LMFDB - Embedded newform 3525.2.a.v.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,2,Mod(1,3525)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.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:	$28.1472667125$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2379008.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} + 23x - 4 $ x^5 - 10x^3 + 23x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.62251$ of defining polynomial
Character	$\chi$	$=$	3525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.62251 q^{2} -1.00000 q^{3} +4.87757 q^{4} -2.62251 q^{6} -4.04637 q^{7} +7.54645 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.62251 q^{2} -1.00000 q^{3} +4.87757 q^{4} -2.62251 q^{6} -4.04637 q^{7} +7.54645 q^{8} +1.00000 q^{9} +1.06521 q^{11} -4.87757 q^{12} -5.43487 q^{13} -10.6117 q^{14} +10.0355 q^{16} -0.622511 q^{17} +2.62251 q^{18} -3.36664 q^{19} +4.04637 q^{21} +2.79352 q^{22} -6.98915 q^{23} -7.54645 q^{24} -14.2530 q^{26} -1.00000 q^{27} -19.7365 q^{28} -1.81236 q^{29} -9.05436 q^{31} +11.2254 q^{32} -1.06521 q^{33} -1.63254 q^{34} +4.87757 q^{36} +6.78267 q^{37} -8.82904 q^{38} +5.43487 q^{39} -9.27487 q^{41} +10.6117 q^{42} +2.90510 q^{43} +5.19563 q^{44} -18.3291 q^{46} -1.00000 q^{47} -10.0355 q^{48} +9.37313 q^{49} +0.622511 q^{51} -26.5089 q^{52} +14.1493 q^{53} -2.62251 q^{54} -30.5358 q^{56} +3.36664 q^{57} -4.75293 q^{58} +6.44588 q^{59} +9.22316 q^{61} -23.7452 q^{62} -4.04637 q^{63} +9.36761 q^{64} -2.79352 q^{66} +3.56447 q^{67} -3.03634 q^{68} +6.98915 q^{69} -4.04417 q^{71} +7.54645 q^{72} -11.3987 q^{73} +17.7876 q^{74} -16.4210 q^{76} -4.31023 q^{77} +14.2530 q^{78} +1.36746 q^{79} +1.00000 q^{81} -24.3234 q^{82} -6.90741 q^{83} +19.7365 q^{84} +7.61867 q^{86} +1.81236 q^{87} +8.03854 q^{88} +1.13344 q^{89} +21.9915 q^{91} -34.0900 q^{92} +9.05436 q^{93} -2.62251 q^{94} -11.2254 q^{96} +0.379692 q^{97} +24.5812 q^{98} +1.06521 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9} - 2 q^{11} - 10 q^{12} - 7 q^{13} - 8 q^{14} + 8 q^{16} + 10 q^{17} + 2 q^{19} + 10 q^{21} + 4 q^{22} - 3 q^{23} - 16 q^{26} - 5 q^{27} - 12 q^{28} - 2 q^{29} - 6 q^{31} + 20 q^{32} + 2 q^{33} - 20 q^{34} + 10 q^{36} - 8 q^{37} + 8 q^{38} + 7 q^{39} + 8 q^{42} - 13 q^{43} + 4 q^{44} - 12 q^{46} - 5 q^{47} - 8 q^{48} + 13 q^{49} - 10 q^{51} - 34 q^{52} + 10 q^{53} - 28 q^{56} - 2 q^{57} + 4 q^{58} - 11 q^{59} + 11 q^{61} - 16 q^{62} - 10 q^{63} - 20 q^{64} - 4 q^{66} - 24 q^{67} + 20 q^{68} + 3 q^{69} - 11 q^{71} - 17 q^{73} + 12 q^{74} - 24 q^{76} + 12 q^{77} + 16 q^{78} - 5 q^{79} + 5 q^{81} - 64 q^{82} + 12 q^{84} + 12 q^{86} + 2 q^{87} + 4 q^{88} - 3 q^{89} + 22 q^{91} - 34 q^{92} + 6 q^{93} - 20 q^{96} + 14 q^{97} + 36 q^{98} - 2 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9 - 2 * q^11 - 10 * q^12 - 7 * q^13 - 8 * q^14 + 8 * q^16 + 10 * q^17 + 2 * q^19 + 10 * q^21 + 4 * q^22 - 3 * q^23 - 16 * q^26 - 5 * q^27 - 12 * q^28 - 2 * q^29 - 6 * q^31 + 20 * q^32 + 2 * q^33 - 20 * q^34 + 10 * q^36 - 8 * q^37 + 8 * q^38 + 7 * q^39 + 8 * q^42 - 13 * q^43 + 4 * q^44 - 12 * q^46 - 5 * q^47 - 8 * q^48 + 13 * q^49 - 10 * q^51 - 34 * q^52 + 10 * q^53 - 28 * q^56 - 2 * q^57 + 4 * q^58 - 11 * q^59 + 11 * q^61 - 16 * q^62 - 10 * q^63 - 20 * q^64 - 4 * q^66 - 24 * q^67 + 20 * q^68 + 3 * q^69 - 11 * q^71 - 17 * q^73 + 12 * q^74 - 24 * q^76 + 12 * q^77 + 16 * q^78 - 5 * q^79 + 5 * q^81 - 64 * q^82 + 12 * q^84 + 12 * q^86 + 2 * q^87 + 4 * q^88 - 3 * q^89 + 22 * q^91 - 34 * q^92 + 6 * q^93 - 20 * q^96 + 14 * q^97 + 36 * q^98 - 2 * q^99

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 $

$2$	2.62251	1.85440	0.927198	−	0.374572i	$-0.122210\pi$
$2$	2.62251	1.85440	0.927198	+	0.374572i	$0.122210\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.87757	2.43878
$5$	0	0
$5$	0	0
$6$	−2.62251	−1.07064
$7$	−4.04637	−1.52939	−0.764693	−	0.644395i	$-0.777109\pi$
$7$	−4.04637	−1.52939	−0.764693	+	0.644395i	$0.777109\pi$
$8$	7.54645	2.66807
$9$	1.00000	0.333333
$10$	0	0
$11$	1.06521	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$11$	1.06521	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$12$	−4.87757	−1.40803
$13$	−5.43487	−1.50736	−0.753681	−	0.657241i	$-0.771723\pi$
$13$	−5.43487	−1.50736	−0.753681	+	0.657241i	$0.771723\pi$
$14$	−10.6117	−2.83609
$15$	0	0
$16$	10.0355	2.50888
$17$	−0.622511	−0.150981	−0.0754906	−	0.997147i	$-0.524052\pi$
$17$	−0.622511	−0.150981	−0.0754906	+	0.997147i	$0.524052\pi$
$18$	2.62251	0.618132
$19$	−3.36664	−0.772359	−0.386180	−	0.922424i	$-0.626206\pi$
$19$	−3.36664	−0.772359	−0.386180	+	0.922424i	$0.626206\pi$
$20$	0	0
$21$	4.04637	0.882991
$22$	2.79352	0.595581
$23$	−6.98915	−1.45734	−0.728669	−	0.684866i	$-0.759861\pi$
$23$	−6.98915	−1.45734	−0.728669	+	0.684866i	$0.759861\pi$
$24$	−7.54645	−1.54041
$25$	0	0
$26$	−14.2530	−2.79524
$27$	−1.00000	−0.192450
$28$	−19.7365	−3.72984
$29$	−1.81236	−0.336546	−0.168273	−	0.985740i	$-0.553819\pi$
$29$	−1.81236	−0.336546	−0.168273	+	0.985740i	$0.553819\pi$
$30$	0	0
$31$	−9.05436	−1.62621	−0.813105	−	0.582117i	$-0.802225\pi$
$31$	−9.05436	−1.62621	−0.813105	+	0.582117i	$0.802225\pi$
$32$	11.2254	1.98438
$33$	−1.06521	−0.185429
$34$	−1.63254	−0.279979
$35$	0	0
$36$	4.87757	0.812928
$37$	6.78267	1.11506	0.557532	−	0.830155i	$-0.311748\pi$
$37$	6.78267	1.11506	0.557532	+	0.830155i	$0.311748\pi$
$38$	−8.82904	−1.43226
$39$	5.43487	0.870276
$40$	0	0
$41$	−9.27487	−1.44849	−0.724245	−	0.689542i	$-0.757812\pi$
$41$	−9.27487	−1.44849	−0.724245	+	0.689542i	$0.757812\pi$
