LMFDB - Embedded newform 5445.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.00000 q^{7} +2.23607 q^{8} -0.618034 q^{10} -2.47214 q^{13} +1.23607 q^{14} +1.85410 q^{16} +3.47214 q^{17} -1.61803 q^{20} -3.76393 q^{23} +1.00000 q^{25} +1.52786 q^{26} +3.23607 q^{28} +8.47214 q^{29} -6.70820 q^{31} -5.61803 q^{32} -2.14590 q^{34} -2.00000 q^{35} -0.472136 q^{37} +2.23607 q^{40} +6.00000 q^{41} -6.00000 q^{43} +2.32624 q^{46} -8.23607 q^{47} -3.00000 q^{49} -0.618034 q^{50} +4.00000 q^{52} +5.94427 q^{53} -4.47214 q^{56} -5.23607 q^{58} +6.94427 q^{59} +3.47214 q^{61} +4.14590 q^{62} -0.236068 q^{64} -2.47214 q^{65} +11.4164 q^{67} -5.61803 q^{68} +1.23607 q^{70} +4.47214 q^{71} +4.94427 q^{73} +0.291796 q^{74} -2.70820 q^{79} +1.85410 q^{80} -3.70820 q^{82} +3.47214 q^{85} +3.70820 q^{86} -8.47214 q^{89} +4.94427 q^{91} +6.09017 q^{92} +5.09017 q^{94} +2.47214 q^{97} +1.85410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7	$ 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + q^{10} + 4 q^{13} - 2 q^{14} - 3 q^{16} - 2 q^{17} - q^{20} - 12 q^{23} + 2 q^{25} + 12 q^{26} + 2 q^{28} + 8 q^{29} - 9 q^{32} - 11 q^{34} - 4 q^{35} + 8 q^{37} + 12 q^{41} - 12 q^{43} - 11 q^{46} - 12 q^{47} - 6 q^{49} + q^{50} + 8 q^{52} - 6 q^{53} - 6 q^{58} - 4 q^{59} - 2 q^{61} + 15 q^{62} + 4 q^{64} + 4 q^{65} - 4 q^{67} - 9 q^{68} - 2 q^{70} - 8 q^{73} + 14 q^{74} + 8 q^{79} - 3 q^{80} + 6 q^{82} - 2 q^{85} - 6 q^{86} - 8 q^{89} - 8 q^{91} + q^{92} - q^{94} - 4 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7 + q^10 + 4 * q^13 - 2 * q^14 - 3 * q^16 - 2 * q^17 - q^20 - 12 * q^23 + 2 * q^25 + 12 * q^26 + 2 * q^28 + 8 * q^29 - 9 * q^32 - 11 * q^34 - 4 * q^35 + 8 * q^37 + 12 * q^41 - 12 * q^43 - 11 * q^46 - 12 * q^47 - 6 * q^49 + q^50 + 8 * q^52 - 6 * q^53 - 6 * q^58 - 4 * q^59 - 2 * q^61 + 15 * q^62 + 4 * q^64 + 4 * q^65 - 4 * q^67 - 9 * q^68 - 2 * q^70 - 8 * q^73 + 14 * q^74 + 8 * q^79 - 3 * q^80 + 6 * q^82 - 2 * q^85 - 6 * q^86 - 8 * q^89 - 8 * q^91 + q^92 - q^94 - 4 * q^97 - 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−0.618034	−0.195440
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	1.23607	0.330353
$15$	0	0
$16$	1.85410	0.463525
$17$	3.47214	0.842117	0.421058	−	0.907034i	$-0.361659\pi$
$17$	3.47214	0.842117	0.421058	+	0.907034i	$0.361659\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	0	0
$23$	−3.76393	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$23$	−3.76393	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.52786	0.299639
$27$	0	0
$28$	3.23607	0.611559
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−2.14590	−0.368018
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−0.472136	−0.0776187	−0.0388093	−	0.999247i	$-0.512356\pi$
$37$	−0.472136	−0.0776187	−0.0388093	+	0.999247i	$0.512356\pi$
$38$	0	0
$39$	0	0
$40$	2.23607	0.353553
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	2.32624	0.342985
$47$	−8.23607	−1.20135	−0.600677	−	0.799492i	$-0.705102\pi$
$47$	−8.23607	−1.20135	−0.600677	+	0.799492i	$0.705102\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−0.618034	−0.0874032
$51$	0	0
$52$	4.00000	0.554700
$53$	5.94427	0.816509	0.408254	−	0.912868i	$-0.366138\pi$
$53$	5.94427	0.816509	0.408254	+	0.912868i	$0.366138\pi$
$54$	0	0
$55$	0	0
$56$	−4.47214	−0.597614
$57$	0	0
$58$	−5.23607	−0.687529
$59$	6.94427	0.904067	0.452034	−	0.892001i	$-0.350699\pi$
$59$	6.94427	0.904067	0.452034	+	0.892001i	$0.350699\pi$
$60$	0	0
$61$	3.47214	0.444561	0.222281	−	0.974983i	$-0.428650\pi$
$61$	3.47214	0.444561	0.222281	+	0.974983i	$0.428650\pi$
$62$	4.14590	0.526530
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−2.47214	−0.306631
$66$	0	0
$67$	11.4164	1.39474	0.697368	−	0.716713i	$-0.254354\pi$
$67$	11.4164	1.39474	0.697368	+	0.716713i	$0.254354\pi$
$68$	−5.61803	−0.681287
$69$	0	0
$70$	1.23607	0.147738
$71$	4.47214	0.530745	0.265372	−	0.964146i	$-0.414505\pi$
$71$	4.47214	0.530745	0.265372	+	0.964146i	$0.414505\pi$
$72$	0	0
$73$	4.94427	0.578683	0.289342	−	0.957226i	$-0.406564\pi$
$73$	4.94427	0.578683	0.289342	+	0.957226i	$0.406564\pi$
$74$	0.291796	0.0339206
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.70820	−0.304697	−0.152348	−	0.988327i	$-0.548684\pi$
$79$	−2.70820	−0.304697	−0.152348	+	0.988327i	$0.548684\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	−3.70820	−0.409503
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	3.47214	0.376606
$86$	3.70820	0.399866
$87$	0	0
$88$	0	0
$89$	−8.47214	−0.898045	−0.449022	−	0.893521i	$-0.648228\pi$
$89$	−8.47214	−0.898045	−0.449022	+	0.893521i	$0.648228\pi$
