LMFDB - Embedded newform 2025.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,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, 2, names="a")

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$16.1697064093$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11661.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 3 $ x^4 - 2x^3 - 4x^2 + 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.63372$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.63372 q^{2} +0.669052 q^{4} -0.505348 q^{7} -2.17440 q^{8} +O(q^{10})$	$q+1.63372 q^{2} +0.669052 q^{4} -0.505348 q^{7} -2.17440 q^{8} -3.10020 q^{11} +6.23927 q^{13} -0.825599 q^{14} -4.89047 q^{16} -6.10020 q^{17} -5.57022 q^{19} -5.06487 q^{22} -3.82560 q^{23} +10.1932 q^{26} -0.338104 q^{28} -2.45932 q^{29} +4.22858 q^{31} -3.64088 q^{32} -9.96604 q^{34} -6.72677 q^{37} -9.10020 q^{38} +5.44185 q^{41} -1.32741 q^{43} -2.07420 q^{44} -6.24997 q^{46} -3.70792 q^{47} -6.74462 q^{49} +4.17440 q^{52} +2.54205 q^{53} +1.09883 q^{56} -4.01785 q^{58} -2.88232 q^{59} -2.84345 q^{61} +6.90833 q^{62} +3.83276 q^{64} -2.40652 q^{67} -4.08135 q^{68} -5.54205 q^{71} +11.7988 q^{73} -10.9897 q^{74} -3.72677 q^{76} +1.56668 q^{77} +3.40298 q^{79} +8.89047 q^{82} -13.9012 q^{83} -2.16862 q^{86} +6.74108 q^{88} -3.38513 q^{89} -3.15301 q^{91} -2.55953 q^{92} -6.05772 q^{94} -11.0756 q^{97} -11.0188 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 + q^11 + 2 * q^13 - 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 + 3 * q^22 - 15 * q^23 + 10 * q^26 - 4 * q^28 - q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 - q^37 - 23 * q^38 + 5 * q^41 - 10 * q^43 - 22 * q^44 - 20 * q^47 - 3 * q^49 + 17 * q^52 - 20 * q^53 + 30 * q^56 - 18 * q^58 - 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 + 17 * q^67 - 34 * q^68 + 8 * q^71 + 2 * q^73 - 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 + 12 * q^82 - 30 * q^83 + 34 * q^86 + 9 * q^88 + 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 - 19 * q^97 - 13 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.63372	1.15522	0.577608	−	0.816314i	$-0.303986\pi$
$2$	1.63372	1.15522	0.577608	+	0.816314i	$0.303986\pi$
$3$	0	0
$3$	0	0
$4$	0.669052	0.334526
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.505348	−0.191004	−0.0955019	−	0.995429i	$-0.530446\pi$
$7$	−0.505348	−0.191004	−0.0955019	+	0.995429i	$0.530446\pi$
$8$	−2.17440	−0.768767
$9$	0	0
$10$	0	0
$11$	−3.10020	−0.934746	−0.467373	−	0.884060i	$-0.654800\pi$
$11$	−3.10020	−0.934746	−0.467373	+	0.884060i	$0.654800\pi$
$12$	0	0
$13$	6.23927	1.73046	0.865232	−	0.501372i	$-0.167171\pi$
$13$	6.23927	1.73046	0.865232	+	0.501372i	$0.167171\pi$
$14$	−0.825599	−0.220651
$15$	0	0
$16$	−4.89047	−1.22262
$17$	−6.10020	−1.47952	−0.739758	−	0.672873i	$-0.765060\pi$
$17$	−6.10020	−1.47952	−0.739758	+	0.672873i	$0.765060\pi$
$18$	0	0
$19$	−5.57022	−1.27790	−0.638948	−	0.769250i	$-0.720630\pi$
$19$	−5.57022	−1.27790	−0.638948	+	0.769250i	$0.720630\pi$
$20$	0	0
$21$	0	0
$22$	−5.06487	−1.07983
$23$	−3.82560	−0.797693	−0.398846	−	0.917018i	$-0.630589\pi$
$23$	−3.82560	−0.797693	−0.398846	+	0.917018i	$0.630589\pi$
$24$	0	0
$25$	0	0
$26$	10.1932	1.99906
$27$	0	0
$28$	−0.338104	−0.0638957
$29$	−2.45932	−0.456685	−0.228342	−	0.973581i	$-0.573331\pi$
$29$	−2.45932	−0.456685	−0.228342	+	0.973581i	$0.573331\pi$
$30$	0	0
$31$	4.22858	0.759475	0.379738	−	0.925094i	$-0.376014\pi$
$31$	4.22858	0.759475	0.379738	+	0.925094i	$0.376014\pi$
$32$	−3.64088	−0.643623
$33$	0	0
$34$	−9.96604	−1.70916
$35$	0	0
$36$	0	0
$37$	−6.72677	−1.10587	−0.552937	−	0.833223i	$-0.686493\pi$
$37$	−6.72677	−1.10587	−0.552937	+	0.833223i	$0.686493\pi$
$38$	−9.10020	−1.47625
$39$	0	0
$40$	0	0
$41$	5.44185	0.849874	0.424937	−	0.905223i	$-0.360296\pi$
$41$	5.44185	0.849874	0.424937	+	0.905223i	$0.360296\pi$
$42$	0	0
$43$	−1.32741	−0.202428	−0.101214	−	0.994865i	$-0.532273\pi$
$43$	−1.32741	−0.202428	−0.101214	+	0.994865i	$0.532273\pi$
$44$	−2.07420	−0.312697
$45$	0	0
$46$	−6.24997	−0.921508
$47$	−3.70792	−0.540856	−0.270428	−	0.962740i	$-0.587165\pi$
$47$	−3.70792	−0.540856	−0.270428	+	0.962740i	$0.587165\pi$
$48$	0	0
$49$	−6.74462	−0.963518
$50$	0	0
$51$	0	0
$52$	4.17440	0.578885
$53$	2.54205	0.349177	0.174589	−	0.984641i	$-0.444140\pi$
$53$	2.54205	0.349177	0.174589	+	0.984641i	$0.444140\pi$
$54$	0	0
$55$	0	0
$56$	1.09883	0.146837
$57$	0	0
$58$	−4.01785	−0.527570
$59$	−2.88232	−0.375246	−0.187623	−	0.982241i	$-0.560078\pi$
$59$	−2.88232	−0.375246	−0.187623	+	0.982241i	$0.560078\pi$
$60$	0	0
$61$	−2.84345	−0.364067	−0.182033	−	0.983292i	$-0.558268\pi$
$61$	−2.84345	−0.364067	−0.182033	+	0.983292i	$0.558268\pi$
$62$	6.90833	0.877358
$63$	0	0
$64$	3.83276	0.479095
$65$	0	0
$66$	0	0
$67$	−2.40652	−0.294003	−0.147002	−	0.989136i	$-0.546962\pi$
$67$	−2.40652	−0.294003	−0.147002	+	0.989136i	$0.546962\pi$
