LMFDB - Embedded newform 2025.4.a.be.1.12

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			1.12
Root			$-4.85133$ 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)

$f(q)$	$=$	$q+4.85133 q^{2} +15.5354 q^{4} +16.2385 q^{7} +36.5569 q^{8} +O(q^{10})$	$q+4.85133 q^{2} +15.5354 q^{4} +16.2385 q^{7} +36.5569 q^{8} -59.5185 q^{11} +27.0790 q^{13} +78.7782 q^{14} +53.0662 q^{16} -78.9077 q^{17} -142.290 q^{19} -288.744 q^{22} -85.6281 q^{23} +131.369 q^{26} +252.272 q^{28} -226.009 q^{29} -150.712 q^{31} -35.0132 q^{32} -382.808 q^{34} +234.171 q^{37} -690.298 q^{38} -73.8580 q^{41} +267.077 q^{43} -924.645 q^{44} -415.410 q^{46} -266.025 q^{47} -79.3120 q^{49} +420.684 q^{52} +603.281 q^{53} +593.628 q^{56} -1096.44 q^{58} -508.721 q^{59} +78.3039 q^{61} -731.155 q^{62} -594.391 q^{64} +488.203 q^{67} -1225.87 q^{68} +73.2487 q^{71} +115.831 q^{73} +1136.04 q^{74} -2210.54 q^{76} -966.489 q^{77} +782.743 q^{79} -358.310 q^{82} +271.465 q^{83} +1295.68 q^{86} -2175.81 q^{88} -211.511 q^{89} +439.722 q^{91} -1330.27 q^{92} -1290.58 q^{94} +1666.10 q^{97} -384.769 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.85133	1.71521	0.857603	−	0.514313i	$-0.171953\pi$
$2$	4.85133	1.71521	0.857603	+	0.514313i	$0.171953\pi$
$3$	0	0
$3$	0	0
$4$	15.5354	1.94193
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.2385	0.876795	0.438398	−	0.898781i	$-0.355546\pi$
$7$	16.2385	0.876795	0.438398	+	0.898781i	$0.355546\pi$
$8$	36.5569	1.61560
$9$	0	0
$10$	0	0
$11$	−59.5185	−1.63141	−0.815704	−	0.578469i	$-0.803650\pi$
$11$	−59.5185	−1.63141	−0.815704	+	0.578469i	$0.803650\pi$
$12$	0	0
$13$	27.0790	0.577721	0.288860	−	0.957371i	$-0.406724\pi$
$13$	27.0790	0.577721	0.288860	+	0.957371i	$0.406724\pi$
$14$	78.7782	1.50388
$15$	0	0
$16$	53.0662	0.829160
$17$	−78.9077	−1.12576	−0.562880	−	0.826538i	$-0.690307\pi$
$17$	−78.9077	−1.12576	−0.562880	+	0.826538i	$0.690307\pi$
$18$	0	0
$19$	−142.290	−1.71809	−0.859044	−	0.511902i	$-0.828941\pi$
$19$	−142.290	−1.71809	−0.859044	+	0.511902i	$0.828941\pi$
$20$	0	0
$21$	0	0
$22$	−288.744	−2.79820
$23$	−85.6281	−0.776291	−0.388145	−	0.921598i	$-0.626884\pi$
$23$	−85.6281	−0.776291	−0.388145	+	0.921598i	$0.626884\pi$
$24$	0	0
$25$	0	0
$26$	131.369	0.990910
$27$	0	0
$28$	252.272	1.70267
$29$	−226.009	−1.44720	−0.723600	−	0.690220i	$-0.757514\pi$
$29$	−226.009	−1.44720	−0.723600	+	0.690220i	$0.757514\pi$
$30$	0	0
$31$	−150.712	−0.873184	−0.436592	−	0.899660i	$-0.643815\pi$
$31$	−150.712	−0.873184	−0.436592	+	0.899660i	$0.643815\pi$
$32$	−35.0132	−0.193422
$33$	0	0
$34$	−382.808	−1.93091
$35$	0	0
$36$	0	0
$37$	234.171	1.04047	0.520237	−	0.854022i	$-0.325844\pi$
$37$	234.171	1.04047	0.520237	+	0.854022i	$0.325844\pi$
$38$	−690.298	−2.94687
$39$	0	0
$40$	0	0
$41$	−73.8580	−0.281334	−0.140667	−	0.990057i	$-0.544925\pi$
$41$	−73.8580	−0.281334	−0.140667	+	0.990057i	$0.544925\pi$
$42$	0	0
$43$	267.077	0.947184	0.473592	−	0.880744i	$-0.342957\pi$
$43$	267.077	0.947184	0.473592	+	0.880744i	$0.342957\pi$
$44$	−924.645	−3.16808
$45$	0	0
$46$	−415.410	−1.33150
$47$	−266.025	−0.825612	−0.412806	−	0.910819i	$-0.635451\pi$
$47$	−266.025	−0.825612	−0.412806	+	0.910819i	$0.635451\pi$
$48$	0	0
$49$	−79.3120	−0.231230
$50$	0	0
$51$	0	0
$52$	420.684	1.12189
$53$	603.281	1.56353	0.781765	−	0.623574i	$-0.214320\pi$
$53$	603.281	1.56353	0.781765	+	0.623574i	$0.214320\pi$
$54$	0	0
$55$	0	0
$56$	593.628	1.41655
$57$	0	0
$58$	−1096.44	−2.48224
$59$	−508.721	−1.12254	−0.561270	−	0.827633i	$-0.689687\pi$
$59$	−508.721	−1.12254	−0.561270	+	0.827633i	$0.689687\pi$
$60$	0	0
$61$	78.3039	0.164357	0.0821786	−	0.996618i	$-0.473812\pi$
$61$	78.3039	0.164357	0.0821786	+	0.996618i	$0.473812\pi$
$62$	−731.155	−1.49769
$63$	0	0
$64$	−594.391	−1.16092
$65$	0	0
$66$	0	0
$67$	488.203	0.890201	0.445101	−	0.895481i	$-0.353168\pi$
$67$	488.203	0.890201	0.445101	+	0.895481i	$0.353168\pi$
$68$	−1225.87	−2.18615
$69$	0	0
$70$	0	0
$71$	73.2487	0.122437	0.0612184	−	0.998124i	$-0.480501\pi$
$71$	73.2487	0.122437	0.0612184	+	0.998124i	$0.480501\pi$
$72$	0	0
$73$	115.831	0.185712	0.0928558	−	0.995680i	$-0.470400\pi$
$73$	115.831	0.185712	0.0928558	+	0.995680i	$0.470400\pi$
$74$	1136.04	1.78463
$75$	0	0
$76$	−2210.54	−3.33640
$77$	−966.489	−1.43041
$78$	0	0
$79$	782.743	1.11475	0.557376	−	0.830260i	$-0.311808\pi$
$79$	782.743	1.11475	0.557376	+	0.830260i	$0.311808\pi$
$80$	0	0
$81$	0	0
$82$	−358.310	−0.482545
$83$	271.465	0.359001	0.179501	−	0.983758i	$-0.442552\pi$
$83$	271.465	0.359001	0.179501	+	0.983758i	$0.442552\pi$
$84$	0	0
$85$	0	0
$86$	1295.68	1.62461
$87$	0	0
