LMFDB - Embedded newform 2025.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,Mod(1,2025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x + 18 $ x^3 - x^2 - 14*x + 18
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32681$ 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)

$f(q)$	$=$	$q+1.32681 q^{2} -6.23958 q^{4} +24.1043 q^{7} -18.8932 q^{8} +O(q^{10})$	$q+1.32681 q^{2} -6.23958 q^{4} +24.1043 q^{7} -18.8932 q^{8} +8.27619 q^{11} -87.1145 q^{13} +31.9818 q^{14} +24.8490 q^{16} -51.9166 q^{17} -88.5107 q^{19} +10.9809 q^{22} +129.245 q^{23} -115.584 q^{26} -150.400 q^{28} -271.109 q^{29} +224.547 q^{31} +184.115 q^{32} -68.8834 q^{34} +70.5268 q^{37} -117.437 q^{38} +366.938 q^{41} +195.547 q^{43} -51.6399 q^{44} +171.483 q^{46} +359.192 q^{47} +238.016 q^{49} +543.558 q^{52} -29.4890 q^{53} -455.407 q^{56} -359.709 q^{58} -858.104 q^{59} -556.811 q^{61} +297.931 q^{62} +45.4941 q^{64} +41.8987 q^{67} +323.938 q^{68} -549.163 q^{71} +185.505 q^{73} +93.5756 q^{74} +552.269 q^{76} +199.492 q^{77} +80.4913 q^{79} +486.857 q^{82} +576.753 q^{83} +259.454 q^{86} -156.364 q^{88} +224.516 q^{89} -2099.83 q^{91} -806.433 q^{92} +476.579 q^{94} +555.016 q^{97} +315.801 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8	$ 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8} - 58 q^{11} + 47 q^{13} - 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 260 q^{22} - 51 q^{23} - 253 q^{26} - 83 q^{28} - 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 414 q^{37} + 397 q^{38} - 179 q^{41} + 836 q^{43} - 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 1335 q^{52} + 505 q^{53} - 15 q^{56} - 1876 q^{58} - 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 40 q^{67} + 830 q^{68} - 452 q^{71} + 710 q^{73} - 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} - 613 q^{82} + 1734 q^{83} - 460 q^{86} + 768 q^{88} + 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 38 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8 - 58 * q^11 + 47 * q^13 - 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 260 * q^22 - 51 * q^23 - 253 * q^26 - 83 * q^28 - 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 414 * q^37 + 397 * q^38 - 179 * q^41 + 836 * q^43 - 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 1335 * q^52 + 505 * q^53 - 15 * q^56 - 1876 * q^58 - 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 40 * q^67 + 830 * q^68 - 452 * q^71 + 710 * q^73 - 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 - 613 * q^82 + 1734 * q^83 - 460 * q^86 + 768 * q^88 + 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 38 * 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 $

$2$	1.32681	0.469098	0.234549	−	0.972104i	$-0.424639\pi$
$2$	1.32681	0.469098	0.234549	+	0.972104i	$0.424639\pi$
$3$	0	0
$3$	0	0
$4$	−6.23958	−0.779947
$5$	0	0
$5$	0	0
$6$	0	0
$7$	24.1043	1.30151	0.650754	−	0.759289i	$-0.274453\pi$
$7$	24.1043	1.30151	0.650754	+	0.759289i	$0.274453\pi$
$8$	−18.8932	−0.834969
$9$	0	0
$10$	0	0
$11$	8.27619	0.226851	0.113426	−	0.993546i	$-0.463818\pi$
$11$	8.27619	0.226851	0.113426	+	0.993546i	$0.463818\pi$
$12$	0	0
$13$	−87.1145	−1.85856	−0.929278	−	0.369382i	$-0.879570\pi$
$13$	−87.1145	−1.85856	−0.929278	+	0.369382i	$0.879570\pi$
$14$	31.9818	0.610534
$15$	0	0
$16$	24.8490	0.388265
$17$	−51.9166	−0.740684	−0.370342	−	0.928895i	$-0.620760\pi$
$17$	−51.9166	−0.740684	−0.370342	+	0.928895i	$0.620760\pi$
$18$	0	0
$19$	−88.5107	−1.06872	−0.534362	−	0.845256i	$-0.679448\pi$
$19$	−88.5107	−1.06872	−0.534362	+	0.845256i	$0.679448\pi$
$20$	0	0
$21$	0	0
$22$	10.9809	0.106415
$23$	129.245	1.17171	0.585856	−	0.810415i	$-0.300758\pi$
$23$	129.245	1.17171	0.585856	+	0.810415i	$0.300758\pi$
$24$	0	0
$25$	0	0
$26$	−115.584	−0.871844
$27$	0	0
$28$	−150.400	−1.01511
$29$	−271.109	−1.73599	−0.867993	−	0.496576i	$-0.834590\pi$
$29$	−271.109	−1.73599	−0.867993	+	0.496576i	$0.834590\pi$
$30$	0	0
$31$	224.547	1.30096	0.650481	−	0.759523i	$-0.274568\pi$
$31$	224.547	1.30096	0.650481	+	0.759523i	$0.274568\pi$
$32$	184.115	1.01710
$33$	0	0
$34$	−68.8834	−0.347453
$35$	0	0
$36$	0	0
$37$	70.5268	0.313366	0.156683	−	0.987649i	$-0.449920\pi$
$37$	70.5268	0.313366	0.156683	+	0.987649i	$0.449920\pi$
$38$	−117.437	−0.501336
$39$	0	0
$40$	0	0
$41$	366.938	1.39771	0.698855	−	0.715263i	$-0.253693\pi$
$41$	366.938	1.39771	0.698855	+	0.715263i	$0.253693\pi$
$42$	0	0
$43$	195.547	0.693504	0.346752	−	0.937957i	$-0.387285\pi$
$43$	195.547	0.693504	0.346752	+	0.937957i	$0.387285\pi$
$44$	−51.6399	−0.176932
$45$	0	0
$46$	171.483	0.549648
$47$	359.192	1.11476	0.557378	−	0.830259i	$-0.311807\pi$
$47$	359.192	1.11476	0.557378	+	0.830259i	$0.311807\pi$
$48$	0	0
$49$	238.016	0.693923
$50$	0	0
$51$	0	0
$52$	543.558	1.44958
$53$	−29.4890	−0.0764270	−0.0382135	−	0.999270i	$-0.512167\pi$
$53$	−29.4890	−0.0764270	−0.0382135	+	0.999270i	$0.512167\pi$
$54$	0	0
$55$	0	0
