LMFDB - Embedded newform 2025.4.a.be.1.9

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			1.9
Root			$-1.66434$ 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)

$f(q)$	$=$	$q+1.66434 q^{2} -5.22997 q^{4} +10.5672 q^{7} -22.0192 q^{8} +O(q^{10})$	$q+1.66434 q^{2} -5.22997 q^{4} +10.5672 q^{7} -22.0192 q^{8} -29.0797 q^{11} +23.5020 q^{13} +17.5874 q^{14} +5.19227 q^{16} -35.6583 q^{17} +23.7670 q^{19} -48.3985 q^{22} +78.9465 q^{23} +39.1153 q^{26} -55.2658 q^{28} -148.199 q^{29} +341.824 q^{31} +184.795 q^{32} -59.3477 q^{34} +337.627 q^{37} +39.5565 q^{38} -354.046 q^{41} -19.5265 q^{43} +152.086 q^{44} +131.394 q^{46} +112.183 q^{47} -231.335 q^{49} -122.914 q^{52} -699.829 q^{53} -232.680 q^{56} -246.654 q^{58} -27.4594 q^{59} -825.326 q^{61} +568.912 q^{62} +266.024 q^{64} +825.314 q^{67} +186.492 q^{68} +337.616 q^{71} -717.566 q^{73} +561.926 q^{74} -124.301 q^{76} -307.289 q^{77} -310.182 q^{79} -589.253 q^{82} +552.139 q^{83} -32.4988 q^{86} +640.311 q^{88} +1085.33 q^{89} +248.349 q^{91} -412.888 q^{92} +186.711 q^{94} -1167.83 q^{97} -385.021 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.66434	0.588434	0.294217	−	0.955739i	$-0.404941\pi$
$2$	1.66434	0.588434	0.294217	+	0.955739i	$0.404941\pi$
$3$	0	0
$3$	0	0
$4$	−5.22997	−0.653746
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.5672	0.570573	0.285286	−	0.958442i	$-0.407911\pi$
$7$	10.5672	0.570573	0.285286	+	0.958442i	$0.407911\pi$
$8$	−22.0192	−0.973120
$9$	0	0
$10$	0	0
$11$	−29.0797	−0.797077	−0.398539	−	0.917152i	$-0.630483\pi$
$11$	−29.0797	−0.797077	−0.398539	+	0.917152i	$0.630483\pi$
$12$	0	0
$13$	23.5020	0.501405	0.250703	−	0.968064i	$-0.419338\pi$
$13$	23.5020	0.501405	0.250703	+	0.968064i	$0.419338\pi$
$14$	17.5874	0.335744
$15$	0	0
$16$	5.19227	0.0811293
$17$	−35.6583	−0.508731	−0.254365	−	0.967108i	$-0.581867\pi$
$17$	−35.6583	−0.508731	−0.254365	+	0.967108i	$0.581867\pi$
$18$	0	0
$19$	23.7670	0.286975	0.143488	−	0.989652i	$-0.454168\pi$
$19$	23.7670	0.286975	0.143488	+	0.989652i	$0.454168\pi$
$20$	0	0
$21$	0	0
$22$	−48.3985	−0.469027
$23$	78.9465	0.715717	0.357858	−	0.933776i	$-0.383507\pi$
$23$	78.9465	0.715717	0.357858	+	0.933776i	$0.383507\pi$
$24$	0	0
$25$	0	0
$26$	39.1153	0.295044
$27$	0	0
$28$	−55.2658	−0.373009
$29$	−148.199	−0.948963	−0.474481	−	0.880266i	$-0.657364\pi$
$29$	−148.199	−0.948963	−0.474481	+	0.880266i	$0.657364\pi$
$30$	0	0
$31$	341.824	1.98043	0.990217	−	0.139537i	$-0.0445616\pi$
$31$	341.824	1.98043	0.990217	+	0.139537i	$0.0445616\pi$
$32$	184.795	1.02086
$33$	0	0
$34$	−59.3477	−0.299354
$35$	0	0
$36$	0	0
$37$	337.627	1.50015	0.750074	−	0.661354i	$-0.230018\pi$
$37$	337.627	1.50015	0.750074	+	0.661354i	$0.230018\pi$
$38$	39.5565	0.168866
$39$	0	0
$40$	0	0
$41$	−354.046	−1.34860	−0.674301	−	0.738457i	$-0.735555\pi$
$41$	−354.046	−1.34860	−0.674301	+	0.738457i	$0.735555\pi$
$42$	0	0
$43$	−19.5265	−0.0692505	−0.0346252	−	0.999400i	$-0.511024\pi$
$43$	−19.5265	−0.0692505	−0.0346252	+	0.999400i	$0.511024\pi$
$44$	152.086	0.521086
$45$	0	0
$46$	131.394	0.421152
$47$	112.183	0.348161	0.174080	−	0.984731i	$-0.444305\pi$
$47$	112.183	0.348161	0.174080	+	0.984731i	$0.444305\pi$
$48$	0	0
$49$	−231.335	−0.674447
$50$	0	0
$51$	0	0
$52$	−122.914	−0.327792
$53$	−699.829	−1.81375	−0.906876	−	0.421398i	$-0.861540\pi$
$53$	−699.829	−1.81375	−0.906876	+	0.421398i	$0.861540\pi$
$54$	0	0
$55$	0	0
$56$	−232.680	−0.555235
$57$	0	0
$58$	−246.654	−0.558402
$59$	−27.4594	−0.0605917	−0.0302958	−	0.999541i	$-0.509645\pi$
$59$	−27.4594	−0.0605917	−0.0302958	+	0.999541i	$0.509645\pi$
$60$	0	0
$61$	−825.326	−1.73233	−0.866166	−	0.499757i	$-0.833423\pi$
$61$	−825.326	−1.73233	−0.866166	+	0.499757i	$0.833423\pi$
$62$	568.912	1.16535
$63$	0	0
$64$	266.024	0.519579
$65$	0	0
$66$	0	0
$67$	825.314	1.50490	0.752449	−	0.658650i	$-0.228872\pi$
$67$	825.314	1.50490	0.752449	+	0.658650i	$0.228872\pi$
$68$	186.492	0.332580
$69$	0	0
$70$	0	0
$71$	337.616	0.564332	0.282166	−	0.959366i	$-0.408947\pi$
$71$	337.616	0.564332	0.282166	+	0.959366i	$0.408947\pi$
$72$	0	0
$73$	−717.566	−1.15048	−0.575238	−	0.817986i	$-0.695091\pi$
$73$	−717.566	−1.15048	−0.575238	+	0.817986i	$0.695091\pi$
$74$	561.926	0.882737
$75$	0	0
$76$	−124.301	−0.187609
$77$	−307.289	−0.454791
$78$	0	0
$79$	−310.182	−0.441749	−0.220875	−	0.975302i	$-0.570891\pi$
$79$	−310.182	−0.441749	−0.220875	+	0.975302i	$0.570891\pi$
$80$	0	0
$81$	0	0
$82$	−589.253	−0.793563
$83$	552.139	0.730182	0.365091	−	0.930972i	$-0.381038\pi$
$83$	552.139	0.730182	0.365091	+	0.930972i	$0.381038\pi$
$84$	0	0
$85$	0	0
$86$	−32.4988	−0.0407493
