LMFDB - Embedded newform 2025.4.a.z.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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 $ x^6 - 2x^5 - 38x^4 + 42x^3 + 393x^2 - 72*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.57457$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.57457 q^{2} +4.77759 q^{4} -14.1597 q^{7} +11.5188 q^{8} +O(q^{10})$	$q-3.57457 q^{2} +4.77759 q^{4} -14.1597 q^{7} +11.5188 q^{8} -63.6638 q^{11} +10.3959 q^{13} +50.6151 q^{14} -79.3954 q^{16} +108.605 q^{17} +18.6145 q^{19} +227.571 q^{22} -128.824 q^{23} -37.1609 q^{26} -67.6494 q^{28} -83.6496 q^{29} -10.4728 q^{31} +191.655 q^{32} -388.217 q^{34} +81.8885 q^{37} -66.5391 q^{38} +307.453 q^{41} +222.439 q^{43} -304.159 q^{44} +460.491 q^{46} +361.931 q^{47} -142.502 q^{49} +49.6673 q^{52} +562.215 q^{53} -163.103 q^{56} +299.012 q^{58} -462.310 q^{59} -649.067 q^{61} +37.4356 q^{62} -49.9208 q^{64} -254.180 q^{67} +518.870 q^{68} +1092.93 q^{71} -1034.52 q^{73} -292.717 q^{74} +88.9326 q^{76} +901.463 q^{77} +1151.85 q^{79} -1099.01 q^{82} +524.320 q^{83} -795.126 q^{86} -733.328 q^{88} -656.485 q^{89} -147.203 q^{91} -615.467 q^{92} -1293.75 q^{94} -1125.51 q^{97} +509.383 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8	$ 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8} - 88 q^{11} - 20 q^{13} - 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 74 q^{22} + 210 q^{23} - 4 q^{26} - 352 q^{28} - 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 204 q^{37} - 20 q^{38} - 344 q^{41} - 512 q^{43} - 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 468 q^{52} + 850 q^{53} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} - 88 q^{67} + 236 q^{68} - 1364 q^{71} - 836 q^{73} - 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} - 1742 q^{82} + 2148 q^{83} - 2872 q^{86} - 1296 q^{88} - 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} + 612 q^{97} + 1982 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8 - 88 * q^11 - 20 * q^13 - 180 * q^14 + 58 * q^16 + 124 * q^17 - 46 * q^19 + 74 * q^22 + 210 * q^23 - 4 * q^26 - 352 * q^28 - 296 * q^29 - 104 * q^31 + 722 * q^32 - 428 * q^34 + 204 * q^37 - 20 * q^38 - 344 * q^41 - 512 * q^43 - 716 * q^44 - 186 * q^46 + 238 * q^47 + 68 * q^49 + 468 * q^52 + 850 * q^53 - 2316 * q^56 - 890 * q^58 - 1840 * q^59 - 364 * q^61 + 1038 * q^62 - 990 * q^64 - 88 * q^67 + 236 * q^68 - 1364 * q^71 - 836 * q^73 - 1316 * q^74 - 2106 * q^76 + 840 * q^77 - 680 * q^79 - 1742 * q^82 + 2148 * q^83 - 2872 * q^86 - 1296 * q^88 - 3000 * q^89 - 3058 * q^91 + 1002 * q^92 - 3662 * q^94 + 612 * q^97 + 1982 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.57457	−1.26380	−0.631902	−	0.775049i	$-0.717725\pi$
$2$	−3.57457	−1.26380	−0.631902	+	0.775049i	$0.717725\pi$
$3$	0	0
$3$	0	0
$4$	4.77759	0.597198
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−14.1597	−0.764554	−0.382277	−	0.924048i	$-0.624860\pi$
$7$	−14.1597	−0.764554	−0.382277	+	0.924048i	$0.624860\pi$
$8$	11.5188	0.509062
$9$	0	0
$10$	0	0
$11$	−63.6638	−1.74503	−0.872516	−	0.488585i	$-0.837513\pi$
$11$	−63.6638	−1.74503	−0.872516	+	0.488585i	$0.837513\pi$
$12$	0	0
$13$	10.3959	0.221793	0.110896	−	0.993832i	$-0.464628\pi$
$13$	10.3959	0.221793	0.110896	+	0.993832i	$0.464628\pi$
$14$	50.6151	0.966246
$15$	0	0
$16$	−79.3954	−1.24055
$17$	108.605	1.54945	0.774724	−	0.632300i	$-0.217889\pi$
$17$	108.605	1.54945	0.774724	+	0.632300i	$0.217889\pi$
$18$	0	0
$19$	18.6145	0.224761	0.112381	−	0.993665i	$-0.464152\pi$
$19$	18.6145	0.224761	0.112381	+	0.993665i	$0.464152\pi$
$20$	0	0
$21$	0	0
$22$	227.571	2.20538
$23$	−128.824	−1.16790	−0.583949	−	0.811791i	$-0.698493\pi$
$23$	−128.824	−1.16790	−0.583949	+	0.811791i	$0.698493\pi$
$24$	0	0
$25$	0	0
$26$	−37.1609	−0.280302
$27$	0	0
$28$	−67.6494	−0.456590
$29$	−83.6496	−0.535632	−0.267816	−	0.963470i	$-0.586302\pi$
$29$	−83.6496	−0.535632	−0.267816	+	0.963470i	$0.586302\pi$
$30$	0	0
$31$	−10.4728	−0.0606762	−0.0303381	−	0.999540i	$-0.509658\pi$
$31$	−10.4728	−0.0606762	−0.0303381	+	0.999540i	$0.509658\pi$
$32$	191.655	1.05875
$33$	0	0
$34$	−388.217	−1.95820
$35$	0	0
$36$	0	0
$37$	81.8885	0.363848	0.181924	−	0.983313i	$-0.441767\pi$
$37$	81.8885	0.363848	0.181924	+	0.983313i	$0.441767\pi$
$38$	−66.5391	−0.284054
$39$	0	0
$40$	0	0
$41$	307.453	1.17112	0.585562	−	0.810628i	$-0.300874\pi$
$41$	307.453	1.17112	0.585562	+	0.810628i	$0.300874\pi$
$42$	0	0
$43$	222.439	0.788876	0.394438	−	0.918922i	$-0.370939\pi$
$43$	222.439	0.788876	0.394438	+	0.918922i	$0.370939\pi$
$44$	−304.159	−1.04213
$45$	0	0
$46$	460.491	1.47599
$47$	361.931	1.12326	0.561628	−	0.827390i	$-0.310175\pi$
$47$	361.931	1.12326	0.561628	+	0.827390i	$0.310175\pi$
$48$	0	0
$49$	−142.502	−0.415457
$50$	0	0
$51$	0	0
$52$	49.6673	0.132454
$53$	562.215	1.45710	0.728548	−	0.684994i	$-0.240195\pi$
$53$	562.215	1.45710	0.728548	+	0.684994i	$0.240195\pi$
$54$	0	0