$42$	10.6117	1.63741
$43$	2.90510	0.443024	0.221512	−	0.975158i	$-0.428901\pi$
$43$	2.90510	0.443024	0.221512	+	0.975158i	$0.428901\pi$
$44$	5.19563	0.783270
$45$	0	0
$46$	−18.3291	−2.70248
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	−10.0355	−1.44850
$49$	9.37313	1.33902
$50$	0	0
$51$	0.622511	0.0871690
$52$	−26.5089	−3.67613
$53$	14.1493	1.94356	0.971778	−	0.235896i	$-0.0758024\pi$
$53$	14.1493	1.94356	0.971778	+	0.235896i	$0.0758024\pi$
$54$	−2.62251	−0.356879
$55$	0	0
$56$	−30.5358	−4.08051
$57$	3.36664	0.445922
$58$	−4.75293	−0.624090
$59$	6.44588	0.839182	0.419591	−	0.907713i	$-0.362174\pi$
$59$	6.44588	0.839182	0.419591	+	0.907713i	$0.362174\pi$
$60$	0	0
$61$	9.22316	1.18091	0.590453	−	0.807072i	$-0.298949\pi$
$61$	9.22316	1.18091	0.590453	+	0.807072i	$0.298949\pi$
$62$	−23.7452	−3.01564
$63$	−4.04637	−0.509795
$64$	9.36761	1.17095
$65$	0	0
$66$	−2.79352	−0.343859
$67$	3.56447	0.435469	0.217734	−	0.976008i	$-0.430133\pi$
$67$	3.56447	0.435469	0.217734	+	0.976008i	$0.430133\pi$
$68$	−3.03634	−0.368210
$69$	6.98915	0.841395
$70$	0	0
$71$	−4.04417	−0.479955	−0.239977	−	0.970779i	$-0.577140\pi$
$71$	−4.04417	−0.479955	−0.239977	+	0.970779i	$0.577140\pi$
$72$	7.54645	0.889358
$73$	−11.3987	−1.33412	−0.667058	−	0.745006i	$-0.732446\pi$
$73$	−11.3987	−1.33412	−0.667058	+	0.745006i	$0.732446\pi$
$74$	17.7876	2.06777
$75$	0	0
$76$	−16.4210	−1.88362
$77$	−4.31023	−0.491196
$78$	14.2530	1.61384
$79$	1.36746	0.153851	0.0769254	−	0.997037i	$-0.475490\pi$
$79$	1.36746	0.153851	0.0769254	+	0.997037i	$0.475490\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−24.3234	−2.68607
$83$	−6.90741	−0.758187	−0.379093	−	0.925358i	$-0.623764\pi$
$83$	−6.90741	−0.758187	−0.379093	+	0.925358i	$0.623764\pi$
$84$	19.7365	2.15342
$85$	0	0
$86$	7.61867	0.821542
$87$	1.81236	0.194305
$88$	8.03854	0.856912
$89$	1.13344	0.120145	0.0600723	−	0.998194i	$-0.480867\pi$
$89$	1.13344	0.120145	0.0600723	+	0.998194i	$0.480867\pi$
$90$	0	0
$91$	21.9915	2.30534
$92$	−34.0900	−3.55413
$93$	9.05436	0.938893
$94$	−2.62251	−0.270491
$95$	0	0
$96$	−11.2254	−1.14568
$97$	0.379692	0.0385519	0.0192760	−	0.999814i	$-0.493864\pi$
$97$	0.379692	0.0385519	0.0192760	+	0.999814i	$0.493864\pi$
$98$	24.5812	2.48307
$99$	1.06521	0.107058
$100$	0	0
$101$	−2.35645	−0.234475	−0.117238	−	0.993104i	$-0.537404\pi$
$101$	−2.35645	−0.234475	−0.117238	+	0.993104i	$0.537404\pi$
$102$	1.63254	0.161646
$103$	6.22619	0.613484	0.306742	−	0.951793i	$-0.400761\pi$
$103$	6.22619	0.613484	0.306742	+	0.951793i	$0.400761\pi$
$104$	−41.0140	−4.02175
$105$	0	0
$106$	37.1067	3.60412
$107$	−7.18764	−0.694856	−0.347428	−	0.937707i	$-0.612945\pi$
$107$	−7.18764	−0.694856	−0.347428	+	0.937707i	$0.612945\pi$
$108$	−4.87757	−0.469344
$109$	−1.64124	−0.157203	−0.0786014	−	0.996906i	$-0.525045\pi$
$109$	−1.64124	−0.157203	−0.0786014	+	0.996906i	$0.525045\pi$
$110$	0	0
$111$	−6.78267	−0.643783
$112$	−40.6075	−3.83704
$113$	−18.9705	−1.78459	−0.892296	−	0.451451i	$-0.850906\pi$
$113$	−18.9705	−1.78459	−0.892296	+	0.451451i	$0.850906\pi$
$114$	8.82904	0.826916
$115$	0	0
$116$	−8.83989	−0.820764
$117$	−5.43487	−0.502454
$118$	16.9044	1.55617
$119$	2.51891	0.230908
$120$	0	0
$121$	−9.86533	−0.896848
$122$	24.1879	2.18987
$123$	9.27487	0.836287
$124$	−44.1632	−3.96597
$125$	0	0
$126$	−10.6117	−0.945362
$127$	−7.62472	−0.676584	−0.338292	−	0.941041i	$-0.609849\pi$
$127$	−7.62472	−0.676584	−0.338292	+	0.941041i	$0.609849\pi$
$128$	2.11594	0.187024
$129$	−2.90510	−0.255780
$130$	0	0
$131$	−20.7871	−1.81618	−0.908090	−	0.418776i	$-0.862459\pi$
$131$	−20.7871	−1.81618	−0.908090	+	0.418776i	$0.862459\pi$
$132$	−5.19563	−0.452221
$133$	13.6227	1.18124
$134$	9.34785	0.807532
$135$	0	0
$136$	−4.69775	−0.402829
$137$	−5.88324	−0.502639	−0.251320	−	0.967904i	$-0.580865\pi$
$137$	−5.88324	−0.502639	−0.251320	+	0.967904i	$0.580865\pi$
$138$	18.3291	1.56028
$139$	−3.12694	−0.265224	−0.132612	−	0.991168i	$-0.542336\pi$
$139$	−3.12694	−0.265224	−0.132612	+	0.991168i	$0.542336\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	−10.6059	−0.890026
$143$	−5.78927	−0.484123
$144$	10.0355	0.836293
$145$	0	0
$146$	−29.8932	−2.47398
$147$	−9.37313	−0.773083
$148$	33.0829	2.71940
$149$	18.3777	1.50556	0.752780	−	0.658272i	$-0.228712\pi$
$149$	18.3777	1.50556	0.752780	+	0.658272i	$0.228712\pi$
$150$	0	0
$151$	−21.1061	−1.71759	−0.858796	−	0.512318i	$-0.828787\pi$
$151$	−21.1061	−1.71759	−0.858796	+	0.512318i	$0.828787\pi$
$152$	−25.4062	−2.06071
$153$	−0.622511	−0.0503271
$154$	−11.3036	−0.910873
$155$	0	0
$156$	26.5089	2.12241
$157$	23.5653	1.88072	0.940359	−	0.340182i	$-0.110489\pi$
$157$	23.5653	1.88072	0.940359	+	0.340182i	$0.110489\pi$
$158$	3.58617	0.285300
$159$	−14.1493	−1.12211
$160$	0	0
$161$	28.2807	2.22883
$162$	2.62251	0.206044
$163$	18.0750	1.41574	0.707871	−	0.706342i	$-0.249656\pi$
$163$	18.0750	1.41574	0.707871	+	0.706342i	$0.249656\pi$
$164$	−45.2388	−3.53255
$165$	0	0
$166$	−18.1148	−1.40598
$167$	−10.6949	−0.827595	−0.413797	−	0.910369i	$-0.635798\pi$
$167$	−10.6949	−0.827595	−0.413797	+	0.910369i	$0.635798\pi$
$168$	30.5358	2.35588
$169$	16.5378	1.27214
$170$	0	0
$171$	−3.36664	−0.257453
$172$	14.1698	1.08044
$173$	5.50791	0.418758	0.209379	−	0.977835i	$-0.432856\pi$
$173$	5.50791	0.418758	0.209379	+	0.977835i	$0.432856\pi$
$174$	4.75293	0.360319
$175$	0	0
$176$	10.6899	0.805783
$177$	−6.44588	−0.484502
$178$	2.97246	0.222796
$179$	17.9076	1.33847	0.669237	−	0.743049i	$-0.266621\pi$
$179$	17.9076	1.33847	0.669237	+	0.743049i	$0.266621\pi$
$180$	0	0
