LMFDB - Embedded newform 2025.2.b.o.649.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.34810603776.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 $ x^8 + 12x^6 + 42x^4 + 49*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.7
Root			$1.63372i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.o.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.63372i q^{2} -0.669052 q^{4} -0.505348i q^{7} +2.17440i q^{8} +O(q^{10})$	$q+1.63372i q^{2} -0.669052 q^{4} -0.505348i q^{7} +2.17440i q^{8} -3.10020 q^{11} -6.23927i q^{13} +0.825599 q^{14} -4.89047 q^{16} -6.10020i q^{17} +5.57022 q^{19} -5.06487i q^{22} +3.82560i q^{23} +10.1932 q^{26} +0.338104i q^{28} +2.45932 q^{29} +4.22858 q^{31} -3.64088i q^{32} +9.96604 q^{34} -6.72677i q^{37} +9.10020i q^{38} +5.44185 q^{41} +1.32741i q^{43} +2.07420 q^{44} -6.24997 q^{46} -3.70792i q^{47} +6.74462 q^{49} +4.17440i q^{52} -2.54205i q^{53} +1.09883 q^{56} +4.01785i q^{58} +2.88232 q^{59} -2.84345 q^{61} +6.90833i q^{62} -3.83276 q^{64} -2.40652i q^{67} +4.08135i q^{68} -5.54205 q^{71} -11.7988i q^{73} +10.9897 q^{74} -3.72677 q^{76} +1.56668i q^{77} -3.40298 q^{79} +8.89047i q^{82} +13.9012i q^{83} -2.16862 q^{86} -6.74108i q^{88} +3.38513 q^{89} -3.15301 q^{91} -2.55953i q^{92} +6.05772 q^{94} -11.0756i q^{97} +11.0188i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 2 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{26} + 2 q^{29} - 8 q^{31} - 18 q^{34} + 10 q^{41} + 44 q^{44} + 6 q^{49} + 60 q^{56} + 34 q^{59} - 26 q^{61} - 38 q^{64} + 16 q^{71} + 80 q^{74} + 22 q^{76} + 14 q^{79} + 68 q^{86} - 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 + 2 * q^11 + 6 * q^14 + 8 * q^16 - 4 * q^19 + 20 * q^26 + 2 * q^29 - 8 * q^31 - 18 * q^34 + 10 * q^41 + 44 * q^44 + 6 * q^49 + 60 * q^56 + 34 * q^59 - 26 * q^61 - 38 * q^64 + 16 * q^71 + 80 * q^74 + 22 * q^76 + 14 * q^79 + 68 * q^86 - 18 * q^89 - 34 * q^91 - 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$.

$n$	$326$	$1702$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.63372i	1.15522i	0.816314	+	0.577608i	$0.196014\pi$
$2$	1.63372i	1.15522i	−0.816314	+	0.577608i	$0.803986\pi$
$3$	0	0
$3$	0	0
$4$	−0.669052	−0.334526
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.505348i	− 0.191004i	−0.995429	−	0.0955019i	$-0.969554\pi$
$7$	− 0.505348i	− 0.191004i	0.995429	−	0.0955019i	$-0.0304456\pi$
$8$	2.17440i	0.768767i
$9$	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
$13$	− 6.23927i	− 1.73046i	−0.501372	−	0.865232i	$-0.667171\pi$
$13$	− 6.23927i	− 1.73046i	0.501372	−	0.865232i	$-0.332829\pi$
$14$	0.825599	0.220651
$15$	0	0
$16$	−4.89047	−1.22262
$17$	− 6.10020i	− 1.47952i	−0.672873	−	0.739758i	$-0.734940\pi$
$17$	− 6.10020i	− 1.47952i	0.672873	−	0.739758i	$-0.265060\pi$
$18$	0	0
$19$	5.57022	1.27790	0.638948	−	0.769250i	$-0.279370\pi$
$19$	5.57022	1.27790	0.638948	+	0.769250i	$0.279370\pi$
$20$	0	0
$21$	0	0
$22$	− 5.06487i	− 1.07983i
$23$	3.82560i	0.797693i	0.917018	+	0.398846i	$0.130589\pi$
$23$	3.82560i	0.797693i	−0.917018	+	0.398846i	$0.869411\pi$
$24$	0	0
$25$	0	0
$26$	10.1932	1.99906
$27$	0	0
$28$	0.338104i	0.0638957i
$29$	2.45932	0.456685	0.228342	−	0.973581i	$-0.426669\pi$
$29$	2.45932	0.456685	0.228342	+	0.973581i	$0.426669\pi$
$30$	0	0
$31$	4.22858	0.759475	0.379738	−	0.925094i	$-0.376014\pi$
$31$	4.22858	0.759475	0.379738	+	0.925094i	$0.376014\pi$
$32$	− 3.64088i	− 0.643623i
$33$	0	0
$34$	9.96604	1.70916
$35$	0	0
$36$	0	0
$37$	− 6.72677i	− 1.10587i	−0.833223	−	0.552937i	$-0.813507\pi$
$37$	− 6.72677i	− 1.10587i	0.833223	−	0.552937i	$-0.186493\pi$
$38$	9.10020i	1.47625i
$39$	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
$43$	1.32741i	0.202428i	0.994865	+	0.101214i	$0.0322726\pi$
$43$	1.32741i	0.202428i	−0.994865	+	0.101214i	$0.967727\pi$
$44$	2.07420	0.312697
$45$	0	0
$46$	−6.24997	−0.921508
$47$	− 3.70792i	− 0.540856i	−0.962740	−	0.270428i	$-0.912835\pi$
$47$	− 3.70792i	− 0.540856i	0.962740	−	0.270428i	$-0.0871652\pi$
$48$	0	0
$49$	6.74462	0.963518
$50$	0	0
$51$	0	0
$52$	4.17440i	0.578885i
$53$	− 2.54205i	− 0.349177i	−0.984641	−	0.174589i	$-0.944140\pi$
$53$	− 2.54205i	− 0.349177i	0.984641	−	0.174589i	$-0.0558596\pi$
$54$	0	0
$55$	0	0
$56$	1.09883	0.146837
$57$	0	0
$58$	4.01785i	0.527570i
$59$	2.88232	0.375246	0.187623	−	0.982241i	$-0.439922\pi$
$59$	2.88232	0.375246	0.187623	+	0.982241i	$0.439922\pi$
$60$	0	0
$61$	−2.84345	−0.364067	−0.182033	−	0.983292i	$-0.558268\pi$
$61$	−2.84345	−0.364067	−0.182033	+	0.983292i	$0.558268\pi$
$62$	6.90833i	0.877358i
$63$	0	0
$64$	−3.83276	−0.479095
$65$	0	0
$66$	0	0
