LMFDB - Embedded newform 2025.4.a.m.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			$1.73205$ 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)

$f(q)$	$=$	$q+2.73205 q^{2} -0.535898 q^{4} -6.92820 q^{7} -23.3205 q^{8} +O(q^{10})$	$q+2.73205 q^{2} -0.535898 q^{4} -6.92820 q^{7} -23.3205 q^{8} +59.4974 q^{11} +25.0718 q^{13} -18.9282 q^{14} -59.4256 q^{16} +112.995 q^{17} -122.779 q^{19} +162.550 q^{22} -97.4256 q^{23} +68.4974 q^{26} +3.71281 q^{28} +126.569 q^{29} -108.215 q^{31} +24.2102 q^{32} +308.708 q^{34} -294.344 q^{37} -335.440 q^{38} +205.862 q^{41} +80.1999 q^{43} -31.8846 q^{44} -266.172 q^{46} -269.713 q^{47} -295.000 q^{49} -13.4359 q^{52} -98.7949 q^{53} +161.569 q^{56} +345.794 q^{58} +304.503 q^{59} +666.554 q^{61} -295.650 q^{62} +541.549 q^{64} +635.933 q^{67} -60.5538 q^{68} +826.482 q^{71} -751.349 q^{73} -804.161 q^{74} +65.7973 q^{76} -412.210 q^{77} +23.8667 q^{79} +562.424 q^{82} +817.230 q^{83} +219.110 q^{86} -1387.51 q^{88} +513.000 q^{89} -173.703 q^{91} +52.2102 q^{92} -736.869 q^{94} -428.102 q^{97} -805.955 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 8 q^{4} - 12 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 8 * q^4 - 12 * q^8	$ 2 q + 2 q^{2} - 8 q^{4} - 12 q^{8} + 22 q^{11} + 64 q^{13} - 24 q^{14} - 8 q^{16} + 32 q^{17} - 10 q^{19} + 190 q^{22} - 84 q^{23} + 40 q^{26} - 48 q^{28} + 170 q^{29} - 258 q^{31} - 104 q^{32} + 368 q^{34} - 76 q^{37} - 418 q^{38} + 578 q^{41} - 380 q^{43} + 248 q^{44} - 276 q^{46} - 484 q^{47} - 590 q^{49} - 304 q^{52} - 544 q^{53} + 240 q^{56} + 314 q^{58} + 706 q^{59} + 668 q^{61} - 186 q^{62} + 224 q^{64} + 1452 q^{67} + 544 q^{68} + 974 q^{71} - 1184 q^{73} - 964 q^{74} - 776 q^{76} - 672 q^{77} + 408 q^{79} + 290 q^{82} - 444 q^{83} + 556 q^{86} - 1812 q^{88} + 1026 q^{89} + 96 q^{91} - 48 q^{92} - 580 q^{94} + 668 q^{97} - 590 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 8 * q^4 - 12 * q^8 + 22 * q^11 + 64 * q^13 - 24 * q^14 - 8 * q^16 + 32 * q^17 - 10 * q^19 + 190 * q^22 - 84 * q^23 + 40 * q^26 - 48 * q^28 + 170 * q^29 - 258 * q^31 - 104 * q^32 + 368 * q^34 - 76 * q^37 - 418 * q^38 + 578 * q^41 - 380 * q^43 + 248 * q^44 - 276 * q^46 - 484 * q^47 - 590 * q^49 - 304 * q^52 - 544 * q^53 + 240 * q^56 + 314 * q^58 + 706 * q^59 + 668 * q^61 - 186 * q^62 + 224 * q^64 + 1452 * q^67 + 544 * q^68 + 974 * q^71 - 1184 * q^73 - 964 * q^74 - 776 * q^76 - 672 * q^77 + 408 * q^79 + 290 * q^82 - 444 * q^83 + 556 * q^86 - 1812 * q^88 + 1026 * q^89 + 96 * q^91 - 48 * q^92 - 580 * q^94 + 668 * q^97 - 590 * 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 $

$2$	2.73205	0.965926	0.482963	−	0.875641i	$-0.339561\pi$
$2$	2.73205	0.965926	0.482963	+	0.875641i	$0.339561\pi$
$3$	0	0
$3$	0	0
$4$	−0.535898	−0.0669873
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.92820	−0.374088	−0.187044	−	0.982352i	$-0.559891\pi$
$7$	−6.92820	−0.374088	−0.187044	+	0.982352i	$0.559891\pi$
$8$	−23.3205	−1.03063
$9$	0	0
$10$	0	0
$11$	59.4974	1.63083	0.815416	−	0.578876i	$-0.196508\pi$
$11$	59.4974	1.63083	0.815416	+	0.578876i	$0.196508\pi$
$12$	0	0
$13$	25.0718	0.534897	0.267449	−	0.963572i	$-0.413819\pi$
$13$	25.0718	0.534897	0.267449	+	0.963572i	$0.413819\pi$
$14$	−18.9282	−0.361341
$15$	0	0
$16$	−59.4256	−0.928525
$17$	112.995	1.61208	0.806038	−	0.591864i	$-0.201608\pi$
$17$	112.995	1.61208	0.806038	+	0.591864i	$0.201608\pi$
$18$	0	0
$19$	−122.779	−1.48250	−0.741251	−	0.671228i	$-0.765767\pi$
$19$	−122.779	−1.48250	−0.741251	+	0.671228i	$0.765767\pi$
$20$	0	0
$21$	0	0
$22$	162.550	1.57526
$23$	−97.4256	−0.883246	−0.441623	−	0.897201i	$-0.645597\pi$
$23$	−97.4256	−0.883246	−0.441623	+	0.897201i	$0.645597\pi$
$24$	0	0
$25$	0	0
$26$	68.4974	0.516671
$27$	0	0
$28$	3.71281	0.0250591
$29$	126.569	0.810459	0.405230	−	0.914215i	$-0.367192\pi$
$29$	126.569	0.810459	0.405230	+	0.914215i	$0.367192\pi$
$30$	0	0
$31$	−108.215	−0.626970	−0.313485	−	0.949593i	$-0.601496\pi$
$31$	−108.215	−0.626970	−0.313485	+	0.949593i	$0.601496\pi$
$32$	24.2102	0.133744
$33$	0	0
$34$	308.708	1.55714
$35$	0	0
$36$	0	0
$37$	−294.344	−1.30783	−0.653916	−	0.756567i	$-0.726875\pi$
$37$	−294.344	−1.30783	−0.653916	+	0.756567i	$0.726875\pi$
$38$	−335.440	−1.43199
$39$	0	0
$40$	0	0
$41$	205.862	0.784151	0.392075	−	0.919933i	$-0.371757\pi$
$41$	205.862	0.784151	0.392075	+	0.919933i	$0.371757\pi$
$42$	0	0
$43$	80.1999	0.284427	0.142214	−	0.989836i	$-0.454578\pi$
$43$	80.1999	0.284427	0.142214	+	0.989836i	$0.454578\pi$
$44$	−31.8846	−0.109245
$45$	0	0
$46$	−266.172	−0.853150
$47$	−269.713	−0.837057	−0.418528	−	0.908204i	$-0.637454\pi$
$47$	−269.713	−0.837057	−0.418528	+	0.908204i	$0.637454\pi$
$48$	0	0
$49$	−295.000	−0.860058
$50$	0	0
$51$	0	0
$52$	−13.4359	−0.0358313
$53$	−98.7949	−0.256048	−0.128024	−	0.991771i	$-0.540863\pi$
$53$	−98.7949	−0.256048	−0.128024	+	0.991771i	$0.540863\pi$
$54$	0	0
$55$	0	0
$56$	161.569	0.385546
$57$	0	0