$87$	0	0
$88$	640.311	0.775652
$89$	1085.33	1.29264	0.646319	−	0.763067i	$-0.276308\pi$
$89$	1085.33	1.29264	0.646319	+	0.763067i	$0.276308\pi$
$90$	0	0
$91$	248.349	0.286088
$92$	−412.888	−0.467897
$93$	0	0
$94$	186.711	0.204870
$95$	0	0
$96$	0	0
$97$	−1167.83	−1.22242	−0.611212	−	0.791467i	$-0.709318\pi$
$97$	−1167.83	−1.22242	−0.611212	+	0.791467i	$0.709318\pi$
$98$	−385.021	−0.396867
$99$	0	0
$100$	0	0
$101$	−1280.08	−1.26111	−0.630557	−	0.776143i	$-0.717173\pi$
$101$	−1280.08	−1.26111	−0.630557	+	0.776143i	$0.717173\pi$
$102$	0	0
$103$	−124.950	−0.119531	−0.0597653	−	0.998212i	$-0.519035\pi$
$103$	−124.950	−0.119531	−0.0597653	+	0.998212i	$0.519035\pi$
$104$	−517.494	−0.487927
$105$	0	0
$106$	−1164.75	−1.06727
$107$	1350.01	1.21973	0.609863	−	0.792507i	$-0.291225\pi$
$107$	1350.01	1.21973	0.609863	+	0.792507i	$0.291225\pi$
$108$	0	0
$109$	−1143.61	−1.00494	−0.502468	−	0.864596i	$-0.667574\pi$
$109$	−1143.61	−1.00494	−0.502468	+	0.864596i	$0.667574\pi$
$110$	0	0
$111$	0	0
$112$	54.8675	0.0462901
$113$	−659.034	−0.548644	−0.274322	−	0.961638i	$-0.588453\pi$
$113$	−659.034	−0.548644	−0.274322	+	0.961638i	$0.588453\pi$
$114$	0	0
$115$	0	0
$116$	775.077	0.620380
$117$	0	0
$118$	−45.7018	−0.0356542
$119$	−376.807	−0.290268
$120$	0	0
$121$	−485.373	−0.364668
$122$	−1373.63	−1.01936
$123$	0	0
$124$	−1787.73	−1.29470
$125$	0	0
$126$	0	0
$127$	1564.04	1.09281	0.546403	−	0.837522i	$-0.315997\pi$
$127$	1564.04	1.09281	0.546403	+	0.837522i	$0.315997\pi$
$128$	−1035.61	−0.715121
$129$	0	0
$130$	0	0
$131$	−2179.50	−1.45361	−0.726807	−	0.686842i	$-0.758997\pi$
$131$	−2179.50	−1.45361	−0.726807	+	0.686842i	$0.758997\pi$
$132$	0	0
$133$	251.150	0.163740
$134$	1373.60	0.885533
$135$	0	0
$136$	785.168	0.495056
$137$	−1053.89	−0.657228	−0.328614	−	0.944464i	$-0.606581\pi$
$137$	−1053.89	−0.657228	−0.328614	+	0.944464i	$0.606581\pi$
$138$	0	0
$139$	486.606	0.296931	0.148465	−	0.988918i	$-0.452567\pi$
$139$	486.606	0.296931	0.148465	+	0.988918i	$0.452567\pi$
$140$	0	0
$141$	0	0
$142$	561.908	0.332072
$143$	−683.429	−0.399659
$144$	0	0
$145$	0	0
$146$	−1194.28	−0.676979
$147$	0	0
$148$	−1765.78	−0.980715
$149$	3246.66	1.78508	0.892540	−	0.450969i	$-0.148922\pi$
$149$	3246.66	1.78508	0.892540	+	0.450969i	$0.148922\pi$
$150$	0	0
$151$	−1529.75	−0.824431	−0.412216	−	0.911086i	$-0.635245\pi$
$151$	−1529.75	−0.824431	−0.412216	+	0.911086i	$0.635245\pi$
$152$	−523.331	−0.279261
$153$	0	0
$154$	−511.435	−0.267614
$155$	0	0
$156$	0	0
$157$	731.552	0.371874	0.185937	−	0.982562i	$-0.440468\pi$
$157$	731.552	0.371874	0.185937	+	0.982562i	$0.440468\pi$
$158$	−516.249	−0.259940
$159$	0	0
$160$	0	0
$161$	834.240	0.408368
$162$	0	0
$163$	132.833	0.0638298	0.0319149	−	0.999491i	$-0.489839\pi$
$163$	132.833	0.0638298	0.0319149	+	0.999491i	$0.489839\pi$
$164$	1851.65	0.881643
$165$	0	0
$166$	918.947	0.429663
$167$	−1368.15	−0.633954	−0.316977	−	0.948433i	$-0.602668\pi$
$167$	−1368.15	−0.633954	−0.316977	+	0.948433i	$0.602668\pi$
$168$	0	0
$169$	−1644.66	−0.748593
$170$	0	0
$171$	0	0
$172$	102.123	0.0452722
$173$	−2117.40	−0.930535	−0.465268	−	0.885170i	$-0.654042\pi$
$173$	−2117.40	−0.930535	−0.465268	+	0.885170i	$0.654042\pi$
$174$	0	0
$175$	0	0
$176$	−150.990	−0.0646663
$177$	0	0
$178$	1806.36	0.760632
$179$	−2224.74	−0.928966	−0.464483	−	0.885582i	$-0.653760\pi$
$179$	−2224.74	−0.928966	−0.464483	+	0.885582i	$0.653760\pi$
$180$	0	0
$181$	−723.066	−0.296934	−0.148467	−	0.988917i	$-0.547434\pi$
$181$	−723.066	−0.296934	−0.148467	+	0.988917i	$0.547434\pi$
$182$	413.337	0.168344
$183$	0	0
$184$	−1738.34	−0.696478
$185$	0	0
$186$	0	0
$187$	1036.93	0.405498
$188$	−586.713	−0.227609
$189$	0	0
$190$	0	0
$191$	251.361	0.0952242	0.0476121	−	0.998866i	$-0.484839\pi$
$191$	251.361	0.0952242	0.0476121	+	0.998866i	$0.484839\pi$
$192$	0	0
$193$	−380.769	−0.142012	−0.0710061	−	0.997476i	$-0.522621\pi$
$193$	−380.769	−0.142012	−0.0710061	+	0.997476i	$0.522621\pi$
$194$	−1943.67	−0.719315
$195$	0	0
$196$	1209.88	0.440917
$197$	−4369.98	−1.58045	−0.790225	−	0.612817i	$-0.790036\pi$
$197$	−4369.98	−1.58045	−0.790225	+	0.612817i	$0.790036\pi$
$198$	0	0
$199$	−985.541	−0.351071	−0.175536	−	0.984473i	$-0.556166\pi$
$199$	−985.541	−0.351071	−0.175536	+	0.984473i	$0.556166\pi$
$200$	0	0
$201$	0	0
$202$	−2130.49	−0.742082
$203$	−1566.04	−0.541452
$204$	0	0
$205$	0	0
$206$	−207.959	−0.0703358
$207$	0	0
$208$	122.029	0.0406787
$209$	−691.137	−0.228741
$210$	0	0
$211$	961.328	0.313652	0.156826	−	0.987626i	$-0.449874\pi$
$211$	961.328	0.313652	0.156826	+	0.987626i	$0.449874\pi$
$212$	3660.08	1.18573