$181$	10.7919	0.802154	0.401077	−	0.916044i	$-0.368636\pi$
$181$	10.7919	0.802154	0.401077	+	0.916044i	$0.368636\pi$
$182$	57.6730	4.27501
$183$	−9.22316	−0.681796
$184$	−52.7433	−3.88828
$185$	0	0
$186$	23.7452	1.74108
$187$	−0.663105	−0.0484910
$188$	−4.87757	−0.355733
$189$	4.04637	0.294330
$190$	0	0
$191$	−26.2640	−1.90040	−0.950199	−	0.311645i	$-0.899120\pi$
$191$	−26.2640	−1.90040	−0.950199	+	0.311645i	$0.899120\pi$
$192$	−9.36761	−0.676049
$193$	−2.58171	−0.185836	−0.0929179	−	0.995674i	$-0.529619\pi$
$193$	−2.58171	−0.185836	−0.0929179	+	0.995674i	$0.529619\pi$
$194$	0.995748	0.0714905
$195$	0	0
$196$	45.7181	3.26558
$197$	8.24282	0.587277	0.293638	−	0.955917i	$-0.405134\pi$
$197$	8.24282	0.587277	0.293638	+	0.955917i	$0.405134\pi$
$198$	2.79352	0.198527
$199$	27.9478	1.98117	0.990583	−	0.136913i	$-0.0437180\pi$
$199$	27.9478	1.98117	0.990583	+	0.136913i	$0.0437180\pi$
$200$	0	0
$201$	−3.56447	−0.251418
$202$	−6.17981	−0.434810
$203$	7.33348	0.514709
$204$	3.03634	0.212586
$205$	0	0
$206$	16.3282	1.13764
$207$	−6.98915	−0.485779
$208$	−54.5417	−3.78179
$209$	−3.58617	−0.248061
$210$	0	0
$211$	21.0703	1.45054	0.725270	−	0.688464i	$-0.241715\pi$
$211$	21.0703	1.45054	0.725270	+	0.688464i	$0.241715\pi$
$212$	69.0142	4.73991
$213$	4.04417	0.277102
$214$	−18.8497	−1.28854
$215$	0	0
$216$	−7.54645	−0.513471
$217$	36.6373	2.48710
$218$	−4.30418	−0.291516
$219$	11.3987	0.770252
$220$	0	0
$221$	3.38327	0.227583
$222$	−17.7876	−1.19383
$223$	16.5423	1.10775	0.553876	−	0.832599i	$-0.313148\pi$
$223$	16.5423	1.10775	0.553876	+	0.832599i	$0.313148\pi$
$224$	−45.4220	−3.03489
$225$	0	0
$226$	−49.7503	−3.30934
$227$	4.55207	0.302132	0.151066	−	0.988524i	$-0.451729\pi$
$227$	4.55207	0.302132	0.151066	+	0.988524i	$0.451729\pi$
$228$	16.4210	1.08751
$229$	−15.5224	−1.02575	−0.512874	−	0.858464i	$-0.671419\pi$
$229$	−15.5224	−1.02575	−0.512874	+	0.858464i	$0.671419\pi$
$230$	0	0
$231$	4.31023	0.283592
$232$	−13.6769	−0.897930
$233$	16.4928	1.08048	0.540240	−	0.841511i	$-0.318333\pi$
$233$	16.4928	1.08048	0.540240	+	0.841511i	$0.318333\pi$
$234$	−14.2530	−0.931748
$235$	0	0
$236$	31.4402	2.04658
$237$	−1.36746	−0.0888258
$238$	6.60588	0.428196
$239$	9.29922	0.601517	0.300758	−	0.953700i	$-0.402760\pi$
$239$	9.29922	0.601517	0.300758	+	0.953700i	$0.402760\pi$
$240$	0	0
$241$	−17.9144	−1.15397	−0.576983	−	0.816756i	$-0.695770\pi$
$241$	−17.9144	−1.15397	−0.576983	+	0.816756i	$0.695770\pi$
$242$	−25.8719	−1.66311
$243$	−1.00000	−0.0641500
$244$	44.9866	2.87997
$245$	0	0
$246$	24.3234	1.55081
$247$	18.2972	1.16422
$248$	−68.3283	−4.33885
$249$	6.90741	0.437739
$250$	0	0
$251$	−2.27922	−0.143863	−0.0719316	−	0.997410i	$-0.522916\pi$
$251$	−2.27922	−0.143863	−0.0719316	+	0.997410i	$0.522916\pi$
$252$	−19.7365	−1.24328
$253$	−7.44490	−0.468057
$254$	−19.9959	−1.25465
$255$	0	0
$256$	−13.1862	−0.824135
$257$	18.0256	1.12441	0.562203	−	0.826999i	$-0.309954\pi$
$257$	18.0256	1.12441	0.562203	+	0.826999i	$0.309954\pi$
$258$	−7.61867	−0.474318
$259$	−27.4452	−1.70536
$260$	0	0
$261$	−1.81236	−0.112182
$262$	−54.5145	−3.36791
$263$	−31.5708	−1.94674	−0.973370	−	0.229239i	$-0.926376\pi$
$263$	−31.5708	−1.94674	−0.973370	+	0.229239i	$0.926376\pi$
$264$	−8.03854	−0.494738
$265$	0	0
$266$	35.7256	2.19048
$267$	−1.13344	−0.0693655
$268$	17.3859	1.06201
$269$	−7.01791	−0.427890	−0.213945	−	0.976846i	$-0.568631\pi$
$269$	−7.01791	−0.427890	−0.213945	+	0.976846i	$0.568631\pi$
$270$	0	0
$271$	10.9074	0.662578	0.331289	−	0.943529i	$-0.392517\pi$
$271$	10.9074	0.662578	0.331289	+	0.943529i	$0.392517\pi$
$272$	−6.24723	−0.378794
$273$	−21.9915	−1.33099
$274$	−15.4289	−0.932092
$275$	0	0
$276$	34.0900	2.05198
$277$	−23.1409	−1.39040	−0.695200	−	0.718817i	$-0.744684\pi$
$277$	−23.1409	−1.39040	−0.695200	+	0.718817i	$0.744684\pi$
$278$	−8.20045	−0.491830
$279$	−9.05436	−0.542070
$280$	0	0
$281$	−5.96479	−0.355830	−0.177915	−	0.984046i	$-0.556935\pi$
$281$	−5.96479	−0.355830	−0.177915	+	0.984046i	$0.556935\pi$
$282$	2.62251	0.156168
$283$	−27.8856	−1.65762	−0.828812	−	0.559527i	$-0.810983\pi$
$283$	−27.8856	−1.65762	−0.828812	+	0.559527i	$0.810983\pi$
$284$	−19.7257	−1.17051
$285$	0	0
$286$	−15.1824	−0.897756
$287$	37.5296	2.21530
$288$	11.2254	0.661461
$289$	−16.6125	−0.977205
$290$	0	0
$291$	−0.379692	−0.0222580
$292$	−55.5978	−3.25362
$293$	5.74678	0.335730	0.167865	−	0.985810i	$-0.446313\pi$
$293$	5.74678	0.335730	0.167865	+	0.985810i	$0.446313\pi$
$294$	−24.5812	−1.43360
$295$	0	0
$296$	51.1851	2.97507
$297$	−1.06521	−0.0618097
$298$	48.1957	2.79190
$299$	37.9851	2.19674
$300$	0	0
$301$	−11.7551	−0.677555
$302$	−55.3510	−3.18509
$303$	2.35645	0.135374
$304$	−33.7860	−1.93776
$305$	0	0
$306$	−1.63254	−0.0933263
$307$	−19.0346	−1.08636	−0.543181	−	0.839616i	$-0.682780\pi$
$307$	−19.0346	−1.08636	−0.543181	+	0.839616i	$0.682780\pi$
$308$	−21.0234	−1.19792
$309$	−6.22619	−0.354195
$310$	0	0
$311$	20.4957	1.16221	0.581103	−	0.813830i	$-0.302621\pi$
$311$	20.4957	1.16221	0.581103	+	0.813830i	$0.302621\pi$
$312$	41.0140	2.32196
$313$	−10.7257	−0.606255	−0.303127	−	0.952950i	$-0.598031\pi$
$313$	−10.7257	−0.606255	−0.303127	+	0.952950i	$0.598031\pi$
$314$	61.8004	3.48760
$315$	0	0
$316$	6.66986	0.375209
$317$	15.3040	0.859561	0.429780	−	0.902933i	$-0.358591\pi$
$317$	15.3040	0.859561	0.429780	+	0.902933i	$0.358591\pi$
$318$	−37.1067	−2.08084
$319$	−1.93054	−0.108089
$320$	0	0
$321$	7.18764	0.401175
$322$	74.1665	4.13314
$323$	2.09577	0.116612
$324$	4.87757	0.270976
$325$	0	0
$326$	47.4019	2.62535
$327$	1.64124	0.0907611