$88$	−2175.81	−2.63571
$89$	−211.511	−0.251911	−0.125955	−	0.992036i	$-0.540200\pi$
$89$	−211.511	−0.251911	−0.125955	+	0.992036i	$0.540200\pi$
$90$	0	0
$91$	439.722	0.506543
$92$	−1330.27	−1.50750
$93$	0	0
$94$	−1290.58	−1.41609
$95$	0	0
$96$	0	0
$97$	1666.10	1.74399	0.871994	−	0.489516i	$-0.162826\pi$
$97$	1666.10	1.74399	0.871994	+	0.489516i	$0.162826\pi$
$98$	−384.769	−0.396607
$99$	0	0
$100$	0	0
$101$	456.583	0.449819	0.224910	−	0.974380i	$-0.427791\pi$
$101$	456.583	0.449819	0.224910	+	0.974380i	$0.427791\pi$
$102$	0	0
$103$	165.932	0.158736	0.0793678	−	0.996845i	$-0.474710\pi$
$103$	165.932	0.158736	0.0793678	+	0.996845i	$0.474710\pi$
$104$	989.925	0.933367
$105$	0	0
$106$	2926.72	2.68177
$107$	−452.071	−0.408443	−0.204221	−	0.978925i	$-0.565466\pi$
$107$	−452.071	−0.408443	−0.204221	+	0.978925i	$0.565466\pi$
$108$	0	0
$109$	−463.243	−0.407070	−0.203535	−	0.979068i	$-0.565243\pi$
$109$	−463.243	−0.407070	−0.203535	+	0.979068i	$0.565243\pi$
$110$	0	0
$111$	0	0
$112$	861.715	0.727004
$113$	−950.082	−0.790940	−0.395470	−	0.918479i	$-0.629418\pi$
$113$	−950.082	−0.790940	−0.395470	+	0.918479i	$0.629418\pi$
$114$	0	0
$115$	0	0
$116$	−3511.14	−2.81036
$117$	0	0
$118$	−2467.97	−1.92539
$119$	−1281.34	−0.987062
$120$	0	0
$121$	2211.45	1.66149
$122$	379.878	0.281906
$123$	0	0
$124$	−2341.38	−1.69566
$125$	0	0
$126$	0	0
$127$	210.383	0.146996	0.0734979	−	0.997295i	$-0.476584\pi$
$127$	210.383	0.146996	0.0734979	+	0.997295i	$0.476584\pi$
$128$	−2603.48	−1.79779
$129$	0	0
$130$	0	0
$131$	924.776	0.616779	0.308389	−	0.951260i	$-0.400210\pi$
$131$	924.776	0.616779	0.308389	+	0.951260i	$0.400210\pi$
$132$	0	0
$133$	−2310.58	−1.50641
$134$	2368.44	1.52688
$135$	0	0
$136$	−2884.62	−1.81878
$137$	1937.67	1.20837	0.604185	−	0.796844i	$-0.293499\pi$
$137$	1937.67	1.20837	0.604185	+	0.796844i	$0.293499\pi$
$138$	0	0
$139$	−1970.51	−1.20242	−0.601211	−	0.799090i	$-0.705315\pi$
$139$	−1970.51	−1.20242	−0.601211	+	0.799090i	$0.705315\pi$
$140$	0	0
$141$	0	0
$142$	355.354	0.210004
$143$	−1611.70	−0.942498
$144$	0	0
$145$	0	0
$146$	561.933	0.318533
$147$	0	0
$148$	3637.95	2.02053
$149$	−1596.33	−0.877695	−0.438848	−	0.898562i	$-0.644613\pi$
$149$	−1596.33	−0.877695	−0.438848	+	0.898562i	$0.644613\pi$
$150$	0	0
$151$	−3672.97	−1.97948	−0.989741	−	0.142870i	$-0.954367\pi$
$151$	−3672.97	−1.97948	−0.989741	+	0.142870i	$0.954367\pi$
$152$	−5201.70	−2.77575
$153$	0	0
$154$	−4688.76	−2.45345
$155$	0	0
$156$	0	0
$157$	1051.56	0.534544	0.267272	−	0.963621i	$-0.413878\pi$
$157$	1051.56	0.534544	0.267272	+	0.963621i	$0.413878\pi$
$158$	3797.35	1.91203
$159$	0	0
$160$	0	0
$161$	−1390.47	−0.680648
$162$	0	0
$163$	−1998.77	−0.960463	−0.480232	−	0.877142i	$-0.659447\pi$
$163$	−1998.77	−0.960463	−0.480232	+	0.877142i	$0.659447\pi$
$164$	−1147.42	−0.546330
$165$	0	0
$166$	1316.97	0.615761
$167$	−2481.03	−1.14963	−0.574813	−	0.818285i	$-0.694925\pi$
$167$	−2481.03	−1.14963	−0.574813	+	0.818285i	$0.694925\pi$
$168$	0	0
$169$	−1463.73	−0.666239
$170$	0	0
$171$	0	0
$172$	4149.16	1.83936
$173$	2626.07	1.15408	0.577042	−	0.816714i	$-0.304207\pi$
$173$	2626.07	1.15408	0.577042	+	0.816714i	$0.304207\pi$
$174$	0	0
$175$	0	0
$176$	−3158.42	−1.35270
$177$	0	0
$178$	−1026.11	−0.432079
$179$	3231.72	1.34944	0.674720	−	0.738074i	$-0.264264\pi$
$179$	3231.72	1.34944	0.674720	+	0.738074i	$0.264264\pi$
$180$	0	0
$181$	337.854	0.138743	0.0693714	−	0.997591i	$-0.477901\pi$
$181$	337.854	0.138743	0.0693714	+	0.997591i	$0.477901\pi$
$182$	2133.24	0.868825
$183$	0	0
$184$	−3130.30	−1.25418
$185$	0	0
$186$	0	0
$187$	4696.47	1.83658
$188$	−4132.82	−1.60328
$189$	0	0
$190$	0	0
$191$	−1010.51	−0.382815	−0.191407	−	0.981511i	$-0.561305\pi$
$191$	−1010.51	−0.382815	−0.191407	+	0.981511i	$0.561305\pi$
$192$	0	0
$193$	−1492.16	−0.556518	−0.278259	−	0.960506i	$-0.589757\pi$
$193$	−1492.16	−0.556518	−0.278259	+	0.960506i	$0.589757\pi$
$194$	8082.81	2.99130
$195$	0	0
$196$	−1232.15	−0.449033
$197$	3313.50	1.19836	0.599181	−	0.800613i	$-0.295493\pi$
$197$	3313.50	1.19836	0.599181	+	0.800613i	$0.295493\pi$
$198$	0	0
$199$	2165.17	0.771279	0.385640	−	0.922649i	$-0.373981\pi$
$199$	2165.17	0.771279	0.385640	+	0.922649i	$0.373981\pi$
$200$	0	0
$201$	0	0
$202$	2215.04	0.771532
$203$	−3670.04	−1.26890
$204$	0	0
$205$	0	0
$206$	804.991	0.272264
$207$	0	0
$208$	1436.98	0.479023
$209$	8468.91	2.80290
$210$	0	0
$211$	−2559.70	−0.835152	−0.417576	−	0.908642i	$-0.637120\pi$
$211$	−2559.70	−0.835152	−0.417576	+	0.908642i	$0.637120\pi$
$212$	9372.24	3.03626
$213$	0	0
$214$	−2193.15	−0.700564
$215$	0	0
$216$	0	0
$217$	−2447.34	−0.765604
