LMFDB - Embedded newform 4225.2.a.ca.1.4

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			1.4
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	$-0.741457\pi$
$2$	−1.94561	−1.37576	−0.687878	+	0.725827i	$0.741457\pi$
$3$	−0.752344	−0.434366	−0.217183	−	0.976131i	$-0.569687\pi$
$3$	−0.752344	−0.434366	−0.217183	+	0.976131i	$0.569687\pi$
$4$	1.78540	0.892702
$5$	0	0
$5$	0	0
$6$	1.46377	0.597582
$7$	1.49584	0.565376	0.282688	−	0.959212i	$-0.408774\pi$
$7$	1.49584	0.565376	0.282688	+	0.959212i	$0.408774\pi$
$8$	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	$-0.354982\pi$
$17$	3.62825	0.879980	0.439990	+	0.898003i	$0.354982\pi$
$18$	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	$-0.468156\pi$
$23$	0.957968	0.199750	0.0998750	+	0.995000i	$0.468156\pi$
$24$	−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	$-0.873047\pi$
$37$	−11.2107	−1.84303	−0.921515	+	0.388344i	$0.873047\pi$
$38$	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	$-0.305953\pi$
$43$	7.50898	1.14511	0.572554	+	0.819867i	$0.305953\pi$
$44$	7.19703	1.08499
$45$	0	0
$46$	−1.86383	−0.274807
$47$	6.85846	1.00041	0.500204	−	0.865907i	$-0.333258\pi$
$47$	6.85846	1.00041	0.500204	+	0.865907i	$0.333258\pi$
$48$	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	$-0.586025\pi$
$53$	−3.88728	−0.533958	−0.266979	+	0.963702i	$0.586025\pi$
$54$	−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	$-0.746479\pi$
$67$	−11.4471	−1.39848	−0.699242	+	0.714885i	$0.746479\pi$
$68$	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	$-0.452186\pi$
$73$	2.55718	0.299295	0.149647	+	0.988739i	$0.452186\pi$
$74$	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	$-0.385128\pi$
$83$	6.43379	0.706200	0.353100	+	0.935586i	$0.385128\pi$
$84$	−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	$-0.214024\pi$
$97$	15.4104	1.56469	0.782344	+	0.622847i	$0.214024\pi$
$98$	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	$-0.640926\pi$
$103$	−8.69574	−0.856817	−0.428409	+	0.903585i	$0.640926\pi$
$104$	0	0
$105$	0	0
$106$	7.56313	0.734596
$107$	10.8203	1.04604	0.523020	−	0.852321i	$-0.324805\pi$
$107$	10.8203	1.04604	0.523020	+	0.852321i	$0.324805\pi$
$108$	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	$-0.887907\pi$
$113$	−19.9556	−1.87727	−0.938633	+	0.344917i	$0.887907\pi$
$114$	−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	$-0.415500\pi$
$127$	5.91326	0.524717	0.262359	+	0.964970i	$0.415500\pi$
$128$	−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	$-0.531295\pi$
$137$	−2.29782	−0.196316	−0.0981580	+	0.995171i	$0.531295\pi$
$138$	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	$-0.437571\pi$
$157$	4.88349	0.389745	0.194873	+	0.980829i	$0.437571\pi$
$158$	−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	$-0.537247\pi$
$163$	−2.98107	−0.233495	−0.116748	+	0.993162i	$0.537247\pi$
$164$	−20.2549	−1.58164
$165$	0	0
$166$	−12.5177	−0.971559
$167$	−13.7982	−1.06774	−0.533870	−	0.845566i	$-0.679263\pi$
$167$	−13.7982	−1.06774	−0.533870	+	0.845566i	$0.679263\pi$
$168$	−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	$-0.530681\pi$
$173$	−2.53163	−0.192476	−0.0962380	+	0.995358i	$0.530681\pi$
$174$	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	$-0.504448\pi$
$193$	−0.388224	−0.0279450	−0.0139725	+	0.999902i	$0.504448\pi$
$194$	−29.9826	−2.15263
$195$	0	0
$196$	−8.50290	−0.607350