$55$	0	0
$56$	−163.103	−0.389206
$57$	0	0
$58$	299.012	0.676934
$59$	−462.310	−1.02013	−0.510065	−	0.860136i	$-0.670378\pi$
$59$	−462.310	−1.02013	−0.510065	+	0.860136i	$0.670378\pi$
$60$	0	0
$61$	−649.067	−1.36237	−0.681184	−	0.732112i	$-0.738535\pi$
$61$	−649.067	−1.36237	−0.681184	+	0.732112i	$0.738535\pi$
$62$	37.4356	0.0766827
$63$	0	0
$64$	−49.9208	−0.0975015
$65$	0	0
$66$	0	0
$67$	−254.180	−0.463478	−0.231739	−	0.972778i	$-0.574442\pi$
$67$	−254.180	−0.463478	−0.231739	+	0.972778i	$0.574442\pi$
$68$	518.870	0.925328
$69$	0	0
$70$	0	0
$71$	1092.93	1.82685	0.913426	−	0.407006i	$-0.133427\pi$
$71$	1092.93	1.82685	0.913426	+	0.407006i	$0.133427\pi$
$72$	0	0
$73$	−1034.52	−1.65865	−0.829324	−	0.558769i	$-0.811274\pi$
$73$	−1034.52	−1.65865	−0.829324	+	0.558769i	$0.811274\pi$
$74$	−292.717	−0.459833
$75$	0	0
$76$	88.9326	0.134227
$77$	901.463	1.33417
$78$	0	0
$79$	1151.85	1.64043	0.820213	−	0.572059i	$-0.193855\pi$
$79$	1151.85	1.64043	0.820213	+	0.572059i	$0.193855\pi$
$80$	0	0
$81$	0	0
$82$	−1099.01	−1.48007
$83$	524.320	0.693393	0.346697	−	0.937977i	$-0.387303\pi$
$83$	524.320	0.693393	0.346697	+	0.937977i	$0.387303\pi$
$84$	0	0
$85$	0	0
$86$	−795.126	−0.996984
$87$	0	0
$88$	−733.328	−0.888330
$89$	−656.485	−0.781879	−0.390940	−	0.920416i	$-0.627850\pi$
$89$	−656.485	−0.781879	−0.390940	+	0.920416i	$0.627850\pi$
$90$	0	0
$91$	−147.203	−0.169573
$92$	−615.467	−0.697466
$93$	0	0
$94$	−1293.75	−1.41957
$95$	0	0
$96$	0	0
$97$	−1125.51	−1.17813	−0.589063	−	0.808087i	$-0.700503\pi$
$97$	−1125.51	−1.17813	−0.589063	+	0.808087i	$0.700503\pi$
$98$	509.383	0.525056
$99$	0	0
$100$	0	0
$101$	380.324	0.374690	0.187345	−	0.982294i	$-0.440012\pi$
$101$	380.324	0.374690	0.187345	+	0.982294i	$0.440012\pi$
$102$	0	0
$103$	−764.277	−0.731131	−0.365566	−	0.930786i	$-0.619124\pi$
$103$	−764.277	−0.731131	−0.365566	+	0.930786i	$0.619124\pi$
$104$	119.748	0.112906
$105$	0	0
$106$	−2009.68	−1.84148
$107$	−816.083	−0.737325	−0.368662	−	0.929563i	$-0.620184\pi$
$107$	−816.083	−0.737325	−0.368662	+	0.929563i	$0.620184\pi$
$108$	0	0
$109$	606.775	0.533198	0.266599	−	0.963808i	$-0.414100\pi$
$109$	606.775	0.533198	0.266599	+	0.963808i	$0.414100\pi$
$110$	0	0
$111$	0	0
$112$	1124.22	0.948470
$113$	−27.3226	−0.0227460	−0.0113730	−	0.999935i	$-0.503620\pi$
$113$	−27.3226	−0.0227460	−0.0113730	+	0.999935i	$0.503620\pi$
$114$	0	0
$115$	0	0
$116$	−399.643	−0.319879
$117$	0	0
$118$	1652.56	1.28924
$119$	−1537.82	−1.18464
$120$	0	0
$121$	2722.08	2.04514
$122$	2320.14	1.72177
$123$	0	0
$124$	−50.0345	−0.0362357
$125$	0	0
$126$	0	0
$127$	1206.22	0.842794	0.421397	−	0.906876i	$-0.361540\pi$
$127$	1206.22	0.842794	0.421397	+	0.906876i	$0.361540\pi$
$128$	−1354.79	−0.935529
$129$	0	0
$130$	0	0
$131$	1532.00	1.02177	0.510885	−	0.859649i	$-0.329318\pi$
$131$	1532.00	1.02177	0.510885	+	0.859649i	$0.329318\pi$
$132$	0	0
$133$	−263.577	−0.171842
$134$	908.585	0.585745
$135$	0	0
$136$	1251.00	0.788765
$137$	2472.12	1.54166	0.770830	−	0.637041i	$-0.219842\pi$
$137$	2472.12	1.54166	0.770830	+	0.637041i	$0.219842\pi$
$138$	0	0
$139$	12.1267	0.00739982	0.00369991	−	0.999993i	$-0.498822\pi$
$139$	12.1267	0.00739982	0.00369991	+	0.999993i	$0.498822\pi$
$140$	0	0
$141$	0	0
$142$	−3906.75	−2.30878
$143$	−661.843	−0.387036
$144$	0	0
$145$	0	0
$146$	3697.96	2.09620
$147$	0	0
$148$	391.230	0.217290
$149$	3136.51	1.72452	0.862258	−	0.506469i	$-0.169049\pi$
$149$	3136.51	1.72452	0.862258	+	0.506469i	$0.169049\pi$
$150$	0	0
$151$	−2700.08	−1.45516	−0.727580	−	0.686023i	$-0.759355\pi$
$151$	−2700.08	−1.45516	−0.727580	+	0.686023i	$0.759355\pi$
$152$	214.416	0.114418
$153$	0	0
$154$	−3222.35	−1.68613
$155$	0	0
$156$	0	0
$157$	−442.627	−0.225003	−0.112502	−	0.993652i	$-0.535886\pi$
$157$	−442.627	−0.225003	−0.112502	+	0.993652i	$0.535886\pi$
$158$	−4117.39	−2.07317
$159$	0	0
$160$	0	0
$161$	1824.11	0.892921
$162$	0	0
$163$	2116.29	1.01694	0.508469	−	0.861080i	$-0.330212\pi$
$163$	2116.29	1.01694	0.508469	+	0.861080i	$0.330212\pi$
$164$	1468.88	0.699393
$165$	0	0
$166$	−1874.22	−0.876312
$167$	1294.62	0.599886	0.299943	−	0.953957i	$-0.403032\pi$
$167$	1294.62	0.599886	0.299943	+	0.953957i	$0.403032\pi$
$168$	0	0
$169$	−2088.93	−0.950808
$170$	0	0
$171$	0	0
$172$	1062.72	0.471116
$173$	2922.17	1.28421	0.642106	−	0.766616i	$-0.278061\pi$
$173$	2922.17	1.28421	0.642106	+	0.766616i	$0.278061\pi$
$174$	0	0
$175$	0	0
$176$	5054.61	2.16480
$177$	0	0
$178$	2346.65	0.988141
$179$	−3955.51	−1.65167	−0.825834	−	0.563913i	$-0.809295\pi$
$179$	−3955.51	−1.65167	−0.825834	+	0.563913i	$0.809295\pi$
$180$	0	0
$181$	−2694.00	−1.10632	−0.553159	−	0.833076i	$-0.686578\pi$
$181$	−2694.00	−1.10632	−0.553159	+	0.833076i	$0.686578\pi$
$182$	526.189	0.214306
$183$	0	0
$184$	−1483.89	−0.594532
$185$	0	0
$186$	0	0
$187$	−6914.22	−2.70384
$188$	1729.15	0.670806
$189$	0	0
$190$	0	0
$191$	856.346	0.324414	0.162207	−	0.986757i	$-0.448139\pi$