$213$	0	0
$214$	2246.88	0.717728
$215$	0	0
$216$	0	0
$217$	3612.11	1.12998
$218$	−1903.36	−0.591338
$219$	0	0
$220$	0	0
$221$	−838.041	−0.255080
$222$	0	0
$223$	−4793.73	−1.43952	−0.719758	−	0.694225i	$-0.755747\pi$
$223$	−4793.73	−1.43952	−0.719758	+	0.694225i	$0.755747\pi$
$224$	1952.76	0.582474
$225$	0	0
$226$	−1096.86	−0.322840
$227$	−4231.85	−1.23735	−0.618673	−	0.785648i	$-0.712329\pi$
$227$	−4231.85	−1.23735	−0.618673	+	0.785648i	$0.712329\pi$
$228$	0	0
$229$	−1602.70	−0.462487	−0.231243	−	0.972896i	$-0.574279\pi$
$229$	−1602.70	−0.462487	−0.231243	+	0.972896i	$0.574279\pi$
$230$	0	0
$231$	0	0
$232$	3263.23	0.923454
$233$	−6094.84	−1.71368	−0.856838	−	0.515586i	$-0.827574\pi$
$233$	−6094.84	−1.71368	−0.856838	+	0.515586i	$0.827574\pi$
$234$	0	0
$235$	0	0
$236$	143.612	0.0396115
$237$	0	0
$238$	−627.136	−0.170803
$239$	3878.44	1.04969	0.524844	−	0.851198i	$-0.324124\pi$
$239$	3878.44	1.04969	0.524844	+	0.851198i	$0.324124\pi$
$240$	0	0
$241$	6506.79	1.73917	0.869583	−	0.493787i	$-0.164388\pi$
$241$	6506.79	1.73917	0.869583	+	0.493787i	$0.164388\pi$
$242$	−807.826	−0.214583
$243$	0	0
$244$	4316.43	1.13250
$245$	0	0
$246$	0	0
$247$	558.571	0.143891
$248$	−7526.69	−1.92720
$249$	0	0
$250$	0	0
$251$	−7716.46	−1.94047	−0.970236	−	0.242161i	$-0.922144\pi$
$251$	−7716.46	−1.94047	−0.970236	+	0.242161i	$0.922144\pi$
$252$	0	0
$253$	−2295.74	−0.570482
$254$	2603.10	0.643044
$255$	0	0
$256$	−3851.80	−0.940380
$257$	−2537.75	−0.615955	−0.307977	−	0.951394i	$-0.599652\pi$
$257$	−2537.75	−0.615955	−0.307977	+	0.951394i	$0.599652\pi$
$258$	0	0
$259$	3567.75	0.855943
$260$	0	0
$261$	0	0
$262$	−3627.43	−0.855356
$263$	−6200.33	−1.45372	−0.726861	−	0.686784i	$-0.759022\pi$
$263$	−6200.33	−1.45372	−0.726861	+	0.686784i	$0.759022\pi$
$264$	0	0
$265$	0	0
$266$	417.999	0.0963503
$267$	0	0
$268$	−4316.37	−0.983821
$269$	−1737.04	−0.393715	−0.196858	−	0.980432i	$-0.563074\pi$
$269$	−1737.04	−0.393715	−0.196858	+	0.980432i	$0.563074\pi$
$270$	0	0
$271$	−2510.25	−0.562683	−0.281341	−	0.959608i	$-0.590779\pi$
$271$	−2510.25	−0.562683	−0.281341	+	0.959608i	$0.590779\pi$
$272$	−185.148	−0.0412729
$273$	0	0
$274$	−1754.04	−0.386735
$275$	0	0
$276$	0	0
$277$	−979.454	−0.212454	−0.106227	−	0.994342i	$-0.533877\pi$
$277$	−979.454	−0.212454	−0.106227	+	0.994342i	$0.533877\pi$
$278$	809.879	0.174724
$279$	0	0
$280$	0	0
$281$	1682.91	0.357273	0.178636	−	0.983915i	$-0.442831\pi$
$281$	1682.91	0.357273	0.178636	+	0.983915i	$0.442831\pi$
$282$	0	0
$283$	2418.33	0.507967	0.253984	−	0.967209i	$-0.418259\pi$
$283$	2418.33	0.507967	0.253984	+	0.967209i	$0.418259\pi$
$284$	−1765.72	−0.368930
$285$	0	0
$286$	−1137.46	−0.235173
$287$	−3741.26	−0.769475
$288$	0	0
$289$	−3641.48	−0.741193
$290$	0	0
$291$	0	0
$292$	3752.85	0.752119
$293$	−1799.28	−0.358755	−0.179378	−	0.983780i	$-0.557408\pi$
$293$	−1799.28	−0.358755	−0.179378	+	0.983780i	$0.557408\pi$
$294$	0	0
$295$	0	0
$296$	−7434.26	−1.45982
$297$	0	0
$298$	5403.55	1.05040
$299$	1855.40	0.358864
$300$	0	0
$301$	−206.340	−0.0395124
$302$	−2546.02	−0.485123
$303$	0	0
$304$	123.405	0.0232821
$305$	0	0
$306$	0	0
$307$	−7372.67	−1.37062	−0.685311	−	0.728251i	$-0.740334\pi$
$307$	−7372.67	−1.37062	−0.685311	+	0.728251i	$0.740334\pi$
$308$	1607.11	0.297317
$309$	0	0
$310$	0	0
$311$	5708.03	1.04075	0.520374	−	0.853938i	$-0.325792\pi$
$311$	5708.03	1.04075	0.520374	+	0.853938i	$0.325792\pi$
$312$	0	0
$313$	710.348	0.128279	0.0641394	−	0.997941i	$-0.479570\pi$
$313$	710.348	0.128279	0.0641394	+	0.997941i	$0.479570\pi$
$314$	1217.55	0.218823
$315$	0	0
$316$	1622.24	0.288792
$317$	8298.66	1.47034	0.735172	−	0.677881i	$-0.237101\pi$
$317$	8298.66	1.47034	0.735172	+	0.677881i	$0.237101\pi$
$318$	0	0
$319$	4309.59	0.756397
$320$	0	0
$321$	0	0
$322$	1388.46	0.240298
$323$	−847.493	−0.145993
$324$	0	0
$325$	0	0
$326$	221.079	0.0375596
$327$	0	0
$328$	7795.80	1.31235
$329$	1185.45	0.198651
$330$	0	0
$331$	4337.91	0.720342	0.360171	−	0.932886i	$-0.382718\pi$
$331$	4337.91	0.720342	0.360171	+	0.932886i	$0.382718\pi$
$332$	−2887.67	−0.477353
$333$	0	0
$334$	−2277.06	−0.373040
$335$	0	0
$336$	0	0
$337$	−5074.79	−0.820301	−0.410151	−	0.912018i	$-0.634524\pi$
$337$	−5074.79	−0.820301	−0.410151	+	0.912018i	$0.634524\pi$
$338$	−2737.27	−0.440497
$339$	0	0
$340$	0	0
$341$	−9940.14	−1.57856
$342$	0	0
$343$	−6069.09	−0.955394
$344$	429.959	0.0673890
$345$	0	0
$346$	−3524.07	−0.547558
$347$	−4535.03	−0.701594	−0.350797	−	0.936452i	$-0.614089\pi$
$347$	−4535.03	−0.701594	−0.350797	+	0.936452i	$0.614089\pi$
$348$	0	0
$349$	2083.73	0.319598	0.159799	−	0.987150i	$-0.448915\pi$