$328$	−69.9923	−3.86468
$329$	4.04637	0.223084
$330$	0	0
$331$	2.99785	0.164777	0.0823884	−	0.996600i	$-0.473745\pi$
$331$	2.99785	0.164777	0.0823884	+	0.996600i	$0.473745\pi$
$332$	−33.6913	−1.84905
$333$	6.78267	0.371688
$334$	−28.0475	−1.53469
$335$	0	0
$336$	40.6075	2.21532
$337$	−19.0655	−1.03857	−0.519283	−	0.854603i	$-0.673801\pi$
$337$	−19.0655	−1.03857	−0.519283	+	0.854603i	$0.673801\pi$
$338$	43.3706	2.35905
$339$	18.9705	1.03033
$340$	0	0
$341$	−9.64478	−0.522294
$342$	−8.82904	−0.477420
$343$	−9.60259	−0.518491
$344$	21.9232	1.18202
$345$	0	0
$346$	14.4445	0.776544
$347$	16.4884	0.885144	0.442572	−	0.896733i	$-0.354066\pi$
$347$	16.4884	0.885144	0.442572	+	0.896733i	$0.354066\pi$
$348$	8.83989	0.473868
$349$	−1.59753	−0.0855136	−0.0427568	−	0.999086i	$-0.513614\pi$
$349$	−1.59753	−0.0855136	−0.0427568	+	0.999086i	$0.513614\pi$
$350$	0	0
$351$	5.43487	0.290092
$352$	11.9574	0.637329
$353$	33.3276	1.77385	0.886925	−	0.461914i	$-0.152837\pi$
$353$	33.3276	1.77385	0.886925	+	0.461914i	$0.152837\pi$
$354$	−16.9044	−0.898458
$355$	0	0
$356$	5.52844	0.293006
$357$	−2.51891	−0.133315
$358$	46.9628	2.48206
$359$	17.8278	0.940916	0.470458	−	0.882422i	$-0.344089\pi$
$359$	17.8278	0.940916	0.470458	+	0.882422i	$0.344089\pi$
$360$	0	0
$361$	−7.66576	−0.403461
$362$	28.3018	1.48751
$363$	9.86533	0.517796
$364$	107.265	5.62222
$365$	0	0
$366$	−24.1879	−1.26432
$367$	18.3580	0.958279	0.479140	−	0.877739i	$-0.340949\pi$
$367$	18.3580	0.958279	0.479140	+	0.877739i	$0.340949\pi$
$368$	−70.1397	−3.65629
$369$	−9.27487	−0.482830
$370$	0	0
$371$	−57.2534	−2.97245
$372$	44.1632	2.28976
$373$	18.7164	0.969098	0.484549	−	0.874764i	$-0.338984\pi$
$373$	18.7164	0.969098	0.484549	+	0.874764i	$0.338984\pi$
$374$	−1.73900	−0.0899215
$375$	0	0
$376$	−7.54645	−0.389178
$377$	9.84993	0.507297
$378$	10.6117	0.545805
$379$	0.826389	0.0424488	0.0212244	−	0.999775i	$-0.493244\pi$
$379$	0.826389	0.0424488	0.0212244	+	0.999775i	$0.493244\pi$
$380$	0	0
$381$	7.62472	0.390626
$382$	−68.8777	−3.52409
$383$	−7.84844	−0.401037	−0.200518	−	0.979690i	$-0.564263\pi$
$383$	−7.84844	−0.401037	−0.200518	+	0.979690i	$0.564263\pi$
$384$	−2.11594	−0.107978
$385$	0	0
$386$	−6.77057	−0.344613
$387$	2.90510	0.147675
$388$	1.85198	0.0940198
$389$	−13.8834	−0.703916	−0.351958	−	0.936016i	$-0.614484\pi$
$389$	−13.8834	−0.703916	−0.351958	+	0.936016i	$0.614484\pi$
$390$	0	0
$391$	4.35082	0.220031
$392$	70.7339	3.57260
$393$	20.7871	1.04857
$394$	21.6169	1.08904
$395$	0	0
$396$	5.19563	0.261090
$397$	31.5279	1.58234	0.791169	−	0.611597i	$-0.209473\pi$
$397$	31.5279	1.58234	0.791169	+	0.611597i	$0.209473\pi$
$398$	73.2934	3.67387
$399$	−13.6227	−0.681986
$400$	0	0
$401$	−13.8502	−0.691645	−0.345823	−	0.938300i	$-0.612400\pi$
$401$	−13.8502	−0.691645	−0.345823	+	0.938300i	$0.612400\pi$
$402$	−9.34785	−0.466229
$403$	49.2092	2.45129
$404$	−11.4937	−0.571835
$405$	0	0
$406$	19.2321	0.954474
$407$	7.22496	0.358128
$408$	4.69775	0.232573
$409$	−8.88463	−0.439317	−0.219658	−	0.975577i	$-0.570494\pi$
$409$	−8.88463	−0.439317	−0.219658	+	0.975577i	$0.570494\pi$
$410$	0	0
$411$	5.88324	0.290199
$412$	30.3686	1.49616
$413$	−26.0824	−1.28343
$414$	−18.3291	−0.900827
$415$	0	0
$416$	−61.0084	−2.99118
$417$	3.12694	0.153127
$418$	−9.40477	−0.460002
$419$	−32.8260	−1.60366	−0.801828	−	0.597555i	$-0.796139\pi$
$419$	−32.8260	−1.60366	−0.801828	+	0.597555i	$0.796139\pi$
$420$	0	0
$421$	−0.0200640	−0.000977858 0	−0.000488929	−	1.00000i	$-0.500156\pi$
$421$	−0.0200640	−0.000977858 0	−0.000488929		1.00000i	$0.500156\pi$
$422$	55.2572	2.68988
$423$	−1.00000	−0.0486217
$424$	106.777	5.18555
$425$	0	0
$426$	10.6059	0.513857
$427$	−37.3204	−1.80606
$428$	−35.0582	−1.69460
$429$	5.78927	0.279509
$430$	0	0
$431$	3.94217	0.189888	0.0949438	−	0.995483i	$-0.469733\pi$
$431$	3.94217	0.189888	0.0949438	+	0.995483i	$0.469733\pi$
$432$	−10.0355	−0.482834
$433$	−23.4722	−1.12800	−0.564001	−	0.825774i	$-0.690738\pi$
$433$	−23.4722	−1.12800	−0.564001	+	0.825774i	$0.690738\pi$
$434$	96.0818	4.61207
$435$	0	0
$436$	−8.00528	−0.383383
$437$	23.5299	1.12559
$438$	29.8932	1.42835
$439$	26.0774	1.24461	0.622303	−	0.782777i	$-0.286197\pi$
$439$	26.0774	1.24461	0.622303	+	0.782777i	$0.286197\pi$
$440$	0	0
$441$	9.37313	0.446340
$442$	8.87266	0.422029
$443$	−3.24610	−0.154227	−0.0771133	−	0.997022i	$-0.524570\pi$
$443$	−3.24610	−0.154227	−0.0771133	+	0.997022i	$0.524570\pi$
$444$	−33.0829	−1.57005
$445$	0	0
$446$	43.3822	2.05421
$447$	−18.3777	−0.869235
$448$	−37.9049	−1.79084
$449$	5.62718	0.265563	0.132781	−	0.991145i	$-0.457609\pi$
$449$	5.62718	0.265563	0.132781	+	0.991145i	$0.457609\pi$
$450$	0	0
$451$	−9.87967	−0.465215
$452$	−92.5297	−4.35223
$453$	21.1061	0.991652
$454$	11.9379	0.560272
$455$	0	0
$456$	25.4062	1.18975
$457$	24.1194	1.12826	0.564128	−	0.825687i	$-0.309212\pi$
$457$	24.1194	1.12826	0.564128	+	0.825687i	$0.309212\pi$
$458$	−40.7076	−1.90214
$459$	0.622511	0.0290563
$460$	0	0
$461$	−30.8280	−1.43580	−0.717901	−	0.696145i	$-0.754897\pi$
$461$	−30.8280	−1.43580	−0.717901	+	0.696145i	$0.754897\pi$
$462$	11.3036	0.525893
$463$	−16.0566	−0.746214	−0.373107	−	0.927788i	$-0.621708\pi$
$463$	−16.0566	−0.746214	−0.373107	+	0.927788i	$0.621708\pi$
$464$	−18.1880	−0.844355
$465$	0	0
$466$	43.2526	2.00364
$467$	4.55418	0.210742	0.105371	−	0.994433i	$-0.466397\pi$
$467$	4.55418	0.210742	0.105371	+	0.994433i	$0.466397\pi$
$468$	−26.5089	−1.22538
$469$	−14.4232	−0.666000
$470$	0	0
$471$	−23.5653	−1.08583
$472$	48.6435	2.23900
$473$	3.09454	0.142287
$474$	−3.58617	−0.164718