$56$	−455.407	−1.08672
$57$	0	0
$58$	−359.709	−0.814347
$59$	−858.104	−1.89349	−0.946743	−	0.321991i	$-0.895648\pi$
$59$	−858.104	−1.89349	−0.946743	+	0.321991i	$0.895648\pi$
$60$	0	0
$61$	−556.811	−1.16873	−0.584364	−	0.811492i	$-0.698656\pi$
$61$	−556.811	−1.16873	−0.584364	+	0.811492i	$0.698656\pi$
$62$	297.931	0.610278
$63$	0	0
$64$	45.4941	0.0888557
$65$	0	0
$66$	0	0
$67$	41.8987	0.0763991	0.0381995	−	0.999270i	$-0.487838\pi$
$67$	41.8987	0.0763991	0.0381995	+	0.999270i	$0.487838\pi$
$68$	323.938	0.577695
$69$	0	0
$70$	0	0
$71$	−549.163	−0.917939	−0.458970	−	0.888452i	$-0.651781\pi$
$71$	−549.163	−0.917939	−0.458970	+	0.888452i	$0.651781\pi$
$72$	0	0
$73$	185.505	0.297420	0.148710	−	0.988881i	$-0.452488\pi$
$73$	185.505	0.297420	0.148710	+	0.988881i	$0.452488\pi$
$74$	93.5756	0.146999
$75$	0	0
$76$	552.269	0.833548
$77$	199.492	0.295249
$78$	0	0
$79$	80.4913	0.114633	0.0573163	−	0.998356i	$-0.481746\pi$
$79$	80.4913	0.114633	0.0573163	+	0.998356i	$0.481746\pi$
$80$	0	0
$81$	0	0
$82$	486.857	0.655663
$83$	576.753	0.762734	0.381367	−	0.924424i	$-0.375453\pi$
$83$	576.753	0.762734	0.381367	+	0.924424i	$0.375453\pi$
$84$	0	0
$85$	0	0
$86$	259.454	0.325321
$87$	0	0
$88$	−156.364	−0.189414
$89$	224.516	0.267401	0.133700	−	0.991022i	$-0.457314\pi$
$89$	224.516	0.267401	0.133700	+	0.991022i	$0.457314\pi$
$90$	0	0
$91$	−2099.83	−2.41893
$92$	−806.433	−0.913874
$93$	0	0
$94$	476.579	0.522930
$95$	0	0
$96$	0	0
$97$	555.016	0.580963	0.290481	−	0.956881i	$-0.406185\pi$
$97$	555.016	0.580963	0.290481	+	0.956881i	$0.406185\pi$
$98$	315.801	0.325518
$99$	0	0
$100$	0	0
$101$	227.000	0.223637	0.111818	−	0.993729i	$-0.464333\pi$
$101$	227.000	0.223637	0.111818	+	0.993729i	$0.464333\pi$
$102$	0	0
$103$	−383.524	−0.366891	−0.183446	−	0.983030i	$-0.558725\pi$
$103$	−383.524	−0.366891	−0.183446	+	0.983030i	$0.558725\pi$
$104$	1645.87	1.55184
$105$	0	0
$106$	−39.1263	−0.0358517
$107$	1775.32	1.60399	0.801993	−	0.597334i	$-0.203773\pi$
$107$	1775.32	1.60399	0.801993	+	0.597334i	$0.203773\pi$
$108$	0	0
$109$	1530.50	1.34491	0.672454	−	0.740139i	$-0.265240\pi$
$109$	1530.50	1.34491	0.672454	+	0.740139i	$0.265240\pi$
$110$	0	0
$111$	0	0
$112$	598.966	0.505330
$113$	840.782	0.699948	0.349974	−	0.936759i	$-0.386190\pi$
$113$	840.782	0.699948	0.349974	+	0.936759i	$0.386190\pi$
$114$	0	0
$115$	0	0
$116$	1691.60	1.35398
$117$	0	0
$118$	−1138.54	−0.888230
$119$	−1251.41	−0.964007
$120$	0	0
$121$	−1262.50	−0.948538
$122$	−738.782	−0.548247
$123$	0	0
$124$	−1401.08	−1.01468
$125$	0	0
$126$	0	0
$127$	−1038.45	−0.725572	−0.362786	−	0.931873i	$-0.618174\pi$
$127$	−1038.45	−0.725572	−0.362786	+	0.931873i	$0.618174\pi$
$128$	−1412.56	−0.975422
$129$	0	0
$130$	0	0
$131$	−803.410	−0.535834	−0.267917	−	0.963442i	$-0.586335\pi$
$131$	−803.410	−0.535834	−0.267917	+	0.963442i	$0.586335\pi$
$132$	0	0
$133$	−2133.49	−1.39095
$134$	55.5915	0.0358386
$135$	0	0
$136$	980.871	0.618449
$137$	−869.152	−0.542019	−0.271010	−	0.962577i	$-0.587358\pi$
$137$	−869.152	−0.542019	−0.271010	+	0.962577i	$0.587358\pi$
$138$	0	0
$139$	−196.398	−0.119844	−0.0599218	−	0.998203i	$-0.519085\pi$
$139$	−196.398	−0.119844	−0.0599218	+	0.998203i	$0.519085\pi$
$140$	0	0
$141$	0	0
$142$	−728.635	−0.430603
$143$	−720.976	−0.421616
$144$	0	0
$145$	0	0
$146$	246.129	0.139519
$147$	0	0
$148$	−440.057	−0.244409
$149$	754.283	0.414720	0.207360	−	0.978265i	$-0.433513\pi$
$149$	754.283	0.414720	0.207360	+	0.978265i	$0.433513\pi$
$150$	0	0
$151$	2057.64	1.10893	0.554464	−	0.832208i	$-0.312923\pi$
$151$	2057.64	1.10893	0.554464	+	0.832208i	$0.312923\pi$
$152$	1672.25	0.892351
$153$	0	0
$154$	264.687	0.138501
$155$	0	0
$156$	0	0
$157$	3358.58	1.70729	0.853644	−	0.520857i	$-0.174387\pi$
$157$	3358.58	1.70729	0.853644	+	0.520857i	$0.174387\pi$
$158$	106.797	0.0537739
$159$	0	0
$160$	0	0
$161$	3115.35	1.52499
$162$	0	0
$163$	710.376	0.341356	0.170678	−	0.985327i	$-0.445404\pi$
$163$	710.376	0.341356	0.170678	+	0.985327i	$0.445404\pi$
$164$	−2289.54	−1.09014
$165$	0	0
$166$	765.241	0.357797
$167$	1148.27	0.532070	0.266035	−	0.963963i	$-0.414286\pi$
$167$	1148.27	0.532070	0.266035	+	0.963963i	$0.414286\pi$
$168$	0	0
$169$	5391.94	2.45423
$170$	0	0
$171$	0	0
$172$	−1220.13	−0.540897
$173$	2925.29	1.28558	0.642790	−	0.766042i	$-0.277777\pi$
$173$	2925.29	1.28558	0.642790	+	0.766042i	$0.277777\pi$
$174$	0	0
$175$	0	0
$176$	205.655	0.0880785
$177$	0	0
$178$	297.890	0.125437
$179$	3422.02	1.42890	0.714452	−	0.699685i	$-0.246676\pi$
$179$	3422.02	1.42890	0.714452	+	0.699685i	$0.246676\pi$
$180$	0	0
$181$	−1151.58	−0.472907	−0.236454	−	0.971643i	$-0.575985\pi$
$181$	−1151.58	−0.472907	−0.236454	+	0.971643i	$0.575985\pi$
$182$	−2786.07	−1.13471
$183$	0	0
$184$	−2441.85	−0.978344
$185$	0	0
$186$	0	0
$187$	−429.672	−0.168025
$188$	−2241.21	−0.869451
$189$	0	0
$190$	0	0
$191$	932.120	0.353120	0.176560	−	0.984290i	$-0.443503\pi$
$191$	932.120	0.353120	0.176560	+	0.984290i	$0.443503\pi$