$68$	−4.08135	−0.494937
$69$	0	0
$70$	0	0
$71$	−5.54205	−0.657720	−0.328860	−	0.944379i	$-0.606664\pi$
$71$	−5.54205	−0.657720	−0.328860	+	0.944379i	$0.606664\pi$
$72$	0	0
$73$	11.7988	1.38095	0.690473	−	0.723359i	$-0.257403\pi$
$73$	11.7988	1.38095	0.690473	+	0.723359i	$0.257403\pi$
$74$	−10.9897	−1.27752
$75$	0	0
$76$	−3.72677	−0.427490
$77$	1.56668	0.178540
$78$	0	0
$79$	3.40298	0.382865	0.191433	−	0.981506i	$-0.438687\pi$
$79$	3.40298	0.382865	0.191433	+	0.981506i	$0.438687\pi$
$80$	0	0
$81$	0	0
$82$	8.89047	0.981789
$83$	−13.9012	−1.52585	−0.762926	−	0.646486i	$-0.776238\pi$
$83$	−13.9012	−1.52585	−0.762926	+	0.646486i	$0.776238\pi$
$84$	0	0
$85$	0	0
$86$	−2.16862	−0.233848
$87$	0	0
$88$	6.74108	0.718602
$89$	−3.38513	−0.358823	−0.179411	−	0.983774i	$-0.557419\pi$
$89$	−3.38513	−0.358823	−0.179411	+	0.983774i	$0.557419\pi$
$90$	0	0
$91$	−3.15301	−0.330525
$92$	−2.55953	−0.266849
$93$	0	0
$94$	−6.05772	−0.624806
$95$	0	0
$96$	0	0
$97$	−11.0756	−1.12455	−0.562277	−	0.826949i	$-0.690074\pi$
$97$	−11.0756	−1.12455	−0.562277	+	0.826949i	$0.690074\pi$
$98$	−11.0188	−1.11307
$99$	0	0
$100$	0	0
$101$	17.3690	1.72828	0.864141	−	0.503249i	$-0.167862\pi$
$101$	17.3690	1.72828	0.864141	+	0.503249i	$0.167862\pi$
$102$	0	0
$103$	−0.832756	−0.0820539	−0.0410269	−	0.999158i	$-0.513063\pi$
$103$	−0.832756	−0.0820539	−0.0410269	+	0.999158i	$0.513063\pi$
$104$	−13.5667	−1.33032
$105$	0	0
$106$	4.15301	0.403376
$107$	−11.0684	−1.07002	−0.535012	−	0.844844i	$-0.679693\pi$
$107$	−11.0684	−1.07002	−0.535012	+	0.844844i	$0.679693\pi$
$108$	0	0
$109$	−4.65836	−0.446190	−0.223095	−	0.974797i	$-0.571616\pi$
$109$	−4.65836	−0.446190	−0.223095	+	0.974797i	$0.571616\pi$
$110$	0	0
$111$	0	0
$112$	2.47139	0.233525
$113$	−11.9942	−1.12832	−0.564160	−	0.825665i	$-0.690800\pi$
$113$	−11.9942	−1.12832	−0.564160	+	0.825665i	$0.690800\pi$
$114$	0	0
$115$	0	0
$116$	−1.64542	−0.152773
$117$	0	0
$118$	−4.70892	−0.433491
$119$	3.08273	0.282593
$120$	0	0
$121$	−1.38874	−0.126249
$122$	−4.64542	−0.420576
$123$	0	0
$124$	2.82914	0.254064
$125$	0	0
$126$	0	0
$127$	−3.22858	−0.286490	−0.143245	−	0.989687i	$-0.545754\pi$
$127$	−3.22858	−0.286490	−0.143245	+	0.989687i	$0.545754\pi$
$128$	13.5434	1.19708
$129$	0	0
$130$	0	0
$131$	9.38513	0.819982	0.409991	−	0.912090i	$-0.365532\pi$
$131$	9.38513	0.819982	0.409991	+	0.912090i	$0.365532\pi$
$132$	0	0
$133$	2.81490	0.244083
$134$	−3.93159	−0.339637
$135$	0	0
$136$	13.2643	1.13740
$137$	−2.30955	−0.197319	−0.0986593	−	0.995121i	$-0.531455\pi$
$137$	−2.30955	−0.197319	−0.0986593	+	0.995121i	$0.531455\pi$
$138$	0	0
$139$	10.8940	0.924018	0.462009	−	0.886875i	$-0.347129\pi$
$139$	10.8940	0.924018	0.462009	+	0.886875i	$0.347129\pi$
$140$	0	0
$141$	0	0
$142$	−9.05418	−0.759810
$143$	−19.3430	−1.61754
$144$	0	0
$145$	0	0
$146$	19.2760	1.59529
$147$	0	0
$148$	−4.50056	−0.369944
$149$	16.3430	1.33887	0.669436	−	0.742870i	$-0.266536\pi$
$149$	16.3430	1.33887	0.669436	+	0.742870i	$0.266536\pi$
$150$	0	0
$151$	22.7827	1.85403	0.927015	−	0.375025i	$-0.122366\pi$
$151$	22.7827	1.85403	0.927015	+	0.375025i	$0.122366\pi$
$152$	12.1119	0.982404
$153$	0	0
$154$	2.55953	0.206252
$155$	0	0
$156$	0	0
$157$	12.4607	0.994472	0.497236	−	0.867615i	$-0.334348\pi$
$157$	12.4607	0.994472	0.497236	+	0.867615i	$0.334348\pi$
$158$	5.55953	0.442292
$159$	0	0
$160$	0	0
$161$	1.93326	0.152362
$162$	0	0
$163$	−7.57384	−0.593229	−0.296614	−	0.954997i	$-0.595858\pi$
$163$	−7.57384	−0.593229	−0.296614	+	0.954997i	$0.595858\pi$
$164$	3.64088	0.284305
$165$	0	0
$166$	−22.7107	−1.76269
$167$	2.97674	0.230347	0.115174	−	0.993345i	$-0.463258\pi$
$167$	2.97674	0.230347	0.115174	+	0.993345i	$0.463258\pi$
$168$	0	0
$169$	25.9285	1.99450
$170$	0	0
$171$	0	0
$172$	−0.888105	−0.0677174
$173$	15.8530	1.20528	0.602640	−	0.798013i	$-0.294116\pi$
$173$	15.8530	1.20528	0.602640	+	0.798013i	$0.294116\pi$
$174$	0	0
$175$	0	0
$176$	15.1615	1.14284
$177$	0	0
$178$	−5.53036	−0.414518
$179$	−17.0841	−1.27693	−0.638463	−	0.769653i	$-0.720429\pi$
$179$	−17.0841	−1.27693	−0.638463	+	0.769653i	$0.720429\pi$
$180$	0	0
$181$	13.3690	0.993712	0.496856	−	0.867833i	$-0.334488\pi$
$181$	13.3690	0.993712	0.496856	+	0.867833i	$0.334488\pi$
$182$	−5.15114	−0.381828
$183$	0	0
$184$	8.31839	0.613240
$185$	0	0
$186$	0	0
$187$	18.9119	1.38297
$188$	−2.48079	−0.180930
$189$	0	0
$190$	0	0
$191$	−25.3372	−1.83334	−0.916669	−	0.399647i	$-0.869133\pi$
$191$	−25.3372	−1.83334	−0.916669	+	0.399647i	$0.869133\pi$
$192$	0	0
$193$	−9.55953	−0.688110	−0.344055	−	0.938950i	$-0.611801\pi$
$193$	−9.55953	−0.688110	−0.344055	+	0.938950i	$0.611801\pi$
$194$	−18.0944	−1.29910
$195$	0	0
$196$	−4.51250	−0.322322
$197$	2.06841	0.147368	0.0736842	−	0.997282i	$-0.476524\pi$
$197$	2.06841	0.147368	0.0736842	+	0.997282i	$0.476524\pi$
$198$	0	0
$199$	13.0970	0.928419	0.464210	−	0.885725i	$-0.346338\pi$