$90$	0	0
$91$	4.94427	0.518301
$92$	6.09017	0.634944
$93$	0	0
$94$	5.09017	0.525011
$95$	0	0
$96$	0	0
$97$	2.47214	0.251007	0.125504	−	0.992093i	$-0.459945\pi$
$97$	2.47214	0.251007	0.125504	+	0.992093i	$0.459945\pi$
$98$	1.85410	0.187293
$99$	0	0
$100$	−1.61803	−0.161803
$101$	−18.9443	−1.88503	−0.942513	−	0.334170i	$-0.891544\pi$
$101$	−18.9443	−1.88503	−0.942513	+	0.334170i	$0.891544\pi$
$102$	0	0
$103$	−12.4721	−1.22892	−0.614458	−	0.788950i	$-0.710625\pi$
$103$	−12.4721	−1.22892	−0.614458	+	0.788950i	$0.710625\pi$
$104$	−5.52786	−0.542052
$105$	0	0
$106$	−3.67376	−0.356827
$107$	6.23607	0.602863	0.301432	−	0.953488i	$-0.402535\pi$
$107$	6.23607	0.602863	0.301432	+	0.953488i	$0.402535\pi$
$108$	0	0
$109$	18.9443	1.81453	0.907266	−	0.420557i	$-0.138165\pi$
$109$	18.9443	1.81453	0.907266	+	0.420557i	$0.138165\pi$
$110$	0	0
$111$	0	0
$112$	−3.70820	−0.350392
$113$	−15.4721	−1.45550	−0.727748	−	0.685845i	$-0.759433\pi$
$113$	−15.4721	−1.45550	−0.727748	+	0.685845i	$0.759433\pi$
$114$	0	0
$115$	−3.76393	−0.350988
$116$	−13.7082	−1.27277
$117$	0	0
$118$	−4.29180	−0.395092
$119$	−6.94427	−0.636580
$120$	0	0
$121$	0	0
$122$	−2.14590	−0.194280
$123$	0	0
$124$	10.8541	0.974727
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.47214	−0.574309	−0.287155	−	0.957884i	$-0.592709\pi$
$127$	−6.47214	−0.574309	−0.287155	+	0.957884i	$0.592709\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	1.52786	0.134003
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	−7.05573	−0.609522
$135$	0	0
$136$	7.76393	0.665752
$137$	−16.4164	−1.40255	−0.701274	−	0.712892i	$-0.747385\pi$
$137$	−16.4164	−1.40255	−0.701274	+	0.712892i	$0.747385\pi$
$138$	0	0
$139$	8.70820	0.738620	0.369310	−	0.929306i	$-0.379594\pi$
$139$	8.70820	0.738620	0.369310	+	0.929306i	$0.379594\pi$
$140$	3.23607	0.273498
$141$	0	0
$142$	−2.76393	−0.231944
$143$	0	0
$144$	0	0
$145$	8.47214	0.703573
$146$	−3.05573	−0.252894
$147$	0	0
$148$	0.763932	0.0627948
$149$	−1.52786	−0.125167	−0.0625837	−	0.998040i	$-0.519934\pi$
$149$	−1.52786	−0.125167	−0.0625837	+	0.998040i	$0.519934\pi$
$150$	0	0
$151$	6.70820	0.545906	0.272953	−	0.962027i	$-0.412000\pi$
$151$	6.70820	0.545906	0.272953	+	0.962027i	$0.412000\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.70820	−0.538816
$156$	0	0
$157$	11.4164	0.911129	0.455564	−	0.890203i	$-0.349438\pi$
$157$	11.4164	0.911129	0.455564	+	0.890203i	$0.349438\pi$
$158$	1.67376	0.133157
$159$	0	0
$160$	−5.61803	−0.444145
$161$	7.52786	0.593279
$162$	0	0
$163$	−11.4164	−0.894202	−0.447101	−	0.894483i	$-0.647544\pi$
$163$	−11.4164	−0.894202	−0.447101	+	0.894483i	$0.647544\pi$
$164$	−9.70820	−0.758083
$165$	0	0
$166$	0	0
$167$	−11.2918	−0.873785	−0.436893	−	0.899514i	$-0.643921\pi$
$167$	−11.2918	−0.873785	−0.436893	+	0.899514i	$0.643921\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	−2.14590	−0.164583
$171$	0	0
$172$	9.70820	0.740244
$173$	5.05573	0.384380	0.192190	−	0.981358i	$-0.438441\pi$
$173$	5.05573	0.384380	0.192190	+	0.981358i	$0.438441\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	5.23607	0.392460
$179$	−26.4721	−1.97862	−0.989310	−	0.145827i	$-0.953416\pi$
$179$	−26.4721	−1.97862	−0.989310	+	0.145827i	$0.953416\pi$
$180$	0	0
$181$	18.9443	1.40812	0.704058	−	0.710142i	$-0.251369\pi$
$181$	18.9443	1.40812	0.704058	+	0.710142i	$0.251369\pi$
$182$	−3.05573	−0.226506
$183$	0	0
$184$	−8.41641	−0.620466
$185$	−0.472136	−0.0347121
$186$	0	0
$187$	0	0
$188$	13.3262	0.971916
$189$	0	0
$190$	0	0
$191$	−18.9443	−1.37076	−0.685380	−	0.728186i	$-0.740364\pi$
$191$	−18.9443	−1.37076	−0.685380	+	0.728186i	$0.740364\pi$
$192$	0	0
$193$	−2.47214	−0.177948	−0.0889741	−	0.996034i	$-0.528359\pi$
$193$	−2.47214	−0.177948	−0.0889741	+	0.996034i	$0.528359\pi$
$194$	−1.52786	−0.109694
$195$	0	0
$196$	4.85410	0.346722
$197$	22.9443	1.63471	0.817356	−	0.576133i	$-0.195439\pi$
$197$	22.9443	1.63471	0.817356	+	0.576133i	$0.195439\pi$
$198$	0	0
$199$	−19.1803	−1.35966	−0.679829	−	0.733371i	$-0.737946\pi$
$199$	−19.1803	−1.35966	−0.679829	+	0.733371i	$0.737946\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	11.7082	0.823786
$203$	−16.9443	−1.18925
$204$	0	0
$205$	6.00000	0.419058
$206$	7.70820	0.537056
$207$	0	0
$208$	−4.58359	−0.317815
$209$	0	0
$210$	0	0
$211$	−16.7082	−1.15024	−0.575120	−	0.818069i	$-0.695045\pi$
$211$	−16.7082	−1.15024	−0.575120	+	0.818069i	$0.695045\pi$
$212$	−9.61803	−0.660569
$213$	0	0
$214$	−3.85410	−0.263461
$215$	−6.00000	−0.409197
$216$	0	0
$217$	13.4164	0.910765
$218$	−11.7082	−0.792980
$219$	0	0