$349$	2083.73	0.319598	0.159799	+	0.987150i	$0.448915\pi$
$350$	0	0
$351$	0	0
$352$	−5373.78	−0.813704
$353$	−8423.78	−1.27012	−0.635060	−	0.772462i	$-0.719025\pi$
$353$	−8423.78	−1.27012	−0.635060	+	0.772462i	$0.719025\pi$
$354$	0	0
$355$	0	0
$356$	−5676.24	−0.845057
$357$	0	0
$358$	−3702.73	−0.546635
$359$	9286.16	1.36519	0.682597	−	0.730795i	$-0.260850\pi$
$359$	9286.16	1.36519	0.682597	+	0.730795i	$0.260850\pi$
$360$	0	0
$361$	−6294.13	−0.917645
$362$	−1203.43	−0.174726
$363$	0	0
$364$	−1298.86	−0.187029
$365$	0	0
$366$	0	0
$367$	152.466	0.0216857	0.0108429	−	0.999941i	$-0.496549\pi$
$367$	152.466	0.0216857	0.0108429	+	0.999941i	$0.496549\pi$
$368$	409.912	0.0580656
$369$	0	0
$370$	0	0
$371$	−7395.20	−1.03488
$372$	0	0
$373$	6828.93	0.947959	0.473979	−	0.880536i	$-0.342817\pi$
$373$	6828.93	0.947959	0.473979	+	0.880536i	$0.342817\pi$
$374$	1725.81	0.238608
$375$	0	0
$376$	−2470.18	−0.338802
$377$	−3482.97	−0.475815
$378$	0	0
$379$	−6963.21	−0.943736	−0.471868	−	0.881669i	$-0.656420\pi$
$379$	−6963.21	−0.943736	−0.471868	+	0.881669i	$0.656420\pi$
$380$	0	0
$381$	0	0
$382$	418.350	0.0560331
$383$	−367.854	−0.0490769	−0.0245385	−	0.999699i	$-0.507812\pi$
$383$	−367.854	−0.0490769	−0.0245385	+	0.999699i	$0.507812\pi$
$384$	0	0
$385$	0	0
$386$	−633.730	−0.0835648
$387$	0	0
$388$	6107.71	0.799154
$389$	5530.31	0.720817	0.360408	−	0.932795i	$-0.382637\pi$
$389$	5530.31	0.720817	0.360408	+	0.932795i	$0.382637\pi$
$390$	0	0
$391$	−2815.10	−0.364107
$392$	5093.81	0.656318
$393$	0	0
$394$	−7273.15	−0.929990
$395$	0	0
$396$	0	0
$397$	9871.38	1.24794	0.623968	−	0.781450i	$-0.285520\pi$
$397$	9871.38	1.24794	0.623968	+	0.781450i	$0.285520\pi$
$398$	−1640.28	−0.206582
$399$	0	0
$400$	0	0
$401$	1901.95	0.236855	0.118427	−	0.992963i	$-0.462215\pi$
$401$	1901.95	0.236855	0.118427	+	0.992963i	$0.462215\pi$
$402$	0	0
$403$	8033.54	0.993000
$404$	6694.76	0.824447
$405$	0	0
$406$	−2606.43	−0.318609
$407$	−9818.07	−1.19573
$408$	0	0
$409$	4211.88	0.509203	0.254601	−	0.967046i	$-0.418056\pi$
$409$	4211.88	0.509203	0.254601	+	0.967046i	$0.418056\pi$
$410$	0	0
$411$	0	0
$412$	653.482	0.0781426
$413$	−290.168	−0.0345719
$414$	0	0
$415$	0	0
$416$	4343.05	0.511864
$417$	0	0
$418$	−1150.29	−0.134599
$419$	16728.5	1.95045	0.975226	−	0.221211i	$-0.0710010\pi$
$419$	16728.5	1.95045	0.975226	+	0.221211i	$0.0710010\pi$
$420$	0	0
$421$	13140.8	1.52125	0.760623	−	0.649194i	$-0.224894\pi$
$421$	13140.8	1.52125	0.760623	+	0.649194i	$0.224894\pi$
$422$	1599.98	0.184563
$423$	0	0
$424$	15409.7	1.76500
$425$	0	0
$426$	0	0
$427$	−8721.35	−0.988421
$428$	−7060.52	−0.797391
$429$	0	0
$430$	0	0
$431$	5801.99	0.648426	0.324213	−	0.945984i	$-0.394900\pi$
$431$	5801.99	0.648426	0.324213	+	0.945984i	$0.394900\pi$
$432$	0	0
$433$	9599.49	1.06541	0.532705	−	0.846301i	$-0.321176\pi$
$433$	9599.49	1.06541	0.532705	+	0.846301i	$0.321176\pi$
$434$	6011.78	0.664919
$435$	0	0
$436$	5981.04	0.656972
$437$	1876.32	0.205393
$438$	0	0
$439$	14037.4	1.52612	0.763062	−	0.646325i	$-0.223695\pi$
$439$	14037.4	1.52612	0.763062	+	0.646325i	$0.223695\pi$
$440$	0	0
$441$	0	0
$442$	−1394.79	−0.150098
$443$	−7130.50	−0.764741	−0.382370	−	0.924009i	$-0.624892\pi$
$443$	−7130.50	−0.764741	−0.382370	+	0.924009i	$0.624892\pi$
$444$	0	0
$445$	0	0
$446$	−7978.41	−0.847060
$447$	0	0
$448$	2811.12	0.296457
$449$	−15299.1	−1.60804	−0.804019	−	0.594604i	$-0.797309\pi$
$449$	−15299.1	−1.60804	−0.804019	+	0.594604i	$0.797309\pi$
$450$	0	0
$451$	10295.5	1.07494
$452$	3446.73	0.358673
$453$	0	0
$454$	−7043.24	−0.728096
$455$	0	0
$456$	0	0
$457$	1920.17	0.196546	0.0982731	−	0.995159i	$-0.468668\pi$
$457$	1920.17	0.196546	0.0982731	+	0.995159i	$0.468668\pi$
$458$	−2667.44	−0.272143
$459$	0	0
$460$	0	0
$461$	−6689.53	−0.675840	−0.337920	−	0.941175i	$-0.609723\pi$
$461$	−6689.53	−0.675840	−0.337920	+	0.941175i	$0.609723\pi$
$462$	0	0
$463$	−15446.2	−1.55042	−0.775209	−	0.631705i	$-0.782356\pi$
$463$	−15446.2	−1.55042	−0.775209	+	0.631705i	$0.782356\pi$
$464$	−769.491	−0.0769887
$465$	0	0
$466$	−10143.9	−1.00838
$467$	6054.68	0.599951	0.299976	−	0.953947i	$-0.403021\pi$
$467$	6054.68	0.599951	0.299976	+	0.953947i	$0.403021\pi$
$468$	0	0
$469$	8721.22	0.858654
$470$	0	0
$471$	0	0
$472$	604.633	0.0589629
$473$	567.826	0.0551980
$474$	0	0
$475$	0	0
$476$	1970.69	0.189761
$477$	0	0
$478$	6455.05	0.617672
$479$	−3056.28	−0.291534	−0.145767	−	0.989319i	$-0.546565\pi$
$479$	−3056.28	−0.291534	−0.145767	+	0.989319i	$0.546565\pi$
$480$	0	0
$481$	7934.88	0.752182
$482$	10829.5	1.02338
$483$	0	0
$484$	2538.48	0.238400
$485$	0	0
$486$	0	0