$218$	−2247.34	−0.698208
$219$	0	0
$220$	0	0
$221$	−2136.74	−0.650375
$222$	0	0
$223$	1414.44	0.424743	0.212371	−	0.977189i	$-0.431881\pi$
$223$	1414.44	0.424743	0.212371	+	0.977189i	$0.431881\pi$
$224$	−568.561	−0.169592
$225$	0	0
$226$	−4609.16	−1.35662
$227$	−1627.06	−0.475736	−0.237868	−	0.971298i	$-0.576449\pi$
$227$	−1627.06	−0.475736	−0.237868	+	0.971298i	$0.576449\pi$
$228$	0	0
$229$	−6619.56	−1.91019	−0.955093	−	0.296305i	$-0.904246\pi$
$229$	−6619.56	−1.91019	−0.955093	+	0.296305i	$0.904246\pi$
$230$	0	0
$231$	0	0
$232$	−8262.18	−2.33810
$233$	1911.28	0.537390	0.268695	−	0.963225i	$-0.413408\pi$
$233$	1911.28	0.537390	0.268695	+	0.963225i	$0.413408\pi$
$234$	0	0
$235$	0	0
$236$	−7903.20	−2.17989
$237$	0	0
$238$	−6216.21	−1.69301
$239$	−2886.69	−0.781274	−0.390637	−	0.920545i	$-0.627745\pi$
$239$	−2886.69	−0.781274	−0.390637	+	0.920545i	$0.627745\pi$
$240$	0	0
$241$	−1288.65	−0.344437	−0.172219	−	0.985059i	$-0.555094\pi$
$241$	−1288.65	−0.344437	−0.172219	+	0.985059i	$0.555094\pi$
$242$	10728.5	2.84980
$243$	0	0
$244$	1216.48	0.319170
$245$	0	0
$246$	0	0
$247$	−3853.09	−0.992575
$248$	−5509.57	−1.41072
$249$	0	0
$250$	0	0
$251$	−4356.21	−1.09546	−0.547732	−	0.836654i	$-0.684509\pi$
$251$	−4356.21	−1.09546	−0.547732	+	0.836654i	$0.684509\pi$
$252$	0	0
$253$	5096.45	1.26645
$254$	1020.64	0.252128
$255$	0	0
$256$	−7875.23	−1.92266
$257$	3236.52	0.785559	0.392780	−	0.919633i	$-0.371513\pi$
$257$	3236.52	0.785559	0.392780	+	0.919633i	$0.371513\pi$
$258$	0	0
$259$	3802.58	0.912282
$260$	0	0
$261$	0	0
$262$	4486.40	1.05790
$263$	1381.54	0.323913	0.161956	−	0.986798i	$-0.448220\pi$
$263$	1381.54	0.323913	0.161956	+	0.986798i	$0.448220\pi$
$264$	0	0
$265$	0	0
$266$	−11209.4	−2.58380
$267$	0	0
$268$	7584.45	1.72871
$269$	−721.063	−0.163435	−0.0817175	−	0.996656i	$-0.526041\pi$
$269$	−721.063	−0.163435	−0.0817175	+	0.996656i	$0.526041\pi$
$270$	0	0
$271$	3816.82	0.855554	0.427777	−	0.903884i	$-0.359297\pi$
$271$	3816.82	0.855554	0.427777	+	0.903884i	$0.359297\pi$
$272$	−4187.34	−0.933436
$273$	0	0
$274$	9400.30	2.07260
$275$	0	0
$276$	0	0
$277$	1928.79	0.418374	0.209187	−	0.977876i	$-0.432918\pi$
$277$	1928.79	0.418374	0.209187	+	0.977876i	$0.432918\pi$
$278$	−9559.61	−2.06240
$279$	0	0
$280$	0	0
$281$	−4943.85	−1.04956	−0.524778	−	0.851239i	$-0.675852\pi$
$281$	−4943.85	−1.04956	−0.524778	+	0.851239i	$0.675852\pi$
$282$	0	0
$283$	−6343.58	−1.33246	−0.666231	−	0.745745i	$-0.732094\pi$
$283$	−6343.58	−1.33246	−0.666231	+	0.745745i	$0.732094\pi$
$284$	1137.95	0.237764
$285$	0	0
$286$	−7818.90	−1.61658
$287$	−1199.34	−0.246672
$288$	0	0
$289$	1313.43	0.267337
$290$	0	0
$291$	0	0
$292$	1799.48	0.360639
$293$	−1506.86	−0.300449	−0.150225	−	0.988652i	$-0.548000\pi$
$293$	−1506.86	−0.300449	−0.150225	+	0.988652i	$0.548000\pi$
$294$	0	0
$295$	0	0
$296$	8560.58	1.68099
$297$	0	0
$298$	−7744.34	−1.50543
$299$	−2318.72	−0.448479
$300$	0	0
$301$	4336.93	0.830486
$302$	−17818.8	−3.39522
$303$	0	0
$304$	−7550.82	−1.42457
$305$	0	0
$306$	0	0
$307$	−8239.13	−1.53170	−0.765850	−	0.643019i	$-0.777681\pi$
$307$	−8239.13	−1.53170	−0.765850	+	0.643019i	$0.777681\pi$
$308$	−15014.8	−2.77776
$309$	0	0
$310$	0	0
$311$	805.896	0.146940	0.0734698	−	0.997297i	$-0.476593\pi$
$311$	805.896	0.146940	0.0734698	+	0.997297i	$0.476593\pi$
$312$	0	0
$313$	−4061.41	−0.733432	−0.366716	−	0.930333i	$-0.619518\pi$
$313$	−4061.41	−0.733432	−0.366716	+	0.930333i	$0.619518\pi$
$314$	5101.46	0.916853
$315$	0	0
$316$	12160.2	2.16477
$317$	1007.61	0.178527	0.0892633	−	0.996008i	$-0.471549\pi$
$317$	1007.61	0.178527	0.0892633	+	0.996008i	$0.471549\pi$
$318$	0	0
$319$	13451.7	2.36097
$320$	0	0
$321$	0	0
$322$	−6745.63	−1.16745
$323$	11227.8	1.93416
$324$	0	0
$325$	0	0
$326$	−9696.68	−1.64739
$327$	0	0
$328$	−2700.02	−0.454523
$329$	−4319.85	−0.723893
$330$	0	0
$331$	9294.41	1.54340	0.771702	−	0.635984i	$-0.219405\pi$
$331$	9294.41	1.54340	0.771702	+	0.635984i	$0.219405\pi$
$332$	4217.32	0.697155
$333$	0	0
$334$	−12036.3	−1.97185
$335$	0	0
$336$	0	0
$337$	3395.70	0.548889	0.274445	−	0.961603i	$-0.411506\pi$
$337$	3395.70	0.548889	0.274445	+	0.961603i	$0.411506\pi$
$338$	−7101.02	−1.14274
$339$	0	0
$340$	0	0
$341$	8970.16	1.42452
$342$	0	0
$343$	−6857.70	−1.07954
$344$	9763.52	1.53027
$345$	0	0
$346$	12740.0	1.97949
$347$	−8173.62	−1.26450	−0.632252	−	0.774763i	$-0.717869\pi$
$347$	−8173.62	−1.26450	−0.632252	+	0.774763i	$0.717869\pi$
$348$	0	0
$349$	−5575.91	−0.855219	−0.427609	−	0.903964i	$-0.640644\pi$
$349$	−5575.91	−0.855219	−0.427609	+	0.903964i	$0.640644\pi$
$350$	0	0
$351$	0	0
