LMFDB - Embedded newform 2025.4.a.bk.1.1

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 - 91x^14 + 3268x^12 - 59128x^10 + 571975x^8 - 2881141x^6 + 6555196x^4 - 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.05435$ 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)

$f(q)$	$=$	$q-5.05435 q^{2} +17.5465 q^{4} +21.0117 q^{7} -48.2513 q^{8} +O(q^{10})$	$q-5.05435 q^{2} +17.5465 q^{4} +21.0117 q^{7} -48.2513 q^{8} -28.5629 q^{11} -10.0704 q^{13} -106.201 q^{14} +103.507 q^{16} -82.7541 q^{17} +1.91981 q^{19} +144.367 q^{22} +170.564 q^{23} +50.8993 q^{26} +368.682 q^{28} -256.428 q^{29} +48.0525 q^{31} -137.151 q^{32} +418.268 q^{34} +161.834 q^{37} -9.70338 q^{38} +279.348 q^{41} +269.073 q^{43} -501.179 q^{44} -862.088 q^{46} -9.58841 q^{47} +98.4924 q^{49} -176.700 q^{52} +35.7716 q^{53} -1013.84 q^{56} +1296.08 q^{58} -562.771 q^{59} -79.3242 q^{61} -242.874 q^{62} -134.846 q^{64} +466.752 q^{67} -1452.04 q^{68} +316.854 q^{71} -633.515 q^{73} -817.968 q^{74} +33.6858 q^{76} -600.156 q^{77} -791.315 q^{79} -1411.92 q^{82} +228.078 q^{83} -1359.99 q^{86} +1378.20 q^{88} +53.9091 q^{89} -211.596 q^{91} +2992.79 q^{92} +48.4632 q^{94} -96.6940 q^{97} -497.815 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 54 q^{4}+O(q^{10}) $ 16 * q + 54 * q^4	$ 16 q + 54 q^{4} - 90 q^{11} - 102 q^{14} + 146 q^{16} + 4 q^{19} - 468 q^{26} - 516 q^{29} + 38 q^{31} + 212 q^{34} - 576 q^{41} - 1644 q^{44} - 290 q^{46} - 4 q^{49} - 2430 q^{56} - 2202 q^{59} + 20 q^{61} - 322 q^{64} - 2952 q^{71} - 4080 q^{74} - 396 q^{76} - 218 q^{79} - 6108 q^{86} - 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) $ 16 * q + 54 * q^4 - 90 * q^11 - 102 * q^14 + 146 * q^16 + 4 * q^19 - 468 * q^26 - 516 * q^29 + 38 * q^31 + 212 * q^34 - 576 * q^41 - 1644 * q^44 - 290 * q^46 - 4 * q^49 - 2430 * q^56 - 2202 * q^59 + 20 * q^61 - 322 * q^64 - 2952 * q^71 - 4080 * q^74 - 396 * q^76 - 218 * q^79 - 6108 * q^86 - 4074 * q^89 - 942 * q^91 - 1078 * q^94

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 $

$2$	−5.05435	−1.78698	−0.893492	−	0.449080i	$-0.851752\pi$
$2$	−5.05435	−1.78698	−0.893492	+	0.449080i	$0.851752\pi$
$3$	0	0
$3$	0	0
$4$	17.5465	2.19331
$5$	0	0
$5$	0	0
$6$	0	0
$7$	21.0117	1.13453	0.567263	−	0.823537i	$-0.308002\pi$
$7$	21.0117	1.13453	0.567263	+	0.823537i	$0.308002\pi$
$8$	−48.2513	−2.13243
$9$	0	0
$10$	0	0
$11$	−28.5629	−0.782914	−0.391457	−	0.920196i	$-0.628029\pi$
$11$	−28.5629	−0.782914	−0.391457	+	0.920196i	$0.628029\pi$
$12$	0	0
$13$	−10.0704	−0.214848	−0.107424	−	0.994213i	$-0.534260\pi$
$13$	−10.0704	−0.214848	−0.107424	+	0.994213i	$0.534260\pi$
$14$	−106.201	−2.02738
$15$	0	0
$16$	103.507	1.61730
$17$	−82.7541	−1.18064	−0.590318	−	0.807171i	$-0.700998\pi$
$17$	−82.7541	−1.18064	−0.590318	+	0.807171i	$0.700998\pi$
$18$	0	0
$19$	1.91981	0.0231807	0.0115904	−	0.999933i	$-0.496311\pi$
$19$	1.91981	0.0231807	0.0115904	+	0.999933i	$0.496311\pi$
$20$	0	0
$21$	0	0
$22$	144.367	1.39905
$23$	170.564	1.54630	0.773151	−	0.634222i	$-0.218679\pi$
$23$	170.564	1.54630	0.773151	+	0.634222i	$0.218679\pi$
$24$	0	0
$25$	0	0
$26$	50.8993	0.383930
$27$	0	0
$28$	368.682	2.48837
$29$	−256.428	−1.64199	−0.820993	−	0.570939i	$-0.806579\pi$
$29$	−256.428	−1.64199	−0.820993	+	0.570939i	$0.806579\pi$
$30$	0	0
$31$	48.0525	0.278403	0.139201	−	0.990264i	$-0.455546\pi$
$31$	48.0525	0.278403	0.139201	+	0.990264i	$0.455546\pi$
$32$	−137.151	−0.757660
$33$	0	0
$34$	418.268	2.10978
$35$	0	0
$36$	0	0
$37$	161.834	0.719065	0.359533	−	0.933132i	$-0.382936\pi$
$37$	161.834	0.719065	0.359533	+	0.933132i	$0.382936\pi$
$38$	−9.70338	−0.0414236
$39$	0	0
$40$	0	0
$41$	279.348	1.06407	0.532034	−	0.846723i	$-0.321428\pi$
$41$	279.348	1.06407	0.532034	+	0.846723i	$0.321428\pi$
$42$	0	0
$43$	269.073	0.954261	0.477130	−	0.878833i	$-0.341677\pi$
$43$	269.073	0.954261	0.477130	+	0.878833i	$0.341677\pi$
$44$	−501.179	−1.71717
$45$	0	0
$46$	−862.088	−2.76322
$47$	−9.58841	−0.0297577	−0.0148789	−	0.999889i	$-0.504736\pi$
$47$	−9.58841	−0.0297577	−0.0148789	+	0.999889i	$0.504736\pi$
$48$	0	0
$49$	98.4924	0.287150
$50$	0	0
$51$	0	0
$52$	−176.700	−0.471229
$53$	35.7716	0.0927095	0.0463547	−	0.998925i	$-0.485240\pi$
$53$	35.7716	0.0927095	0.0463547	+	0.998925i	$0.485240\pi$
$54$	0	0
$55$	0	0
$56$	−1013.84	−2.41929
$57$	0	0
$58$	1296.08	2.93420
$59$	−562.771	−1.24181	−0.620903	−	0.783888i	$-0.713234\pi$
$59$	−562.771	−1.24181	−0.620903	+	0.783888i	$0.713234\pi$
$60$	0	0
$61$	−79.3242	−0.166499	−0.0832494	−	0.996529i	$-0.526530\pi$
$61$	−79.3242	−0.166499	−0.0832494	+	0.996529i	$0.526530\pi$
$62$	−242.874	−0.497501
$63$	0	0
$64$	−134.846	−0.263372
$65$	0	0
$66$	0	0
$67$	466.752	0.851087	0.425544	−	0.904938i	$-0.360083\pi$
$67$	466.752	0.851087	0.425544	+	0.904938i	$0.360083\pi$
$68$	−1452.04	−2.58950
$69$	0	0
$70$	0	0