$192$	0	0
$193$	4272.81	1.59359	0.796797	−	0.604247i	$-0.206526\pi$
$193$	4272.81	1.59359	0.796797	+	0.604247i	$0.206526\pi$
$194$	736.400	0.272528
$195$	0	0
$196$	−1485.12	−0.541224
$197$	1924.15	0.695888	0.347944	−	0.937515i	$-0.386880\pi$
$197$	1924.15	0.695888	0.347944	+	0.937515i	$0.386880\pi$
$198$	0	0
$199$	1738.84	0.619414	0.309707	−	0.950832i	$-0.399769\pi$
$199$	1738.84	0.619414	0.309707	+	0.950832i	$0.399769\pi$
$200$	0	0
$201$	0	0
$202$	301.185	0.104907
$203$	−6534.87	−2.25940
$204$	0	0
$205$	0	0
$206$	−508.864	−0.172108
$207$	0	0
$208$	−2164.71	−0.721613
$209$	−732.531	−0.242441
$210$	0	0
$211$	3202.35	1.04483	0.522414	−	0.852692i	$-0.325031\pi$
$211$	3202.35	1.04483	0.522414	+	0.852692i	$0.325031\pi$
$212$	183.999	0.0596090
$213$	0	0
$214$	2355.51	0.752426
$215$	0	0
$216$	0	0
$217$	5412.54	1.69321
$218$	2030.68	0.630893
$219$	0	0
$220$	0	0
$221$	4522.69	1.37660
$222$	0	0
$223$	1404.99	0.421906	0.210953	−	0.977496i	$-0.432343\pi$
$223$	1404.99	0.421906	0.210953	+	0.977496i	$0.432343\pi$
$224$	4437.97	1.32377
$225$	0	0
$226$	1115.56	0.328344
$227$	−6238.35	−1.82402	−0.912012	−	0.410163i	$-0.865472\pi$
$227$	−6238.35	−1.82402	−0.912012	+	0.410163i	$0.865472\pi$
$228$	0	0
$229$	6630.31	1.91329	0.956644	−	0.291259i	$-0.0940742\pi$
$229$	6630.31	1.91329	0.956644	+	0.291259i	$0.0940742\pi$
$230$	0	0
$231$	0	0
$232$	5122.11	1.44949
$233$	−2453.48	−0.689840	−0.344920	−	0.938632i	$-0.612094\pi$
$233$	−2453.48	−0.689840	−0.344920	+	0.938632i	$0.612094\pi$
$234$	0	0
$235$	0	0
$236$	5354.21	1.47682
$237$	0	0
$238$	−1660.38	−0.452213
$239$	−6949.74	−1.88093	−0.940463	−	0.339895i	$-0.889608\pi$
$239$	−6949.74	−1.88093	−0.940463	+	0.339895i	$0.889608\pi$
$240$	0	0
$241$	6335.19	1.69330	0.846651	−	0.532149i	$-0.178615\pi$
$241$	6335.19	1.69330	0.846651	+	0.532149i	$0.178615\pi$
$242$	−1675.10	−0.444957
$243$	0	0
$244$	3474.27	0.911546
$245$	0	0
$246$	0	0
$247$	7710.57	1.98628
$248$	−4242.41	−1.08626
$249$	0	0
$250$	0	0
$251$	−4022.65	−1.01158	−0.505792	−	0.862656i	$-0.668800\pi$
$251$	−4022.65	−1.01158	−0.505792	+	0.862656i	$0.668800\pi$
$252$	0	0
$253$	1069.65	0.265805
$254$	−1377.83	−0.340364
$255$	0	0
$256$	−2238.15	−0.546424
$257$	4997.74	1.21304	0.606519	−	0.795069i	$-0.292565\pi$
$257$	4997.74	1.21304	0.606519	+	0.795069i	$0.292565\pi$
$258$	0	0
$259$	1700.00	0.407848
$260$	0	0
$261$	0	0
$262$	−1065.97	−0.251359
$263$	3992.47	0.936069	0.468035	−	0.883710i	$-0.344962\pi$
$263$	3992.47	0.936069	0.468035	+	0.883710i	$0.344962\pi$
$264$	0	0
$265$	0	0
$266$	−2830.73	−0.652493
$267$	0	0
$268$	−261.430	−0.0595872
$269$	−2188.89	−0.496131	−0.248065	−	0.968743i	$-0.579795\pi$
$269$	−2188.89	−0.496131	−0.248065	+	0.968743i	$0.579795\pi$
$270$	0	0
$271$	4280.26	0.959437	0.479718	−	0.877423i	$-0.340739\pi$
$271$	4280.26	0.959437	0.479718	+	0.877423i	$0.340739\pi$
$272$	−1290.08	−0.287582
$273$	0	0
$274$	−1153.20	−0.254260
$275$	0	0
$276$	0	0
$277$	3879.97	0.841606	0.420803	−	0.907152i	$-0.361748\pi$
$277$	3879.97	0.841606	0.420803	+	0.907152i	$0.361748\pi$
$278$	−260.583	−0.0562184
$279$	0	0
$280$	0	0
$281$	−6397.43	−1.35814	−0.679072	−	0.734071i	$-0.737618\pi$
$281$	−6397.43	−1.35814	−0.679072	+	0.734071i	$0.737618\pi$
$282$	0	0
$283$	342.869	0.0720193	0.0360096	−	0.999351i	$-0.488535\pi$
$283$	342.869	0.0720193	0.0360096	+	0.999351i	$0.488535\pi$
$284$	3426.55	0.715944
$285$	0	0
$286$	−956.598	−0.197779
$287$	8844.78	1.81913
$288$	0	0
$289$	−2217.66	−0.451387
$290$	0	0
$291$	0	0
$292$	−1157.47	−0.231972
$293$	7333.43	1.46220	0.731098	−	0.682272i	$-0.239008\pi$
$293$	7333.43	1.46220	0.731098	+	0.682272i	$0.239008\pi$
$294$	0	0
$295$	0	0
$296$	−1332.48	−0.261651
$297$	0	0
$298$	1000.79	0.194544
$299$	−11259.1	−2.17769
$300$	0	0
$301$	4713.52	0.902601
$302$	2730.09	0.520195
$303$	0	0
$304$	−2199.40	−0.414948
$305$	0	0
$306$	0	0
$307$	7965.33	1.48080	0.740399	−	0.672167i	$-0.234636\pi$
$307$	7965.33	1.48080	0.740399	+	0.672167i	$0.234636\pi$
$308$	−1244.74	−0.230279
$309$	0	0
$310$	0	0
$311$	−2186.19	−0.398609	−0.199305	−	0.979938i	$-0.563868\pi$
$311$	−2186.19	−0.398609	−0.199305	+	0.979938i	$0.563868\pi$
$312$	0	0
$313$	−38.6303	−0.00697609	−0.00348805	−	0.999994i	$-0.501110\pi$
$313$	−38.6303	−0.00697609	−0.00348805	+	0.999994i	$0.501110\pi$
$314$	4456.20	0.800885
$315$	0	0
$316$	−502.232	−0.0894074
$317$	−9535.81	−1.68954	−0.844770	−	0.535129i	$-0.820263\pi$
$317$	−9535.81	−1.68954	−0.844770	+	0.535129i	$0.820263\pi$
$318$	0	0
$319$	−2243.75	−0.393811
$320$	0	0
$321$	0	0
$322$	4133.47	0.715371
$323$	4595.18	0.791587
$324$	0	0
$325$	0	0
$326$	942.533	0.160129
$327$	0	0
$328$	−6932.63	−1.16704
$329$	8658.06	1.45086
$330$	0	0
$331$	−3010.58	−0.499929	−0.249964	−	0.968255i	$-0.580419\pi$
$331$	−3010.58	−0.499929	−0.249964	+	0.968255i	$0.580419\pi$
$332$	−3598.70	−0.594892
$333$	0	0
$334$	1523.53	0.249593
$335$	0	0
$336$	0	0
$337$	−2812.72	−0.454655	−0.227327	−	0.973818i	$-0.572999\pi$
