LMFDB - Embedded newform 6615.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6615,2,Mod(1,6615)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6615, 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("6615.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6615 = 3^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6615.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:	$52.8210409371$
Analytic rank:	$0$
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:	no (minimal twist has level 945)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6615.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -2.23607 q^{8} +O(q^{10})$	$q+1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -2.23607 q^{8} +1.61803 q^{10} -4.23607 q^{11} -1.38197 q^{13} -4.85410 q^{16} +1.61803 q^{17} +7.09017 q^{19} +0.618034 q^{20} -6.85410 q^{22} -5.38197 q^{23} +1.00000 q^{25} -2.23607 q^{26} +9.56231 q^{29} -6.70820 q^{31} -3.38197 q^{32} +2.61803 q^{34} +6.70820 q^{37} +11.4721 q^{38} -2.23607 q^{40} +8.09017 q^{41} -9.94427 q^{43} -2.61803 q^{44} -8.70820 q^{46} +11.0000 q^{47} +1.61803 q^{50} -0.854102 q^{52} +4.38197 q^{53} -4.23607 q^{55} +15.4721 q^{58} +1.29180 q^{59} +13.8541 q^{61} -10.8541 q^{62} +4.23607 q^{64} -1.38197 q^{65} +11.3262 q^{67} +1.00000 q^{68} -3.85410 q^{71} -13.9443 q^{73} +10.8541 q^{74} +4.38197 q^{76} +5.61803 q^{79} -4.85410 q^{80} +13.0902 q^{82} +10.7082 q^{83} +1.61803 q^{85} -16.0902 q^{86} +9.47214 q^{88} +1.47214 q^{89} -3.32624 q^{92} +17.7984 q^{94} +7.09017 q^{95} +3.32624 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 2 q^{5}+O(q^{10}) $ 2 * q + q^2 - q^4 + 2 * q^5	$ 2 q + q^{2} - q^{4} + 2 q^{5} + q^{10} - 4 q^{11} - 5 q^{13} - 3 q^{16} + q^{17} + 3 q^{19} - q^{20} - 7 q^{22} - 13 q^{23} + 2 q^{25} - q^{29} - 9 q^{32} + 3 q^{34} + 14 q^{38} + 5 q^{41} - 2 q^{43} - 3 q^{44} - 4 q^{46} + 22 q^{47} + q^{50} + 5 q^{52} + 11 q^{53} - 4 q^{55} + 22 q^{58} + 16 q^{59} + 21 q^{61} - 15 q^{62} + 4 q^{64} - 5 q^{65} + 7 q^{67} + 2 q^{68} - q^{71} - 10 q^{73} + 15 q^{74} + 11 q^{76} + 9 q^{79} - 3 q^{80} + 15 q^{82} + 8 q^{83} + q^{85} - 21 q^{86} + 10 q^{88} - 6 q^{89} + 9 q^{92} + 11 q^{94} + 3 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q + q^2 - q^4 + 2 * q^5 + q^10 - 4 * q^11 - 5 * q^13 - 3 * q^16 + q^17 + 3 * q^19 - q^20 - 7 * q^22 - 13 * q^23 + 2 * q^25 - q^29 - 9 * q^32 + 3 * q^34 + 14 * q^38 + 5 * q^41 - 2 * q^43 - 3 * q^44 - 4 * q^46 + 22 * q^47 + q^50 + 5 * q^52 + 11 * q^53 - 4 * q^55 + 22 * q^58 + 16 * q^59 + 21 * q^61 - 15 * q^62 + 4 * q^64 - 5 * q^65 + 7 * q^67 + 2 * q^68 - q^71 - 10 * q^73 + 15 * q^74 + 11 * q^76 + 9 * q^79 - 3 * q^80 + 15 * q^82 + 8 * q^83 + q^85 - 21 * q^86 + 10 * q^88 - 6 * q^89 + 9 * q^92 + 11 * q^94 + 3 * q^95 - 9 * q^97

