092\pi$
$158$	−1100.13	−0.553935
$159$	0	0
$160$	0	0
$161$	−2769.21	−1.35555
$162$	0	0
$163$	71.8259	0.0345143	0.0172572	−	0.999851i	$-0.494507\pi$
$163$	71.8259	0.0345143	0.0172572	+	0.999851i	$0.494507\pi$
$164$	−525.284	−0.250108
$165$	0	0
$166$	1116.59	0.522073
$167$	−1156.15	−0.535724	−0.267862	−	0.963457i	$-0.586317\pi$
$167$	−1156.15	−0.535724	−0.267862	+	0.963457i	$0.586317\pi$
$168$	0	0
$169$	3537.64	1.61022
$170$	0	0
$171$	0	0
$172$	−204.648	−0.0907224
$173$	−1052.75	−0.462655	−0.231328	−	0.972876i	$-0.574307\pi$
$173$	−1052.75	−0.462655	−0.231328	+	0.972876i	$0.574307\pi$
$174$	0	0
$175$	0	0
$176$	690.907	0.295904
$177$	0	0
$178$	719.851	0.303118
$179$	584.650	0.244127	0.122064	−	0.992522i	$-0.461049\pi$
$179$	584.650	0.244127	0.122064	+	0.992522i	$0.461049\pi$
$180$	0	0
$181$	4393.89	1.80439	0.902197	−	0.431324i	$-0.141953\pi$
$181$	4393.89	1.80439	0.902197	+	0.431324i	$0.141953\pi$
$182$	−2265.24	−0.922586
$183$	0	0
$184$	2360.09	0.945587
$185$	0	0
$186$	0	0
$187$	3611.63	1.41235
$188$	387.552	0.150346
$189$	0	0
$190$	0	0
$191$	−1139.65	−0.431739	−0.215870	−	0.976422i	$-0.569259\pi$
$191$	−1139.65	−0.431739	−0.215870	+	0.976422i	$0.569259\pi$
$192$	0	0
$193$	−4293.74	−1.60140	−0.800700	−	0.599066i	$-0.795539\pi$
$193$	−4293.74	−1.60140	−0.800700	+	0.599066i	$0.795539\pi$
$194$	142.457	0.0527205
$195$	0	0
$196$	−965.460	−0.351844
$197$	−1972.02	−0.713202	−0.356601	−	0.934257i	$-0.616064\pi$
$197$	−1972.02	−0.713202	−0.356601	+	0.934257i	$0.616064\pi$
$198$	0	0
$199$	−1051.17	−0.374451	−0.187225	−	0.982317i	$-0.559949\pi$
$199$	−1051.17	−0.374451	−0.187225	+	0.982317i	$0.559949\pi$
$200$	0	0
$201$	0	0
$202$	−2250.55	−0.783902
$203$	−1868.53	−0.646034
$204$	0	0
$205$	0	0
$206$	−1079.50	−0.365110
$207$	0	0
$208$	−1828.74	−0.609616
$209$	−2079.53	−0.688249
$210$	0	0
$211$	−607.849	−0.198322	−0.0991612	−	0.995071i	$-0.531616\pi$
$211$	−607.849	−0.198322	−0.0991612	+	0.995071i	$0.531616\pi$
$212$	−1928.90	−0.624892
$213$	0	0
$214$	−2681.56	−0.856579
$215$	0	0
$216$	0	0
$217$	844.877	0.264304
$218$	−2614.30	−0.812213
$219$	0	0
$220$	0	0
$221$	−9559.51	−2.90969
$222$	0	0
$223$	−5.02179	−0.00150800	−0.000754000	−	1.00000i	$-0.500240\pi$
$223$	−5.02179	−0.00150800	−0.000754000		1.00000i	$0.500240\pi$
$224$	4121.77	1.22945
$225$	0	0
$226$	−457.402	−0.134628
$227$	−1262.76	−0.369218	−0.184609	−	0.982812i	$-0.559102\pi$
$227$	−1262.76	−0.369218	−0.184609	+	0.982812i	$0.559102\pi$
$228$	0	0
$229$	−851.178	−0.245622	−0.122811	−	0.992430i	$-0.539191\pi$
$229$	−851.178	−0.245622	−0.122811	+	0.992430i	$0.539191\pi$
$230$	0	0
$231$	0	0
$232$	1592.48	0.450651
$233$	−5158.49	−1.45040	−0.725201	−	0.688537i	$-0.758253\pi$
$233$	−5158.49	−1.45040	−0.725201	+	0.688537i	$0.758253\pi$
$234$	0	0
$235$	0	0
$236$	−1244.19	−0.343177
$237$	0	0
$238$	3776.10	1.02844
$239$	−5632.22	−1.52434	−0.762172	−	0.647375i	$-0.775867\pi$
$239$	−5632.22	−1.52434	−0.762172	+	0.647375i	$0.775867\pi$
$240$	0	0
$241$	−628.544	−0.168000	−0.0840002	−	0.996466i	$-0.526770\pi$
$241$	−628.544	−0.168000	−0.0840002	+	0.996466i	$0.526770\pi$
$242$	−686.511	−0.182358
$243$	0	0
$244$	−3064.74	−0.804098
$245$	0	0
$246$	0	0
$247$	5504.24	1.41792
$248$	−720.056	−0.184369
$249$	0	0
$250$	0	0
$251$	1147.14	0.288474	0.144237	−	0.989543i	$-0.453927\pi$
$251$	1147.14	0.288474	0.144237	+	0.989543i	$0.453927\pi$
$252$	0	0
$253$	−3548.19	−0.881710
$254$	−1083.41	−0.267635
$255$	0	0
$256$	−2314.01	−0.564943
$257$	−5152.15	−1.25051	−0.625257	−	0.780419i	$-0.715006\pi$
$257$	−5152.15	−1.25051	−0.625257	+	0.780419i	$0.715006\pi$
$258$	0	0
$259$	−5722.17	−1.37281
$260$	0	0
$261$	0	0
$262$	3575.96	0.843219
$263$	−7224.08	−1.69375	−0.846875	−	0.531793i	$-0.821519\pi$
$263$	−7224.08	−1.69375	−0.846875	+	0.531793i	$0.821519\pi$
$264$	0	0
$265$	0	0
$266$	−2174.23	−0.501167
$267$	0	0
$268$	−1164.99	−0.265535