$67$	− 2.40652i	− 0.294003i	−0.989136	−	0.147002i	$-0.953038\pi$
$67$	− 2.40652i	− 0.294003i	0.989136	−	0.147002i	$-0.0469622\pi$
$68$	4.08135i	0.494937i
$69$	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
$73$	− 11.7988i	− 1.38095i	−0.723359	−	0.690473i	$-0.757403\pi$
$73$	− 11.7988i	− 1.38095i	0.723359	−	0.690473i	$-0.242597\pi$
$74$	10.9897	1.27752
$75$	0	0
$76$	−3.72677	−0.427490
$77$	1.56668i	0.178540i
$78$	0	0
$79$	−3.40298	−0.382865	−0.191433	−	0.981506i	$-0.561313\pi$
$79$	−3.40298	−0.382865	−0.191433	+	0.981506i	$0.561313\pi$
$80$	0	0
$81$	0	0
$82$	8.89047i	0.981789i
$83$	13.9012i	1.52585i	0.646486	+	0.762926i	$0.276238\pi$
$83$	13.9012i	1.52585i	−0.646486	+	0.762926i	$0.723762\pi$
$84$	0	0
$85$	0	0
$86$	−2.16862	−0.233848
$87$	0	0
$88$	− 6.74108i	− 0.718602i
$89$	3.38513	0.358823	0.179411	−	0.983774i	$-0.442581\pi$
$89$	3.38513	0.358823	0.179411	+	0.983774i	$0.442581\pi$
$90$	0	0
$91$	−3.15301	−0.330525
$92$	− 2.55953i	− 0.266849i
$93$	0	0
$94$	6.05772	0.624806
$95$	0	0
$96$	0	0
$97$	− 11.0756i	− 1.12455i	−0.826949	−	0.562277i	$-0.809926\pi$
$97$	− 11.0756i	− 1.12455i	0.826949	−	0.562277i	$-0.190074\pi$
$98$	11.0188i	1.11307i
$99$	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
$103$	0.832756i	0.0820539i	0.999158	+	0.0410269i	$0.0130629\pi$
$103$	0.832756i	0.0820539i	−0.999158	+	0.0410269i	$0.986937\pi$
$104$	13.5667	1.33032
$105$	0	0
$106$	4.15301	0.403376
$107$	− 11.0684i	− 1.07002i	−0.844844	−	0.535012i	$-0.820307\pi$
$107$	− 11.0684i	− 1.07002i	0.844844	−	0.535012i	$-0.179693\pi$
$108$	0	0
$109$	4.65836	0.446190	0.223095	−	0.974797i	$-0.428384\pi$
$109$	4.65836	0.446190	0.223095	+	0.974797i	$0.428384\pi$
$110$	0	0
$111$	0	0
$112$	2.47139i	0.233525i
$113$	11.9942i	1.12832i	0.825665	+	0.564160i	$0.190800\pi$
$113$	11.9942i	1.12832i	−0.825665	+	0.564160i	$0.809200\pi$
$114$	0	0
$115$	0	0
$116$	−1.64542	−0.152773
$117$	0	0
$118$	4.70892i	0.433491i
$119$	−3.08273	−0.282593
$120$	0	0
$121$	−1.38874	−0.126249
$122$	− 4.64542i	− 0.420576i
$123$	0	0
$124$	−2.82914	−0.254064
$125$	0	0
$126$	0	0
$127$	− 3.22858i	− 0.286490i	−0.989687	−	0.143245i	$-0.954246\pi$
$127$	− 3.22858i	− 0.286490i	0.989687	−	0.143245i	$-0.0457537\pi$
$128$	− 13.5434i	− 1.19708i
$129$	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
$133$	− 2.81490i	− 0.244083i
$134$	3.93159	0.339637
$135$	0	0
$136$	13.2643	1.13740
$137$	− 2.30955i	− 0.197319i	−0.995121	−	0.0986593i	$-0.968545\pi$
$137$	− 2.30955i	− 0.197319i	0.995121	−	0.0986593i	$-0.0314554\pi$
$138$	0	0
$139$	−10.8940	−0.924018	−0.462009	−	0.886875i	$-0.652871\pi$
$139$	−10.8940	−0.924018	−0.462009	+	0.886875i	$0.652871\pi$
$140$	0	0
$141$	0	0
$142$	− 9.05418i	− 0.759810i
$143$	19.3430i	1.61754i
$144$	0	0
$145$	0	0
$146$	19.2760	1.59529
$147$	0	0
$148$	4.50056i	0.369944i
$149$	−16.3430	−1.33887	−0.669436	−	0.742870i	$-0.733464\pi$
$149$	−16.3430	−1.33887	−0.669436	+	0.742870i	$0.733464\pi$
$150$	0	0
$151$	22.7827	1.85403	0.927015	−	0.375025i	$-0.122366\pi$
$151$	22.7827	1.85403	0.927015	+	0.375025i	$0.122366\pi$
$152$	12.1119i	0.982404i
$153$	0	0
$154$	−2.55953	−0.206252
$155$	0	0
$156$	0	0
$157$	12.4607i	0.994472i	0.867615	+	0.497236i	$0.165652\pi$
$157$	12.4607i	0.994472i	−0.867615	+	0.497236i	$0.834348\pi$
$158$	− 5.55953i	− 0.442292i
$159$	0	0
$160$	0	0
$161$	1.93326	0.152362
$162$	0	0
$163$	7.57384i	0.593229i	0.954997	+	0.296614i	$0.0958576\pi$
$163$	7.57384i	0.593229i	−0.954997	+	0.296614i	$0.904142\pi$
$164$	−3.64088	−0.284305
$165$	0	0
$166$	−22.7107	−1.76269
$167$	2.97674i	0.230347i	0.993345	+	0.115174i	$0.0367424\pi$
$167$	2.97674i	0.230347i	−0.993345	+	0.115174i	$0.963258\pi$
$168$	0	0
$169$	−25.9285	−1.99450
$170$	0	0
$171$	0	0
$172$	− 0.888105i	− 0.0677174i
$173$	− 15.8530i	− 1.20528i	−0.798013	−	0.602640i	$-0.794116\pi$
$173$	− 15.8530i	− 1.20528i	0.798013	−	0.602640i	$-0.205884\pi$
$174$	0	0
$175$	0	0
$176$	15.1615	1.14284
$177$	0	0
$178$	5.53036i	0.414518i
$179$	17.0841	1.27693	0.638463	−	0.769653i	$-0.279571\pi$
$179$	17.0841	1.27693	0.638463	+	0.769653i	$0.279571\pi$
$180$	0	0
$181$	13.3690	0.993712	0.496856	−	0.867833i	$-0.334488\pi$
$181$	13.3690	0.993712	0.496856	+	0.867833i	$0.334488\pi$
$182$	− 5.15114i	− 0.381828i
$183$	0	0
$184$	−8.31839	−0.613240
$185$	0	0
$186$	0	0
$187$	18.9119i	1.38297i
$188$	2.48079i	0.180930i
$189$	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
$193$	9.55953i	0.688110i	0.938950	+	0.344055i	$0.111801\pi$
$193$	9.55953i	0.688110i	−0.938950	+	0.344055i	$0.888199\pi$
$194$	18.0944	1.29910
$195$	0	0
$196$	−4.51250	−0.322322