$58$	345.794	0.782843
$59$	304.503	0.671913	0.335956	−	0.941878i	$-0.390941\pi$
$59$	304.503	0.671913	0.335956	+	0.941878i	$0.390941\pi$
$60$	0	0
$61$	666.554	1.39907	0.699537	−	0.714597i	$-0.253390\pi$
$61$	666.554	1.39907	0.699537	+	0.714597i	$0.253390\pi$
$62$	−295.650	−0.605606
$63$	0	0
$64$	541.549	1.05771
$65$	0	0
$66$	0	0
$67$	635.933	1.15958	0.579788	−	0.814767i	$-0.303135\pi$
$67$	635.933	1.15958	0.579788	+	0.814767i	$0.303135\pi$
$68$	−60.5538	−0.107989
$69$	0	0
$70$	0	0
$71$	826.482	1.38148	0.690742	−	0.723101i	$-0.257284\pi$
$71$	826.482	1.38148	0.690742	+	0.723101i	$0.257284\pi$
$72$	0	0
$73$	−751.349	−1.20464	−0.602320	−	0.798255i	$-0.705757\pi$
$73$	−751.349	−1.20464	−0.602320	+	0.798255i	$0.705757\pi$
$74$	−804.161	−1.26327
$75$	0	0
$76$	65.7973	0.0993088
$77$	−412.210	−0.610074
$78$	0	0
$79$	23.8667	0.0339901	0.0169950	−	0.999856i	$-0.494590\pi$
$79$	23.8667	0.0339901	0.0169950	+	0.999856i	$0.494590\pi$
$80$	0	0
$81$	0	0
$82$	562.424	0.757431
$83$	817.230	1.08076	0.540378	−	0.841423i	$-0.318281\pi$
$83$	817.230	1.08076	0.540378	+	0.841423i	$0.318281\pi$
$84$	0	0
$85$	0	0
$86$	219.110	0.274736
$87$	0	0
$88$	−1387.51	−1.68078
$89$	513.000	0.610988	0.305494	−	0.952194i	$-0.401178\pi$
$89$	513.000	0.610988	0.305494	+	0.952194i	$0.401178\pi$
$90$	0	0
$91$	−173.703	−0.200099
$92$	52.2102	0.0591662
$93$	0	0
$94$	−736.869	−0.808535
$95$	0	0
$96$	0	0
$97$	−428.102	−0.448116	−0.224058	−	0.974576i	$-0.571930\pi$
$97$	−428.102	−0.448116	−0.224058	+	0.974576i	$0.571930\pi$
$98$	−805.955	−0.830753
$99$	0	0
$100$	0	0
$101$	912.087	0.898575	0.449287	−	0.893387i	$-0.351678\pi$
$101$	912.087	0.898575	0.449287	+	0.893387i	$0.351678\pi$
$102$	0	0
$103$	−278.497	−0.266419	−0.133210	−	0.991088i	$-0.542528\pi$
$103$	−278.497	−0.266419	−0.133210	+	0.991088i	$0.542528\pi$
$104$	−584.687	−0.551282
$105$	0	0
$106$	−269.913	−0.247323
$107$	1512.33	1.36638	0.683191	−	0.730240i	$-0.260592\pi$
$107$	1512.33	1.36638	0.683191	+	0.730240i	$0.260592\pi$
$108$	0	0
$109$	1991.53	1.75004	0.875020	−	0.484087i	$-0.160848\pi$
$109$	1991.53	1.75004	0.875020	+	0.484087i	$0.160848\pi$
$110$	0	0
$111$	0	0
$112$	411.713	0.347350
$113$	507.902	0.422827	0.211413	−	0.977397i	$-0.432193\pi$
$113$	507.902	0.422827	0.211413	+	0.977397i	$0.432193\pi$
$114$	0	0
$115$	0	0
$116$	−67.8282	−0.0542905
$117$	0	0
$118$	831.917	0.649018
$119$	−782.851	−0.603058
$120$	0	0
$121$	2208.94	1.65961
$122$	1821.06	1.35140
$123$	0	0
$124$	57.9925	0.0419990
$125$	0	0
$126$	0	0
$127$	−1845.91	−1.28975	−0.644873	−	0.764290i	$-0.723090\pi$
$127$	−1845.91	−1.28975	−0.644873	+	0.764290i	$0.723090\pi$
$128$	1285.86	0.887928
$129$	0	0
$130$	0	0
$131$	2564.71	1.71053	0.855265	−	0.518190i	$-0.173394\pi$
$131$	2564.71	1.71053	0.855265	+	0.518190i	$0.173394\pi$
$132$	0	0
$133$	850.641	0.554586
$134$	1737.40	1.12006
$135$	0	0
$136$	−2635.10	−1.66145
$137$	−506.780	−0.316037	−0.158019	−	0.987436i	$-0.550511\pi$
$137$	−506.780	−0.316037	−0.158019	+	0.987436i	$0.550511\pi$
$138$	0	0
$139$	2222.65	1.35628	0.678138	−	0.734935i	$-0.262787\pi$
$139$	2222.65	1.35628	0.678138	+	0.734935i	$0.262787\pi$
$140$	0	0
$141$	0	0
$142$	2257.99	1.33441
$143$	1491.71	0.872327
$144$	0	0
$145$	0	0
$146$	−2052.72	−1.16359
$147$	0	0
$148$	157.738	0.0876081
$149$	510.092	0.280459	0.140229	−	0.990119i	$-0.455216\pi$
$149$	510.092	0.280459	0.140229	+	0.990119i	$0.455216\pi$
$150$	0	0
$151$	675.261	0.363920	0.181960	−	0.983306i	$-0.441756\pi$
$151$	675.261	0.363920	0.181960	+	0.983306i	$0.441756\pi$
$152$	2863.28	1.52791
$153$	0	0
$154$	−1126.18	−0.589286
$155$	0	0
$156$	0	0
$157$	3627.98	1.84423	0.922115	−	0.386915i	$-0.126459\pi$
$157$	3627.98	1.84423	0.922115	+	0.386915i	$0.126459\pi$
$158$	65.2051	0.0328319
$159$	0	0
$160$	0	0
$161$	674.985	0.330411
$162$	0	0
$163$	119.559	0.0574514	0.0287257	−	0.999587i	$-0.490855\pi$
$163$	119.559	0.0574514	0.0287257	+	0.999587i	$0.490855\pi$
$164$	−110.321	−0.0525281
$165$	0	0
$166$	2232.72	1.04393
$167$	3068.99	1.42207	0.711035	−	0.703157i	$-0.248227\pi$
$167$	3068.99	1.42207	0.711035	+	0.703157i	$0.248227\pi$
$168$	0	0
$169$	−1568.41	−0.713885
$170$	0	0
$171$	0	0
$172$	−42.9790	−0.0190530
$173$	−2112.61	−0.928432	−0.464216	−	0.885722i	$-0.653664\pi$
$173$	−2112.61	−0.928432	−0.464216	+	0.885722i	$0.653664\pi$
$174$	0	0
$175$	0	0
$176$	−3535.67	−1.51427
$177$	0	0
$178$	1401.54	0.590169
$179$	−1624.79	−0.678449	−0.339225	−	0.940705i	$-0.610165\pi$
$179$	−1624.79	−0.678449	−0.339225	+	0.940705i	$0.610165\pi$
$180$	0	0
$181$	−2545.41	−1.04530	−0.522649	−	0.852548i	$-0.675056\pi$
$181$	−2545.41	−1.04530	−0.522649	+	0.852548i	$0.675056\pi$
$182$	−474.564	−0.193280
$183$	0	0
$184$	2272.02	0.910300
$185$	0	0
$186$	0	0
$187$	6722.90	2.62902
$188$	144.539	0.0560722
$189$	0	0
$190$	0	0
$191$	4805.65	1.82055	0.910274	−	0.414006i	$-0.135871\pi$
$191$	4805.65	1.82055	0.910274	+	0.414006i	$0.135871\pi$
$192$	0	0