$71$	316.854	0.529628	0.264814	−	0.964299i	$-0.414689\pi$
$71$	316.854	0.529628	0.264814	+	0.964299i	$0.414689\pi$
$72$	0	0
$73$	−633.515	−1.01572	−0.507859	−	0.861440i	$-0.669563\pi$
$73$	−633.515	−1.01572	−0.507859	+	0.861440i	$0.669563\pi$
$74$	−817.968	−1.28496
$75$	0	0
$76$	33.6858	0.0508425
$77$	−600.156	−0.888236
$78$	0	0
$79$	−791.315	−1.12696	−0.563481	−	0.826129i	$-0.690538\pi$
$79$	−791.315	−1.12696	−0.563481	+	0.826129i	$0.690538\pi$
$80$	0	0
$81$	0	0
$82$	−1411.92	−1.90147
$83$	228.078	0.301624	0.150812	−	0.988562i	$-0.451811\pi$
$83$	228.078	0.301624	0.150812	+	0.988562i	$0.451811\pi$
$84$	0	0
$85$	0	0
$86$	−1359.99	−1.70525
$87$	0	0
$88$	1378.20	1.66950
$89$	53.9091	0.0642062	0.0321031	−	0.999485i	$-0.489780\pi$
$89$	53.9091	0.0642062	0.0321031	+	0.999485i	$0.489780\pi$
$90$	0	0
$91$	−211.596	−0.243751
$92$	2992.79	3.39152
$93$	0	0
$94$	48.4632	0.0531766
$95$	0	0
$96$	0	0
$97$	−96.6940	−0.101214	−0.0506071	−	0.998719i	$-0.516116\pi$
$97$	−96.6940	−0.101214	−0.0506071	+	0.998719i	$0.516116\pi$
$98$	−497.815	−0.513132
$99$	0	0
$100$	0	0
$101$	−1083.42	−1.06737	−0.533686	−	0.845683i	$-0.679193\pi$
$101$	−1083.42	−1.06737	−0.533686	+	0.845683i	$0.679193\pi$
$102$	0	0
$103$	430.322	0.411659	0.205830	−	0.978588i	$-0.434011\pi$
$103$	430.322	0.411659	0.205830	+	0.978588i	$0.434011\pi$
$104$	485.909	0.458148
$105$	0	0
$106$	−180.802	−0.165670
$107$	12.9348	0.0116865	0.00584323	−	0.999983i	$-0.498140\pi$
$107$	12.9348	0.0116865	0.00584323	+	0.999983i	$0.498140\pi$
$108$	0	0
$109$	−20.1087	−0.0176703	−0.00883517	−	0.999961i	$-0.502812\pi$
$109$	−20.1087	−0.0176703	−0.00883517	+	0.999961i	$0.502812\pi$
$110$	0	0
$111$	0	0
$112$	2174.86	1.83487
$113$	1130.64	0.941254	0.470627	−	0.882332i	$-0.344028\pi$
$113$	1130.64	0.941254	0.470627	+	0.882332i	$0.344028\pi$
$114$	0	0
$115$	0	0
$116$	−4499.42	−3.60138
$117$	0	0
$118$	2844.44	2.21909
$119$	−1738.81	−1.33946
$120$	0	0
$121$	−515.159	−0.387046
$122$	400.933	0.297531
$123$	0	0
$124$	843.152	0.610623
$125$	0	0
$126$	0	0
$127$	−277.149	−0.193646	−0.0968228	−	0.995302i	$-0.530868\pi$
$127$	−277.149	−0.193646	−0.0968228	+	0.995302i	$0.530868\pi$
$128$	1778.77	1.22830
$129$	0	0
$130$	0	0
$131$	−502.599	−0.335208	−0.167604	−	0.985854i	$-0.553603\pi$
$131$	−502.599	−0.335208	−0.167604	+	0.985854i	$0.553603\pi$
$132$	0	0
$133$	40.3384	0.0262991
$134$	−2359.13	−1.52088
$135$	0	0
$136$	3992.99	2.51762
$137$	−2221.12	−1.38513	−0.692566	−	0.721354i	$-0.743520\pi$
$137$	−2221.12	−1.38513	−0.692566	+	0.721354i	$0.743520\pi$
$138$	0	0
$139$	2632.79	1.60655	0.803273	−	0.595611i	$-0.203090\pi$
$139$	2632.79	1.60655	0.803273	+	0.595611i	$0.203090\pi$
$140$	0	0
$141$	0	0
$142$	−1601.49	−0.946437
$143$	287.640	0.168208
$144$	0	0
$145$	0	0
$146$	3202.01	1.81507
$147$	0	0
$148$	2839.62	1.57713
$149$	−2565.71	−1.41068	−0.705340	−	0.708869i	$-0.749206\pi$
$149$	−2565.71	−1.41068	−0.705340	+	0.708869i	$0.749206\pi$
$150$	0	0
$151$	−1504.49	−0.810818	−0.405409	−	0.914135i	$-0.632871\pi$
$151$	−1504.49	−0.810818	−0.405409	+	0.914135i	$0.632871\pi$
$152$	−92.6331	−0.0494312
$153$	0	0
$154$	3033.40	1.58726
$155$	0	0
$156$	0	0
$157$	−1198.19	−0.609082	−0.304541	−	0.952499i	$-0.598503\pi$
$157$	−1198.19	−0.609082	−0.304541	+	0.952499i	$0.598503\pi$
$158$	3999.59	2.01386
$159$	0	0
$160$	0	0
$161$	3583.83	1.75432
$162$	0	0
$163$	2204.91	1.05952	0.529760	−	0.848148i	$-0.322282\pi$
$163$	2204.91	1.05952	0.529760	+	0.848148i	$0.322282\pi$
$164$	4901.57	2.33383
$165$	0	0
$166$	−1152.78	−0.538997
$167$	−1132.11	−0.524584	−0.262292	−	0.964989i	$-0.584478\pi$
$167$	−1132.11	−0.524584	−0.262292	+	0.964989i	$0.584478\pi$
$168$	0	0
$169$	−2095.59	−0.953840
$170$	0	0
$171$	0	0
$172$	4721.28	2.09299
$173$	−1854.00	−0.814778	−0.407389	−	0.913255i	$-0.633561\pi$
$173$	−1854.00	−0.814778	−0.407389	+	0.913255i	$0.633561\pi$
$174$	0	0
$175$	0	0
$176$	−2956.47	−1.26620
$177$	0	0
$178$	−272.476	−0.114735
$179$	3205.81	1.33862	0.669312	−	0.742981i	$-0.266589\pi$
$179$	3205.81	1.33862	0.669312	+	0.742981i	$0.266589\pi$
$180$	0	0
$181$	2278.79	0.935806	0.467903	−	0.883780i	$-0.345010\pi$
$181$	2278.79	0.935806	0.467903	+	0.883780i	$0.345010\pi$
$182$	1069.48	0.435579
$183$	0	0
$184$	−8229.91	−3.29737
$185$	0	0
$186$	0	0
$187$	2363.70	0.924336
$188$	−168.243	−0.0652679
$189$	0	0
$190$	0	0
$191$	3388.65	1.28374	0.641870	−	0.766814i	$-0.278159\pi$
$191$	3388.65	1.28374	0.641870	+	0.766814i	$0.278159\pi$
$192$	0	0
$193$	−2778.22	−1.03617	−0.518085	−	0.855329i	$-0.673355\pi$
$193$	−2778.22	−1.03617	−0.518085	+	0.855329i	$0.673355\pi$
$194$	488.725	0.180868
$195$	0	0
$196$	1728.19	0.629808
$197$	−2941.71	−1.06390	−0.531950	−	0.846776i	$-0.678541\pi$
$197$	−2941.71	−1.06390	−0.531950	+	0.846776i	$0.678541\pi$
$198$	0	0
$199$	−4929.28	−1.75592	−0.877959	−	0.478737i	$-0.841095\pi$
$199$	−4929.28	−1.75592	−0.877959	+	0.478737i	$0.841095\pi$