$337$	−2812.72	−0.454655	−0.227327	+	0.973818i	$0.572999\pi$
$338$	7154.07	1.15127
$339$	0	0
$340$	0	0
$341$	1858.39	0.295125
$342$	0	0
$343$	−2530.57	−0.398361
$344$	−3694.51	−0.579054
$345$	0	0
$346$	3881.30	0.603063
$347$	139.354	0.0215588	0.0107794	−	0.999942i	$-0.496569\pi$
$347$	139.354	0.0215588	0.0107794	+	0.999942i	$0.496569\pi$
$348$	0	0
$349$	−5210.29	−0.799141	−0.399571	−	0.916702i	$-0.630841\pi$
$349$	−5210.29	−0.799141	−0.399571	+	0.916702i	$0.630841\pi$
$350$	0	0
$351$	0	0
$352$	1523.77	0.230731
$353$	3410.46	0.514223	0.257112	−	0.966382i	$-0.417229\pi$
$353$	3410.46	0.514223	0.257112	+	0.966382i	$0.417229\pi$
$354$	0	0
$355$	0	0
$356$	−1400.89	−0.208558
$357$	0	0
$358$	4540.36	0.670295
$359$	8131.85	1.19549	0.597747	−	0.801685i	$-0.296063\pi$
$359$	8131.85	1.19549	0.597747	+	0.801685i	$0.296063\pi$
$360$	0	0
$361$	975.143	0.142170
$362$	−1527.93	−0.221840
$363$	0	0
$364$	13102.1	1.88663
$365$	0	0
$366$	0	0
$367$	−132.151	−0.0187963	−0.00939816	−	0.999956i	$-0.502992\pi$
$367$	−132.151	−0.0187963	−0.00939816	+	0.999956i	$0.502992\pi$
$368$	3211.60	0.454935
$369$	0	0
$370$	0	0
$371$	−710.812	−0.0994703
$372$	0	0
$373$	−9348.88	−1.29777	−0.648883	−	0.760888i	$-0.724764\pi$
$373$	−9348.88	−1.29777	−0.648883	+	0.760888i	$0.724764\pi$
$374$	−570.092	−0.0788203
$375$	0	0
$376$	−6786.29	−0.930787
$377$	23617.5	3.22643
$378$	0	0
$379$	6164.17	0.835441	0.417720	−	0.908576i	$-0.362829\pi$
$379$	6164.17	0.835441	0.417720	+	0.908576i	$0.362829\pi$
$380$	0	0
$381$	0	0
$382$	1236.75	0.165648
$383$	2601.27	0.347047	0.173523	−	0.984830i	$-0.444485\pi$
$383$	2601.27	0.347047	0.173523	+	0.984830i	$0.444485\pi$
$384$	0	0
$385$	0	0
$386$	5669.20	0.747552
$387$	0	0
$388$	−3463.07	−0.453120
$389$	−4250.57	−0.554017	−0.277008	−	0.960867i	$-0.589343\pi$
$389$	−4250.57	−0.554017	−0.277008	+	0.960867i	$0.589343\pi$
$390$	0	0
$391$	−6709.95	−0.867869
$392$	−4496.88	−0.579405
$393$	0	0
$394$	2552.98	0.326440
$395$	0	0
$396$	0	0
$397$	6088.35	0.769687	0.384843	−	0.922982i	$-0.374255\pi$
$397$	6088.35	0.769687	0.384843	+	0.922982i	$0.374255\pi$
$398$	2307.11	0.290566
$399$	0	0
$400$	0	0
$401$	−7725.79	−0.962114	−0.481057	−	0.876689i	$-0.659747\pi$
$401$	−7725.79	−0.962114	−0.481057	+	0.876689i	$0.659747\pi$
$402$	0	0
$403$	−19561.3	−2.41791
$404$	−1416.38	−0.174425
$405$	0	0
$406$	−8670.53	−1.05988
$407$	583.693	0.0710875
$408$	0	0
$409$	16321.1	1.97318	0.986588	−	0.163232i	$-0.0521920\pi$
$409$	16321.1	1.97318	0.986588	+	0.163232i	$0.0521920\pi$
$410$	0	0
$411$	0	0
$412$	2393.03	0.286156
$413$	−20684.0	−2.46439
$414$	0	0
$415$	0	0
$416$	−16039.1	−1.89034
$417$	0	0
$418$	−971.929	−0.113729
$419$	−1152.46	−0.134370	−0.0671851	−	0.997741i	$-0.521402\pi$
$419$	−1152.46	−0.134370	−0.0671851	+	0.997741i	$0.521402\pi$
$420$	0	0
$421$	−3531.23	−0.408792	−0.204396	−	0.978888i	$-0.565523\pi$
$421$	−3531.23	−0.408792	−0.204396	+	0.978888i	$0.565523\pi$
$422$	4248.91	0.490127
$423$	0	0
$424$	557.142	0.0638142
$425$	0	0
$426$	0	0
$427$	−13421.5	−1.52111
$428$	−11077.2	−1.25102
$429$	0	0
$430$	0	0
$431$	−10230.6	−1.14337	−0.571683	−	0.820475i	$-0.693709\pi$
$431$	−10230.6	−1.14337	−0.571683	+	0.820475i	$0.693709\pi$
$432$	0	0
$433$	7311.31	0.811453	0.405726	−	0.913995i	$-0.367019\pi$
$433$	7311.31	0.811453	0.405726	+	0.913995i	$0.367019\pi$
$434$	7181.40	0.794282
$435$	0	0
$436$	−9549.65	−1.04896
$437$	−11439.5	−1.25224
$438$	0	0
$439$	7053.18	0.766811	0.383405	−	0.923580i	$-0.374751\pi$
$439$	7053.18	0.766811	0.383405	+	0.923580i	$0.374751\pi$
$440$	0	0
$441$	0	0
$442$	6000.75	0.645761
$443$	−167.790	−0.0179954	−0.00899770	−	0.999960i	$-0.502864\pi$
$443$	−167.790	−0.0179954	−0.00899770	+	0.999960i	$0.502864\pi$
$444$	0	0
$445$	0	0
$446$	1864.15	0.197915
$447$	0	0
$448$	1096.60	0.115646
$449$	−3949.94	−0.415165	−0.207582	−	0.978218i	$-0.566560\pi$
$449$	−3949.94	−0.415165	−0.207582	+	0.978218i	$0.566560\pi$
$450$	0	0
$451$	3036.85	0.317072
$452$	−5246.13	−0.545922
$453$	0	0
$454$	−8277.09	−0.855646
$455$	0	0
$456$	0	0
$457$	7447.60	0.762328	0.381164	−	0.924507i	$-0.375523\pi$
$457$	7447.60	0.762328	0.381164	+	0.924507i	$0.375523\pi$
$458$	8797.15	0.897519
$459$	0	0
$460$	0	0
$461$	−10887.6	−1.09997	−0.549987	−	0.835173i	$-0.685367\pi$
$461$	−10887.6	−1.09997	−0.549987	+	0.835173i	$0.685367\pi$
$462$	0	0
$463$	3171.27	0.318318	0.159159	−	0.987253i	$-0.449122\pi$
$463$	3171.27	0.318318	0.159159	+	0.987253i	$0.449122\pi$
$464$	−6736.77	−0.674023
$465$	0	0
$466$	−3255.29	−0.323602
$467$	−16348.5	−1.61996	−0.809978	−	0.586461i	$-0.800521\pi$
$467$	−16348.5	−1.61996	−0.809978	+	0.586461i	$0.800521\pi$
$468$	0	0
$469$	1009.94	0.0994340
$470$	0	0
$471$	0	0
$472$	16212.3	1.58100
$473$	1618.39	0.157322
$474$	0	0
$475$	0	0
$476$	7808.29	0.751874
$477$	0	0
$478$	−9220.98	−0.882338
$479$	−9413.69	−0.897959	−0.448980	−	0.893542i	$-0.648212\pi$