$193$	−3577.84	−1.33440	−0.667198	−	0.744880i	$-0.732507\pi$
$193$	−3577.84	−1.33440	−0.667198	+	0.744880i	$0.732507\pi$
$194$	−1169.60	−0.432846
$195$	0	0
$196$	158.090	0.0576130
$197$	3788.88	1.37029	0.685144	−	0.728408i	$-0.259739\pi$
$197$	3788.88	1.37029	0.685144	+	0.728408i	$0.259739\pi$
$198$	0	0
$199$	2048.31	0.729652	0.364826	−	0.931076i	$-0.381129\pi$
$199$	2048.31	0.729652	0.364826	+	0.931076i	$0.381129\pi$
$200$	0	0
$201$	0	0
$202$	2491.87	0.867956
$203$	−876.897	−0.303183
$204$	0	0
$205$	0	0
$206$	−760.869	−0.257341
$207$	0	0
$208$	−1489.91	−0.496666
$209$	−7305.06	−2.41771
$210$	0	0
$211$	2381.79	0.777107	0.388553	−	0.921426i	$-0.372975\pi$
$211$	2381.79	0.777107	0.388553	+	0.921426i	$0.372975\pi$
$212$	52.9440	0.0171519
$213$	0	0
$214$	4131.77	1.31982
$215$	0	0
$216$	0	0
$217$	749.738	0.234542
$218$	5440.97	1.69041
$219$	0	0
$220$	0	0
$221$	2832.98	0.862295
$222$	0	0
$223$	557.154	0.167309	0.0836543	−	0.996495i	$-0.473341\pi$
$223$	557.154	0.167309	0.0836543	+	0.996495i	$0.473341\pi$
$224$	−167.733	−0.0500320
$225$	0	0
$226$	1387.62	0.408419
$227$	−5178.09	−1.51402	−0.757008	−	0.653405i	$-0.773340\pi$
$227$	−5178.09	−1.51402	−0.757008	+	0.653405i	$0.773340\pi$
$228$	0	0
$229$	−918.276	−0.264984	−0.132492	−	0.991184i	$-0.542298\pi$
$229$	−918.276	−0.264984	−0.132492	+	0.991184i	$0.542298\pi$
$230$	0	0
$231$	0	0
$232$	−2951.66	−0.835284
$233$	−5896.92	−1.65803	−0.829013	−	0.559230i	$-0.811097\pi$
$233$	−5896.92	−1.65803	−0.829013	+	0.559230i	$0.811097\pi$
$234$	0	0
$235$	0	0
$236$	−163.182	−0.0450096
$237$	0	0
$238$	−2138.79	−0.582509
$239$	−526.840	−0.142588	−0.0712938	−	0.997455i	$-0.522713\pi$
$239$	−526.840	−0.142588	−0.0712938	+	0.997455i	$0.522713\pi$
$240$	0	0
$241$	−5823.19	−1.55645	−0.778226	−	0.627984i	$-0.783880\pi$
$241$	−5823.19	−1.55645	−0.778226	+	0.627984i	$0.783880\pi$
$242$	6034.95	1.60306
$243$	0	0
$244$	−357.205	−0.0937201
$245$	0	0
$246$	0	0
$247$	−3078.30	−0.792986
$248$	2523.64	0.646174
$249$	0	0
$250$	0	0
$251$	3244.98	0.816022	0.408011	−	0.912977i	$-0.366222\pi$
$251$	3244.98	0.816022	0.408011	+	0.912977i	$0.366222\pi$
$252$	0	0
$253$	−5796.57	−1.44042
$254$	−5043.11	−1.24580
$255$	0	0
$256$	−819.364	−0.200040
$257$	−1261.40	−0.306163	−0.153082	−	0.988214i	$-0.548920\pi$
$257$	−1261.40	−0.306163	−0.153082	+	0.988214i	$0.548920\pi$
$258$	0	0
$259$	2039.27	0.489244
$260$	0	0
$261$	0	0
$262$	7006.91	1.65225
$263$	1924.28	0.451165	0.225582	−	0.974224i	$-0.427572\pi$
$263$	1924.28	0.451165	0.225582	+	0.974224i	$0.427572\pi$
$264$	0	0
$265$	0	0
$266$	2323.99	0.535689
$267$	0	0
$268$	−340.796	−0.0776769
$269$	4762.53	1.07947	0.539733	−	0.841836i	$-0.318525\pi$
$269$	4762.53	1.07947	0.539733	+	0.841836i	$0.318525\pi$
$270$	0	0
$271$	1595.10	0.357547	0.178774	−	0.983890i	$-0.442787\pi$
$271$	1595.10	0.357547	0.178774	+	0.983890i	$0.442787\pi$
$272$	−6714.79	−1.49685
$273$	0	0
$274$	−1384.55	−0.305269
$275$	0	0
$276$	0	0
$277$	−1049.58	−0.227666	−0.113833	−	0.993500i	$-0.536313\pi$
$277$	−1049.58	−0.227666	−0.113833	+	0.993500i	$0.536313\pi$
$278$	6072.38	1.31006
$279$	0	0
$280$	0	0
$281$	−7576.04	−1.60836	−0.804179	−	0.594387i	$-0.797395\pi$
$281$	−7576.04	−1.60836	−0.804179	+	0.594387i	$0.797395\pi$
$282$	0	0
$283$	5852.93	1.22940	0.614701	−	0.788760i	$-0.289277\pi$
$283$	5852.93	1.22940	0.614701	+	0.788760i	$0.289277\pi$
$284$	−442.910	−0.0925419
$285$	0	0
$286$	4075.42	0.842604
$287$	−1426.25	−0.293341
$288$	0	0
$289$	7854.84	1.59879
$290$	0	0
$291$	0	0
$292$	402.647	0.0806956
$293$	−3539.60	−0.705752	−0.352876	−	0.935670i	$-0.614796\pi$
$293$	−3539.60	−0.705752	−0.352876	+	0.935670i	$0.614796\pi$
$294$	0	0
$295$	0	0
$296$	6864.24	1.34789
$297$	0	0
$298$	1393.60	0.270902
$299$	−2442.64	−0.472446
$300$	0	0
$301$	−555.641	−0.106401
$302$	1844.85	0.351520
$303$	0	0
$304$	7296.25	1.37654
$305$	0	0
$306$	0	0
$307$	293.303	0.0545267	0.0272634	−	0.999628i	$-0.491321\pi$
$307$	293.303	0.0545267	0.0272634	+	0.999628i	$0.491321\pi$
$308$	220.903	0.0408672
$309$	0	0
$310$	0	0
$311$	−4591.90	−0.837243	−0.418621	−	0.908161i	$-0.637487\pi$
$311$	−4591.90	−0.837243	−0.418621	+	0.908161i	$0.637487\pi$
$312$	0	0
$313$	−7741.06	−1.39792	−0.698962	−	0.715159i	$-0.746354\pi$
$313$	−7741.06	−1.39792	−0.698962	+	0.715159i	$0.746354\pi$
$314$	9911.82	1.78139
$315$	0	0
$316$	−12.7901	−0.00227690
$317$	−1917.32	−0.339709	−0.169854	−	0.985469i	$-0.554330\pi$
$317$	−1917.32	−0.339709	−0.169854	+	0.985469i	$0.554330\pi$
$318$	0	0
$319$	7530.54	1.32172
$320$	0	0
$321$	0	0
$322$	1844.09	0.319153
$323$	−13873.4	−2.38990
$324$	0	0
$325$	0	0
$326$	326.641	0.0554938
$327$	0	0
$328$	−4800.80	−0.808170
$329$	1868.63	0.313133
$330$	0	0
$331$	8256.53	1.37106	0.685529	−	0.728046i	$-0.259571\pi$
$331$	8256.53	1.37106	0.685529	+	0.728046i	$0.259571\pi$
$332$	−437.952	−0.0723969
$333$	0	0
$334$	8384.63	1.37361
$335$	0	0
$336$	0	0
$337$	8518.68	1.37698	0.688490	−	0.725246i	$-0.258274\pi$