$269$	−3918.24	−0.888102	−0.444051	−	0.896001i	$-0.646459\pi$
$269$	−3918.24	−0.888102	−0.444051	+	0.896001i	$0.646459\pi$
$270$	0	0
$271$	4049.81	0.907781	0.453891	−	0.891057i	$-0.350036\pi$
$271$	4049.81	0.907781	0.453891	+	0.891057i	$0.350036\pi$
$272$	3048.46	0.679559
$273$	0	0
$274$	−2611.62	−0.575817
$275$	0	0
$276$	0	0
$277$	−4030.73	−0.874306	−0.437153	−	0.899387i	$-0.644013\pi$
$277$	−4030.73	−0.874306	−0.437153	+	0.899387i	$0.644013\pi$
$278$	−2810.72	−0.606387
$279$	0	0
$280$	0	0
$281$	−498.421	−0.105813	−0.0529063	−	0.998599i	$-0.516848\pi$
$281$	−498.421	−0.105813	−0.0529063	+	0.998599i	$0.516848\pi$
$282$	0	0
$283$	1703.67	0.357853	0.178926	−	0.983862i	$-0.442738\pi$
$283$	1703.67	0.357853	0.178926	+	0.983862i	$0.442738\pi$
$284$	6228.90	1.30147
$285$	0	0
$286$	−2902.45	−0.600090
$287$	1890.16	0.388755
$288$	0	0
$289$	11022.5	2.24353
$290$	0	0
$291$	0	0
$292$	−5776.40	−1.15766
$293$	−7727.41	−1.54075	−0.770376	−	0.637590i	$-0.779931\pi$
$293$	−7727.41	−1.54075	−0.770376	+	0.637590i	$0.779931\pi$
$294$	0	0
$295$	0	0
$296$	4876.79	0.957627
$297$	0	0
$298$	1483.58	0.288394
$299$	9391.58	1.81649
$300$	0	0
$301$	736.396	0.141014
$302$	−671.227	−0.127897
$303$	0	0
$304$	−1755.26	−0.331155
$305$	0	0
$306$	0	0
$307$	2063.95	0.383700	0.191850	−	0.981424i	$-0.438551\pi$
$307$	2063.95	0.383700	0.191850	+	0.981424i	$0.438551\pi$
$308$	−3964.21	−0.733382
$309$	0	0
$310$	0	0
$311$	−4238.94	−0.772888	−0.386444	−	0.922313i	$-0.626297\pi$
$311$	−4238.94	−0.772888	−0.386444	+	0.922313i	$0.626297\pi$
$312$	0	0
$313$	9996.83	1.80529	0.902643	−	0.430390i	$-0.141624\pi$
$313$	9996.83	1.80529	0.902643	+	0.430390i	$0.141624\pi$
$314$	1245.04	0.223763
$315$	0	0
$316$	5095.87	0.907169
$317$	−201.958	−0.0357827	−0.0178913	−	0.999840i	$-0.505695\pi$
$317$	−201.958	−0.0357827	−0.0178913	+	0.999840i	$0.505695\pi$
$318$	0	0
$319$	−2394.15	−0.420209
$320$	0	0
$321$	0	0
$322$	−3709.77	−0.642041
$323$	−9175.43	−1.58060
$324$	0	0
$325$	0	0
$326$	96.2215	0.0163473
$327$	0	0
$328$	−1610.91	−0.271182
$329$	−1394.55	−0.233690
$330$	0	0
$331$	3982.51	0.661324	0.330662	−	0.943749i	$-0.392728\pi$
$331$	3982.51	0.661324	0.330662	+	0.943749i	$0.392728\pi$
$332$	−5172.11	−0.854989
$333$	0	0
$334$	−1548.84	−0.253739
$335$	0	0
$336$	0	0
$337$	−6393.58	−1.03347	−0.516737	−	0.856144i	$-0.672853\pi$
$337$	−6393.58	−1.03347	−0.516737	+	0.856144i	$0.672853\pi$
$338$	4739.20	0.762659
$339$	0	0
$340$	0	0
$341$	1082.54	0.171915
$342$	0	0
$343$	−4184.78	−0.658767
$344$	−627.602	−0.0983664
$345$	0	0
$346$	−1410.32	−0.219131
$347$	−6004.43	−0.928918	−0.464459	−	0.885594i	$-0.653751\pi$
$347$	−6004.43	−0.928918	−0.464459	+	0.885594i	$0.653751\pi$
$348$	0	0
$349$	8385.71	1.28618	0.643090	−	0.765791i	$-0.277652\pi$
$349$	8385.71	1.28618	0.643090	+	0.765791i	$0.277652\pi$
$350$	0	0
$351$	0	0
$352$	5281.23	0.799689
$353$	3002.77	0.452752	0.226376	−	0.974040i	$-0.427312\pi$
$353$	3002.77	0.452752	0.226376	+	0.974040i	$0.427312\pi$
$354$	0	0
$355$	0	0
$356$	−3334.39	−0.496412
$357$	0	0
$358$	783.226	0.115628
$359$	3376.84	0.496442	0.248221	−	0.968703i	$-0.420154\pi$
$359$	3376.84	0.496442	0.248221	+	0.968703i	$0.420154\pi$
$360$	0	0
$361$	−1575.91	−0.229758
$362$	5886.27	0.854629
$363$	0	0
$364$	10492.7	1.51090
$365$	0	0
$366$	0	0
$367$	−11872.1	−1.68861	−0.844307	−	0.535860i	$-0.819987\pi$
$367$	−11872.1	−1.68861	−0.844307	+	0.535860i	$0.819987\pi$
$368$	−2994.91	−0.424241
$369$	0	0
$370$	0	0
$371$	6940.85	0.971297
$372$	0	0
$373$	2105.97	0.292340	0.146170	−	0.989259i	$-0.453305\pi$
$373$	2105.97	0.292340	0.146170	+	0.989259i	$0.453305\pi$
$374$	4838.32	0.668940
$375$	0	0
$376$	1188.52	0.163014
$377$	6336.99	0.865707
$378$	0	0
$379$	4282.09	0.580359	0.290180	−	0.956972i	$-0.406285\pi$
$379$	4282.09	0.580359	0.290180	+	0.956972i	$0.406285\pi$
$380$	0	0
$381$	0	0
$382$	−1526.73	−0.204488