$200$	0	0
$201$	0	0
$202$	5475.99	1.90737
$203$	−5388.00	−1.86288
$204$	0	0
$205$	0	0
$206$	−2175.00	−0.735628
$207$	0	0
$208$	−1042.36	−0.347474
$209$	−54.8353	−0.0181485
$210$	0	0
$211$	5418.51	1.76789	0.883947	−	0.467587i	$-0.154877\pi$
$211$	5418.51	1.76789	0.883947	+	0.467587i	$0.154877\pi$
$212$	627.665	0.203341
$213$	0	0
$214$	−65.3768	−0.0208835
$215$	0	0
$216$	0	0
$217$	1009.67	0.315855
$218$	101.637	0.0315766
$219$	0	0
$220$	0	0
$221$	833.366	0.253657
$222$	0	0
$223$	4084.35	1.22650	0.613248	−	0.789891i	$-0.289863\pi$
$223$	4084.35	1.22650	0.613248	+	0.789891i	$0.289863\pi$
$224$	−2881.78	−0.859586
$225$	0	0
$226$	−5714.65	−1.68201
$227$	−221.829	−0.0648604	−0.0324302	−	0.999474i	$-0.510325\pi$
$227$	−221.829	−0.0648604	−0.0324302	+	0.999474i	$0.510325\pi$
$228$	0	0
$229$	−1598.07	−0.461152	−0.230576	−	0.973054i	$-0.574061\pi$
$229$	−1598.07	−0.461152	−0.230576	+	0.973054i	$0.574061\pi$
$230$	0	0
$231$	0	0
$232$	12373.0	3.50141
$233$	6620.40	1.86144	0.930722	−	0.365727i	$-0.119180\pi$
$233$	6620.40	1.86144	0.930722	+	0.365727i	$0.119180\pi$
$234$	0	0
$235$	0	0
$236$	−9874.65	−2.72366
$237$	0	0
$238$	8788.53	2.39360
$239$	3557.20	0.962744	0.481372	−	0.876516i	$-0.340139\pi$
$239$	3557.20	0.962744	0.481372	+	0.876516i	$0.340139\pi$
$240$	0	0
$241$	2967.34	0.793126	0.396563	−	0.918007i	$-0.370203\pi$
$241$	2967.34	0.793126	0.396563	+	0.918007i	$0.370203\pi$
$242$	2603.79	0.691645
$243$	0	0
$244$	−1391.86	−0.365183
$245$	0	0
$246$	0	0
$247$	−19.3332	−0.00498034
$248$	−2318.59	−0.593673
$249$	0	0
$250$	0	0
$251$	−903.562	−0.227220	−0.113610	−	0.993525i	$-0.536241\pi$
$251$	−903.562	−0.227220	−0.113610	+	0.993525i	$0.536241\pi$
$252$	0	0
$253$	−4871.79	−1.21062
$254$	1400.81	0.346041
$255$	0	0
$256$	−7911.76	−1.93158
$257$	−1484.49	−0.360312	−0.180156	−	0.983638i	$-0.557660\pi$
$257$	−1484.49	−0.360312	−0.180156	+	0.983638i	$0.557660\pi$
$258$	0	0
$259$	3400.42	0.815799
$260$	0	0
$261$	0	0
$262$	2540.31	0.599011
$263$	2716.65	0.636943	0.318472	−	0.947932i	$-0.396830\pi$
$263$	2716.65	0.636943	0.318472	+	0.947932i	$0.396830\pi$
$264$	0	0
$265$	0	0
$266$	−203.885	−0.0469961
$267$	0	0
$268$	8189.85	1.86670
$269$	7492.51	1.69824	0.849120	−	0.528201i	$-0.177133\pi$
$269$	7492.51	1.69824	0.849120	+	0.528201i	$0.177133\pi$
$270$	0	0
$271$	2013.02	0.451226	0.225613	−	0.974217i	$-0.427561\pi$
$271$	2013.02	0.451226	0.225613	+	0.974217i	$0.427561\pi$
$272$	−8565.63	−1.90944
$273$	0	0
$274$	11226.3	2.47521
$275$	0	0
$276$	0	0
$277$	−3570.97	−0.774580	−0.387290	−	0.921958i	$-0.626589\pi$
$277$	−3570.97	−0.774580	−0.387290	+	0.921958i	$0.626589\pi$
$278$	−13307.0	−2.87087
$279$	0	0
$280$	0	0
$281$	−6761.40	−1.43541	−0.717707	−	0.696346i	$-0.754808\pi$
$281$	−6761.40	−1.43541	−0.717707	+	0.696346i	$0.754808\pi$
$282$	0	0
$283$	−4012.29	−0.842776	−0.421388	−	0.906880i	$-0.638457\pi$
$283$	−4012.29	−0.842776	−0.421388	+	0.906880i	$0.638457\pi$
$284$	5559.66	1.16164
$285$	0	0
$286$	−1453.83	−0.300584
$287$	5869.58	1.20721
$288$	0	0
$289$	1935.23	0.393901
$290$	0	0
$291$	0	0
$292$	−11116.0	−2.22778
$293$	−2446.05	−0.487712	−0.243856	−	0.969811i	$-0.578412\pi$
$293$	−2446.05	−0.487712	−0.243856	+	0.969811i	$0.578412\pi$
$294$	0	0
$295$	0	0
$296$	−7808.72	−1.53335
$297$	0	0
$298$	12968.0	2.52086
$299$	−1717.64	−0.332220
$300$	0	0
$301$	5653.68	1.08263
$302$	7604.22	1.44892
$303$	0	0
$304$	198.714	0.0374902
$305$	0	0
$306$	0	0
$307$	−5539.06	−1.02974	−0.514871	−	0.857268i	$-0.672160\pi$
$307$	−5539.06	−1.02974	−0.514871	+	0.857268i	$0.672160\pi$
$308$	−10530.6	−1.94818
$309$	0	0
$310$	0	0
$311$	−3525.20	−0.642752	−0.321376	−	0.946952i	$-0.604145\pi$
$311$	−3525.20	−0.642752	−0.321376	+	0.946952i	$0.604145\pi$
$312$	0	0
$313$	−6910.59	−1.24795	−0.623977	−	0.781443i	$-0.714484\pi$
$313$	−6910.59	−1.24795	−0.623977	+	0.781443i	$0.714484\pi$
$314$	6056.07	1.08842
$315$	0	0
$316$	−13884.8	−2.47177
$317$	4209.66	0.745861	0.372930	−	0.927859i	$-0.378353\pi$
$317$	4209.66	0.745861	0.372930	+	0.927859i	$0.378353\pi$
$318$	0	0
$319$	7324.35	1.28553
$320$	0	0
$321$	0	0
$322$	−18114.0	−3.13494
$323$	−158.872	−0.0273680
$324$	0	0
$325$	0	0
$326$	−11144.4	−1.89334
$327$	0	0
$328$	−13478.9	−2.26905
$329$	−201.469	−0.0337609
$330$	0	0
$331$	−1696.24	−0.281673	−0.140836	−	0.990033i	$-0.544979\pi$
$331$	−1696.24	−0.281673	−0.140836	+	0.990033i	$0.544979\pi$
$332$	4001.96	0.661554
$333$	0	0
$334$	5722.10	0.937423
$335$	0	0
$336$	0	0
$337$	−7205.22	−1.16467	−0.582334	−	0.812950i	$-0.697860\pi$
$337$	−7205.22	−1.16467	−0.582334	+	0.812950i	$0.697860\pi$
$338$	10591.8	1.70450
$339$	0	0
$340$	0	0
$341$	−1372.52	−0.217965
$342$	0	0
$343$	−5137.53	−0.808747
$344$	−12983.1	−2.03489
$345$	0	0
$346$	9370.75	1.45600
$347$	6496.64	1.00507	0.502533	−	0.864558i	$-0.332401\pi$
$347$	6496.64	1.00507	0.502533	+	0.864558i	$0.332401\pi$
$348$	0	0
