LMFDB - Embedded newform 4225.2.a.ca.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$1.94561$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.94561 q^{2} +0.752344 q^{3} +1.78540 q^{4} +1.46377 q^{6} -1.49584 q^{7} -0.417520 q^{8} -2.43398 q^{9} +O(q^{10})$	\(q+1.94561 q^{2} +0.752344 q^{3} +1.78540 q^{4} +1.46377 q^{6} -1.49584 q^{7} -0.417520 q^{8} -2.43398 q^{9} +4.03104 q^{11} +1.34324 q^{12} -2.91033 q^{14} -4.38314 q^{16} -3.62825 q^{17} -4.73558 q^{18} -6.32047 q^{19} -1.12539 q^{21} +7.84283 q^{22} -0.957968 q^{23} -0.314119 q^{24} -4.08822 q^{27} -2.67069 q^{28} +1.88869 q^{29} +2.33142 q^{31} -7.69285 q^{32} +3.03273 q^{33} -7.05917 q^{34} -4.34563 q^{36} +11.2107 q^{37} -12.2972 q^{38} -11.3447 q^{41} -2.18957 q^{42} -7.50898 q^{43} +7.19703 q^{44} -1.86383 q^{46} -6.85846 q^{47} -3.29763 q^{48} -4.76245 q^{49} -2.72969 q^{51} +3.88728 q^{53} -7.95409 q^{54} +0.624544 q^{56} -4.75517 q^{57} +3.67465 q^{58} -5.27215 q^{59} -4.11814 q^{61} +4.53604 q^{62} +3.64085 q^{63} -6.20102 q^{64} +5.90051 q^{66} +11.4471 q^{67} -6.47789 q^{68} -0.720722 q^{69} -13.6363 q^{71} +1.01623 q^{72} -2.55718 q^{73} +21.8117 q^{74} -11.2846 q^{76} -6.02980 q^{77} +10.7412 q^{79} +4.22618 q^{81} -22.0724 q^{82} -6.43379 q^{83} -2.00928 q^{84} -14.6096 q^{86} +1.42094 q^{87} -1.68304 q^{88} -8.98382 q^{89} -1.71036 q^{92} +1.75403 q^{93} -13.3439 q^{94} -5.78767 q^{96} -15.4104 q^{97} -9.26588 q^{98} -9.81146 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * 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 $

$2$	1.94561	1.37576	0.687878	−	0.725827i	$-0.258543\pi$
$2$	1.94561	1.37576	0.687878	+	0.725827i	$0.258543\pi$
$3$	0.752344	0.434366	0.217183	−	0.976131i	$-0.430313\pi$
$3$	0.752344	0.434366	0.217183	+	0.976131i	$0.430313\pi$
$4$	1.78540	0.892702
$5$	0	0
$5$	0	0
$6$	1.46377	0.597582
$7$	−1.49584	−0.565376	−0.282688	−	0.959212i	$-0.591226\pi$
$7$	−1.49584	−0.565376	−0.282688	+	0.959212i	$0.591226\pi$
$8$	−0.417520	−0.147615
$9$	−2.43398	−0.811326
$10$	0	0
$11$	4.03104	1.21540	0.607702	−	0.794165i	$-0.292092\pi$
$11$	4.03104	1.21540	0.607702	+	0.794165i	$0.292092\pi$
$12$	1.34324	0.387760
$13$	0	0
$13$	0	0
$14$	−2.91033	−0.777819
$15$	0	0
$16$	−4.38314	−1.09578
$17$	−3.62825	−0.879980	−0.439990	−	0.898003i	$-0.645018\pi$
$17$	−3.62825	−0.879980	−0.439990	+	0.898003i	$0.645018\pi$
$18$	−4.73558	−1.11619
$19$	−6.32047	−1.45001	−0.725007	−	0.688741i	$-0.758164\pi$
$19$	−6.32047	−1.45001	−0.725007	+	0.688741i	$0.758164\pi$
$20$	0	0
$21$	−1.12539	−0.245580
$22$	7.84283	1.67210
$23$	−0.957968	−0.199750	−0.0998750	−	0.995000i	$-0.531844\pi$
$23$	−0.957968	−0.199750	−0.0998750	+	0.995000i	$0.531844\pi$
$24$	−0.314119	−0.0641192
$25$	0	0
$26$	0	0
$27$	−4.08822	−0.786779
$28$	−2.67069	−0.504712
$29$	1.88869	0.350720	0.175360	−	0.984504i	$-0.443891\pi$
$29$	1.88869	0.350720	0.175360	+	0.984504i	$0.443891\pi$
$30$	0	0
$31$	2.33142	0.418735	0.209368	−	0.977837i	$-0.432859\pi$
$31$	2.33142	0.418735	0.209368	+	0.977837i	$0.432859\pi$
$32$	−7.69285	−1.35992
$33$	3.03273	0.527930
$34$	−7.05917	−1.21064
$35$	0	0
$36$	−4.34563	−0.724272
$37$	11.2107	1.84303	0.921515	−	0.388344i	$-0.126953\pi$
$37$	11.2107	1.84303	0.921515	+	0.388344i	$0.126953\pi$
$38$	−12.2972	−1.99487
$39$	0	0
$40$	0	0
$41$	−11.3447	−1.77175	−0.885873	−	0.463928i	$-0.846439\pi$
$41$	−11.3447	−1.77175	−0.885873	+	0.463928i	$0.846439\pi$
$42$	−2.18957	−0.337858
$43$	−7.50898	−1.14511	−0.572554	−	0.819867i	$-0.694047\pi$
$43$	−7.50898	−1.14511	−0.572554	+	0.819867i	$0.694047\pi$
$44$	7.19703	1.08499
$45$	0	0
$46$	−1.86383	−0.274807
$47$	−6.85846	−1.00041	−0.500204	−	0.865907i	$-0.666742\pi$
$47$	−6.85846	−1.00041	−0.500204	+	0.865907i	$0.666742\pi$
$48$	−3.29763	−0.475972
$49$	−4.76245	−0.680350
$50$	0	0
$51$	−2.72969	−0.382234
$52$	0	0
$53$	3.88728	0.533958	0.266979	−	0.963702i	$-0.413975\pi$
$53$	3.88728	0.533958	0.266979	+	0.963702i	$0.413975\pi$
$54$	−7.95409	−1.08242
$55$	0	0
$56$	0.624544	0.0834582
$57$	−4.75517	−0.629838
$58$	3.67465	0.482505
$59$	−5.27215	−0.686376	−0.343188	−	0.939267i	$-0.611507\pi$
$59$	−5.27215	−0.686376	−0.343188	+	0.939267i	$0.611507\pi$
$60$	0	0
$61$	−4.11814	−0.527274	−0.263637	−	0.964622i	$-0.584922\pi$
$61$	−4.11814	−0.527274	−0.263637	+	0.964622i	$0.584922\pi$
$62$	4.53604	0.576077
$63$	3.64085	0.458704
$64$	−6.20102	−0.775127
$65$	0	0
$66$	5.90051	0.726303
$67$	11.4471	1.39848	0.699242	−	0.714885i	$-0.253521\pi$
$67$	11.4471	1.39848	0.699242	+	0.714885i	$0.253521\pi$
$68$	−6.47789	−0.785560
$69$	−0.720722	−0.0867647
$70$	0	0
$71$	−13.6363	−1.61833	−0.809167	−	0.587578i	$-0.800081\pi$
$71$	−13.6363	−1.61833	−0.809167	+	0.587578i	$0.800081\pi$
$72$	1.01623	0.119764
$73$	−2.55718	−0.299295	−0.149647	−	0.988739i	$-0.547814\pi$
$73$	−2.55718	−0.299295	−0.149647	+	0.988739i	$0.547814\pi$
$74$	21.8117	2.53556
$75$	0	0
$76$	−11.2846	−1.29443
$77$	−6.02980	−0.687160
$78$	0	0
$79$	10.7412	1.20848	0.604239	−	0.796803i	$-0.293477\pi$
$79$	10.7412	1.20848	0.604239	+	0.796803i	$0.293477\pi$
$80$	0	0
$81$	4.22618	0.469576
$82$	−22.0724	−2.43749
$83$	−6.43379	−0.706200	−0.353100	−	0.935586i	$-0.614872\pi$
$83$	−6.43379	−0.706200	−0.353100	+	0.935586i	$0.614872\pi$