$199$	13.0970	0.928419	0.464210	+	0.885725i	$0.346338\pi$
$200$	0	0
$201$	0	0
$202$	28.3762	1.99654
$203$	1.24281	0.0872285
$204$	0	0
$205$	0	0
$206$	−1.36049	−0.0947900
$207$	0	0
$208$	−30.5130	−2.11570
$209$	17.2688	1.19451
$210$	0	0
$211$	11.1119	0.764974	0.382487	−	0.923961i	$-0.375068\pi$
$211$	11.1119	0.764974	0.382487	+	0.923961i	$0.375068\pi$
$212$	1.70076	0.116809
$213$	0	0
$214$	−18.0827	−1.23611
$215$	0	0
$216$	0	0
$217$	−2.13690	−0.145063
$218$	−7.61046	−0.515446
$219$	0	0
$220$	0	0
$221$	−38.0608	−2.56025
$222$	0	0
$223$	−3.89401	−0.260762	−0.130381	−	0.991464i	$-0.541620\pi$
$223$	−3.89401	−0.260762	−0.130381	+	0.991464i	$0.541620\pi$
$224$	1.83991	0.122934
$225$	0	0
$226$	−19.5952	−1.30346
$227$	−12.8081	−0.850105	−0.425053	−	0.905169i	$-0.639744\pi$
$227$	−12.8081	−0.850105	−0.425053	+	0.905169i	$0.639744\pi$
$228$	0	0
$229$	6.65295	0.439639	0.219820	−	0.975541i	$-0.429453\pi$
$229$	6.65295	0.439639	0.219820	+	0.975541i	$0.429453\pi$
$230$	0	0
$231$	0	0
$232$	5.34755	0.351084
$233$	3.65836	0.239667	0.119833	−	0.992794i	$-0.461764\pi$
$233$	3.65836	0.239667	0.119833	+	0.992794i	$0.461764\pi$
$234$	0	0
$235$	0	0
$236$	−1.92842	−0.125530
$237$	0	0
$238$	5.03632	0.326456
$239$	15.6915	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$239$	15.6915	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$240$	0	0
$241$	11.2250	0.723063	0.361532	−	0.932360i	$-0.382254\pi$
$241$	11.2250	0.723063	0.361532	+	0.932360i	$0.382254\pi$
$242$	−2.26882	−0.145845
$243$	0	0
$244$	−1.90242	−0.121790
$245$	0	0
$246$	0	0
$247$	−34.7541	−2.21135
$248$	−9.19462	−0.583859
$249$	0	0
$250$	0	0
$251$	−6.94042	−0.438075	−0.219038	−	0.975716i	$-0.570292\pi$
$251$	−6.94042	−0.438075	−0.219038	+	0.975716i	$0.570292\pi$
$252$	0	0
$253$	11.8601	0.745640
$254$	−5.27460	−0.330958
$255$	0	0
$256$	14.4607	0.903794
$257$	−18.3327	−1.14356	−0.571781	−	0.820406i	$-0.693747\pi$
$257$	−18.3327	−1.14356	−0.571781	+	0.820406i	$0.693747\pi$
$258$	0	0
$259$	3.39936	0.211226
$260$	0	0
$261$	0	0
$262$	15.3327	0.947257
$263$	−16.0766	−0.991328	−0.495664	−	0.868514i	$-0.665075\pi$
$263$	−16.0766	−0.991328	−0.495664	+	0.868514i	$0.665075\pi$
$264$	0	0
$265$	0	0
$266$	4.59877	0.281969
$267$	0	0
$268$	−1.61009	−0.0983517
$269$	18.2004	1.10970	0.554849	−	0.831951i	$-0.312776\pi$
$269$	18.2004	1.10970	0.554849	+	0.831951i	$0.312776\pi$
$270$	0	0
$271$	−2.48571	−0.150996	−0.0754979	−	0.997146i	$-0.524055\pi$
$271$	−2.48571	−0.150996	−0.0754979	+	0.997146i	$0.524055\pi$
$272$	29.8329	1.80888
$273$	0	0
$274$	−3.77317	−0.227946
$275$	0	0
$276$	0	0
$277$	7.66726	0.460681	0.230341	−	0.973110i	$-0.426016\pi$
$277$	7.66726	0.460681	0.230341	+	0.973110i	$0.426016\pi$
$278$	17.7978	1.06744
$279$	0	0
$280$	0	0
$281$	−0.273230	−0.0162996	−0.00814978	−	0.999967i	$-0.502594\pi$
$281$	−0.273230	−0.0162996	−0.00814978	+	0.999967i	$0.502594\pi$
$282$	0	0
$283$	3.37089	0.200379	0.100189	−	0.994968i	$-0.468055\pi$
$283$	3.37089	0.200379	0.100189	+	0.994968i	$0.468055\pi$
$284$	−3.70792	−0.220025
$285$	0	0
$286$	−31.6011	−1.86861
$287$	−2.75003	−0.162329
$288$	0	0
$289$	20.2125	1.18897
$290$	0	0
$291$	0	0
$292$	7.89401	0.461962
$293$	5.64404	0.329728	0.164864	−	0.986316i	$-0.447281\pi$
$293$	5.64404	0.329728	0.164864	+	0.986316i	$0.447281\pi$
$294$	0	0
$295$	0	0
$296$	14.6267	0.850159
$297$	0	0
$298$	26.7000	1.54669
$299$	−23.8690	−1.38038
$300$	0	0
$301$	0.670803	0.0386645
$302$	37.2206	2.14181
$303$	0	0
$304$	27.2410	1.56238
$305$	0	0
$306$	0	0
$307$	5.44105	0.310537	0.155269	−	0.987872i	$-0.450376\pi$
$307$	5.44105	0.310537	0.155269	+	0.987872i	$0.450376\pi$
$308$	1.04819	0.0597263
$309$	0	0
$310$	0	0
$311$	19.0797	1.08191	0.540955	−	0.841052i	$-0.318063\pi$
$311$	19.0797	1.08191	0.540955	+	0.841052i	$0.318063\pi$
$312$	0	0
$313$	−9.14231	−0.516754	−0.258377	−	0.966044i	$-0.583188\pi$
$313$	−9.14231	−0.516754	−0.258377	+	0.966044i	$0.583188\pi$
$314$	20.3573	1.14883
$315$	0	0
$316$	2.27677	0.128078
$317$	−14.2367	−0.799614	−0.399807	−	0.916599i	$-0.630923\pi$
$317$	−14.2367	−0.799614	−0.399807	+	0.916599i	$0.630923\pi$
$318$	0	0
$319$	7.62440	0.426884
$320$	0	0
$321$	0	0
$322$	3.15841	0.176011
$323$	33.9795	1.89067
$324$	0	0
$325$	0	0
$326$	−12.3736	−0.685308
$327$	0	0
$328$	−11.8328	−0.653355
$329$	1.87379	0.103305
$330$	0	0
$331$	−12.2000	−0.670574	−0.335287	−	0.942116i	$-0.608833\pi$
$331$	−12.2000	−0.670574	−0.335287	+	0.942116i	$0.608833\pi$
$332$	−9.30061	−0.510437
$333$	0	0
$334$	4.86317	0.266101
$335$	0	0
$336$	0	0
$337$	4.58986	0.250026	0.125013	−	0.992155i	$-0.460103\pi$
$337$	4.58986	0.250026	0.125013	+	0.992155i	$0.460103\pi$
$338$	42.3601	2.30408
$339$	0	0
$340$	0	0
$341$	−13.1094	−0.709917
$342$	0	0
$343$	6.94582	0.375039
$344$	2.88632	0.155620
$345$	0	0
$346$	25.8994	1.39236
$347$	−33.4603	−1.79624	−0.898121	−	0.439748i	$-0.855068\pi$
$347$	−33.4603	−1.79624	−0.898121	+	0.439748i	$0.855068\pi$