$349$	−6464.51	−0.991510	−0.495755	−	0.868462i	$-0.665109\pi$
$349$	−6464.51	−0.991510	−0.495755	+	0.868462i	$0.665109\pi$
$350$	0	0
$351$	0	0
$352$	3917.44	0.593183
$353$	−11849.0	−1.78657	−0.893284	−	0.449493i	$-0.851605\pi$
$353$	−11849.0	−1.78657	−0.893284	+	0.449493i	$0.851605\pi$
$354$	0	0
$355$	0	0
$356$	945.915	0.140824
$357$	0	0
$358$	−16203.3	−2.39210
$359$	−11702.9	−1.72048	−0.860242	−	0.509886i	$-0.829688\pi$
$359$	−11702.9	−1.72048	−0.860242	+	0.509886i	$0.829688\pi$
$360$	0	0
$361$	−6855.31	−0.999463
$362$	−11517.8	−1.67227
$363$	0	0
$364$	−3712.77	−0.534621
$365$	0	0
$366$	0	0
$367$	2580.51	0.367034	0.183517	−	0.983017i	$-0.441252\pi$
$367$	2580.51	0.367034	0.183517	+	0.983017i	$0.441252\pi$
$368$	17654.5	2.50083
$369$	0	0
$370$	0	0
$371$	751.622	0.105181
$372$	0	0
$373$	−4499.56	−0.624607	−0.312304	−	0.949982i	$-0.601101\pi$
$373$	−4499.56	−0.624607	−0.312304	+	0.949982i	$0.601101\pi$
$374$	−11947.0	−1.65177
$375$	0	0
$376$	462.653	0.0634561
$377$	2582.34	0.352777
$378$	0	0
$379$	−5888.24	−0.798044	−0.399022	−	0.916941i	$-0.630650\pi$
$379$	−5888.24	−0.798044	−0.399022	+	0.916941i	$0.630650\pi$
$380$	0	0
$381$	0	0
$382$	−17127.4	−2.29402
$383$	8020.16	1.07000	0.535001	−	0.844851i	$-0.320311\pi$
$383$	8020.16	1.07000	0.535001	+	0.844851i	$0.320311\pi$
$384$	0	0
$385$	0	0
$386$	14042.1	1.85162
$387$	0	0
$388$	−1696.64	−0.221994
$389$	−167.380	−0.0218162	−0.0109081	−	0.999941i	$-0.503472\pi$
$389$	−167.380	−0.0218162	−0.0109081	+	0.999941i	$0.503472\pi$
$390$	0	0
$391$	−14114.8	−1.82562
$392$	−4752.38	−0.612325
$393$	0	0
$394$	14868.4	1.90117
$395$	0	0
$396$	0	0
$397$	−5893.09	−0.745002	−0.372501	−	0.928032i	$-0.621500\pi$
$397$	−5893.09	−0.745002	−0.372501	+	0.928032i	$0.621500\pi$
$398$	24914.3	3.13779
$399$	0	0
$400$	0	0
$401$	−2632.86	−0.327877	−0.163938	−	0.986471i	$-0.552420\pi$
$401$	−2632.86	−0.327877	−0.163938	+	0.986471i	$0.552420\pi$
$402$	0	0
$403$	−483.908	−0.0598143
$404$	−19010.2	−2.34108
$405$	0	0
$406$	27232.9	3.32893
$407$	−4622.47	−0.562966
$408$	0	0
$409$	−13690.3	−1.65512	−0.827558	−	0.561381i	$-0.810270\pi$
$409$	−13690.3	−1.65512	−0.827558	+	0.561381i	$0.810270\pi$
$410$	0	0
$411$	0	0
$412$	7550.64	0.902896
$413$	−11824.8	−1.40886
$414$	0	0
$415$	0	0
$416$	1381.17	0.162782
$417$	0	0
$418$	277.157	0.0324311
$419$	258.973	0.0301949	0.0150974	−	0.999886i	$-0.495194\pi$
$419$	258.973	0.0301949	0.0150974	+	0.999886i	$0.495194\pi$
$420$	0	0
$421$	13643.1	1.57939	0.789696	−	0.613499i	$-0.210238\pi$
$421$	13643.1	1.57939	0.789696	+	0.613499i	$0.210238\pi$
$422$	−27387.1	−3.15920
$423$	0	0
$424$	−1726.02	−0.197696
$425$	0	0
$426$	0	0
$427$	−1666.74	−0.188897
$428$	226.959	0.0256320
$429$	0	0
$430$	0	0
$431$	−13258.5	−1.48176	−0.740881	−	0.671636i	$-0.765592\pi$
$431$	−13258.5	−1.48176	−0.740881	+	0.671636i	$0.765592\pi$
$432$	0	0
$433$	−14980.2	−1.66259	−0.831295	−	0.555832i	$-0.812400\pi$
$433$	−14980.2	−1.66259	−0.831295	+	0.555832i	$0.812400\pi$
$434$	−5103.20	−0.564428
$435$	0	0
$436$	−352.837	−0.0387565
$437$	327.449	0.0358444
$438$	0	0
$439$	−7362.59	−0.800450	−0.400225	−	0.916417i	$-0.631068\pi$
$439$	−7362.59	−0.800450	−0.400225	+	0.916417i	$0.631068\pi$
$440$	0	0
$441$	0	0
$442$	−4212.13	−0.453282
$443$	1587.81	0.170291	0.0851456	−	0.996369i	$-0.472864\pi$
$443$	1587.81	0.170291	0.0851456	+	0.996369i	$0.472864\pi$
$444$	0	0
$445$	0	0
$446$	−20643.8	−2.19173
$447$	0	0
$448$	−2833.35	−0.298802
$449$	−7331.63	−0.770604	−0.385302	−	0.922791i	$-0.625903\pi$
$449$	−7331.63	−0.770604	−0.385302	+	0.922791i	$0.625903\pi$
$450$	0	0
$451$	−7978.99	−0.833074
$452$	19838.8	2.06446
$453$	0	0
$454$	1121.20	0.115905
$455$	0	0
$456$	0	0
$457$	−6296.83	−0.644537	−0.322268	−	0.946648i	$-0.604445\pi$
$457$	−6296.83	−0.644537	−0.322268	+	0.946648i	$0.604445\pi$
$458$	8077.23	0.824070
$459$	0	0
$460$	0	0
$461$	6335.46	0.640068	0.320034	−	0.947406i	$-0.396306\pi$
$461$	6335.46	0.640068	0.320034	+	0.947406i	$0.396306\pi$
$462$	0	0
$463$	2295.72	0.230435	0.115217	−	0.993340i	$-0.463244\pi$
$463$	2295.72	0.230435	0.115217	+	0.993340i	$0.463244\pi$
$464$	−26542.2	−2.65558
$465$	0	0
$466$	−33461.8	−3.32637
$467$	6182.82	0.612648	0.306324	−	0.951927i	$-0.400901\pi$
$467$	6182.82	0.612648	0.306324	+	0.951927i	$0.400901\pi$
$468$	0	0
$469$	9807.26	0.965581
$470$	0	0
$471$	0	0
$472$	27154.4	2.64806
$473$	−7685.51	−0.747104
$474$	0	0
$475$	0	0
$476$	−30509.9	−2.93786
$477$	0	0
$478$	−17979.3	−1.72041
$479$	3898.36	0.371860	0.185930	−	0.982563i	$-0.440470\pi$
$479$	3898.36	0.371860	0.185930	+	0.982563i	$0.440470\pi$
$480$	0	0
$481$	−1629.74	−0.154490
$482$	−14998.0	−1.41730
$483$	0	0
$484$	−9039.22	−0.848913
$485$	0	0
$486$	0	0
$487$	−14776.3	−1.37490	−0.687450	−	0.726232i	$-0.741270\pi$
$487$	−14776.3	−1.37490	−0.687450	+	0.726232i	$0.741270\pi$
$488$	3827.49	0.355046
$489$	0	0
$490$	0	0