$84$	−2.00928	−0.219230
$85$	0	0
$86$	−14.6096	−1.57539
$87$	1.42094	0.152341
$88$	−1.68304	−0.179412
$89$	−8.98382	−0.952283	−0.476142	−	0.879369i	$-0.657965\pi$
$89$	−8.98382	−0.952283	−0.476142	+	0.879369i	$0.657965\pi$
$90$	0	0
$91$	0	0
$92$	−1.71036	−0.178317
$93$	1.75403	0.181884
$94$	−13.3439	−1.37632
$95$	0	0
$96$	−5.78767	−0.590702
$97$	−15.4104	−1.56469	−0.782344	−	0.622847i	$-0.785976\pi$
$97$	−15.4104	−1.56469	−0.782344	+	0.622847i	$0.785976\pi$
$98$	−9.26588	−0.935995
$99$	−9.81146	−0.986088
$100$	0	0
$101$	8.00801	0.796827	0.398413	−	0.917206i	$-0.369561\pi$
$101$	8.00801	0.796827	0.398413	+	0.917206i	$0.369561\pi$
$102$	−5.31092	−0.525860
$103$	8.69574	0.856817	0.428409	−	0.903585i	$-0.359074\pi$
$103$	8.69574	0.856817	0.428409	+	0.903585i	$0.359074\pi$
$104$	0	0
$105$	0	0
$106$	7.56313	0.734596
$107$	−10.8203	−1.04604	−0.523020	−	0.852321i	$-0.675195\pi$
$107$	−10.8203	−1.04604	−0.523020	+	0.852321i	$0.675195\pi$
$108$	−7.29913	−0.702359
$109$	6.43106	0.615984	0.307992	−	0.951389i	$-0.400343\pi$
$109$	6.43106	0.615984	0.307992	+	0.951389i	$0.400343\pi$
$110$	0	0
$111$	8.43431	0.800550
$112$	6.55649	0.619530
$113$	19.9556	1.87727	0.938633	−	0.344917i	$-0.112093\pi$
$113$	19.9556	1.87727	0.938633	+	0.344917i	$0.112093\pi$
$114$	−9.25171	−0.866502
$115$	0	0
$116$	3.37207	0.313089
$117$	0	0
$118$	−10.2576	−0.944285
$119$	5.42729	0.497519
$120$	0	0
$121$	5.24926	0.477206
$122$	−8.01231	−0.725400
$123$	−8.53513	−0.769586
$124$	4.16253	0.373806
$125$	0	0
$126$	7.08368	0.631064
$127$	−5.91326	−0.524717	−0.262359	−	0.964970i	$-0.584500\pi$
$127$	−5.91326	−0.524717	−0.262359	+	0.964970i	$0.584500\pi$
$128$	3.32093	0.293531
$129$	−5.64934	−0.497396
$130$	0	0
$131$	−3.25156	−0.284090	−0.142045	−	0.989860i	$-0.545368\pi$
$131$	−3.25156	−0.284090	−0.142045	+	0.989860i	$0.545368\pi$
$132$	5.41465	0.471284
$133$	9.45443	0.819803
$134$	22.2716	1.92397
$135$	0	0
$136$	1.51487	0.129899
$137$	2.29782	0.196316	0.0981580	−	0.995171i	$-0.468705\pi$
$137$	2.29782	0.196316	0.0981580	+	0.995171i	$0.468705\pi$
$138$	−1.40224	−0.119367
$139$	−1.71897	−0.145801	−0.0729006	−	0.997339i	$-0.523226\pi$
$139$	−1.71897	−0.145801	−0.0729006	+	0.997339i	$0.523226\pi$
$140$	0	0
$141$	−5.15992	−0.434544
$142$	−26.5310	−2.22643
$143$	0	0
$144$	10.6685	0.889039
$145$	0	0
$146$	−4.97527	−0.411756
$147$	−3.58300	−0.295521
$148$	20.0157	1.64528
$149$	0.985658	0.0807482	0.0403741	−	0.999185i	$-0.487145\pi$
$149$	0.985658	0.0807482	0.0403741	+	0.999185i	$0.487145\pi$
$150$	0	0
$151$	6.98938	0.568788	0.284394	−	0.958708i	$-0.408208\pi$
$151$	6.98938	0.568788	0.284394	+	0.958708i	$0.408208\pi$
$152$	2.63892	0.214045
$153$	8.83108	0.713951
$154$	−11.7317	−0.945363
$155$	0	0
$156$	0	0
$157$	−4.88349	−0.389745	−0.194873	−	0.980829i	$-0.562429\pi$
$157$	−4.88349	−0.389745	−0.194873	+	0.980829i	$0.562429\pi$
$158$	20.8982	1.66257
$159$	2.92457	0.231933
$160$	0	0
$161$	1.43297	0.112934
$162$	8.22251	0.646021
$163$	2.98107	0.233495	0.116748	−	0.993162i	$-0.462753\pi$
$163$	2.98107	0.233495	0.116748	+	0.993162i	$0.462753\pi$
$164$	−20.2549	−1.58164
$165$	0	0
$166$	−12.5177	−0.971559
$167$	13.7982	1.06774	0.533870	−	0.845566i	$-0.320737\pi$
$167$	13.7982	1.06774	0.533870	+	0.845566i	$0.320737\pi$
$168$	0.469872	0.0362514
$169$	0	0
$170$	0	0
$171$	15.3839	1.17643
$172$	−13.4066	−1.02224
$173$	2.53163	0.192476	0.0962380	−	0.995358i	$-0.469319\pi$
$173$	2.53163	0.192476	0.0962380	+	0.995358i	$0.469319\pi$
$174$	2.76460	0.209584
$175$	0	0
$176$	−17.6686	−1.33182
$177$	−3.96647	−0.298138
$178$	−17.4790	−1.31011
$179$	−4.57523	−0.341969	−0.170984	−	0.985274i	$-0.554695\pi$
$179$	−4.57523	−0.341969	−0.170984	+	0.985274i	$0.554695\pi$
$180$	0	0
$181$	−8.35744	−0.621204	−0.310602	−	0.950540i	$-0.600531\pi$
$181$	−8.35744	−0.621204	−0.310602	+	0.950540i	$0.600531\pi$
$182$	0	0
$183$	−3.09826	−0.229030
$184$	0.399970	0.0294862
$185$	0	0
$186$	3.41266	0.250229
$187$	−14.6256	−1.06953
$188$	−12.2451	−0.893067
$189$	6.11534	0.444826
$190$	0	0
$191$	13.2106	0.955885	0.477942	−	0.878391i	$-0.341383\pi$
$191$	13.2106	0.955885	0.477942	+	0.878391i	$0.341383\pi$
$192$	−4.66530	−0.336689
$193$	0.388224	0.0279450	0.0139725	−	0.999902i	$-0.495552\pi$
$193$	0.388224	0.0279450	0.0139725	+	0.999902i	$0.495552\pi$
$194$	−29.9826	−2.15263
$195$	0	0
$196$	−8.50290	−0.607350
$197$	−14.0420	−1.00045	−0.500227	−	0.865894i	$-0.666750\pi$
$197$	−14.0420	−1.00045	−0.500227	+	0.865894i	$0.666750\pi$
$198$	−19.0893	−1.35662
$199$	10.4092	0.737889	0.368945	−	0.929451i	$-0.379719\pi$
$199$	10.4092	0.737889	0.368945	+	0.929451i	$0.379719\pi$
$200$	0	0
$201$	8.61215	0.607454
$202$	15.5805	1.09624
$203$	−2.82518	−0.198289
$204$	−4.87361	−0.341221
$205$	0	0
$206$	16.9185	1.17877
$207$	2.33167	0.162062
$208$	0	0
$209$	−25.4780	−1.76235
$210$	0	0
$211$	8.21986	0.565879	0.282939	−	0.959138i	$-0.408690\pi$
$211$	8.21986	0.565879	0.282939	+	0.959138i	$0.408690\pi$
$212$	6.94036	0.476666
$213$	−10.2592	−0.702950
$214$	−21.0521	−1.43909
$215$	0	0
$216$	1.70691	0.116141
$217$	−3.48744	−0.236743
$218$	12.5123	0.847443
$219$	−1.92388	−0.130004
$220$	0	0
$221$	0	0
$222$	16.4099	1.10136
$223$	−22.1274	−1.48176	−0.740882	−	0.671635i	$-0.765592\pi$