$487$	2335.83	0.217344	0.108672	−	0.994078i	$-0.465340\pi$
$487$	2335.83	0.217344	0.108672	+	0.994078i	$0.465340\pi$
$488$	18173.0	1.68577
$489$	0	0
$490$	0	0
$491$	−5497.00	−0.505247	−0.252624	−	0.967565i	$-0.581293\pi$
$491$	−5497.00	−0.505247	−0.252624	+	0.967565i	$0.581293\pi$
$492$	0	0
$493$	5284.54	0.482766
$494$	929.654	0.0846703
$495$	0	0
$496$	1774.84	0.160671
$497$	3567.64	0.321993
$498$	0	0
$499$	17483.4	1.56846	0.784231	−	0.620469i	$-0.213058\pi$
$499$	17483.4	1.56846	0.784231	+	0.620469i	$0.213058\pi$
$500$	0	0
$501$	0	0
$502$	−12842.8	−1.14184
$503$	−6699.56	−0.593874	−0.296937	−	0.954897i	$-0.595965\pi$
$503$	−6699.56	−0.593874	−0.296937	+	0.954897i	$0.595965\pi$
$504$	0	0
$505$	0	0
$506$	−3820.89	−0.335691
$507$	0	0
$508$	−8179.89	−0.714417
$509$	−2632.87	−0.229273	−0.114637	−	0.993408i	$-0.536570\pi$
$509$	−2632.87	−0.229273	−0.114637	+	0.993408i	$0.536570\pi$
$510$	0	0
$511$	−7582.63	−0.656430
$512$	1874.14	0.161770
$513$	0	0
$514$	−4223.68	−0.362448
$515$	0	0
$516$	0	0
$517$	−3262.24	−0.277511
$518$	5937.96	0.503666
$519$	0	0
$520$	0	0
$521$	18292.5	1.53821	0.769105	−	0.639122i	$-0.220702\pi$
$521$	18292.5	1.53821	0.769105	+	0.639122i	$0.220702\pi$
$522$	0	0
$523$	−16446.5	−1.37506	−0.687530	−	0.726156i	$-0.741305\pi$
$523$	−16446.5	−1.37506	−0.687530	+	0.726156i	$0.741305\pi$
$524$	11398.7	0.950294
$525$	0	0
$526$	−10319.5	−0.855419
$527$	−12188.9	−1.00751
$528$	0	0
$529$	−5934.45	−0.487750
$530$	0	0
$531$	0	0
$532$	−1313.50	−0.107044
$533$	−8320.77	−0.676196
$534$	0	0
$535$	0	0
$536$	−18172.7	−1.46445
$537$	0	0
$538$	−2891.03	−0.231675
$539$	6727.15	0.537586
$540$	0	0
$541$	13884.4	1.10340	0.551698	−	0.834044i	$-0.313980\pi$
$541$	13884.4	1.10340	0.551698	+	0.834044i	$0.313980\pi$
$542$	−4177.92	−0.331102
$543$	0	0
$544$	−6589.49	−0.519342
$545$	0	0
$546$	0	0
$547$	−6783.10	−0.530209	−0.265104	−	0.964220i	$-0.585406\pi$
$547$	−6783.10	−0.530209	−0.265104	+	0.964220i	$0.585406\pi$
$548$	5511.83	0.429660
$549$	0	0
$550$	0	0
$551$	−3522.26	−0.272329
$552$	0	0
$553$	−3277.74	−0.252050
$554$	−1630.15	−0.125015
$555$	0	0
$556$	−2544.93	−0.194117
$557$	8088.38	0.615288	0.307644	−	0.951501i	$-0.400459\pi$
$557$	8088.38	0.615288	0.307644	+	0.951501i	$0.400459\pi$
$558$	0	0
$559$	−458.912	−0.0347226
$560$	0	0
$561$	0	0
$562$	2800.93	0.210231
$563$	167.599	0.0125461	0.00627307	−	0.999980i	$-0.498003\pi$
$563$	167.599	0.0125461	0.00627307	+	0.999980i	$0.498003\pi$
$564$	0	0
$565$	0	0
$566$	4024.93	0.298905
$567$	0	0
$568$	−7434.02	−0.549163
$569$	−7306.31	−0.538306	−0.269153	−	0.963097i	$-0.586744\pi$
$569$	−7306.31	−0.538306	−0.269153	+	0.963097i	$0.586744\pi$
$570$	0	0
$571$	23222.3	1.70197	0.850985	−	0.525190i	$-0.176006\pi$
$571$	23222.3	1.70197	0.850985	+	0.525190i	$0.176006\pi$
$572$	3574.31	0.261275
$573$	0	0
$574$	−6226.73	−0.452785
$575$	0	0
$576$	0	0
$577$	5447.52	0.393038	0.196519	−	0.980500i	$-0.437036\pi$
$577$	5447.52	0.393038	0.196519	+	0.980500i	$0.437036\pi$
$578$	−6060.67	−0.436143
$579$	0	0
$580$	0	0
$581$	5834.53	0.416622
$582$	0	0
$583$	20350.8	1.44570
$584$	15800.2	1.11955
$585$	0	0
$586$	−2994.62	−0.211104
$587$	−12872.4	−0.905112	−0.452556	−	0.891736i	$-0.649488\pi$
$587$	−12872.4	−0.905112	−0.452556	+	0.891736i	$0.649488\pi$
$588$	0	0
$589$	8124.14	0.568335
$590$	0	0
$591$	0	0
$592$	1753.05	0.121706
$593$	−16563.3	−1.14700	−0.573501	−	0.819205i	$-0.694415\pi$
$593$	−16563.3	−1.14700	−0.573501	+	0.819205i	$0.694415\pi$
$594$	0	0
$595$	0	0
$596$	−16979.9	−1.16699
$597$	0	0
$598$	3088.02	0.211168
$599$	−15762.4	−1.07518	−0.537590	−	0.843207i	$-0.680665\pi$
$599$	−15762.4	−1.07518	−0.537590	+	0.843207i	$0.680665\pi$
$600$	0	0
$601$	−12635.0	−0.857560	−0.428780	−	0.903409i	$-0.641056\pi$
$601$	−12635.0	−0.857560	−0.428780	+	0.903409i	$0.641056\pi$
$602$	−343.420	−0.0232504
$603$	0	0
$604$	8000.53	0.538968
$605$	0	0
$606$	0	0
$607$	19345.7	1.29360	0.646801	−	0.762658i	$-0.276106\pi$
$607$	19345.7	1.29360	0.646801	+	0.762658i	$0.276106\pi$
$608$	4392.03	0.292961
$609$	0	0
$610$	0	0
$611$	2636.52	0.174570
$612$	0	0
$613$	−4801.16	−0.316341	−0.158170	−	0.987412i	$-0.550560\pi$
$613$	−4801.16	−0.316341	−0.158170	+	0.987412i	$0.550560\pi$
$614$	−12270.7	−0.806520
$615$	0	0
$616$	6766.26	0.442566
$617$	−22036.9	−1.43788	−0.718939	−	0.695073i	$-0.755372\pi$
$617$	−22036.9	−1.43788	−0.718939	+	0.695073i	$0.755372\pi$
$618$	0	0
$619$	−1454.10	−0.0944187	−0.0472094	−	0.998885i	$-0.515033\pi$
$619$	−1454.10	−0.0944187	−0.0472094	+	0.998885i	$0.515033\pi$
$620$	0	0
$621$	0	0
$622$	9500.12	0.612412
$623$	11468.9	0.737544
$624$	0	0
$625$	0	0