$491$	3637.24	0.334310	0.167155	−	0.985931i	$-0.446542\pi$
$491$	3637.24	0.334310	0.167155	+	0.985931i	$0.446542\pi$
$492$	0	0
$493$	21220.5	1.93859
$494$	97.7169	0.00889978
$495$	0	0
$496$	4973.77	0.450260
$497$	6657.64	0.600877
$498$	0	0
$499$	−3257.75	−0.292258	−0.146129	−	0.989266i	$-0.546681\pi$
$499$	−3257.75	−0.292258	−0.146129	+	0.989266i	$0.546681\pi$
$500$	0	0
$501$	0	0
$502$	4566.92	0.406039
$503$	20857.4	1.84888	0.924440	−	0.381327i	$-0.124533\pi$
$503$	20857.4	1.84888	0.924440	+	0.381327i	$0.124533\pi$
$504$	0	0
$505$	0	0
$506$	24623.8	2.16336
$507$	0	0
$508$	−4862.99	−0.424725
$509$	−19907.5	−1.73356	−0.866781	−	0.498688i	$-0.833815\pi$
$509$	−19907.5	−1.73356	−0.866781	+	0.498688i	$0.833815\pi$
$510$	0	0
$511$	−13311.2	−1.15236
$512$	25758.7	2.22340
$513$	0	0
$514$	7503.15	0.643871
$515$	0	0
$516$	0	0
$517$	273.873	0.0232977
$518$	−17186.9	−1.45782
$519$	0	0
$520$	0	0
$521$	−4264.69	−0.358617	−0.179308	−	0.983793i	$-0.557386\pi$
$521$	−4264.69	−0.358617	−0.179308	+	0.983793i	$0.557386\pi$
$522$	0	0
$523$	3687.86	0.308334	0.154167	−	0.988045i	$-0.450731\pi$
$523$	3687.86	0.308334	0.154167	+	0.988045i	$0.450731\pi$
$524$	−8818.84	−0.735215
$525$	0	0
$526$	−13730.9	−1.13821
$527$	−3976.54	−0.328692
$528$	0	0
$529$	16924.9	1.39105
$530$	0	0
$531$	0	0
$532$	707.798	0.0576822
$533$	−2813.14	−0.228613
$534$	0	0
$535$	0	0
$536$	−22521.4	−1.81488
$537$	0	0
$538$	−37869.8	−3.03473
$539$	−2813.23	−0.224813
$540$	0	0
$541$	21551.0	1.71266	0.856331	−	0.516428i	$-0.172739\pi$
$541$	21551.0	1.71266	0.856331	+	0.516428i	$0.172739\pi$
$542$	−10174.5	−0.806334
$543$	0	0
$544$	11349.8	0.894521
$545$	0	0
$546$	0	0
$547$	−19907.0	−1.55606	−0.778029	−	0.628229i	$-0.783780\pi$
$547$	−19907.0	−1.55606	−0.778029	+	0.628229i	$0.783780\pi$
$548$	−38972.8	−3.03802
$549$	0	0
$550$	0	0
$551$	−492.293	−0.0380624
$552$	0	0
$553$	−16626.9	−1.27857
$554$	18048.9	1.38416
$555$	0	0
$556$	46196.1	3.52365
$557$	1747.00	0.132896	0.0664479	−	0.997790i	$-0.478833\pi$
$557$	1747.00	0.132896	0.0664479	+	0.997790i	$0.478833\pi$
$558$	0	0
$559$	−2709.67	−0.205021
$560$	0	0
$561$	0	0
$562$	34174.5	2.56506
$563$	−2122.04	−0.158851	−0.0794257	−	0.996841i	$-0.525309\pi$
$563$	−2122.04	−0.158851	−0.0794257	+	0.996841i	$0.525309\pi$
$564$	0	0
$565$	0	0
$566$	20279.5	1.50603
$567$	0	0
$568$	−15288.6	−1.12939
$569$	10753.1	0.792254	0.396127	−	0.918196i	$-0.370354\pi$
$569$	10753.1	0.792254	0.396127	+	0.918196i	$0.370354\pi$
$570$	0	0
$571$	725.745	0.0531900	0.0265950	−	0.999646i	$-0.491534\pi$
$571$	725.745	0.0531900	0.0265950	+	0.999646i	$0.491534\pi$
$572$	5047.07	0.368931
$573$	0	0
$574$	−29666.9	−2.15727
$575$	0	0
$576$	0	0
$577$	23839.3	1.72001	0.860003	−	0.510290i	$-0.170462\pi$
$577$	23839.3	1.72001	0.860003	+	0.510290i	$0.170462\pi$
$578$	−9781.36	−0.703894
$579$	0	0
$580$	0	0
$581$	4792.30	0.342200
$582$	0	0
$583$	−1021.74	−0.0725835
$584$	30567.9	2.16594
$585$	0	0
$586$	12363.2	0.871534
$587$	7777.68	0.546881	0.273440	−	0.961889i	$-0.411838\pi$
$587$	7777.68	0.546881	0.273440	+	0.961889i	$0.411838\pi$
$588$	0	0
$589$	92.2515	0.00645357
$590$	0	0
$591$	0	0
$592$	16751.0	1.16294
$593$	6677.79	0.462435	0.231217	−	0.972902i	$-0.425729\pi$
$593$	6677.79	0.462435	0.231217	+	0.972902i	$0.425729\pi$
$594$	0	0
$595$	0	0
$596$	−45019.2	−3.09406
$597$	0	0
$598$	8681.57	0.593672
$599$	12946.9	0.883132	0.441566	−	0.897229i	$-0.354423\pi$
$599$	12946.9	0.883132	0.441566	+	0.897229i	$0.354423\pi$
$600$	0	0
$601$	−4366.09	−0.296334	−0.148167	−	0.988962i	$-0.547337\pi$
$601$	−4366.09	−0.296334	−0.148167	+	0.988962i	$0.547337\pi$
$602$	−28575.7	−1.93465
$603$	0	0
$604$	−26398.5	−1.77838
$605$	0	0
$606$	0	0
$607$	−24177.5	−1.61669	−0.808347	−	0.588706i	$-0.799638\pi$
$607$	−24177.5	−1.61669	−0.808347	+	0.588706i	$0.799638\pi$
$608$	−263.304	−0.0175631
$609$	0	0
$610$	0	0
$611$	96.5591	0.00639339
$612$	0	0
$613$	7772.63	0.512127	0.256063	−	0.966660i	$-0.417574\pi$
$613$	7772.63	0.512127	0.256063	+	0.966660i	$0.417574\pi$
$614$	27996.3	1.84013
$615$	0	0
$616$	28958.3	1.89410
$617$	626.260	0.0408627	0.0204313	−	0.999791i	$-0.493496\pi$
$617$	626.260	0.0408627	0.0204313	+	0.999791i	$0.493496\pi$
$618$	0	0
$619$	22835.1	1.48275	0.741373	−	0.671093i	$-0.234175\pi$
$619$	22835.1	1.48275	0.741373	+	0.671093i	$0.234175\pi$
$620$	0	0
$621$	0	0
$622$	17817.6	1.14859
$623$	1132.72	0.0728436
$624$	0	0
$625$	0	0
$626$	34928.6	2.23007
$627$	0	0
$628$	−21024.0	−1.33591
$629$	−13392.5	−0.848954
$630$	0	0
$631$	−26862.1	−1.69471	−0.847355	−	0.531027i	$-0.821806\pi$
$631$	−26862.1	−1.69471	−0.847355	+	0.531027i	$0.821806\pi$
$632$	38182.0	2.40316
$633$	0	0
$634$	−21277.1	−1.33284
$635$	0	0
$636$	0	0
$637$	−991.857	−0.0616936
$638$	−37019.8	−2.29723
$639$	0	0
$640$	0	0
$641$	8609.98	0.530536	0.265268	−	0.964175i	$-0.414540\pi$
$641$	8609.98	0.530536	0.265268	+	0.964175i	$0.414540\pi$