$475$	0	0
$476$	12.2862	0.563136
$477$	14.1493	0.647852
$478$	24.3873	1.11545
$479$	−9.92365	−0.453423	−0.226711	−	0.973962i	$-0.572797\pi$
$479$	−9.92365	−0.453423	−0.226711	+	0.973962i	$0.572797\pi$
$480$	0	0
$481$	−36.8629	−1.68080
$482$	−46.9806	−2.13991
$483$	−28.2807	−1.28682
$484$	−48.1188	−2.18722
$485$	0	0
$486$	−2.62251	−0.118960
$487$	−0.596860	−0.0270463	−0.0135231	−	0.999909i	$-0.504305\pi$
$487$	−0.596860	−0.0270463	−0.0135231	+	0.999909i	$0.504305\pi$
$488$	69.6021	3.15074
$489$	−18.0750	−0.817379
$490$	0	0
$491$	1.88300	0.0849785	0.0424893	−	0.999097i	$-0.486471\pi$
$491$	1.88300	0.0849785	0.0424893	+	0.999097i	$0.486471\pi$
$492$	45.2388	2.03952
$493$	1.12821	0.0508122
$494$	47.9847	2.15893
$495$	0	0
$496$	−90.8652	−4.07997
$497$	16.3642	0.734035
$498$	18.1148	0.811742
$499$	−40.0807	−1.79426	−0.897130	−	0.441767i	$-0.854352\pi$
$499$	−40.0807	−1.79426	−0.897130	+	0.441767i	$0.854352\pi$
$500$	0	0
$501$	10.6949	0.477812
$502$	−5.97728	−0.266779
$503$	0.688851	0.0307144	0.0153572	−	0.999882i	$-0.495111\pi$
$503$	0.688851	0.0307144	0.0153572	+	0.999882i	$0.495111\pi$
$504$	−30.5358	−1.36017
$505$	0	0
$506$	−19.5243	−0.867963
$507$	−16.5378	−0.734470
$508$	−37.1901	−1.65004
$509$	5.21645	0.231215	0.115608	−	0.993295i	$-0.463119\pi$
$509$	5.21645	0.231215	0.115608	+	0.993295i	$0.463119\pi$
$510$	0	0
$511$	46.1233	2.04038
$512$	−38.8127	−1.71530
$513$	3.36664	0.148641
$514$	47.2723	2.08509
$515$	0	0
$516$	−14.1698	−0.623792
$517$	−1.06521	−0.0468478
$518$	−71.9754	−3.16242
$519$	−5.50791	−0.241770
$520$	0	0
$521$	33.2981	1.45882	0.729408	−	0.684079i	$-0.239796\pi$
$521$	33.2981	1.45882	0.729408	+	0.684079i	$0.239796\pi$
$522$	−4.75293	−0.208030
$523$	−18.3292	−0.801481	−0.400740	−	0.916192i	$-0.631247\pi$
$523$	−18.3292	−0.801481	−0.400740	+	0.916192i	$0.631247\pi$
$524$	−101.391	−4.42927
$525$	0	0
$526$	−82.7949	−3.61003
$527$	5.63644	0.245527
$528$	−10.6899	−0.465219
$529$	25.8482	1.12383
$530$	0	0
$531$	6.44588	0.279727
$532$	66.4455	2.88078
$533$	50.4077	2.18340
$534$	−2.97246	−0.128631
$535$	0	0
$536$	26.8991	1.16186
$537$	−17.9076	−0.772768
$538$	−18.4046	−0.793477
$539$	9.98434	0.430056
$540$	0	0
$541$	4.11424	0.176885	0.0884424	−	0.996081i	$-0.471811\pi$
$541$	4.11424	0.176885	0.0884424	+	0.996081i	$0.471811\pi$
$542$	28.6048	1.22868
$543$	−10.7919	−0.463124
$544$	−6.98792	−0.299605
$545$	0	0
$546$	−57.6730	−2.46818
$547$	−39.8151	−1.70237	−0.851186	−	0.524864i	$-0.824116\pi$
$547$	−39.8151	−1.70237	−0.851186	+	0.524864i	$0.824116\pi$
$548$	−28.6959	−1.22583
$549$	9.22316	0.393635
$550$	0	0
$551$	6.10155	0.259935
$552$	52.7433	2.24490
$553$	−5.53324	−0.235297
$554$	−60.6872	−2.57835
$555$	0	0
$556$	−15.2519	−0.646824
$557$	−2.57444	−0.109082	−0.0545412	−	0.998512i	$-0.517370\pi$
$557$	−2.57444	−0.109082	−0.0545412	+	0.998512i	$0.517370\pi$
$558$	−23.7452	−1.00521
$559$	−15.7889	−0.667798
$560$	0	0
$561$	0.663105	0.0279963
$562$	−15.6427	−0.659849
$563$	−1.54727	−0.0652097	−0.0326048	−	0.999468i	$-0.510380\pi$
$563$	−1.54727	−0.0652097	−0.0326048	+	0.999468i	$0.510380\pi$
$564$	4.87757	0.205383
$565$	0	0
$566$	−73.1302	−3.07389
$567$	−4.04637	−0.169932
$568$	−30.5191	−1.28055
$569$	−0.220395	−0.00923946	−0.00461973	−	0.999989i	$-0.501471\pi$
$569$	−0.220395	−0.00923946	−0.00461973	+	0.999989i	$0.501471\pi$
$570$	0	0
$571$	−43.0259	−1.80058	−0.900288	−	0.435295i	$-0.856644\pi$
$571$	−43.0259	−1.80058	−0.900288	+	0.435295i	$0.856644\pi$
$572$	−28.2375	−1.18067
$573$	26.2640	1.09719
$574$	98.4217	4.10804
$575$	0	0
$576$	9.36761	0.390317
$577$	10.6351	0.442745	0.221372	−	0.975189i	$-0.428946\pi$
$577$	10.6351	0.442745	0.221372	+	0.975189i	$0.428946\pi$
$578$	−43.5664	−1.81212
$579$	2.58171	0.107292
$580$	0	0
$581$	27.9500	1.15956
$582$	−0.995748	−0.0412751
$583$	15.0720	0.624217
$584$	−86.0196	−3.55952
$585$	0	0
$586$	15.0710	0.622577
$587$	−20.1513	−0.831731	−0.415866	−	0.909426i	$-0.636521\pi$
$587$	−20.1513	−0.831731	−0.415866	+	0.909426i	$0.636521\pi$
$588$	−45.7181	−1.88538
$589$	30.4827	1.25602
$590$	0	0
$591$	−8.24282	−0.339064
$592$	68.0676	2.79756
$593$	−14.5554	−0.597719	−0.298860	−	0.954297i	$-0.596606\pi$
$593$	−14.5554	−0.597719	−0.298860	+	0.954297i	$0.596606\pi$
$594$	−2.79352	−0.114620
$595$	0	0
$596$	89.6384	3.67173
$597$	−27.9478	−1.14383
$598$	99.6164	4.07362
$599$	2.29983	0.0939687	0.0469844	−	0.998896i	$-0.485039\pi$
$599$	2.29983	0.0939687	0.0469844	+	0.998896i	$0.485039\pi$
$600$	0	0
$601$	18.2841	0.745823	0.372911	−	0.927867i	$-0.378360\pi$
$601$	18.2841	0.745823	0.372911	+	0.927867i	$0.378360\pi$
$602$	−30.8280	−1.25645
$603$	3.56447	0.145156
$604$	−102.946	−4.18883
$605$	0	0
$606$	6.17981	0.251038
$607$	−19.1692	−0.778055	−0.389027	−	0.921226i	$-0.627189\pi$
$607$	−19.1692	−0.778055	−0.389027	+	0.921226i	$0.627189\pi$
$608$	−37.7917	−1.53266
$609$	−7.33348	−0.297167
$610$	0	0
$611$	5.43487	0.219871
$612$	−3.03634	−0.122737
$613$	−3.08118	−0.124448	−0.0622238	−	0.998062i	$-0.519819\pi$
$613$	−3.08118	−0.124448	−0.0622238	+	0.998062i	$0.519819\pi$
$614$	−49.9185	−2.01454
$615$	0	0
$616$	−32.5270	−1.31055
$617$	23.4010	0.942088	0.471044	−	0.882110i	$-0.343877\pi$
$617$	23.4010	0.942088	0.471044	+	0.882110i	$0.343877\pi$
$618$	−16.3282	−0.656818
$619$	23.3937	0.940273	0.470136	−	0.882594i	$-0.344205\pi$
$619$	23.3937	0.940273	0.470136	+	0.882594i	$0.344205\pi$
$620$	0	0
$621$	6.98915	0.280465
$622$	53.7503	2.15519
$623$	−4.58633	−0.183747
$624$	54.5417	2.18342
$625$	0	0
$626$	−28.1284	−1.12424
$627$	3.58617	0.143218
$628$	114.942	4.58667