$352$	2083.93	0.315551
$353$	−3223.77	−0.486074	−0.243037	−	0.970017i	$-0.578144\pi$
$353$	−3223.77	−0.486074	−0.243037	+	0.970017i	$0.578144\pi$
$354$	0	0
$355$	0	0
$356$	−3285.91	−0.489193
$357$	0	0
$358$	15678.1	2.31457
$359$	2593.48	0.381278	0.190639	−	0.981660i	$-0.438944\pi$
$359$	2593.48	0.381278	0.190639	+	0.981660i	$0.438944\pi$
$360$	0	0
$361$	13387.6	1.95182
$362$	1639.04	0.237973
$363$	0	0
$364$	6831.27	0.983670
$365$	0	0
$366$	0	0
$367$	1768.17	0.251492	0.125746	−	0.992062i	$-0.459868\pi$
$367$	1768.17	0.251492	0.125746	+	0.992062i	$0.459868\pi$
$368$	−4543.96	−0.643669
$369$	0	0
$370$	0	0
$371$	9796.37	1.37089
$372$	0	0
$373$	−3029.49	−0.420539	−0.210269	−	0.977643i	$-0.567434\pi$
$373$	−3029.49	−0.420539	−0.210269	+	0.977643i	$0.567434\pi$
$374$	22784.1	3.15010
$375$	0	0
$376$	−9725.06	−1.33386
$377$	−6120.10	−0.836077
$378$	0	0
$379$	9435.84	1.27886	0.639428	−	0.768851i	$-0.279171\pi$
$379$	9435.84	1.27886	0.639428	+	0.768851i	$0.279171\pi$
$380$	0	0
$381$	0	0
$382$	−4902.30	−0.656606
$383$	9698.66	1.29394	0.646969	−	0.762516i	$-0.276036\pi$
$383$	9698.66	1.29394	0.646969	+	0.762516i	$0.276036\pi$
$384$	0	0
$385$	0	0
$386$	−7238.96	−0.954542
$387$	0	0
$388$	25883.6	3.38670
$389$	−12758.7	−1.66296	−0.831481	−	0.555553i	$-0.812507\pi$
$389$	−12758.7	−1.66296	−0.831481	+	0.555553i	$0.812507\pi$
$390$	0	0
$391$	6756.72	0.873918
$392$	−2899.40	−0.373576
$393$	0	0
$394$	16074.9	2.05544
$395$	0	0
$396$	0	0
$397$	14088.8	1.78111	0.890553	−	0.454879i	$-0.150318\pi$
$397$	14088.8	1.78111	0.890553	+	0.454879i	$0.150318\pi$
$398$	10503.9	1.32290
$399$	0	0
$400$	0	0
$401$	7165.27	0.892310	0.446155	−	0.894956i	$-0.352793\pi$
$401$	7165.27	0.892310	0.446155	+	0.894956i	$0.352793\pi$
$402$	0	0
$403$	−4081.14	−0.504457
$404$	7093.22	0.873517
$405$	0	0
$406$	−17804.6	−2.17642
$407$	−13937.5	−1.69744
$408$	0	0
$409$	2039.65	0.246587	0.123293	−	0.992370i	$-0.460654\pi$
$409$	2039.65	0.246587	0.123293	+	0.992370i	$0.460654\pi$
$410$	0	0
$411$	0	0
$412$	2577.83	0.308253
$413$	−8260.85	−0.984237
$414$	0	0
$415$	0	0
$416$	−948.123	−0.111744
$417$	0	0
$418$	41085.5	4.80755
$419$	−13193.7	−1.53831	−0.769156	−	0.639061i	$-0.779323\pi$
$419$	−13193.7	−1.53831	−0.769156	+	0.639061i	$0.779323\pi$
$420$	0	0
$421$	11242.2	1.30145	0.650724	−	0.759315i	$-0.274466\pi$
$421$	11242.2	1.30145	0.650724	+	0.759315i	$0.274466\pi$
$422$	−12418.0	−1.43246
$423$	0	0
$424$	22054.1	2.52604
$425$	0	0
$426$	0	0
$427$	1271.54	0.144108
$428$	−7023.13	−0.793167
$429$	0	0
$430$	0	0
$431$	5546.88	0.619916	0.309958	−	0.950750i	$-0.399685\pi$
$431$	5546.88	0.619916	0.309958	+	0.950750i	$0.399685\pi$
$432$	0	0
$433$	−4484.74	−0.497744	−0.248872	−	0.968536i	$-0.580060\pi$
$433$	−4484.74	−0.497744	−0.248872	+	0.968536i	$0.580060\pi$
$434$	−11872.8	−1.31317
$435$	0	0
$436$	−7196.68	−0.790501
$437$	12184.1	1.33374
$438$	0	0
$439$	5269.72	0.572916	0.286458	−	0.958093i	$-0.407522\pi$
$439$	5269.72	0.572916	0.286458	+	0.958093i	$0.407522\pi$
$440$	0	0
$441$	0	0
$442$	−10366.1	−1.11553
$443$	−7401.33	−0.793787	−0.396894	−	0.917865i	$-0.629912\pi$
$443$	−7401.33	−0.793787	−0.396894	+	0.917865i	$0.629912\pi$
$444$	0	0
$445$	0	0
$446$	6861.90	0.728521
$447$	0	0
$448$	−9652.00	−1.01789
$449$	938.733	0.0986672	0.0493336	−	0.998782i	$-0.484290\pi$
$449$	938.733	0.0986672	0.0493336	+	0.998782i	$0.484290\pi$
$450$	0	0
$451$	4395.91	0.458970
$452$	−14759.9	−1.53595
$453$	0	0
$454$	−7893.43	−0.815985
$455$	0	0
$456$	0	0
$457$	−4307.16	−0.440876	−0.220438	−	0.975401i	$-0.570749\pi$
$457$	−4307.16	−0.440876	−0.220438	+	0.975401i	$0.570749\pi$
$458$	−32113.7	−3.27636
$459$	0	0
$460$	0	0
$461$	−5242.78	−0.529676	−0.264838	−	0.964293i	$-0.585318\pi$
$461$	−5242.78	−0.529676	−0.264838	+	0.964293i	$0.585318\pi$
$462$	0	0
$463$	−8549.55	−0.858167	−0.429083	−	0.903265i	$-0.641163\pi$
$463$	−8549.55	−0.858167	−0.429083	+	0.903265i	$0.641163\pi$
$464$	−11993.4	−1.19996
$465$	0	0
$466$	9272.23	0.921734
$467$	−12706.2	−1.25904	−0.629522	−	0.776983i	$-0.716749\pi$
$467$	−12706.2	−1.25904	−0.629522	+	0.776983i	$0.716749\pi$
$468$	0	0
$469$	7927.67	0.780524
$470$	0	0
$471$	0	0
$472$	−18597.3	−1.81358
$473$	−15896.0	−1.54524
$474$	0	0
$475$	0	0
$476$	−19906.2	−1.91680
$477$	0	0
$478$	−14004.3	−1.34005
$479$	18356.2	1.75097	0.875485	−	0.483246i	$-0.160542\pi$
$479$	18356.2	1.75097	0.875485	+	0.483246i	$0.160542\pi$
$480$	0	0
$481$	6341.13	0.601103
$482$	−6251.68	−0.590780
$483$	0	0
$484$	34355.8	3.22650
$485$	0	0
$486$	0	0
$487$	−15195.3	−1.41389	−0.706945	−	0.707268i	$-0.749927\pi$
$487$	−15195.3	−1.41389	−0.706945	+	0.707268i	$0.749927\pi$