$197$	2.06841i	0.147368i	0.997282	+	0.0736842i	$0.0234757\pi$
$197$	2.06841i	0.147368i	−0.997282	+	0.0736842i	$0.976524\pi$
$198$	0	0
$199$	−13.0970	−0.928419	−0.464210	−	0.885725i	$-0.653662\pi$
$199$	−13.0970	−0.928419	−0.464210	+	0.885725i	$0.653662\pi$
$200$	0	0
$201$	0	0
$202$	28.3762i	1.99654i
$203$	− 1.24281i	− 0.0872285i
$204$	0	0
$205$	0	0
$206$	−1.36049	−0.0947900
$207$	0	0
$208$	30.5130i	2.11570i
$209$	−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	$0.375068\pi$
$212$	1.70076i	0.116809i
$213$	0	0
$214$	18.0827	1.23611
$215$	0	0
$216$	0	0
$217$	− 2.13690i	− 0.145063i
$218$	7.61046i	0.515446i
$219$	0	0
$220$	0	0
$221$	−38.0608	−2.56025
$222$	0	0
$223$	3.89401i	0.260762i	0.991464	+	0.130381i	$0.0416201\pi$
$223$	3.89401i	0.260762i	−0.991464	+	0.130381i	$0.958380\pi$
$224$	−1.83991	−0.122934
$225$	0	0
$226$	−19.5952	−1.30346
$227$	− 12.8081i	− 0.850105i	−0.905169	−	0.425053i	$-0.860256\pi$
$227$	− 12.8081i	− 0.850105i	0.905169	−	0.425053i	$-0.139744\pi$
$228$	0	0
$229$	−6.65295	−0.439639	−0.219820	−	0.975541i	$-0.570547\pi$
$229$	−6.65295	−0.439639	−0.219820	+	0.975541i	$0.570547\pi$
$230$	0	0
$231$	0	0
$232$	5.34755i	0.351084i
$233$	− 3.65836i	− 0.239667i	−0.992794	−	0.119833i	$-0.961764\pi$
$233$	− 3.65836i	− 0.239667i	0.992794	−	0.119833i	$-0.0382360\pi$
$234$	0	0
$235$	0	0
$236$	−1.92842	−0.125530
$237$	0	0
$238$	− 5.03632i	− 0.326456i
$239$	−15.6915	−1.01500	−0.507500	−	0.861652i	$-0.669430\pi$
$239$	−15.6915	−1.01500	−0.507500	+	0.861652i	$0.669430\pi$
$240$	0	0
$241$	11.2250	0.723063	0.361532	−	0.932360i	$-0.382254\pi$
$241$	11.2250	0.723063	0.361532	+	0.932360i	$0.382254\pi$
$242$	− 2.26882i	− 0.145845i
$243$	0	0
$244$	1.90242	0.121790
$245$	0	0
$246$	0	0
$247$	− 34.7541i	− 2.21135i
$248$	9.19462i	0.583859i
$249$	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
$253$	− 11.8601i	− 0.745640i
$254$	5.27460	0.330958
$255$	0	0
$256$	14.4607	0.903794
$257$	− 18.3327i	− 1.14356i	−0.820406	−	0.571781i	$-0.806253\pi$
$257$	− 18.3327i	− 1.14356i	0.820406	−	0.571781i	$-0.193747\pi$
$258$	0	0
$259$	−3.39936	−0.211226
$260$	0	0
$261$	0	0
$262$	15.3327i	0.947257i
$263$	16.0766i	0.991328i	0.868514	+	0.495664i	$0.165075\pi$
$263$	16.0766i	0.991328i	−0.868514	+	0.495664i	$0.834925\pi$
$264$	0	0
$265$	0	0
$266$	4.59877	0.281969
$267$	0	0
$268$	1.61009i	0.0983517i
$269$	−18.2004	−1.10970	−0.554849	−	0.831951i	$-0.687224\pi$
$269$	−18.2004	−1.10970	−0.554849	+	0.831951i	$0.687224\pi$
$270$	0	0
$271$	−2.48571	−0.150996	−0.0754979	−	0.997146i	$-0.524055\pi$
$271$	−2.48571	−0.150996	−0.0754979	+	0.997146i	$0.524055\pi$
$272$	29.8329i	1.80888i
$273$	0	0
$274$	3.77317	0.227946
$275$	0	0
$276$	0	0
$277$	7.66726i	0.460681i	0.973110	+	0.230341i	$0.0739840\pi$
$277$	7.66726i	0.460681i	−0.973110	+	0.230341i	$0.926016\pi$
$278$	− 17.7978i	− 1.06744i
$279$	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
$283$	− 3.37089i	− 0.200379i	−0.994968	−	0.100189i	$-0.968055\pi$
$283$	− 3.37089i	− 0.200379i	0.994968	−	0.100189i	$-0.0319448\pi$
$284$	3.70792	0.220025
$285$	0	0
$286$	−31.6011	−1.86861
$287$	− 2.75003i	− 0.162329i
$288$	0	0
$289$	−20.2125	−1.18897
$290$	0	0
$291$	0	0
$292$	7.89401i	0.461962i
$293$	− 5.64404i	− 0.329728i	−0.986316	−	0.164864i	$-0.947281\pi$
$293$	− 5.64404i	− 0.329728i	0.986316	−	0.164864i	$-0.0527186\pi$
$294$	0	0
$295$	0	0
$296$	14.6267	0.850159
$297$	0	0
$298$	− 26.7000i	− 1.54669i
$299$	23.8690	1.38038
$300$	0	0
$301$	0.670803	0.0386645
$302$	37.2206i	2.14181i
$303$	0	0
$304$	−27.2410	−1.56238
$305$	0	0
$306$	0	0
$307$	5.44105i	0.310537i	0.987872	+	0.155269i	$0.0496243\pi$
$307$	5.44105i	0.310537i	−0.987872	+	0.155269i	$0.950376\pi$
$308$	− 1.04819i	− 0.0597263i
$309$	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
$313$	9.14231i	0.516754i	0.966044	+	0.258377i	$0.0831877\pi$
$313$	9.14231i	0.516754i	−0.966044	+	0.258377i	$0.916812\pi$
$314$	−20.3573	−1.14883
$315$	0	0
$316$	2.27677	0.128078
$317$	− 14.2367i	− 0.799614i	−0.916599	−	0.399807i	$-0.869077\pi$
$317$	− 14.2367i	− 0.799614i	0.916599	−	0.399807i	$-0.130923\pi$
$318$	0	0
$319$	−7.62440	−0.426884
$320$	0	0
$321$	0	0
$322$	3.15841i	0.176011i
$323$	− 33.9795i	− 1.89067i
$324$	0	0
$325$	0	0
$326$	−12.3736	−0.685308
$327$	0	0
$328$	11.8328i	0.653355i
$329$	−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	$0.608833\pi$
$332$	− 9.30061i	− 0.510437i
$333$	0	0
$334$	−4.86317	−0.266101
$335$	0	0
$336$	0	0
$337$	4.58986i	0.250026i	0.992155	+	0.125013i	$0.0398972\pi$
$337$	4.58986i	0.250026i	−0.992155	+	0.125013i	$0.960103\pi$
$338$	− 42.3601i	− 2.30408i
$339$	0	0
$340$	0	0