$220$	0	0
$221$	−8.58359	−0.577395
$222$	0	0
$223$	−15.5279	−1.03982	−0.519911	−	0.854220i	$-0.674035\pi$
$223$	−15.5279	−1.03982	−0.519911	+	0.854220i	$0.674035\pi$
$224$	11.2361	0.750741
$225$	0	0
$226$	9.56231	0.636075
$227$	15.1803	1.00755	0.503777	−	0.863834i	$-0.331943\pi$
$227$	15.1803	1.00755	0.503777	+	0.863834i	$0.331943\pi$
$228$	0	0
$229$	0.527864	0.0348822	0.0174411	−	0.999848i	$-0.494448\pi$
$229$	0.527864	0.0348822	0.0174411	+	0.999848i	$0.494448\pi$
$230$	2.32624	0.153388
$231$	0	0
$232$	18.9443	1.24375
$233$	−27.3607	−1.79246	−0.896229	−	0.443592i	$-0.853704\pi$
$233$	−27.3607	−1.79246	−0.896229	+	0.443592i	$0.853704\pi$
$234$	0	0
$235$	−8.23607	−0.537262
$236$	−11.2361	−0.731406
$237$	0	0
$238$	4.29180	0.278196
$239$	16.3607	1.05828	0.529142	−	0.848533i	$-0.322514\pi$
$239$	16.3607	1.05828	0.529142	+	0.848533i	$0.322514\pi$
$240$	0	0
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	0	0
$243$	0	0
$244$	−5.61803	−0.359658
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	−15.0000	−0.952501
$249$	0	0
$250$	−0.618034	−0.0390879
$251$	−19.4164	−1.22555	−0.612776	−	0.790256i	$-0.709947\pi$
$251$	−19.4164	−1.22555	−0.612776	+	0.790256i	$0.709947\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	−6.56231	−0.410144
$257$	8.52786	0.531954	0.265977	−	0.963979i	$-0.414306\pi$
$257$	8.52786	0.531954	0.265977	+	0.963979i	$0.414306\pi$
$258$	0	0
$259$	0.944272	0.0586742
$260$	4.00000	0.248069
$261$	0	0
$262$	2.76393	0.170756
$263$	0.708204	0.0436697	0.0218349	−	0.999762i	$-0.493049\pi$
$263$	0.708204	0.0436697	0.0218349	+	0.999762i	$0.493049\pi$
$264$	0	0
$265$	5.94427	0.365154
$266$	0	0
$267$	0	0
$268$	−18.4721	−1.12837
$269$	17.8885	1.09068	0.545342	−	0.838214i	$-0.316400\pi$
$269$	17.8885	1.09068	0.545342	+	0.838214i	$0.316400\pi$
$270$	0	0
$271$	−1.76393	−0.107151	−0.0535756	−	0.998564i	$-0.517062\pi$
$271$	−1.76393	−0.107151	−0.0535756	+	0.998564i	$0.517062\pi$
$272$	6.43769	0.390343
$273$	0	0
$274$	10.1459	0.612936
$275$	0	0
$276$	0	0
$277$	6.94427	0.417241	0.208620	−	0.977997i	$-0.433103\pi$
$277$	6.94427	0.417241	0.208620	+	0.977997i	$0.433103\pi$
$278$	−5.38197	−0.322789
$279$	0	0
$280$	−4.47214	−0.267261
$281$	−16.3607	−0.975996	−0.487998	−	0.872845i	$-0.662273\pi$
$281$	−16.3607	−0.975996	−0.487998	+	0.872845i	$0.662273\pi$
$282$	0	0
$283$	−30.3607	−1.80476	−0.902378	−	0.430946i	$-0.858180\pi$
$283$	−30.3607	−1.80476	−0.902378	+	0.430946i	$0.858180\pi$
$284$	−7.23607	−0.429382
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−4.94427	−0.290840
$290$	−5.23607	−0.307472
$291$	0	0
$292$	−8.00000	−0.468165
$293$	−5.94427	−0.347268	−0.173634	−	0.984810i	$-0.555551\pi$
$293$	−5.94427	−0.347268	−0.173634	+	0.984810i	$0.555551\pi$
$294$	0	0
$295$	6.94427	0.404311
$296$	−1.05573	−0.0613629
$297$	0	0
$298$	0.944272	0.0547002
$299$	9.30495	0.538119
$300$	0	0
$301$	12.0000	0.691669
$302$	−4.14590	−0.238570
$303$	0	0
$304$	0	0
$305$	3.47214	0.198814
$306$	0	0
$307$	−14.3607	−0.819607	−0.409804	−	0.912174i	$-0.634403\pi$
$307$	−14.3607	−0.819607	−0.409804	+	0.912174i	$0.634403\pi$
$308$	0	0
$309$	0	0
$310$	4.14590	0.235471
$311$	−26.4721	−1.50110	−0.750549	−	0.660815i	$-0.770211\pi$
$311$	−26.4721	−1.50110	−0.750549	+	0.660815i	$0.770211\pi$
$312$	0	0
$313$	−3.52786	−0.199407	−0.0997033	−	0.995017i	$-0.531789\pi$
$313$	−3.52786	−0.199407	−0.0997033	+	0.995017i	$0.531789\pi$
$314$	−7.05573	−0.398178
$315$	0	0
$316$	4.38197	0.246505
$317$	−10.0557	−0.564786	−0.282393	−	0.959299i	$-0.591128\pi$
$317$	−10.0557	−0.564786	−0.282393	+	0.959299i	$0.591128\pi$
$318$	0	0
$319$	0	0
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	−4.65248	−0.259272
$323$	0	0
$324$	0	0
$325$	−2.47214	−0.137129
$326$	7.05573	0.390781
$327$	0	0
$328$	13.4164	0.740797
$329$	16.4721	0.908138
$330$	0	0
$331$	19.7639	1.08632	0.543162	−	0.839628i	$-0.317227\pi$
$331$	19.7639	1.08632	0.543162	+	0.839628i	$0.317227\pi$
$332$	0	0
$333$	0	0
$334$	6.97871	0.381858
$335$	11.4164	0.623745
$336$	0	0
$337$	−24.0000	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$338$	4.25735	0.231570
$339$	0	0
$340$	−5.61803	−0.304681
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	−13.4164	−0.723364
$345$	0	0
$346$	−3.12461	−0.167980
$347$	−8.12461	−0.436152	−0.218076	−	0.975932i	$-0.569978\pi$
$347$	−8.12461	−0.436152	−0.218076	+	0.975932i	$0.569978\pi$
$348$	0	0
$349$	−3.58359	−0.191825	−0.0959126	−	0.995390i	$-0.530577\pi$
$349$	−3.58359	−0.191825	−0.0959126	+	0.995390i	$0.530577\pi$
$350$	1.23607	0.0660706
$351$	0	0
$352$	0	0