$488$	2862.55	0.265536
$489$	0	0
$490$	0	0
$491$	−7943.91	−0.730150	−0.365075	−	0.930978i	$-0.618957\pi$
$491$	−7943.91	−0.730150	−0.365075	+	0.930978i	$0.618957\pi$
$492$	0	0
$493$	17833.8	1.62920
$494$	−18692.6	−1.70247
$495$	0	0
$496$	−7997.73	−0.724009
$497$	1189.45	0.107352
$498$	0	0
$499$	−19157.9	−1.71869	−0.859343	−	0.511400i	$-0.829127\pi$
$499$	−19157.9	−1.71869	−0.859343	+	0.511400i	$0.829127\pi$
$500$	0	0
$501$	0	0
$502$	−21133.4	−1.87895
$503$	−11203.5	−0.993119	−0.496559	−	0.868003i	$-0.665403\pi$
$503$	−11203.5	−0.993119	−0.496559	+	0.868003i	$0.665403\pi$
$504$	0	0
$505$	0	0
$506$	24724.6	2.17222
$507$	0	0
$508$	3268.39	0.285455
$509$	−7088.04	−0.617233	−0.308617	−	0.951186i	$-0.599866\pi$
$509$	−7088.04	−0.617233	−0.308617	+	0.951186i	$0.599866\pi$
$510$	0	0
$511$	1880.91	0.162831
$512$	−17377.5	−1.49997
$513$	0	0
$514$	15701.5	1.34740
$515$	0	0
$516$	0	0
$517$	15833.4	1.34691
$518$	18447.6	1.56475
$519$	0	0
$520$	0	0
$521$	9739.55	0.818997	0.409499	−	0.912311i	$-0.365704\pi$
$521$	9739.55	0.818997	0.409499	+	0.912311i	$0.365704\pi$
$522$	0	0
$523$	14153.1	1.18331	0.591656	−	0.806190i	$-0.298474\pi$
$523$	14153.1	1.18331	0.591656	+	0.806190i	$0.298474\pi$
$524$	14366.8	1.19774
$525$	0	0
$526$	6702.29	0.555577
$527$	11892.4	0.982996
$528$	0	0
$529$	−4834.83	−0.397373
$530$	0	0
$531$	0	0
$532$	−35895.9	−2.92534
$533$	−2000.00	−0.162532
$534$	0	0
$535$	0	0
$536$	17847.2	1.43821
$537$	0	0
$538$	−3498.12	−0.280324
$539$	4720.52	0.377231
$540$	0	0
$541$	−12175.5	−0.967592	−0.483796	−	0.875181i	$-0.660742\pi$
$541$	−12175.5	−0.967592	−0.483796	+	0.875181i	$0.660742\pi$
$542$	18516.7	1.46745
$543$	0	0
$544$	2762.81	0.217747
$545$	0	0
$546$	0	0
$547$	−1219.23	−0.0953029	−0.0476515	−	0.998864i	$-0.515174\pi$
$547$	−1219.23	−0.0953029	−0.0476515	+	0.998864i	$0.515174\pi$
$548$	30102.6	2.34657
$549$	0	0
$550$	0	0
$551$	32158.9	2.48641
$552$	0	0
$553$	12710.5	0.977409
$554$	9357.18	0.717597
$555$	0	0
$556$	−30612.8	−2.33502
$557$	−12672.4	−0.963995	−0.481998	−	0.876172i	$-0.660089\pi$
$557$	−12672.4	−0.963995	−0.481998	+	0.876172i	$0.660089\pi$
$558$	0	0
$559$	7232.19	0.547208
$560$	0	0
$561$	0	0
$562$	−23984.3	−1.80020
$563$	−8233.59	−0.616349	−0.308175	−	0.951330i	$-0.599718\pi$
$563$	−8233.59	−0.616349	−0.308175	+	0.951330i	$0.599718\pi$
$564$	0	0
$565$	0	0
$566$	−30774.8	−2.28545
$567$	0	0
$568$	2677.75	0.197809
$569$	−3949.10	−0.290958	−0.145479	−	0.989361i	$-0.546472\pi$
$569$	−3949.10	−0.290958	−0.145479	+	0.989361i	$0.546472\pi$
$570$	0	0
$571$	−3910.26	−0.286584	−0.143292	−	0.989680i	$-0.545769\pi$
$571$	−3910.26	−0.286584	−0.143292	+	0.989680i	$0.545769\pi$
$572$	−25038.5	−1.83027
$573$	0	0
$574$	−5818.40	−0.423093
$575$	0	0
$576$	0	0
$577$	18788.9	1.35562	0.677810	−	0.735237i	$-0.262929\pi$
$577$	18788.9	1.35562	0.677810	+	0.735237i	$0.262929\pi$
$578$	6371.88	0.458538
$579$	0	0
$580$	0	0
$581$	4408.17	0.314771
$582$	0	0
$583$	−35906.4	−2.55075
$584$	4234.41	0.300036
$585$	0	0
$586$	−7310.27	−0.515332
$587$	−4400.22	−0.309398	−0.154699	−	0.987962i	$-0.549441\pi$
$587$	−4400.22	−0.309398	−0.154699	+	0.987962i	$0.549441\pi$
$588$	0	0
$589$	21444.9	1.50021
$590$	0	0
$591$	0	0
$592$	12426.6	0.862719
$593$	−13311.2	−0.921798	−0.460899	−	0.887453i	$-0.652473\pi$
$593$	−13311.2	−0.921798	−0.460899	+	0.887453i	$0.652473\pi$
$594$	0	0
$595$	0	0
$596$	−24799.7	−1.70442
$597$	0	0
$598$	−11248.9	−0.769234
$599$	15932.0	1.08675	0.543377	−	0.839489i	$-0.317146\pi$
$599$	15932.0	1.08675	0.543377	+	0.839489i	$0.317146\pi$
$600$	0	0
$601$	23509.9	1.59566	0.797828	−	0.602886i	$-0.205982\pi$
$601$	23509.9	1.59566	0.797828	+	0.602886i	$0.205982\pi$
$602$	21039.9	1.42445
$603$	0	0
$604$	−57061.2	−3.84402
$605$	0	0
$606$	0	0
$607$	9752.01	0.652095	0.326048	−	0.945353i	$-0.394283\pi$
$607$	9752.01	0.652095	0.326048	+	0.945353i	$0.394283\pi$
$608$	4982.04	0.332317
$609$	0	0
$610$	0	0
$611$	−7203.71	−0.476973
$612$	0	0
$613$	24076.3	1.58635	0.793176	−	0.608992i	$-0.208426\pi$
$613$	24076.3	1.58635	0.793176	+	0.608992i	$0.208426\pi$
$614$	−39970.8	−2.62718
$615$	0	0
$616$	−35331.8	−2.31097
$617$	6648.49	0.433806	0.216903	−	0.976193i	$-0.430404\pi$
$617$	6648.49	0.433806	0.216903	+	0.976193i	$0.430404\pi$
$618$	0	0
$619$	−8201.66	−0.532557	−0.266278	−	0.963896i	$-0.585794\pi$
$619$	−8201.66	−0.532557	−0.266278	+	0.963896i	$0.585794\pi$
$620$	0	0
$621$	0	0
$622$	3909.67	0.252031
$623$	−3434.61	−0.220874
$624$	0	0
$625$	0	0
$626$	−19703.2	−1.25799
$627$	0	0
$628$	16336.4	1.03805
$629$	−18477.9	−1.17132
$630$	0	0
$631$	28159.1	1.77654	0.888269	−	0.459323i	$-0.151908\pi$