$348$	0	0
$349$	−28.0816	−1.50317	−0.751586	−	0.659636i	$-0.770711\pi$
$349$	−28.0816	−1.50317	−0.751586	+	0.659636i	$0.770711\pi$
$350$	0	0
$351$	0	0
$352$	11.2875	0.601624
$353$	1.84170	0.0980239	0.0490119	−	0.998798i	$-0.484393\pi$
$353$	1.84170	0.0980239	0.0490119	+	0.998798i	$0.484393\pi$
$354$	0	0
$355$	0	0
$356$	−2.26483	−0.120036
$357$	0	0
$358$	−27.9107	−1.47513
$359$	12.1119	0.639241	0.319621	−	0.947546i	$-0.396445\pi$
$359$	12.1119	0.639241	0.319621	+	0.947546i	$0.396445\pi$
$360$	0	0
$361$	12.0274	0.633020
$362$	21.8413	1.14795
$363$	0	0
$364$	−2.10953	−0.110569
$365$	0	0
$366$	0	0
$367$	−14.5738	−0.760744	−0.380372	−	0.924834i	$-0.624204\pi$
$367$	−14.5738	−0.760744	−0.380372	+	0.924834i	$0.624204\pi$
$368$	18.7090	0.975274
$369$	0	0
$370$	0	0
$371$	−1.28462	−0.0666942
$372$	0	0
$373$	−9.44646	−0.489119	−0.244560	−	0.969634i	$-0.578643\pi$
$373$	−9.44646	−0.489119	−0.244560	+	0.969634i	$0.578643\pi$
$374$	30.8968	1.59763
$375$	0	0
$376$	8.06251	0.415792
$377$	−15.3444	−0.790276
$378$	0	0
$379$	−28.5541	−1.46673	−0.733363	−	0.679837i	$-0.762051\pi$
$379$	−28.5541	−1.46673	−0.733363	+	0.679837i	$0.762051\pi$
$380$	0	0
$381$	0	0
$382$	−41.3940	−2.11790
$383$	−1.46541	−0.0748788	−0.0374394	−	0.999299i	$-0.511920\pi$
$383$	−1.46541	−0.0748788	−0.0374394	+	0.999299i	$0.511920\pi$
$384$	0	0
$385$	0	0
$386$	−15.6176	−0.794916
$387$	0	0
$388$	−7.41014	−0.376193
$389$	12.9101	0.654569	0.327284	−	0.944926i	$-0.393866\pi$
$389$	12.9101	0.654569	0.327284	+	0.944926i	$0.393866\pi$
$390$	0	0
$391$	23.3369	1.18020
$392$	14.6655	0.740720
$393$	0	0
$394$	3.37922	0.170242
$395$	0	0
$396$	0	0
$397$	−0.868386	−0.0435831	−0.0217915	−	0.999763i	$-0.506937\pi$
$397$	−0.868386	−0.0435831	−0.0217915	+	0.999763i	$0.506937\pi$
$398$	21.3968	1.07253
$399$	0	0
$400$	0	0
$401$	−33.4125	−1.66854	−0.834270	−	0.551356i	$-0.814111\pi$
$401$	−33.4125	−1.66854	−0.834270	+	0.551356i	$0.814111\pi$
$402$	0	0
$403$	26.3833	1.31424
$404$	11.6208	0.578156
$405$	0	0
$406$	2.03042	0.100768
$407$	20.8544	1.03371
$408$	0	0
$409$	5.05535	0.249971	0.124985	−	0.992159i	$-0.460112\pi$
$409$	5.05535	0.249971	0.124985	+	0.992159i	$0.460112\pi$
$410$	0	0
$411$	0	0
$412$	−0.557157	−0.0274492
$413$	1.45658	0.0716734
$414$	0	0
$415$	0	0
$416$	−22.7165	−1.11377
$417$	0	0
$418$	28.2125	1.37992
$419$	−10.9576	−0.535313	−0.267657	−	0.963514i	$-0.586249\pi$
$419$	−10.9576	−0.535313	−0.267657	+	0.963514i	$0.586249\pi$
$420$	0	0
$421$	−10.6386	−0.518495	−0.259248	−	0.965811i	$-0.583475\pi$
$421$	−10.6386	−0.518495	−0.259248	+	0.965811i	$0.583475\pi$
$422$	18.1538	0.883711
$423$	0	0
$424$	−5.52744	−0.268436
$425$	0	0
$426$	0	0
$427$	1.43693	0.0695381
$428$	−7.40535	−0.357951
$429$	0	0
$430$	0	0
$431$	37.3529	1.79923	0.899613	−	0.436687i	$-0.143848\pi$
$431$	37.3529	1.79923	0.899613	+	0.436687i	$0.143848\pi$
$432$	0	0
$433$	−17.2125	−0.827179	−0.413589	−	0.910464i	$-0.635725\pi$
$433$	−17.2125	−0.827179	−0.413589	+	0.910464i	$0.635725\pi$
$434$	−3.49111	−0.167579
$435$	0	0
$436$	−3.11668	−0.149262
$437$	21.3094	1.01937
$438$	0	0
$439$	−31.7487	−1.51528	−0.757642	−	0.652670i	$-0.773649\pi$
$439$	−31.7487	−1.51528	−0.757642	+	0.652670i	$0.773649\pi$
$440$	0	0
$441$	0	0
$442$	−62.1809	−2.95764
$443$	0.355958	0.0169121	0.00845603	−	0.999964i	$-0.497308\pi$
$443$	0.355958	0.0169121	0.00845603	+	0.999964i	$0.497308\pi$
$444$	0	0
$445$	0	0
$446$	−6.36174	−0.301237
$447$	0	0
$448$	−1.93688	−0.0915088
$449$	7.85632	0.370762	0.185381	−	0.982667i	$-0.440648\pi$
$449$	7.85632	0.370762	0.185381	+	0.982667i	$0.440648\pi$
$450$	0	0
$451$	−16.8708	−0.794416
$452$	−8.02476	−0.377453
$453$	0	0
$454$	−20.9249	−0.982056
$455$	0	0
$456$	0	0
$457$	21.4910	1.00531	0.502654	−	0.864488i	$-0.332357\pi$
$457$	21.4910	1.00531	0.502654	+	0.864488i	$0.332357\pi$
$458$	10.8691	0.507879
$459$	0	0
$460$	0	0
$461$	−40.9927	−1.90922	−0.954611	−	0.297856i	$-0.903729\pi$
$461$	−40.9927	−1.90922	−0.954611	+	0.297856i	$0.903729\pi$
$462$	0	0
$463$	42.1339	1.95813	0.979063	−	0.203555i	$-0.0652497\pi$
$463$	42.1339	1.95813	0.979063	+	0.203555i	$0.0652497\pi$
$464$	12.0273	0.558351
$465$	0	0
$466$	5.97674	0.276867
$467$	22.5376	1.04292	0.521459	−	0.853276i	$-0.325388\pi$
$467$	22.5376	1.04292	0.521459	+	0.853276i	$0.325388\pi$
$468$	0	0
$469$	1.21613	0.0561557
$470$	0	0
$471$	0	0
$472$	6.26732	0.288477
$473$	4.11523	0.189219
$474$	0	0
$475$	0	0
$476$	2.06251	0.0945348
$477$	0	0
$478$	25.6356	1.17255
$479$	33.2880	1.52097	0.760483	−	0.649358i	$-0.224962\pi$
$479$	33.2880	1.52097	0.760483	+	0.649358i	$0.224962\pi$
$480$	0	0
$481$	−41.9702	−1.91367
$482$	18.3385	0.835295
$483$	0	0
$484$	−0.929141	−0.0422337
$485$	0	0
$486$	0	0
$487$	23.7703	1.07713	0.538566	−	0.842583i	$-0.318966\pi$
$487$	23.7703	1.07713	0.538566	+	0.842583i	$0.318966\pi$
$488$	6.18281	0.279882
$489$	0	0
$490$	0	0