$383$	−7268.32	−0.969696	−0.484848	−	0.874598i	$-0.661125\pi$
$383$	−7268.32	−0.969696	−0.484848	+	0.874598i	$0.661125\pi$
$384$	0	0
$385$	0	0
$386$	−5752.11	−0.758483
$387$	0	0
$388$	−659.868	−0.0863395
$389$	9497.12	1.23785	0.618924	−	0.785451i	$-0.287569\pi$
$389$	9497.12	1.23785	0.618924	+	0.785451i	$0.287569\pi$
$390$	0	0
$391$	−15655.5	−2.02490
$392$	−2960.82	−0.381489
$393$	0	0
$394$	−2641.82	−0.337799
$395$	0	0
$396$	0	0
$397$	−1899.74	−0.240164	−0.120082	−	0.992764i	$-0.538316\pi$
$397$	−1899.74	−0.240164	−0.120082	+	0.992764i	$0.538316\pi$
$398$	−1408.20	−0.177354
$399$	0	0
$400$	0	0
$401$	−10018.5	−1.24763	−0.623813	−	0.781574i	$-0.714417\pi$
$401$	−10018.5	−1.24763	−0.623813	+	0.781574i	$0.714417\pi$
$402$	0	0
$403$	−2865.34	−0.354176
$404$	10424.7	1.28378
$405$	0	0
$406$	−2503.17	−0.305986
$407$	−7331.83	−0.892937
$408$	0	0
$409$	−3113.49	−0.376411	−0.188205	−	0.982130i	$-0.560267\pi$
$409$	−3113.49	−0.376411	−0.188205	+	0.982130i	$0.560267\pi$
$410$	0	0
$411$	0	0
$412$	5000.33	0.597933
$413$	4477.03	0.533415
$414$	0	0
$415$	0	0
$416$	−13978.7	−1.64751
$417$	0	0
$418$	−2785.84	−0.325981
$419$	10232.2	1.19303	0.596513	−	0.802603i	$-0.296552\pi$
$419$	10232.2	1.19303	0.596513	+	0.802603i	$0.296552\pi$
$420$	0	0
$421$	−12855.6	−1.48822	−0.744111	−	0.668056i	$-0.767127\pi$
$421$	−12855.6	−1.48822	−0.744111	+	0.668056i	$0.767127\pi$
$422$	−814.304	−0.0939330
$423$	0	0
$424$	−5915.42	−0.677543
$425$	0	0
$426$	0	0
$427$	11028.0	1.24985
$428$	12421.2	1.40280
$429$	0	0
$430$	0	0
$431$	209.970	0.0234661	0.0117330	−	0.999931i	$-0.496265\pi$
$431$	209.970	0.0234661	0.0117330	+	0.999931i	$0.496265\pi$
$432$	0	0
$433$	9664.22	1.07259	0.536296	−	0.844030i	$-0.319823\pi$
$433$	9664.22	1.07259	0.536296	+	0.844030i	$0.319823\pi$
$434$	1131.84	0.125184
$435$	0	0
$436$	12109.6	1.33015
$437$	9014.25	0.986750
$438$	0	0
$439$	4601.37	0.500254	0.250127	−	0.968213i	$-0.419528\pi$
$439$	4601.37	0.500254	0.250127	+	0.968213i	$0.419528\pi$
$440$	0	0
$441$	0	0
$442$	−12806.4	−1.37814
$443$	541.716	0.0580986	0.0290493	−	0.999578i	$-0.490752\pi$
$443$	541.716	0.0580986	0.0290493	+	0.999578i	$0.490752\pi$
$444$	0	0
$445$	0	0
$446$	−6.72744	−0.000714246 0
$447$	0	0
$448$	1207.95	0.127389
$449$	2870.48	0.301707	0.150853	−	0.988556i	$-0.451798\pi$
$449$	2870.48	0.301707	0.150853	+	0.988556i	$0.451798\pi$
$450$	0	0
$451$	2421.86	0.252863
$452$	2118.71	0.220478
$453$	0	0
$454$	−1691.66	−0.174875
$455$	0	0
$456$	0	0
$457$	14156.2	1.44901	0.724506	−	0.689269i	$-0.242068\pi$
$457$	14156.2	1.44901	0.724506	+	0.689269i	$0.242068\pi$
$458$	−1140.28	−0.116336
$459$	0	0
$460$	0	0
$461$	−3381.93	−0.341675	−0.170838	−	0.985299i	$-0.554647\pi$
$461$	−3381.93	−0.341675	−0.170838	+	0.985299i	$0.554647\pi$
$462$	0	0
$463$	9528.59	0.956439	0.478219	−	0.878240i	$-0.341282\pi$
$463$	9528.59	0.956439	0.478219	+	0.878240i	$0.341282\pi$
$464$	−2020.82	−0.202186
$465$	0	0
$466$	−6910.57	−0.686965
$467$	−5461.96	−0.541220	−0.270610	−	0.962689i	$-0.587225\pi$
$467$	−5461.96	−0.541220	−0.270610	+	0.962689i	$0.587225\pi$
$468$	0	0
$469$	4192.06	0.412732
$470$	0	0
$471$	0	0
$472$	−3815.60	−0.372092
$473$	943.545	0.0917215
$474$	0	0
$475$	0	0
$476$	−17491.1	−1.68425
$477$	0	0
$478$	−7545.20	−0.721986
$479$	6405.60	0.611021	0.305511	−	0.952189i	$-0.401173\pi$
$479$	6405.60	0.611021	0.305511	+	0.952189i	$0.401173\pi$
$480$	0	0
$481$	19406.4	1.83961
$482$	−842.029	−0.0795713
$483$	0	0
$484$	3179.96	0.298644
$485$	0	0
$486$	0	0
$487$	−6591.45	−0.613321	−0.306661	−	0.951819i	$-0.599212\pi$
$487$	−6591.45	−0.613321	−0.306661	+	0.951819i	$0.599212\pi$
$488$	−9398.77	−0.871849
$489$	0	0
$490$	0	0
$491$	17916.0	1.64672	0.823360	−	0.567520i	$-0.192097\pi$
$491$	17916.0	1.64672	0.823360	+	0.567520i	$0.192097\pi$
$492$	0	0
$493$	−10563.6	−0.965033
$494$	7373.75	0.671580
$495$	0	0
$496$	913.739	0.0827179
$497$	−22413.8	−2.02293
$498$	0	0