$626$	1182.26	0.0754836
$627$	0	0
$628$	−3825.99	−0.243111
$629$	−12039.2	−0.763171
$630$	0	0
$631$	2680.50	0.169111	0.0845554	−	0.996419i	$-0.473053\pi$
$631$	2680.50	0.169111	0.0845554	+	0.996419i	$0.473053\pi$
$632$	6829.95	0.429875
$633$	0	0
$634$	13811.8	0.865200
$635$	0	0
$636$	0	0
$637$	−5436.83	−0.338171
$638$	7172.63	0.445089
$639$	0	0
$640$	0	0
$641$	22353.3	1.37738	0.688690	−	0.725056i	$-0.258186\pi$
$641$	22353.3	1.37738	0.688690	+	0.725056i	$0.258186\pi$
$642$	0	0
$643$	13116.7	0.804466	0.402233	−	0.915537i	$-0.368234\pi$
$643$	13116.7	0.804466	0.402233	+	0.915537i	$0.368234\pi$
$644$	−4363.05	−0.266969
$645$	0	0
$646$	−1410.52	−0.0859072
$647$	21272.7	1.29261	0.646303	−	0.763081i	$-0.276314\pi$
$647$	21272.7	1.29261	0.646303	+	0.763081i	$0.276314\pi$
$648$	0	0
$649$	798.510	0.0482962
$650$	0	0
$651$	0	0
$652$	−694.711	−0.0417285
$653$	12979.9	0.777857	0.388929	−	0.921268i	$-0.372845\pi$
$653$	12979.9	0.777857	0.388929	+	0.921268i	$0.372845\pi$
$654$	0	0
$655$	0	0
$656$	−1838.30	−0.109411
$657$	0	0
$658$	1973.00	0.116893
$659$	−29728.2	−1.75728	−0.878640	−	0.477484i	$-0.841549\pi$
$659$	−29728.2	−1.75728	−0.878640	+	0.477484i	$0.841549\pi$
$660$	0	0
$661$	−6938.87	−0.408307	−0.204153	−	0.978939i	$-0.565444\pi$
$661$	−6938.87	−0.408307	−0.204153	+	0.978939i	$0.565444\pi$
$662$	7219.76	0.423873
$663$	0	0
$664$	−12157.6	−0.710554
$665$	0	0
$666$	0	0
$667$	−11699.8	−0.679188
$668$	7155.36	0.414445
$669$	0	0
$670$	0	0
$671$	24000.2	1.38080
$672$	0	0
$673$	−13773.6	−0.788907	−0.394453	−	0.918916i	$-0.629066\pi$
$673$	−13773.6	−0.788907	−0.394453	+	0.918916i	$0.629066\pi$
$674$	−8446.19	−0.482693
$675$	0	0
$676$	8601.51	0.489389
$677$	12160.5	0.690348	0.345174	−	0.938539i	$-0.387820\pi$
$677$	12160.5	0.690348	0.345174	+	0.938539i	$0.387820\pi$
$678$	0	0
$679$	−12340.6	−0.697481
$680$	0	0
$681$	0	0
$682$	−16543.8	−0.928877
$683$	−7397.68	−0.414443	−0.207221	−	0.978294i	$-0.566442\pi$
$683$	−7397.68	−0.414443	−0.207221	+	0.978294i	$0.566442\pi$
$684$	0	0
$685$	0	0
$686$	−10101.0	−0.562186
$687$	0	0
$688$	−101.387	−0.00561824
$689$	−16447.3	−0.909425
$690$	0	0
$691$	−22677.5	−1.24847	−0.624235	−	0.781237i	$-0.714589\pi$
$691$	−22677.5	−1.24847	−0.624235	+	0.781237i	$0.714589\pi$
$692$	11073.9	0.608333
$693$	0	0
$694$	−7547.84	−0.412842
$695$	0	0
$696$	0	0
$697$	12624.7	0.686075
$698$	3468.04	0.188062
$699$	0	0
$700$	0	0
$701$	18577.8	1.00096	0.500479	−	0.865748i	$-0.333157\pi$
$701$	18577.8	1.00096	0.500479	+	0.865748i	$0.333157\pi$
$702$	0	0
$703$	8024.38	0.430505
$704$	−7735.90	−0.414144
$705$	0	0
$706$	−14020.1	−0.747382
$707$	−13526.8	−0.719557
$708$	0	0
$709$	9783.86	0.518252	0.259126	−	0.965844i	$-0.416566\pi$
$709$	9783.86	0.518252	0.259126	+	0.965844i	$0.416566\pi$
$710$	0	0
$711$	0	0
$712$	−23898.1	−1.25789
$713$	26985.8	1.41743
$714$	0	0
$715$	0	0
$716$	11635.3	0.607307
$717$	0	0
$718$	15455.4	0.803327
$719$	15405.6	0.799073	0.399537	−	0.916717i	$-0.369171\pi$
$719$	15405.6	0.799073	0.399537	+	0.916717i	$0.369171\pi$
$720$	0	0
$721$	−1320.36	−0.0682009
$722$	−10475.6	−0.539973
$723$	0	0
$724$	3781.61	0.194119
$725$	0	0
$726$	0	0
$727$	14748.9	0.752415	0.376207	−	0.926536i	$-0.377228\pi$
$727$	14748.9	0.752415	0.376207	+	0.926536i	$0.377228\pi$
$728$	−5468.44	−0.278398
$729$	0	0
$730$	0	0
$731$	696.284	0.0352298
$732$	0	0
$733$	22799.2	1.14885	0.574425	−	0.818557i	$-0.305226\pi$
$733$	22799.2	1.14885	0.574425	+	0.818557i	$0.305226\pi$
$734$	253.755	0.0127606
$735$	0	0
$736$	14588.9	0.730646
$737$	−23999.9	−1.19952
$738$	0	0
$739$	−23076.7	−1.14870	−0.574350	−	0.818610i	$-0.694745\pi$
$739$	−23076.7	−1.14870	−0.574350	+	0.818610i	$0.694745\pi$
$740$	0	0
$741$	0	0
$742$	−12308.1	−0.608957
$743$	−7898.32	−0.389988	−0.194994	−	0.980804i	$-0.562469\pi$
$743$	−7898.32	−0.389988	−0.194994	+	0.980804i	$0.562469\pi$
$744$	0	0
$745$	0	0
$746$	11365.7	0.557811
$747$	0	0
$748$	−5423.12	−0.265092
$749$	14265.8	0.695942
$750$	0	0
$751$	2974.35	0.144521	0.0722607	−	0.997386i	$-0.476979\pi$
$751$	2974.35	0.144521	0.0722607	+	0.997386i	$0.476979\pi$
$752$	582.484	0.0282460
$753$	0	0
$754$	−5796.86	−0.279986
$755$	0	0
$756$	0	0
$757$	24466.0	1.17468	0.587340	−	0.809340i	$-0.300175\pi$
$757$	24466.0	1.17468	0.587340	+	0.809340i	$0.300175\pi$
$758$	−11589.2	−0.555326
$759$	0	0
$760$	0	0
$761$	−16290.9	−0.776009	−0.388005	−	0.921657i	$-0.626836\pi$
$761$	−16290.9	−0.776009	−0.388005	+	0.921657i	$0.626836\pi$
$762$	0	0
$763$	−12084.7	−0.573389
$764$	−1314.61	−0.0622524
$765$	0	0
$766$	−612.235	−0.0288785
$767$	−645.349	−0.0303810
$768$	0	0