$197$	14.0420	1.00045	0.500227	−	0.865894i	$-0.333250\pi$
$197$	14.0420	1.00045	0.500227	+	0.865894i	$0.333250\pi$
$198$	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	$-0.234408\pi$
$223$	22.1274	1.48176	0.740882	+	0.671635i	$0.234408\pi$
$224$	11.5073	0.768864
$225$	0	0
$226$	38.8259	2.58266
$227$	−15.5953	−1.03510	−0.517548	−	0.855654i	$-0.673155\pi$
$227$	−15.5953	−1.03510	−0.517548	+	0.855654i	$0.673155\pi$
$228$	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	$-0.645790\pi$
$233$	−13.4988	−0.884334	−0.442167	+	0.896933i	$0.645790\pi$
$234$	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	$-0.385316\pi$
$257$	11.3035	0.705093	0.352546	+	0.935794i	$0.385316\pi$
$258$	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	$-0.685053\pi$
$263$	−17.8119	−1.09833	−0.549163	+	0.835715i	$0.685053\pi$
$264$	−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	$-0.502001\pi$
$277$	−0.209275	−0.0125741	−0.00628704	+	0.999980i	$0.502001\pi$
$278$	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	$-0.158738\pi$
$283$	29.5476	1.75642	0.878211	+	0.478274i	$0.158738\pi$
$284$	−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	$-0.759656\pi$
$293$	−24.9305	−1.45646	−0.728229	+	0.685334i	$0.759656\pi$
$294$	−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	$-0.402760\pi$
$307$	10.5394	0.601515	0.300758	+	0.953701i	$0.402760\pi$
$308$	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	$-0.581763\pi$
$313$	−8.98921	−0.508100	−0.254050	+	0.967191i	$0.581763\pi$
$314$	−9.50138	−0.536194
$315$	0	0
$316$	19.1774	1.07881
$317$	−16.7107	−0.938568	−0.469284	−	0.883047i	$-0.655488\pi$
$317$	−16.7107	−0.938568	−0.469284	+	0.883047i	$0.655488\pi$
$318$	−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	$-0.680037\pi$
$337$	−19.6766	−1.07185	−0.535925	+	0.844266i	$0.680037\pi$
$338$	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	$-0.419921\pi$
$347$	9.27409	0.497859	0.248930	+	0.968522i	$0.419921\pi$
$348$	−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	$-0.563518\pi$
$353$	−7.44869	−0.396454	−0.198227	+	0.980156i	$0.563518\pi$
$354$	−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	$-0.682384\pi$
$367$	−20.7716	−1.08427	−0.542135	+	0.840291i	$0.682384\pi$
$368$	−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	$-0.439637\pi$
$373$	7.28118	0.377005	0.188503	+	0.982073i	$0.439637\pi$
$374$	−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	$-0.740263\pi$
$383$	−26.8173	−1.37030	−0.685149	+	0.728403i	$0.740263\pi$
$384$	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	$-0.549534\pi$
$397$	−6.17624	−0.309976	−0.154988	+	0.987916i	$0.549534\pi$
$398$	−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	$-0.705388\pi$
$433$	−25.0285	−1.20279	−0.601395	+	0.798952i	$0.705388\pi$
$434$	−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	$-0.583546\pi$
$443$	−10.9222	−0.518930	−0.259465	+	0.965753i	$0.583546\pi$
$444$	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	$-0.279654\pi$
$457$	27.2889	1.27652	0.638261	+	0.769820i	$0.279654\pi$
$458$	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	$-0.730412\pi$
$463$	−28.5012	−1.32456	−0.662281	+	0.749255i	$0.730412\pi$
$464$	−8.27837	−0.384314
$465$	0	0
$466$	26.2634	1.21663
$467$	−28.0713	−1.29899	−0.649493	−	0.760368i	$-0.725019\pi$
$467$	−28.0713	−1.29899	−0.649493	+	0.760368i	$0.725019\pi$
$468$	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	$-0.529193\pi$