$353$	−20.5279	−1.09259	−0.546294	−	0.837594i	$-0.683962\pi$
$353$	−20.5279	−1.09259	−0.546294	+	0.837594i	$0.683962\pi$
$354$	0	0
$355$	4.47214	0.237356
$356$	13.7082	0.726533
$357$	0	0
$358$	16.3607	0.864689
$359$	19.5279	1.03064	0.515321	−	0.856997i	$-0.327673\pi$
$359$	19.5279	1.03064	0.515321	+	0.856997i	$0.327673\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−11.7082	−0.615370
$363$	0	0
$364$	−8.00000	−0.419314
$365$	4.94427	0.258795
$366$	0	0
$367$	31.4164	1.63992	0.819962	−	0.572419i	$-0.193995\pi$
$367$	31.4164	1.63992	0.819962	+	0.572419i	$0.193995\pi$
$368$	−6.97871	−0.363791
$369$	0	0
$370$	0.291796	0.0151698
$371$	−11.8885	−0.617222
$372$	0	0
$373$	−37.8885	−1.96179	−0.980897	−	0.194527i	$-0.937683\pi$
$373$	−37.8885	−1.96179	−0.980897	+	0.194527i	$0.937683\pi$
$374$	0	0
$375$	0	0
$376$	−18.4164	−0.949754
$377$	−20.9443	−1.07868
$378$	0	0
$379$	22.5967	1.16072	0.580358	−	0.814361i	$-0.302912\pi$
$379$	22.5967	1.16072	0.580358	+	0.814361i	$0.302912\pi$
$380$	0	0
$381$	0	0
$382$	11.7082	0.599044
$383$	−7.05573	−0.360531	−0.180265	−	0.983618i	$-0.557696\pi$
$383$	−7.05573	−0.360531	−0.180265	+	0.983618i	$0.557696\pi$
$384$	0	0
$385$	0	0
$386$	1.52786	0.0777662
$387$	0	0
$388$	−4.00000	−0.203069
$389$	−8.47214	−0.429554	−0.214777	−	0.976663i	$-0.568903\pi$
$389$	−8.47214	−0.429554	−0.214777	+	0.976663i	$0.568903\pi$
$390$	0	0
$391$	−13.0689	−0.660922
$392$	−6.70820	−0.338815
$393$	0	0
$394$	−14.1803	−0.714395
$395$	−2.70820	−0.136265
$396$	0	0
$397$	−0.944272	−0.0473916	−0.0236958	−	0.999719i	$-0.507543\pi$
$397$	−0.944272	−0.0473916	−0.0236958	+	0.999719i	$0.507543\pi$
$398$	11.8541	0.594192
$399$	0	0
$400$	1.85410	0.0927051
$401$	−17.8885	−0.893311	−0.446656	−	0.894706i	$-0.647385\pi$
$401$	−17.8885	−0.893311	−0.446656	+	0.894706i	$0.647385\pi$
$402$	0	0
$403$	16.5836	0.826088
$404$	30.6525	1.52502
$405$	0	0
$406$	10.4721	0.519723
$407$	0	0
$408$	0	0
$409$	−3.00000	−0.148340	−0.0741702	−	0.997246i	$-0.523631\pi$
$409$	−3.00000	−0.148340	−0.0741702	+	0.997246i	$0.523631\pi$
$410$	−3.70820	−0.183135
$411$	0	0
$412$	20.1803	0.994214
$413$	−13.8885	−0.683411
$414$	0	0
$415$	0	0
$416$	13.8885	0.680942
$417$	0	0
$418$	0	0
$419$	−39.7771	−1.94324	−0.971619	−	0.236552i	$-0.923983\pi$
$419$	−39.7771	−1.94324	−0.971619	+	0.236552i	$0.923983\pi$
$420$	0	0
$421$	7.58359	0.369602	0.184801	−	0.982776i	$-0.440836\pi$
$421$	7.58359	0.369602	0.184801	+	0.982776i	$0.440836\pi$
$422$	10.3262	0.502673
$423$	0	0
$424$	13.2918	0.645507
$425$	3.47214	0.168423
$426$	0	0
$427$	−6.94427	−0.336057
$428$	−10.0902	−0.487727
$429$	0	0
$430$	3.70820	0.178825
$431$	−11.8885	−0.572651	−0.286326	−	0.958132i	$-0.592434\pi$
$431$	−11.8885	−0.572651	−0.286326	+	0.958132i	$0.592434\pi$
$432$	0	0
$433$	−36.4721	−1.75274	−0.876369	−	0.481639i	$-0.840042\pi$
$433$	−36.4721	−1.75274	−0.876369	+	0.481639i	$0.840042\pi$
$434$	−8.29180	−0.398019
$435$	0	0
$436$	−30.6525	−1.46799
$437$	0	0
$438$	0	0
$439$	2.70820	0.129256	0.0646278	−	0.997909i	$-0.479414\pi$
$439$	2.70820	0.129256	0.0646278	+	0.997909i	$0.479414\pi$
$440$	0	0
$441$	0	0
$442$	5.30495	0.252331
$443$	−3.05573	−0.145182	−0.0725910	−	0.997362i	$-0.523127\pi$
$443$	−3.05573	−0.145182	−0.0725910	+	0.997362i	$0.523127\pi$
$444$	0	0
$445$	−8.47214	−0.401618
$446$	9.59675	0.454419
$447$	0	0
$448$	0.472136	0.0223063
$449$	−17.8885	−0.844213	−0.422106	−	0.906546i	$-0.638709\pi$
$449$	−17.8885	−0.844213	−0.422106	+	0.906546i	$0.638709\pi$
$450$	0	0
$451$	0	0
$452$	25.0344	1.17752
$453$	0	0
$454$	−9.38197	−0.440317
$455$	4.94427	0.231791
$456$	0	0
$457$	−16.8328	−0.787406	−0.393703	−	0.919238i	$-0.628806\pi$
$457$	−16.8328	−0.787406	−0.393703	+	0.919238i	$0.628806\pi$
$458$	−0.326238	−0.0152441
$459$	0	0
$460$	6.09017	0.283956
$461$	5.52786	0.257458	0.128729	−	0.991680i	$-0.458910\pi$
$461$	5.52786	0.257458	0.128729	+	0.991680i	$0.458910\pi$
$462$	0	0
$463$	−35.3050	−1.64076	−0.820380	−	0.571819i	$-0.806238\pi$
$463$	−35.3050	−1.64076	−0.820380	+	0.571819i	$0.806238\pi$
$464$	15.7082	0.729235
$465$	0	0
$466$	16.9098	0.783333
$467$	27.6525	1.27960	0.639802	−	0.768540i	$-0.279016\pi$
$467$	27.6525	1.27960	0.639802	+	0.768540i	$0.279016\pi$
$468$	0	0
$469$	−22.8328	−1.05432
$470$	5.09017	0.234792
$471$	0	0
$472$	15.5279	0.714728
$473$	0	0
$474$	0	0
$475$	0	0
$476$	11.2361	0.515004
$477$	0	0
$478$	−10.1115	−0.462487
$479$	19.4164	0.887158	0.443579	−	0.896235i	$-0.353709\pi$
$479$	19.4164	0.887158	0.443579	+	0.896235i	$0.353709\pi$
$480$	0	0
$481$	1.16718	0.0532190
$482$	−9.27051	−0.422260
$483$	0	0
$484$	0	0
$485$	2.47214	0.112254