$499$	−6576.23	−0.589965	−0.294982	−	0.955503i	$-0.595314\pi$
$499$	−6576.23	−0.589965	−0.294982	+	0.955503i	$0.595314\pi$
$500$	0	0
$501$	0	0
$502$	1536.77	0.136632
$503$	7130.81	0.632101	0.316051	−	0.948742i	$-0.397643\pi$
$503$	7130.81	0.632101	0.316051	+	0.948742i	$0.397643\pi$
$504$	0	0
$505$	0	0
$506$	−4753.33	−0.417611
$507$	0	0
$508$	5018.43	0.438301
$509$	−4447.33	−0.387278	−0.193639	−	0.981073i	$-0.562029\pi$
$509$	−4447.33	−0.387278	−0.193639	+	0.981073i	$0.562029\pi$
$510$	0	0
$511$	20785.5	1.79941
$512$	8134.19	0.702117
$513$	0	0
$514$	−6902.07	−0.592291
$515$	0	0
$516$	0	0
$517$	−1786.84	−0.152002
$518$	−7665.71	−0.650216
$519$	0	0
$520$	0	0
$521$	−6657.00	−0.559786	−0.279893	−	0.960031i	$-0.590299\pi$
$521$	−6657.00	−0.559786	−0.279893	+	0.960031i	$0.590299\pi$
$522$	0	0
$523$	7912.15	0.661518	0.330759	−	0.943715i	$-0.392695\pi$
$523$	7912.15	0.661518	0.330759	+	0.943715i	$0.392695\pi$
$524$	−16564.1	−1.38092
$525$	0	0
$526$	−9677.74	−0.802223
$527$	4776.46	0.394812
$528$	0	0
$529$	3213.52	0.264118
$530$	0	0
$531$	0	0
$532$	10071.2	0.820752
$533$	−6410.35	−0.520944
$534$	0	0
$535$	0	0
$536$	−3572.74	−0.287908
$537$	0	0
$538$	−5249.07	−0.420639
$539$	4451.33	0.355719
$540$	0	0
$541$	1311.19	0.104201	0.0521004	−	0.998642i	$-0.483408\pi$
$541$	1311.19	0.104201	0.0521004	+	0.998642i	$0.483408\pi$
$542$	5425.33	0.429959
$543$	0	0
$544$	23302.2	1.83653
$545$	0	0
$546$	0	0
$547$	3263.79	0.255118	0.127559	−	0.991831i	$-0.459286\pi$
$547$	3263.79	0.255118	0.127559	+	0.991831i	$0.459286\pi$
$548$	12097.2	0.943004
$549$	0	0
$550$	0	0
$551$	6082.38	0.470269
$552$	0	0
$553$	−18336.8	−1.41005
$554$	−5399.76	−0.414104
$555$	0	0
$556$	13019.4	0.993068
$557$	−8625.82	−0.656172	−0.328086	−	0.944648i	$-0.606404\pi$
$557$	−8625.82	−0.656172	−0.328086	+	0.944648i	$0.606404\pi$
$558$	0	0
$559$	−2497.44	−0.188963
$560$	0	0
$561$	0	0
$562$	−667.710	−0.0501168
$563$	−19185.5	−1.43619	−0.718094	−	0.695946i	$-0.754985\pi$
$563$	−19185.5	−1.43619	−0.718094	+	0.695946i	$0.754985\pi$
$564$	0	0
$565$	0	0
$566$	2282.31	0.169493
$567$	0	0
$568$	19102.4	1.41113
$569$	−10956.8	−0.807265	−0.403633	−	0.914921i	$-0.632253\pi$
$569$	−10956.8	−0.807265	−0.403633	+	0.914921i	$0.632253\pi$
$570$	0	0
$571$	21998.9	1.61230	0.806151	−	0.591710i	$-0.201547\pi$
$571$	21998.9	1.61230	0.806151	+	0.591710i	$0.201547\pi$
$572$	13444.4	0.982756
$573$	0	0
$574$	2532.15	0.184129
$575$	0	0
$576$	0	0
$577$	−15196.9	−1.09646	−0.548229	−	0.836328i	$-0.684698\pi$
$577$	−15196.9	−1.09646	−0.548229	+	0.836328i	$0.684698\pi$
$578$	14766.2	1.06262
$579$	0	0
$580$	0	0
$581$	18611.1	1.32895
$582$	0	0
$583$	8893.33	0.631774
$584$	−17714.7	−1.25521
$585$	0	0
$586$	−10352.0	−0.729758
$587$	23033.0	1.61955	0.809775	−	0.586741i	$-0.199589\pi$
$587$	23033.0	1.61955	0.809775	+	0.586741i	$0.199589\pi$
$588$	0	0
$589$	−2750.22	−0.192395
$590$	0	0
$591$	0	0
$592$	−6188.56	−0.429642
$593$	−24574.7	−1.70179	−0.850895	−	0.525335i	$-0.823940\pi$
$593$	−24574.7	−1.70179	−0.850895	+	0.525335i	$0.823940\pi$
$594$	0	0
$595$	0	0
$596$	−6872.04	−0.472298
$597$	0	0
$598$	12581.4	0.860356
$599$	9133.20	0.622992	0.311496	−	0.950247i	$-0.399170\pi$
$599$	9133.20	0.622992	0.311496	+	0.950247i	$0.399170\pi$
$600$	0	0
$601$	8987.36	0.609987	0.304993	−	0.952354i	$-0.401346\pi$
$601$	8987.36	0.609987	0.304993	+	0.952354i	$0.401346\pi$
$602$	986.512	0.0667894
$603$	0	0
$604$	3109.17	0.209454
$605$	0	0
$606$	0	0
$607$	6401.40	0.428048	0.214024	−	0.976828i	$-0.431343\pi$
$607$	6401.40	0.428048	0.214024	+	0.976828i	$0.431343\pi$
$608$	−13417.1	−0.894958
$609$	0	0
$610$	0	0
$611$	4729.52	0.313152
$612$	0	0
$613$	−1175.26	−0.0774361	−0.0387181	−	0.999250i	$-0.512327\pi$
$613$	−1175.26	−0.0774361	−0.0387181	+	0.999250i	$0.512327\pi$
$614$	2764.97	0.181735
$615$	0	0
$616$	−12157.2	−0.795175
$617$	18515.7	1.20813	0.604065	−	0.796935i	$-0.293547\pi$