$223$	−22.1274	−1.48176	−0.740882	+	0.671635i	$0.765592\pi$
$224$	11.5073	0.768864
$225$	0	0
$226$	38.8259	2.58266
$227$	15.5953	1.03510	0.517548	−	0.855654i	$-0.326845\pi$
$227$	15.5953	1.03510	0.517548	+	0.855654i	$0.326845\pi$
$228$	−8.48990	−0.562257
$229$	−18.6635	−1.23332	−0.616658	−	0.787231i	$-0.711514\pi$
$229$	−18.6635	−1.23332	−0.616658	+	0.787231i	$0.711514\pi$
$230$	0	0
$231$	−4.53649	−0.298479
$232$	−0.788563	−0.0517717
$233$	13.4988	0.884334	0.442167	−	0.896933i	$-0.354210\pi$
$233$	13.4988	0.884334	0.442167	+	0.896933i	$0.354210\pi$
$234$	0	0
$235$	0	0
$236$	−9.41292	−0.612729
$237$	8.08107	0.524922
$238$	10.5594	0.684465
$239$	−15.2496	−0.986412	−0.493206	−	0.869913i	$-0.664175\pi$
$239$	−15.2496	−0.986412	−0.493206	+	0.869913i	$0.664175\pi$
$240$	0	0
$241$	1.54187	0.0993205	0.0496603	−	0.998766i	$-0.484186\pi$
$241$	1.54187	0.0993205	0.0496603	+	0.998766i	$0.484186\pi$
$242$	10.2130	0.656518
$243$	15.4442	0.990747
$244$	−7.35255	−0.470699
$245$	0	0
$246$	−16.6060	−1.05876
$247$	0	0
$248$	−0.973413	−0.0618118
$249$	−4.84043	−0.306750
$250$	0	0
$251$	2.88959	0.182389	0.0911947	−	0.995833i	$-0.470931\pi$
$251$	2.88959	0.182389	0.0911947	+	0.995833i	$0.470931\pi$
$252$	6.50039	0.409486
$253$	−3.86160	−0.242777
$254$	−11.5049	−0.721883
$255$	0	0
$256$	18.8633	1.17895
$257$	−11.3035	−0.705093	−0.352546	−	0.935794i	$-0.614684\pi$
$257$	−11.3035	−0.705093	−0.352546	+	0.935794i	$0.614684\pi$
$258$	−10.9914	−0.684296
$259$	−16.7695	−1.04200
$260$	0	0
$261$	−4.59702	−0.284548
$262$	−6.32627	−0.390838
$263$	17.8119	1.09833	0.549163	−	0.835715i	$-0.314947\pi$
$263$	17.8119	1.09833	0.549163	+	0.835715i	$0.314947\pi$
$264$	−1.26622	−0.0779307
$265$	0	0
$266$	18.3947	1.12785
$267$	−6.75893	−0.413640
$268$	20.4377	1.24843
$269$	12.1521	0.740924	0.370462	−	0.928848i	$-0.379199\pi$
$269$	12.1521	0.740924	0.370462	+	0.928848i	$0.379199\pi$
$270$	0	0
$271$	−20.7542	−1.26073	−0.630363	−	0.776301i	$-0.717094\pi$
$271$	−20.7542	−1.26073	−0.630363	+	0.776301i	$0.717094\pi$
$272$	15.9031	0.964269
$273$	0	0
$274$	4.47066	0.270083
$275$	0	0
$276$	−1.28678	−0.0774550
$277$	0.209275	0.0125741	0.00628704	−	0.999980i	$-0.497999\pi$
$277$	0.209275	0.0125741	0.00628704	+	0.999980i	$0.497999\pi$
$278$	−3.34445	−0.200587
$279$	−5.67462	−0.339731
$280$	0	0
$281$	−14.7229	−0.878296	−0.439148	−	0.898415i	$-0.644720\pi$
$281$	−14.7229	−0.878296	−0.439148	+	0.898415i	$0.644720\pi$
$282$	−10.0392	−0.597826
$283$	−29.5476	−1.75642	−0.878211	−	0.478274i	$-0.841262\pi$
$283$	−29.5476	−1.75642	−0.878211	+	0.478274i	$0.841262\pi$
$284$	−24.3464	−1.44469
$285$	0	0
$286$	0	0
$287$	16.9699	1.00170
$288$	18.7242	1.10334
$289$	−3.83580	−0.225635
$290$	0	0
$291$	−11.5939	−0.679647
$292$	−4.56559	−0.267181
$293$	24.9305	1.45646	0.728229	−	0.685334i	$-0.240344\pi$
$293$	24.9305	1.45646	0.728229	+	0.685334i	$0.240344\pi$
$294$	−6.97113	−0.406565
$295$	0	0
$296$	−4.68069	−0.272060
$297$	−16.4798	−0.956254
$298$	1.91771	0.111090
$299$	0	0
$300$	0	0
$301$	11.2323	0.647416
$302$	13.5986	0.782513
$303$	6.02478	0.346115
$304$	27.7035	1.58890
$305$	0	0
$306$	17.1819	0.982221
$307$	−10.5394	−0.601515	−0.300758	−	0.953701i	$-0.597240\pi$
$307$	−10.5394	−0.601515	−0.300758	+	0.953701i	$0.597240\pi$
$308$	−10.7656	−0.613429
$309$	6.54219	0.372172
$310$	0	0
$311$	−4.37575	−0.248126	−0.124063	−	0.992274i	$-0.539592\pi$
$311$	−4.37575	−0.248126	−0.124063	+	0.992274i	$0.539592\pi$
$312$	0	0
$313$	8.98921	0.508100	0.254050	−	0.967191i	$-0.418237\pi$
$313$	8.98921	0.508100	0.254050	+	0.967191i	$0.418237\pi$
$314$	−9.50138	−0.536194
$315$	0	0
$316$	19.1774	1.07881
$317$	16.7107	0.938568	0.469284	−	0.883047i	$-0.344512\pi$
$317$	16.7107	0.938568	0.469284	+	0.883047i	$0.344512\pi$
$318$	5.69008	0.319084
$319$	7.61336	0.426266
$320$	0	0
$321$	−8.14060	−0.454364
$322$	2.78800	0.155369
$323$	22.9322	1.27598
$324$	7.54544	0.419191
$325$	0	0
$326$	5.80000	0.321232
$327$	4.83837	0.267563
$328$	4.73664	0.261537
$329$	10.2592	0.565607
$330$	0	0
$331$	−23.4754	−1.29032	−0.645162	−	0.764046i	$-0.723210\pi$
$331$	−23.4754	−1.29032	−0.645162	+	0.764046i	$0.723210\pi$
$332$	−11.4869	−0.630426
$333$	−27.2866	−1.49530
$334$	26.8460	1.46895
$335$	0	0
$336$	4.93274	0.269103
$337$	19.6766	1.07185	0.535925	−	0.844266i	$-0.319963\pi$
$337$	19.6766	1.07185	0.535925	+	0.844266i	$0.319963\pi$
$338$	0	0
$339$	15.0135	0.815421
$340$	0	0
$341$	9.39804	0.508932
$342$	29.9311	1.61849
$343$	17.5948	0.950029
$344$	3.13515	0.169036
$345$	0	0
$346$	4.92556	0.264800
$347$	−9.27409	−0.497859	−0.248930	−	0.968522i	$-0.580079\pi$
$347$	−9.27409	−0.497859	−0.248930	+	0.968522i	$0.580079\pi$
$348$	2.53696	0.135995
$349$	16.3908	0.877378	0.438689	−	0.898639i	$-0.355443\pi$
$349$	16.3908	0.877378	0.438689	+	0.898639i	$0.355443\pi$
$350$	0	0
$351$	0	0
$352$	−31.0102	−1.65285
$353$	7.44869	0.396454	0.198227	−	0.980156i	$-0.436482\pi$
$353$	7.44869	0.396454	0.198227	+	0.980156i	$0.436482\pi$
$354$	−7.71722	−0.410165
$355$	0	0
$356$	−16.0398	−0.850105
$357$	4.08319	0.216106
$358$	−8.90163	−0.470466
$359$	24.4832	1.29217	0.646087	−	0.763264i	$-0.276404\pi$
$359$	24.4832	1.29217	0.646087	+	0.763264i	$0.276404\pi$