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	7.76393	0.351457
$489$	0	0
$490$	1.85410	0.0837598
$491$	−7.52786	−0.339728	−0.169864	−	0.985468i	$-0.554333\pi$
$491$	−7.52786	−0.339728	−0.169864	+	0.985468i	$0.554333\pi$
$492$	0	0
$493$	29.4164	1.32485
$494$	0	0
$495$	0	0
$496$	−12.4377	−0.558469
$497$	−8.94427	−0.401205
$498$	0	0
$499$	−4.94427	−0.221336	−0.110668	−	0.993857i	$-0.535299\pi$
$499$	−4.94427	−0.221336	−0.110668	+	0.993857i	$0.535299\pi$
$500$	−1.61803	−0.0723607
$501$	0	0
$502$	12.0000	0.535586
$503$	−27.7639	−1.23793	−0.618966	−	0.785418i	$-0.712448\pi$
$503$	−27.7639	−1.23793	−0.618966	+	0.785418i	$0.712448\pi$
$504$	0	0
$505$	−18.9443	−0.843009
$506$	0	0
$507$	0	0
$508$	10.4721	0.464626
$509$	30.3607	1.34571	0.672857	−	0.739773i	$-0.265067\pi$
$509$	30.3607	1.34571	0.672857	+	0.739773i	$0.265067\pi$
$510$	0	0
$511$	−9.88854	−0.437443
$512$	−18.7082	−0.826794
$513$	0	0
$514$	−5.27051	−0.232472
$515$	−12.4721	−0.549588
$516$	0	0
$517$	0	0
$518$	−0.583592	−0.0256416
$519$	0	0
$520$	−5.52786	−0.242413
$521$	26.4721	1.15977	0.579883	−	0.814700i	$-0.303098\pi$
$521$	26.4721	1.15977	0.579883	+	0.814700i	$0.303098\pi$
$522$	0	0
$523$	−6.47214	−0.283007	−0.141503	−	0.989938i	$-0.545194\pi$
$523$	−6.47214	−0.283007	−0.141503	+	0.989938i	$0.545194\pi$
$524$	7.23607	0.316109
$525$	0	0
$526$	−0.437694	−0.0190844
$527$	−23.2918	−1.01461
$528$	0	0
$529$	−8.83282	−0.384035
$530$	−3.67376	−0.159578
$531$	0	0
$532$	0	0
$533$	−14.8328	−0.642481
$534$	0	0
$535$	6.23607	0.269609
$536$	25.5279	1.10264
$537$	0	0
$538$	−11.0557	−0.476646
$539$	0	0
$540$	0	0
$541$	−32.8328	−1.41159	−0.705797	−	0.708415i	$-0.749411\pi$
$541$	−32.8328	−1.41159	−0.705797	+	0.708415i	$0.749411\pi$
$542$	1.09017	0.0468268
$543$	0	0
$544$	−19.5066	−0.836338
$545$	18.9443	0.811483
$546$	0	0
$547$	39.4164	1.68532	0.842662	−	0.538443i	$-0.180987\pi$
$547$	39.4164	1.68532	0.842662	+	0.538443i	$0.180987\pi$
$548$	26.5623	1.13469
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	5.41641	0.230329
$554$	−4.29180	−0.182341
$555$	0	0
$556$	−14.0902	−0.597556
$557$	0.0557281	0.00236127	0.00118064	−	0.999999i	$-0.499624\pi$
$557$	0.0557281	0.00236127	0.00118064	+	0.999999i	$0.499624\pi$
$558$	0	0
$559$	14.8328	0.627361
$560$	−3.70820	−0.156700
$561$	0	0
$562$	10.1115	0.426526
$563$	−35.7771	−1.50782	−0.753912	−	0.656975i	$-0.771836\pi$
$563$	−35.7771	−1.50782	−0.753912	+	0.656975i	$0.771836\pi$
$564$	0	0
$565$	−15.4721	−0.650918
$566$	18.7639	0.788707
$567$	0	0
$568$	10.0000	0.419591
$569$	2.36068	0.0989648	0.0494824	−	0.998775i	$-0.484243\pi$
$569$	2.36068	0.0989648	0.0494824	+	0.998775i	$0.484243\pi$
$570$	0	0
$571$	36.7082	1.53619	0.768095	−	0.640336i	$-0.221205\pi$
$571$	36.7082	1.53619	0.768095	+	0.640336i	$0.221205\pi$
$572$	0	0
$573$	0	0
$574$	7.41641	0.309555
$575$	−3.76393	−0.156967
$576$	0	0
$577$	−16.0000	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$578$	3.05573	0.127102
$579$	0	0
$580$	−13.7082	−0.569202
$581$	0	0
$582$	0	0
$583$	0	0
$584$	11.0557	0.457489
$585$	0	0
$586$	3.67376	0.151762
$587$	−39.6525	−1.63663	−0.818316	−	0.574768i	$-0.805092\pi$
$587$	−39.6525	−1.63663	−0.818316	+	0.574768i	$0.805092\pi$
$588$	0	0
$589$	0	0
$590$	−4.29180	−0.176690
$591$	0	0
$592$	−0.875388	−0.0359782
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−6.94427	−0.284687
$596$	2.47214	0.101263
$597$	0	0
$598$	−5.75078	−0.235167
$599$	−0.111456	−0.00455398	−0.00227699	−	0.999997i	$-0.500725\pi$
$599$	−0.111456	−0.00455398	−0.00227699	+	0.999997i	$0.500725\pi$
$600$	0	0
$601$	−36.8328	−1.50244	−0.751221	−	0.660051i	$-0.770535\pi$
$601$	−36.8328	−1.50244	−0.751221	+	0.660051i	$0.770535\pi$
$602$	−7.41641	−0.302270
$603$	0	0
$604$	−10.8541	−0.441647
$605$	0	0
$606$	0	0
$607$	−6.47214	−0.262696	−0.131348	−	0.991336i	$-0.541931\pi$
$607$	−6.47214	−0.262696	−0.131348	+	0.991336i	$0.541931\pi$
$608$	0	0
$609$	0	0
$610$	−2.14590	−0.0868849
$611$	20.3607	0.823705
$612$	0	0
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	8.87539	0.358182
$615$	0	0
$616$	0	0
$617$	−26.9443	−1.08474	−0.542368	−	0.840141i	$-0.682472\pi$
$617$	−26.9443	−1.08474	−0.542368	+	0.840141i	$0.682472\pi$
$618$	0	0
$619$	−4.94427	−0.198727	−0.0993635	−	0.995051i	$-0.531681\pi$
$619$	−4.94427	−0.198727	−0.0993635	+	0.995051i	$0.531681\pi$
$620$	10.8541	0.435911
$621$	0	0
$622$	16.3607	0.656003
$623$	16.9443	0.678858
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.18034	0.0871439
$627$	0	0
$628$	−18.4721	−0.737118
$629$	−1.63932	−0.0653640
$630$	0	0
$631$	42.7082	1.70019	0.850093	−	0.526632i	$-0.176545\pi$