$191$	856.346	0.324414	0.162207	+	0.986757i	$0.448139\pi$
$192$	0	0
$193$	1735.21	0.647168	0.323584	−	0.946199i	$-0.395112\pi$
$193$	1735.21	0.647168	0.323584	+	0.946199i	$0.395112\pi$
$194$	4023.22	1.48892
$195$	0	0
$196$	−680.814	−0.248110
$197$	2943.95	1.06471	0.532354	−	0.846522i	$-0.321308\pi$
$197$	2943.95	1.06471	0.532354	+	0.846522i	$0.321308\pi$
$198$	0	0
$199$	−4172.06	−1.48618	−0.743088	−	0.669193i	$-0.766640\pi$
$199$	−4172.06	−1.48618	−0.743088	+	0.669193i	$0.766640\pi$
$200$	0	0
$201$	0	0
$202$	−1359.50	−0.473534
$203$	1184.46	0.409520
$204$	0	0
$205$	0	0
$206$	2731.97	0.924006
$207$	0	0
$208$	−825.387	−0.275146
$209$	−1185.07	−0.392216
$210$	0	0
$211$	3851.04	1.25648	0.628239	−	0.778021i	$-0.283776\pi$
$211$	3851.04	1.25648	0.628239	+	0.778021i	$0.283776\pi$
$212$	2686.03	0.870176
$213$	0	0
$214$	2917.15	0.931833
$215$	0	0
$216$	0	0
$217$	148.291	0.0463902
$218$	−2168.96	−0.673857
$219$	0	0
$220$	0	0
$221$	1129.05	0.343656
$222$	0	0
$223$	−820.714	−0.246453	−0.123227	−	0.992379i	$-0.539324\pi$
$223$	−820.714	−0.246453	−0.123227	+	0.992379i	$0.539324\pi$
$224$	−2713.78	−0.809473
$225$	0	0
$226$	97.6667	0.0287464
$227$	−5261.23	−1.53833	−0.769163	−	0.639053i	$-0.779326\pi$
$227$	−5261.23	−1.53833	−0.769163	+	0.639053i	$0.779326\pi$
$228$	0	0
$229$	19.5989	0.00565560	0.00282780	−	0.999996i	$-0.499100\pi$
$229$	19.5989	0.00565560	0.00282780	+	0.999996i	$0.499100\pi$
$230$	0	0
$231$	0	0
$232$	−963.539	−0.272670
$233$	−3181.79	−0.894619	−0.447309	−	0.894379i	$-0.647618\pi$
$233$	−3181.79	−0.894619	−0.447309	+	0.894379i	$0.647618\pi$
$234$	0	0
$235$	0	0
$236$	−2208.72	−0.609219
$237$	0	0
$238$	5497.05	1.49715
$239$	−1634.10	−0.442265	−0.221133	−	0.975244i	$-0.570975\pi$
$239$	−1634.10	−0.442265	−0.221133	+	0.975244i	$0.570975\pi$
$240$	0	0
$241$	1414.71	0.378131	0.189065	−	0.981965i	$-0.439454\pi$
$241$	1414.71	0.378131	0.189065	+	0.981965i	$0.439454\pi$
$242$	−9730.28	−2.58465
$243$	0	0
$244$	−3100.97	−0.813604
$245$	0	0
$246$	0	0
$247$	193.515	0.0498505
$248$	−120.633	−0.0308879
$249$	0	0
$250$	0	0
$251$	−5932.53	−1.49186	−0.745932	−	0.666022i	$-0.767995\pi$
$251$	−5932.53	−1.49186	−0.745932	+	0.666022i	$0.767995\pi$
$252$	0	0
$253$	8201.42	2.03802
$254$	−4311.73	−1.06513
$255$	0	0
$256$	5242.17	1.27983
$257$	7001.29	1.69933	0.849666	−	0.527321i	$-0.176803\pi$
$257$	7001.29	1.69933	0.849666	+	0.527321i	$0.176803\pi$
$258$	0	0
$259$	−1159.52	−0.278182
$260$	0	0
$261$	0	0
$262$	−5476.26	−1.29132
$263$	3654.54	0.856839	0.428420	−	0.903580i	$-0.359071\pi$
$263$	3654.54	0.856839	0.428420	+	0.903580i	$0.359071\pi$
$264$	0	0
$265$	0	0
$266$	942.176	0.217175
$267$	0	0
$268$	−1214.37	−0.276788
$269$	−2589.46	−0.586922	−0.293461	−	0.955971i	$-0.594807\pi$
$269$	−2589.46	−0.586922	−0.293461	+	0.955971i	$0.594807\pi$
$270$	0	0
$271$	4854.10	1.08806	0.544032	−	0.839064i	$-0.316897\pi$
$271$	4854.10	1.08806	0.544032	+	0.839064i	$0.316897\pi$
$272$	−8622.74	−1.92217
$273$	0	0
$274$	−8836.77	−1.94835
$275$	0	0
$276$	0	0
$277$	7515.15	1.63012	0.815058	−	0.579380i	$-0.196705\pi$
$277$	7515.15	1.63012	0.815058	+	0.579380i	$0.196705\pi$
$278$	−43.3478	−0.00935191
$279$	0	0
$280$	0	0
$281$	3612.07	0.766825	0.383412	−	0.923577i	$-0.374749\pi$
$281$	3612.07	0.766825	0.383412	+	0.923577i	$0.374749\pi$
$282$	0	0
$283$	−5166.91	−1.08530	−0.542652	−	0.839958i	$-0.682580\pi$
$283$	−5166.91	−1.08530	−0.542652	+	0.839958i	$0.682580\pi$
$284$	5221.55	1.09099
$285$	0	0
$286$	2365.81	0.489137
$287$	−4353.45	−0.895387
$288$	0	0
$289$	6882.07	1.40079
$290$	0	0
$291$	0	0
$292$	−4942.50	−0.990541
$293$	−4226.35	−0.842683	−0.421342	−	0.906902i	$-0.638441\pi$
$293$	−4226.35	−0.842683	−0.421342	+	0.906902i	$0.638441\pi$
$294$	0	0
$295$	0	0
$296$	943.255	0.185221
$297$	0	0
$298$	−11211.7	−2.17945
$299$	−1339.24	−0.259031
$300$	0	0
$301$	−3149.68	−0.603139
$302$	9651.62	1.83903
$303$	0	0
$304$	−1477.91	−0.278828
$305$	0	0
$306$	0	0
$307$	−5734.25	−1.06603	−0.533015	−	0.846106i	$-0.678941\pi$
$307$	−5734.25	−1.06603	−0.533015	+	0.846106i	$0.678941\pi$
$308$	4306.82	0.796765
$309$	0	0
$310$	0	0
$311$	−6620.39	−1.20710	−0.603549	−	0.797326i	$-0.706247\pi$
$311$	−6620.39	−1.20710	−0.603549	+	0.797326i	$0.706247\pi$
$312$	0	0
$313$	−93.4309	−0.0168723	−0.00843615	−	0.999964i	$-0.502685\pi$
$313$	−93.4309	−0.0168723	−0.00843615	+	0.999964i	$0.502685\pi$
$314$	1582.20	0.284360
$315$	0	0
$316$	5503.08	0.979659
$317$	1993.17	0.353147	0.176573	−	0.984287i	$-0.443499\pi$
$317$	1993.17	0.353147	0.176573	+	0.984287i	$0.443499\pi$
$318$	0	0
$319$	5325.45	0.934696
$320$	0	0
$321$	0	0
$322$	−6520.43	−1.12848
$323$	2021.63	0.348256
$324$	0	0
$325$	0	0
$326$	−7564.85	−1.28521
$327$	0	0
$328$	3541.47	0.596174
$329$	−5124.84	−0.858790
$330$	0	0
$331$	−2389.18	−0.396740	−0.198370	−	0.980127i	$-0.563565\pi$
$331$	−2389.18	−0.396740	−0.198370	+	0.980127i	$0.563565\pi$
$332$	2504.99	0.414093
$333$	0	0
$334$	−4627.73	−0.758138