$629$	−4.22229	−0.168354
$630$	0	0
$631$	43.5952	1.73550	0.867748	−	0.497004i	$-0.165567\pi$
$631$	43.5952	1.73550	0.867748	+	0.497004i	$0.165567\pi$
$632$	10.3194	0.410485
$633$	−21.0703	−0.837470
$634$	40.1350	1.59397
$635$	0	0
$636$	−69.0142	−2.73659
$637$	−50.9418	−2.01839
$638$	−5.06286	−0.200441
$639$	−4.04417	−0.159985
$640$	0	0
$641$	−17.2320	−0.680624	−0.340312	−	0.940313i	$-0.610533\pi$
$641$	−17.2320	−0.680624	−0.340312	+	0.940313i	$0.610533\pi$
$642$	18.8497	0.743937
$643$	−15.2177	−0.600127	−0.300064	−	0.953919i	$-0.597008\pi$
$643$	−15.2177	−0.600127	−0.300064	+	0.953919i	$0.597008\pi$
$644$	137.941	5.43564
$645$	0	0
$646$	5.49618	0.216244
$647$	−25.4839	−1.00188	−0.500939	−	0.865483i	$-0.667012\pi$
$647$	−25.4839	−1.00188	−0.500939	+	0.865483i	$0.667012\pi$
$648$	7.54645	0.296453
$649$	6.86620	0.269522
$650$	0	0
$651$	−36.6373	−1.43593
$652$	88.1620	3.45269
$653$	−30.2168	−1.18248	−0.591238	−	0.806497i	$-0.701360\pi$
$653$	−30.2168	−1.18248	−0.591238	+	0.806497i	$0.701360\pi$
$654$	4.30418	0.168307
$655$	0	0
$656$	−93.0781	−3.63409
$657$	−11.3987	−0.444705
$658$	10.6117	0.413686
$659$	−35.8277	−1.39565	−0.697825	−	0.716268i	$-0.745849\pi$
$659$	−35.8277	−1.39565	−0.697825	+	0.716268i	$0.745849\pi$
$660$	0	0
$661$	12.0278	0.467829	0.233915	−	0.972257i	$-0.424846\pi$
$661$	12.0278	0.467829	0.233915	+	0.972257i	$0.424846\pi$
$662$	7.86189	0.305561
$663$	−3.38327	−0.131395
$664$	−52.1264	−2.02290
$665$	0	0
$666$	17.7876	0.689257
$667$	12.6668	0.490462
$668$	−52.1650	−2.01832
$669$	−16.5423	−0.639560
$670$	0	0
$671$	9.82459	0.379274
$672$	45.4220	1.75219
$673$	−26.3250	−1.01475	−0.507377	−	0.861724i	$-0.669385\pi$
$673$	−26.3250	−1.01475	−0.507377	+	0.861724i	$0.669385\pi$
$674$	−49.9995	−1.92591
$675$	0	0
$676$	80.6642	3.10247
$677$	4.72902	0.181751	0.0908754	−	0.995862i	$-0.471033\pi$
$677$	4.72902	0.181751	0.0908754	+	0.995862i	$0.471033\pi$
$678$	49.7503	1.91065
$679$	−1.53638	−0.0589607
$680$	0	0
$681$	−4.55207	−0.174436
$682$	−25.2935	−0.968540
$683$	18.5856	0.711157	0.355578	−	0.934646i	$-0.384284\pi$
$683$	18.5856	0.711157	0.355578	+	0.934646i	$0.384284\pi$
$684$	−16.4210	−0.627872
$685$	0	0
$686$	−25.1829	−0.961487
$687$	15.5224	0.592216
$688$	29.1542	1.11149
$689$	−76.8996	−2.92964
$690$	0	0
$691$	−1.43503	−0.0545912	−0.0272956	−	0.999627i	$-0.508690\pi$
$691$	−1.43503	−0.0545912	−0.0272956	+	0.999627i	$0.508690\pi$
$692$	26.8652	1.02126
$693$	−4.31023	−0.163732
$694$	43.2410	1.64141
$695$	0	0
$696$	13.6769	0.518420
$697$	5.77371	0.218695
$698$	−4.18953	−0.158576
$699$	−16.4928	−0.623816
$700$	0	0
$701$	30.4561	1.15031	0.575156	−	0.818044i	$-0.304941\pi$
$701$	30.4561	1.15031	0.575156	+	0.818044i	$0.304941\pi$
$702$	14.2530	0.537945
$703$	−22.8348	−0.861230
$704$	9.97846	0.376077
$705$	0	0
$706$	87.4020	3.28942
$707$	9.53507	0.358603
$708$	−31.4402	−1.18159
$709$	−7.08455	−0.266066	−0.133033	−	0.991112i	$-0.542472\pi$
$709$	−7.08455	−0.266066	−0.133033	+	0.991112i	$0.542472\pi$
$710$	0	0
$711$	1.36746	0.0512836
$712$	8.55346	0.320554
$713$	63.2822	2.36994
$714$	−6.60588	−0.247219
$715$	0	0
$716$	87.3453	3.26425
$717$	−9.29922	−0.347286
$718$	46.7536	1.74483
$719$	−8.30106	−0.309577	−0.154789	−	0.987948i	$-0.549470\pi$
$719$	−8.30106	−0.309577	−0.154789	+	0.987948i	$0.549470\pi$
$720$	0	0
$721$	−25.1935	−0.938254
$722$	−20.1035	−0.748176
$723$	17.9144	0.666242
$724$	52.6381	1.95628
$725$	0	0
$726$	25.8719	0.960198
$727$	−42.3188	−1.56952	−0.784759	−	0.619801i	$-0.787214\pi$
$727$	−42.3188	−1.56952	−0.784759	+	0.619801i	$0.787214\pi$
$728$	165.958	6.15081
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.80846	−0.0668883
$732$	−44.9866	−1.66275
$733$	22.5912	0.834426	0.417213	−	0.908809i	$-0.363007\pi$
$733$	22.5912	0.834426	0.417213	+	0.908809i	$0.363007\pi$
$734$	48.1440	1.77703
$735$	0	0
$736$	−78.4558	−2.89192
$737$	3.79690	0.139861
$738$	−24.3234	−0.895358
$739$	−0.854126	−0.0314195	−0.0157098	−	0.999877i	$-0.505001\pi$
$739$	−0.854126	−0.0314195	−0.0157098	+	0.999877i	$0.505001\pi$
$740$	0	0
$741$	−18.2972	−0.672166
$742$	−150.148	−5.51209
$743$	−11.2060	−0.411107	−0.205554	−	0.978646i	$-0.565899\pi$
$743$	−11.2060	−0.411107	−0.205554	+	0.978646i	$0.565899\pi$
$744$	68.3283	2.50504
$745$	0	0
$746$	49.0839	1.79709
$747$	−6.90741	−0.252729
$748$	−3.23434	−0.118259
$749$	29.0839	1.06270
$750$	0	0
$751$	−14.7484	−0.538178	−0.269089	−	0.963115i	$-0.586723\pi$
$751$	−14.7484	−0.538178	−0.269089	+	0.963115i	$0.586723\pi$
$752$	−10.0355	−0.365958
$753$	2.27922	0.0830594
$754$	25.8315	0.940729
$755$	0	0
$756$	19.7365	0.717808
$757$	6.87522	0.249884	0.124942	−	0.992164i	$-0.460125\pi$
$757$	6.87522	0.249884	0.124942	+	0.992164i	$0.460125\pi$
$758$	2.16722	0.0787168
$759$	7.44490	0.270233
$760$	0	0
$761$	−20.3226	−0.736695	−0.368347	−	0.929688i	$-0.620076\pi$
$761$	−20.3226	−0.736695	−0.368347	+	0.929688i	$0.620076\pi$
$762$	19.9959	0.724375
$763$	6.64109	0.240424
$764$	−128.104	−4.63466
$765$	0	0
$766$	−20.5826	−0.743681
$767$	−35.0325	−1.26495
$768$	13.1862	0.475814
$769$	−46.3403	−1.67108	−0.835538	−	0.549433i	$-0.814844\pi$
$769$	−46.3403	−1.67108	−0.835538	+	0.549433i	$0.814844\pi$
$770$	0	0
$771$	−18.0256	−0.649176
$772$	−12.5925	−0.453213
$773$	−42.9549	−1.54498	−0.772489	−	0.635028i	$-0.780989\pi$
$773$	−42.9549	−1.54498	−0.772489	+	0.635028i	$0.780989\pi$
$774$	7.61867	0.273847
$775$	0	0
$776$	2.86533	0.102859
$777$	27.4452	0.984592
$778$	−36.4094	−1.30534
$779$	31.2251	1.11876
$780$	0	0
$781$	−4.30788	−0.154148
$782$	11.4101	0.408024