$341$	−13.1094	−0.709917
$342$	0	0
$343$	− 6.94582i	− 0.375039i
$344$	−2.88632	−0.155620
$345$	0	0
$346$	25.8994	1.39236
$347$	− 33.4603i	− 1.79624i	−0.439748	−	0.898121i	$-0.644932\pi$
$347$	− 33.4603i	− 1.79624i	0.439748	−	0.898121i	$-0.355068\pi$
$348$	0	0
$349$	28.0816	1.50317	0.751586	−	0.659636i	$-0.229289\pi$
$349$	28.0816	1.50317	0.751586	+	0.659636i	$0.229289\pi$
$350$	0	0
$351$	0	0
$352$	11.2875i	0.601624i
$353$	− 1.84170i	− 0.0980239i	−0.998798	−	0.0490119i	$-0.984393\pi$
$353$	− 1.84170i	− 0.0980239i	0.998798	−	0.0490119i	$-0.0156072\pi$
$354$	0	0
$355$	0	0
$356$	−2.26483	−0.120036
$357$	0	0
$358$	27.9107i	1.47513i
$359$	−12.1119	−0.639241	−0.319621	−	0.947546i	$-0.603555\pi$
$359$	−12.1119	−0.639241	−0.319621	+	0.947546i	$0.603555\pi$
$360$	0	0
$361$	12.0274	0.633020
$362$	21.8413i	1.14795i
$363$	0	0
$364$	2.10953	0.110569
$365$	0	0
$366$	0	0
$367$	− 14.5738i	− 0.760744i	−0.924834	−	0.380372i	$-0.875796\pi$
$367$	− 14.5738i	− 0.760744i	0.924834	−	0.380372i	$-0.124204\pi$
$368$	− 18.7090i	− 0.975274i
$369$	0	0
$370$	0	0
$371$	−1.28462	−0.0666942
$372$	0	0
$373$	9.44646i	0.489119i	0.969634	+	0.244560i	$0.0786434\pi$
$373$	9.44646i	0.489119i	−0.969634	+	0.244560i	$0.921357\pi$
$374$	−30.8968	−1.59763
$375$	0	0
$376$	8.06251	0.415792
$377$	− 15.3444i	− 0.790276i
$378$	0	0
$379$	28.5541	1.46673	0.733363	−	0.679837i	$-0.237949\pi$
$379$	28.5541	1.46673	0.733363	+	0.679837i	$0.237949\pi$
$380$	0	0
$381$	0	0
$382$	− 41.3940i	− 2.11790i
$383$	1.46541i	0.0748788i	0.999299	+	0.0374394i	$0.0119201\pi$
$383$	1.46541i	0.0748788i	−0.999299	+	0.0374394i	$0.988080\pi$
$384$	0	0
$385$	0	0
$386$	−15.6176	−0.794916
$387$	0	0
$388$	7.41014i	0.376193i
$389$	−12.9101	−0.654569	−0.327284	−	0.944926i	$-0.606134\pi$
$389$	−12.9101	−0.654569	−0.327284	+	0.944926i	$0.606134\pi$
$390$	0	0
$391$	23.3369	1.18020
$392$	14.6655i	0.740720i
$393$	0	0
$394$	−3.37922	−0.170242
$395$	0	0
$396$	0	0
$397$	− 0.868386i	− 0.0435831i	−0.999763	−	0.0217915i	$-0.993063\pi$
$397$	− 0.868386i	− 0.0435831i	0.999763	−	0.0217915i	$-0.00693701\pi$
$398$	− 21.3968i	− 1.07253i
$399$	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
$403$	− 26.3833i	− 1.31424i
$404$	−11.6208	−0.578156
$405$	0	0
$406$	2.03042	0.100768
$407$	20.8544i	1.03371i
$408$	0	0
$409$	−5.05535	−0.249971	−0.124985	−	0.992159i	$-0.539888\pi$
$409$	−5.05535	−0.249971	−0.124985	+	0.992159i	$0.539888\pi$
$410$	0	0
$411$	0	0
$412$	− 0.557157i	− 0.0274492i
$413$	− 1.45658i	− 0.0716734i
$414$	0	0
$415$	0	0
$416$	−22.7165	−1.11377
$417$	0	0
$418$	− 28.2125i	− 1.37992i
$419$	10.9576	0.535313	0.267657	−	0.963514i	$-0.413751\pi$
$419$	10.9576	0.535313	0.267657	+	0.963514i	$0.413751\pi$
$420$	0	0
$421$	−10.6386	−0.518495	−0.259248	−	0.965811i	$-0.583475\pi$
$421$	−10.6386	−0.518495	−0.259248	+	0.965811i	$0.583475\pi$
$422$	18.1538i	0.883711i
$423$	0	0
$424$	5.52744	0.268436
$425$	0	0
$426$	0	0
$427$	1.43693i	0.0695381i
$428$	7.40535i	0.357951i
$429$	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
$433$	17.2125i	0.827179i	0.910464	+	0.413589i	$0.135725\pi$
$433$	17.2125i	0.827179i	−0.910464	+	0.413589i	$0.864275\pi$
$434$	3.49111	0.167579
$435$	0	0
$436$	−3.11668	−0.149262
$437$	21.3094i	1.01937i
$438$	0	0
$439$	31.7487	1.51528	0.757642	−	0.652670i	$-0.226351\pi$
$439$	31.7487	1.51528	0.757642	+	0.652670i	$0.226351\pi$
$440$	0	0
$441$	0	0
$442$	− 62.1809i	− 2.95764i
$443$	− 0.355958i	− 0.0169121i	−0.999964	−	0.00845603i	$-0.997308\pi$
$443$	− 0.355958i	− 0.0169121i	0.999964	−	0.00845603i	$-0.00269167\pi$
$444$	0	0
$445$	0	0
$446$	−6.36174	−0.301237
$447$	0	0
$448$	1.93688i	0.0915088i
$449$	−7.85632	−0.370762	−0.185381	−	0.982667i	$-0.559352\pi$
$449$	−7.85632	−0.370762	−0.185381	+	0.982667i	$0.559352\pi$
$450$	0	0
$451$	−16.8708	−0.794416
$452$	− 8.02476i	− 0.377453i
$453$	0	0
$454$	20.9249	0.982056
$455$	0	0
$456$	0	0
$457$	21.4910i	1.00531i	0.864488	+	0.502654i	$0.167643\pi$
$457$	21.4910i	1.00531i	−0.864488	+	0.502654i	$0.832357\pi$
$458$	− 10.8691i	− 0.507879i
$459$	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
$463$	− 42.1339i	− 1.95813i	−0.203555	−	0.979063i	$-0.565250\pi$
$463$	− 42.1339i	− 1.95813i	0.203555	−	0.979063i	$-0.434750\pi$
$464$	−12.0273	−0.558351
$465$	0	0
$466$	5.97674	0.276867
$467$	22.5376i	1.04292i	0.853276	+	0.521459i	$0.174612\pi$
$467$	22.5376i	1.04292i	−0.853276	+	0.521459i	$0.825388\pi$
$468$	0	0
$469$	−1.21613	−0.0561557
$470$	0	0
$471$	0	0
$472$	6.26732i	0.288477i
$473$	− 4.11523i	− 0.189219i
$474$	0	0
$475$	0	0
$476$	2.06251	0.0945348
$477$	0	0
$478$	− 25.6356i	− 1.17255i