$335$	0	0
$336$	0	0
$337$	−10285.5	−1.66257	−0.831286	−	0.555845i	$-0.812395\pi$
$337$	−10285.5	−1.66257	−0.831286	+	0.555845i	$0.812395\pi$
$338$	7467.02	1.20163
$339$	0	0
$340$	0	0
$341$	666.735	0.105882
$342$	0	0
$343$	6874.58	1.08219
$344$	2562.23	0.401587
$345$	0	0
$346$	−10445.5	−1.62299
$347$	4230.21	0.654436	0.327218	−	0.944949i	$-0.393889\pi$
$347$	4230.21	0.654436	0.327218	+	0.944949i	$0.393889\pi$
$348$	0	0
$349$	−11210.9	−1.71950	−0.859752	−	0.510712i	$-0.829382\pi$
$349$	−11210.9	−1.71950	−0.859752	+	0.510712i	$0.829382\pi$
$350$	0	0
$351$	0	0
$352$	−12201.5	−1.84756
$353$	−11383.0	−1.71631	−0.858154	−	0.513393i	$-0.828388\pi$
$353$	−11383.0	−1.71631	−0.858154	+	0.513393i	$0.828388\pi$
$354$	0	0
$355$	0	0
$356$	−3136.41	−0.466937
$357$	0	0
$358$	14139.3	2.08738
$359$	−8942.57	−1.31468	−0.657341	−	0.753593i	$-0.728319\pi$
$359$	−8942.57	−1.31468	−0.657341	+	0.753593i	$0.728319\pi$
$360$	0	0
$361$	−6512.50	−0.949482
$362$	9629.90	1.39817
$363$	0	0
$364$	−703.277	−0.101268
$365$	0	0
$366$	0	0
$367$	444.170	0.0631757	0.0315878	−	0.999501i	$-0.489944\pi$
$367$	444.170	0.0631757	0.0315878	+	0.999501i	$0.489944\pi$
$368$	10228.0	1.44884
$369$	0	0
$370$	0	0
$371$	−7960.82	−1.11403
$372$	0	0
$373$	−3416.09	−0.474205	−0.237103	−	0.971485i	$-0.576198\pi$
$373$	−3416.09	−0.474205	−0.237103	+	0.971485i	$0.576198\pi$
$374$	24715.4	3.41712
$375$	0	0
$376$	4168.99	0.571807
$377$	−869.613	−0.118799
$378$	0	0
$379$	598.277	0.0810855	0.0405428	−	0.999178i	$-0.487091\pi$
$379$	598.277	0.0810855	0.0405428	+	0.999178i	$0.487091\pi$
$380$	0	0
$381$	0	0
$382$	−3061.07	−0.409995
$383$	−9057.17	−1.20836	−0.604178	−	0.796850i	$-0.706498\pi$
$383$	−9057.17	−1.20836	−0.604178	+	0.796850i	$0.706498\pi$
$384$	0	0
$385$	0	0
$386$	−6202.65	−0.817893
$387$	0	0
$388$	−5377.22	−0.703575
$389$	−631.406	−0.0822971	−0.0411485	−	0.999153i	$-0.513102\pi$
$389$	−631.406	−0.0822971	−0.0411485	+	0.999153i	$0.513102\pi$
$390$	0	0
$391$	−13990.9	−1.80960
$392$	−1641.44	−0.211493
$393$	0	0
$394$	−10523.4	−1.34558
$395$	0	0
$396$	0	0
$397$	4615.39	0.583475	0.291738	−	0.956498i	$-0.405767\pi$
$397$	4615.39	0.583475	0.291738	+	0.956498i	$0.405767\pi$
$398$	14913.3	1.87824
$399$	0	0
$400$	0	0
$401$	−4691.20	−0.584207	−0.292104	−	0.956387i	$-0.594355\pi$
$401$	−4691.20	−0.584207	−0.292104	+	0.956387i	$0.594355\pi$
$402$	0	0
$403$	−108.874	−0.0134575
$404$	1817.03	0.223764
$405$	0	0
$406$	−4233.93	−0.517552
$407$	−5213.34	−0.634927
$408$	0	0
$409$	−15563.3	−1.88155	−0.940776	−	0.339028i	$-0.889902\pi$
$409$	−15563.3	−1.88155	−0.940776	+	0.339028i	$0.889902\pi$
$410$	0	0
$411$	0	0
$412$	−3651.40	−0.436630
$413$	6546.19	0.779944
$414$	0	0
$415$	0	0
$416$	1992.42	0.234824
$417$	0	0
$418$	4236.13	0.495684
$419$	−4278.44	−0.498843	−0.249422	−	0.968395i	$-0.580240\pi$
$419$	−4278.44	−0.498843	−0.249422	+	0.968395i	$0.580240\pi$
$420$	0	0
$421$	−97.7063	−0.0113110	−0.00565548	−	0.999984i	$-0.501800\pi$
$421$	−97.7063	−0.0113110	−0.00565548	+	0.999984i	$0.501800\pi$
$422$	−13765.8	−1.58794
$423$	0	0
$424$	6476.02	0.741753
$425$	0	0
$426$	0	0
$427$	9190.62	1.04160
$428$	−3898.91	−0.440329
$429$	0	0
$430$	0	0
$431$	−11932.9	−1.33361	−0.666806	−	0.745231i	$-0.732339\pi$
$431$	−11932.9	−1.33361	−0.666806	+	0.745231i	$0.732339\pi$
$432$	0	0
$433$	−6304.32	−0.699691	−0.349845	−	0.936807i	$-0.613766\pi$
$433$	−6304.32	−0.699691	−0.349845	+	0.936807i	$0.613766\pi$
$434$	−530.079	−0.0586281
$435$	0	0
$436$	2898.92	0.318425
$437$	−2398.00	−0.262498
$438$	0	0
$439$	−12922.7	−1.40493	−0.702466	−	0.711717i	$-0.747918\pi$
$439$	−12922.7	−1.40493	−0.702466	+	0.711717i	$0.747918\pi$
$440$	0	0
$441$	0	0
$442$	−4035.87	−0.434314
$443$	−3824.41	−0.410165	−0.205083	−	0.978745i	$-0.565746\pi$
$443$	−3824.41	−0.410165	−0.205083	+	0.978745i	$0.565746\pi$
$444$	0	0
$445$	0	0
$446$	2933.70	0.311468
$447$	0	0
$448$	706.865	0.0745452
$449$	530.656	0.0557756	0.0278878	−	0.999611i	$-0.491122\pi$
$449$	530.656	0.0557756	0.0278878	+	0.999611i	$0.491122\pi$
$450$	0	0
$451$	−19573.6	−2.04365
$452$	−130.536	−0.0135839
$453$	0	0
$454$	18806.6	1.94414
$455$	0	0
$456$	0	0
$457$	15593.6	1.59614	0.798072	−	0.602562i	$-0.205853\pi$
$457$	15593.6	1.59614	0.798072	+	0.602562i	$0.205853\pi$
$458$	−70.0577	−0.00714756
$459$	0	0
$460$	0	0
$461$	7274.61	0.734951	0.367475	−	0.930033i	$-0.380222\pi$
$461$	7274.61	0.734951	0.367475	+	0.930033i	$0.380222\pi$
$462$	0	0
$463$	−2685.14	−0.269523	−0.134761	−	0.990878i	$-0.543027\pi$
$463$	−2685.14	−0.269523	−0.134761	+	0.990878i	$0.543027\pi$
$464$	6641.39	0.664480
$465$	0	0
$466$	11373.6	1.13062
$467$	−8844.77	−0.876418	−0.438209	−	0.898873i	$-0.644387\pi$
$467$	−8844.77	−0.876418	−0.438209	+	0.898873i	$0.644387\pi$
$468$	0	0
$469$	3599.12	0.354354
$470$	0	0
$471$	0	0
$472$	−5325.24	−0.519309
$473$	−14161.3	−1.37662
$474$	0	0
$475$	0	0
$476$	−7347.07	−0.707463
$477$	0	0
$478$	5841.23	0.558936