$783$	1.81236	0.0647684
$784$	94.0643	3.35944
$785$	0	0
$786$	54.5145	1.94447
$787$	−34.2414	−1.22058	−0.610288	−	0.792180i	$-0.708946\pi$
$787$	−34.2414	−1.22058	−0.610288	+	0.792180i	$0.708946\pi$
$788$	40.2049	1.43224
$789$	31.5708	1.12395
$790$	0	0
$791$	76.7616	2.72933
$792$	8.03854	0.285637
$793$	−50.1267	−1.78005
$794$	82.6822	2.93428
$795$	0	0
$796$	136.317	4.83163
$797$	15.9243	0.564067	0.282034	−	0.959405i	$-0.408991\pi$
$797$	15.9243	0.564067	0.282034	+	0.959405i	$0.408991\pi$
$798$	−35.7256	−1.26467
$799$	0.622511	0.0220229
$800$	0	0
$801$	1.13344	0.0400482
$802$	−36.3223	−1.28258
$803$	−12.1420	−0.428481
$804$	−17.3859	−0.613154
$805$	0	0
$806$	129.052	4.54566
$807$	7.01791	0.247042
$808$	−17.7828	−0.625598
$809$	49.7199	1.74806	0.874030	−	0.485872i	$-0.161498\pi$
$809$	49.7199	1.74806	0.874030	+	0.485872i	$0.161498\pi$
$810$	0	0
$811$	−34.2032	−1.20104	−0.600519	−	0.799611i	$-0.705039\pi$
$811$	−34.2032	−1.20104	−0.600519	+	0.799611i	$0.705039\pi$
$812$	35.7695	1.25526
$813$	−10.9074	−0.382540
$814$	18.9475	0.664111
$815$	0	0
$816$	6.24723	0.218697
$817$	−9.78043	−0.342174
$818$	−23.3000	−0.814667
$819$	21.9915	0.768445
$820$	0	0
$821$	−11.2369	−0.392170	−0.196085	−	0.980587i	$-0.562823\pi$
$821$	−11.2369	−0.392170	−0.196085	+	0.980587i	$0.562823\pi$
$822$	15.4289	0.538144
$823$	−8.98439	−0.313176	−0.156588	−	0.987664i	$-0.550050\pi$
$823$	−8.98439	−0.313176	−0.156588	+	0.987664i	$0.550050\pi$
$824$	46.9856	1.63682
$825$	0	0
$826$	−68.4014	−2.37999
$827$	5.82971	0.202719	0.101359	−	0.994850i	$-0.467681\pi$
$827$	5.82971	0.202719	0.101359	+	0.994850i	$0.467681\pi$
$828$	−34.0900	−1.18471
$829$	11.0970	0.385415	0.192707	−	0.981256i	$-0.438273\pi$
$829$	11.0970	0.385415	0.192707	+	0.981256i	$0.438273\pi$
$830$	0	0
$831$	23.1409	0.802748
$832$	−50.9117	−1.76505
$833$	−5.83488	−0.202167
$834$	8.20045	0.283958
$835$	0	0
$836$	−17.4918	−0.604966
$837$	9.05436	0.312964
$838$	−86.0866	−2.97381
$839$	−11.0725	−0.382266	−0.191133	−	0.981564i	$-0.561216\pi$
$839$	−11.0725	−0.382266	−0.191133	+	0.981564i	$0.561216\pi$
$840$	0	0
$841$	−25.7154	−0.886737
$842$	−0.0526180	−0.00181334
$843$	5.96479	0.205438
$844$	102.772	3.53755
$845$	0	0
$846$	−2.62251	−0.0901638
$847$	39.9188	1.37163
$848$	141.996	4.87615
$849$	27.8856	0.957030
$850$	0	0
$851$	−47.4051	−1.62503
$852$	19.7257	0.675791
$853$	−16.1835	−0.554114	−0.277057	−	0.960854i	$-0.589359\pi$
$853$	−16.1835	−0.554114	−0.277057	+	0.960854i	$0.589359\pi$
$854$	−97.8731	−3.34915
$855$	0	0
$856$	−54.2412	−1.85393
$857$	−14.2969	−0.488372	−0.244186	−	0.969728i	$-0.578521\pi$
$857$	−14.2969	−0.488372	−0.244186	+	0.969728i	$0.578521\pi$
$858$	15.1824	0.518319
$859$	39.5481	1.34937	0.674683	−	0.738108i	$-0.264281\pi$
$859$	39.5481	1.34937	0.674683	+	0.738108i	$0.264281\pi$
$860$	0	0
$861$	−37.5296	−1.27900
$862$	10.3384	0.352127
$863$	−17.2103	−0.585846	−0.292923	−	0.956136i	$-0.594628\pi$
$863$	−17.2103	−0.585846	−0.292923	+	0.956136i	$0.594628\pi$
$864$	−11.2254	−0.381895
$865$	0	0
$866$	−61.5561	−2.09176
$867$	16.6125	0.564189
$868$	178.701	6.06550
$869$	1.45663	0.0494127
$870$	0	0
$871$	−19.3724	−0.656409
$872$	−12.3856	−0.419428
$873$	0.379692	0.0128506
$874$	61.7075	2.08729
$875$	0	0
$876$	55.5978	1.87848
$877$	−2.26904	−0.0766200	−0.0383100	−	0.999266i	$-0.512197\pi$
$877$	−2.26904	−0.0766200	−0.0383100	+	0.999266i	$0.512197\pi$
$878$	68.3882	2.30799
$879$	−5.74678	−0.193834
$880$	0	0
$881$	−49.9360	−1.68239	−0.841194	−	0.540734i	$-0.818147\pi$
$881$	−49.9360	−1.68239	−0.841194	+	0.540734i	$0.818147\pi$
$882$	24.5812	0.827690
$883$	−9.58156	−0.322445	−0.161223	−	0.986918i	$-0.551544\pi$
$883$	−9.58156	−0.322445	−0.161223	+	0.986918i	$0.551544\pi$
$884$	16.5021	0.555026
$885$	0	0
$886$	−8.51292	−0.285997
$887$	26.9929	0.906331	0.453166	−	0.891426i	$-0.350295\pi$
$887$	26.9929	0.906331	0.453166	+	0.891426i	$0.350295\pi$
$888$	−51.1851	−1.71766
$889$	30.8524	1.03476
$890$	0	0
$891$	1.06521	0.0356858
$892$	80.6859	2.70156
$893$	3.36664	0.112660
$894$	−48.1957	−1.61191
$895$	0	0
$896$	−8.56187	−0.286032
$897$	−37.9851	−1.26829
$898$	14.7573	0.492459
$899$	16.4097	0.547295
$900$	0	0
$901$	−8.80811	−0.293441
$902$	−25.9095	−0.862693
$903$	11.7551	0.391186
$904$	−143.160	−4.76142
$905$	0	0
$906$	55.3510	1.83891
$907$	22.7943	0.756874	0.378437	−	0.925627i	$-0.376462\pi$
$907$	22.7943	0.756874	0.378437	+	0.925627i	$0.376462\pi$
$908$	22.2030	0.736834
$909$	−2.35645	−0.0781585
$910$	0	0
$911$	42.6779	1.41398	0.706992	−	0.707222i	$-0.250052\pi$
$911$	42.6779	1.41398	0.706992	+	0.707222i	$0.250052\pi$
$912$	33.7860	1.11876
$913$	−7.35783	−0.243509
$914$	63.2533	2.09223
$915$	0	0
$916$	−75.7115	−2.50158
$917$	84.1124	2.77764
$918$	1.63254	0.0538820
$919$	−51.7511	−1.70711	−0.853556	−	0.521001i	$-0.825559\pi$
$919$	−51.7511	−1.70711	−0.853556	+	0.521001i	$0.825559\pi$
$920$	0	0
$921$	19.0346	0.627211
$922$	−80.8467	−2.66254
$923$	21.9795	0.723465
$924$	21.0234	0.691620
$925$	0	0
$926$	−42.1087	−1.38378
$927$	6.22619	0.204495
$928$	−20.3444	−0.667837
$929$	−2.60660	−0.0855197	−0.0427598	−	0.999085i	$-0.513615\pi$
$929$	−2.60660	−0.0855197	−0.0427598	+	0.999085i	$0.513615\pi$
$930$	0	0
$931$	−31.5559	−1.03420
$932$	80.4448	2.63506
$933$	−20.4957	−0.671000
$934$	11.9434	0.390799
$935$	0	0
$936$	−41.0140	−1.34058
$937$	7.08882	0.231582	0.115791	−	0.993274i	$-0.463060\pi$
$937$	7.08882	0.231582	0.115791	+	0.993274i	$0.463060\pi$
$938$	−37.8249	−1.23503
$939$	10.7257	0.350021
$940$	0	0
$941$	6.65293	0.216879	0.108440	−	0.994103i	$-0.465415\pi$