$360$	0	0
$361$	20.9483	1.10254
$362$	−16.2603	−0.854624
$363$	3.94925	0.207282
$364$	0	0
$365$	0	0
$366$	−6.02801	−0.315089
$367$	20.7716	1.08427	0.542135	−	0.840291i	$-0.317616\pi$
$367$	20.7716	1.08427	0.542135	+	0.840291i	$0.317616\pi$
$368$	4.19891	0.218883
$369$	27.6128	1.43746
$370$	0	0
$371$	−5.81476	−0.301887
$372$	3.13165	0.162369
$373$	−7.28118	−0.377005	−0.188503	−	0.982073i	$-0.560363\pi$
$373$	−7.28118	−0.377005	−0.188503	+	0.982073i	$0.560363\pi$
$374$	−28.4558	−1.47141
$375$	0	0
$376$	2.86354	0.147676
$377$	0	0
$378$	11.8981	0.611971
$379$	0.0816022	0.00419163	0.00209581	−	0.999998i	$-0.499333\pi$
$379$	0.0816022	0.00419163	0.00209581	+	0.999998i	$0.499333\pi$
$380$	0	0
$381$	−4.44881	−0.227919
$382$	25.7027	1.31506
$383$	26.8173	1.37030	0.685149	−	0.728403i	$-0.259737\pi$
$383$	26.8173	1.37030	0.685149	+	0.728403i	$0.259737\pi$
$384$	2.49848	0.127500
$385$	0	0
$386$	0.755333	0.0384454
$387$	18.2767	0.929056
$388$	−27.5138	−1.39680
$389$	10.4993	0.532335	0.266167	−	0.963927i	$-0.414243\pi$
$389$	10.4993	0.532335	0.266167	+	0.963927i	$0.414243\pi$
$390$	0	0
$391$	3.47575	0.175776
$392$	1.98842	0.100430
$393$	−2.44629	−0.123399
$394$	−27.3204	−1.37638
$395$	0	0
$396$	−17.5174	−0.880283
$397$	6.17624	0.309976	0.154988	−	0.987916i	$-0.450466\pi$
$397$	6.17624	0.309976	0.154988	+	0.987916i	$0.450466\pi$
$398$	20.2523	1.01515
$399$	7.11299	0.356095
$400$	0	0
$401$	−33.0284	−1.64936	−0.824679	−	0.565600i	$-0.808644\pi$
$401$	−33.0284	−1.64936	−0.824679	+	0.565600i	$0.808644\pi$
$402$	16.7559	0.835708
$403$	0	0
$404$	14.2975	0.711329
$405$	0	0
$406$	−5.49670	−0.272797
$407$	45.1908	2.24002
$408$	1.13970	0.0564236
$409$	−0.499565	−0.0247019	−0.0123509	−	0.999924i	$-0.503932\pi$
$409$	−0.499565	−0.0247019	−0.0123509	+	0.999924i	$0.503932\pi$
$410$	0	0
$411$	1.72875	0.0852730
$412$	15.5254	0.764883
$413$	7.88631	0.388060
$414$	4.53653	0.222958
$415$	0	0
$416$	0	0
$417$	−1.29326	−0.0633311
$418$	−49.5704	−2.42457
$419$	−7.59931	−0.371251	−0.185625	−	0.982621i	$-0.559431\pi$
$419$	−7.59931	−0.371251	−0.185625	+	0.982621i	$0.559431\pi$
$420$	0	0
$421$	−31.3622	−1.52850	−0.764251	−	0.644919i	$-0.776891\pi$
$421$	−31.3622	−1.52850	−0.764251	+	0.644919i	$0.776891\pi$
$422$	15.9927	0.778511
$423$	16.6933	0.811658
$424$	−1.62301	−0.0788205
$425$	0	0
$426$	−19.9605	−0.967087
$427$	6.16010	0.298108
$428$	−19.3186	−0.933802
$429$	0	0
$430$	0	0
$431$	−17.3814	−0.837230	−0.418615	−	0.908164i	$-0.637484\pi$
$431$	−17.3814	−0.837230	−0.418615	+	0.908164i	$0.637484\pi$
$432$	17.9193	0.862140
$433$	25.0285	1.20279	0.601395	−	0.798952i	$-0.294612\pi$
$433$	25.0285	1.20279	0.601395	+	0.798952i	$0.294612\pi$
$434$	−6.78520	−0.325700
$435$	0	0
$436$	11.4820	0.549890
$437$	6.05480	0.289641
$438$	−3.74312	−0.178853
$439$	0.477761	0.0228023	0.0114011	−	0.999935i	$-0.496371\pi$
$439$	0.477761	0.0228023	0.0114011	+	0.999935i	$0.496371\pi$
$440$	0	0
$441$	11.5917	0.551986
$442$	0	0
$443$	10.9222	0.518930	0.259465	−	0.965753i	$-0.416454\pi$
$443$	10.9222	0.518930	0.259465	+	0.965753i	$0.416454\pi$
$444$	15.0587	0.714652
$445$	0	0
$446$	−43.0514	−2.03854
$447$	0.741554	0.0350743
$448$	9.27575	0.438238
$449$	−9.89236	−0.466849	−0.233425	−	0.972375i	$-0.574993\pi$
$449$	−9.89236	−0.466849	−0.233425	+	0.972375i	$0.574993\pi$
$450$	0	0
$451$	−45.7309	−2.15339
$452$	35.6288	1.67584
$453$	5.25842	0.247062
$454$	30.3424	1.42404
$455$	0	0
$456$	1.98538	0.0929738
$457$	−27.2889	−1.27652	−0.638261	−	0.769820i	$-0.720346\pi$
$457$	−27.2889	−1.27652	−0.638261	+	0.769820i	$0.720346\pi$
$458$	−36.3119	−1.69674
$459$	14.8331	0.692350
$460$	0	0
$461$	−18.2645	−0.850662	−0.425331	−	0.905038i	$-0.639842\pi$
$461$	−18.2645	−0.850662	−0.425331	+	0.905038i	$0.639842\pi$
$462$	−8.82624	−0.410634
$463$	28.5012	1.32456	0.662281	−	0.749255i	$-0.269588\pi$
$463$	28.5012	1.32456	0.662281	+	0.749255i	$0.269588\pi$
$464$	−8.27837	−0.384314
$465$	0	0
$466$	26.2634	1.21663
$467$	28.0713	1.29899	0.649493	−	0.760368i	$-0.274981\pi$
$467$	28.0713	1.29899	0.649493	+	0.760368i	$0.274981\pi$
$468$	0	0
$469$	−17.1230	−0.790669
$470$	0	0
$471$	−3.67407	−0.169292
$472$	2.20123	0.101320
$473$	−30.2690	−1.39177
$474$	15.7226	0.722164
$475$	0	0
$476$	9.68992	0.444137
$477$	−9.46154	−0.433214
$478$	−29.6697	−1.35706
$479$	−7.04393	−0.321846	−0.160923	−	0.986967i	$-0.551447\pi$
$479$	−7.04393	−0.321846	−0.160923	+	0.986967i	$0.551447\pi$
$480$	0	0
$481$	0	0
$482$	2.99988	0.136641
$483$	1.07809	0.0490546
$484$	9.37206	0.426003
$485$	0	0
$486$	30.0484	1.36302
$487$	4.04221	0.183170	0.0915849	−	0.995797i	$-0.470807\pi$
$487$	4.04221	0.183170	0.0915849	+	0.995797i	$0.470807\pi$
$488$	1.71941	0.0778338
$489$	2.24279	0.101422
$490$	0	0
$491$	−1.29885	−0.0586161	−0.0293080	−	0.999570i	$-0.509330\pi$
$491$	−1.29885	−0.0586161	−0.0293080	+	0.999570i	$0.509330\pi$
$492$	−15.2387	−0.687011
$493$	−6.85262	−0.308627
$494$	0	0
$495$	0	0
$496$	−10.2189	−0.458844
$497$	20.3978	0.914967
$498$	−9.41759	−0.422012
$499$	−14.0036	−0.626886	−0.313443	−	0.949607i	$-0.601482\pi$
$499$	−14.0036	−0.626886	−0.313443	+	0.949607i	$0.601482\pi$
$500$	0	0
$501$	10.3810	0.463790
$502$	5.62202	0.250923