$642$	0	0
$643$	−17173.4	−1.05327	−0.526636	−	0.850091i	$-0.676547\pi$
$643$	−17173.4	−1.05327	−0.526636	+	0.850091i	$0.676547\pi$
$644$	62883.6	3.84777
$645$	0	0
$646$	802.994	0.0489062
$647$	1258.94	0.0764978	0.0382489	−	0.999268i	$-0.487822\pi$
$647$	1258.94	0.0764978	0.0382489	+	0.999268i	$0.487822\pi$
$648$	0	0
$649$	16074.4	0.972226
$650$	0	0
$651$	0	0
$652$	38688.4	2.32386
$653$	−2540.41	−0.152242	−0.0761209	−	0.997099i	$-0.524254\pi$
$653$	−2540.41	−0.152242	−0.0761209	+	0.997099i	$0.524254\pi$
$654$	0	0
$655$	0	0
$656$	28914.5	1.72092
$657$	0	0
$658$	1018.29	0.0603302
$659$	−26199.8	−1.54871	−0.774356	−	0.632750i	$-0.781926\pi$
$659$	−26199.8	−1.54871	−0.774356	+	0.632750i	$0.781926\pi$
$660$	0	0
$661$	−5443.77	−0.320330	−0.160165	−	0.987090i	$-0.551203\pi$
$661$	−5443.77	−0.320330	−0.160165	+	0.987090i	$0.551203\pi$
$662$	8573.39	0.503345
$663$	0	0
$664$	−11005.0	−0.643190
$665$	0	0
$666$	0	0
$667$	−43737.3	−2.53901
$668$	−19864.6	−1.15058
$669$	0	0
$670$	0	0
$671$	2265.73	0.130354
$672$	0	0
$673$	−16820.8	−0.963440	−0.481720	−	0.876325i	$-0.659988\pi$
$673$	−16820.8	−0.963440	−0.481720	+	0.876325i	$0.659988\pi$
$674$	36417.7	2.08124
$675$	0	0
$676$	−36770.2	−2.09207
$677$	−23199.5	−1.31703	−0.658515	−	0.752568i	$-0.728815\pi$
$677$	−23199.5	−1.31703	−0.658515	+	0.752568i	$0.728815\pi$
$678$	0	0
$679$	−2031.71	−0.114830
$680$	0	0
$681$	0	0
$682$	6937.20	0.389500
$683$	24902.4	1.39512	0.697558	−	0.716528i	$-0.254270\pi$
$683$	24902.4	1.39512	0.697558	+	0.716528i	$0.254270\pi$
$684$	0	0
$685$	0	0
$686$	25966.9	1.44522
$687$	0	0
$688$	27850.9	1.54332
$689$	−360.234	−0.0199185
$690$	0	0
$691$	8467.45	0.466161	0.233080	−	0.972457i	$-0.425119\pi$
$691$	8467.45	0.466161	0.233080	+	0.972457i	$0.425119\pi$
$692$	−32531.1	−1.78706
$693$	0	0
$694$	−32836.3	−1.79604
$695$	0	0
$696$	0	0
$697$	−23117.2	−1.25628
$698$	32673.9	1.77181
$699$	0	0
$700$	0	0
$701$	5972.30	0.321784	0.160892	−	0.986972i	$-0.448563\pi$
$701$	5972.30	0.321784	0.160892	+	0.986972i	$0.448563\pi$
$702$	0	0
$703$	310.691	0.0166685
$704$	3851.61	0.206197
$705$	0	0
$706$	59889.0	3.19257
$707$	−22764.6	−1.21096
$708$	0	0
$709$	−7549.78	−0.399913	−0.199956	−	0.979805i	$-0.564080\pi$
$709$	−7549.78	−0.399913	−0.199956	+	0.979805i	$0.564080\pi$
$710$	0	0
$711$	0	0
$712$	−2601.18	−0.136915
$713$	8196.00	0.430495
$714$	0	0
$715$	0	0
$716$	56250.7	2.93602
$717$	0	0
$718$	59150.4	3.07448
$719$	−31449.7	−1.63126	−0.815631	−	0.578573i	$-0.803610\pi$
$719$	−31449.7	−1.63126	−0.815631	+	0.578573i	$0.803610\pi$
$720$	0	0
$721$	9041.81	0.467038
$722$	34649.2	1.78602
$723$	0	0
$724$	39984.7	2.05251
$725$	0	0
$726$	0	0
$727$	−7962.03	−0.406183	−0.203092	−	0.979160i	$-0.565099\pi$
$727$	−7962.03	−0.406183	−0.203092	+	0.979160i	$0.565099\pi$
$728$	10209.8	0.519780
$729$	0	0
$730$	0	0
$731$	−22266.9	−1.12663
$732$	0	0
$733$	11003.2	0.554451	0.277226	−	0.960805i	$-0.410585\pi$
$733$	11003.2	0.554451	0.277226	+	0.960805i	$0.410585\pi$
$734$	−13042.8	−0.655884
$735$	0	0
$736$	−23393.0	−1.17157
$737$	−13331.8	−0.666328
$738$	0	0
$739$	24719.1	1.23046	0.615228	−	0.788349i	$-0.289064\pi$
$739$	24719.1	1.23046	0.615228	+	0.788349i	$0.289064\pi$
$740$	0	0
$741$	0	0
$742$	−3798.96	−0.187957
$743$	13326.8	0.658024	0.329012	−	0.944326i	$-0.393284\pi$
$743$	13326.8	0.658024	0.329012	+	0.944326i	$0.393284\pi$
$744$	0	0
$745$	0	0
$746$	22742.4	1.11616
$747$	0	0
$748$	41474.6	2.02735
$749$	271.781	0.0132586
$750$	0	0
$751$	−23800.9	−1.15647	−0.578234	−	0.815871i	$-0.696258\pi$
$751$	−23800.9	−1.15647	−0.578234	+	0.815871i	$0.696258\pi$
$752$	−992.468	−0.0481271
$753$	0	0
$754$	−13052.0	−0.630407
$755$	0	0
$756$	0	0
$757$	−20867.5	−1.00191	−0.500953	−	0.865474i	$-0.667017\pi$
$757$	−20867.5	−1.00191	−0.500953	+	0.865474i	$0.667017\pi$
$758$	29761.2	1.42609
$759$	0	0
$760$	0	0
$761$	−15487.5	−0.737742	−0.368871	−	0.929481i	$-0.620256\pi$
$761$	−15487.5	−0.737742	−0.368871	+	0.929481i	$0.620256\pi$
$762$	0	0
$763$	−422.519	−0.0200475
$764$	59458.9	2.81564
$765$	0	0
$766$	−40536.7	−1.91208
$767$	5667.33	0.266800
$768$	0	0
$769$	−18468.0	−0.866024	−0.433012	−	0.901388i	$-0.642549\pi$
$769$	−18468.0	−0.866024	−0.433012	+	0.901388i	$0.642549\pi$
$770$	0	0
$771$	0	0
$772$	−48748.0	−2.27264
$773$	−15379.9	−0.715621	−0.357810	−	0.933794i	$-0.616477\pi$
$773$	−15379.9	−0.715621	−0.357810	+	0.933794i	$0.616477\pi$
$774$	0	0
$775$	0	0
$776$	4665.61	0.215832
$777$	0	0
$778$	845.997	0.0389852
$779$	536.294	0.0246659
$780$	0	0
$781$	−9050.27	−0.414653
$782$	71341.3	3.26235
$783$	0	0
$784$	10194.7	0.464407
$785$	0	0
$786$	0	0
$787$	−38251.8	−1.73257	−0.866284	−	0.499552i	$-0.833498\pi$
$787$	−38251.8	−1.73257	−0.866284	+	0.499552i	$0.833498\pi$
$788$	−51616.7	−2.33346
$789$	0	0
$790$	0	0
$791$	23756.7	1.06788
$792$	0	0
$793$	798.826	0.0357720
$794$	29785.8	1.33131
$795$	0	0