$941$	6.65293	0.216879	0.108440	+	0.994103i	$0.465415\pi$
$942$	−61.8004	−2.01357
$943$	64.8234	2.11094
$944$	64.6877	2.10541
$945$	0	0
$946$	8.11547	0.263857
$947$	11.4998	0.373693	0.186846	−	0.982389i	$-0.440173\pi$
$947$	11.4998	0.373693	0.186846	+	0.982389i	$0.440173\pi$
$948$	−6.66986	−0.216627
$949$	61.9504	2.01099
$950$	0	0
$951$	−15.3040	−0.496268
$952$	19.0089	0.616081
$953$	−0.361935	−0.0117242	−0.00586211	−	0.999983i	$-0.501866\pi$
$953$	−0.361935	−0.0117242	−0.00586211	+	0.999983i	$0.501866\pi$
$954$	37.1067	1.20137
$955$	0	0
$956$	45.3576	1.46697
$957$	1.93054	0.0624055
$958$	−26.0249	−0.840825
$959$	23.8058	0.768729
$960$	0	0
$961$	50.9814	1.64456
$962$	−96.6734	−3.11688
$963$	−7.18764	−0.231619
$964$	−87.3785	−2.81427
$965$	0	0
$966$	−74.1665	−2.38627
$967$	−8.70431	−0.279912	−0.139956	−	0.990158i	$-0.544696\pi$
$967$	−8.70431	−0.279912	−0.139956	+	0.990158i	$0.544696\pi$
$968$	−74.4482	−2.39286
$969$	−2.09577	−0.0673258
$970$	0	0
$971$	−6.67359	−0.214166	−0.107083	−	0.994250i	$-0.534151\pi$
$971$	−6.67359	−0.214166	−0.107083	+	0.994250i	$0.534151\pi$
$972$	−4.87757	−0.156448
$973$	12.6528	0.405630
$974$	−1.56527	−0.0501545
$975$	0	0
$976$	92.5592	2.96275
$977$	57.4227	1.83712	0.918558	−	0.395286i	$-0.129355\pi$
$977$	57.4227	1.83712	0.918558	+	0.395286i	$0.129355\pi$
$978$	−47.4019	−1.51574
$979$	1.20735	0.0385871
$980$	0	0
$981$	−1.64124	−0.0524009
$982$	4.93819	0.157584
$983$	−13.0009	−0.414663	−0.207332	−	0.978271i	$-0.566478\pi$
$983$	−13.0009	−0.414663	−0.207332	+	0.978271i	$0.566478\pi$
$984$	69.9923	2.23127
$985$	0	0
$986$	2.95875	0.0942259
$987$	−4.04637	−0.128797
$988$	89.2459	2.83929
$989$	−20.3042	−0.645636
$990$	0	0
$991$	−31.9573	−1.01516	−0.507578	−	0.861606i	$-0.669459\pi$
$991$	−31.9573	−1.01516	−0.507578	+	0.861606i	$0.669459\pi$
$992$	−101.638	−3.22702
$993$	−2.99785	−0.0951339
$994$	42.9153	1.36119
$995$	0	0
$996$	33.6913	1.06755
$997$	4.01228	0.127070	0.0635351	−	0.997980i	$-0.479763\pi$
$997$	4.01228	0.127070	0.0635351	+	0.997980i	$0.479763\pi$
$998$	−105.112	−3.32727
$999$	−6.78267	−0.214594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.v.1.5		5
5.4	even	2		705.2.a.l.1.1	✓	5
15.14	odd	2		2115.2.a.q.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.l.1.1	✓	5	5.4	even	2
2115.2.a.q.1.5		5	15.14	odd	2
3525.2.a.v.1.5		5	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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62251 q^{2} -1.00000 q^{3} +4.87757 q^{4} -2.62251 q^{6} -4.04637 q^{7} +7.54645 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.62251 q^{2} -1.00000 q^{3} +4.87757 q^{4} -2.62251 q^{6} -4.04637 q^{7} +7.54645 q^{8} +1.00000 q^{9} +1.06521 q^{11} -4.87757 q^{12} -5.43487 q^{13} -10.6117 q^{14} +10.0355 q^{16} -0.622511 q^{17} +2.62251 q^{18} -3.36664 q^{19} +4.04637 q^{21} +2.79352 q^{22} -6.98915 q^{23} -7.54645 q^{24} -14.2530 q^{26} -1.00000 q^{27} -19.7365 q^{28} -1.81236 q^{29} -9.05436 q^{31} +11.2254 q^{32} -1.06521 q^{33} -1.63254 q^{34} +4.87757 q^{36} +6.78267 q^{37} -8.82904 q^{38} +5.43487 q^{39} -9.27487 q^{41} +10.6117 q^{42} +2.90510 q^{43} +5.19563 q^{44} -18.3291 q^{46} -1.00000 q^{47} -10.0355 q^{48} +9.37313 q^{49} +0.622511 q^{51} -26.5089 q^{52} +14.1493 q^{53} -2.62251 q^{54} -30.5358 q^{56} +3.36664 q^{57} -4.75293 q^{58} +6.44588 q^{59} +9.22316 q^{61} -23.7452 q^{62} -4.04637 q^{63} +9.36761 q^{64} -2.79352 q^{66} +3.56447 q^{67} -3.03634 q^{68} +6.98915 q^{69} -4.04417 q^{71} +7.54645 q^{72} -11.3987 q^{73} +17.7876 q^{74} -16.4210 q^{76} -4.31023 q^{77} +14.2530 q^{78} +1.36746 q^{79} +1.00000 q^{81} -24.3234 q^{82} -6.90741 q^{83} +19.7365 q^{84} +7.61867 q^{86} +1.81236 q^{87} +8.03854 q^{88} +1.13344 q^{89} +21.9915 q^{91} -34.0900 q^{92} +9.05436 q^{93} -2.62251 q^{94} -11.2254 q^{96} +0.379692 q^{97} +24.5812 q^{98} +1.06521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9} - 2 q^{11} - 10 q^{12} - 7 q^{13} - 8 q^{14} + 8 q^{16} + 10 q^{17} + 2 q^{19} + 10 q^{21} + 4 q^{22} - 3 q^{23} - 16 q^{26} - 5 q^{27} - 12 q^{28} - 2 q^{29} - 6 q^{31} + 20 q^{32} + 2 q^{33} - 20 q^{34} + 10 q^{36} - 8 q^{37} + 8 q^{38} + 7 q^{39} + 8 q^{42} - 13 q^{43} + 4 q^{44} - 12 q^{46} - 5 q^{47} - 8 q^{48} + 13 q^{49} - 10 q^{51} - 34 q^{52} + 10 q^{53} - 28 q^{56} - 2 q^{57} + 4 q^{58} - 11 q^{59} + 11 q^{61} - 16 q^{62} - 10 q^{63} - 20 q^{64} - 4 q^{66} - 24 q^{67} + 20 q^{68} + 3 q^{69} - 11 q^{71} - 17 q^{73} + 12 q^{74} - 24 q^{76} + 12 q^{77} + 16 q^{78} - 5 q^{79} + 5 q^{81} - 64 q^{82} + 12 q^{84} + 12 q^{86} + 2 q^{87} + 4 q^{88} - 3 q^{89} + 22 q^{91} - 34 q^{92} + 6 q^{93} - 20 q^{96} + 14 q^{97} + 36 q^{98} - 2 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9 - 2 * q^11 - 10 * q^12 - 7 * q^13 - 8 * q^14 + 8 * q^16 + 10 * q^17 + 2 * q^19 + 10 * q^21 + 4 * q^22 - 3 * q^23 - 16 * q^26 - 5 * q^27 - 12 * q^28 - 2 * q^29 - 6 * q^31 + 20 * q^32 + 2 * q^33 - 20 * q^34 + 10 * q^36 - 8 * q^37 + 8 * q^38 + 7 * q^39 + 8 * q^42 - 13 * q^43 + 4 * q^44 - 12 * q^46 - 5 * q^47 - 8 * q^48 + 13 * q^49 - 10 * q^51 - 34 * q^52 + 10 * q^53 - 28 * q^56 - 2 * q^57 + 4 * q^58 - 11 * q^59 + 11 * q^61 - 16 * q^62 - 10 * q^63 - 20 * q^64 - 4 * q^66 - 24 * q^67 + 20 * q^68 + 3 * q^69 - 11 * q^71 - 17 * q^73 + 12 * q^74 - 24 * q^76 + 12 * q^77 + 16 * q^78 - 5 * q^79 + 5 * q^81 - 64 * q^82 + 12 * q^84 + 12 * q^86 + 2 * q^87 + 4 * q^88 - 3 * q^89 + 22 * q^91 - 34 * q^92 + 6 * q^93 - 20 * q^96 + 14 * q^97 + 36 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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