$491$	4.60563	0.207849	0.103925	−	0.994585i	$-0.466860\pi$
$491$	4.60563	0.207849	0.103925	+	0.994585i	$0.466860\pi$
$492$	0	0
$493$	15.0024	0.675673
$494$	−56.7787	−2.55459
$495$	0	0
$496$	−20.6797	−0.928548
$497$	2.80067	0.125627
$498$	0	0
$499$	−18.8976	−0.845971	−0.422985	−	0.906137i	$-0.639018\pi$
$499$	−18.8976	−0.845971	−0.422985	+	0.906137i	$0.639018\pi$
$500$	0	0
$501$	0	0
$502$	−11.3387	−0.506072
$503$	−35.7581	−1.59438	−0.797188	−	0.603731i	$-0.793680\pi$
$503$	−35.7581	−1.59438	−0.797188	+	0.603731i	$0.793680\pi$
$504$	0	0
$505$	0	0
$506$	19.3762	0.861376
$507$	0	0
$508$	−2.16009	−0.0958384
$509$	−24.4067	−1.08181	−0.540904	−	0.841084i	$-0.681918\pi$
$509$	−24.4067	−1.08181	−0.540904	+	0.841084i	$0.681918\pi$
$510$	0	0
$511$	−5.96250	−0.263766
$512$	−3.46207	−0.153003
$513$	0	0
$514$	−29.9506	−1.32106
$515$	0	0
$516$	0	0
$517$	11.4953	0.505563
$518$	5.55362	0.244012
$519$	0	0
$520$	0	0
$521$	−33.3968	−1.46314	−0.731571	−	0.681766i	$-0.761212\pi$
$521$	−33.3968	−1.46314	−0.731571	+	0.681766i	$0.761212\pi$
$522$	0	0
$523$	−37.3654	−1.63388	−0.816938	−	0.576726i	$-0.804330\pi$
$523$	−37.3654	−1.63388	−0.816938	+	0.576726i	$0.804330\pi$
$524$	6.27914	0.274305
$525$	0	0
$526$	−26.2648	−1.14520
$527$	−25.7952	−1.12366
$528$	0	0
$529$	−8.36479	−0.363686
$530$	0	0
$531$	0	0
$532$	1.88332	0.0816521
$533$	33.9532	1.47068
$534$	0	0
$535$	0	0
$536$	5.23274	0.226020
$537$	0	0
$538$	29.7344	1.28194
$539$	20.9097	0.900645
$540$	0	0
$541$	28.2560	1.21482	0.607409	−	0.794389i	$-0.292209\pi$
$541$	28.2560	1.21482	0.607409	+	0.794389i	$0.292209\pi$
$542$	−4.06096	−0.174433
$543$	0	0
$544$	22.2101	0.952250
$545$	0	0
$546$	0	0
$547$	−38.5452	−1.64807	−0.824036	−	0.566537i	$-0.808283\pi$
$547$	−38.5452	−1.64807	−0.824036	+	0.566537i	$0.808283\pi$
$548$	−1.54521	−0.0660082
$549$	0	0
$550$	0	0
$551$	13.6990	0.583596
$552$	0	0
$553$	−1.71969	−0.0731286
$554$	12.5262	0.532187
$555$	0	0
$556$	7.28866	0.309108
$557$	27.4125	1.16151	0.580753	−	0.814080i	$-0.302758\pi$
$557$	27.4125	1.16151	0.580753	+	0.814080i	$0.302758\pi$
$558$	0	0
$559$	−8.28206	−0.350294
$560$	0	0
$561$	0	0
$562$	−0.446383	−0.0188295
$563$	27.6392	1.16485	0.582427	−	0.812883i	$-0.302103\pi$
$563$	27.6392	1.16485	0.582427	+	0.812883i	$0.302103\pi$
$564$	0	0
$565$	0	0
$566$	5.50710	0.231481
$567$	0	0
$568$	12.0506	0.505634
$569$	14.7161	0.616933	0.308467	−	0.951235i	$-0.400184\pi$
$569$	14.7161	0.616933	0.308467	+	0.951235i	$0.400184\pi$
$570$	0	0
$571$	−28.3006	−1.18434	−0.592172	−	0.805812i	$-0.701729\pi$
$571$	−28.3006	−1.18434	−0.592172	+	0.805812i	$0.701729\pi$
$572$	−12.9415	−0.541111
$573$	0	0
$574$	−4.49279	−0.187525
$575$	0	0
$576$	0	0
$577$	−40.7976	−1.69843	−0.849214	−	0.528049i	$-0.822924\pi$
$577$	−40.7976	−1.69843	−0.849214	+	0.528049i	$0.822924\pi$
$578$	33.0216	1.37352
$579$	0	0
$580$	0	0
$581$	7.02493	0.291443
$582$	0	0
$583$	−7.88087	−0.326392
$584$	−25.6553	−1.06162
$585$	0	0
$586$	9.22080	0.380908
$587$	−2.78033	−0.114756	−0.0573782	−	0.998353i	$-0.518274\pi$
$587$	−2.78033	−0.114756	−0.0573782	+	0.998353i	$0.518274\pi$
$588$	0	0
$589$	−23.5541	−0.970531
$590$	0	0
$591$	0	0
$592$	32.8971	1.35206
$593$	14.8084	0.608109	0.304055	−	0.952655i	$-0.401659\pi$
$593$	14.8084	0.608109	0.304055	+	0.952655i	$0.401659\pi$
$594$	0	0
$595$	0	0
$596$	10.9343	0.447888
$597$	0	0
$598$	−38.9953	−1.59464
$599$	16.3430	0.667758	0.333879	−	0.942616i	$-0.391642\pi$
$599$	16.3430	0.667758	0.333879	+	0.942616i	$0.391642\pi$
$600$	0	0
$601$	−6.62371	−0.270187	−0.135093	−	0.990833i	$-0.543133\pi$
$601$	−6.62371	−0.270187	−0.135093	+	0.990833i	$0.543133\pi$
$602$	1.09591	0.0446658
$603$	0	0
$604$	15.2428	0.620221
$605$	0	0
$606$	0	0
$607$	30.3094	1.23022	0.615110	−	0.788442i	$-0.289112\pi$
$607$	30.3094	1.23022	0.615110	+	0.788442i	$0.289112\pi$
$608$	20.2805	0.822483
$609$	0	0
$610$	0	0
$611$	−23.1347	−0.935931
$612$	0	0
$613$	14.7803	0.596969	0.298484	−	0.954415i	$-0.403519\pi$
$613$	14.7803	0.596969	0.298484	+	0.954415i	$0.403519\pi$
$614$	8.88918	0.358738
$615$	0	0
$616$	−3.40660	−0.137256
$617$	33.8512	1.36280	0.681399	−	0.731912i	$-0.261372\pi$
$617$	33.8512	1.36280	0.681399	+	0.731912i	$0.261372\pi$
$618$	0	0
$619$	11.6887	0.469807	0.234903	−	0.972019i	$-0.424523\pi$
$619$	11.6887	0.469807	0.234903	+	0.972019i	$0.424523\pi$
$620$	0	0
$621$	0	0
$622$	31.1709	1.24984
$623$	1.71067	0.0685364
$624$	0	0
$625$	0	0
$626$	−14.9360	−0.596963
$627$	0	0
$628$	8.33686	0.332677
$629$	41.0347	1.63616
$630$	0	0
$631$	−38.1357	−1.51816	−0.759078	−	0.650999i	$-0.774350\pi$
$631$	−38.1357	−1.51816	−0.759078	+	0.650999i	$0.774350\pi$
$632$	−7.39944	−0.294334
$633$	0	0
$634$	−23.2589	−0.923728
$635$	0	0
$636$	0	0
$637$	−42.0816	−1.66733
$638$	12.4562	0.493144
$639$	0	0
$640$	0	0
$641$	34.7654	1.37315	0.686576	−	0.727058i	$-0.259113\pi$
$641$	34.7654	1.37315	0.686576	+	0.727058i	$0.259113\pi$
$642$	0	0