$479$	−9413.69	−0.897959	−0.448980	+	0.893542i	$0.648212\pi$
$480$	0	0
$481$	−6143.91	−0.582408
$482$	8405.59	0.794324
$483$	0	0
$484$	7877.50	0.739810
$485$	0	0
$486$	0	0
$487$	−13482.3	−1.25450	−0.627250	−	0.778818i	$-0.715820\pi$
$487$	−13482.3	−1.25450	−0.627250	+	0.778818i	$0.715820\pi$
$488$	10519.9	0.975852
$489$	0	0
$490$	0	0
$491$	9219.74	0.847416	0.423708	−	0.905799i	$-0.360728\pi$
$491$	9219.74	0.847416	0.423708	+	0.905799i	$0.360728\pi$
$492$	0	0
$493$	14075.0	1.28582
$494$	10230.4	0.931760
$495$	0	0
$496$	5579.76	0.505118
$497$	−13237.2	−1.19471
$498$	0	0
$499$	104.346	0.00936108	0.00468054	−	0.999989i	$-0.498510\pi$
$499$	104.346	0.00936108	0.00468054	+	0.999989i	$0.498510\pi$
$500$	0	0
$501$	0	0
$502$	−5337.29	−0.474531
$503$	490.652	0.0434933	0.0217466	−	0.999764i	$-0.493077\pi$
$503$	490.652	0.0434933	0.0217466	+	0.999764i	$0.493077\pi$
$504$	0	0
$505$	0	0
$506$	1419.23	0.124688
$507$	0	0
$508$	6479.50	0.565908
$509$	3412.81	0.297191	0.148595	−	0.988898i	$-0.452525\pi$
$509$	3412.81	0.297191	0.148595	+	0.988898i	$0.452525\pi$
$510$	0	0
$511$	4471.45	0.387095
$512$	8330.89	0.719095
$513$	0	0
$514$	6631.05	0.569033
$515$	0	0
$516$	0	0
$517$	2972.74	0.252884
$518$	2255.57	0.191321
$519$	0	0
$520$	0	0
$521$	3486.31	0.293163	0.146582	−	0.989199i	$-0.453173\pi$
$521$	3486.31	0.293163	0.146582	+	0.989199i	$0.453173\pi$
$522$	0	0
$523$	−14465.6	−1.20943	−0.604717	−	0.796440i	$-0.706714\pi$
$523$	−14465.6	−1.20943	−0.604717	+	0.796440i	$0.706714\pi$
$524$	5012.94	0.417923
$525$	0	0
$526$	5297.24	0.439108
$527$	−11657.7	−0.963602
$528$	0	0
$529$	4537.21	0.372911
$530$	0	0
$531$	0	0
$532$	13312.1	1.08487
$533$	−31965.6	−2.59772
$534$	0	0
$535$	0	0
$536$	−791.600	−0.0637909
$537$	0	0
$538$	−2904.24	−0.232734
$539$	1969.86	0.157417
$540$	0	0
$541$	6602.78	0.524724	0.262362	−	0.964970i	$-0.415499\pi$
$541$	6602.78	0.524724	0.262362	+	0.964970i	$0.415499\pi$
$542$	5679.09	0.450070
$543$	0	0
$544$	−9558.65	−0.753353
$545$	0	0
$546$	0	0
$547$	−6358.56	−0.497024	−0.248512	−	0.968629i	$-0.579942\pi$
$547$	−6358.56	−0.497024	−0.248512	+	0.968629i	$0.579942\pi$
$548$	5423.14	0.422746
$549$	0	0
$550$	0	0
$551$	23996.0	1.85529
$552$	0	0
$553$	1940.18	0.149195
$554$	5147.98	0.394795
$555$	0	0
$556$	1225.44	0.0934717
$557$	−20782.3	−1.58092	−0.790461	−	0.612513i	$-0.790159\pi$
$557$	−20782.3	−1.58092	−0.790461	+	0.612513i	$0.790159\pi$
$558$	0	0
$559$	−17035.0	−1.28892
$560$	0	0
$561$	0	0
$562$	−8488.17	−0.637103
$563$	16470.0	1.23291	0.616453	−	0.787392i	$-0.288569\pi$
$563$	16470.0	1.23291	0.616453	+	0.787392i	$0.288569\pi$
$564$	0	0
$565$	0	0
$566$	454.922	0.0337841
$567$	0	0
$568$	10375.4	0.766451
$569$	−5425.39	−0.399726	−0.199863	−	0.979824i	$-0.564050\pi$
$569$	−5425.39	−0.399726	−0.199863	+	0.979824i	$0.564050\pi$
$570$	0	0
$571$	−14689.7	−1.07661	−0.538306	−	0.842749i	$-0.680936\pi$
$571$	−14689.7	−1.07661	−0.538306	+	0.842749i	$0.680936\pi$
$572$	4498.59	0.328838
$573$	0	0
$574$	11735.3	0.853350
$575$	0	0
$576$	0	0
$577$	17933.1	1.29387	0.646935	−	0.762545i	$-0.276051\pi$
$577$	17933.1	1.29387	0.646935	+	0.762545i	$0.276051\pi$
$578$	−2942.41	−0.211745
$579$	0	0
$580$	0	0
$581$	13902.2	0.992704
$582$	0	0
$583$	−244.057	−0.0173376
$584$	−3504.78	−0.248337
$585$	0	0
$586$	9730.06	0.685913
$587$	4801.16	0.337589	0.168795	−	0.985651i	$-0.446013\pi$
$587$	4801.16	0.337589	0.168795	+	0.985651i	$0.446013\pi$
$588$	0	0
$589$	−19874.8	−1.39037
$590$	0	0
$591$	0	0
$592$	1752.52	0.121669
$593$	−16803.8	−1.16366	−0.581830	−	0.813311i	$-0.697663\pi$
$593$	−16803.8	−1.16366	−0.581830	+	0.813311i	$0.697663\pi$
$594$	0	0
$595$	0	0
$596$	−4706.41	−0.323460
$597$	0	0
$598$	−14938.7	−1.02155
$599$	17220.2	1.17462	0.587310	−	0.809362i	$-0.300187\pi$
$599$	17220.2	1.17462	0.587310	+	0.809362i	$0.300187\pi$
$600$	0	0
$601$	21536.7	1.46173	0.730865	−	0.682522i	$-0.239117\pi$
$601$	21536.7	1.46173	0.730865	+	0.682522i	$0.239117\pi$
$602$	6253.94	0.423408
$603$	0	0
$604$	−12838.8	−0.864905
$605$	0	0
$606$	0	0
$607$	24050.9	1.60823	0.804117	−	0.594472i	$-0.202639\pi$
$607$	24050.9	1.60823	0.804117	+	0.594472i	$0.202639\pi$
$608$	−16296.2	−1.08700
$609$	0	0
$610$	0	0
$611$	−31290.8	−2.07184
$612$	0	0
$613$	3554.49	0.234200	0.117100	−	0.993120i	$-0.462640\pi$
$613$	3554.49	0.234200	0.117100	+	0.993120i	$0.462640\pi$
$614$	10568.5	0.694639
$615$	0	0
$616$	−3769.03	−0.246524
$617$	9747.42	0.636007	0.318003	−	0.948090i	$-0.396988\pi$
$617$	9747.42	0.636007	0.318003	+	0.948090i	$0.396988\pi$
$618$	0	0
$619$	−20765.2	−1.34834	−0.674171	−	0.738576i	$-0.735499\pi$
$619$	−20765.2	−1.34834	−0.674171	+	0.738576i	$0.735499\pi$
$620$	0	0
$621$	0	0
$622$	−2900.66	−0.186987
$623$	5411.80	0.348024
$624$	0	0
$625$	0	0
$626$	−51.2551	−0.00327247
$627$	0	0
$628$	−20956.2	−1.33159
$629$	−3661.51	−0.232105
$630$	0	0
$631$	23740.6	1.49778	0.748890	−	0.662694i	$-0.230587\pi$
$631$	23740.6	1.49778	0.748890	+	0.662694i	$0.230587\pi$