$487$	−4.04221	−0.183170	−0.0915849	+	0.995797i	$0.529193\pi$
$488$	−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	$-0.233144\pi$
$503$	33.3519	1.48709	0.743543	+	0.668688i	$0.233144\pi$
$504$	−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	$-0.860638\pi$
$523$	−41.4243	−1.81136	−0.905679	+	0.423965i	$0.860638\pi$
$524$	−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	$-0.585147\pi$
$547$	−12.3638	−0.528636	−0.264318	+	0.964436i	$0.585147\pi$
$548$	−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	$-0.311479\pi$
$557$	26.3496	1.11647	0.558233	+	0.829684i	$0.311479\pi$
$558$	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	$-0.479382\pi$
$563$	3.07168	0.129456	0.0647279	+	0.997903i	$0.479382\pi$
$564$	−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	$-0.583019\pi$
$577$	−12.3883	−0.515733	−0.257866	+	0.966181i	$0.583019\pi$
$578$	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	$-0.461706\pi$
$587$	5.81541	0.240027	0.120014	+	0.992772i	$0.461706\pi$
$588$	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	$-0.211526\pi$
$593$	38.3395	1.57441	0.787207	+	0.616688i	$0.211526\pi$
$594$	−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	$-0.520313\pi$
$607$	−3.14241	−0.127547	−0.0637733	+	0.997964i	$0.520313\pi$
$608$	−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	$-0.358712\pi$
$613$	21.2647	0.858873	0.429437	+	0.903097i	$0.358712\pi$
$614$	−20.5056	−0.827538
$615$	0	0
$616$	2.51756	0.101435
$617$	−20.8157	−0.838010	−0.419005	−	0.907984i	$-0.637621\pi$
$617$	−20.8157	−0.838010	−0.419005	+	0.907984i	$0.637621\pi$
$618$	−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	$-0.285664\pi$
$643$	31.6264	1.24723	0.623613	+	0.781734i	$0.285664\pi$
$644$	2.55843	0.100816
$645$	0	0
$646$	44.6172	1.75544
$647$	−35.7326	−1.40479	−0.702397	−	0.711785i	$-0.747887\pi$
$647$	−35.7326	−1.40479	−0.702397	+	0.711785i	$0.747887\pi$
$648$	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	$-0.497415\pi$
$653$	0.415100	0.0162441	0.00812206	+	0.999967i	$0.497415\pi$
$654$	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	$-0.388078\pi$
$673$	17.8697	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$674$	38.2829	1.47460
$675$	0	0
$676$	0	0
$677$	4.13330	0.158856	0.0794279	−	0.996841i	$-0.474691\pi$
$677$	4.13330	0.158856	0.0794279	+	0.996841i	$0.474691\pi$
$678$	−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	$-0.326304\pi$
$683$	27.1275	1.03800	0.519002	+	0.854773i	$0.326304\pi$
$684$	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	$-0.317641\pi$
$727$	29.2316	1.08414	0.542069	+	0.840334i	$0.317641\pi$
$728$	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	$-0.631669\pi$
$733$	−21.7650	−0.803909	−0.401954	+	0.915660i	$0.631669\pi$
$734$	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	$-0.525001\pi$
$743$	−4.27738	−0.156922	−0.0784608	+	0.996917i	$0.525001\pi$
$744$	−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	$-0.694485\pi$
$757$	−31.5680	−1.14736	−0.573680	+	0.819080i	$0.694485\pi$
$758$	−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	$-0.557245\pi$
$773$	−9.94629	−0.357743	−0.178872	+	0.983872i	$0.557245\pi$
$774$	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	$-0.848731\pi$
$787$	−49.8898	−1.77838	−0.889189	+	0.457540i	$0.848731\pi$
$788$	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	$-0.311977\pi$