$643$	2.68515	0.105892	0.0529461	−	0.998597i	$-0.483139\pi$
$643$	2.68515	0.105892	0.0529461	+	0.998597i	$0.483139\pi$
$644$	1.29345	0.0509692
$645$	0	0
$646$	55.5131	2.18413
$647$	−40.5103	−1.59262	−0.796311	−	0.604887i	$-0.793218\pi$
$647$	−40.5103	−1.59262	−0.796311	+	0.604887i	$0.793218\pi$
$648$	0	0
$649$	8.93578	0.350760
$650$	0	0
$651$	0	0
$652$	−5.06729	−0.198451
$653$	−13.3354	−0.521856	−0.260928	−	0.965358i	$-0.584029\pi$
$653$	−13.3354	−0.521856	−0.260928	+	0.965358i	$0.584029\pi$
$654$	0	0
$655$	0	0
$656$	−26.6132	−1.03907
$657$	0	0
$658$	3.06126	0.119340
$659$	−31.1543	−1.21360	−0.606800	−	0.794855i	$-0.707547\pi$
$659$	−31.1543	−1.21360	−0.606800	+	0.794855i	$0.707547\pi$
$660$	0	0
$661$	6.31788	0.245737	0.122869	−	0.992423i	$-0.460791\pi$
$661$	6.31788	0.245737	0.122869	+	0.992423i	$0.460791\pi$
$662$	−19.9315	−0.774659
$663$	0	0
$664$	30.2267	1.17302
$665$	0	0
$666$	0	0
$667$	9.40838	0.364294
$668$	1.99160	0.0770571
$669$	0	0
$670$	0	0
$671$	8.81528	0.340310
$672$	0	0
$673$	−6.59220	−0.254110	−0.127055	−	0.991896i	$-0.540553\pi$
$673$	−6.59220	−0.254110	−0.127055	+	0.991896i	$0.540553\pi$
$674$	7.49857	0.288834
$675$	0	0
$676$	17.3476	0.667214
$677$	−34.8947	−1.34111	−0.670556	−	0.741859i	$-0.733944\pi$
$677$	−34.8947	−1.34111	−0.670556	+	0.741859i	$0.733944\pi$
$678$	0	0
$679$	5.59702	0.214794
$680$	0	0
$681$	0	0
$682$	−21.4172	−0.820108
$683$	−26.0958	−0.998528	−0.499264	−	0.866450i	$-0.666396\pi$
$683$	−26.0958	−0.998528	−0.499264	+	0.866450i	$0.666396\pi$
$684$	0	0
$685$	0	0
$686$	11.3476	0.433252
$687$	0	0
$688$	6.49165	0.247492
$689$	15.8606	0.604239
$690$	0	0
$691$	−29.3058	−1.11485	−0.557423	−	0.830229i	$-0.688210\pi$
$691$	−29.3058	−1.11485	−0.557423	+	0.830229i	$0.688210\pi$
$692$	10.6065	0.403197
$693$	0	0
$694$	−54.6648	−2.07505
$695$	0	0
$696$	0	0
$697$	−33.1964	−1.25740
$698$	−45.8775	−1.73649
$699$	0	0
$700$	0	0
$701$	−15.3891	−0.581239	−0.290620	−	0.956839i	$-0.593861\pi$
$701$	−15.3891	−0.581239	−0.290620	+	0.956839i	$0.593861\pi$
$702$	0	0
$703$	37.4696	1.41319
$704$	−11.8823	−0.447832
$705$	0	0
$706$	3.00883	0.113239
$707$	−8.77741	−0.330108
$708$	0	0
$709$	−7.73991	−0.290678	−0.145339	−	0.989382i	$-0.546427\pi$
$709$	−7.73991	−0.290678	−0.145339	+	0.989382i	$0.546427\pi$
$710$	0	0
$711$	0	0
$712$	7.36062	0.275851
$713$	−16.1768	−0.605828
$714$	0	0
$715$	0	0
$716$	−11.4302	−0.427165
$717$	0	0
$718$	19.7875	0.738462
$719$	15.1316	0.564313	0.282156	−	0.959368i	$-0.408950\pi$
$719$	15.1316	0.564313	0.282156	+	0.959368i	$0.408950\pi$
$720$	0	0
$721$	0.420832	0.0156726
$722$	19.6494	0.731275
$723$	0	0
$724$	8.94457	0.332422
$725$	0	0
$726$	0	0
$727$	0.161876	0.00600366	0.00300183	−	0.999995i	$-0.499044\pi$
$727$	0.161876	0.00600366	0.00300183	+	0.999995i	$0.499044\pi$
$728$	6.85590	0.254097
$729$	0	0
$730$	0	0
$731$	8.09746	0.299495
$732$	0	0
$733$	−50.0332	−1.84802	−0.924009	−	0.382371i	$-0.875107\pi$
$733$	−50.0332	−1.84802	−0.924009	+	0.382371i	$0.875107\pi$
$734$	−23.8095	−0.878825
$735$	0	0
$736$	13.9285	0.513413
$737$	7.46070	0.274818
$738$	0	0
$739$	30.5505	1.12382	0.561909	−	0.827199i	$-0.310067\pi$
$739$	30.5505	1.12382	0.561909	+	0.827199i	$0.310067\pi$
$740$	0	0
$741$	0	0
$742$	−2.09872	−0.0770463
$743$	−5.96684	−0.218902	−0.109451	−	0.993992i	$-0.534909\pi$
$743$	−5.96684	−0.218902	−0.109451	+	0.993992i	$0.534909\pi$
$744$	0	0
$745$	0	0
$746$	−15.4329	−0.565039
$747$	0	0
$748$	12.6530	0.462640
$749$	5.59340	0.204379
$750$	0	0
$751$	34.3976	1.25519	0.627593	−	0.778542i	$-0.284040\pi$
$751$	34.3976	1.25519	0.627593	+	0.778542i	$0.284040\pi$
$752$	18.1335	0.661260
$753$	0	0
$754$	−25.0685	−0.912941
$755$	0	0
$756$	0	0
$757$	40.6873	1.47881	0.739403	−	0.673263i	$-0.235108\pi$
$757$	40.6873	1.47881	0.739403	+	0.673263i	$0.235108\pi$
$758$	−46.6495	−1.69439
$759$	0	0
$760$	0	0
$761$	−28.2596	−1.02441	−0.512204	−	0.858864i	$-0.671171\pi$
$761$	−28.2596	−1.02441	−0.512204	+	0.858864i	$0.671171\pi$
$762$	0	0
$763$	2.35409	0.0852239
$764$	−16.9519	−0.613299
$765$	0	0
$766$	−2.39407	−0.0865013
$767$	−17.9836	−0.649350
$768$	0	0
$769$	−46.9036	−1.69139	−0.845694	−	0.533668i	$-0.820813\pi$
$769$	−46.9036	−1.69139	−0.845694	+	0.533668i	$0.820813\pi$
$770$	0	0
$771$	0	0
$772$	−6.39582	−0.230191
$773$	−9.19641	−0.330772	−0.165386	−	0.986229i	$-0.552887\pi$
$773$	−9.19641	−0.330772	−0.165386	+	0.986229i	$0.552887\pi$
$774$	0	0
$775$	0	0
$776$	24.0827	0.864520
$777$	0	0
$778$	21.0916	0.756169
$779$	−30.3123	−1.08605
$780$	0	0
$781$	17.1815	0.614802
$782$	38.1261	1.36339
$783$	0	0
$784$	32.9844	1.17801
$785$	0	0
$786$	0	0
$787$	5.74637	0.204836	0.102418	−	0.994741i	$-0.467342\pi$
$787$	5.74637	0.204836	0.102418	+	0.994741i	$0.467342\pi$
$788$	1.38388	0.0492986
$789$	0	0
$790$	0	0
$791$	6.06126	0.215514
$792$	0	0
$793$	−17.7411	−0.630004
$794$	−1.41870	−0.0503479
$795$	0	0
$796$	8.76255	0.310580