$631$	42.7082	1.70019	0.850093	+	0.526632i	$0.176545\pi$
$632$	−6.05573	−0.240884
$633$	0	0
$634$	6.21478	0.246821
$635$	−6.47214	−0.256839
$636$	0	0
$637$	7.41641	0.293849
$638$	0	0
$639$	0	0
$640$	11.3820	0.449912
$641$	4.58359	0.181041	0.0905205	−	0.995895i	$-0.471147\pi$
$641$	4.58359	0.181041	0.0905205	+	0.995895i	$0.471147\pi$
$642$	0	0
$643$	−37.4164	−1.47556	−0.737780	−	0.675042i	$-0.764126\pi$
$643$	−37.4164	−1.47556	−0.737780	+	0.675042i	$0.764126\pi$
$644$	−12.1803	−0.479973
$645$	0	0
$646$	0	0
$647$	25.6525	1.00850	0.504251	−	0.863557i	$-0.331768\pi$
$647$	25.6525	1.00850	0.504251	+	0.863557i	$0.331768\pi$
$648$	0	0
$649$	0	0
$650$	1.52786	0.0599278
$651$	0	0
$652$	18.4721	0.723425
$653$	−23.8885	−0.934831	−0.467415	−	0.884038i	$-0.654815\pi$
$653$	−23.8885	−0.934831	−0.467415	+	0.884038i	$0.654815\pi$
$654$	0	0
$655$	−4.47214	−0.174741
$656$	11.1246	0.434343
$657$	0	0
$658$	−10.1803	−0.396871
$659$	−32.8328	−1.27898	−0.639492	−	0.768797i	$-0.720855\pi$
$659$	−32.8328	−1.27898	−0.639492	+	0.768797i	$0.720855\pi$
$660$	0	0
$661$	1.05573	0.0410631	0.0205315	−	0.999789i	$-0.493464\pi$
$661$	1.05573	0.0410631	0.0205315	+	0.999789i	$0.493464\pi$
$662$	−12.2148	−0.474741
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−31.8885	−1.23473
$668$	18.2705	0.706907
$669$	0	0
$670$	−7.05573	−0.272587
$671$	0	0
$672$	0	0
$673$	45.4164	1.75067	0.875337	−	0.483513i	$-0.160640\pi$
$673$	45.4164	1.75067	0.875337	+	0.483513i	$0.160640\pi$
$674$	14.8328	0.571339
$675$	0	0
$676$	11.1459	0.428688
$677$	15.8885	0.610646	0.305323	−	0.952249i	$-0.401236\pi$
$677$	15.8885	0.610646	0.305323	+	0.952249i	$0.401236\pi$
$678$	0	0
$679$	−4.94427	−0.189744
$680$	7.76393	0.297733
$681$	0	0
$682$	0	0
$683$	35.7771	1.36897	0.684486	−	0.729026i	$-0.260027\pi$
$683$	35.7771	1.36897	0.684486	+	0.729026i	$0.260027\pi$
$684$	0	0
$685$	−16.4164	−0.627239
$686$	−12.3607	−0.471933
$687$	0	0
$688$	−11.1246	−0.424122
$689$	−14.6950	−0.559837
$690$	0	0
$691$	14.5967	0.555286	0.277643	−	0.960684i	$-0.410447\pi$
$691$	14.5967	0.555286	0.277643	+	0.960684i	$0.410447\pi$
$692$	−8.18034	−0.310970
$693$	0	0
$694$	5.02129	0.190605
$695$	8.70820	0.330321
$696$	0	0
$697$	20.8328	0.789099
$698$	2.21478	0.0838307
$699$	0	0
$700$	3.23607	0.122312
$701$	−44.3607	−1.67548	−0.837740	−	0.546070i	$-0.816123\pi$
$701$	−44.3607	−1.67548	−0.837740	+	0.546070i	$0.816123\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	12.6869	0.477478
$707$	37.8885	1.42495
$708$	0	0
$709$	33.4721	1.25707	0.628536	−	0.777780i	$-0.283654\pi$
$709$	33.4721	1.25707	0.628536	+	0.777780i	$0.283654\pi$
$710$	−2.76393	−0.103729
$711$	0	0
$712$	−18.9443	−0.709967
$713$	25.2492	0.945591
$714$	0	0
$715$	0	0
$716$	42.8328	1.60074
$717$	0	0
$718$	−12.0689	−0.450407
$719$	19.3050	0.719953	0.359977	−	0.932961i	$-0.382785\pi$
$719$	19.3050	0.719953	0.359977	+	0.932961i	$0.382785\pi$
$720$	0	0
$721$	24.9443	0.928973
$722$	11.7426	0.437016
$723$	0	0
$724$	−30.6525	−1.13919
$725$	8.47214	0.314647
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	11.0557	0.409753
$729$	0	0
$730$	−3.05573	−0.113098
$731$	−20.8328	−0.770530
$732$	0	0
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	−19.4164	−0.716673
$735$	0	0
$736$	21.1459	0.779448
$737$	0	0
$738$	0	0
$739$	−15.2918	−0.562518	−0.281259	−	0.959632i	$-0.590752\pi$
$739$	−15.2918	−0.562518	−0.281259	+	0.959632i	$0.590752\pi$
$740$	0.763932	0.0280827
$741$	0	0
$742$	7.34752	0.269736
$743$	46.5967	1.70947	0.854734	−	0.519066i	$-0.173720\pi$
$743$	46.5967	1.70947	0.854734	+	0.519066i	$0.173720\pi$
$744$	0	0
$745$	−1.52786	−0.0559766
$746$	23.4164	0.857336
$747$	0	0
$748$	0	0
$749$	−12.4721	−0.455722
$750$	0	0
$751$	41.5410	1.51585	0.757927	−	0.652340i	$-0.226212\pi$
$751$	41.5410	1.51585	0.757927	+	0.652340i	$0.226212\pi$
$752$	−15.2705	−0.556858
$753$	0	0
$754$	12.9443	0.471403
$755$	6.70820	0.244137
$756$	0	0
$757$	10.9443	0.397776	0.198888	−	0.980022i	$-0.436267\pi$
$757$	10.9443	0.397776	0.198888	+	0.980022i	$0.436267\pi$
$758$	−13.9656	−0.507252
$759$	0	0
$760$	0	0
$761$	−4.58359	−0.166155	−0.0830775	−	0.996543i	$-0.526475\pi$
$761$	−4.58359	−0.166155	−0.0830775	+	0.996543i	$0.526475\pi$
$762$	0	0
$763$	−37.8885	−1.37166
$764$	30.6525	1.10897
$765$	0	0
$766$	4.36068	0.157558
$767$	−17.1672	−0.619871
$768$	0	0
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	20.0557	0.721354	0.360677	−	0.932691i	$-0.382546\pi$
$773$	20.0557	0.721354	0.360677	+	0.932691i	$0.382546\pi$
$774$	0	0
$775$	−6.70820	−0.240966
$776$	5.52786	0.198439
$777$	0	0