$796$	−86491.5	−3.85127
$797$	38981.7	1.73250	0.866249	−	0.499612i	$-0.166524\pi$
$797$	38981.7	1.73250	0.866249	+	0.499612i	$0.166524\pi$
$798$	0	0
$799$	793.480	0.0351330
$800$	0	0
$801$	0	0
$802$	13307.4	0.585911
$803$	18095.1	0.795219
$804$	0	0
$805$	0	0
$806$	2445.84	0.106887
$807$	0	0
$808$	52276.5	2.27609
$809$	−4490.50	−0.195152	−0.0975758	−	0.995228i	$-0.531109\pi$
$809$	−4490.50	−0.195152	−0.0975758	+	0.995228i	$0.531109\pi$
$810$	0	0
$811$	33791.9	1.46312	0.731562	−	0.681775i	$-0.238792\pi$
$811$	33791.9	1.46312	0.731562	+	0.681775i	$0.238792\pi$
$812$	−94540.5	−4.08586
$813$	0	0
$814$	23363.6	1.00601
$815$	0	0
$816$	0	0
$817$	516.568	0.0221205
$818$	69195.7	2.95766
$819$	0	0
$820$	0	0
$821$	6765.56	0.287600	0.143800	−	0.989607i	$-0.454068\pi$
$821$	6765.56	0.287600	0.143800	+	0.989607i	$0.454068\pi$
$822$	0	0
$823$	39347.9	1.66656	0.833282	−	0.552848i	$-0.186459\pi$
$823$	39347.9	1.66656	0.833282	+	0.552848i	$0.186459\pi$
$824$	−20763.6	−0.877832
$825$	0	0
$826$	59766.6	2.51761
$827$	632.071	0.0265771	0.0132885	−	0.999912i	$-0.495770\pi$
$827$	632.071	0.0265771	0.0132885	+	0.999912i	$0.495770\pi$
$828$	0	0
$829$	−19585.8	−0.820560	−0.410280	−	0.911959i	$-0.634569\pi$
$829$	−19585.8	−0.820560	−0.410280	+	0.911959i	$0.634569\pi$
$830$	0	0
$831$	0	0
$832$	1357.96	0.0565849
$833$	−8150.64	−0.339019
$834$	0	0
$835$	0	0
$836$	−962.167	−0.0398053
$837$	0	0
$838$	−1308.94	−0.0539577
$839$	11672.6	0.480315	0.240157	−	0.970734i	$-0.422801\pi$
$839$	11672.6	0.480315	0.240157	+	0.970734i	$0.422801\pi$
$840$	0	0
$841$	41366.5	1.69611
$842$	−68957.0	−2.82235
$843$	0	0
$844$	95075.8	3.87754
$845$	0	0
$846$	0	0
$847$	−10824.4	−0.439114
$848$	3702.61	0.149939
$849$	0	0
$850$	0	0
$851$	27603.1	1.11189
$852$	0	0
$853$	−22100.9	−0.887129	−0.443565	−	0.896242i	$-0.646286\pi$
$853$	−22100.9	−0.887129	−0.443565	+	0.896242i	$0.646286\pi$
$854$	8424.28	0.337556
$855$	0	0
$856$	−624.118	−0.0249205
$857$	−29897.9	−1.19171	−0.595853	−	0.803093i	$-0.703186\pi$
$857$	−29897.9	−1.19171	−0.595853	+	0.803093i	$0.703186\pi$
$858$	0	0
$859$	14083.5	0.559397	0.279698	−	0.960088i	$-0.409766\pi$
$859$	14083.5	0.559397	0.279698	+	0.960088i	$0.409766\pi$
$860$	0	0
$861$	0	0
$862$	67013.1	2.64788
$863$	−11367.8	−0.448396	−0.224198	−	0.974544i	$-0.571976\pi$
$863$	−11367.8	−0.448396	−0.224198	+	0.974544i	$0.571976\pi$
$864$	0	0
$865$	0	0
$866$	75715.1	2.97102
$867$	0	0
$868$	17716.1	0.692768
$869$	22602.3	0.882313
$870$	0	0
$871$	−4700.38	−0.182854
$872$	970.272	0.0376807
$873$	0	0
$874$	−1655.04	−0.0640534
$875$	0	0
$876$	0	0
$877$	44820.8	1.72576	0.862879	−	0.505411i	$-0.168659\pi$
$877$	44820.8	1.72576	0.862879	+	0.505411i	$0.168659\pi$
$878$	37213.1	1.43039
$879$	0	0
$880$	0	0
$881$	11986.7	0.458391	0.229196	−	0.973380i	$-0.426390\pi$
$881$	11986.7	0.458391	0.229196	+	0.973380i	$0.426390\pi$
$882$	0	0
$883$	−14223.4	−0.542078	−0.271039	−	0.962568i	$-0.587367\pi$
$883$	−14223.4	−0.542078	−0.271039	+	0.962568i	$0.587367\pi$
$884$	14622.6	0.556349
$885$	0	0
$886$	−8025.33	−0.304307
$887$	−23140.0	−0.875947	−0.437973	−	0.898988i	$-0.644304\pi$
$887$	−23140.0	−0.875947	−0.437973	+	0.898988i	$0.644304\pi$
$888$	0	0
$889$	−5823.38	−0.219696
$890$	0	0
$891$	0	0
$892$	71666.0	2.69008
$893$	−18.4079	−0.000689806 0
$894$	0	0
$895$	0	0
$896$	37375.0	1.39354
$897$	0	0
$898$	37056.7	1.37706
$899$	−12322.0	−0.457133
$900$	0	0
$901$	−2960.24	−0.109456
$902$	40328.6	1.48869
$903$	0	0
$904$	−54554.8	−2.00715
$905$	0	0
$906$	0	0
$907$	−24525.8	−0.897867	−0.448934	−	0.893565i	$-0.648196\pi$
$907$	−24525.8	−0.897867	−0.448934	+	0.893565i	$0.648196\pi$
$908$	−3892.32	−0.142259
$909$	0	0
$910$	0	0
$911$	−8453.86	−0.307452	−0.153726	−	0.988114i	$-0.549127\pi$
$911$	−8453.86	−0.307452	−0.153726	+	0.988114i	$0.549127\pi$
$912$	0	0
$913$	−6514.57	−0.236145
$914$	31826.4	1.15178
$915$	0	0
$916$	−28040.6	−1.01145
$917$	−10560.5	−0.380302
$918$	0	0
$919$	29199.7	1.04811	0.524053	−	0.851686i	$-0.324419\pi$
$919$	29199.7	1.04811	0.524053	+	0.851686i	$0.324419\pi$
$920$	0	0
$921$	0	0
$922$	−32021.6	−1.14379
$923$	−3190.84	−0.113790
$924$	0	0
$925$	0	0
$926$	−11603.4	−0.411783
$927$	0	0
$928$	35169.5	1.24407
$929$	−50606.8	−1.78725	−0.893624	−	0.448816i	$-0.851846\pi$
$929$	−50606.8	−1.78725	−0.893624	+	0.448816i	$0.851846\pi$
$930$	0	0
$931$	189.086	0.00665634
$932$	116165.	4.08272
$933$	0	0
$934$	−31250.1	−1.09479
$935$	0	0
$936$	0	0
$937$	30351.9	1.05822	0.529110	−	0.848553i	$-0.322526\pi$
$937$	30351.9	1.05822	0.529110	+	0.848553i	$0.322526\pi$
$938$	−49569.4	−1.72548
$939$	0	0
$940$	0	0
$941$	−22500.5	−0.779486	−0.389743	−	0.920924i	$-0.627436\pi$
$941$	−22500.5	−0.779486	−0.389743	+	0.920924i	$0.627436\pi$
$942$	0	0
$943$	47646.5	1.64537
$944$	−58250.8	−2.00837
$945$	0	0
$946$	38845.3	1.33506
$947$	36629.5	1.25691	0.628457	−	0.777844i	$-0.283687\pi$