$479$	−12334.3	−1.17656	−0.588278	−	0.808659i	$-0.700194\pi$
$479$	−12334.3	−1.17656	−0.588278	+	0.808659i	$0.700194\pi$
$480$	0	0
$481$	851.306	0.0806990
$482$	−5056.99	−0.477883
$483$	0	0
$484$	13005.0	1.22135
$485$	0	0
$486$	0	0
$487$	−3540.31	−0.329419	−0.164709	−	0.986342i	$-0.552669\pi$
$487$	−3540.31	−0.329419	−0.164709	+	0.986342i	$0.552669\pi$
$488$	−7476.45	−0.693530
$489$	0	0
$490$	0	0
$491$	−7711.25	−0.708765	−0.354383	−	0.935100i	$-0.615309\pi$
$491$	−7711.25	−0.708765	−0.354383	+	0.935100i	$0.615309\pi$
$492$	0	0
$493$	−9084.77	−0.829934
$494$	−691.734	−0.0630012
$495$	0	0
$496$	831.488	0.0752720
$497$	−15475.5	−1.39673
$498$	0	0
$499$	−12162.6	−1.09113	−0.545563	−	0.838070i	$-0.683684\pi$
$499$	−12162.6	−1.09113	−0.545563	+	0.838070i	$0.683684\pi$
$500$	0	0
$501$	0	0
$502$	21206.3	1.88542
$503$	−8796.85	−0.779786	−0.389893	−	0.920860i	$-0.627488\pi$
$503$	−8796.85	−0.779786	−0.389893	+	0.920860i	$0.627488\pi$
$504$	0	0
$505$	0	0
$506$	−29316.6	−2.57566
$507$	0	0
$508$	5762.83	0.503315
$509$	21948.2	1.91127	0.955634	−	0.294558i	$-0.0951723\pi$
$509$	21948.2	1.91127	0.955634	+	0.294558i	$0.0951723\pi$
$510$	0	0
$511$	14648.5	1.26813
$512$	−7900.20	−0.681919
$513$	0	0
$514$	−25026.6	−2.14762
$515$	0	0
$516$	0	0
$517$	−23041.9	−1.96012
$518$	4144.79	0.351567
$519$	0	0
$520$	0	0
$521$	11232.4	0.944531	0.472265	−	0.881456i	$-0.343436\pi$
$521$	11232.4	0.944531	0.472265	+	0.881456i	$0.343436\pi$
$522$	0	0
$523$	1962.88	0.164112	0.0820560	−	0.996628i	$-0.473851\pi$
$523$	1962.88	0.164112	0.0820560	+	0.996628i	$0.473851\pi$
$524$	7319.28	0.610199
$525$	0	0
$526$	−13063.4	−1.08288
$527$	−1137.39	−0.0940146
$528$	0	0
$529$	4428.60	0.363985
$530$	0	0
$531$	0	0
$532$	−1259.26	−0.102624
$533$	3196.25	0.259747
$534$	0	0
$535$	0	0
$536$	−2927.84	−0.235939
$537$	0	0
$538$	9256.21	0.741754
$539$	9072.20	0.724986
$540$	0	0
$541$	2816.86	0.223856	0.111928	−	0.993716i	$-0.464297\pi$
$541$	2816.86	0.223856	0.111928	+	0.993716i	$0.464297\pi$
$542$	−17351.3	−1.37510
$543$	0	0
$544$	20814.7	1.64048
$545$	0	0
$546$	0	0
$547$	−4528.91	−0.354007	−0.177004	−	0.984210i	$-0.556640\pi$
$547$	−4528.91	−0.354007	−0.177004	+	0.984210i	$0.556640\pi$
$548$	11810.8	0.920676
$549$	0	0
$550$	0	0
$551$	−1557.10	−0.120389
$552$	0	0
$553$	−16309.9	−1.25419
$554$	−26863.5	−2.06014
$555$	0	0
$556$	57.9364	0.00441916
$557$	−6267.47	−0.476771	−0.238385	−	0.971171i	$-0.576618\pi$
$557$	−6267.47	−0.476771	−0.238385	+	0.971171i	$0.576618\pi$
$558$	0	0
$559$	2312.46	0.174967
$560$	0	0
$561$	0	0
$562$	−12911.6	−0.969116
$563$	9794.00	0.733158	0.366579	−	0.930387i	$-0.380529\pi$
$563$	9794.00	0.733158	0.366579	+	0.930387i	$0.380529\pi$
$564$	0	0
$565$	0	0
$566$	18469.5	1.37161
$567$	0	0
$568$	12589.2	0.929981
$569$	7599.88	0.559936	0.279968	−	0.960009i	$-0.409676\pi$
$569$	7599.88	0.559936	0.279968	+	0.960009i	$0.409676\pi$
$570$	0	0
$571$	−12094.1	−0.886382	−0.443191	−	0.896427i	$-0.646154\pi$
$571$	−12094.1	−0.886382	−0.443191	+	0.896427i	$0.646154\pi$
$572$	−3162.01	−0.231137
$573$	0	0
$574$	15561.7	1.13159
$575$	0	0
$576$	0	0
$577$	−3113.85	−0.224665	−0.112332	−	0.993671i	$-0.535832\pi$
$577$	−3113.85	−0.224665	−0.112332	+	0.993671i	$0.535832\pi$
$578$	−24600.5	−1.77032
$579$	0	0
$580$	0	0
$581$	−7424.24	−0.530137
$582$	0	0
$583$	−35792.7	−2.54268
$584$	−11916.4	−0.844354
$585$	0	0
$586$	15107.4	1.06499
$587$	−5947.91	−0.418222	−0.209111	−	0.977892i	$-0.567057\pi$
$587$	−5947.91	−0.418222	−0.209111	+	0.977892i	$0.567057\pi$
$588$	0	0
$589$	−194.945	−0.0136377
$590$	0	0
$591$	0	0
$592$	−6501.57	−0.451373
$593$	−22153.0	−1.53409	−0.767043	−	0.641596i	$-0.778273\pi$
$593$	−22153.0	−1.53409	−0.767043	+	0.641596i	$0.778273\pi$
$594$	0	0
$595$	0	0
$596$	14984.9	1.02988
$597$	0	0
$598$	4787.22	0.327364
$599$	−26876.4	−1.83329	−0.916643	−	0.399708i	$-0.869112\pi$
$599$	−26876.4	−1.83329	−0.916643	+	0.399708i	$0.869112\pi$
$600$	0	0
$601$	25840.2	1.75381	0.876907	−	0.480659i	$-0.159603\pi$
$601$	25840.2	1.75381	0.876907	+	0.480659i	$0.159603\pi$
$602$	11258.8	0.762249
$603$	0	0
$604$	−12899.8	−0.869019
$605$	0	0
$606$	0	0
$607$	2768.74	0.185139	0.0925697	−	0.995706i	$-0.470492\pi$
$607$	2768.74	0.185139	0.0925697	+	0.995706i	$0.470492\pi$
$608$	3567.56	0.237967
$609$	0	0
$610$	0	0
$611$	3762.60	0.249130
$612$	0	0
$613$	−10177.3	−0.670563	−0.335282	−	0.942118i	$-0.608831\pi$
$613$	−10177.3	−0.670563	−0.335282	+	0.942118i	$0.608831\pi$
$614$	20497.5	1.34725
$615$	0	0
$616$	10383.7	0.679176
$617$	−8417.23	−0.549213	−0.274607	−	0.961557i	$-0.588548\pi$
$617$	−8417.23	−0.549213	−0.274607	+	0.961557i	$0.588548\pi$
$618$	0	0
$619$	−13280.3	−0.862324	−0.431162	−	0.902274i	$-0.641896\pi$
$619$	−13280.3	−0.862324	−0.431162	+	0.902274i	$0.641896\pi$
$620$	0	0
$621$	0	0
$622$	23665.1	1.52553
$623$	9295.66	0.597789
$624$	0	0
$625$	0	0
$626$	333.976	0.0213233
$627$	0	0
$628$	−2114.69	−0.134371
$629$	8893.52	0.563764