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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	0	0
$10$	1.61803	0.511667
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	0	0
$13$	−1.38197	−0.383288	−0.191644	−	0.981464i	$-0.561382\pi$
$13$	−1.38197	−0.383288	−0.191644	+	0.981464i	$0.561382\pi$
$14$	0	0
$15$	0	0
$16$	−4.85410	−1.21353
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	0	0
$19$	7.09017	1.62660	0.813298	−	0.581847i	$-0.197670\pi$
$19$	7.09017	1.62660	0.813298	+	0.581847i	$0.197670\pi$
$20$	0.618034	0.138197
$21$	0	0
$22$	−6.85410	−1.46130
$23$	−5.38197	−1.12222	−0.561109	−	0.827742i	$-0.689625\pi$
$23$	−5.38197	−1.12222	−0.561109	+	0.827742i	$0.689625\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.23607	−0.438529
$27$	0	0
$28$	0	0
$29$	9.56231	1.77568	0.887838	−	0.460157i	$-0.152207\pi$
$29$	9.56231	1.77568	0.887838	+	0.460157i	$0.152207\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	2.61803	0.448989
$35$	0	0
$36$	0	0
$37$	6.70820	1.10282	0.551411	−	0.834234i	$-0.314090\pi$
$37$	6.70820	1.10282	0.551411	+	0.834234i	$0.314090\pi$
$38$	11.4721	1.86103
$39$	0	0
$40$	−2.23607	−0.353553
$41$	8.09017	1.26347	0.631736	−	0.775183i	$-0.282343\pi$
$41$	8.09017	1.26347	0.631736	+	0.775183i	$0.282343\pi$
$42$	0	0
$43$	−9.94427	−1.51649	−0.758244	−	0.651971i	$-0.773942\pi$
$43$	−9.94427	−1.51649	−0.758244	+	0.651971i	$0.773942\pi$
$44$	−2.61803	−0.394683
$45$	0	0
$46$	−8.70820	−1.28395
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	0	0
$49$	0	0
$50$	1.61803	0.228825
$51$	0	0
$52$	−0.854102	−0.118443
$53$	4.38197	0.601909	0.300955	−	0.953638i	$-0.402695\pi$
$53$	4.38197	0.601909	0.300955	+	0.953638i	$0.402695\pi$
$54$	0	0
$55$	−4.23607	−0.571191
$56$	0	0
$57$	0	0
$58$	15.4721	2.03159
$59$	1.29180	0.168178	0.0840888	−	0.996458i	$-0.473202\pi$
$59$	1.29180	0.168178	0.0840888	+	0.996458i	$0.473202\pi$
$60$	0	0
$61$	13.8541	1.77384	0.886918	−	0.461927i	$-0.152842\pi$
$61$	13.8541	1.77384	0.886918	+	0.461927i	$0.152842\pi$
$62$	−10.8541	−1.37847
$63$	0	0
$64$	4.23607	0.529508
$65$	−1.38197	−0.171412
$66$	0	0
$67$	11.3262	1.38372	0.691860	−	0.722032i	$-0.256791\pi$
$67$	11.3262	1.38372	0.691860	+	0.722032i	$0.256791\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	−3.85410	−0.457398	−0.228699	−	0.973497i	$-0.573447\pi$
$71$	−3.85410	−0.457398	−0.228699	+	0.973497i	$0.573447\pi$
$72$	0	0
$73$	−13.9443	−1.63205	−0.816027	−	0.578014i	$-0.803828\pi$