$337$	8518.68	1.37698	0.688490	+	0.725246i	$0.258274\pi$
$338$	−4284.96	−0.689560
$339$	0	0
$340$	0	0
$341$	−6438.54	−1.02248
$342$	0	0
$343$	4420.19	0.695825
$344$	−1870.30	−0.293139
$345$	0	0
$346$	−5771.76	−0.896796
$347$	−9627.63	−1.48945	−0.744724	−	0.667373i	$-0.767419\pi$
$347$	−9627.63	−1.48945	−0.744724	+	0.667373i	$0.767419\pi$
$348$	0	0
$349$	−7591.79	−1.16441	−0.582205	−	0.813042i	$-0.697810\pi$
$349$	−7591.79	−1.16441	−0.582205	+	0.813042i	$0.697810\pi$
$350$	0	0
$351$	0	0
$352$	1440.45	0.218114
$353$	−1387.23	−0.209163	−0.104582	−	0.994516i	$-0.533350\pi$
$353$	−1387.23	−0.209163	−0.104582	+	0.994516i	$0.533350\pi$
$354$	0	0
$355$	0	0
$356$	−274.916	−0.0409284
$357$	0	0
$358$	−4439.01	−0.655332
$359$	6335.93	0.931469	0.465735	−	0.884924i	$-0.345790\pi$
$359$	6335.93	0.931469	0.465735	+	0.884924i	$0.345790\pi$
$360$	0	0
$361$	8215.79	1.19781
$362$	−6954.19	−1.00968
$363$	0	0
$364$	93.0869	0.0134041
$365$	0	0
$366$	0	0
$367$	9190.27	1.30716	0.653581	−	0.756857i	$-0.273266\pi$
$367$	9190.27	1.30716	0.653581	+	0.756857i	$0.273266\pi$
$368$	5789.58	0.820116
$369$	0	0
$370$	0	0
$371$	684.471	0.0957843
$372$	0	0
$373$	12685.4	1.76092	0.880462	−	0.474116i	$-0.157232\pi$
$373$	12685.4	1.76092	0.880462	+	0.474116i	$0.157232\pi$
$374$	18367.3	2.53944
$375$	0	0
$376$	6289.84	0.862696
$377$	3173.32	0.433512
$378$	0	0
$379$	−13119.0	−1.77804	−0.889022	−	0.457865i	$-0.848614\pi$
$379$	−13119.0	−1.77804	−0.889022	+	0.457865i	$0.848614\pi$
$380$	0	0
$381$	0	0
$382$	13129.3	1.75851
$383$	1734.25	0.231374	0.115687	−	0.993286i	$-0.463093\pi$
$383$	1734.25	0.231374	0.115687	+	0.993286i	$0.463093\pi$
$384$	0	0
$385$	0	0
$386$	−9774.84	−1.28893
$387$	0	0
$388$	229.419	0.0300181
$389$	9363.99	1.22050	0.610248	−	0.792210i	$-0.291070\pi$
$389$	9363.99	1.22050	0.610248	+	0.792210i	$0.291070\pi$
$390$	0	0
$391$	−11008.6	−1.42386
$392$	6879.55	0.886402
$393$	0	0
$394$	10351.4	1.32360
$395$	0	0
$396$	0	0
$397$	7661.94	0.968619	0.484309	−	0.874897i	$-0.339071\pi$
$397$	7661.94	0.968619	0.484309	+	0.874897i	$0.339071\pi$
$398$	5596.08	0.704789
$399$	0	0
$400$	0	0
$401$	−145.649	−0.0181381	−0.00906906	−	0.999959i	$-0.502887\pi$
$401$	−145.649	−0.0181381	−0.00906906	+	0.999959i	$0.502887\pi$
$402$	0	0
$403$	−2713.15	−0.335364
$404$	−488.786	−0.0601931
$405$	0	0
$406$	−2395.73	−0.292852
$407$	−17512.7	−2.13285
$408$	0	0
$409$	8890.72	1.07486	0.537430	−	0.843308i	$-0.319395\pi$
$409$	8890.72	1.07486	0.537430	+	0.843308i	$0.319395\pi$
$410$	0	0
$411$	0	0
$412$	149.246	0.0178467
$413$	−2109.66	−0.251354
$414$	0	0
$415$	0	0
$416$	606.994	0.0715393
$417$	0	0
$418$	−19957.8	−2.33533
$419$	−3168.40	−0.369419	−0.184709	−	0.982793i	$-0.559134\pi$
$419$	−3168.40	−0.369419	−0.184709	+	0.982793i	$0.559134\pi$
$420$	0	0
$421$	−8495.03	−0.983426	−0.491713	−	0.870757i	$-0.663629\pi$
$421$	−8495.03	−0.983426	−0.491713	+	0.870757i	$0.663629\pi$
$422$	6507.18	0.750627
$423$	0	0
$424$	2303.95	0.263891
$425$	0	0
$426$	0	0
$427$	−4618.02	−0.523376
$428$	−810.457	−0.0915302
$429$	0	0
$430$	0	0
$431$	2072.30	0.231599	0.115799	−	0.993273i	$-0.463057\pi$
$431$	2072.30	0.231599	0.115799	+	0.993273i	$0.463057\pi$
$432$	0	0
$433$	−2906.44	−0.322574	−0.161287	−	0.986908i	$-0.551565\pi$
$433$	−2906.44	−0.322574	−0.161287	+	0.986908i	$0.551565\pi$
$434$	2048.32	0.226550
$435$	0	0
$436$	−1067.26	−0.117230
$437$	11961.9	1.30941
$438$	0	0
$439$	1931.08	0.209944	0.104972	−	0.994475i	$-0.466525\pi$
$439$	1931.08	0.209944	0.104972	+	0.994475i	$0.466525\pi$
$440$	0	0
$441$	0	0
$442$	7739.86	0.832913
$443$	244.624	0.0262358	0.0131179	−	0.999914i	$-0.495824\pi$
$443$	244.624	0.0262358	0.0131179	+	0.999914i	$0.495824\pi$
$444$	0	0
$445$	0	0
$446$	1522.17	0.161608
$447$	0	0
$448$	−3751.96	−0.395677
$449$	13276.7	1.39547	0.697734	−	0.716357i	$-0.254192\pi$
$449$	13276.7	1.39547	0.697734	+	0.716357i	$0.254192\pi$
$450$	0	0
$451$	12248.2	1.27882
$452$	−272.184	−0.0283240
$453$	0	0
$454$	−14146.8	−1.46243
$455$	0	0
$456$	0	0
$457$	−12514.2	−1.28094	−0.640470	−	0.767984i	$-0.721260\pi$
$457$	−12514.2	−1.28094	−0.640470	+	0.767984i	$0.721260\pi$
$458$	−2508.78	−0.255955
$459$	0	0
$460$	0	0
$461$	4490.89	0.453712	0.226856	−	0.973928i	$-0.427155\pi$
$461$	4490.89	0.453712	0.226856	+	0.973928i	$0.427155\pi$
$462$	0	0
$463$	6092.89	0.611578	0.305789	−	0.952099i	$-0.401080\pi$
$463$	6092.89	0.611578	0.305789	+	0.952099i	$0.401080\pi$
$464$	−7521.46	−0.752532
$465$	0	0
$466$	−16110.7	−1.60153
$467$	5193.68	0.514635	0.257318	−	0.966327i	$-0.417161\pi$
$467$	5193.68	0.514635	0.257318	+	0.966327i	$0.417161\pi$
$468$	0	0
$469$	−4405.88	−0.433783
$470$	0	0
$471$	0	0
$472$	−7101.15	−0.692494
$473$	4771.69	0.463853
$474$	0	0
$475$	0	0
$476$	419.529	0.0403972
$477$	0	0
$478$	−1439.35	−0.137729
$479$	−18882.5	−1.80118	−0.900590	−	0.434669i	$-0.856865\pi$
$479$	−18882.5	−1.80118	−0.900590	+	0.434669i	$0.856865\pi$
$480$	0	0
$481$	−7379.72	−0.699556