$632$	−1520.74	−0.0957147
$633$	0	0
$634$	−12652.2	−0.792560
$635$	0	0
$636$	0	0
$637$	−20734.6	−1.28970
$638$	−2977.02	−0.184736
$639$	0	0
$640$	0	0
$641$	26108.7	1.60879	0.804394	−	0.594097i	$-0.202490\pi$
$641$	26108.7	1.60879	0.804394	+	0.594097i	$0.202490\pi$
$642$	0	0
$643$	24017.9	1.47305	0.736526	−	0.676409i	$-0.236465\pi$
$643$	24017.9	1.47305	0.736526	+	0.676409i	$0.236465\pi$
$644$	−19438.5	−1.18941
$645$	0	0
$646$	6096.92	0.371331
$647$	−18588.3	−1.12949	−0.564747	−	0.825264i	$-0.691026\pi$
$647$	−18588.3	−1.12949	−0.564747	+	0.825264i	$0.691026\pi$
$648$	0	0
$649$	−7101.83	−0.429540
$650$	0	0
$651$	0	0
$652$	−4432.45	−0.266239
$653$	−12171.9	−0.729441	−0.364721	−	0.931117i	$-0.618835\pi$
$653$	−12171.9	−0.729441	−0.364721	+	0.931117i	$0.618835\pi$
$654$	0	0
$655$	0	0
$656$	9118.04	0.542682
$657$	0	0
$658$	11487.6	0.680597
$659$	8589.64	0.507746	0.253873	−	0.967238i	$-0.418295\pi$
$659$	8589.64	0.507746	0.253873	+	0.967238i	$0.418295\pi$
$660$	0	0
$661$	24995.3	1.47081	0.735405	−	0.677628i	$-0.236992\pi$
$661$	24995.3	1.47081	0.735405	+	0.677628i	$0.236992\pi$
$662$	−3994.46	−0.234515
$663$	0	0
$664$	−10896.7	−0.636859
$665$	0	0
$666$	0	0
$667$	−35039.4	−2.03408
$668$	−7164.71	−0.414987
$669$	0	0
$670$	0	0
$671$	−4608.28	−0.265127
$672$	0	0
$673$	2540.17	0.145492	0.0727461	−	0.997350i	$-0.476824\pi$
$673$	2540.17	0.145492	0.0727461	+	0.997350i	$0.476824\pi$
$674$	−3731.94	−0.213277
$675$	0	0
$676$	−33643.4	−1.91417
$677$	−4197.79	−0.238307	−0.119154	−	0.992876i	$-0.538018\pi$
$677$	−4197.79	−0.238307	−0.119154	+	0.992876i	$0.538018\pi$
$678$	0	0
$679$	13378.3	0.756128
$680$	0	0
$681$	0	0
$682$	2465.73	0.138442
$683$	−8523.02	−0.477488	−0.238744	−	0.971083i	$-0.576736\pi$
$683$	−8523.02	−0.477488	−0.238744	+	0.971083i	$0.576736\pi$
$684$	0	0
$685$	0	0
$686$	−3357.58	−0.186870
$687$	0	0
$688$	4859.15	0.269263
$689$	2568.92	0.142044
$690$	0	0
$691$	−20292.0	−1.11714	−0.558571	−	0.829457i	$-0.688650\pi$
$691$	−20292.0	−1.11714	−0.558571	+	0.829457i	$0.688650\pi$
$692$	−18252.6	−1.00269
$693$	0	0
$694$	184.896	0.0101132
$695$	0	0
$696$	0	0
$697$	−19050.2	−1.03526
$698$	−6913.05	−0.374875
$699$	0	0
$700$	0	0
$701$	11223.6	0.604721	0.302361	−	0.953194i	$-0.402225\pi$
$701$	11223.6	0.604721	0.302361	+	0.953194i	$0.402225\pi$
$702$	0	0
$703$	−6242.38	−0.334901
$704$	376.518	0.0201570
$705$	0	0
$706$	4525.03	0.241221
$707$	5471.66	0.291065
$708$	0	0
$709$	−19931.0	−1.05575	−0.527873	−	0.849324i	$-0.677010\pi$
$709$	−19931.0	−1.05575	−0.527873	+	0.849324i	$0.677010\pi$
$710$	0	0
$711$	0	0
$712$	−4241.83	−0.223271
$713$	29021.5	1.52435
$714$	0	0
$715$	0	0
$716$	−21352.0	−1.11447
$717$	0	0
$718$	10789.4	0.560804
$719$	3186.13	0.165261	0.0826305	−	0.996580i	$-0.473668\pi$
$719$	3186.13	0.165261	0.0826305	+	0.996580i	$0.473668\pi$
$720$	0	0
$721$	−9244.58	−0.477512
$722$	1293.83	0.0666916
$723$	0	0
$724$	7185.37	0.368843
$725$	0	0
$726$	0	0
$727$	23088.2	1.17785	0.588923	−	0.808189i	$-0.299552\pi$
$727$	23088.2	1.17785	0.588923	+	0.808189i	$0.299552\pi$
$728$	39672.5	2.01973
$729$	0	0
$730$	0	0
$731$	−10152.2	−0.513667
$732$	0	0
$733$	29801.4	1.50169	0.750846	−	0.660477i	$-0.229646\pi$
$733$	29801.4	1.50169	0.750846	+	0.660477i	$0.229646\pi$
$734$	−175.340	−0.00881731
$735$	0	0
$736$	23796.0	1.19175
$737$	346.761	0.0173312
$738$	0	0
$739$	39741.5	1.97823	0.989117	−	0.147134i	$-0.0470047\pi$
$739$	39741.5	1.97823	0.989117	+	0.147134i	$0.0470047\pi$
$740$	0	0
$741$	0	0
$742$	−943.111	−0.0466613
$743$	9713.53	0.479616	0.239808	−	0.970820i	$-0.422915\pi$
$743$	9713.53	0.479616	0.239808	+	0.970820i	$0.422915\pi$
$744$	0	0
$745$	0	0
$746$	−12404.2	−0.608779
$747$	0	0
$748$	2680.97	0.131051
$749$	42792.7	2.08760
$750$	0	0
$751$	−2509.28	−0.121924	−0.0609619	−	0.998140i	$-0.519417\pi$
$751$	−2509.28	−0.121924	−0.0609619	+	0.998140i	$0.519417\pi$
$752$	8925.55	0.432821
$753$	0	0
$754$	31335.9	1.51351
$755$	0	0
$756$	0	0
$757$	−9705.73	−0.465998	−0.232999	−	0.972477i	$-0.574854\pi$
$757$	−9705.73	−0.465998	−0.232999	+	0.972477i	$0.574854\pi$
$758$	8178.67	0.391903
$759$	0	0
$760$	0	0
$761$	7778.44	0.370523	0.185262	−	0.982689i	$-0.440687\pi$
$761$	7778.44	0.370523	0.185262	+	0.982689i	$0.440687\pi$
$762$	0	0
$763$	36891.5	1.75041
$764$	−5816.04	−0.275415
$765$	0	0
$766$	3451.39	0.162799
$767$	74753.3	3.51915
$768$	0	0
$769$	1387.90	0.0650833	0.0325416	−	0.999470i	$-0.489640\pi$
$769$	1387.90	0.0650833	0.0325416	+	0.999470i	$0.489640\pi$
$770$	0	0
$771$	0	0
$772$	−26660.5	−1.24292
$773$	20692.0	0.962796	0.481398	−	0.876502i	$-0.340129\pi$
$773$	20692.0	0.962796	0.481398	+	0.876502i	$0.340129\pi$
$774$	0	0
$775$	0	0
$776$	−10486.0	−0.485086
$777$	0	0
$778$	−5639.70	−0.259888
$779$	−32478.0	−1.49377
$780$	0	0
$781$	−4544.98	−0.208236
$782$	−8902.82	−0.407115
$783$	0	0
$784$	5914.45	0.269426
$785$	0	0
$786$	0	0
$787$	−33896.8	−1.53531	−0.767655	−	0.640863i	$-0.778577\pi$