$503$	−33.3519	−1.48709	−0.743543	−	0.668688i	$-0.766856\pi$
$503$	−33.3519	−1.48709	−0.743543	+	0.668688i	$0.766856\pi$
$504$	−1.52013	−0.0677118
$505$	0	0
$506$	−7.51318	−0.334002
$507$	0	0
$508$	−10.5576	−0.468416
$509$	−37.8633	−1.67826	−0.839130	−	0.543930i	$-0.816936\pi$
$509$	−37.8633	−1.67826	−0.839130	+	0.543930i	$0.816936\pi$
$510$	0	0
$511$	3.82514	0.169214
$512$	30.0587	1.32842
$513$	25.8395	1.14084
$514$	−21.9922	−0.970035
$515$	0	0
$516$	−10.0864	−0.444027
$517$	−27.6467	−1.21590
$518$	−32.6269	−1.43354
$519$	1.90465	0.0836050
$520$	0	0
$521$	−7.09897	−0.311012	−0.155506	−	0.987835i	$-0.549701\pi$
$521$	−7.09897	−0.311012	−0.155506	+	0.987835i	$0.549701\pi$
$522$	−8.94401	−0.391469
$523$	41.4243	1.81136	0.905679	−	0.423965i	$-0.139362\pi$
$523$	41.4243	1.81136	0.905679	+	0.423965i	$0.139362\pi$
$524$	−5.80535	−0.253608
$525$	0	0
$526$	34.6549	1.51103
$527$	−8.45897	−0.368479
$528$	−13.2929	−0.578498
$529$	−22.0823	−0.960100
$530$	0	0
$531$	12.8323	0.556874
$532$	16.8800	0.731840
$533$	0	0
$534$	−13.1502	−0.569067
$535$	0	0
$536$	−4.77938	−0.206438
$537$	−3.44215	−0.148540
$538$	23.6432	1.01933
$539$	−19.1976	−0.826900
$540$	0	0
$541$	−22.7596	−0.978511	−0.489256	−	0.872140i	$-0.662731\pi$
$541$	−22.7596	−0.978511	−0.489256	+	0.872140i	$0.662731\pi$
$542$	−40.3795	−1.73445
$543$	−6.28768	−0.269830
$544$	27.9116	1.19670
$545$	0	0
$546$	0	0
$547$	12.3638	0.528636	0.264318	−	0.964436i	$-0.414853\pi$
$547$	12.3638	0.528636	0.264318	+	0.964436i	$0.414853\pi$
$548$	4.10254	0.175252
$549$	10.0235	0.427791
$550$	0	0
$551$	−11.9374	−0.508549
$552$	0.300915	0.0128078
$553$	−16.0671	−0.683244
$554$	0.407167	0.0172989
$555$	0	0
$556$	−3.06906	−0.130157
$557$	−26.3496	−1.11647	−0.558233	−	0.829684i	$-0.688521\pi$
$557$	−26.3496	−1.11647	−0.558233	+	0.829684i	$0.688521\pi$
$558$	−11.0406	−0.467386
$559$	0	0
$560$	0	0
$561$	−11.0035	−0.464568
$562$	−28.6451	−1.20832
$563$	−3.07168	−0.129456	−0.0647279	−	0.997903i	$-0.520618\pi$
$563$	−3.07168	−0.129456	−0.0647279	+	0.997903i	$0.520618\pi$
$564$	−9.21255	−0.387918
$565$	0	0
$566$	−57.4881	−2.41641
$567$	−6.32171	−0.265487
$568$	5.69344	0.238891
$569$	39.8064	1.66877	0.834386	−	0.551181i	$-0.185823\pi$
$569$	39.8064	1.66877	0.834386	+	0.551181i	$0.185823\pi$
$570$	0	0
$571$	35.5678	1.48847	0.744233	−	0.667920i	$-0.232815\pi$
$571$	35.5678	1.48847	0.744233	+	0.667920i	$0.232815\pi$
$572$	0	0
$573$	9.93891	0.415204
$574$	33.0168	1.37810
$575$	0	0
$576$	15.0931	0.628881
$577$	12.3883	0.515733	0.257866	−	0.966181i	$-0.416981\pi$
$577$	12.3883	0.515733	0.257866	+	0.966181i	$0.416981\pi$
$578$	−7.46298	−0.310419
$579$	0.292078	0.0121383
$580$	0	0
$581$	9.62394	0.399268
$582$	−22.5573	−0.935028
$583$	15.6698	0.648975
$584$	1.06767	0.0441805
$585$	0	0
$586$	48.5052	2.00373
$587$	−5.81541	−0.240027	−0.120014	−	0.992772i	$-0.538294\pi$
$587$	−5.81541	−0.240027	−0.120014	+	0.992772i	$0.538294\pi$
$588$	−6.39711	−0.263812
$589$	−14.7357	−0.607172
$590$	0	0
$591$	−10.5645	−0.434564
$592$	−49.1381	−2.01956
$593$	−38.3395	−1.57441	−0.787207	−	0.616688i	$-0.788474\pi$
$593$	−38.3395	−1.57441	−0.787207	+	0.616688i	$0.788474\pi$
$594$	−32.0632	−1.31557
$595$	0	0
$596$	1.75980	0.0720841
$597$	7.83131	0.320514
$598$	0	0
$599$	−2.97330	−0.121486	−0.0607430	−	0.998153i	$-0.519347\pi$
$599$	−2.97330	−0.121486	−0.0607430	+	0.998153i	$0.519347\pi$
$600$	0	0
$601$	−9.28075	−0.378570	−0.189285	−	0.981922i	$-0.560617\pi$
$601$	−9.28075	−0.378570	−0.189285	+	0.981922i	$0.560617\pi$
$602$	21.8536	0.890686
$603$	−27.8619	−1.13463
$604$	12.4789	0.507758
$605$	0	0
$606$	11.7219	0.476169
$607$	3.14241	0.127547	0.0637733	−	0.997964i	$-0.479687\pi$
$607$	3.14241	0.127547	0.0637733	+	0.997964i	$0.479687\pi$
$608$	48.6224	1.97190
$609$	−2.12551	−0.0861299
$610$	0	0
$611$	0	0
$612$	15.7671	0.637345
$613$	−21.2647	−0.858873	−0.429437	−	0.903097i	$-0.641288\pi$
$613$	−21.2647	−0.858873	−0.429437	+	0.903097i	$0.641288\pi$
$614$	−20.5056	−0.827538
$615$	0	0
$616$	2.51756	0.101435
$617$	20.8157	0.838010	0.419005	−	0.907984i	$-0.362379\pi$
$617$	20.8157	0.838010	0.419005	+	0.907984i	$0.362379\pi$
$618$	12.7286	0.512018
$619$	−19.9078	−0.800161	−0.400080	−	0.916480i	$-0.631018\pi$
$619$	−19.9078	−0.800161	−0.400080	+	0.916480i	$0.631018\pi$
$620$	0	0
$621$	3.91639	0.157159
$622$	−8.51351	−0.341361
$623$	13.4384	0.538398
$624$	0	0
$625$	0	0
$626$	17.4895	0.699022
$627$	−19.1683	−0.765507
$628$	−8.71901	−0.347926
$629$	−40.6753	−1.62183
$630$	0	0
$631$	16.2459	0.646741	0.323370	−	0.946272i	$-0.395184\pi$
$631$	16.2459	0.646741	0.323370	+	0.946272i	$0.395184\pi$
$632$	−4.48466	−0.178390
$633$	6.18417	0.245799
$634$	32.5126	1.29124
$635$	0	0
$636$	5.22154	0.207048
$637$	0	0
$638$	14.8126	0.586438
$639$	33.1905	1.31300
$640$	0	0
$641$	7.51357	0.296768	0.148384	−	0.988930i	$-0.452593\pi$
$641$	7.51357	0.296768	0.148384	+	0.988930i	$0.452593\pi$
$642$	−15.8385	−0.625094
$643$	−31.6264	−1.24723	−0.623613	−	0.781734i	$-0.714336\pi$
$643$	−31.6264	−1.24723	−0.623613	+	0.781734i	$0.714336\pi$
$644$	2.55843	0.100816
$645$	0	0
$646$	44.6172	1.75544
$647$	35.7326	1.40479	0.702397	−	0.711785i	$-0.252113\pi$
$647$	35.7326	1.40479	0.702397	+	0.711785i	$0.252113\pi$
$648$	−1.76451	−0.0693167