$630$	0	0
$631$	−14454.1	−0.911903	−0.455951	−	0.890005i	$-0.650701\pi$
$631$	−14454.1	−0.911903	−0.455951	+	0.890005i	$0.650701\pi$
$632$	13267.9	0.835078
$633$	0	0
$634$	−7124.73	−0.446308
$635$	0	0
$636$	0	0
$637$	−1481.43	−0.0921453
$638$	−19036.2	−1.18127
$639$	0	0
$640$	0	0
$641$	21064.9	1.29799	0.648997	−	0.760791i	$-0.275189\pi$
$641$	21064.9	1.29799	0.648997	+	0.760791i	$0.275189\pi$
$642$	0	0
$643$	7128.81	0.437221	0.218610	−	0.975812i	$-0.429848\pi$
$643$	7128.81	0.437221	0.218610	+	0.975812i	$0.429848\pi$
$644$	8714.86	0.533251
$645$	0	0
$646$	−7226.48	−0.440127
$647$	−1364.92	−0.0829376	−0.0414688	−	0.999140i	$-0.513204\pi$
$647$	−1364.92	−0.0829376	−0.0414688	+	0.999140i	$0.513204\pi$
$648$	0	0
$649$	29432.4	1.78016
$650$	0	0
$651$	0	0
$652$	10110.8	0.607313
$653$	−12160.6	−0.728763	−0.364381	−	0.931250i	$-0.618720\pi$
$653$	−12160.6	−0.728763	−0.364381	+	0.931250i	$0.618720\pi$
$654$	0	0
$655$	0	0
$656$	−24410.3	−1.45284
$657$	0	0
$658$	18319.1	1.08534
$659$	−6168.57	−0.364633	−0.182317	−	0.983240i	$-0.558360\pi$
$659$	−6168.57	−0.364633	−0.182317	+	0.983240i	$0.558360\pi$
$660$	0	0
$661$	18137.7	1.06728	0.533642	−	0.845711i	$-0.320823\pi$
$661$	18137.7	1.06728	0.533642	+	0.845711i	$0.320823\pi$
$662$	8540.29	0.501402
$663$	0	0
$664$	6039.52	0.352980
$665$	0	0
$666$	0	0
$667$	10776.1	0.625564
$668$	6185.18	0.358251
$669$	0	0
$670$	0	0
$671$	41322.1	2.37738
$672$	0	0
$673$	21702.1	1.24302	0.621512	−	0.783404i	$-0.286519\pi$
$673$	21702.1	1.24302	0.621512	+	0.783404i	$0.286519\pi$
$674$	36766.3	2.10116
$675$	0	0
$676$	−9980.02	−0.567821
$677$	10470.6	0.594412	0.297206	−	0.954813i	$-0.403945\pi$
$677$	10470.6	0.594412	0.297206	+	0.954813i	$0.403945\pi$
$678$	0	0
$679$	15936.9	0.900742
$680$	0	0
$681$	0	0
$682$	−2383.29	−0.133814
$683$	15386.0	0.861974	0.430987	−	0.902358i	$-0.358166\pi$
$683$	15386.0	0.861974	0.430987	+	0.902358i	$0.358166\pi$
$684$	0	0
$685$	0	0
$686$	−24573.7	−1.36768
$687$	0	0
$688$	−17660.7	−0.978643
$689$	5844.73	0.323174
$690$	0	0
$691$	15191.1	0.836322	0.418161	−	0.908373i	$-0.362675\pi$
$691$	15191.1	0.836322	0.418161	+	0.908373i	$0.362675\pi$
$692$	13960.9	0.766929
$693$	0	0
$694$	−15121.2	−0.827079
$695$	0	0
$696$	0	0
$697$	33390.9	1.81459
$698$	40074.3	2.17311
$699$	0	0
$700$	0	0
$701$	−5181.93	−0.279199	−0.139600	−	0.990208i	$-0.544582\pi$
$701$	−5181.93	−0.279199	−0.139600	+	0.990208i	$0.544582\pi$
$702$	0	0
$703$	1524.32	0.0817791
$704$	3178.15	0.170143
$705$	0	0
$706$	40689.4	2.16907
$707$	−5385.30	−0.286471
$708$	0	0
$709$	−8786.53	−0.465423	−0.232712	−	0.972546i	$-0.574760\pi$
$709$	−8786.53	−0.465423	−0.232712	+	0.972546i	$0.574760\pi$
$710$	0	0
$711$	0	0
$712$	−7561.89	−0.398025
$713$	1349.14	0.0708636
$714$	0	0
$715$	0	0
$716$	−18897.8	−0.986373
$717$	0	0
$718$	31965.9	1.66150
$719$	−14433.0	−0.748625	−0.374312	−	0.927303i	$-0.622121\pi$
$719$	−14433.0	−0.748625	−0.374312	+	0.927303i	$0.622121\pi$
$720$	0	0
$721$	10822.0	0.558989
$722$	23279.4	1.19996
$723$	0	0
$724$	−12870.8	−0.660691
$725$	0	0
$726$	0	0
$727$	−26698.9	−1.36205	−0.681023	−	0.732262i	$-0.738465\pi$
$727$	−26698.9	−1.36205	−0.681023	+	0.732262i	$0.738465\pi$
$728$	−1695.60	−0.0863230
$729$	0	0
$730$	0	0
$731$	24158.1	1.22232
$732$	0	0
$733$	7733.84	0.389708	0.194854	−	0.980832i	$-0.437577\pi$
$733$	7733.84	0.389708	0.194854	+	0.980832i	$0.437577\pi$
$734$	−1587.72	−0.0798416
$735$	0	0
$736$	−24689.7	−1.23651
$737$	16182.1	0.808784
$738$	0	0
$739$	28142.2	1.40085	0.700426	−	0.713725i	$-0.252994\pi$
$739$	28142.2	1.40085	0.700426	+	0.713725i	$0.252994\pi$
$740$	0	0
$741$	0	0
$742$	28456.5	1.40791
$743$	−22769.1	−1.12425	−0.562125	−	0.827052i	$-0.690016\pi$
$743$	−22769.1	−1.12425	−0.562125	+	0.827052i	$0.690016\pi$
$744$	0	0
$745$	0	0
$746$	12211.1	0.599302
$747$	0	0
$748$	−33033.3	−1.61473
$749$	11555.5	0.563725
$750$	0	0
$751$	509.837	0.0247726	0.0123863	−	0.999923i	$-0.496057\pi$
$751$	509.837	0.0247726	0.0123863	+	0.999923i	$0.496057\pi$
$752$	−28735.6	−1.39346
$753$	0	0
$754$	3108.50	0.150139
$755$	0	0
$756$	0	0
$757$	28105.6	1.34943	0.674713	−	0.738080i	$-0.264267\pi$
$757$	28105.6	1.34943	0.674713	+	0.738080i	$0.264267\pi$
$758$	−2138.58	−0.102476
$759$	0	0
$760$	0	0
$761$	8695.64	0.414214	0.207107	−	0.978318i	$-0.433595\pi$
$761$	8695.64	0.414214	0.207107	+	0.978318i	$0.433595\pi$
$762$	0	0
$763$	−8591.78	−0.407659
$764$	4091.27	0.193739
$765$	0	0
$766$	32375.5	1.52712
$767$	−4806.13	−0.226257
$768$	0	0
$769$	5461.87	0.256125	0.128062	−	0.991766i	$-0.459124\pi$
$769$	5461.87	0.256125	0.128062	+	0.991766i	$0.459124\pi$
$770$	0	0
$771$	0	0
$772$	8290.13	0.386487
$773$	−37952.9	−1.76594	−0.882970	−	0.469430i	$-0.844459\pi$
$773$	−37952.9	−1.76594	−0.882970	+	0.469430i	$0.844459\pi$
$774$	0	0
$775$	0	0
$776$	−12964.5	−0.599740
$777$	0	0
$778$	2257.01	0.104007
$779$	5723.09	0.263223
$780$	0	0
$781$	−69579.8	−3.18791
$782$	50011.7	2.28697
$783$	0	0