$73$	−13.9443	−1.63205	−0.816027	+	0.578014i	$0.803828\pi$
$74$	10.8541	1.26176
$75$	0	0
$76$	4.38197	0.502646
$77$	0	0
$78$	0	0
$79$	5.61803	0.632078	0.316039	−	0.948746i	$-0.397647\pi$
$79$	5.61803	0.632078	0.316039	+	0.948746i	$0.397647\pi$
$80$	−4.85410	−0.542705
$81$	0	0
$82$	13.0902	1.44557
$83$	10.7082	1.17538	0.587689	−	0.809087i	$-0.300038\pi$
$83$	10.7082	1.17538	0.587689	+	0.809087i	$0.300038\pi$
$84$	0	0
$85$	1.61803	0.175500
$86$	−16.0902	−1.73505
$87$	0	0
$88$	9.47214	1.00973
$89$	1.47214	0.156046	0.0780230	−	0.996952i	$-0.475139\pi$
$89$	1.47214	0.156046	0.0780230	+	0.996952i	$0.475139\pi$
$90$	0	0
$91$	0	0
$92$	−3.32624	−0.346784
$93$	0	0
$94$	17.7984	1.83576
$95$	7.09017	0.727436
$96$	0	0
$97$	3.32624	0.337728	0.168864	−	0.985639i	$-0.445990\pi$
$97$	3.32624	0.337728	0.168864	+	0.985639i	$0.445990\pi$
$98$	0	0
$99$	0	0
$100$	0.618034	0.0618034
$101$	5.52786	0.550043	0.275022	−	0.961438i	$-0.411315\pi$
$101$	5.52786	0.550043	0.275022	+	0.961438i	$0.411315\pi$
$102$	0	0
$103$	−10.6180	−1.04623	−0.523113	−	0.852263i	$-0.675229\pi$
$103$	−10.6180	−1.04623	−0.523113	+	0.852263i	$0.675229\pi$
$104$	3.09017	0.303016
$105$	0	0
$106$	7.09017	0.688658
$107$	2.70820	0.261812	0.130906	−	0.991395i	$-0.458211\pi$
$107$	2.70820	0.261812	0.130906	+	0.991395i	$0.458211\pi$
$108$	0	0
$109$	12.5623	1.20325	0.601625	−	0.798778i	$-0.294520\pi$
$109$	12.5623	1.20325	0.601625	+	0.798778i	$0.294520\pi$
$110$	−6.85410	−0.653513
$111$	0	0
$112$	0	0
$113$	−5.85410	−0.550708	−0.275354	−	0.961343i	$-0.588795\pi$
$113$	−5.85410	−0.550708	−0.275354	+	0.961343i	$0.588795\pi$
$114$	0	0
$115$	−5.38197	−0.501871
$116$	5.90983	0.548714
$117$	0	0
$118$	2.09017	0.192416
$119$	0	0
$120$	0	0
$121$	6.94427	0.631297
$122$	22.4164	2.02949
$123$	0	0
$124$	−4.14590	−0.372313
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.0901699	0.00800129	0.00400064	−	0.999992i	$-0.498727\pi$
$127$	0.0901699	0.00800129	0.00400064	+	0.999992i	$0.498727\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	−2.23607	−0.196116
$131$	4.38197	0.382854	0.191427	−	0.981507i	$-0.438688\pi$
$131$	4.38197	0.382854	0.191427	+	0.981507i	$0.438688\pi$
$132$	0	0
$133$	0	0
$134$	18.3262	1.58315
$135$	0	0
$136$	−3.61803	−0.310244
$137$	−13.8885	−1.18658	−0.593289	−	0.804989i	$-0.702171\pi$
$137$	−13.8885	−1.18658	−0.593289	+	0.804989i	$0.702171\pi$
$138$	0	0
$139$	−3.18034	−0.269753	−0.134876	−	0.990862i	$-0.543064\pi$
$139$	−3.18034	−0.269753	−0.134876	+	0.990862i	$0.543064\pi$
$140$	0	0
$141$	0	0
$142$	−6.23607	−0.523319
$143$	5.85410	0.489545
$144$	0	0
$145$	9.56231	0.794106
$146$	−22.5623	−1.86727