$787$	−33896.8	−1.53531	−0.767655	+	0.640863i	$0.778577\pi$
$788$	−12005.9	−0.542756
$789$	0	0
$790$	0	0
$791$	20266.4	0.910988
$792$	0	0
$793$	48506.4	2.17215
$794$	8078.08	0.361058
$795$	0	0
$796$	−10849.7	−0.483110
$797$	−11954.5	−0.531307	−0.265653	−	0.964069i	$-0.585588\pi$
$797$	−11954.5	−0.531307	−0.265653	+	0.964069i	$0.585588\pi$
$798$	0	0
$799$	−18648.0	−0.825682
$800$	0	0
$801$	0	0
$802$	−10250.7	−0.451325
$803$	1535.27	0.0674702
$804$	0	0
$805$	0	0
$806$	−25954.1	−1.13424
$807$	0	0
$808$	−4288.75	−0.186730
$809$	−27377.3	−1.18978	−0.594891	−	0.803807i	$-0.702805\pi$
$809$	−27377.3	−1.18978	−0.594891	+	0.803807i	$0.702805\pi$
$810$	0	0
$811$	−713.034	−0.0308730	−0.0154365	−	0.999881i	$-0.504914\pi$
$811$	−713.034	−0.0308730	−0.0154365	+	0.999881i	$0.504914\pi$
$812$	40774.9	1.76221
$813$	0	0
$814$	774.449	0.0333470
$815$	0	0
$816$	0	0
$817$	−17308.0	−0.741164
$818$	21655.0	0.925612
$819$	0	0
$820$	0	0
$821$	−6384.12	−0.271385	−0.135693	−	0.990751i	$-0.543326\pi$
$821$	−6384.12	−0.271385	−0.135693	+	0.990751i	$0.543326\pi$
$822$	0	0
$823$	−19636.3	−0.831686	−0.415843	−	0.909436i	$-0.636513\pi$
$823$	−19636.3	−0.831686	−0.415843	+	0.909436i	$0.636513\pi$
$824$	7246.00	0.306343
$825$	0	0
$826$	−27443.7	−1.15604
$827$	43578.4	1.83237	0.916185	−	0.400756i	$-0.131253\pi$
$827$	43578.4	1.83237	0.916185	+	0.400756i	$0.131253\pi$
$828$	0	0
$829$	−20418.8	−0.855458	−0.427729	−	0.903907i	$-0.640686\pi$
$829$	−20418.8	−0.855458	−0.427729	+	0.903907i	$0.640686\pi$
$830$	0	0
$831$	0	0
$832$	−3963.20	−0.165143
$833$	−12357.0	−0.513978
$834$	0	0
$835$	0	0
$836$	4570.69	0.189092
$837$	0	0
$838$	−1529.09	−0.0630328
$839$	10962.0	0.451073	0.225536	−	0.974235i	$-0.427587\pi$
$839$	10962.0	0.451073	0.225536	+	0.974235i	$0.427587\pi$
$840$	0	0
$841$	49110.9	2.01365
$842$	−4685.26	−0.191763
$843$	0	0
$844$	−19981.3	−0.814911
$845$	0	0
$846$	0	0
$847$	−30431.8	−1.23453
$848$	−732.772	−0.0296739
$849$	0	0
$850$	0	0
$851$	9115.22	0.367175
$852$	0	0
$853$	5246.41	0.210590	0.105295	−	0.994441i	$-0.466421\pi$
$853$	5246.41	0.210590	0.105295	+	0.994441i	$0.466421\pi$
$854$	−17807.8	−0.713548
$855$	0	0
$856$	−33541.4	−1.33928
$857$	−27283.1	−1.08748	−0.543741	−	0.839253i	$-0.682993\pi$
$857$	−27283.1	−1.08748	−0.543741	+	0.839253i	$0.682993\pi$
$858$	0	0
$859$	3581.05	0.142240	0.0711198	−	0.997468i	$-0.477343\pi$
$859$	3581.05	0.142240	0.0711198	+	0.997468i	$0.477343\pi$
$860$	0	0
$861$	0	0
$862$	−13574.0	−0.536350
$863$	−41710.9	−1.64526	−0.822629	−	0.568579i	$-0.807493\pi$
$863$	−41710.9	−1.64526	−0.822629	+	0.568579i	$0.807493\pi$
$864$	0	0
$865$	0	0
$866$	9700.71	0.380651
$867$	0	0
$868$	−33771.9	−1.32062
$869$	666.161	0.0260046
$870$	0	0
$871$	−3649.98	−0.141992
$872$	−28916.0	−1.12296
$873$	0	0
$874$	−15178.1	−0.587422
$875$	0	0
$876$	0	0
$877$	−21533.6	−0.829119	−0.414560	−	0.910022i	$-0.636064\pi$
$877$	−21533.6	−0.829119	−0.414560	+	0.910022i	$0.636064\pi$
$878$	9358.22	0.359709
$879$	0	0
$880$	0	0
$881$	30919.1	1.18240	0.591199	−	0.806526i	$-0.298655\pi$
$881$	30919.1	1.18240	0.591199	+	0.806526i	$0.298655\pi$
$882$	0	0
$883$	−5341.72	−0.203582	−0.101791	−	0.994806i	$-0.532457\pi$
$883$	−5341.72	−0.203582	−0.101791	+	0.994806i	$0.532457\pi$
$884$	−28219.7	−1.07368
$885$	0	0
$886$	−222.626	−0.00844160
$887$	−11210.3	−0.424358	−0.212179	−	0.977231i	$-0.568056\pi$
$887$	−11210.3	−0.424358	−0.212179	+	0.977231i	$0.568056\pi$
$888$	0	0
$889$	−25031.1	−0.944337
$890$	0	0
$891$	0	0
$892$	−8766.54	−0.329064
$893$	−31792.3	−1.19137
$894$	0	0
$895$	0	0
$896$	−34048.8	−1.26952
$897$	0	0
$898$	−5240.81	−0.194753
$899$	−60876.6	−2.25845
$900$	0	0
$901$	1530.97	0.0566083
$902$	4029.32	0.148738
$903$	0	0
$904$	−15885.1	−0.584435
$905$	0	0
$906$	0	0
$907$	−20690.8	−0.757473	−0.378737	−	0.925505i	$-0.623641\pi$
$907$	−20690.8	−0.757473	−0.378737	+	0.925505i	$0.623641\pi$
$908$	38924.6	1.42264
$909$	0	0
$910$	0	0
$911$	19471.6	0.708147	0.354073	−	0.935218i	$-0.384796\pi$
$911$	19471.6	0.708147	0.354073	+	0.935218i	$0.384796\pi$
$912$	0	0
$913$	4773.32	0.173027
$914$	9881.53	0.357606
$915$	0	0
$916$	−41370.3	−1.49226
$917$	−19365.6	−0.697393
$918$	0	0
$919$	−30002.0	−1.07690	−0.538452	−	0.842656i	$-0.680991\pi$
$919$	−30002.0	−1.07690	−0.538452	+	0.842656i	$0.680991\pi$
$920$	0	0
$921$	0	0
$922$	−14445.8	−0.515996
$923$	47840.1	1.70604
$924$	0	0
$925$	0	0
$926$	4207.66	0.149322
$927$	0	0
$928$	−49915.3	−1.76568
$929$	22262.8	0.786241	0.393121	−	0.919487i	$-0.371395\pi$
$929$	22262.8	0.786241	0.393121	+	0.919487i	$0.371395\pi$
$930$	0	0
$931$	−21066.9	−0.741612
$932$	15308.7	0.538039
$933$	0	0
$934$	−21691.4	−0.759917
$935$	0	0
$936$	0	0
$937$	−23361.5	−0.814500	−0.407250	−	0.913317i	$-0.633512\pi$
$937$	−23361.5	−0.814500	−0.407250	+	0.913317i	$0.633512\pi$
$938$	1339.99	0.0466443
$939$	0	0
$940$	0	0
$941$	39330.9	1.36254	0.681271	−	0.732031i	$-0.261428\pi$
$941$	39330.9	1.36254	0.681271	+	0.732031i	$0.261428\pi$