$784$	11314.0	0.515396
$785$	0	0
$786$	0	0
$787$	18215.1	0.825032	0.412516	−	0.910950i	$-0.364650\pi$
$787$	18215.1	0.825032	0.412516	+	0.910950i	$0.364650\pi$
$788$	14065.0	0.635842
$789$	0	0
$790$	0	0
$791$	386.881	0.0173905
$792$	0	0
$793$	−6747.64	−0.302164
$794$	−16498.1	−0.737398
$795$	0	0
$796$	−19932.4	−0.887542
$797$	−28549.5	−1.26885	−0.634427	−	0.772983i	$-0.718764\pi$
$797$	−28549.5	−1.26885	−0.634427	+	0.772983i	$0.718764\pi$
$798$	0	0
$799$	39307.5	1.74043
$800$	0	0
$801$	0	0
$802$	16769.0	0.738323
$803$	65861.4	2.89439
$804$	0	0
$805$	0	0
$806$	389.177	0.0170077
$807$	0	0
$808$	4380.87	0.190741
$809$	2685.20	0.116695	0.0583476	−	0.998296i	$-0.481417\pi$
$809$	2685.20	0.116695	0.0583476	+	0.998296i	$0.481417\pi$
$810$	0	0
$811$	−12491.4	−0.540854	−0.270427	−	0.962740i	$-0.587165\pi$
$811$	−12491.4	−0.540854	−0.270427	+	0.962740i	$0.587165\pi$
$812$	5658.84	0.244565
$813$	0	0
$814$	18635.5	0.802423
$815$	0	0
$816$	0	0
$817$	4140.61	0.177309
$818$	55632.1	2.37791
$819$	0	0
$820$	0	0
$821$	23517.5	0.999715	0.499858	−	0.866108i	$-0.333386\pi$
$821$	23517.5	0.999715	0.499858	+	0.866108i	$0.333386\pi$
$822$	0	0
$823$	18126.2	0.767727	0.383863	−	0.923390i	$-0.374593\pi$
$823$	18126.2	0.767727	0.383863	+	0.923390i	$0.374593\pi$
$824$	−8803.53	−0.372191
$825$	0	0
$826$	−23399.8	−0.985696
$827$	45677.1	1.92062	0.960308	−	0.278940i	$-0.0899832\pi$
$827$	45677.1	1.92062	0.960308	+	0.278940i	$0.0899832\pi$
$828$	0	0
$829$	9130.02	0.382507	0.191254	−	0.981541i	$-0.438745\pi$
$829$	9130.02	0.382507	0.191254	+	0.981541i	$0.438745\pi$
$830$	0	0
$831$	0	0
$832$	−518.972	−0.0216251
$833$	−15476.4	−0.643729
$834$	0	0
$835$	0	0
$836$	−5661.78	−0.234231
$837$	0	0
$838$	15293.6	0.630440
$839$	−18206.4	−0.749172	−0.374586	−	0.927192i	$-0.622215\pi$
$839$	−18206.4	−0.749172	−0.374586	+	0.927192i	$0.622215\pi$
$840$	0	0
$841$	−17391.8	−0.713098
$842$	349.258	0.0142948
$843$	0	0
$844$	18398.7	0.750366
$845$	0	0
$846$	0	0
$847$	−38543.9	−1.56362
$848$	−44637.2	−1.80761
$849$	0	0
$850$	0	0
$851$	−10549.2	−0.424938
$852$	0	0
$853$	−29792.9	−1.19589	−0.597943	−	0.801539i	$-0.704015\pi$
$853$	−29792.9	−1.19589	−0.597943	+	0.801539i	$0.704015\pi$
$854$	−32852.6	−1.31638
$855$	0	0
$856$	−9400.27	−0.375344
$857$	21499.5	0.856954	0.428477	−	0.903553i	$-0.359050\pi$
$857$	21499.5	0.856954	0.428477	+	0.903553i	$0.359050\pi$
$858$	0	0
$859$	−30413.5	−1.20803	−0.604013	−	0.796974i	$-0.706433\pi$
$859$	−30413.5	−1.20803	−0.604013	+	0.796974i	$0.706433\pi$
$860$	0	0
$861$	0	0
$862$	42655.0	1.68542
$863$	−7021.71	−0.276966	−0.138483	−	0.990365i	$-0.544223\pi$
$863$	−7021.71	−0.276966	−0.138483	+	0.990365i	$0.544223\pi$
$864$	0	0
$865$	0	0
$866$	22535.3	0.884272
$867$	0	0
$868$	708.475	0.0277042
$869$	−73331.3	−2.86260
$870$	0	0
$871$	−2642.43	−0.102796
$872$	6989.30	0.271431
$873$	0	0
$874$	8571.82	0.331746
$875$	0	0
$876$	0	0
$877$	12018.7	0.462764	0.231382	−	0.972863i	$-0.425675\pi$
$877$	12018.7	0.462764	0.231382	+	0.972863i	$0.425675\pi$
$878$	46193.1	1.77556
$879$	0	0
$880$	0	0
$881$	21990.9	0.840968	0.420484	−	0.907300i	$-0.361860\pi$
$881$	21990.9	0.840968	0.420484	+	0.907300i	$0.361860\pi$
$882$	0	0
$883$	6023.11	0.229551	0.114776	−	0.993391i	$-0.463385\pi$
$883$	6023.11	0.229551	0.114776	+	0.993391i	$0.463385\pi$
$884$	5394.13	0.205231
$885$	0	0
$886$	13670.6	0.518368
$887$	8578.09	0.324717	0.162359	−	0.986732i	$-0.448090\pi$
$887$	8578.09	0.324717	0.162359	+	0.986732i	$0.448090\pi$
$888$	0	0
$889$	−17079.8	−0.644362
$890$	0	0
$891$	0	0
$892$	−3921.03	−0.147181
$893$	6737.17	0.252465
$894$	0	0
$895$	0	0
$896$	19183.5	0.715263
$897$	0	0
$898$	−1896.87	−0.0704893
$899$	876.041	0.0325001
$900$	0	0
$901$	61059.4	2.25770
$902$	69967.3	2.58277
$903$	0	0
$904$	−314.723	−0.0115791
$905$	0	0
$906$	0	0
$907$	−9340.62	−0.341952	−0.170976	−	0.985275i	$-0.554692\pi$
$907$	−9340.62	−0.341952	−0.170976	+	0.985275i	$0.554692\pi$
$908$	−25136.0	−0.918685
$909$	0	0
$910$	0	0
$911$	16352.9	0.594727	0.297363	−	0.954764i	$-0.403893\pi$
$911$	16352.9	0.594727	0.297363	+	0.954764i	$0.403893\pi$
$912$	0	0
$913$	−33380.2	−1.20999
$914$	−55740.5	−2.01721
$915$	0	0
$916$	93.6354	0.00337751
$917$	−21692.8	−0.781198
$918$	0	0
$919$	1321.13	0.0474211	0.0237105	−	0.999719i	$-0.492452\pi$
$919$	1321.13	0.0474211	0.0237105	+	0.999719i	$0.492452\pi$
$920$	0	0
$921$	0	0
$922$	−26003.6	−0.928833
$923$	11362.0	0.405182
$924$	0	0
$925$	0	0
$926$	9598.23	0.340623
$927$	0	0
$928$	−16031.8	−0.567102
$929$	−38318.9	−1.35329	−0.676643	−	0.736311i	$-0.736566\pi$
$929$	−38318.9	−1.35329	−0.676643	+	0.736311i	$0.736566\pi$
$930$	0	0
$931$	−2652.60	−0.0933787
$932$	−15201.3	−0.534265
$933$	0	0
$934$	31616.3	1.10762
$935$	0	0
$936$	0	0
$937$	24307.8	0.847494	0.423747	−	0.905781i	$-0.360715\pi$
$937$	24307.8	0.847494	0.423747	+	0.905781i	$0.360715\pi$
$938$	−12865.3	−0.447834
$939$	0	0
$940$	0	0