$631$	28159.1	1.77654	0.888269	+	0.459323i	$0.151908\pi$
$632$	28614.6	1.80100
$633$	0	0
$634$	4888.25	0.306210
$635$	0	0
$636$	0	0
$637$	−2147.69	−0.133586
$638$	65258.6	4.04955
$639$	0	0
$640$	0	0
$641$	−11556.0	−0.712065	−0.356032	−	0.934474i	$-0.615871\pi$
$641$	−11556.0	−0.712065	−0.356032	+	0.934474i	$0.615871\pi$
$642$	0	0
$643$	13803.7	0.846604	0.423302	−	0.905989i	$-0.360871\pi$
$643$	13803.7	0.846604	0.423302	+	0.905989i	$0.360871\pi$
$644$	−21601.5	−1.32177
$645$	0	0
$646$	54469.9	3.31747
$647$	24257.1	1.47395	0.736974	−	0.675921i	$-0.236254\pi$
$647$	24257.1	1.47395	0.736974	+	0.675921i	$0.236254\pi$
$648$	0	0
$649$	30278.3	1.83132
$650$	0	0
$651$	0	0
$652$	−31051.7	−1.86515
$653$	6498.20	0.389425	0.194712	−	0.980860i	$-0.437623\pi$
$653$	6498.20	0.389425	0.194712	+	0.980860i	$0.437623\pi$
$654$	0	0
$655$	0	0
$656$	−3919.37	−0.233271
$657$	0	0
$658$	−20957.0	−1.24162
$659$	−3802.73	−0.224785	−0.112393	−	0.993664i	$-0.535851\pi$
$659$	−3802.73	−0.224785	−0.112393	+	0.993664i	$0.535851\pi$
$660$	0	0
$661$	9674.09	0.569257	0.284628	−	0.958638i	$-0.408130\pi$
$661$	9674.09	0.569257	0.284628	+	0.958638i	$0.408130\pi$
$662$	45090.3	2.64726
$663$	0	0
$664$	9923.91	0.580003
$665$	0	0
$666$	0	0
$667$	19352.7	1.12345
$668$	−38543.8	−2.23249
$669$	0	0
$670$	0	0
$671$	−4660.53	−0.268134
$672$	0	0
$673$	−2032.94	−0.116440	−0.0582198	−	0.998304i	$-0.518542\pi$
$673$	−2032.94	−0.116440	−0.0582198	+	0.998304i	$0.518542\pi$
$674$	16473.7	0.941458
$675$	0	0
$676$	−22739.6	−1.29379
$677$	−8999.00	−0.510871	−0.255435	−	0.966826i	$-0.582219\pi$
$677$	−8999.00	−0.510871	−0.255435	+	0.966826i	$0.582219\pi$
$678$	0	0
$679$	27054.9	1.52912
$680$	0	0
$681$	0	0
$682$	43517.2	2.44334
$683$	23826.0	1.33481	0.667407	−	0.744693i	$-0.267404\pi$
$683$	23826.0	1.33481	0.667407	+	0.744693i	$0.267404\pi$
$684$	0	0
$685$	0	0
$686$	−33269.0	−1.85163
$687$	0	0
$688$	14172.8	0.785367
$689$	16336.3	0.903283
$690$	0	0
$691$	−5878.98	−0.323657	−0.161828	−	0.986819i	$-0.551739\pi$
$691$	−5878.98	−0.323657	−0.161828	+	0.986819i	$0.551739\pi$
$692$	40797.2	2.24115
$693$	0	0
$694$	−39653.0	−2.16888
$695$	0	0
$696$	0	0
$697$	5827.96	0.316714
$698$	−27050.6	−1.46688
$699$	0	0
$700$	0	0
$701$	24961.0	1.34489	0.672443	−	0.740148i	$-0.265245\pi$
$701$	24961.0	1.34489	0.672443	+	0.740148i	$0.265245\pi$
$702$	0	0
$703$	−33320.3	−1.78762
$704$	35377.2	1.89393
$705$	0	0
$706$	−15639.6	−0.833717
$707$	7414.22	0.394399
$708$	0	0
$709$	−11508.9	−0.609625	−0.304813	−	0.952412i	$-0.598594\pi$
$709$	−11508.9	−0.609625	−0.304813	+	0.952412i	$0.598594\pi$
$710$	0	0
$711$	0	0
$712$	−7732.17	−0.406988
$713$	12905.2	0.677845
$714$	0	0
$715$	0	0
$716$	50206.1	2.62052
$717$	0	0
$718$	12581.8	0.653970
$719$	−16503.6	−0.856022	−0.428011	−	0.903773i	$-0.640786\pi$
$719$	−16503.6	−0.856022	−0.428011	+	0.903773i	$0.640786\pi$
$720$	0	0
$721$	2694.48	0.139179
$722$	64947.5	3.34778
$723$	0	0
$724$	5248.70	0.269429
$725$	0	0
$726$	0	0
$727$	38448.0	1.96143	0.980713	−	0.195452i	$-0.0626173\pi$
$727$	38448.0	1.96143	0.980713	+	0.195452i	$0.0626173\pi$
$728$	16074.9	0.818372
$729$	0	0
$730$	0	0
$731$	−21074.5	−1.06630
$732$	0	0
$733$	−34441.4	−1.73550	−0.867750	−	0.497001i	$-0.834435\pi$
$733$	−34441.4	−1.73550	−0.867750	+	0.497001i	$0.834435\pi$
$734$	8577.96	0.431360
$735$	0	0
$736$	2998.11	0.150152
$737$	−29057.1	−1.45228
$738$	0	0
$739$	4619.83	0.229964	0.114982	−	0.993368i	$-0.463319\pi$
$739$	4619.83	0.229964	0.114982	+	0.993368i	$0.463319\pi$
$740$	0	0
$741$	0	0
$742$	47525.4	2.35137
$743$	593.074	0.0292837	0.0146418	−	0.999893i	$-0.495339\pi$
$743$	593.074	0.0292837	0.0146418	+	0.999893i	$0.495339\pi$
$744$	0	0
$745$	0	0
$746$	−14697.1	−0.721310
$747$	0	0
$748$	72961.6	3.56650
$749$	−7340.95	−0.358121
$750$	0	0
$751$	−22199.6	−1.07866	−0.539331	−	0.842094i	$-0.681323\pi$
$751$	−22199.6	−1.07866	−0.539331	+	0.842094i	$0.681323\pi$
$752$	−14117.0	−0.684565
$753$	0	0
$754$	−29690.6	−1.43404
$755$	0	0
$756$	0	0
$757$	−9329.48	−0.447934	−0.223967	−	0.974597i	$-0.571901\pi$
$757$	−9329.48	−0.447934	−0.223967	+	0.974597i	$0.571901\pi$
$758$	45776.4	2.19350
$759$	0	0
$760$	0	0
$761$	−22871.8	−1.08949	−0.544745	−	0.838602i	$-0.683373\pi$
$761$	−22871.8	−1.08949	−0.544745	+	0.838602i	$0.683373\pi$
$762$	0	0
$763$	−7522.35	−0.356917
$764$	−15698.7	−0.743400
$765$	0	0
$766$	47051.4	2.21937
$767$	−13775.7	−0.648514
$768$	0	0
$769$	−12832.3	−0.601749	−0.300874	−	0.953664i	$-0.597279\pi$
$769$	−12832.3	−0.601749	−0.300874	+	0.953664i	$0.597279\pi$
$770$	0	0
$771$	0	0
$772$	−23181.3	−1.08072