$797$	7.07450	0.250592	0.125296	−	0.992119i	$-0.460012\pi$
$797$	7.07450	0.250592	0.125296	+	0.992119i	$0.460012\pi$
$798$	0	0
$799$	22.6191	0.800205
$800$	0	0
$801$	0	0
$802$	−54.5868	−1.92753
$803$	−36.5787	−1.29083
$804$	0	0
$805$	0	0
$806$	43.1029	1.51824
$807$	0	0
$808$	−37.7672	−1.32865
$809$	−38.1075	−1.33979	−0.669894	−	0.742457i	$-0.733660\pi$
$809$	−38.1075	−1.33979	−0.669894	+	0.742457i	$0.733660\pi$
$810$	0	0
$811$	−1.44105	−0.0506022	−0.0253011	−	0.999680i	$-0.508054\pi$
$811$	−1.44105	−0.0506022	−0.0253011	+	0.999680i	$0.508054\pi$
$812$	0.831508	0.0291802
$813$	0	0
$814$	34.0702	1.19416
$815$	0	0
$816$	0	0
$817$	7.39396	0.258682
$818$	8.25904	0.288771
$819$	0	0
$820$	0	0
$821$	−22.5143	−0.785753	−0.392876	−	0.919591i	$-0.628520\pi$
$821$	−22.5143	−0.785753	−0.392876	+	0.919591i	$0.628520\pi$
$822$	0	0
$823$	−41.4589	−1.44517	−0.722583	−	0.691284i	$-0.757045\pi$
$823$	−41.4589	−1.44517	−0.722583	+	0.691284i	$0.757045\pi$
$824$	1.81075	0.0630803
$825$	0	0
$826$	2.37964	0.0827984
$827$	−27.8133	−0.967164	−0.483582	−	0.875299i	$-0.660664\pi$
$827$	−27.8133	−0.967164	−0.483582	+	0.875299i	$0.660664\pi$
$828$	0	0
$829$	20.7232	0.719745	0.359872	−	0.933002i	$-0.382820\pi$
$829$	20.7232	0.719745	0.359872	+	0.933002i	$0.382820\pi$
$830$	0	0
$831$	0	0
$832$	23.9136	0.829056
$833$	41.1436	1.42554
$834$	0	0
$835$	0	0
$836$	11.5537	0.399595
$837$	0	0
$838$	−17.9017	−0.618403
$839$	−18.1451	−0.626437	−0.313218	−	0.949681i	$-0.601407\pi$
$839$	−18.1451	−0.626437	−0.313218	+	0.949681i	$0.601407\pi$
$840$	0	0
$841$	−22.9517	−0.791439
$842$	−17.3806	−0.598975
$843$	0	0
$844$	7.43444	0.255904
$845$	0	0
$846$	0	0
$847$	0.701798	0.0241141
$848$	−12.4318	−0.426911
$849$	0	0
$850$	0	0
$851$	25.7339	0.882148
$852$	0	0
$853$	−4.00708	−0.137200	−0.0685999	−	0.997644i	$-0.521853\pi$
$853$	−4.00708	−0.137200	−0.0685999	+	0.997644i	$0.521853\pi$
$854$	2.34755	0.0803316
$855$	0	0
$856$	24.0672	0.822599
$857$	9.08971	0.310499	0.155249	−	0.987875i	$-0.450382\pi$
$857$	9.08971	0.310499	0.155249	+	0.987875i	$0.450382\pi$
$858$	0	0
$859$	16.3870	0.559116	0.279558	−	0.960129i	$-0.409812\pi$
$859$	16.3870	0.559116	0.279558	+	0.960129i	$0.409812\pi$
$860$	0	0
$861$	0	0
$862$	61.0243	2.07850
$863$	−23.7967	−0.810050	−0.405025	−	0.914306i	$-0.632737\pi$
$863$	−23.7967	−0.810050	−0.405025	+	0.914306i	$0.632737\pi$
$864$	0	0
$865$	0	0
$866$	−28.1204	−0.955571
$867$	0	0
$868$	−1.42970	−0.0485272
$869$	−10.5499	−0.357882
$870$	0	0
$871$	−15.0149	−0.508762
$872$	10.1291	0.343016
$873$	0	0
$874$	34.8137	1.17759
$875$	0	0
$876$	0	0
$877$	36.0744	1.21815	0.609073	−	0.793114i	$-0.291542\pi$
$877$	36.0744	1.21815	0.609073	+	0.793114i	$0.291542\pi$
$878$	−51.8687	−1.75048
$879$	0	0
$880$	0	0
$881$	35.4575	1.19459	0.597297	−	0.802020i	$-0.296241\pi$
$881$	35.4575	1.19459	0.597297	+	0.802020i	$0.296241\pi$
$882$	0	0
$883$	−39.1320	−1.31690	−0.658448	−	0.752626i	$-0.728787\pi$
$883$	−39.1320	−1.31690	−0.658448	+	0.752626i	$0.728787\pi$
$884$	−25.4647	−0.856470
$885$	0	0
$886$	0.581537	0.0195371
$887$	−51.2833	−1.72192	−0.860962	−	0.508670i	$-0.830137\pi$
$887$	−51.2833	−1.72192	−0.860962	+	0.508670i	$0.830137\pi$
$888$	0	0
$889$	1.63156	0.0547207
$890$	0	0
$891$	0	0
$892$	−2.60530	−0.0872318
$893$	20.6539	0.691158
$894$	0	0
$895$	0	0
$896$	−6.84415	−0.228647
$897$	0	0
$898$	12.8350	0.428311
$899$	−10.3994	−0.346841
$900$	0	0
$901$	−15.5070	−0.516614
$902$	−27.5623	−0.917723
$903$	0	0
$904$	26.0802	0.867416
$905$	0	0
$906$	0	0
$907$	47.8588	1.58913	0.794563	−	0.607181i	$-0.207700\pi$
$907$	47.8588	1.58913	0.794563	+	0.607181i	$0.207700\pi$
$908$	−8.56930	−0.284382
$909$	0	0
$910$	0	0
$911$	−18.0431	−0.597793	−0.298897	−	0.954285i	$-0.596619\pi$
$911$	−18.0431	−0.597793	−0.298897	+	0.954285i	$0.596619\pi$
$912$	0	0
$913$	43.0964	1.42628
$914$	35.1104	1.16135
$915$	0	0
$916$	4.45117	0.147071
$917$	−4.74276	−0.156620
$918$	0	0
$919$	10.3976	0.342984	0.171492	−	0.985185i	$-0.445141\pi$
$919$	10.3976	0.342984	0.171492	+	0.985185i	$0.445141\pi$
$920$	0	0
$921$	0	0
$922$	−66.9708	−2.20557
$923$	−34.5784	−1.13816
$924$	0	0
$925$	0	0
$926$	68.8351	2.26206
$927$	0	0
$928$	8.95410	0.293933
$929$	36.0216	1.18183	0.590915	−	0.806734i	$-0.298767\pi$
$929$	36.0216	1.18183	0.590915	+	0.806734i	$0.298767\pi$
$930$	0	0
$931$	37.5691	1.23128
$932$	2.44763	0.0801748
$933$	0	0
$934$	36.8203	1.20480
$935$	0	0
$936$	0	0
$937$	24.0326	0.785111	0.392555	−	0.919728i	$-0.371591\pi$
$937$	24.0326	0.785111	0.392555	+	0.919728i	$0.371591\pi$
$938$	1.98682	0.0648720
$939$	0	0
$940$	0	0
$941$	−16.6676	−0.543348	−0.271674	−	0.962389i	$-0.587577\pi$
$941$	−16.6676	−0.543348	−0.271674	+	0.962389i	$0.587577\pi$
$942$	0	0
$943$	−20.8183	−0.677938
$944$	14.0959	0.458783
$945$	0	0
$946$	6.72315	0.218589
$947$	27.5400	0.894928	0.447464	−	0.894302i	$-0.352327\pi$
$947$	27.5400	0.894928	0.447464	+	0.894302i	$0.352327\pi$