$617$	18515.7	1.20813	0.604065	+	0.796935i	$0.293547\pi$
$618$	0	0
$619$	17015.5	1.10487	0.552433	−	0.833557i	$-0.313699\pi$
$619$	17015.5	1.10487	0.552433	+	0.833557i	$0.313699\pi$
$620$	0	0
$621$	0	0
$622$	−5678.69	−0.366069
$623$	11998.3	0.771594
$624$	0	0
$625$	0	0
$626$	13392.3	0.855051
$627$	0	0
$628$	−5767.10	−0.366453
$629$	−32350.0	−2.05068
$630$	0	0
$631$	−16438.1	−1.03707	−0.518535	−	0.855056i	$-0.673523\pi$
$631$	−16438.1	−1.03707	−0.518535	+	0.855056i	$0.673523\pi$
$632$	15627.7	0.983604
$633$	0	0
$634$	−270.553	−0.0169480
$635$	0	0
$636$	0	0
$637$	−11782.1	−0.732846
$638$	−3207.32	−0.199027
$639$	0	0
$640$	0	0
$641$	5023.32	0.309531	0.154766	−	0.987951i	$-0.450538\pi$
$641$	5023.32	0.309531	0.154766	+	0.987951i	$0.450538\pi$
$642$	0	0
$643$	−17520.4	−1.07455	−0.537277	−	0.843406i	$-0.680547\pi$
$643$	−17520.4	−1.07455	−0.537277	+	0.843406i	$0.680547\pi$
$644$	17183.9	1.05146
$645$	0	0
$646$	−12291.9	−0.748633
$647$	−10298.5	−0.625775	−0.312888	−	0.949790i	$-0.601296\pi$
$647$	−10298.5	−0.625775	−0.312888	+	0.949790i	$0.601296\pi$
$648$	0	0
$649$	5736.43	0.346956
$650$	0	0
$651$	0	0
$652$	−445.704	−0.0267716
$653$	11208.7	0.671716	0.335858	−	0.941913i	$-0.390974\pi$
$653$	11208.7	0.671716	0.335858	+	0.941913i	$0.390974\pi$
$654$	0	0
$655$	0	0
$656$	2044.22	0.121667
$657$	0	0
$658$	−1868.21	−0.110684
$659$	13207.9	0.780741	0.390371	−	0.920658i	$-0.372347\pi$
$659$	13207.9	0.780741	0.390371	+	0.920658i	$0.372347\pi$
$660$	0	0
$661$	−6004.29	−0.353313	−0.176656	−	0.984273i	$-0.556528\pi$
$661$	−6004.29	−0.353313	−0.176656	+	0.984273i	$0.556528\pi$
$662$	5335.16	0.313228
$663$	0	0
$664$	−15861.5	−0.927027
$665$	0	0
$666$	0	0
$667$	10378.0	0.602458
$668$	7174.32	0.415543
$669$	0	0
$670$	0	0
$671$	14130.2	0.812954
$672$	0	0
$673$	8504.38	0.487102	0.243551	−	0.969888i	$-0.421688\pi$
$673$	8504.38	0.487102	0.243551	+	0.969888i	$0.421688\pi$
$674$	−8565.15	−0.489492
$675$	0	0
$676$	−21952.3	−1.24899
$677$	−26155.7	−1.48485	−0.742426	−	0.669928i	$-0.766325\pi$
$677$	−26155.7	−1.48485	−0.742426	+	0.669928i	$0.766325\pi$
$678$	0	0
$679$	2374.44	0.134201
$680$	0	0
$681$	0	0
$682$	1450.23	0.0814253
$683$	15736.7	0.881625	0.440812	−	0.897599i	$-0.354690\pi$
$683$	15736.7	0.881625	0.440812	+	0.897599i	$0.354690\pi$
$684$	0	0
$685$	0	0
$686$	−5606.14	−0.312017
$687$	0	0
$688$	796.416	0.0441324
$689$	−23539.5	−1.30157
$690$	0	0
$691$	33323.6	1.83457	0.917287	−	0.398226i	$-0.130374\pi$
$691$	33323.6	1.83457	0.917287	+	0.398226i	$0.130374\pi$
$692$	6532.69	0.358867
$693$	0	0
$694$	−8043.83	−0.439971
$695$	0	0
$696$	0	0
$697$	10685.9	0.580713
$698$	11233.9	0.609183
$699$	0	0
$700$	0	0
$701$	14146.2	0.762187	0.381094	−	0.924536i	$-0.375548\pi$
$701$	14146.2	0.762187	0.381094	+	0.924536i	$0.375548\pi$
$702$	0	0
$703$	18626.7	0.999314
$704$	1547.74	0.0828591
$705$	0	0
$706$	4022.66	0.214440
$707$	−37511.8	−1.99544
$708$	0	0
$709$	−22082.9	−1.16973	−0.584867	−	0.811129i	$-0.698853\pi$
$709$	−22082.9	−1.16973	−0.584867	+	0.811129i	$0.698853\pi$
$710$	0	0
$711$	0	0
$712$	−10225.7	−0.538238
$713$	−4692.55	−0.246476
$714$	0	0
$715$	0	0
$716$	−3627.95	−0.189362
$717$	0	0
$718$	4523.78	0.235134
$719$	−8590.31	−0.445570	−0.222785	−	0.974868i	$-0.571515\pi$
$719$	−8590.31	−0.445570	−0.222785	+	0.974868i	$0.571515\pi$
$720$	0	0
$721$	−17993.0	−0.929394
$722$	−2111.17	−0.108822
$723$	0	0
$724$	−27265.6	−1.39961
$725$	0	0
$726$	0	0
$727$	889.078	0.0453564	0.0226782	−	0.999743i	$-0.492781\pi$
$727$	889.078	0.0453564	0.0226782	+	0.999743i	$0.492781\pi$
$728$	32178.5	1.63821
$729$	0	0
$730$	0	0
$731$	4163.17	0.210643
$732$	0	0
$733$	−10141.8	−0.511044	−0.255522	−	0.966803i	$-0.582247\pi$
$733$	−10141.8	−0.511044	−0.255522	+	0.966803i	$0.582247\pi$
$734$	−15904.5	−0.799791
$735$	0	0
$736$	−22892.8	−1.14652
$737$	5371.30	0.268459
$738$	0	0
$739$	−7179.60	−0.357383	−0.178691	−	0.983905i	$-0.557186\pi$