$649$	−21.2522	−0.834223
$650$	0	0
$651$	−2.62376	−0.102833
$652$	5.32241	0.208442
$653$	−0.415100	−0.0162441	−0.00812206	−	0.999967i	$-0.502585\pi$
$653$	−0.415100	−0.0162441	−0.00812206	+	0.999967i	$0.502585\pi$
$654$	9.41359	0.368101
$655$	0	0
$656$	49.7254	1.94145
$657$	6.22411	0.242826
$658$	19.9604	0.778137
$659$	−26.6367	−1.03762	−0.518808	−	0.854891i	$-0.673624\pi$
$659$	−26.6367	−1.03762	−0.518808	+	0.854891i	$0.673624\pi$
$660$	0	0
$661$	44.5829	1.73407	0.867037	−	0.498245i	$-0.166022\pi$
$661$	44.5829	1.73407	0.867037	+	0.498245i	$0.166022\pi$
$662$	−45.6740	−1.77517
$663$	0	0
$664$	2.68623	0.104246
$665$	0	0
$666$	−53.0892	−2.05716
$667$	−1.80930	−0.0700564
$668$	24.6354	0.953174
$669$	−16.6475	−0.643628
$670$	0	0
$671$	−16.6004	−0.640851
$672$	8.65745	0.333968
$673$	−17.8697	−0.688828	−0.344414	−	0.938818i	$-0.611922\pi$
$673$	−17.8697	−0.688828	−0.344414	+	0.938818i	$0.611922\pi$
$674$	38.2829	1.47460
$675$	0	0
$676$	0	0
$677$	−4.13330	−0.158856	−0.0794279	−	0.996841i	$-0.525309\pi$
$677$	−4.13330	−0.158856	−0.0794279	+	0.996841i	$0.525309\pi$
$678$	29.2104	1.12182
$679$	23.0515	0.884636
$680$	0	0
$681$	11.7330	0.449611
$682$	18.2849	0.700166
$683$	−27.1275	−1.03800	−0.519002	−	0.854773i	$-0.673696\pi$
$683$	−27.1275	−1.03800	−0.519002	+	0.854773i	$0.673696\pi$
$684$	27.4664	1.05021
$685$	0	0
$686$	34.2326	1.30701
$687$	−14.0414	−0.535711
$688$	32.9129	1.25479
$689$	0	0
$690$	0	0
$691$	−20.7057	−0.787682	−0.393841	−	0.919178i	$-0.628854\pi$
$691$	−20.7057	−0.787682	−0.393841	+	0.919178i	$0.628854\pi$
$692$	4.51998	0.171824
$693$	14.6764	0.557510
$694$	−18.0438	−0.684932
$695$	0	0
$696$	−0.593271	−0.0224879
$697$	41.1614	1.55910
$698$	31.8901	1.20706
$699$	10.1557	0.384125
$700$	0	0
$701$	−34.5162	−1.30366	−0.651830	−	0.758365i	$-0.725999\pi$
$701$	−34.5162	−1.30366	−0.651830	+	0.758365i	$0.725999\pi$
$702$	0	0
$703$	−70.8569	−2.67242
$704$	−24.9965	−0.942092
$705$	0	0
$706$	14.4923	0.545424
$707$	−11.9787	−0.450506
$708$	−7.08176	−0.266149
$709$	45.5082	1.70910	0.854548	−	0.519372i	$-0.173834\pi$
$709$	45.5082	1.70910	0.854548	+	0.519372i	$0.173834\pi$
$710$	0	0
$711$	−26.1438	−0.980469
$712$	3.75092	0.140572
$713$	−2.23342	−0.0836424
$714$	7.94431	0.297308
$715$	0	0
$716$	−8.16864	−0.305276
$717$	−11.4729	−0.428464
$718$	47.6348	1.77772
$719$	−30.0159	−1.11941	−0.559703	−	0.828693i	$-0.689085\pi$
$719$	−30.0159	−1.11941	−0.559703	+	0.828693i	$0.689085\pi$
$720$	0	0
$721$	−13.0075	−0.484424
$722$	40.7573	1.51683
$723$	1.16002	0.0431415
$724$	−14.9214	−0.554550
$725$	0	0
$726$	7.68371	0.285169
$727$	−29.2316	−1.08414	−0.542069	−	0.840334i	$-0.682359\pi$
$727$	−29.2316	−1.08414	−0.542069	+	0.840334i	$0.682359\pi$
$728$	0	0
$729$	−1.05918	−0.0392289
$730$	0	0
$731$	27.2444	1.00767
$732$	−5.53165	−0.204456
$733$	21.7650	0.803909	0.401954	−	0.915660i	$-0.368331\pi$
$733$	21.7650	0.803909	0.401954	+	0.915660i	$0.368331\pi$
$734$	40.4135	1.49169
$735$	0	0
$736$	7.36950	0.271643
$737$	46.1436	1.69972
$738$	53.7237	1.97760
$739$	−49.5948	−1.82437	−0.912187	−	0.409774i	$-0.865608\pi$
$739$	−49.5948	−1.82437	−0.912187	+	0.409774i	$0.865608\pi$
$740$	0	0
$741$	0	0
$742$	−11.3133	−0.415323
$743$	4.27738	0.156922	0.0784608	−	0.996917i	$-0.474999\pi$
$743$	4.27738	0.156922	0.0784608	+	0.996917i	$0.474999\pi$
$744$	−0.732342	−0.0268490
$745$	0	0
$746$	−14.1663	−0.518667
$747$	15.6597	0.572959
$748$	−26.1126	−0.954772
$749$	16.1855	0.591405
$750$	0	0
$751$	−5.12194	−0.186902	−0.0934511	−	0.995624i	$-0.529790\pi$
$751$	−5.12194	−0.186902	−0.0934511	+	0.995624i	$0.529790\pi$
$752$	30.0616	1.09623
$753$	2.17397	0.0792238
$754$	0	0
$755$	0	0
$756$	10.9184	0.397097
$757$	31.5680	1.14736	0.573680	−	0.819080i	$-0.305515\pi$
$757$	31.5680	1.14736	0.573680	+	0.819080i	$0.305515\pi$
$758$	0.158766	0.00576665
$759$	−2.90526	−0.105454
$760$	0	0
$761$	13.2971	0.482019	0.241010	−	0.970523i	$-0.422521\pi$
$761$	13.2971	0.482019	0.241010	+	0.970523i	$0.422521\pi$
$762$	−8.65566	−0.313561
$763$	−9.61986	−0.348262
$764$	23.5862	0.853320
$765$	0	0
$766$	52.1760	1.88520
$767$	0	0
$768$	14.1917	0.512098
$769$	−2.15235	−0.0776158	−0.0388079	−	0.999247i	$-0.512356\pi$
$769$	−2.15235	−0.0776158	−0.0388079	+	0.999247i	$0.512356\pi$
$770$	0	0
$771$	−8.50412	−0.306268
$772$	0.693137	0.0249465
$773$	9.94629	0.357743	0.178872	−	0.983872i	$-0.442755\pi$
$773$	9.94629	0.357743	0.178872	+	0.983872i	$0.442755\pi$
$774$	35.5593	1.27815
$775$	0	0
$776$	6.43414	0.230972
$777$	−12.6164	−0.452611
$778$	20.4275	0.732362
$779$	71.7039	2.56906
$780$	0	0
$781$	−54.9686	−1.96693
$782$	6.76245	0.241825
$783$	−7.72137	−0.275939
$784$	20.8745	0.745518
$785$	0	0
$786$	−4.75954	−0.169767
$787$	49.8898	1.77838	0.889189	−	0.457540i	$-0.151269\pi$
$787$	49.8898	1.77838	0.889189	+	0.457540i	$0.151269\pi$
$788$	−25.0707	−0.893108
$789$	13.4006	0.477076
$790$	0	0
$791$	−29.8505	−1.06136
$792$	4.09648	0.145562
$793$	0	0
$794$	12.0166	0.426452
$795$	0	0
$796$	18.5846	0.658715
$797$	−31.4459	−1.11387	−0.556936	−	0.830555i	$-0.688023\pi$
$797$	−31.4459	−1.11387	−0.556936	+	0.830555i	$0.688023\pi$
$798$	13.8391	0.489899
$799$	24.8842	0.880340
$800$	0	0
$801$	21.8664	0.772612