$773$	−10821.5	−0.503523	−0.251762	−	0.967789i	$-0.581010\pi$
$773$	−10821.5	−0.503523	−0.251762	+	0.967789i	$0.581010\pi$
$774$	0	0
$775$	0	0
$776$	60907.5	2.81759
$777$	0	0
$778$	−61896.8	−2.85232
$779$	10509.3	0.483356
$780$	0	0
$781$	−4359.65	−0.199745
$782$	32779.1	1.49895
$783$	0	0
$784$	−4208.79	−0.191727
$785$	0	0
$786$	0	0
$787$	13986.3	0.633492	0.316746	−	0.948510i	$-0.397410\pi$
$787$	13986.3	0.633492	0.316746	+	0.948510i	$0.397410\pi$
$788$	51476.7	2.32714
$789$	0	0
$790$	0	0
$791$	−15427.9	−0.693492
$792$	0	0
$793$	2120.39	0.0949525
$794$	68349.7	3.05496
$795$	0	0
$796$	33636.8	1.49777
$797$	−20139.2	−0.895063	−0.447532	−	0.894268i	$-0.647697\pi$
$797$	−20139.2	−0.895063	−0.447532	+	0.894268i	$0.647697\pi$
$798$	0	0
$799$	20991.4	0.929442
$800$	0	0
$801$	0	0
$802$	34761.1	1.53050
$803$	−6894.06	−0.302971
$804$	0	0
$805$	0	0
$806$	−19799.0	−0.865247
$807$	0	0
$808$	16691.3	0.726729
$809$	−4443.65	−0.193115	−0.0965576	−	0.995327i	$-0.530783\pi$
$809$	−4443.65	−0.193115	−0.0965576	+	0.995327i	$0.530783\pi$
$810$	0	0
$811$	3871.68	0.167636	0.0838181	−	0.996481i	$-0.473289\pi$
$811$	3871.68	0.167636	0.0838181	+	0.996481i	$0.473289\pi$
$812$	−57015.6	−2.46411
$813$	0	0
$814$	−67615.5	−2.91145
$815$	0	0
$816$	0	0
$817$	−38002.5	−1.62734
$818$	9895.00	0.422947
$819$	0	0
$820$	0	0
$821$	33193.2	1.41103	0.705513	−	0.708697i	$-0.250717\pi$
$821$	33193.2	1.41103	0.705513	+	0.708697i	$0.250717\pi$
$822$	0	0
$823$	−6144.77	−0.260259	−0.130129	−	0.991497i	$-0.541539\pi$
$823$	−6144.77	−0.260259	−0.130129	+	0.991497i	$0.541539\pi$
$824$	6065.96	0.256454
$825$	0	0
$826$	−40076.1	−1.68817
$827$	138.554	0.00582585	0.00291293	−	0.999996i	$-0.499073\pi$
$827$	138.554	0.00582585	0.00291293	+	0.999996i	$0.499073\pi$
$828$	0	0
$829$	−21615.2	−0.905580	−0.452790	−	0.891617i	$-0.649571\pi$
$829$	−21615.2	−0.905580	−0.452790	+	0.891617i	$0.649571\pi$
$830$	0	0
$831$	0	0
$832$	−16095.5	−0.670687
$833$	6258.33	0.260310
$834$	0	0
$835$	0	0
$836$	131568.	5.44304
$837$	0	0
$838$	−64006.8	−2.63852
$839$	−12969.8	−0.533690	−0.266845	−	0.963739i	$-0.585981\pi$
$839$	−12969.8	−0.533690	−0.266845	+	0.963739i	$0.585981\pi$
$840$	0	0
$841$	26691.0	1.09439
$842$	54539.5	2.23225
$843$	0	0
$844$	−39766.1	−1.62181
$845$	0	0
$846$	0	0
$847$	35910.5	1.45679
$848$	32013.9	1.29642
$849$	0	0
$850$	0	0
$851$	−20051.6	−0.807710
$852$	0	0
$853$	−39544.6	−1.58732	−0.793658	−	0.608364i	$-0.791826\pi$
$853$	−39544.6	−1.58732	−0.793658	+	0.608364i	$0.791826\pi$
$854$	6168.64	0.247174
$855$	0	0
$856$	−16526.3	−0.659881
$857$	−14838.2	−0.591439	−0.295719	−	0.955275i	$-0.595559\pi$
$857$	−14838.2	−0.591439	−0.295719	+	0.955275i	$0.595559\pi$
$858$	0	0
$859$	−24489.7	−0.972733	−0.486366	−	0.873755i	$-0.661678\pi$
$859$	−24489.7	−0.972733	−0.486366	+	0.873755i	$0.661678\pi$
$860$	0	0
$861$	0	0
$862$	26909.8	1.06328
$863$	10016.3	0.395084	0.197542	−	0.980294i	$-0.436704\pi$
$863$	10016.3	0.395084	0.197542	+	0.980294i	$0.436704\pi$
$864$	0	0
$865$	0	0
$866$	−21757.0	−0.853732
$867$	0	0
$868$	−38020.4	−1.48675
$869$	−46587.6	−1.81862
$870$	0	0
$871$	13220.1	0.514288
$872$	−16934.7	−0.657663
$873$	0	0
$874$	59108.9	2.28763
$875$	0	0
$876$	0	0
$877$	33914.1	1.30581	0.652907	−	0.757438i	$-0.273549\pi$
$877$	33914.1	1.30581	0.652907	+	0.757438i	$0.273549\pi$
$878$	25565.2	0.982668
$879$	0	0
$880$	0	0
$881$	−37237.1	−1.42401	−0.712004	−	0.702176i	$-0.752212\pi$
$881$	−37237.1	−1.42401	−0.712004	+	0.702176i	$0.752212\pi$
$882$	0	0
$883$	−36907.3	−1.40660	−0.703302	−	0.710892i	$-0.748292\pi$
$883$	−36907.3	−1.40660	−0.703302	+	0.710892i	$0.748292\pi$
$884$	−33195.2	−1.26298
$885$	0	0
$886$	−35906.3	−1.36151
$887$	−15583.0	−0.589884	−0.294942	−	0.955515i	$-0.595300\pi$
$887$	−15583.0	−0.589884	−0.294942	+	0.955515i	$0.595300\pi$
$888$	0	0
$889$	3416.30	0.128885
$890$	0	0
$891$	0	0
$892$	21973.9	0.824820
$893$	37852.9	1.41847
$894$	0	0
$895$	0	0
$896$	−42276.6	−1.57630
$897$	0	0
$898$	4554.11	0.169234
$899$	34062.3	1.26367
$900$	0	0
$901$	−47603.5	−1.76016
$902$	21326.0	0.787228
$903$	0	0
$904$	−34732.0	−1.27784
$905$	0	0
$906$	0	0
$907$	−27118.6	−0.992788	−0.496394	−	0.868097i	$-0.665343\pi$
$907$	−27118.6	−0.992788	−0.496394	+	0.868097i	$0.665343\pi$
$908$	−25277.1	−0.923845
$909$	0	0
$910$	0	0
$911$	20733.5	0.754042	0.377021	−	0.926205i	$-0.376948\pi$
$911$	20733.5	0.754042	0.377021	+	0.926205i	$0.376948\pi$
$912$	0	0
$913$	−16157.2	−0.585678
$914$	−20895.5	−0.756193
$915$	0	0
$916$	−102838.	−3.70945
$917$	15016.9	0.540789
$918$	0	0
$919$	−48056.4	−1.72495	−0.862477	−	0.506096i	$-0.831088\pi$