$739$	−7179.60	−0.357383	−0.178691	+	0.983905i	$0.557186\pi$
$740$	0	0
$741$	0	0
$742$	9298.31	0.460043
$743$	−5272.29	−0.260325	−0.130162	−	0.991493i	$-0.541550\pi$
$743$	−5272.29	−0.260325	−0.130162	+	0.991493i	$0.541550\pi$
$744$	0	0
$745$	0	0
$746$	2821.26	0.138463
$747$	0	0
$748$	−22411.4	−1.09551
$749$	−44695.8	−2.18044
$750$	0	0
$751$	−13797.9	−0.670429	−0.335214	−	0.942142i	$-0.608809\pi$
$751$	−13797.9	−0.670429	−0.335214	+	0.942142i	$0.608809\pi$
$752$	−1508.21	−0.0731367
$753$	0	0
$754$	8489.35	0.410032
$755$	0	0
$756$	0	0
$757$	26668.3	1.28042	0.640208	−	0.768202i	$-0.278848\pi$
$757$	26668.3	1.28042	0.640208	+	0.768202i	$0.278848\pi$
$758$	5736.50	0.274880
$759$	0	0
$760$	0	0
$761$	−15573.5	−0.741840	−0.370920	−	0.928665i	$-0.620958\pi$
$761$	−15573.5	−0.741840	−0.370920	+	0.928665i	$0.620958\pi$
$762$	0	0
$763$	−43574.6	−2.06750
$764$	7071.92	0.334886
$765$	0	0
$766$	−9737.00	−0.459285
$767$	−15183.6	−0.714793
$768$	0	0
$769$	10429.8	0.489088	0.244544	−	0.969638i	$-0.421362\pi$
$769$	10429.8	0.489088	0.244544	+	0.969638i	$0.421362\pi$
$770$	0	0
$771$	0	0
$772$	26644.1	1.24215
$773$	7082.64	0.329554	0.164777	−	0.986331i	$-0.447310\pi$
$773$	7082.64	0.329554	0.164777	+	0.986331i	$0.447310\pi$
$774$	0	0
$775$	0	0
$776$	−2023.64	−0.0936142
$777$	0	0
$778$	12722.8	0.586292
$779$	−6152.80	−0.282987
$780$	0	0
$781$	−28718.8	−1.31580
$782$	−20972.9	−0.959067
$783$	0	0
$784$	3757.23	0.171156
$785$	0	0
$786$	0	0
$787$	−32634.6	−1.47814	−0.739072	−	0.673627i	$-0.764736\pi$
$787$	−32634.6	−1.47814	−0.739072	+	0.673627i	$0.764736\pi$
$788$	12237.1	0.553208
$789$	0	0
$790$	0	0
$791$	−7623.89	−0.342698
$792$	0	0
$793$	−37400.9	−1.67483
$794$	−2544.99	−0.113751
$795$	0	0
$796$	6522.88	0.290449
$797$	8175.60	0.363356	0.181678	−	0.983358i	$-0.441847\pi$
$797$	8175.60	0.363356	0.181678	+	0.983358i	$0.441847\pi$
$798$	0	0
$799$	−7884.00	−0.349081
$800$	0	0
$801$	0	0
$802$	−13421.2	−0.590922
$803$	26632.5	1.17041
$804$	0	0
$805$	0	0
$806$	−3838.56	−0.167751
$807$	0	0
$808$	31969.8	1.39195
$809$	15466.3	0.672146	0.336073	−	0.941836i	$-0.390901\pi$
$809$	15466.3	0.672146	0.336073	+	0.941836i	$0.390901\pi$
$810$	0	0
$811$	45091.6	1.95238	0.976191	−	0.216914i	$-0.0695992\pi$
$811$	45091.6	1.95238	0.976191	+	0.216914i	$0.0695992\pi$
$812$	11594.9	0.501108
$813$	0	0
$814$	−9822.08	−0.422928
$815$	0	0
$816$	0	0
$817$	−2397.10	−0.102648
$818$	−4170.98	−0.178282
$819$	0	0
$820$	0	0
$821$	41893.8	1.78088	0.890440	−	0.455100i	$-0.150396\pi$
$821$	41893.8	1.78088	0.890440	+	0.455100i	$0.150396\pi$
$822$	0	0
$823$	−4098.52	−0.173591	−0.0867955	−	0.996226i	$-0.527663\pi$
$823$	−4098.52	−0.173591	−0.0867955	+	0.996226i	$0.527663\pi$
$824$	15334.7	0.648313
$825$	0	0
$826$	5997.65	0.252645
$827$	−24781.0	−1.04198	−0.520991	−	0.853562i	$-0.674438\pi$
$827$	−24781.0	−1.04198	−0.520991	+	0.853562i	$0.674438\pi$
$828$	0	0
$829$	−14621.0	−0.612555	−0.306278	−	0.951942i	$-0.599084\pi$
$829$	−14621.0	−0.612555	−0.306278	+	0.951942i	$0.599084\pi$
$830$	0	0
$831$	0	0
$832$	−4096.67	−0.170705
$833$	19640.4	0.816928
$834$	0	0
$835$	0	0
$836$	12904.2	0.533852
$837$	0	0
$838$	13707.6	0.565062
$839$	−21337.7	−0.878021	−0.439011	−	0.898482i	$-0.644671\pi$
$839$	−21337.7	−0.878021	−0.439011	+	0.898482i	$0.644671\pi$
$840$	0	0
$841$	−17386.4	−0.712878
$842$	−17221.9	−0.704878
$843$	0	0
$844$	3771.91	0.153832
$845$	0	0
$846$	0	0
$847$	−11442.6	−0.464196
$848$	7506.57	0.303982
$849$	0	0
$850$	0	0
$851$	31781.7	1.28021
$852$	0	0
$853$	25699.9	1.03159	0.515797	−	0.856711i	$-0.327496\pi$
$853$	25699.9	1.03159	0.515797	+	0.856711i	$0.327496\pi$
$854$	14773.7	0.591974
$855$	0	0
$856$	38092.5	1.52100
$857$	−693.306	−0.0276346	−0.0138173	−	0.999905i	$-0.504398\pi$
$857$	−693.306	−0.0276346	−0.0138173	+	0.999905i	$0.504398\pi$
$858$	0	0
$859$	33929.9	1.34770	0.673849	−	0.738869i	$-0.264640\pi$