$482$	−15909.3	−1.50342
$483$	0	0
$484$	−1183.77	−0.111173
$485$	0	0
$486$	0	0
$487$	17619.1	1.63942	0.819708	−	0.572781i	$-0.194136\pi$
$487$	17619.1	1.63942	0.819708	+	0.572781i	$0.194136\pi$
$488$	−15544.4	−1.44193
$489$	0	0
$490$	0	0
$491$	−916.267	−0.0842170	−0.0421085	−	0.999113i	$-0.513408\pi$
$491$	−916.267	−0.0842170	−0.0421085	+	0.999113i	$0.513408\pi$
$492$	0	0
$493$	14301.7	1.30652
$494$	−8410.08	−0.765966
$495$	0	0
$496$	6430.77	0.582157
$497$	−5726.03	−0.516796
$498$	0	0
$499$	12463.4	1.11812	0.559058	−	0.829129i	$-0.311163\pi$
$499$	12463.4	1.11812	0.559058	+	0.829129i	$0.311163\pi$
$500$	0	0
$501$	0	0
$502$	8865.46	0.788217
$503$	−49.4842	−0.00438646	−0.00219323	−	0.999998i	$-0.500698\pi$
$503$	−49.4842	−0.00438646	−0.00219323	+	0.999998i	$0.500698\pi$
$504$	0	0
$505$	0	0
$506$	−15836.5	−1.39134
$507$	0	0
$508$	989.219	0.0863966
$509$	−6155.01	−0.535985	−0.267992	−	0.963421i	$-0.586360\pi$
$509$	−6155.01	−0.535985	−0.267992	+	0.963421i	$0.586360\pi$
$510$	0	0
$511$	5205.50	0.450641
$512$	−12525.4	−1.08115
$513$	0	0
$514$	−3446.21	−0.295731
$515$	0	0
$516$	0	0
$517$	−16047.2	−1.36510
$518$	5571.39	0.472573
$519$	0	0
$520$	0	0
$521$	−14718.1	−1.23764	−0.618821	−	0.785532i	$-0.712390\pi$
$521$	−14718.1	−1.23764	−0.618821	+	0.785532i	$0.712390\pi$
$522$	0	0
$523$	8318.46	0.695489	0.347744	−	0.937589i	$-0.386948\pi$
$523$	8318.46	0.695489	0.347744	+	0.937589i	$0.386948\pi$
$524$	−1374.42	−0.114584
$525$	0	0
$526$	5257.24	0.435792
$527$	−12227.8	−1.01072
$528$	0	0
$529$	−2675.25	−0.219877
$530$	0	0
$531$	0	0
$532$	−455.857	−0.0371502
$533$	5161.32	0.419440
$534$	0	0
$535$	0	0
$536$	−14830.3	−1.19510
$537$	0	0
$538$	13011.5	1.04268
$539$	−17551.7	−1.40261
$540$	0	0
$541$	19372.0	1.53949	0.769746	−	0.638350i	$-0.220383\pi$
$541$	19372.0	1.53949	0.769746	+	0.638350i	$0.220383\pi$
$542$	4357.89	0.345364
$543$	0	0
$544$	2735.63	0.215605
$545$	0	0
$546$	0	0
$547$	989.727	0.0773632	0.0386816	−	0.999252i	$-0.487684\pi$
$547$	989.727	0.0773632	0.0386816	+	0.999252i	$0.487684\pi$
$548$	271.582	0.0211705
$549$	0	0
$550$	0	0
$551$	−15540.1	−1.20151
$552$	0	0
$553$	−165.353	−0.0127153
$554$	−2867.52	−0.219908
$555$	0	0
$556$	−1191.11	−0.0908533
$557$	−4616.60	−0.351188	−0.175594	−	0.984463i	$-0.556185\pi$
$557$	−4616.60	−0.351188	−0.175594	+	0.984463i	$0.556185\pi$
$558$	0	0
$559$	2010.76	0.152139
$560$	0	0
$561$	0	0
$562$	−20698.1	−1.55355
$563$	−13509.6	−1.01130	−0.505649	−	0.862739i	$-0.668747\pi$
$563$	−13509.6	−1.01130	−0.505649	+	0.862739i	$0.668747\pi$
$564$	0	0
$565$	0	0
$566$	15990.5	1.18751
$567$	0	0
$568$	−19274.0	−1.42380
$569$	−11388.1	−0.839039	−0.419520	−	0.907746i	$-0.637801\pi$
$569$	−11388.1	−0.839039	−0.419520	+	0.907746i	$0.637801\pi$
$570$	0	0
$571$	9417.39	0.690202	0.345101	−	0.938566i	$-0.387845\pi$
$571$	9417.39	0.690202	0.345101	+	0.938566i	$0.387845\pi$
$572$	−799.404	−0.0584349
$573$	0	0
$574$	−3896.59	−0.283346
$575$	0	0
$576$	0	0
$577$	3751.39	0.270663	0.135332	−	0.990800i	$-0.456790\pi$
$577$	3751.39	0.270663	0.135332	+	0.990800i	$0.456790\pi$
$578$	21459.8	1.54431
$579$	0	0
$580$	0	0
$581$	−5661.94	−0.404297
$582$	0	0
$583$	−5878.04	−0.417571
$584$	17521.8	1.24154
$585$	0	0
$586$	−9670.35	−0.681704
$587$	9293.55	0.653468	0.326734	−	0.945116i	$-0.394052\pi$
$587$	9293.55	0.653468	0.326734	+	0.945116i	$0.394052\pi$
$588$	0	0
$589$	13286.6	0.929484
$590$	0	0
$591$	0	0
$592$	17491.5	1.21436
$593$	−1527.88	−0.105805	−0.0529027	−	0.998600i	$-0.516847\pi$
$593$	−1527.88	−0.105805	−0.0529027	+	0.998600i	$0.516847\pi$
$594$	0	0
$595$	0	0
$596$	−273.358	−0.0187872
$597$	0	0
$598$	−6673.40	−0.456347
$599$	1670.40	0.113941	0.0569704	−	0.998376i	$-0.481856\pi$
$599$	1670.40	0.113941	0.0569704	+	0.998376i	$0.481856\pi$
$600$	0	0
$601$	17393.9	1.18055	0.590276	−	0.807201i	$-0.299019\pi$
$601$	17393.9	1.18055	0.590276	+	0.807201i	$0.299019\pi$
$602$	−1518.04	−0.102775
$603$	0	0
$604$	−361.871	−0.0243780
$605$	0	0
$606$	0	0
$607$	−4081.36	−0.272911	−0.136456	−	0.990646i	$-0.543571\pi$
$607$	−4081.36	−0.272911	−0.136456	+	0.990646i	$0.543571\pi$
$608$	−2972.52	−0.198276
$609$	0	0
$610$	0	0
$611$	−6762.18	−0.447739
$612$	0	0
$613$	−2300.45	−0.151573	−0.0757866	−	0.997124i	$-0.524147\pi$
$613$	−2300.45	−0.151573	−0.0757866	+	0.997124i	$0.524147\pi$
$614$	801.320	0.0526688
$615$	0	0
$616$	9612.95	0.628761
$617$	2905.84	0.189602	0.0948012	−	0.995496i	$-0.469778\pi$
$617$	2905.84	0.189602	0.0948012	+	0.995496i	$0.469778\pi$
$618$	0	0
$619$	17855.5	1.15941	0.579703	−	0.814828i	$-0.303169\pi$
$619$	17855.5	1.15941	0.579703	+	0.814828i	$0.303169\pi$
$620$	0	0
$621$	0	0
$622$	−12545.3	−0.808714
$623$	−3554.17	−0.228563
$624$	0	0
$625$	0	0
$626$	−21149.0	−1.35029
$627$	0	0
$628$	−1944.23	−0.123540
$629$	−33259.3	−2.10832
$630$	0	0
$631$	−24250.9	−1.52998	−0.764988	−	0.644045i	$-0.777255\pi$
$631$	−24250.9	−1.52998	−0.764988	+	0.644045i	$0.777255\pi$
$632$	−556.584	−0.0350312