$948$	0	0
$949$	73.6160	2.38968
$950$	0	0
$951$	0	0
$952$	−6.70308	−0.217248
$953$	18.1344	0.587432	0.293716	−	0.955893i	$-0.405108\pi$
$953$	18.1344	0.587432	0.293716	+	0.955893i	$0.405108\pi$
$954$	0	0
$955$	0	0
$956$	10.4984	0.339544
$957$	0	0
$958$	54.3833	1.75705
$959$	1.16713	0.0376886
$960$	0	0
$961$	−13.1191	−0.423198
$962$	−68.5676	−2.21071
$963$	0	0
$964$	7.51009	0.241884
$965$	0	0
$966$	0	0
$967$	36.1875	1.16371	0.581855	−	0.813292i	$-0.302327\pi$
$967$	36.1875	1.16371	0.581855	+	0.813292i	$0.302327\pi$
$968$	3.01968	0.0970562
$969$	0	0
$970$	0	0
$971$	34.6173	1.11092	0.555461	−	0.831542i	$-0.312542\pi$
$971$	34.6173	1.11092	0.555461	+	0.831542i	$0.312542\pi$
$972$	0	0
$973$	−5.50527	−0.176491
$974$	38.8340	1.24432
$975$	0	0
$976$	13.9058	0.445115
$977$	29.4331	0.941650	0.470825	−	0.882227i	$-0.343956\pi$
$977$	29.4331	0.941650	0.470825	+	0.882227i	$0.343956\pi$
$978$	0	0
$979$	10.4946	0.335408
$980$	0	0
$981$	0	0
$982$	7.52432	0.240111
$983$	−24.4911	−0.781145	−0.390573	−	0.920572i	$-0.627723\pi$
$983$	−24.4911	−0.781145	−0.390573	+	0.920572i	$0.627723\pi$
$984$	0	0
$985$	0	0
$986$	24.5097	0.780549
$987$	0	0
$988$	−23.2523	−0.739755
$989$	5.07813	0.161475
$990$	0	0
$991$	−13.2821	−0.421919	−0.210959	−	0.977495i	$-0.567659\pi$
$991$	−13.2821	−0.421919	−0.210959	+	0.977495i	$0.567659\pi$
$992$	−15.3957	−0.488815
$993$	0	0
$994$	4.57551	0.145126
$995$	0	0
$996$	0	0
$997$	39.1530	1.23999	0.619994	−	0.784606i	$-0.287135\pi$
$997$	39.1530	1.23999	0.619994	+	0.784606i	$0.287135\pi$
$998$	−30.8734	−0.977280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.a.p.1.4		4
3.2	odd	2		2025.2.a.y.1.1		4
5.2	odd	4		2025.2.b.o.649.7		8
5.3	odd	4		2025.2.b.o.649.2		8
5.4	even	2		2025.2.a.z.1.1		4
9.2	odd	6		225.2.e.c.76.4	✓	8
9.4	even	3		675.2.e.e.451.1		8
9.5	odd	6		225.2.e.c.151.4	yes	8
9.7	even	3		675.2.e.e.226.1		8
15.2	even	4		2025.2.b.n.649.2		8
15.8	even	4		2025.2.b.n.649.7		8
15.14	odd	2		2025.2.a.q.1.4		4
45.2	even	12		225.2.k.c.49.2		16
45.4	even	6		675.2.e.c.451.4		8
45.7	odd	12		675.2.k.c.199.7		16
45.13	odd	12		675.2.k.c.424.7		16
45.14	odd	6		225.2.e.e.151.1	yes	8
45.22	odd	12		675.2.k.c.424.2		16
45.23	even	12		225.2.k.c.124.2		16
45.29	odd	6		225.2.e.e.76.1	yes	8
45.32	even	12		225.2.k.c.124.7		16
45.34	even	6		675.2.e.c.226.4		8
45.38	even	12		225.2.k.c.49.7		16
45.43	odd	12		675.2.k.c.199.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.e.c.76.4	✓	8	9.2	odd	6
225.2.e.c.151.4	yes	8	9.5	odd	6
225.2.e.e.76.1	yes	8	45.29	odd	6
225.2.e.e.151.1	yes	8	45.14	odd	6
225.2.k.c.49.2		16	45.2	even	12
225.2.k.c.49.7		16	45.38	even	12
225.2.k.c.124.2		16	45.23	even	12
225.2.k.c.124.7		16	45.32	even	12
675.2.e.c.226.4		8	45.34	even	6
675.2.e.c.451.4		8	45.4	even	6
675.2.e.e.226.1		8	9.7	even	3
675.2.e.e.451.1		8	9.4	even	3
675.2.k.c.199.2		16	45.43	odd	12
675.2.k.c.199.7		16	45.7	odd	12
675.2.k.c.424.2		16	45.22	odd	12
675.2.k.c.424.7		16	45.13	odd	12
2025.2.a.p.1.4		4	1.1	even	1	trivial
2025.2.a.q.1.4		4	15.14	odd	2
2025.2.a.y.1.1		4	3.2	odd	2
2025.2.a.z.1.1		4	5.4	even	2
2025.2.b.n.649.2		8	15.2	even	4
2025.2.b.n.649.7		8	15.8	even	4
2025.2.b.o.649.2		8	5.3	odd	4
2025.2.b.o.649.7		8	5.2	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.63372 q^{2} +0.669052 q^{4} -0.505348 q^{7} -2.17440 q^{8} +O(q^{10})\)	\(q+1.63372 q^{2} +0.669052 q^{4} -0.505348 q^{7} -2.17440 q^{8} -3.10020 q^{11} +6.23927 q^{13} -0.825599 q^{14} -4.89047 q^{16} -6.10020 q^{17} -5.57022 q^{19} -5.06487 q^{22} -3.82560 q^{23} +10.1932 q^{26} -0.338104 q^{28} -2.45932 q^{29} +4.22858 q^{31} -3.64088 q^{32} -9.96604 q^{34} -6.72677 q^{37} -9.10020 q^{38} +5.44185 q^{41} -1.32741 q^{43} -2.07420 q^{44} -6.24997 q^{46} -3.70792 q^{47} -6.74462 q^{49} +4.17440 q^{52} +2.54205 q^{53} +1.09883 q^{56} -4.01785 q^{58} -2.88232 q^{59} -2.84345 q^{61} +6.90833 q^{62} +3.83276 q^{64} -2.40652 q^{67} -4.08135 q^{68} -5.54205 q^{71} +11.7988 q^{73} -10.9897 q^{74} -3.72677 q^{76} +1.56668 q^{77} +3.40298 q^{79} +8.89047 q^{82} -13.9012 q^{83} -2.16862 q^{86} +6.74108 q^{88} -3.38513 q^{89} -3.15301 q^{91} -2.55953 q^{92} -6.05772 q^{94} -11.0756 q^{97} -11.0188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 + q^11 + 2 * q^13 - 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 + 3 * q^22 - 15 * q^23 + 10 * q^26 - 4 * q^28 - q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 - q^37 - 23 * q^38 + 5 * q^41 - 10 * q^43 - 22 * q^44 - 20 * q^47 - 3 * q^49 + 17 * q^52 - 20 * q^53 + 30 * q^56 - 18 * q^58 - 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 + 17 * q^67 - 34 * q^68 + 8 * q^71 + 2 * q^73 - 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 + 12 * q^82 - 30 * q^83 + 34 * q^86 + 9 * q^88 + 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 - 19 * q^97 - 13 * q^98

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