$859$	33929.9	1.34770	0.673849	+	0.738869i	$0.264640\pi$
$860$	0	0
$861$	0	0
$862$	281.286	0.0111144
$863$	36133.8	1.42527	0.712636	−	0.701534i	$-0.247501\pi$
$863$	36133.8	1.42527	0.712636	+	0.701534i	$0.247501\pi$
$864$	0	0
$865$	0	0
$866$	12946.7	0.508020
$867$	0	0
$868$	−5242.75	−0.205012
$869$	−23494.9	−0.917159
$870$	0	0
$871$	−14217.1	−0.553075
$872$	37137.0	1.44222
$873$	0	0
$874$	12075.9	0.467362
$875$	0	0
$876$	0	0
$877$	12692.0	0.488685	0.244343	−	0.969689i	$-0.421428\pi$
$877$	12692.0	0.488685	0.244343	+	0.969689i	$0.421428\pi$
$878$	6164.23	0.236939
$879$	0	0
$880$	0	0
$881$	44595.2	1.70539	0.852697	−	0.522406i	$-0.174965\pi$
$881$	44595.2	1.70539	0.852697	+	0.522406i	$0.174965\pi$
$882$	0	0
$883$	27617.1	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$883$	27617.1	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$884$	59320.0	2.25695
$885$	0	0
$886$	725.709	0.0275177
$887$	3788.16	0.143398	0.0716989	−	0.997426i	$-0.477158\pi$
$887$	3788.16	0.143398	0.0716989	+	0.997426i	$0.477158\pi$
$888$	0	0
$889$	−18058.1	−0.681270
$890$	0	0
$891$	0	0
$892$	31.1619	0.00116971
$893$	4539.50	0.170110
$894$	0	0
$895$	0	0
$896$	−31355.9	−1.16912
$897$	0	0
$898$	3845.44	0.142900
$899$	−3166.31	−0.117466
$900$	0	0
$901$	39239.7	1.45090
$902$	3244.45	0.119765
$903$	0	0
$904$	6497.55	0.239054
$905$	0	0
$906$	0	0
$907$	14327.0	0.524497	0.262249	−	0.965000i	$-0.415536\pi$
$907$	14327.0	0.524497	0.262249	+	0.965000i	$0.415536\pi$
$908$	7835.86	0.286390
$909$	0	0
$910$	0	0
$911$	28460.6	1.03506	0.517531	−	0.855665i	$-0.326851\pi$
$911$	28460.6	1.03506	0.517531	+	0.855665i	$0.326851\pi$
$912$	0	0
$913$	23846.4	0.864404
$914$	18964.3	0.686306
$915$	0	0
$916$	5281.85	0.190521
$917$	59603.4	2.14643
$918$	0	0
$919$	4778.56	0.171524	0.0857618	−	0.996316i	$-0.472668\pi$
$919$	4778.56	0.171524	0.0857618	+	0.996316i	$0.472668\pi$
$920$	0	0
$921$	0	0
$922$	−4530.61	−0.161830
$923$	76015.0	2.71079
$924$	0	0
$925$	0	0
$926$	12765.0	0.453005
$927$	0	0
$928$	−15447.0	−0.546414
$929$	−36209.7	−1.27880	−0.639398	−	0.768876i	$-0.720817\pi$
$929$	−36209.7	−1.27880	−0.639398	+	0.768876i	$0.720817\pi$
$930$	0	0
$931$	−11308.7	−0.398096
$932$	32010.2	1.12503
$933$	0	0
$934$	−7317.12	−0.256342
$935$	0	0
$936$	0	0
$937$	16152.5	0.563158	0.281579	−	0.959538i	$-0.409142\pi$
$937$	16152.5	0.563158	0.281579	+	0.959538i	$0.409142\pi$
$938$	5615.90	0.195486
$939$	0	0
$940$	0	0
$941$	23287.1	0.806736	0.403368	−	0.915038i	$-0.367839\pi$
$941$	23287.1	0.806736	0.403368	+	0.915038i	$0.367839\pi$
$942$	0	0
$943$	−10498.2	−0.362532
$944$	4841.93	0.166940
$945$	0	0
$946$	1264.02	0.0434427
$947$	15940.2	0.546977	0.273489	−	0.961875i	$-0.411822\pi$
$947$	15940.2	0.546977	0.273489	+	0.961875i	$0.411822\pi$
$948$	0	0
$949$	−70492.8	−2.41127
$950$	0	0
$951$	0	0
$952$	−53640.8	−1.82616
$953$	−35813.5	−1.21733	−0.608663	−	0.793429i	$-0.708294\pi$
$953$	−35813.5	−1.21733	−0.608663	+	0.793429i	$0.708294\pi$
$954$	0	0
$955$	0	0
$956$	34949.8	1.18238
$957$	0	0
$958$	8581.26	0.289403
$959$	−43530.0	−1.46575
$960$	0	0
$961$	−28359.3	−0.951942
$962$	25997.7	0.871311
$963$	0	0
$964$	3900.33	0.130312
$965$	0	0
$966$	0	0
$967$	10857.7	0.361075	0.180537	−	0.983568i	$-0.442216\pi$
$967$	10857.7	0.361075	0.180537	+	0.983568i	$0.442216\pi$
$968$	9752.12	0.323807
$969$	0	0
$970$	0	0
$971$	−15611.5	−0.515961	−0.257981	−	0.966150i	$-0.583057\pi$
$971$	−15611.5	−0.515961	−0.257981	+	0.966150i	$0.583057\pi$
$972$	0	0
$973$	−46848.5	−1.54357
$974$	−8830.24	−0.290492
$975$	0	0
$976$	11926.9	0.391158
$977$	−43213.6	−1.41507	−0.707536	−	0.706677i	$-0.750193\pi$
$977$	−43213.6	−1.41507	−0.707536	+	0.706677i	$0.750193\pi$
$978$	0	0
$979$	15373.5	0.501878
$980$	0	0
$981$	0	0
$982$	24001.2	0.779948
$983$	−37431.3	−1.21452	−0.607260	−	0.794504i	$-0.707731\pi$
$983$	−37431.3	−1.21452	−0.607260	+	0.794504i	$0.707731\pi$