$778$	5.23607	0.187722
$779$	0	0
$780$	0	0
$781$	0	0
$782$	8.07701	0.288833
$783$	0	0
$784$	−5.56231	−0.198654
$785$	11.4164	0.407469
$786$	0	0
$787$	6.58359	0.234680	0.117340	−	0.993092i	$-0.462563\pi$
$787$	6.58359	0.234680	0.117340	+	0.993092i	$0.462563\pi$
$788$	−37.1246	−1.32251
$789$	0	0
$790$	1.67376	0.0595498
$791$	30.9443	1.10025
$792$	0	0
$793$	−8.58359	−0.304812
$794$	0.583592	0.0207109
$795$	0	0
$796$	31.0344	1.09999
$797$	−1.05573	−0.0373958	−0.0186979	−	0.999825i	$-0.505952\pi$
$797$	−1.05573	−0.0373958	−0.0186979	+	0.999825i	$0.505952\pi$
$798$	0	0
$799$	−28.5967	−1.01168
$800$	−5.61803	−0.198627
$801$	0	0
$802$	11.0557	0.390391
$803$	0	0
$804$	0	0
$805$	7.52786	0.265322
$806$	−10.2492	−0.361014
$807$	0	0
$808$	−42.3607	−1.49024
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	46.1246	1.61965	0.809827	−	0.586669i	$-0.199561\pi$
$811$	46.1246	1.61965	0.809827	+	0.586669i	$0.199561\pi$
$812$	27.4164	0.962127
$813$	0	0
$814$	0	0
$815$	−11.4164	−0.399899
$816$	0	0
$817$	0	0
$818$	1.85410	0.0648272
$819$	0	0
$820$	−9.70820	−0.339025
$821$	−23.8885	−0.833716	−0.416858	−	0.908972i	$-0.636869\pi$
$821$	−23.8885	−0.833716	−0.416858	+	0.908972i	$0.636869\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	−27.8885	−0.971543
$825$	0	0
$826$	8.58359	0.298661
$827$	3.05573	0.106258	0.0531290	−	0.998588i	$-0.483081\pi$
$827$	3.05573	0.106258	0.0531290	+	0.998588i	$0.483081\pi$
$828$	0	0
$829$	28.4164	0.986943	0.493471	−	0.869762i	$-0.335728\pi$
$829$	28.4164	0.986943	0.493471	+	0.869762i	$0.335728\pi$
$830$	0	0
$831$	0	0
$832$	0.583592	0.0202324
$833$	−10.4164	−0.360907
$834$	0	0
$835$	−11.2918	−0.390769
$836$	0	0
$837$	0	0
$838$	24.5836	0.849226
$839$	−6.36068	−0.219595	−0.109798	−	0.993954i	$-0.535020\pi$
$839$	−6.36068	−0.219595	−0.109798	+	0.993954i	$0.535020\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	−4.68692	−0.161522
$843$	0	0
$844$	27.0344	0.930564
$845$	−6.88854	−0.236973
$846$	0	0
$847$	0	0
$848$	11.0213	0.378473
$849$	0	0
$850$	−2.14590	−0.0736037
$851$	1.77709	0.0609178
$852$	0	0
$853$	−32.8328	−1.12417	−0.562087	−	0.827078i	$-0.690001\pi$
$853$	−32.8328	−1.12417	−0.562087	+	0.827078i	$0.690001\pi$
$854$	4.29180	0.146862
$855$	0	0
$856$	13.9443	0.476605
$857$	31.4721	1.07507	0.537534	−	0.843242i	$-0.319356\pi$
$857$	31.4721	1.07507	0.537534	+	0.843242i	$0.319356\pi$
$858$	0	0
$859$	31.7771	1.08422	0.542110	−	0.840307i	$-0.317626\pi$
$859$	31.7771	1.08422	0.542110	+	0.840307i	$0.317626\pi$
$860$	9.70820	0.331047
$861$	0	0
$862$	7.34752	0.250258
$863$	27.0557	0.920988	0.460494	−	0.887663i	$-0.347672\pi$
$863$	27.0557	0.920988	0.460494	+	0.887663i	$0.347672\pi$
$864$	0	0
$865$	5.05573	0.171900
$866$	22.5410	0.765975
$867$	0	0
$868$	−21.7082	−0.736824
$869$	0	0
$870$	0	0
$871$	−28.2229	−0.956297
$872$	42.3607	1.43451
$873$	0	0
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	−26.8328	−0.906080	−0.453040	−	0.891490i	$-0.649660\pi$
$877$	−26.8328	−0.906080	−0.453040	+	0.891490i	$0.649660\pi$
$878$	−1.67376	−0.0564867
$879$	0	0
$880$	0	0
$881$	2.94427	0.0991950	0.0495975	−	0.998769i	$-0.484206\pi$
$881$	2.94427	0.0991950	0.0495975	+	0.998769i	$0.484206\pi$
$882$	0	0
$883$	−36.3607	−1.22363	−0.611817	−	0.790999i	$-0.709561\pi$
$883$	−36.3607	−1.22363	−0.611817	+	0.790999i	$0.709561\pi$
$884$	13.8885	0.467122
$885$	0	0
$886$	1.88854	0.0634469
$887$	−39.7771	−1.33558	−0.667792	−	0.744348i	$-0.732760\pi$
$887$	−39.7771	−1.33558	−0.667792	+	0.744348i	$0.732760\pi$
$888$	0	0
$889$	12.9443	0.434137
$890$	5.23607	0.175513
$891$	0	0
$892$	25.1246	0.841234
$893$	0	0
$894$	0	0
$895$	−26.4721	−0.884866
$896$	−22.7639	−0.760490
$897$	0	0
$898$	11.0557	0.368934
$899$	−56.8328	−1.89548
$900$	0	0
$901$	20.6393	0.687595
$902$	0	0
$903$	0	0
$904$	−34.5967	−1.15067
$905$	18.9443	0.629729
$906$	0	0
$907$	14.8328	0.492516	0.246258	−	0.969204i	$-0.420799\pi$
$907$	14.8328	0.492516	0.246258	+	0.969204i	$0.420799\pi$
$908$	−24.5623	−0.815129
$909$	0	0
$910$	−3.05573	−0.101296
$911$	20.8328	0.690222	0.345111	−	0.938562i	$-0.387841\pi$
$911$	20.8328	0.690222	0.345111	+	0.938562i	$0.387841\pi$
$912$	0	0
$913$	0	0
$914$	10.4033	0.344109
$915$	0	0
$916$	−0.854102	−0.0282203
$917$	8.94427	0.295366
$918$	0	0
$919$	−34.8328	−1.14903	−0.574514	−	0.818495i	$-0.694809\pi$
$919$	−34.8328	−1.14903	−0.574514	+	0.818495i	$0.694809\pi$
$920$	−8.41641	−0.277481
$921$	0	0
$922$	−3.41641	−0.112513
$923$	−11.0557	−0.363904
$924$	0	0
$925$	−0.472136	−0.0155237
$926$	21.8197	0.717039
$927$	0	0
$928$	−47.5967	−1.56244
$929$	50.7214	1.66411	0.832057	−	0.554690i	$-0.187163\pi$