$947$	36629.5	1.25691	0.628457	+	0.777844i	$0.283687\pi$
$948$	0	0
$949$	6379.75	0.218225
$950$	0	0
$951$	0	0
$952$	83899.6	2.85630
$953$	−29632.2	−1.00722	−0.503611	−	0.863931i	$-0.667995\pi$
$953$	−29632.2	−1.00722	−0.503611	+	0.863931i	$0.667995\pi$
$954$	0	0
$955$	0	0
$956$	62416.3	2.11160
$957$	0	0
$958$	−19703.7	−0.664507
$959$	−46669.6	−1.57147
$960$	0	0
$961$	−27482.0	−0.922492
$962$	8237.27	0.276071
$963$	0	0
$964$	52066.4	1.73957
$965$	0	0
$966$	0	0
$967$	49714.8	1.65328	0.826639	−	0.562733i	$-0.190250\pi$
$967$	49714.8	1.65328	0.826639	+	0.562733i	$0.190250\pi$
$968$	24857.1	0.825347
$969$	0	0
$970$	0	0
$971$	−22201.1	−0.733745	−0.366872	−	0.930271i	$-0.619571\pi$
$971$	−22201.1	−0.733745	−0.366872	+	0.930271i	$0.619571\pi$
$972$	0	0
$973$	55319.4	1.82267
$974$	74684.4	2.45692
$975$	0	0
$976$	−8210.62	−0.269278
$977$	7907.35	0.258934	0.129467	−	0.991584i	$-0.458673\pi$
$977$	7907.35	0.258934	0.129467	+	0.991584i	$0.458673\pi$
$978$	0	0
$979$	−1539.80	−0.0502679
$980$	0	0
$981$	0	0
$982$	−18383.9	−0.597406
$983$	−8184.84	−0.265571	−0.132785	−	0.991145i	$-0.542392\pi$
$983$	−8184.84	−0.265571	−0.132785	+	0.991145i	$0.542392\pi$
$984$	0	0
$985$	0	0
$986$	−107256.	−3.46422
$987$	0	0
$988$	−339.230	−0.0109234
$989$	45894.0	1.47558
$990$	0	0
$991$	−4964.85	−0.159146	−0.0795729	−	0.996829i	$-0.525356\pi$
$991$	−4964.85	−0.159146	−0.0795729	+	0.996829i	$0.525356\pi$
$992$	−6590.46	−0.210935
$993$	0	0
$994$	−33650.0	−1.07376
$995$	0	0
$996$	0	0
$997$	7098.29	0.225481	0.112741	−	0.993624i	$-0.464037\pi$
$997$	7098.29	0.225481	0.112741	+	0.993624i	$0.464037\pi$
$998$	16465.8	0.522261
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bk.1.1		16
3.2	odd	2		2025.4.a.bl.1.16		16
5.2	odd	4		405.4.b.e.244.1		16
5.3	odd	4		405.4.b.e.244.16		16
5.4	even	2	inner	2025.4.a.bk.1.16		16
9.4	even	3		225.4.e.g.151.16		32
9.7	even	3		225.4.e.g.76.16		32
15.2	even	4		405.4.b.f.244.16		16
15.8	even	4		405.4.b.f.244.1		16
15.14	odd	2		2025.4.a.bl.1.1		16
45.2	even	12		135.4.j.a.64.16		32
45.4	even	6		225.4.e.g.151.1		32
45.7	odd	12		45.4.j.a.4.1	✓	32
45.13	odd	12		45.4.j.a.34.1	yes	32
45.22	odd	12		45.4.j.a.34.16	yes	32
45.23	even	12		135.4.j.a.19.16		32
45.32	even	12		135.4.j.a.19.1		32
45.34	even	6		225.4.e.g.76.1		32
45.38	even	12		135.4.j.a.64.1		32
45.43	odd	12		45.4.j.a.4.16	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.1	✓	32	45.7	odd	12
45.4.j.a.4.16	yes	32	45.43	odd	12
45.4.j.a.34.1	yes	32	45.13	odd	12
45.4.j.a.34.16	yes	32	45.22	odd	12
135.4.j.a.19.1		32	45.32	even	12
135.4.j.a.19.16		32	45.23	even	12
135.4.j.a.64.1		32	45.38	even	12
135.4.j.a.64.16		32	45.2	even	12
225.4.e.g.76.1		32	45.34	even	6
225.4.e.g.76.16		32	9.7	even	3
225.4.e.g.151.1		32	45.4	even	6
225.4.e.g.151.16		32	9.4	even	3
405.4.b.e.244.1		16	5.2	odd	4
405.4.b.e.244.16		16	5.3	odd	4
405.4.b.f.244.1		16	15.8	even	4
405.4.b.f.244.16		16	15.2	even	4
2025.4.a.bk.1.1		16	1.1	even	1	trivial
2025.4.a.bk.1.16		16	5.4	even	2	inner
2025.4.a.bl.1.1		16	15.14	odd	2
2025.4.a.bl.1.16		16	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-5.05435 q^{2} +17.5465 q^{4} +21.0117 q^{7} -48.2513 q^{8} +O(q^{10})\)	\(q-5.05435 q^{2} +17.5465 q^{4} +21.0117 q^{7} -48.2513 q^{8} -28.5629 q^{11} -10.0704 q^{13} -106.201 q^{14} +103.507 q^{16} -82.7541 q^{17} +1.91981 q^{19} +144.367 q^{22} +170.564 q^{23} +50.8993 q^{26} +368.682 q^{28} -256.428 q^{29} +48.0525 q^{31} -137.151 q^{32} +418.268 q^{34} +161.834 q^{37} -9.70338 q^{38} +279.348 q^{41} +269.073 q^{43} -501.179 q^{44} -862.088 q^{46} -9.58841 q^{47} +98.4924 q^{49} -176.700 q^{52} +35.7716 q^{53} -1013.84 q^{56} +1296.08 q^{58} -562.771 q^{59} -79.3242 q^{61} -242.874 q^{62} -134.846 q^{64} +466.752 q^{67} -1452.04 q^{68} +316.854 q^{71} -633.515 q^{73} -817.968 q^{74} +33.6858 q^{76} -600.156 q^{77} -791.315 q^{79} -1411.92 q^{82} +228.078 q^{83} -1359.99 q^{86} +1378.20 q^{88} +53.9091 q^{89} -211.596 q^{91} +2992.79 q^{92} +48.4632 q^{94} -96.6940 q^{97} -497.815 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 54 q^{4}+O(q^{10}) \) 16 * q + 54 * q^4	\( 16 q + 54 q^{4} - 90 q^{11} - 102 q^{14} + 146 q^{16} + 4 q^{19} - 468 q^{26} - 516 q^{29} + 38 q^{31} + 212 q^{34} - 576 q^{41} - 1644 q^{44} - 290 q^{46} - 4 q^{49} - 2430 q^{56} - 2202 q^{59} + 20 q^{61} - 322 q^{64} - 2952 q^{71} - 4080 q^{74} - 396 q^{76} - 218 q^{79} - 6108 q^{86} - 4074 q^{89} - 942 q^{91} - 1078 q^{94}+O(q^{100}) \) 16 * q + 54 * q^4 - 90 * q^11 - 102 * q^14 + 146 * q^16 + 4 * q^19 - 468 * q^26 - 516 * q^29 + 38 * q^31 + 212 * q^34 - 576 * q^41 - 1644 * q^44 - 290 * q^46 - 4 * q^49 - 2430 * q^56 - 2202 * q^59 + 20 * q^61 - 322 * q^64 - 2952 * q^71 - 4080 * q^74 - 396 * q^76 - 218 * q^79 - 6108 * q^86 - 4074 * q^89 - 942 * q^91 - 1078 * q^94

Introduction

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