$633$	0	0
$634$	−5238.23	−0.328133
$635$	0	0
$636$	0	0
$637$	−7396.18	−0.460043
$638$	20573.8	1.27669
$639$	0	0
$640$	0	0
$641$	1499.84	0.0924181	0.0462091	−	0.998932i	$-0.485286\pi$
$641$	1499.84	0.0924181	0.0462091	+	0.998932i	$0.485286\pi$
$642$	0	0
$643$	−23120.1	−1.41799	−0.708996	−	0.705213i	$-0.750851\pi$
$643$	−23120.1	−1.41799	−0.708996	+	0.705213i	$0.750851\pi$
$644$	−361.723	−0.0221334
$645$	0	0
$646$	−37903.0	−2.30847
$647$	−10703.6	−0.650391	−0.325195	−	0.945647i	$-0.605430\pi$
$647$	−10703.6	−0.650391	−0.325195	+	0.945647i	$0.605430\pi$
$648$	0	0
$649$	18117.1	1.09578
$650$	0	0
$651$	0	0
$652$	−64.0714	−0.00384851
$653$	18109.1	1.08525	0.542623	−	0.839976i	$-0.317431\pi$
$653$	18109.1	1.08525	0.542623	+	0.839976i	$0.317431\pi$
$654$	0	0
$655$	0	0
$656$	−12233.5	−0.728104
$657$	0	0
$658$	5105.18	0.302463
$659$	25113.4	1.48449	0.742244	−	0.670130i	$-0.233762\pi$
$659$	25113.4	1.48449	0.742244	+	0.670130i	$0.233762\pi$
$660$	0	0
$661$	−18593.6	−1.09411	−0.547055	−	0.837096i	$-0.684251\pi$
$661$	−18593.6	−1.09411	−0.547055	+	0.837096i	$0.684251\pi$
$662$	22557.3	1.32434
$663$	0	0
$664$	−19058.2	−1.11386
$665$	0	0
$666$	0	0
$667$	−12331.1	−0.715834
$668$	−1644.67	−0.0952606
$669$	0	0
$670$	0	0
$671$	39658.2	2.28165
$672$	0	0
$673$	−4290.38	−0.245739	−0.122869	−	0.992423i	$-0.539210\pi$
$673$	−4290.38	−0.245739	−0.122869	+	0.992423i	$0.539210\pi$
$674$	23273.5	1.33006
$675$	0	0
$676$	840.506	0.0478212
$677$	26950.5	1.52998	0.764988	−	0.644045i	$-0.222745\pi$
$677$	26950.5	1.52998	0.764988	+	0.644045i	$0.222745\pi$
$678$	0	0
$679$	2965.98	0.167635
$680$	0	0
$681$	0	0
$682$	−17590.4	−0.987642
$683$	−18870.9	−1.05721	−0.528607	−	0.848867i	$-0.677285\pi$
$683$	−18870.9	−1.05721	−0.528607	+	0.848867i	$0.677285\pi$
$684$	0	0
$685$	0	0
$686$	12076.2	0.672115
$687$	0	0
$688$	−4765.93	−0.264098
$689$	−2476.97	−0.136959
$690$	0	0
$691$	3696.05	0.203480	0.101740	−	0.994811i	$-0.467559\pi$
$691$	3696.05	0.203480	0.101740	+	0.994811i	$0.467559\pi$
$692$	1132.14	0.0621931
$693$	0	0
$694$	−26303.2	−1.43870
$695$	0	0
$696$	0	0
$697$	23261.3	1.26411
$698$	−20741.1	−1.12473
$699$	0	0
$700$	0	0
$701$	−1123.53	−0.0605349	−0.0302674	−	0.999542i	$-0.509636\pi$
$701$	−1123.53	−0.0605349	−0.0302674	+	0.999542i	$0.509636\pi$
$702$	0	0
$703$	36139.3	1.93886
$704$	32220.7	1.72495
$705$	0	0
$706$	−3789.97	−0.202036
$707$	−6319.12	−0.336146
$708$	0	0
$709$	−26467.7	−1.40200	−0.700998	−	0.713163i	$-0.747262\pi$
$709$	−26467.7	−1.40200	−0.700998	+	0.713163i	$0.747262\pi$
$710$	0	0
$711$	0	0
$712$	−11963.4	−0.629702
$713$	10543.0	0.553768
$714$	0	0
$715$	0	0
$716$	870.722	0.0454475
$717$	0	0
$718$	17310.1	0.899730
$719$	−7882.34	−0.408848	−0.204424	−	0.978882i	$-0.565532\pi$
$719$	−7882.34	−0.408848	−0.204424	+	0.978882i	$0.565532\pi$
$720$	0	0
$721$	1929.49	0.0996641
$722$	22446.0	1.15700
$723$	0	0
$724$	1364.08	0.0700216
$725$	0	0
$726$	0	0
$727$	−20008.9	−1.02076	−0.510379	−	0.859950i	$-0.670495\pi$
$727$	−20008.9	−1.02076	−0.510379	+	0.859950i	$0.670495\pi$
$728$	4050.83	0.206228
$729$	0	0
$730$	0	0
$731$	9062.18	0.458518
$732$	0	0
$733$	36293.5	1.82883	0.914414	−	0.404781i	$-0.132652\pi$
$733$	36293.5	1.82883	0.914414	+	0.404781i	$0.132652\pi$
$734$	25108.3	1.26262
$735$	0	0
$736$	−2358.70	−0.118129
$737$	37836.4	1.89107
$738$	0	0
$739$	−27475.5	−1.36766	−0.683832	−	0.729639i	$-0.739688\pi$
$739$	−27475.5	−1.36766	−0.683832	+	0.729639i	$0.739688\pi$
$740$	0	0
$741$	0	0
$742$	1870.01	0.0925205
$743$	38148.9	1.88364	0.941822	−	0.336113i	$-0.109112\pi$
$743$	38148.9	1.88364	0.941822	+	0.336113i	$0.109112\pi$
$744$	0	0
$745$	0	0
$746$	34657.1	1.70092
$747$	0	0
$748$	−3602.79	−0.176111
$749$	−10477.8	−0.511146
$750$	0	0
$751$	28223.1	1.37134	0.685669	−	0.727914i	$-0.259510\pi$
$751$	28223.1	1.37134	0.685669	+	0.727914i	$0.259510\pi$
$752$	16027.9	0.777228
$753$	0	0
$754$	8669.67	0.418741
$755$	0	0
$756$	0	0
$757$	−34786.4	−1.67019	−0.835095	−	0.550106i	$-0.814587\pi$
$757$	−34786.4	−1.67019	−0.835095	+	0.550106i	$0.814587\pi$
$758$	−35841.8	−1.71746
$759$	0	0
$760$	0	0
$761$	−1165.96	−0.0555403	−0.0277702	−	0.999614i	$-0.508841\pi$
$761$	−1165.96	−0.0555403	−0.0277702	+	0.999614i	$0.508841\pi$
$762$	0	0
$763$	−13797.7	−0.654668
$764$	−2575.34	−0.121954
$765$	0	0
$766$	4738.07	0.223490
$767$	7634.43	0.359404
$768$	0	0
$769$	−26345.3	−1.23542	−0.617710	−	0.786406i	$-0.711939\pi$
$769$	−26345.3	−1.23542	−0.617710	+	0.786406i	$0.711939\pi$
$770$	0	0
$771$	0	0
$772$	1917.36	0.0893876
$773$	25867.0	1.20358	0.601792	−	0.798653i	$-0.294454\pi$
$773$	25867.0	1.20358	0.601792	+	0.798653i	$0.294454\pi$
$774$	0	0
$775$	0	0
$776$	9983.56	0.461842
$777$	0	0
$778$	25582.9	1.17891
$779$	−25275.6	−1.16250
$780$	0	0
$781$	49173.5	2.25297
$782$	−30076.0	−1.37534
$783$	0	0
$784$	17530.6	0.798586
$785$	0	0
$786$	0	0
$787$	8070.97	0.365564	0.182782	−	0.983153i	$-0.441490\pi$