$479$	−33.2880	−1.52097	−0.760483	−	0.649358i	$-0.775038\pi$
$479$	−33.2880	−1.52097	−0.760483	+	0.649358i	$0.775038\pi$
$480$	0	0
$481$	−41.9702	−1.91367
$482$	18.3385i	0.835295i
$483$	0	0
$484$	0.929141	0.0422337
$485$	0	0
$486$	0	0
$487$	23.7703i	1.07713i	0.842583	+	0.538566i	$0.181034\pi$
$487$	23.7703i	1.07713i	−0.842583	+	0.538566i	$0.818966\pi$
$488$	− 6.18281i	− 0.279882i
$489$	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
$493$	− 15.0024i	− 0.675673i
$494$	56.7787	2.55459
$495$	0	0
$496$	−20.6797	−0.928548
$497$	2.80067i	0.125627i
$498$	0	0
$499$	18.8976	0.845971	0.422985	−	0.906137i	$-0.360982\pi$
$499$	18.8976	0.845971	0.422985	+	0.906137i	$0.360982\pi$
$500$	0	0
$501$	0	0
$502$	− 11.3387i	− 0.506072i
$503$	35.7581i	1.59438i	0.603731	+	0.797188i	$0.293680\pi$
$503$	35.7581i	1.59438i	−0.603731	+	0.797188i	$0.706320\pi$
$504$	0	0
$505$	0	0
$506$	19.3762	0.861376
$507$	0	0
$508$	2.16009i	0.0958384i
$509$	24.4067	1.08181	0.540904	−	0.841084i	$-0.318082\pi$
$509$	24.4067	1.08181	0.540904	+	0.841084i	$0.318082\pi$
$510$	0	0
$511$	−5.96250	−0.263766
$512$	− 3.46207i	− 0.153003i
$513$	0	0
$514$	29.9506	1.32106
$515$	0	0
$516$	0	0
$517$	11.4953i	0.505563i
$518$	− 5.55362i	− 0.244012i
$519$	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
$523$	37.3654i	1.63388i	0.576726	+	0.816938i	$0.304330\pi$
$523$	37.3654i	1.63388i	−0.576726	+	0.816938i	$0.695670\pi$
$524$	−6.27914	−0.274305
$525$	0	0
$526$	−26.2648	−1.14520
$527$	− 25.7952i	− 1.12366i
$528$	0	0
$529$	8.36479	0.363686
$530$	0	0
$531$	0	0
$532$	1.88332i	0.0816521i
$533$	− 33.9532i	− 1.47068i
$534$	0	0
$535$	0	0
$536$	5.23274	0.226020
$537$	0	0
$538$	− 29.7344i	− 1.28194i
$539$	−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	$0.292209\pi$
$542$	− 4.06096i	− 0.174433i
$543$	0	0
$544$	−22.2101	−0.952250
$545$	0	0
$546$	0	0
$547$	− 38.5452i	− 1.64807i	−0.566537	−	0.824036i	$-0.691717\pi$
$547$	− 38.5452i	− 1.64807i	0.566537	−	0.824036i	$-0.308283\pi$
$548$	1.54521i	0.0660082i
$549$	0	0
$550$	0	0
$551$	13.6990	0.583596
$552$	0	0
$553$	1.71969i	0.0731286i
$554$	−12.5262	−0.532187
$555$	0	0
$556$	7.28866	0.309108
$557$	27.4125i	1.16151i	0.814080	+	0.580753i	$0.197242\pi$
$557$	27.4125i	1.16151i	−0.814080	+	0.580753i	$0.802758\pi$
$558$	0	0
$559$	8.28206	0.350294
$560$	0	0
$561$	0	0
$562$	− 0.446383i	− 0.0188295i
$563$	− 27.6392i	− 1.16485i	−0.812883	−	0.582427i	$-0.802103\pi$
$563$	− 27.6392i	− 1.16485i	0.812883	−	0.582427i	$-0.197897\pi$
$564$	0	0
$565$	0	0
$566$	5.50710	0.231481
$567$	0	0
$568$	− 12.0506i	− 0.505634i
$569$	−14.7161	−0.616933	−0.308467	−	0.951235i	$-0.599816\pi$
$569$	−14.7161	−0.616933	−0.308467	+	0.951235i	$0.599816\pi$
$570$	0	0
$571$	−28.3006	−1.18434	−0.592172	−	0.805812i	$-0.701729\pi$
$571$	−28.3006	−1.18434	−0.592172	+	0.805812i	$0.701729\pi$
$572$	− 12.9415i	− 0.541111i
$573$	0	0
$574$	4.49279	0.187525
$575$	0	0
$576$	0	0
$577$	− 40.7976i	− 1.69843i	−0.528049	−	0.849214i	$-0.677076\pi$
$577$	− 40.7976i	− 1.69843i	0.528049	−	0.849214i	$-0.322924\pi$
$578$	− 33.0216i	− 1.37352i
$579$	0	0
$580$	0	0
$581$	7.02493	0.291443
$582$	0	0
$583$	7.88087i	0.326392i
$584$	25.6553	1.06162
$585$	0	0
$586$	9.22080	0.380908
$587$	− 2.78033i	− 0.114756i	−0.998353	−	0.0573782i	$-0.981726\pi$
$587$	− 2.78033i	− 0.114756i	0.998353	−	0.0573782i	$-0.0182741\pi$
$588$	0	0
$589$	23.5541	0.970531
$590$	0	0
$591$	0	0
$592$	32.8971i	1.35206i
$593$	− 14.8084i	− 0.608109i	−0.952655	−	0.304055i	$-0.901659\pi$
$593$	− 14.8084i	− 0.608109i	0.952655	−	0.304055i	$-0.0983405\pi$
$594$	0	0
$595$	0	0
$596$	10.9343	0.447888
$597$	0	0
$598$	38.9953i	1.59464i
$599$	−16.3430	−0.667758	−0.333879	−	0.942616i	$-0.608358\pi$
$599$	−16.3430	−0.667758	−0.333879	+	0.942616i	$0.608358\pi$
$600$	0	0
$601$	−6.62371	−0.270187	−0.135093	−	0.990833i	$-0.543133\pi$
$601$	−6.62371	−0.270187	−0.135093	+	0.990833i	$0.543133\pi$
$602$	1.09591i	0.0446658i
$603$	0	0
$604$	−15.2428	−0.620221
$605$	0	0
$606$	0	0
$607$	30.3094i	1.23022i	0.788442	+	0.615110i	$0.210888\pi$
$607$	30.3094i	1.23022i	−0.788442	+	0.615110i	$0.789112\pi$
$608$	− 20.2805i	− 0.822483i
$609$	0	0
$610$	0	0
$611$	−23.1347	−0.935931
$612$	0	0
$613$	− 14.7803i	− 0.596969i	−0.954415	−	0.298484i	$-0.903519\pi$
$613$	− 14.7803i	− 0.596969i	0.954415	−	0.298484i	$-0.0964811\pi$
$614$	−8.88918	−0.358738
$615$	0	0
$616$	−3.40660	−0.137256
$617$	33.8512i	1.36280i	0.731912	+	0.681399i	$0.238628\pi$
$617$	33.8512i	1.36280i	−0.731912	+	0.681399i	$0.761372\pi$
$618$	0	0
$619$	−11.6887	−0.469807	−0.234903	−	0.972019i	$-0.575477\pi$
$619$	−11.6887	−0.469807	−0.234903	+	0.972019i	$0.575477\pi$