$919$	−48056.4	−1.72495	−0.862477	+	0.506096i	$0.831088\pi$
$920$	0	0
$921$	0	0
$922$	−25434.5	−0.908502
$923$	1983.50	0.0707343
$924$	0	0
$925$	0	0
$926$	−41476.7	−1.47193
$927$	0	0
$928$	7913.29	0.279921
$929$	−35677.5	−1.26000	−0.630001	−	0.776595i	$-0.716945\pi$
$929$	−35677.5	−1.26000	−0.630001	+	0.776595i	$0.716945\pi$
$930$	0	0
$931$	11285.3	0.397274
$932$	29692.5	1.04357
$933$	0	0
$934$	−61642.1	−2.15952
$935$	0	0
$936$	0	0
$937$	−33519.2	−1.16865	−0.584324	−	0.811520i	$-0.698640\pi$
$937$	−33519.2	−1.16865	−0.584324	+	0.811520i	$0.698640\pi$
$938$	38459.8	1.33876
$939$	0	0
$940$	0	0
$941$	52275.0	1.81096	0.905482	−	0.424384i	$-0.139509\pi$
$941$	52275.0	1.81096	0.905482	+	0.424384i	$0.139509\pi$
$942$	0	0
$943$	6324.32	0.218397
$944$	−26995.9	−0.930765
$945$	0	0
$946$	−77116.9	−2.65041
$947$	28468.6	0.976881	0.488440	−	0.872597i	$-0.337566\pi$
$947$	28468.6	0.976881	0.488440	+	0.872597i	$0.337566\pi$
$948$	0	0
$949$	3136.58	0.107289
$950$	0	0
$951$	0	0
$952$	−46841.9	−1.59470
$953$	−492.737	−0.0167485	−0.00837424	−	0.999965i	$-0.502666\pi$
$953$	−492.737	−0.0167485	−0.00837424	+	0.999965i	$0.502666\pi$
$954$	0	0
$955$	0	0
$956$	−44846.0	−1.51718
$957$	0	0
$958$	89051.8	3.00327
$959$	31464.9	1.05949
$960$	0	0
$961$	−7076.84	−0.237550
$962$	30762.9	1.03102
$963$	0	0
$964$	−20019.8	−0.668873
$965$	0	0
$966$	0	0
$967$	−23206.3	−0.771732	−0.385866	−	0.922555i	$-0.626097\pi$
$967$	−23206.3	−0.771732	−0.385866	+	0.922555i	$0.626097\pi$
$968$	80843.6	2.68431
$969$	0	0
$970$	0	0
$971$	47923.0	1.58385	0.791927	−	0.610615i	$-0.209078\pi$
$971$	47923.0	1.58385	0.791927	+	0.610615i	$0.209078\pi$
$972$	0	0
$973$	−31998.1	−1.05428
$974$	−73717.5	−2.42511
$975$	0	0
$976$	4155.29	0.136278
$977$	−24328.5	−0.796661	−0.398330	−	0.917242i	$-0.630410\pi$
$977$	−24328.5	−0.796661	−0.398330	+	0.917242i	$0.630410\pi$
$978$	0	0
$979$	12588.8	0.410969
$980$	0	0
$981$	0	0
$982$	−38538.5	−1.25236
$983$	−25372.5	−0.823253	−0.411627	−	0.911353i	$-0.635039\pi$
$983$	−25372.5	−0.823253	−0.411627	+	0.911353i	$0.635039\pi$
$984$	0	0
$985$	0	0
$986$	86517.9	2.79441
$987$	0	0
$988$	−59859.4	−1.92751
$989$	−22869.3	−0.735290
$990$	0	0
$991$	32297.1	1.03527	0.517635	−	0.855602i	$-0.326813\pi$
$991$	32297.1	1.03527	0.517635	+	0.855602i	$0.326813\pi$
$992$	5276.91	0.168893
$993$	0	0
$994$	5770.40	0.184131
$995$	0	0
$996$	0	0
$997$	25457.2	0.808663	0.404332	−	0.914612i	$-0.367504\pi$
$997$	25457.2	0.808663	0.404332	+	0.914612i	$0.367504\pi$
$998$	−92941.2	−2.94790
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.be.1.12		12
3.2	odd	2		2025.4.a.bi.1.1		12
5.4	even	2		2025.4.a.bj.1.1		12
9.2	odd	6		225.4.e.e.76.12	✓	24
9.5	odd	6		225.4.e.e.151.12	yes	24
15.14	odd	2		2025.4.a.bf.1.12		12
45.2	even	12		225.4.k.e.49.2		48
45.14	odd	6		225.4.e.f.151.1	yes	24
45.23	even	12		225.4.k.e.124.2		48
45.29	odd	6		225.4.e.f.76.1	yes	24
45.32	even	12		225.4.k.e.124.23		48
45.38	even	12		225.4.k.e.49.23		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.e.e.76.12	✓	24	9.2	odd	6
225.4.e.e.151.12	yes	24	9.5	odd	6
225.4.e.f.76.1	yes	24	45.29	odd	6
225.4.e.f.151.1	yes	24	45.14	odd	6
225.4.k.e.49.2		48	45.2	even	12
225.4.k.e.49.23		48	45.38	even	12
225.4.k.e.124.2		48	45.23	even	12
225.4.k.e.124.23		48	45.32	even	12
2025.4.a.be.1.12		12	1.1	even	1	trivial
2025.4.a.bf.1.12		12	15.14	odd	2
2025.4.a.bi.1.1		12	3.2	odd	2
2025.4.a.bj.1.1		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)

\(f(q)\)	\(=\)	\(q+4.85133 q^{2} +15.5354 q^{4} +16.2385 q^{7} +36.5569 q^{8} +O(q^{10})\)	\(q+4.85133 q^{2} +15.5354 q^{4} +16.2385 q^{7} +36.5569 q^{8} -59.5185 q^{11} +27.0790 q^{13} +78.7782 q^{14} +53.0662 q^{16} -78.9077 q^{17} -142.290 q^{19} -288.744 q^{22} -85.6281 q^{23} +131.369 q^{26} +252.272 q^{28} -226.009 q^{29} -150.712 q^{31} -35.0132 q^{32} -382.808 q^{34} +234.171 q^{37} -690.298 q^{38} -73.8580 q^{41} +267.077 q^{43} -924.645 q^{44} -415.410 q^{46} -266.025 q^{47} -79.3120 q^{49} +420.684 q^{52} +603.281 q^{53} +593.628 q^{56} -1096.44 q^{58} -508.721 q^{59} +78.3039 q^{61} -731.155 q^{62} -594.391 q^{64} +488.203 q^{67} -1225.87 q^{68} +73.2487 q^{71} +115.831 q^{73} +1136.04 q^{74} -2210.54 q^{76} -966.489 q^{77} +782.743 q^{79} -358.310 q^{82} +271.465 q^{83} +1295.68 q^{86} -2175.81 q^{88} -211.511 q^{89} +439.722 q^{91} -1330.27 q^{92} -1290.58 q^{94} +1666.10 q^{97} -384.769 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) 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

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Database

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