$802$	−64.2604	−2.26911
$803$	−10.3081	−0.363764
$804$	15.3762	0.542276
$805$	0	0
$806$	0	0
$807$	9.14254	0.321832
$808$	−3.34350	−0.117624
$809$	−22.3034	−0.784147	−0.392073	−	0.919934i	$-0.628242\pi$
$809$	−22.3034	−0.784147	−0.392073	+	0.919934i	$0.628242\pi$
$810$	0	0
$811$	24.3536	0.855171	0.427586	−	0.903975i	$-0.359364\pi$
$811$	24.3536	0.855171	0.427586	+	0.903975i	$0.359364\pi$
$812$	−5.04408	−0.177013
$813$	−15.6143	−0.547616
$814$	87.9237	3.08173
$815$	0	0
$816$	11.9646	0.418846
$817$	47.4603	1.66042
$818$	−0.971959	−0.0339837
$819$	0	0
$820$	0	0
$821$	23.7924	0.830359	0.415179	−	0.909740i	$-0.363719\pi$
$821$	23.7924	0.830359	0.415179	+	0.909740i	$0.363719\pi$
$822$	3.36348	0.117315
$823$	−38.3919	−1.33826	−0.669129	−	0.743146i	$-0.733333\pi$
$823$	−38.3919	−1.33826	−0.669129	+	0.743146i	$0.733333\pi$
$824$	−3.63064	−0.126479
$825$	0	0
$826$	15.3437	0.533876
$827$	−26.5289	−0.922499	−0.461250	−	0.887270i	$-0.652599\pi$
$827$	−26.5289	−0.922499	−0.461250	+	0.887270i	$0.652599\pi$
$828$	4.16298	0.144673
$829$	−30.8503	−1.07148	−0.535738	−	0.844384i	$-0.679967\pi$
$829$	−30.8503	−1.07148	−0.535738	+	0.844384i	$0.679967\pi$
$830$	0	0
$831$	0.157447	0.00546176
$832$	0	0
$833$	17.2794	0.598695
$834$	−2.51618	−0.0871281
$835$	0	0
$836$	−45.4886	−1.57326
$837$	−9.53136	−0.329452
$838$	−14.7853	−0.510750
$839$	−31.1915	−1.07685	−0.538425	−	0.842673i	$-0.680980\pi$
$839$	−31.1915	−1.07685	−0.538425	+	0.842673i	$0.680980\pi$
$840$	0	0
$841$	−25.4329	−0.876995
$842$	−61.0187	−2.10284
$843$	−11.0767	−0.381502
$844$	14.6758	0.505161
$845$	0	0
$846$	32.4787	1.11664
$847$	−7.85208	−0.269801
$848$	−17.0385	−0.585103
$849$	−22.2300	−0.762930
$850$	0	0
$851$	−10.7395	−0.368145
$852$	−18.3169	−0.627525
$853$	3.48801	0.119427	0.0597136	−	0.998216i	$-0.480981\pi$
$853$	3.48801	0.119427	0.0597136	+	0.998216i	$0.480981\pi$
$854$	11.9852	0.410124
$855$	0	0
$856$	4.51769	0.154412
$857$	−28.0577	−0.958432	−0.479216	−	0.877697i	$-0.659079\pi$
$857$	−28.0577	−0.958432	−0.479216	+	0.877697i	$0.659079\pi$
$858$	0	0
$859$	5.47126	0.186677	0.0933386	−	0.995634i	$-0.470246\pi$
$859$	5.47126	0.186677	0.0933386	+	0.995634i	$0.470246\pi$
$860$	0	0
$861$	12.7672	0.435105
$862$	−33.8174	−1.15182
$863$	−4.15653	−0.141490	−0.0707449	−	0.997494i	$-0.522538\pi$
$863$	−4.15653	−0.141490	−0.0707449	+	0.997494i	$0.522538\pi$
$864$	31.4501	1.06995
$865$	0	0
$866$	48.6957	1.65475
$867$	−2.88584	−0.0980083
$868$	−6.22649	−0.211341
$869$	43.2981	1.46879
$870$	0	0
$871$	0	0
$872$	−2.68509	−0.0909287
$873$	37.5085	1.26947
$874$	11.7803	0.398475
$875$	0	0
$876$	−3.43490	−0.116054
$877$	−52.4590	−1.77141	−0.885707	−	0.464246i	$-0.846326\pi$
$877$	−52.4590	−1.77141	−0.885707	+	0.464246i	$0.846326\pi$
$878$	0.929538	0.0313704
$879$	18.7564	0.632636
$880$	0	0
$881$	44.6024	1.50269	0.751346	−	0.659909i	$-0.229405\pi$
$881$	44.6024	1.50269	0.751346	+	0.659909i	$0.229405\pi$
$882$	22.5530	0.759397
$883$	−21.4813	−0.722904	−0.361452	−	0.932391i	$-0.617719\pi$
$883$	−21.4813	−0.722904	−0.361452	+	0.932391i	$0.617719\pi$
$884$	0	0
$885$	0	0
$886$	21.2504	0.713921
$887$	−23.5601	−0.791069	−0.395535	−	0.918451i	$-0.629441\pi$
$887$	−23.5601	−0.791069	−0.395535	+	0.918451i	$0.629441\pi$
$888$	−3.52149	−0.118174
$889$	8.84532	0.296662
$890$	0	0
$891$	17.0359	0.570724
$892$	−39.5064	−1.32277
$893$	43.3487	1.45061
$894$	1.44278	0.0482537
$895$	0	0
$896$	−4.96759	−0.165956
$897$	0	0
$898$	−19.2467	−0.642270
$899$	4.40332	0.146859
$900$	0	0
$901$	−14.1040	−0.469873
$902$	−88.9746	−2.96253
$903$	8.45052	0.281216
$904$	−8.33186	−0.277114
$905$	0	0
$906$	10.2308	0.339897
$907$	5.51921	0.183262	0.0916312	−	0.995793i	$-0.470792\pi$
$907$	5.51921	0.183262	0.0916312	+	0.995793i	$0.470792\pi$
$908$	27.8439	0.924033
$909$	−19.4913	−0.646486
$910$	0	0
$911$	−59.4196	−1.96866	−0.984330	−	0.176336i	$-0.943575\pi$
$911$	−59.4196	−1.96866	−0.984330	+	0.176336i	$0.943575\pi$
$912$	20.8426	0.690166
$913$	−25.9348	−0.858318
$914$	−53.0936	−1.75618
$915$	0	0
$916$	−33.3218	−1.10098
$917$	4.86383	0.160618
$918$	28.8594	0.952504
$919$	−3.24068	−0.106900	−0.0534501	−	0.998571i	$-0.517022\pi$
$919$	−3.24068	−0.106900	−0.0534501	+	0.998571i	$0.517022\pi$
$920$	0	0
$921$	−7.92926	−0.261278
$922$	−35.5356	−1.17030
$923$	0	0
$924$	−8.09946	−0.266453
$925$	0	0
$926$	55.4522	1.82227
$927$	−21.1653	−0.695158
$928$	−14.5294	−0.476950
$929$	16.5394	0.542639	0.271319	−	0.962489i	$-0.412540\pi$
$929$	16.5394	0.542639	0.271319	+	0.962489i	$0.412540\pi$
$930$	0	0
$931$	30.1009	0.986518
$932$	24.1008	0.789447
$933$	−3.29207	−0.107778
$934$	54.6159	1.78709
$935$	0	0
$936$	0	0
$937$	22.2410	0.726581	0.363291	−	0.931676i	$-0.381653\pi$
$937$	22.2410	0.726581	0.363291	+	0.931676i	$0.381653\pi$
$938$	−33.3148	−1.08777
$939$	6.76299	0.220702
$940$	0	0
$941$	−21.1247	−0.688646	−0.344323	−	0.938851i	$-0.611891\pi$
$941$	−21.1247	−0.688646	−0.344323	+	0.938851i	$0.611891\pi$
$942$	−7.14831	−0.232905
$943$	10.8679	0.353906
$944$	23.1086	0.752120
$945$	0	0
$946$	−58.8917	−1.91473
$947$	−36.7247	−1.19339	−0.596696	−	0.802468i	$-0.703520\pi$
$947$	−36.7247	−1.19339	−0.596696	+	0.802468i	$0.703520\pi$
$948$	14.4280	0.468599
$949$	0	0
$950$	0	0