$620$	0	0
$621$	0	0
$622$	31.1709i	1.24984i
$623$	− 1.71067i	− 0.0685364i
$624$	0	0
$625$	0	0
$626$	−14.9360	−0.596963
$627$	0	0
$628$	− 8.33686i	− 0.332677i
$629$	−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	$0.774350\pi$
$632$	− 7.39944i	− 0.294334i
$633$	0	0
$634$	23.2589	0.923728
$635$	0	0
$636$	0	0
$637$	− 42.0816i	− 1.66733i
$638$	− 12.4562i	− 0.493144i
$639$	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
$643$	− 2.68515i	− 0.105892i	−0.998597	−	0.0529461i	$-0.983139\pi$
$643$	− 2.68515i	− 0.105892i	0.998597	−	0.0529461i	$-0.0168611\pi$
$644$	−1.29345	−0.0509692
$645$	0	0
$646$	55.5131	2.18413
$647$	− 40.5103i	− 1.59262i	−0.604887	−	0.796311i	$-0.706782\pi$
$647$	− 40.5103i	− 1.59262i	0.604887	−	0.796311i	$-0.293218\pi$
$648$	0	0
$649$	−8.93578	−0.350760
$650$	0	0
$651$	0	0
$652$	− 5.06729i	− 0.198451i
$653$	13.3354i	0.521856i	0.965358	+	0.260928i	$0.0840286\pi$
$653$	13.3354i	0.521856i	−0.965358	+	0.260928i	$0.915971\pi$
$654$	0	0
$655$	0	0
$656$	−26.6132	−1.03907
$657$	0	0
$658$	− 3.06126i	− 0.119340i
$659$	31.1543	1.21360	0.606800	−	0.794855i	$-0.292453\pi$
$659$	31.1543	1.21360	0.606800	+	0.794855i	$0.292453\pi$
$660$	0	0
$661$	6.31788	0.245737	0.122869	−	0.992423i	$-0.460791\pi$
$661$	6.31788	0.245737	0.122869	+	0.992423i	$0.460791\pi$
$662$	− 19.9315i	− 0.774659i
$663$	0	0
$664$	−30.2267	−1.17302
$665$	0	0
$666$	0	0
$667$	9.40838i	0.364294i
$668$	− 1.99160i	− 0.0770571i
$669$	0	0
$670$	0	0
$671$	8.81528	0.340310
$672$	0	0
$673$	6.59220i	0.254110i	0.991896	+	0.127055i	$0.0405525\pi$
$673$	6.59220i	0.254110i	−0.991896	+	0.127055i	$0.959447\pi$
$674$	−7.49857	−0.288834
$675$	0	0
$676$	17.3476	0.667214
$677$	− 34.8947i	− 1.34111i	−0.741859	−	0.670556i	$-0.766056\pi$
$677$	− 34.8947i	− 1.34111i	0.741859	−	0.670556i	$-0.233944\pi$
$678$	0	0
$679$	−5.59702	−0.214794
$680$	0	0
$681$	0	0
$682$	− 21.4172i	− 0.820108i
$683$	26.0958i	0.998528i	0.866450	+	0.499264i	$0.166396\pi$
$683$	26.0958i	0.998528i	−0.866450	+	0.499264i	$0.833604\pi$
$684$	0	0
$685$	0	0
$686$	11.3476	0.433252
$687$	0	0
$688$	− 6.49165i	− 0.247492i
$689$	−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	$0.688210\pi$
$692$	10.6065i	0.403197i
$693$	0	0
$694$	54.6648	2.07505
$695$	0	0
$696$	0	0
$697$	− 33.1964i	− 1.25740i
$698$	45.8775i	1.73649i
$699$	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
$703$	− 37.4696i	− 1.41319i
$704$	11.8823	0.447832
$705$	0	0
$706$	3.00883	0.113239
$707$	− 8.77741i	− 0.330108i
$708$	0	0
$709$	7.73991	0.290678	0.145339	−	0.989382i	$-0.453573\pi$
$709$	7.73991	0.290678	0.145339	+	0.989382i	$0.453573\pi$
$710$	0	0
$711$	0	0
$712$	7.36062i	0.275851i
$713$	16.1768i	0.605828i
$714$	0	0
$715$	0	0
$716$	−11.4302	−0.427165
$717$	0	0
$718$	− 19.7875i	− 0.738462i
$719$	−15.1316	−0.564313	−0.282156	−	0.959368i	$-0.591050\pi$
$719$	−15.1316	−0.564313	−0.282156	+	0.959368i	$0.591050\pi$
$720$	0	0
$721$	0.420832	0.0156726
$722$	19.6494i	0.731275i
$723$	0	0
$724$	−8.94457	−0.332422
$725$	0	0
$726$	0	0
$727$	0.161876i	0.00600366i	0.999995	+	0.00300183i	$0.000955513\pi$
$727$	0.161876i	0.00600366i	−0.999995	+	0.00300183i	$0.999044\pi$
$728$	− 6.85590i	− 0.254097i
$729$	0	0
$730$	0	0
$731$	8.09746	0.299495
$732$	0	0
$733$	50.0332i	1.84802i	0.382371	+	0.924009i	$0.375107\pi$
$733$	50.0332i	1.84802i	−0.382371	+	0.924009i	$0.624893\pi$
$734$	23.8095	0.878825
$735$	0	0
$736$	13.9285	0.513413
$737$	7.46070i	0.274818i
$738$	0	0
$739$	−30.5505	−1.12382	−0.561909	−	0.827199i	$-0.689933\pi$
$739$	−30.5505	−1.12382	−0.561909	+	0.827199i	$0.689933\pi$
$740$	0	0
$741$	0	0
$742$	− 2.09872i	− 0.0770463i
$743$	5.96684i	0.218902i	0.993992	+	0.109451i	$0.0349093\pi$
$743$	5.96684i	0.218902i	−0.993992	+	0.109451i	$0.965091\pi$
$744$	0	0
$745$	0	0
$746$	−15.4329	−0.565039
$747$	0	0
$748$	− 12.6530i	− 0.462640i
$749$	−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	$0.284040\pi$
$752$	18.1335i	0.661260i
$753$	0	0
$754$	25.0685	0.912941
$755$	0	0
$756$	0	0
$757$	40.6873i	1.47881i	0.673263	+	0.739403i	$0.264892\pi$
$757$	40.6873i	1.47881i	−0.673263	+	0.739403i	$0.735108\pi$
$758$	46.6495i	1.69439i
$759$	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
$763$	− 2.35409i	− 0.0852239i
$764$	16.9519	0.613299
$765$	0	0
$766$	−2.39407	−0.0865013
$767$	− 17.9836i	− 0.649350i
$768$	0	0
$769$	46.9036	1.69139	0.845694	−	0.533668i	$-0.179187\pi$
$769$	46.9036	1.69139	0.845694	+	0.533668i	$0.179187\pi$
$770$	0	0
$771$	0	0
$772$	− 6.39582i	− 0.230191i
$773$	9.19641i	0.330772i	0.986229	+	0.165386i	$0.0528870\pi$