$797$	31.4459	1.11387	0.556936	+	0.830555i	$0.311977\pi$
$798$	−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	$-0.266667\pi$
$823$	38.3919	1.33826	0.669129	+	0.743146i	$0.266667\pi$
$824$	−3.63064	−0.126479
$825$	0	0
$826$	15.3437	0.533876
$827$	26.5289	0.922499	0.461250	−	0.887270i	$-0.347401\pi$
$827$	26.5289	0.922499	0.461250	+	0.887270i	$0.347401\pi$
$828$	−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	$-0.519019\pi$
$853$	−3.48801	−0.119427	−0.0597136	+	0.998216i	$0.519019\pi$
$854$	11.9852	0.410124
$855$	0	0
$856$	4.51769	0.154412
$857$	28.0577	0.958432	0.479216	−	0.877697i	$-0.340921\pi$
$857$	28.0577	0.958432	0.479216	+	0.877697i	$0.340921\pi$
$858$	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	$-0.477462\pi$
$863$	4.15653	0.141490	0.0707449	+	0.997494i	$0.477462\pi$
$864$	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	$-0.153674\pi$
$877$	52.4590	1.77141	0.885707	+	0.464246i	$0.153674\pi$
$878$	−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	$-0.382281\pi$
$883$	21.4813	0.722904	0.361452	+	0.932391i	$0.382281\pi$
$884$	0	0
$885$	0	0
$886$	21.2504	0.713921
$887$	23.5601	0.791069	0.395535	−	0.918451i	$-0.370559\pi$
$887$	23.5601	0.791069	0.395535	+	0.918451i	$0.370559\pi$
$888$	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	$-0.529208\pi$
$907$	−5.51921	−0.183262	−0.0916312	+	0.995793i	$0.529208\pi$
$908$	−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	$-0.618347\pi$
$937$	−22.2410	−0.726581	−0.363291	+	0.931676i	$0.618347\pi$
$938$	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	$-0.296480\pi$
$947$	36.7247	1.19339	0.596696	+	0.802468i	$0.296480\pi$
$948$	−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	$-0.663298\pi$
$953$	−30.3031	−0.981613	−0.490806	+	0.871269i	$0.663298\pi$
$954$	−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	$-0.337920\pi$
$967$	30.3173	0.974939	0.487469	+	0.873140i	$0.337920\pi$
$968$	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	$-0.604049\pi$
$977$	−20.0726	−0.642179	−0.321089	+	0.947049i	$0.604049\pi$
$978$	−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	$-0.314483\pi$
$983$	34.5118	1.10076	0.550378	+	0.834915i	$0.314483\pi$
$984$	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	$-0.772273\pi$
$997$	−47.6670	−1.50963	−0.754814	+	0.655939i	$0.772273\pi$
$998$	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.1.4		18
5.2	odd	4		845.2.b.g.339.4	✓	18
5.3	odd	4		845.2.b.g.339.15	yes	18
5.4	even	2	inner	4225.2.a.ca.1.15		18
13.12	even	2		4225.2.a.cb.1.15		18
65.2	even	12		845.2.l.g.654.29		72
65.3	odd	12		845.2.n.i.529.4		36
65.7	even	12		845.2.l.g.699.8		72
65.8	even	4		845.2.d.e.844.7		36
65.12	odd	4		845.2.b.h.339.15	yes	18
65.17	odd	12		845.2.n.h.484.15		36
65.18	even	4		845.2.d.e.844.29		36
65.22	odd	12		845.2.n.i.484.4		36
65.23	odd	12		845.2.n.h.529.15		36
65.28	even	12		845.2.l.g.654.8		72
65.32	even	12		845.2.l.g.699.30		72
65.33	even	12		845.2.l.g.699.29		72
65.37	even	12		845.2.l.g.654.7		72
65.38	odd	4		845.2.b.h.339.4	yes	18
65.42	odd	12		845.2.n.i.529.15		36
65.43	odd	12		845.2.n.h.484.4		36
65.47	even	4		845.2.d.e.844.30		36
65.48	odd	12		845.2.n.i.484.15		36
65.57	even	4		845.2.d.e.844.8		36
65.58	even	12		845.2.l.g.699.7		72
65.62	odd	12		845.2.n.h.529.4		36
65.63	even	12		845.2.l.g.654.30		72
65.64	even	2		4225.2.a.cb.1.4		18

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