$769$	−24906.0	−1.16792	−0.583962	−	0.811781i	$-0.698498\pi$
$769$	−24906.0	−1.16792	−0.583962	+	0.811781i	$0.698498\pi$
$770$	0	0
$771$	0	0
$772$	1991.41	0.0928399
$773$	21955.7	1.02160	0.510798	−	0.859701i	$-0.329350\pi$
$773$	21955.7	1.02160	0.510798	+	0.859701i	$0.329350\pi$
$774$	0	0
$775$	0	0
$776$	25714.6	1.18956
$777$	0	0
$778$	9204.32	0.424153
$779$	−8414.62	−0.387015
$780$	0	0
$781$	−9817.75	−0.449817
$782$	−4685.29	−0.214253
$783$	0	0
$784$	−1201.16	−0.0547174
$785$	0	0
$786$	0	0
$787$	3382.17	0.153191	0.0765955	−	0.997062i	$-0.475595\pi$
$787$	3382.17	0.153191	0.0765955	+	0.997062i	$0.475595\pi$
$788$	22854.9	1.03321
$789$	0	0
$790$	0	0
$791$	−6964.12	−0.313041
$792$	0	0
$793$	−19396.8	−0.868600
$794$	16429.4	0.734328
$795$	0	0
$796$	5154.35	0.229511
$797$	18016.8	0.800736	0.400368	−	0.916354i	$-0.368882\pi$
$797$	18016.8	0.800736	0.400368	+	0.916354i	$0.368882\pi$
$798$	0	0
$799$	−4000.26	−0.177120
$800$	0	0
$801$	0	0
$802$	3165.49	0.139373
$803$	20866.6	0.917018
$804$	0	0
$805$	0	0
$806$	13370.6	0.584315
$807$	0	0
$808$	28186.3	1.22721
$809$	−18030.0	−0.783559	−0.391780	−	0.920059i	$-0.628140\pi$
$809$	−18030.0	−0.783559	−0.391780	+	0.920059i	$0.628140\pi$
$810$	0	0
$811$	−35444.8	−1.53469	−0.767347	−	0.641233i	$-0.778423\pi$
$811$	−35444.8	−1.53469	−0.767347	+	0.641233i	$0.778423\pi$
$812$	8190.36	0.353972
$813$	0	0
$814$	−16340.6	−0.703610
$815$	0	0
$816$	0	0
$817$	−464.088	−0.0198732
$818$	7010.00	0.299632
$819$	0	0
$820$	0	0
$821$	22609.7	0.961127	0.480563	−	0.876960i	$-0.340432\pi$
$821$	22609.7	0.961127	0.480563	+	0.876960i	$0.340432\pi$
$822$	0	0
$823$	−33852.1	−1.43379	−0.716895	−	0.697181i	$-0.754437\pi$
$823$	−33852.1	−1.43379	−0.716895	+	0.697181i	$0.754437\pi$
$824$	2751.29	0.116318
$825$	0	0
$826$	−482.938	−0.0203433
$827$	−29173.2	−1.22667	−0.613333	−	0.789824i	$-0.710172\pi$
$827$	−29173.2	−1.22667	−0.613333	+	0.789824i	$0.710172\pi$
$828$	0	0
$829$	20552.8	0.861073	0.430537	−	0.902573i	$-0.358324\pi$
$829$	20552.8	0.861073	0.430537	+	0.902573i	$0.358324\pi$
$830$	0	0
$831$	0	0
$832$	6252.09	0.260519
$833$	8249.03	0.343112
$834$	0	0
$835$	0	0
$836$	3614.62	0.149539
$837$	0	0
$838$	27841.9	1.14771
$839$	−13847.0	−0.569789	−0.284895	−	0.958559i	$-0.591959\pi$
$839$	−13847.0	−0.569789	−0.284895	+	0.958559i	$0.591959\pi$
$840$	0	0
$841$	−2425.97	−0.0994696
$842$	21870.8	0.895152
$843$	0	0
$844$	−5027.71	−0.205049
$845$	0	0
$846$	0	0
$847$	−5129.01	−0.208069
$848$	−3633.70	−0.147148
$849$	0	0
$850$	0	0
$851$	26654.4	1.07368
$852$	0	0
$853$	−37952.3	−1.52340	−0.761701	−	0.647929i	$-0.775635\pi$
$853$	−37952.3	−1.52340	−0.761701	+	0.647929i	$0.775635\pi$
$854$	−14515.3	−0.581620
$855$	0	0
$856$	−29726.2	−1.18694
$857$	−5920.24	−0.235976	−0.117988	−	0.993015i	$-0.537644\pi$
$857$	−5920.24	−0.235976	−0.117988	+	0.993015i	$0.537644\pi$
$858$	0	0
$859$	−20480.3	−0.813480	−0.406740	−	0.913544i	$-0.633335\pi$
$859$	−20480.3	−0.813480	−0.406740	+	0.913544i	$0.633335\pi$
$860$	0	0
$861$	0	0
$862$	9656.49	0.381556
$863$	21117.9	0.832980	0.416490	−	0.909140i	$-0.363260\pi$
$863$	21117.9	0.832980	0.416490	+	0.909140i	$0.363260\pi$
$864$	0	0
$865$	0	0
$866$	15976.8	0.626923
$867$	0	0
$868$	−18891.2	−0.738720
$869$	9019.99	0.352108
$870$	0	0
$871$	19396.5	0.754564
$872$	25181.4	0.977923
$873$	0	0
$874$	3122.84	0.120860
$875$	0	0
$876$	0	0
$877$	−28403.2	−1.09362	−0.546812	−	0.837255i	$-0.684159\pi$
$877$	−28403.2	−1.09362	−0.546812	+	0.837255i	$0.684159\pi$
$878$	23363.0	0.898023
$879$	0	0
$880$	0	0
$881$	−14925.4	−0.570772	−0.285386	−	0.958413i	$-0.592122\pi$
$881$	−14925.4	−0.570772	−0.285386	+	0.958413i	$0.592122\pi$
$882$	0	0
$883$	10817.4	0.412271	0.206136	−	0.978523i	$-0.433911\pi$
$883$	10817.4	0.412271	0.206136	+	0.978523i	$0.433911\pi$
$884$	4382.92	0.166758
$885$	0	0
$886$	−11867.6	−0.449999
$887$	14211.3	0.537958	0.268979	−	0.963146i	$-0.413314\pi$
$887$	14211.3	0.537958	0.268979	+	0.963146i	$0.413314\pi$
$888$	0	0
$889$	16527.5	0.623525
$890$	0	0
$891$	0	0
$892$	25071.1	0.941078
$893$	2666.25	0.0999135
$894$	0	0
$895$	0	0
$896$	−10943.4	−0.408029
$897$	0	0
$898$	−25462.9	−0.946223
$899$	−50658.1	−1.87936
$900$	0	0
$901$	24954.7	0.922711
$902$	17135.3	0.632531
$903$	0	0
$904$	14511.4	0.533896
$905$	0	0
$906$	0	0
$907$	28242.5	1.03393	0.516967	−	0.856006i	$-0.327061\pi$
$907$	28242.5	1.03393	0.516967	+	0.856006i	$0.327061\pi$
$908$	22132.4	0.808910
$909$	0	0
$910$	0	0
$911$	30760.6	1.11871	0.559355	−	0.828928i	$-0.311049\pi$
$911$	30760.6	1.11871	0.559355	+	0.828928i	$0.311049\pi$
$912$	0	0
$913$	−16056.0	−0.582011
$914$	3195.81	0.115654
$915$	0	0