$773$	9.19641i	0.330772i	−0.986229	+	0.165386i	$0.947113\pi$
$774$	0	0
$775$	0	0
$776$	24.0827	0.864520
$777$	0	0
$778$	− 21.0916i	− 0.756169i
$779$	30.3123	1.08605
$780$	0	0
$781$	17.1815	0.614802
$782$	38.1261i	1.36339i
$783$	0	0
$784$	−32.9844	−1.17801
$785$	0	0
$786$	0	0
$787$	5.74637i	0.204836i	0.994741	+	0.102418i	$0.0326579\pi$
$787$	5.74637i	0.204836i	−0.994741	+	0.102418i	$0.967342\pi$
$788$	− 1.38388i	− 0.0492986i
$789$	0	0
$790$	0	0
$791$	6.06126	0.215514
$792$	0	0
$793$	17.7411i	0.630004i
$794$	1.41870	0.0503479
$795$	0	0
$796$	8.76255	0.310580
$797$	7.07450i	0.250592i	0.992119	+	0.125296i	$0.0399880\pi$
$797$	7.07450i	0.250592i	−0.992119	+	0.125296i	$0.960012\pi$
$798$	0	0
$799$	−22.6191	−0.800205
$800$	0	0
$801$	0	0
$802$	− 54.5868i	− 1.92753i
$803$	36.5787i	1.29083i
$804$	0	0
$805$	0	0
$806$	43.1029	1.51824
$807$	0	0
$808$	37.7672i	1.32865i
$809$	38.1075	1.33979	0.669894	−	0.742457i	$-0.266340\pi$
$809$	38.1075	1.33979	0.669894	+	0.742457i	$0.266340\pi$
$810$	0	0
$811$	−1.44105	−0.0506022	−0.0253011	−	0.999680i	$-0.508054\pi$
$811$	−1.44105	−0.0506022	−0.0253011	+	0.999680i	$0.508054\pi$
$812$	0.831508i	0.0291802i
$813$	0	0
$814$	−34.0702	−1.19416
$815$	0	0
$816$	0	0
$817$	7.39396i	0.258682i
$818$	− 8.25904i	− 0.288771i
$819$	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
$823$	41.4589i	1.44517i	0.691284	+	0.722583i	$0.257045\pi$
$823$	41.4589i	1.44517i	−0.691284	+	0.722583i	$0.742955\pi$
$824$	−1.81075	−0.0630803
$825$	0	0
$826$	2.37964	0.0827984
$827$	− 27.8133i	− 0.967164i	−0.875299	−	0.483582i	$-0.839336\pi$
$827$	− 27.8133i	− 0.967164i	0.875299	−	0.483582i	$-0.160664\pi$
$828$	0	0
$829$	−20.7232	−0.719745	−0.359872	−	0.933002i	$-0.617180\pi$
$829$	−20.7232	−0.719745	−0.359872	+	0.933002i	$0.617180\pi$
$830$	0	0
$831$	0	0
$832$	23.9136i	0.829056i
$833$	− 41.1436i	− 1.42554i
$834$	0	0
$835$	0	0
$836$	11.5537	0.399595
$837$	0	0
$838$	17.9017i	0.618403i
$839$	18.1451	0.626437	0.313218	−	0.949681i	$-0.398593\pi$
$839$	18.1451	0.626437	0.313218	+	0.949681i	$0.398593\pi$
$840$	0	0
$841$	−22.9517	−0.791439
$842$	− 17.3806i	− 0.598975i
$843$	0	0
$844$	−7.43444	−0.255904
$845$	0	0
$846$	0	0
$847$	0.701798i	0.0241141i
$848$	12.4318i	0.426911i
$849$	0	0
$850$	0	0
$851$	25.7339	0.882148
$852$	0	0
$853$	4.00708i	0.137200i	0.997644	+	0.0685999i	$0.0218532\pi$
$853$	4.00708i	0.137200i	−0.997644	+	0.0685999i	$0.978147\pi$
$854$	−2.34755	−0.0803316
$855$	0	0
$856$	24.0672	0.822599
$857$	9.08971i	0.310499i	0.987875	+	0.155249i	$0.0496181\pi$
$857$	9.08971i	0.310499i	−0.987875	+	0.155249i	$0.950382\pi$
$858$	0	0
$859$	−16.3870	−0.559116	−0.279558	−	0.960129i	$-0.590188\pi$
$859$	−16.3870	−0.559116	−0.279558	+	0.960129i	$0.590188\pi$
$860$	0	0
$861$	0	0
$862$	61.0243i	2.07850i
$863$	23.7967i	0.810050i	0.914306	+	0.405025i	$0.132737\pi$
$863$	23.7967i	0.810050i	−0.914306	+	0.405025i	$0.867263\pi$
$864$	0	0
$865$	0	0
$866$	−28.1204	−0.955571
$867$	0	0
$868$	1.42970i	0.0485272i
$869$	10.5499	0.357882
$870$	0	0
$871$	−15.0149	−0.508762
$872$	10.1291i	0.343016i
$873$	0	0
$874$	−34.8137	−1.17759
$875$	0	0
$876$	0	0
$877$	36.0744i	1.21815i	0.793114	+	0.609073i	$0.208458\pi$
$877$	36.0744i	1.21815i	−0.793114	+	0.609073i	$0.791542\pi$
$878$	51.8687i	1.75048i
$879$	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
$883$	39.1320i	1.31690i	0.752626	+	0.658448i	$0.228787\pi$
$883$	39.1320i	1.31690i	−0.752626	+	0.658448i	$0.771213\pi$
$884$	25.4647	0.856470
$885$	0	0
$886$	0.581537	0.0195371
$887$	− 51.2833i	− 1.72192i	−0.508670	−	0.860962i	$-0.669863\pi$
$887$	− 51.2833i	− 1.72192i	0.508670	−	0.860962i	$-0.330137\pi$
$888$	0	0
$889$	−1.63156	−0.0547207
$890$	0	0
$891$	0	0
$892$	− 2.60530i	− 0.0872318i
$893$	− 20.6539i	− 0.691158i
$894$	0	0
$895$	0	0
$896$	−6.84415	−0.228647
$897$	0	0
$898$	− 12.8350i	− 0.428311i
$899$	10.3994	0.346841
$900$	0	0
$901$	−15.5070	−0.516614
$902$	− 27.5623i	− 0.917723i
$903$	0	0
$904$	−26.0802	−0.867416
$905$	0	0
$906$	0	0
$907$	47.8588i	1.58913i	0.607181	+	0.794563i	$0.292300\pi$
$907$	47.8588i	1.58913i	−0.607181	+	0.794563i	$0.707700\pi$
$908$	8.56930i	0.284382i
$909$	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
$913$	− 43.0964i	− 1.42628i
$914$	−35.1104	−1.16135
$915$	0	0
$916$	4.45117	0.147071
$917$	− 4.74276i	− 0.156620i
$918$	0	0
$919$	−10.3976	−0.342984	−0.171492	−	0.985185i	$-0.554859\pi$
$919$	−10.3976	−0.342984	−0.171492	+	0.985185i	$0.554859\pi$
$920$	0	0
$921$	0	0
$922$	− 66.9708i	− 2.20557i
$923$	34.5784i	1.13816i
$924$	0	0
$925$	0	0
$926$	68.8351	2.26206
$927$	0	0
$928$	− 8.95410i	− 0.293933i