$929$	50.7214	1.66411	0.832057	+	0.554690i	$0.187163\pi$
$930$	0	0
$931$	0	0
$932$	44.2705	1.45013
$933$	0	0
$934$	−17.0902	−0.559207
$935$	0	0
$936$	0	0
$937$	−58.2492	−1.90292	−0.951460	−	0.307774i	$-0.900416\pi$
$937$	−58.2492	−1.90292	−0.951460	+	0.307774i	$0.900416\pi$
$938$	14.1115	0.460755
$939$	0	0
$940$	13.3262	0.434654
$941$	58.4721	1.90614	0.953069	−	0.302754i	$-0.0979062\pi$
$941$	58.4721	1.90614	0.953069	+	0.302754i	$0.0979062\pi$
$942$	0	0
$943$	−22.5836	−0.735423
$944$	12.8754	0.419058
$945$	0	0
$946$	0	0
$947$	−16.3475	−0.531223	−0.265612	−	0.964080i	$-0.585574\pi$
$947$	−16.3475	−0.531223	−0.265612	+	0.964080i	$0.585574\pi$
$948$	0	0
$949$	−12.2229	−0.396773
$950$	0	0
$951$	0	0
$952$	−15.5279	−0.503261
$953$	−19.8885	−0.644253	−0.322127	−	0.946697i	$-0.604398\pi$
$953$	−19.8885	−0.644253	−0.322127	+	0.946697i	$0.604398\pi$
$954$	0	0
$955$	−18.9443	−0.613022
$956$	−26.4721	−0.856170
$957$	0	0
$958$	−12.0000	−0.387702
$959$	32.8328	1.06023
$960$	0	0
$961$	14.0000	0.451613
$962$	−0.721360	−0.0232576
$963$	0	0
$964$	−24.2705	−0.781700
$965$	−2.47214	−0.0795809
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	0	0
$970$	−1.52786	−0.0490568
$971$	9.30495	0.298610	0.149305	−	0.988791i	$-0.452296\pi$
$971$	9.30495	0.298610	0.149305	+	0.988791i	$0.452296\pi$
$972$	0	0
$973$	−17.4164	−0.558344
$974$	−1.23607	−0.0396062
$975$	0	0
$976$	6.43769	0.206066
$977$	−12.5279	−0.400802	−0.200401	−	0.979714i	$-0.564224\pi$
$977$	−12.5279	−0.400802	−0.200401	+	0.979714i	$0.564224\pi$
$978$	0	0
$979$	0	0
$980$	4.85410	0.155059
$981$	0	0
$982$	4.65248	0.148466
$983$	−57.4296	−1.83172	−0.915859	−	0.401499i	$-0.868489\pi$
$983$	−57.4296	−1.83172	−0.915859	+	0.401499i	$0.868489\pi$
$984$	0	0
$985$	22.9443	0.731065
$986$	−18.1803	−0.578980
$987$	0	0
$988$	0	0
$989$	22.5836	0.718116
$990$	0	0
$991$	5.29180	0.168099	0.0840497	−	0.996462i	$-0.473215\pi$
$991$	5.29180	0.168099	0.0840497	+	0.996462i	$0.473215\pi$
$992$	37.6869	1.19656
$993$	0	0
$994$	5.52786	0.175333
$995$	−19.1803	−0.608058
$996$	0	0
$997$	−14.5836	−0.461867	−0.230933	−	0.972970i	$-0.574178\pi$
$997$	−14.5836	−0.461867	−0.230933	+	0.972970i	$0.574178\pi$
$998$	3.05573	0.0967274
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.w.1.1	yes	2
3.2	odd	2		5445.2.a.n.1.2	✓	2
11.10	odd	2		5445.2.a.p.1.2	yes	2
33.32	even	2		5445.2.a.v.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5445.2.a.n.1.2	✓	2	3.2	odd	2
5445.2.a.p.1.2	yes	2	11.10	odd	2
5445.2.a.v.1.1	yes	2	33.32	even	2
5445.2.a.w.1.1	yes	2	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.00000 q^{7} +2.23607 q^{8} -0.618034 q^{10} -2.47214 q^{13} +1.23607 q^{14} +1.85410 q^{16} +3.47214 q^{17} -1.61803 q^{20} -3.76393 q^{23} +1.00000 q^{25} +1.52786 q^{26} +3.23607 q^{28} +8.47214 q^{29} -6.70820 q^{31} -5.61803 q^{32} -2.14590 q^{34} -2.00000 q^{35} -0.472136 q^{37} +2.23607 q^{40} +6.00000 q^{41} -6.00000 q^{43} +2.32624 q^{46} -8.23607 q^{47} -3.00000 q^{49} -0.618034 q^{50} +4.00000 q^{52} +5.94427 q^{53} -4.47214 q^{56} -5.23607 q^{58} +6.94427 q^{59} +3.47214 q^{61} +4.14590 q^{62} -0.236068 q^{64} -2.47214 q^{65} +11.4164 q^{67} -5.61803 q^{68} +1.23607 q^{70} +4.47214 q^{71} +4.94427 q^{73} +0.291796 q^{74} -2.70820 q^{79} +1.85410 q^{80} -3.70820 q^{82} +3.47214 q^{85} +3.70820 q^{86} -8.47214 q^{89} +4.94427 q^{91} +6.09017 q^{92} +5.09017 q^{94} +2.47214 q^{97} +1.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + q^{10} + 4 q^{13} - 2 q^{14} - 3 q^{16} - 2 q^{17} - q^{20} - 12 q^{23} + 2 q^{25} + 12 q^{26} + 2 q^{28} + 8 q^{29} - 9 q^{32} - 11 q^{34} - 4 q^{35} + 8 q^{37} + 12 q^{41} - 12 q^{43} - 11 q^{46} - 12 q^{47} - 6 q^{49} + q^{50} + 8 q^{52} - 6 q^{53} - 6 q^{58} - 4 q^{59} - 2 q^{61} + 15 q^{62} + 4 q^{64} + 4 q^{65} - 4 q^{67} - 9 q^{68} - 2 q^{70} - 8 q^{73} + 14 q^{74} + 8 q^{79} - 3 q^{80} + 6 q^{82} - 2 q^{85} - 6 q^{86} - 8 q^{89} - 8 q^{91} + q^{92} - q^{94} - 4 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7 + q^10 + 4 * q^13 - 2 * q^14 - 3 * q^16 - 2 * q^17 - q^20 - 12 * q^23 + 2 * q^25 + 12 * q^26 + 2 * q^28 + 8 * q^29 - 9 * q^32 - 11 * q^34 - 4 * q^35 + 8 * q^37 + 12 * q^41 - 12 * q^43 - 11 * q^46 - 12 * q^47 - 6 * q^49 + q^50 + 8 * q^52 - 6 * q^53 - 6 * q^58 - 4 * q^59 - 2 * q^61 + 15 * q^62 + 4 * q^64 + 4 * q^65 - 4 * q^67 - 9 * q^68 - 2 * q^70 - 8 * q^73 + 14 * q^74 + 8 * q^79 - 3 * q^80 + 6 * q^82 - 2 * q^85 - 6 * q^86 - 8 * q^89 - 8 * q^91 + q^92 - q^94 - 4 * q^97 - 3 * q^98

Introduction

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