$787$	8070.97	0.365564	0.182782	+	0.983153i	$0.441490\pi$
$788$	−2030.46	−0.0917919
$789$	0	0
$790$	0	0
$791$	−3518.85	−0.158174
$792$	0	0
$793$	16711.7	0.748361
$794$	20932.8	0.935614
$795$	0	0
$796$	−1097.68	−0.0488774
$797$	−12901.1	−0.573375	−0.286688	−	0.958024i	$-0.592554\pi$
$797$	−12901.1	−0.573375	−0.286688	+	0.958024i	$0.592554\pi$
$798$	0	0
$799$	−30476.2	−1.34940
$800$	0	0
$801$	0	0
$802$	−397.922	−0.0175201
$803$	−44703.3	−1.96456
$804$	0	0
$805$	0	0
$806$	−7412.48	−0.323937
$807$	0	0
$808$	−21270.3	−0.926098
$809$	−26916.2	−1.16974	−0.584872	−	0.811126i	$-0.698855\pi$
$809$	−26916.2	−1.16974	−0.584872	+	0.811126i	$0.698855\pi$
$810$	0	0
$811$	−28082.5	−1.21592	−0.607959	−	0.793969i	$-0.708011\pi$
$811$	−28082.5	−1.21592	−0.607959	+	0.793969i	$0.708011\pi$
$812$	469.928	0.0203094
$813$	0	0
$814$	−47845.5	−2.06018
$815$	0	0
$816$	0	0
$817$	−9846.90	−0.421664
$818$	24289.9	1.03823
$819$	0	0
$820$	0	0
$821$	46373.7	1.97132	0.985659	−	0.168750i	$-0.0539731\pi$
$821$	46373.7	1.97132	0.985659	+	0.168750i	$0.0539731\pi$
$822$	0	0
$823$	7804.55	0.330558	0.165279	−	0.986247i	$-0.447148\pi$
$823$	7804.55	0.330558	0.165279	+	0.986247i	$0.447148\pi$
$824$	6494.70	0.274580
$825$	0	0
$826$	−5763.69	−0.242790
$827$	−23964.1	−1.00764	−0.503818	−	0.863810i	$-0.668072\pi$
$827$	−23964.1	−1.00764	−0.503818	+	0.863810i	$0.668072\pi$
$828$	0	0
$829$	−4583.86	−0.192044	−0.0960218	−	0.995379i	$-0.530612\pi$
$829$	−4583.86	−0.192044	−0.0960218	+	0.995379i	$0.530612\pi$
$830$	0	0
$831$	0	0
$832$	13577.6	0.565767
$833$	−33333.5	−1.38648
$834$	0	0
$835$	0	0
$836$	3914.77	0.161956
$837$	0	0
$838$	−8656.23	−0.356831
$839$	26928.3	1.10807	0.554034	−	0.832494i	$-0.313088\pi$
$839$	26928.3	1.10807	0.554034	+	0.832494i	$0.313088\pi$
$840$	0	0
$841$	−8369.23	−0.343156
$842$	−23208.9	−0.949917
$843$	0	0
$844$	−1276.40	−0.0520563
$845$	0	0
$846$	0	0
$847$	−15304.0	−0.620841
$848$	5870.95	0.237747
$849$	0	0
$850$	0	0
$851$	28676.6	1.15514
$852$	0	0
$853$	7013.39	0.281517	0.140759	−	0.990044i	$-0.455046\pi$
$853$	7013.39	0.281517	0.140759	+	0.990044i	$0.455046\pi$
$854$	−12616.7	−0.505543
$855$	0	0
$856$	−35268.4	−1.40823
$857$	20665.0	0.823692	0.411846	−	0.911253i	$-0.364884\pi$
$857$	20665.0	0.823692	0.411846	+	0.911253i	$0.364884\pi$
$858$	0	0
$859$	−20255.9	−0.804565	−0.402283	−	0.915516i	$-0.631783\pi$
$859$	−20255.9	−0.804565	−0.402283	+	0.915516i	$0.631783\pi$
$860$	0	0
$861$	0	0
$862$	5661.62	0.223707
$863$	−16293.0	−0.642666	−0.321333	−	0.946966i	$-0.604131\pi$
$863$	−16293.0	−0.642666	−0.321333	+	0.946966i	$0.604131\pi$
$864$	0	0
$865$	0	0
$866$	−7940.54	−0.311583
$867$	0	0
$868$	−401.783	−0.0157113
$869$	1420.01	0.0554321
$870$	0	0
$871$	15944.0	0.620254
$872$	−46443.6	−1.80364
$873$	0	0
$874$	32680.4	1.26480
$875$	0	0
$876$	0	0
$877$	−28264.1	−1.08827	−0.544134	−	0.838998i	$-0.683142\pi$
$877$	−28264.1	−1.08827	−0.544134	+	0.838998i	$0.683142\pi$
$878$	5275.80	0.202790
$879$	0	0
$880$	0	0
$881$	−24644.6	−0.942447	−0.471224	−	0.882014i	$-0.656188\pi$
$881$	−24644.6	−0.942447	−0.471224	+	0.882014i	$0.656188\pi$
$882$	0	0
$883$	−36026.4	−1.37303	−0.686515	−	0.727116i	$-0.740860\pi$
$883$	−36026.4	−1.37303	−0.686515	+	0.727116i	$0.740860\pi$
$884$	−1518.19	−0.0577628
$885$	0	0
$886$	668.326	0.0253418
$887$	−14174.7	−0.536572	−0.268286	−	0.963339i	$-0.586457\pi$
$887$	−14174.7	−0.536572	−0.268286	+	0.963339i	$0.586457\pi$
$888$	0	0
$889$	12788.8	0.482478
$890$	0	0
$891$	0	0
$892$	−298.578	−0.0112075
$893$	33115.2	1.24094
$894$	0	0
$895$	0	0
$896$	−8908.67	−0.332163
$897$	0	0
$898$	36272.5	1.34792
$899$	−13696.7	−0.508133
$900$	0	0
$901$	−11163.3	−0.412768
$902$	33462.8	1.23524
$903$	0	0
$904$	−11844.5	−0.435778
$905$	0	0
$906$	0	0
$907$	−36005.1	−1.31812	−0.659058	−	0.752092i	$-0.729045\pi$
$907$	−36005.1	−1.31812	−0.659058	+	0.752092i	$0.729045\pi$
$908$	2774.93	0.101420
$909$	0	0
$910$	0	0
$911$	−1202.49	−0.0437325	−0.0218662	−	0.999761i	$-0.506961\pi$
$911$	−1202.49	−0.0437325	−0.0218662	+	0.999761i	$0.506961\pi$
$912$	0	0
$913$	48623.1	1.76253
$914$	−34189.4	−1.23729
$915$	0	0
$916$	492.103	0.0177506
$917$	−17768.8	−0.639889
$918$	0	0
$919$	1151.63	0.0413370	0.0206685	−	0.999786i	$-0.493421\pi$
$919$	1151.63	0.0413370	0.0206685	+	0.999786i	$0.493421\pi$
$920$	0	0
$921$	0	0
$922$	12269.3	0.438253
$923$	20721.4	0.738952
$924$	0	0
$925$	0	0
$926$	16646.1	0.590739
$927$	0	0
$928$	3064.27	0.108394
$929$	−5397.06	−0.190605	−0.0953023	−	0.995448i	$-0.530382\pi$
$929$	−5397.06	−0.190605	−0.0953023	+	0.995448i	$0.530382\pi$
$930$	0	0
$931$	36219.9	1.27504
$932$	3160.15	0.111067
$933$	0	0
$934$	14189.4	0.497100
$935$	0	0
$936$	0	0
$937$	−45594.6	−1.58966	−0.794830	−	0.606832i	$-0.792440\pi$
$937$	−45594.6	−1.58966	−0.794830	+	0.606832i	$0.792440\pi$
$938$	−12037.1	−0.419003
$939$	0	0
$940$	0	0
$941$	37245.5	1.29030	0.645149	−	0.764057i	$-0.276795\pi$
$941$	37245.5	1.29030	0.645149	+	0.764057i	$0.276795\pi$