$916$	8382.07	0.302349
$917$	−23031.1	−0.829392
$918$	0	0
$919$	9067.61	0.325476	0.162738	−	0.986669i	$-0.447967\pi$
$919$	9067.61	0.325476	0.162738	+	0.986669i	$0.447967\pi$
$920$	0	0
$921$	0	0
$922$	−11133.7	−0.397687
$923$	7934.63	0.282959
$924$	0	0
$925$	0	0
$926$	−25707.7	−0.912318
$927$	0	0
$928$	−27386.5	−0.968757
$929$	−27058.4	−0.955605	−0.477802	−	0.878467i	$-0.658566\pi$
$929$	−27058.4	−0.955605	−0.477802	+	0.878467i	$0.658566\pi$
$930$	0	0
$931$	−5498.15	−0.193550
$932$	31875.8	1.12031
$933$	0	0
$934$	10077.1	0.353032
$935$	0	0
$936$	0	0
$937$	−3320.73	−0.115778	−0.0578888	−	0.998323i	$-0.518437\pi$
$937$	−3320.73	−0.115778	−0.0578888	+	0.998323i	$0.518437\pi$
$938$	14515.1	0.505261
$939$	0	0
$940$	0	0
$941$	13868.0	0.480429	0.240215	−	0.970720i	$-0.422782\pi$
$941$	13868.0	0.480429	0.240215	+	0.970720i	$0.422782\pi$
$942$	0	0
$943$	−27950.7	−0.965217
$944$	−142.577	−0.00491576
$945$	0	0
$946$	945.056	0.0324804
$947$	−30368.2	−1.04206	−0.521032	−	0.853537i	$-0.674453\pi$
$947$	−30368.2	−1.04206	−0.521032	+	0.853537i	$0.674453\pi$
$948$	0	0
$949$	−16864.2	−0.576855
$950$	0	0
$951$	0	0
$952$	8296.99	0.282465
$953$	55107.7	1.87315	0.936576	−	0.350464i	$-0.113976\pi$
$953$	55107.7	1.87315	0.936576	+	0.350464i	$0.113976\pi$
$954$	0	0
$955$	0	0
$956$	−20284.1	−0.686229
$957$	0	0
$958$	−5086.69	−0.171549
$959$	−11136.7	−0.374996
$960$	0	0
$961$	87052.8	2.92212
$962$	13206.4	0.442609
$963$	0	0
$964$	−34030.3	−1.13697
$965$	0	0
$966$	0	0
$967$	2732.45	0.0908684	0.0454342	−	0.998967i	$-0.485533\pi$
$967$	2732.45	0.0908684	0.0454342	+	0.998967i	$0.485533\pi$
$968$	10687.5	0.354865
$969$	0	0
$970$	0	0
$971$	52740.8	1.74308	0.871542	−	0.490321i	$-0.163120\pi$
$971$	52740.8	1.74308	0.871542	+	0.490321i	$0.163120\pi$
$972$	0	0
$973$	5142.04	0.169421
$974$	3887.63	0.127893
$975$	0	0
$976$	−4285.32	−0.140543
$977$	−24987.3	−0.818233	−0.409116	−	0.912482i	$-0.634163\pi$
$977$	−24987.3	−0.818233	−0.409116	+	0.912482i	$0.634163\pi$
$978$	0	0
$979$	−31561.1	−1.03033
$980$	0	0
$981$	0	0
$982$	−9148.89	−0.297304
$983$	11102.8	0.360249	0.180125	−	0.983644i	$-0.442350\pi$
$983$	11102.8	0.360249	0.180125	+	0.983644i	$0.442350\pi$
$984$	0	0
$985$	0	0
$986$	8795.28	0.284076
$987$	0	0
$988$	−2921.31	−0.0940681
$989$	−1541.55	−0.0495637
$990$	0	0
$991$	4790.04	0.153543	0.0767713	−	0.997049i	$-0.475539\pi$
$991$	4790.04	0.153543	0.0767713	+	0.997049i	$0.475539\pi$
$992$	63167.5	2.02174
$993$	0	0
$994$	5937.77	0.189471
$995$	0	0
$996$	0	0
$997$	36275.9	1.15233	0.576164	−	0.817334i	$-0.304549\pi$
$997$	36275.9	1.15233	0.576164	+	0.817334i	$0.304549\pi$
$998$	29098.3	0.922936
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.be.1.9		12
3.2	odd	2		2025.4.a.bi.1.4		12
5.4	even	2		2025.4.a.bj.1.4		12
9.2	odd	6		225.4.e.e.76.9	✓	24
9.5	odd	6		225.4.e.e.151.9	yes	24
15.14	odd	2		2025.4.a.bf.1.9		12
45.2	even	12		225.4.k.e.49.9		48
45.14	odd	6		225.4.e.f.151.4	yes	24
45.23	even	12		225.4.k.e.124.9		48
45.29	odd	6		225.4.e.f.76.4	yes	24
45.32	even	12		225.4.k.e.124.16		48
45.38	even	12		225.4.k.e.49.16		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.9	✓	24	9.2	odd	6
225.4.e.e.151.9	yes	24	9.5	odd	6
225.4.e.f.76.4	yes	24	45.29	odd	6
225.4.e.f.151.4	yes	24	45.14	odd	6
225.4.k.e.49.9		48	45.2	even	12
225.4.k.e.49.16		48	45.38	even	12
225.4.k.e.124.9		48	45.23	even	12
225.4.k.e.124.16		48	45.32	even	12
2025.4.a.be.1.9		12	1.1	even	1	trivial
2025.4.a.bf.1.9		12	15.14	odd	2
2025.4.a.bi.1.4		12	3.2	odd	2
2025.4.a.bj.1.4		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)

\(f(q)\)	\(=\)	\(q+1.66434 q^{2} -5.22997 q^{4} +10.5672 q^{7} -22.0192 q^{8} +O(q^{10})\)	\(q+1.66434 q^{2} -5.22997 q^{4} +10.5672 q^{7} -22.0192 q^{8} -29.0797 q^{11} +23.5020 q^{13} +17.5874 q^{14} +5.19227 q^{16} -35.6583 q^{17} +23.7670 q^{19} -48.3985 q^{22} +78.9465 q^{23} +39.1153 q^{26} -55.2658 q^{28} -148.199 q^{29} +341.824 q^{31} +184.795 q^{32} -59.3477 q^{34} +337.627 q^{37} +39.5565 q^{38} -354.046 q^{41} -19.5265 q^{43} +152.086 q^{44} +131.394 q^{46} +112.183 q^{47} -231.335 q^{49} -122.914 q^{52} -699.829 q^{53} -232.680 q^{56} -246.654 q^{58} -27.4594 q^{59} -825.326 q^{61} +568.912 q^{62} +266.024 q^{64} +825.314 q^{67} +186.492 q^{68} +337.616 q^{71} -717.566 q^{73} +561.926 q^{74} -124.301 q^{76} -307.289 q^{77} -310.182 q^{79} -589.253 q^{82} +552.139 q^{83} -32.4988 q^{86} +640.311 q^{88} +1085.33 q^{89} +248.349 q^{91} -412.888 q^{92} +186.711 q^{94} -1167.83 q^{97} -385.021 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

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