$951$	12.5722	0.407682
$952$	−2.26600	−0.0734416
$953$	30.3031	0.981613	0.490806	−	0.871269i	$-0.336702\pi$
$953$	30.3031	0.981613	0.490806	+	0.871269i	$0.336702\pi$
$954$	−18.4085	−0.595997
$955$	0	0
$956$	−27.2266	−0.880572
$957$	5.72787	0.185156
$958$	−13.7048	−0.442781
$959$	−3.43718	−0.110992
$960$	0	0
$961$	−25.5645	−0.824661
$962$	0	0
$963$	26.3364	0.848679
$964$	2.75286	0.0886637
$965$	0	0
$966$	2.09754	0.0674872
$967$	−30.3173	−0.974939	−0.487469	−	0.873140i	$-0.662080\pi$
$967$	−30.3173	−0.974939	−0.487469	+	0.873140i	$0.662080\pi$
$968$	−2.19167	−0.0704430
$969$	17.2529	0.554244
$970$	0	0
$971$	−12.4585	−0.399813	−0.199906	−	0.979815i	$-0.564064\pi$
$971$	−12.4585	−0.399813	−0.199906	+	0.979815i	$0.564064\pi$
$972$	27.5742	0.884442
$973$	2.57131	0.0824324
$974$	7.86456	0.251997
$975$	0	0
$976$	18.0504	0.577779
$977$	20.0726	0.642179	0.321089	−	0.947049i	$-0.395951\pi$
$977$	20.0726	0.642179	0.321089	+	0.947049i	$0.395951\pi$
$978$	4.36360	0.139532
$979$	−36.2141	−1.15741
$980$	0	0
$981$	−15.6531	−0.499764
$982$	−2.52705	−0.0806414
$983$	−34.5118	−1.10076	−0.550378	−	0.834915i	$-0.685517\pi$
$983$	−34.5118	−1.10076	−0.550378	+	0.834915i	$0.685517\pi$
$984$	3.56358	0.113603
$985$	0	0
$986$	−13.3325	−0.424595
$987$	7.71844	0.245681
$988$	0	0
$989$	7.19336	0.228735
$990$	0	0
$991$	22.6013	0.717953	0.358977	−	0.933347i	$-0.383126\pi$
$991$	22.6013	0.717953	0.358977	+	0.933347i	$0.383126\pi$
$992$	−17.9353	−0.569445
$993$	−17.6616	−0.560473
$994$	39.6862	1.25877
$995$	0	0
$996$	−8.64212	−0.273836
$997$	47.6670	1.50963	0.754814	−	0.655939i	$-0.227727\pi$
$997$	47.6670	1.50963	0.754814	+	0.655939i	$0.227727\pi$
$998$	−27.2455	−0.862441
$999$	−45.8319	−1.45006

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.ca.1.15		18
5.2	odd	4		845.2.b.g.339.15	yes	18
5.3	odd	4		845.2.b.g.339.4	✓	18
5.4	even	2	inner	4225.2.a.ca.1.4		18
13.12	even	2		4225.2.a.cb.1.4		18
65.2	even	12		845.2.l.g.654.8		72
65.3	odd	12		845.2.n.i.529.15		36
65.7	even	12		845.2.l.g.699.29		72
65.8	even	4		845.2.d.e.844.30		36
65.12	odd	4		845.2.b.h.339.4	yes	18
65.17	odd	12		845.2.n.h.484.4		36
65.18	even	4		845.2.d.e.844.8		36
65.22	odd	12		845.2.n.i.484.15		36
65.23	odd	12		845.2.n.h.529.4		36
65.28	even	12		845.2.l.g.654.29		72
65.32	even	12		845.2.l.g.699.7		72
65.33	even	12		845.2.l.g.699.8		72
65.37	even	12		845.2.l.g.654.30		72
65.38	odd	4		845.2.b.h.339.15	yes	18
65.42	odd	12		845.2.n.i.529.4		36
65.43	odd	12		845.2.n.h.484.15		36
65.47	even	4		845.2.d.e.844.7		36
65.48	odd	12		845.2.n.i.484.4		36
65.57	even	4		845.2.d.e.844.29		36
65.58	even	12		845.2.l.g.699.30		72
65.62	odd	12		845.2.n.h.529.15		36
65.63	even	12		845.2.l.g.654.7		72
65.64	even	2		4225.2.a.cb.1.15		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.4	✓	18	5.3	odd	4
845.2.b.g.339.15	yes	18	5.2	odd	4
845.2.b.h.339.4	yes	18	65.12	odd	4
845.2.b.h.339.15	yes	18	65.38	odd	4
845.2.d.e.844.7		36	65.47	even	4
845.2.d.e.844.8		36	65.18	even	4
845.2.d.e.844.29		36	65.57	even	4
845.2.d.e.844.30		36	65.8	even	4
845.2.l.g.654.7		72	65.63	even	12
845.2.l.g.654.8		72	65.2	even	12
845.2.l.g.654.29		72	65.28	even	12
845.2.l.g.654.30		72	65.37	even	12
845.2.l.g.699.7		72	65.32	even	12
845.2.l.g.699.8		72	65.33	even	12
845.2.l.g.699.29		72	65.7	even	12
845.2.l.g.699.30		72	65.58	even	12
845.2.n.h.484.4		36	65.17	odd	12
845.2.n.h.484.15		36	65.43	odd	12
845.2.n.h.529.4		36	65.23	odd	12
845.2.n.h.529.15		36	65.62	odd	12
845.2.n.i.484.4		36	65.48	odd	12
845.2.n.i.484.15		36	65.22	odd	12
845.2.n.i.529.4		36	65.42	odd	12
845.2.n.i.529.15		36	65.3	odd	12
4225.2.a.ca.1.4		18	5.4	even	2	inner
4225.2.a.ca.1.15		18	1.1	even	1	trivial
4225.2.a.cb.1.4		18	13.12	even	2
4225.2.a.cb.1.15		18	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.94561 q^{2} +0.752344 q^{3} +1.78540 q^{4} +1.46377 q^{6} -1.49584 q^{7} -0.417520 q^{8} -2.43398 q^{9} +O(q^{10})\)	\(q+1.94561 q^{2} +0.752344 q^{3} +1.78540 q^{4} +1.46377 q^{6} -1.49584 q^{7} -0.417520 q^{8} -2.43398 q^{9} +4.03104 q^{11} +1.34324 q^{12} -2.91033 q^{14} -4.38314 q^{16} -3.62825 q^{17} -4.73558 q^{18} -6.32047 q^{19} -1.12539 q^{21} +7.84283 q^{22} -0.957968 q^{23} -0.314119 q^{24} -4.08822 q^{27} -2.67069 q^{28} +1.88869 q^{29} +2.33142 q^{31} -7.69285 q^{32} +3.03273 q^{33} -7.05917 q^{34} -4.34563 q^{36} +11.2107 q^{37} -12.2972 q^{38} -11.3447 q^{41} -2.18957 q^{42} -7.50898 q^{43} +7.19703 q^{44} -1.86383 q^{46} -6.85846 q^{47} -3.29763 q^{48} -4.76245 q^{49} -2.72969 q^{51} +3.88728 q^{53} -7.95409 q^{54} +0.624544 q^{56} -4.75517 q^{57} +3.67465 q^{58} -5.27215 q^{59} -4.11814 q^{61} +4.53604 q^{62} +3.64085 q^{63} -6.20102 q^{64} +5.90051 q^{66} +11.4471 q^{67} -6.47789 q^{68} -0.720722 q^{69} -13.6363 q^{71} +1.01623 q^{72} -2.55718 q^{73} +21.8117 q^{74} -11.2846 q^{76} -6.02980 q^{77} +10.7412 q^{79} +4.22618 q^{81} -22.0724 q^{82} -6.43379 q^{83} -2.00928 q^{84} -14.6096 q^{86} +1.42094 q^{87} -1.68304 q^{88} -8.98382 q^{89} -1.71036 q^{92} +1.75403 q^{93} -13.3439 q^{94} -5.78767 q^{96} -15.4104 q^{97} -9.26588 q^{98} -9.81146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * q^99

Introduction

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