$929$	−36.0216	−1.18183	−0.590915	−	0.806734i	$-0.701233\pi$
$929$	−36.0216	−1.18183	−0.590915	+	0.806734i	$0.701233\pi$
$930$	0	0
$931$	37.5691	1.23128
$932$	2.44763i	0.0801748i
$933$	0	0
$934$	−36.8203	−1.20480
$935$	0	0
$936$	0	0
$937$	24.0326i	0.785111i	0.919728	+	0.392555i	$0.128409\pi$
$937$	24.0326i	0.785111i	−0.919728	+	0.392555i	$0.871591\pi$
$938$	− 1.98682i	− 0.0648720i
$939$	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
$943$	20.8183i	0.677938i
$944$	−14.0959	−0.458783
$945$	0	0
$946$	6.72315	0.218589
$947$	27.5400i	0.894928i	0.894302	+	0.447464i	$0.147673\pi$
$947$	27.5400i	0.894928i	−0.894302	+	0.447464i	$0.852327\pi$
$948$	0	0
$949$	−73.6160	−2.38968
$950$	0	0
$951$	0	0
$952$	− 6.70308i	− 0.217248i
$953$	− 18.1344i	− 0.587432i	−0.955893	−	0.293716i	$-0.905108\pi$
$953$	− 18.1344i	− 0.587432i	0.955893	−	0.293716i	$-0.0948920\pi$
$954$	0	0
$955$	0	0
$956$	10.4984	0.339544
$957$	0	0
$958$	− 54.3833i	− 1.75705i
$959$	−1.16713	−0.0376886
$960$	0	0
$961$	−13.1191	−0.423198
$962$	− 68.5676i	− 2.21071i
$963$	0	0
$964$	−7.51009	−0.241884
$965$	0	0
$966$	0	0
$967$	36.1875i	1.16371i	0.813292	+	0.581855i	$0.197673\pi$
$967$	36.1875i	1.16371i	−0.813292	+	0.581855i	$0.802327\pi$
$968$	− 3.01968i	− 0.0970562i
$969$	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
$973$	5.50527i	0.176491i
$974$	−38.8340	−1.24432
$975$	0	0
$976$	13.9058	0.445115
$977$	29.4331i	0.941650i	0.882227	+	0.470825i	$0.156044\pi$
$977$	29.4331i	0.941650i	−0.882227	+	0.470825i	$0.843956\pi$
$978$	0	0
$979$	−10.4946	−0.335408
$980$	0	0
$981$	0	0
$982$	7.52432i	0.240111i
$983$	24.4911i	0.781145i	0.920572	+	0.390573i	$0.127723\pi$
$983$	24.4911i	0.781145i	−0.920572	+	0.390573i	$0.872277\pi$
$984$	0	0
$985$	0	0
$986$	24.5097	0.780549
$987$	0	0
$988$	23.2523i	0.739755i
$989$	−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	$0.567659\pi$
$992$	− 15.3957i	− 0.488815i
$993$	0	0
$994$	−4.57551	−0.145126
$995$	0	0
$996$	0	0
$997$	39.1530i	1.23999i	0.784606	+	0.619994i	$0.212865\pi$
$997$	39.1530i	1.23999i	−0.784606	+	0.619994i	$0.787135\pi$
$998$	30.8734i	0.977280i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.e.c.76.4	✓	8	45.38	even	12
225.2.e.c.151.4	yes	8	45.23	even	12
225.2.e.e.76.1	yes	8	45.2	even	12
225.2.e.e.151.1	yes	8	45.32	even	12
225.2.k.c.49.2		16	9.2	odd	6
225.2.k.c.49.7		16	45.29	odd	6
225.2.k.c.124.2		16	45.14	odd	6
225.2.k.c.124.7		16	9.5	odd	6
675.2.e.c.226.4		8	45.7	odd	12
675.2.e.c.451.4		8	45.22	odd	12
675.2.e.e.226.1		8	45.43	odd	12
675.2.e.e.451.1		8	45.13	odd	12
675.2.k.c.199.2		16	45.34	even	6
675.2.k.c.199.7		16	9.7	even	3
675.2.k.c.424.2		16	9.4	even	3
675.2.k.c.424.7		16	45.4	even	6
2025.2.a.p.1.4		4	5.3	odd	4
2025.2.a.q.1.4		4	15.2	even	4
2025.2.a.y.1.1		4	15.8	even	4
2025.2.a.z.1.1		4	5.2	odd	4
2025.2.b.n.649.2		8	3.2	odd	2
2025.2.b.n.649.7		8	15.14	odd	2
2025.2.b.o.649.2		8	5.4	even	2	inner
2025.2.b.o.649.7		8	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.63372i q^{2} -0.669052 q^{4} -0.505348i q^{7} +2.17440i q^{8} +O(q^{10})\)	\(q+1.63372i q^{2} -0.669052 q^{4} -0.505348i q^{7} +2.17440i q^{8} -3.10020 q^{11} -6.23927i q^{13} +0.825599 q^{14} -4.89047 q^{16} -6.10020i q^{17} +5.57022 q^{19} -5.06487i q^{22} +3.82560i q^{23} +10.1932 q^{26} +0.338104i q^{28} +2.45932 q^{29} +4.22858 q^{31} -3.64088i q^{32} +9.96604 q^{34} -6.72677i q^{37} +9.10020i q^{38} +5.44185 q^{41} +1.32741i q^{43} +2.07420 q^{44} -6.24997 q^{46} -3.70792i q^{47} +6.74462 q^{49} +4.17440i q^{52} -2.54205i q^{53} +1.09883 q^{56} +4.01785i q^{58} +2.88232 q^{59} -2.84345 q^{61} +6.90833i q^{62} -3.83276 q^{64} -2.40652i q^{67} +4.08135i q^{68} -5.54205 q^{71} -11.7988i q^{73} +10.9897 q^{74} -3.72677 q^{76} +1.56668i q^{77} -3.40298 q^{79} +8.89047i q^{82} +13.9012i q^{83} -2.16862 q^{86} -6.74108i q^{88} +3.38513 q^{89} -3.15301 q^{91} -2.55953i q^{92} +6.05772 q^{94} -11.0756i q^{97} +11.0188i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 2 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{26} + 2 q^{29} - 8 q^{31} - 18 q^{34} + 10 q^{41} + 44 q^{44} + 6 q^{49} + 60 q^{56} + 34 q^{59} - 26 q^{61} - 38 q^{64} + 16 q^{71} + 80 q^{74} + 22 q^{76} + 14 q^{79} + 68 q^{86} - 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 + 2 * q^11 + 6 * q^14 + 8 * q^16 - 4 * q^19 + 20 * q^26 + 2 * q^29 - 8 * q^31 - 18 * q^34 + 10 * q^41 + 44 * q^44 + 6 * q^49 + 60 * q^56 + 34 * q^59 - 26 * q^61 - 38 * q^64 + 16 * q^71 + 80 * q^74 + 22 * q^76 + 14 * q^79 + 68 * q^86 - 18 * q^89 - 34 * q^91 - 6 * q^94

Introduction

L-functions

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