$942$	0	0
$943$	−20056.2	−0.692598
$944$	−18095.3	−0.623888
$945$	0	0
$946$	13036.5	0.448048
$947$	4777.54	0.163938	0.0819689	−	0.996635i	$-0.473879\pi$
$947$	4777.54	0.163938	0.0819689	+	0.996635i	$0.473879\pi$
$948$	0	0
$949$	−18837.7	−0.644359
$950$	0	0
$951$	0	0
$952$	18256.5	0.621530
$953$	−56883.2	−1.93350	−0.966752	−	0.255717i	$-0.917689\pi$
$953$	−56883.2	−1.93350	−0.966752	+	0.255717i	$0.917689\pi$
$954$	0	0
$955$	0	0
$956$	282.333	0.00955156
$957$	0	0
$958$	−51588.1	−1.73981
$959$	3511.07	0.118226
$960$	0	0
$961$	−18080.4	−0.606909
$962$	−20161.8	−0.675719
$963$	0	0
$964$	3120.64	0.104262
$965$	0	0
$966$	0	0
$967$	−14050.0	−0.467236	−0.233618	−	0.972329i	$-0.575056\pi$
$967$	−14050.0	−0.467236	−0.233618	+	0.972329i	$0.575056\pi$
$968$	−51513.7	−1.71045
$969$	0	0
$970$	0	0
$971$	−779.492	−0.0257622	−0.0128811	−	0.999917i	$-0.504100\pi$
$971$	−779.492	−0.0257622	−0.0128811	+	0.999917i	$0.504100\pi$
$972$	0	0
$973$	−15398.9	−0.507366
$974$	48136.2	1.58355
$975$	0	0
$976$	−39610.4	−1.29907
$977$	46754.9	1.53103	0.765517	−	0.643415i	$-0.222483\pi$
$977$	46754.9	1.53103	0.765517	+	0.643415i	$0.222483\pi$
$978$	0	0
$979$	30522.2	0.996418
$980$	0	0
$981$	0	0
$982$	−2503.29	−0.0813474
$983$	−3596.10	−0.116681	−0.0583406	−	0.998297i	$-0.518581\pi$
$983$	−3596.10	−0.116681	−0.0583406	+	0.998297i	$0.518581\pi$
$984$	0	0
$985$	0	0
$986$	39072.9	1.26200
$987$	0	0
$988$	1649.66	0.0531200
$989$	−7813.53	−0.251219
$990$	0	0
$991$	−15143.8	−0.485427	−0.242713	−	0.970098i	$-0.578037\pi$
$991$	−15143.8	−0.485427	−0.242713	+	0.970098i	$0.578037\pi$
$992$	−2619.92	−0.0838534
$993$	0	0
$994$	−15643.8	−0.499187
$995$	0	0
$996$	0	0
$997$	−22008.9	−0.699126	−0.349563	−	0.936913i	$-0.613670\pi$
$997$	−22008.9	−0.699126	−0.349563	+	0.936913i	$0.613670\pi$
$998$	34050.7	1.08002
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.m.1.2		2
3.2	odd	2		2025.4.a.i.1.1		2
5.4	even	2		405.4.a.c.1.1	✓	2
15.14	odd	2		405.4.a.f.1.2	yes	2
45.4	even	6		405.4.e.p.136.2		4
45.14	odd	6		405.4.e.o.136.1		4
45.29	odd	6		405.4.e.o.271.1		4
45.34	even	6		405.4.e.p.271.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.c.1.1	✓	2	5.4	even	2
405.4.a.f.1.2	yes	2	15.14	odd	2
405.4.e.o.136.1		4	45.14	odd	6
405.4.e.o.271.1		4	45.29	odd	6
405.4.e.p.136.2		4	45.4	even	6
405.4.e.p.271.2		4	45.34	even	6
2025.4.a.i.1.1		2	3.2	odd	2
2025.4.a.m.1.2		2	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)

\(f(q)\)	\(=\)	\(q+2.73205 q^{2} -0.535898 q^{4} -6.92820 q^{7} -23.3205 q^{8} +O(q^{10})\)	\(q+2.73205 q^{2} -0.535898 q^{4} -6.92820 q^{7} -23.3205 q^{8} +59.4974 q^{11} +25.0718 q^{13} -18.9282 q^{14} -59.4256 q^{16} +112.995 q^{17} -122.779 q^{19} +162.550 q^{22} -97.4256 q^{23} +68.4974 q^{26} +3.71281 q^{28} +126.569 q^{29} -108.215 q^{31} +24.2102 q^{32} +308.708 q^{34} -294.344 q^{37} -335.440 q^{38} +205.862 q^{41} +80.1999 q^{43} -31.8846 q^{44} -266.172 q^{46} -269.713 q^{47} -295.000 q^{49} -13.4359 q^{52} -98.7949 q^{53} +161.569 q^{56} +345.794 q^{58} +304.503 q^{59} +666.554 q^{61} -295.650 q^{62} +541.549 q^{64} +635.933 q^{67} -60.5538 q^{68} +826.482 q^{71} -751.349 q^{73} -804.161 q^{74} +65.7973 q^{76} -412.210 q^{77} +23.8667 q^{79} +562.424 q^{82} +817.230 q^{83} +219.110 q^{86} -1387.51 q^{88} +513.000 q^{89} -173.703 q^{91} +52.2102 q^{92} -736.869 q^{94} -428.102 q^{97} -805.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 8 q^{4} - 12 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 8 * q^4 - 12 * q^8	\( 2 q + 2 q^{2} - 8 q^{4} - 12 q^{8} + 22 q^{11} + 64 q^{13} - 24 q^{14} - 8 q^{16} + 32 q^{17} - 10 q^{19} + 190 q^{22} - 84 q^{23} + 40 q^{26} - 48 q^{28} + 170 q^{29} - 258 q^{31} - 104 q^{32} + 368 q^{34} - 76 q^{37} - 418 q^{38} + 578 q^{41} - 380 q^{43} + 248 q^{44} - 276 q^{46} - 484 q^{47} - 590 q^{49} - 304 q^{52} - 544 q^{53} + 240 q^{56} + 314 q^{58} + 706 q^{59} + 668 q^{61} - 186 q^{62} + 224 q^{64} + 1452 q^{67} + 544 q^{68} + 974 q^{71} - 1184 q^{73} - 964 q^{74} - 776 q^{76} - 672 q^{77} + 408 q^{79} + 290 q^{82} - 444 q^{83} + 556 q^{86} - 1812 q^{88} + 1026 q^{89} + 96 q^{91} - 48 q^{92} - 580 q^{94} + 668 q^{97} - 590 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 8 * q^4 - 12 * q^8 + 22 * q^11 + 64 * q^13 - 24 * q^14 - 8 * q^16 + 32 * q^17 - 10 * q^19 + 190 * q^22 - 84 * q^23 + 40 * q^26 - 48 * q^28 + 170 * q^29 - 258 * q^31 - 104 * q^32 + 368 * q^34 - 76 * q^37 - 418 * q^38 + 578 * q^41 - 380 * q^43 + 248 * q^44 - 276 * q^46 - 484 * q^47 - 590 * q^49 - 304 * q^52 - 544 * q^53 + 240 * q^56 + 314 * q^58 + 706 * q^59 + 668 * q^61 - 186 * q^62 + 224 * q^64 + 1452 * q^67 + 544 * q^68 + 974 * q^71 - 1184 * q^73 - 964 * q^74 - 776 * q^76 - 672 * q^77 + 408 * q^79 + 290 * q^82 - 444 * q^83 + 556 * q^86 - 1812 * q^88 + 1026 * q^89 + 96 * q^91 - 48 * q^92 - 580 * q^94 + 668 * q^97 - 590 * q^98

Introduction

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