$942$	0	0
$943$	47424.8	1.63771
$944$	−21323.0	−0.735175
$945$	0	0
$946$	2147.29	0.0737995
$947$	26869.9	0.922021	0.461011	−	0.887395i	$-0.347487\pi$
$947$	26869.9	0.922021	0.461011	+	0.887395i	$0.347487\pi$
$948$	0	0
$949$	−16160.1	−0.552772
$950$	0	0
$951$	0	0
$952$	23643.2	0.804916
$953$	−16422.6	−0.558218	−0.279109	−	0.960259i	$-0.590039\pi$
$953$	−16422.6	−0.558218	−0.279109	+	0.960259i	$0.590039\pi$
$954$	0	0
$955$	0	0
$956$	43363.5	1.46702
$957$	0	0
$958$	−12490.2	−0.421231
$959$	−20950.3	−0.705442
$960$	0	0
$961$	20630.3	0.692500
$962$	−8151.79	−0.273206
$963$	0	0
$964$	−39528.9	−1.32069
$965$	0	0
$966$	0	0
$967$	−10304.4	−0.342677	−0.171339	−	0.985212i	$-0.554809\pi$
$967$	−10304.4	−0.342677	−0.171339	+	0.985212i	$0.554809\pi$
$968$	23852.7	0.792000
$969$	0	0
$970$	0	0
$971$	−44153.1	−1.45926	−0.729630	−	0.683842i	$-0.760308\pi$
$971$	−44153.1	−1.45926	−0.729630	+	0.683842i	$0.760308\pi$
$972$	0	0
$973$	−4734.03	−0.155977
$974$	−17888.4	−0.588483
$975$	0	0
$976$	−13836.2	−0.453776
$977$	−16115.6	−0.527722	−0.263861	−	0.964561i	$-0.584996\pi$
$977$	−16115.6	−0.527722	−0.263861	+	0.964561i	$0.584996\pi$
$978$	0	0
$979$	1858.14	0.0606602
$980$	0	0
$981$	0	0
$982$	12232.8	0.397521
$983$	−38748.6	−1.25726	−0.628630	−	0.777704i	$-0.716384\pi$
$983$	−38748.6	−1.25726	−0.628630	+	0.777704i	$0.716384\pi$
$984$	0	0
$985$	0	0
$986$	18674.9	0.603174
$987$	0	0
$988$	−48110.7	−1.54920
$989$	25273.5	0.812587
$990$	0	0
$991$	−42906.4	−1.37534	−0.687672	−	0.726022i	$-0.741367\pi$
$991$	−42906.4	−1.37534	−0.687672	+	0.726022i	$0.741367\pi$
$992$	41342.5	1.32321
$993$	0	0
$994$	−17563.2	−0.560434
$995$	0	0
$996$	0	0
$997$	19395.3	0.616104	0.308052	−	0.951369i	$-0.400323\pi$
$997$	19395.3	0.616104	0.308052	+	0.951369i	$0.400323\pi$
$998$	138.447	0.00439126
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.r.1.2		3
3.2	odd	2		2025.4.a.p.1.2		3
5.4	even	2		405.4.a.g.1.2	✓	3
15.14	odd	2		405.4.a.i.1.2	yes	3
45.4	even	6		405.4.e.u.136.2		6
45.14	odd	6		405.4.e.s.136.2		6
45.29	odd	6		405.4.e.s.271.2		6
45.34	even	6		405.4.e.u.271.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.g.1.2	✓	3	5.4	even	2
405.4.a.i.1.2	yes	3	15.14	odd	2
405.4.e.s.136.2		6	45.14	odd	6
405.4.e.s.271.2		6	45.29	odd	6
405.4.e.u.136.2		6	45.4	even	6
405.4.e.u.271.2		6	45.34	even	6
2025.4.a.p.1.2		3	3.2	odd	2
2025.4.a.r.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.32681 q^{2} -6.23958 q^{4} +24.1043 q^{7} -18.8932 q^{8} +O(q^{10})\)	\(q+1.32681 q^{2} -6.23958 q^{4} +24.1043 q^{7} -18.8932 q^{8} +8.27619 q^{11} -87.1145 q^{13} +31.9818 q^{14} +24.8490 q^{16} -51.9166 q^{17} -88.5107 q^{19} +10.9809 q^{22} +129.245 q^{23} -115.584 q^{26} -150.400 q^{28} -271.109 q^{29} +224.547 q^{31} +184.115 q^{32} -68.8834 q^{34} +70.5268 q^{37} -117.437 q^{38} +366.938 q^{41} +195.547 q^{43} -51.6399 q^{44} +171.483 q^{46} +359.192 q^{47} +238.016 q^{49} +543.558 q^{52} -29.4890 q^{53} -455.407 q^{56} -359.709 q^{58} -858.104 q^{59} -556.811 q^{61} +297.931 q^{62} +45.4941 q^{64} +41.8987 q^{67} +323.938 q^{68} -549.163 q^{71} +185.505 q^{73} +93.5756 q^{74} +552.269 q^{76} +199.492 q^{77} +80.4913 q^{79} +486.857 q^{82} +576.753 q^{83} +259.454 q^{86} -156.364 q^{88} +224.516 q^{89} -2099.83 q^{91} -806.433 q^{92} +476.579 q^{94} +555.016 q^{97} +315.801 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8	\( 3 q + q^{2} + 5 q^{4} + 25 q^{7} - 27 q^{8} - 58 q^{11} + 47 q^{13} - 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 260 q^{22} - 51 q^{23} - 253 q^{26} - 83 q^{28} - 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 414 q^{37} + 397 q^{38} - 179 q^{41} + 836 q^{43} - 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 1335 q^{52} + 505 q^{53} - 15 q^{56} - 1876 q^{58} - 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 40 q^{67} + 830 q^{68} - 452 q^{71} + 710 q^{73} - 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} - 613 q^{82} + 1734 q^{83} - 460 q^{86} + 768 q^{88} + 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 38 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 + 25 * q^7 - 27 * q^8 - 58 * q^11 + 47 * q^13 - 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 260 * q^22 - 51 * q^23 - 253 * q^26 - 83 * q^28 - 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 414 * q^37 + 397 * q^38 - 179 * q^41 + 836 * q^43 - 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 1335 * q^52 + 505 * q^53 - 15 * q^56 - 1876 * q^58 - 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 40 * q^67 + 830 * q^68 - 452 * q^71 + 710 * q^73 - 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 - 613 * q^82 + 1734 * q^83 - 460 * q^86 + 768 * q^88 + 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 38 * q^98

Introduction

L-functions

Modular forms

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Database

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