$941$	−23022.5	−0.797568	−0.398784	−	0.917045i	$-0.630568\pi$
$941$	−23022.5	−0.797568	−0.398784	+	0.917045i	$0.630568\pi$
$942$	0	0
$943$	−39607.3	−1.36775
$944$	36705.3	1.26552
$945$	0	0
$946$	50620.8	1.73977
$947$	28324.8	0.971945	0.485972	−	0.873974i	$-0.338466\pi$
$947$	28324.8	0.971945	0.485972	+	0.873974i	$0.338466\pi$
$948$	0	0
$949$	−10754.8	−0.367876
$950$	0	0
$951$	0	0
$952$	−17713.8	−0.603054
$953$	−12986.4	−0.441417	−0.220709	−	0.975340i	$-0.570837\pi$
$953$	−12986.4	−0.441417	−0.220709	+	0.975340i	$0.570837\pi$
$954$	0	0
$955$	0	0
$956$	−7807.08	−0.264120
$957$	0	0
$958$	44090.0	1.48694
$959$	−35004.6	−1.17868
$960$	0	0
$961$	−29681.3	−0.996318
$962$	−3043.06	−0.101988
$963$	0	0
$964$	6758.90	0.225819
$965$	0	0
$966$	0	0
$967$	39251.9	1.30533	0.652666	−	0.757645i	$-0.273650\pi$
$967$	39251.9	1.30533	0.652666	+	0.757645i	$0.273650\pi$
$968$	31355.0	1.04110
$969$	0	0
$970$	0	0
$971$	2404.97	0.0794843	0.0397422	−	0.999210i	$-0.487346\pi$
$971$	2404.97	0.0794843	0.0397422	+	0.999210i	$0.487346\pi$
$972$	0	0
$973$	−171.711	−0.00565756
$974$	12655.1	0.416321
$975$	0	0
$976$	51532.9	1.69009
$977$	−17640.0	−0.577641	−0.288820	−	0.957383i	$-0.593263\pi$
$977$	−17640.0	−0.577641	−0.288820	+	0.957383i	$0.593263\pi$
$978$	0	0
$979$	41794.3	1.36440
$980$	0	0
$981$	0	0
$982$	27564.4	0.895740
$983$	−53222.8	−1.72690	−0.863451	−	0.504433i	$-0.831702\pi$
$983$	−53222.8	−1.72690	−0.863451	+	0.504433i	$0.831702\pi$
$984$	0	0
$985$	0	0
$986$	32474.2	1.04887
$987$	0	0
$988$	924.535	0.0297706
$989$	−28655.5	−0.921327
$990$	0	0
$991$	−33124.1	−1.06178	−0.530889	−	0.847441i	$-0.678142\pi$
$991$	−33124.1	−1.06178	−0.530889	+	0.847441i	$0.678142\pi$
$992$	−2007.15	−0.0642410
$993$	0	0
$994$	55318.5	1.76519
$995$	0	0
$996$	0	0
$997$	11736.9	0.372830	0.186415	−	0.982471i	$-0.440313\pi$
$997$	11736.9	0.372830	0.186415	+	0.982471i	$0.440313\pi$
$998$	43476.1	1.37897
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.z.1.2		6
3.2	odd	2		2025.4.a.y.1.5		6
5.4	even	2		405.4.a.k.1.5	✓	6
15.14	odd	2		405.4.a.l.1.2	yes	6
45.4	even	6		405.4.e.x.136.2		12
45.14	odd	6		405.4.e.w.136.5		12
45.29	odd	6		405.4.e.w.271.5		12
45.34	even	6		405.4.e.x.271.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.5	✓	6	5.4	even	2
405.4.a.l.1.2	yes	6	15.14	odd	2
405.4.e.w.136.5		12	45.14	odd	6
405.4.e.w.271.5		12	45.29	odd	6
405.4.e.x.136.2		12	45.4	even	6
405.4.e.x.271.2		12	45.34	even	6
2025.4.a.y.1.5		6	3.2	odd	2
2025.4.a.z.1.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.57457 q^{2} +4.77759 q^{4} -14.1597 q^{7} +11.5188 q^{8} +O(q^{10})\)	\(q-3.57457 q^{2} +4.77759 q^{4} -14.1597 q^{7} +11.5188 q^{8} -63.6638 q^{11} +10.3959 q^{13} +50.6151 q^{14} -79.3954 q^{16} +108.605 q^{17} +18.6145 q^{19} +227.571 q^{22} -128.824 q^{23} -37.1609 q^{26} -67.6494 q^{28} -83.6496 q^{29} -10.4728 q^{31} +191.655 q^{32} -388.217 q^{34} +81.8885 q^{37} -66.5391 q^{38} +307.453 q^{41} +222.439 q^{43} -304.159 q^{44} +460.491 q^{46} +361.931 q^{47} -142.502 q^{49} +49.6673 q^{52} +562.215 q^{53} -163.103 q^{56} +299.012 q^{58} -462.310 q^{59} -649.067 q^{61} +37.4356 q^{62} -49.9208 q^{64} -254.180 q^{67} +518.870 q^{68} +1092.93 q^{71} -1034.52 q^{73} -292.717 q^{74} +88.9326 q^{76} +901.463 q^{77} +1151.85 q^{79} -1099.01 q^{82} +524.320 q^{83} -795.126 q^{86} -733.328 q^{88} -656.485 q^{89} -147.203 q^{91} -615.467 q^{92} -1293.75 q^{94} -1125.51 q^{97} +509.383 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8	\( 6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8} - 88 q^{11} - 20 q^{13} - 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 74 q^{22} + 210 q^{23} - 4 q^{26} - 352 q^{28} - 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 204 q^{37} - 20 q^{38} - 344 q^{41} - 512 q^{43} - 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 468 q^{52} + 850 q^{53} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} - 88 q^{67} + 236 q^{68} - 1364 q^{71} - 836 q^{73} - 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} - 1742 q^{82} + 2148 q^{83} - 2872 q^{86} - 1296 q^{88} - 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} + 612 q^{97} + 1982 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 34 * q^4 - 40 * q^7 + 66 * q^8 - 88 * q^11 - 20 * q^13 - 180 * q^14 + 58 * q^16 + 124 * q^17 - 46 * q^19 + 74 * q^22 + 210 * q^23 - 4 * q^26 - 352 * q^28 - 296 * q^29 - 104 * q^31 + 722 * q^32 - 428 * q^34 + 204 * q^37 - 20 * q^38 - 344 * q^41 - 512 * q^43 - 716 * q^44 - 186 * q^46 + 238 * q^47 + 68 * q^49 + 468 * q^52 + 850 * q^53 - 2316 * q^56 - 890 * q^58 - 1840 * q^59 - 364 * q^61 + 1038 * q^62 - 990 * q^64 - 88 * q^67 + 236 * q^68 - 1364 * q^71 - 836 * q^73 - 1316 * q^74 - 2106 * q^76 + 840 * q^77 - 680 * q^79 - 1742 * q^82 + 2148 * q^83 - 2872 * q^86 - 1296 * q^88 - 3000 * q^89 - 3058 * q^91 + 1002 * q^92 - 3662 * q^94 + 612 * q^97 + 1982 * q^98

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