$984$	0	0
$985$	0	0
$986$	−14151.5	−0.457076
$987$	0	0
$988$	−34155.7	−1.09983
$989$	−4090.04	−0.131502
$990$	0	0
$991$	−31721.1	−1.01680	−0.508402	−	0.861120i	$-0.669764\pi$
$991$	−31721.1	−1.01680	−0.508402	+	0.861120i	$0.669764\pi$
$992$	6984.54	0.223548
$993$	0	0
$994$	−30026.6	−0.958136
$995$	0	0
$996$	0	0
$997$	32167.9	1.02183	0.510917	−	0.859630i	$-0.329306\pi$
$997$	32167.9	1.02183	0.510917	+	0.859630i	$0.329306\pi$
$998$	−8809.84	−0.279430
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.be.1.8		12
3.2	odd	2		2025.4.a.bi.1.5		12
5.4	even	2		2025.4.a.bj.1.5		12
9.2	odd	6		225.4.e.e.76.8	✓	24
9.5	odd	6		225.4.e.e.151.8	yes	24
15.14	odd	2		2025.4.a.bf.1.8		12
45.2	even	12		225.4.k.e.49.11		48
45.14	odd	6		225.4.e.f.151.5	yes	24
45.23	even	12		225.4.k.e.124.11		48
45.29	odd	6		225.4.e.f.76.5	yes	24
45.32	even	12		225.4.k.e.124.14		48
45.38	even	12		225.4.k.e.49.14		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.8	✓	24	9.2	odd	6
225.4.e.e.151.8	yes	24	9.5	odd	6
225.4.e.f.76.5	yes	24	45.29	odd	6
225.4.e.f.151.5	yes	24	45.14	odd	6
225.4.k.e.49.11		48	45.2	even	12
225.4.k.e.49.14		48	45.38	even	12
225.4.k.e.124.11		48	45.23	even	12
225.4.k.e.124.14		48	45.32	even	12
2025.4.a.be.1.8		12	1.1	even	1	trivial
2025.4.a.bf.1.8		12	15.14	odd	2
2025.4.a.bi.1.5		12	3.2	odd	2
2025.4.a.bj.1.5		12	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.33965 q^{2} -6.20534 q^{4} +22.3290 q^{7} -19.0302 q^{8} +O(q^{10})\)	\(q+1.33965 q^{2} -6.20534 q^{4} +22.3290 q^{7} -19.0302 q^{8} +28.6102 q^{11} -75.7274 q^{13} +29.9131 q^{14} +24.1490 q^{16} +126.236 q^{17} -72.6849 q^{19} +38.3277 q^{22} -124.018 q^{23} -101.448 q^{26} -138.559 q^{28} -83.6816 q^{29} +37.8376 q^{31} +184.593 q^{32} +169.112 q^{34} -256.266 q^{37} -97.3722 q^{38} +84.6503 q^{41} +32.9793 q^{43} -177.536 q^{44} -166.141 q^{46} -62.4545 q^{47} +155.585 q^{49} +469.914 q^{52} +310.844 q^{53} -424.925 q^{56} -112.104 q^{58} +200.503 q^{59} +493.888 q^{61} +50.6891 q^{62} +54.0976 q^{64} +187.741 q^{67} -783.336 q^{68} -1003.80 q^{71} +930.875 q^{73} -343.307 q^{74} +451.034 q^{76} +638.838 q^{77} -821.208 q^{79} +113.402 q^{82} +833.493 q^{83} +44.1807 q^{86} -544.457 q^{88} +537.343 q^{89} -1690.92 q^{91} +769.575 q^{92} -83.6672 q^{94} +106.339 q^{97} +208.430 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} - 29 q^{11} - 24 q^{13} + 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} + 18 q^{22} - 318 q^{23} + 154 q^{26} - 96 q^{28} - 106 q^{29} + 60 q^{31} - 914 q^{32} - 108 q^{34} + 84 q^{37} - 640 q^{38} + 353 q^{41} + 426 q^{43} - 571 q^{44} + 270 q^{46} - 1210 q^{47} + 666 q^{49} + 75 q^{52} - 448 q^{53} + 570 q^{56} - 594 q^{58} - 482 q^{59} + 402 q^{61} - 2544 q^{62} + 975 q^{64} + 201 q^{67} - 3437 q^{68} + 944 q^{71} + 453 q^{73} - 10 q^{74} - 462 q^{76} - 2652 q^{77} + 258 q^{79} - 873 q^{82} - 3012 q^{83} - 1952 q^{86} + 216 q^{88} + 738 q^{89} - 618 q^{91} - 5232 q^{92} + 63 q^{94} + 318 q^{97} - 7511 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 + 48 * q^4 - 6 * q^7 - 45 * q^8 - 29 * q^11 - 24 * q^13 + 69 * q^14 + 192 * q^16 - 79 * q^17 - 75 * q^19 + 18 * q^22 - 318 * q^23 + 154 * q^26 - 96 * q^28 - 106 * q^29 + 60 * q^31 - 914 * q^32 - 108 * q^34 + 84 * q^37 - 640 * q^38 + 353 * q^41 + 426 * q^43 - 571 * q^44 + 270 * q^46 - 1210 * q^47 + 666 * q^49 + 75 * q^52 - 448 * q^53 + 570 * q^56 - 594 * q^58 - 482 * q^59 + 402 * q^61 - 2544 * q^62 + 975 * q^64 + 201 * q^67 - 3437 * q^68 + 944 * q^71 + 453 * q^73 - 10 * q^74 - 462 * q^76 - 2652 * q^77 + 258 * q^79 - 873 * q^82 - 3012 * q^83 - 1952 * q^86 + 216 * q^88 + 738 * q^89 - 618 * q^91 - 5232 * q^92 + 63 * q^94 + 318 * q^97 - 7511 * q^98

Introduction

L-functions

Modular forms

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Database

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