$147$	0	0
$148$	4.14590	0.340791
$149$	15.7984	1.29425	0.647127	−	0.762383i	$-0.275971\pi$
$149$	15.7984	1.29425	0.647127	+	0.762383i	$0.275971\pi$
$150$	0	0
$151$	−0.0557281	−0.00453509	−0.00226754	−	0.999997i	$-0.500722\pi$
$151$	−0.0557281	−0.00453509	−0.00226754	+	0.999997i	$0.500722\pi$
$152$	−15.8541	−1.28594
$153$	0	0
$154$	0	0
$155$	−6.70820	−0.538816
$156$	0	0
$157$	5.23607	0.417884	0.208942	−	0.977928i	$-0.432998\pi$
$157$	5.23607	0.417884	0.208942	+	0.977928i	$0.432998\pi$
$158$	9.09017	0.723175
$159$	0	0
$160$	−3.38197	−0.267368
$161$	0	0
$162$	0	0
$163$	−2.94427	−0.230613	−0.115307	−	0.993330i	$-0.536785\pi$
$163$	−2.94427	−0.230613	−0.115307	+	0.993330i	$0.536785\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	17.3262	1.34478
$167$	−7.47214	−0.578211	−0.289106	−	0.957297i	$-0.593358\pi$
$167$	−7.47214	−0.578211	−0.289106	+	0.957297i	$0.593358\pi$
$168$	0	0
$169$	−11.0902	−0.853090
$170$	2.61803	0.200794
$171$	0	0
$172$	−6.14590	−0.468620
$173$	11.4721	0.872210	0.436105	−	0.899896i	$-0.356358\pi$
$173$	11.4721	0.872210	0.436105	+	0.899896i	$0.356358\pi$
$174$	0	0
$175$	0	0
$176$	20.5623	1.54994
$177$	0	0
$178$	2.38197	0.178536
$179$	7.00000	0.523205	0.261602	−	0.965176i	$-0.415749\pi$
$179$	7.00000	0.523205	0.261602	+	0.965176i	$0.415749\pi$
$180$	0	0
$181$	19.4721	1.44735	0.723676	−	0.690140i	$-0.242451\pi$
$181$	19.4721	1.44735	0.723676	+	0.690140i	$0.242451\pi$
$182$	0	0
$183$	0	0
$184$	12.0344	0.887191
$185$	6.70820	0.493197
$186$	0	0
$187$	−6.85410	−0.501222
$188$	6.79837	0.495822
$189$	0	0
$190$	11.4721	0.832276
$191$	−18.7984	−1.36020	−0.680101	−	0.733118i	$-0.738064\pi$
$191$	−18.7984	−1.36020	−0.680101	+	0.733118i	$0.738064\pi$
$192$	0	0
$193$	−0.854102	−0.0614796	−0.0307398	−	0.999527i	$-0.509786\pi$
$193$	−0.854102	−0.0614796	−0.0307398	+	0.999527i	$0.509786\pi$
$194$	5.38197	0.386403
$195$	0	0
$196$	0	0
$197$	19.4721	1.38733	0.693666	−	0.720297i	$-0.255994\pi$
$197$	19.4721	1.38733	0.693666	+	0.720297i	$0.255994\pi$
$198$	0	0
$199$	−1.61803	−0.114699	−0.0573497	−	0.998354i	$-0.518265\pi$
$199$	−1.61803	−0.114699	−0.0573497	+	0.998354i	$0.518265\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	8.94427	0.629317
$203$	0	0
$204$	0	0
$205$	8.09017	0.565042
$206$	−17.1803	−1.19701
$207$	0	0
$208$	6.70820	0.465130
$209$	−30.0344	−2.07753
$210$	0	0
$211$	−17.1246	−1.17891	−0.589453	−	0.807802i	$-0.700657\pi$
$211$	−17.1246	−1.17891	−0.589453	+	0.807802i	$0.700657\pi$
$212$	2.70820	0.186000
$213$	0	0
$214$	4.38197	0.299545
$215$	−9.94427	−0.678194
$216$	0	0
$217$	0	0
$218$	20.3262	1.37667
$219$	0	0
$220$	−2.61803	−0.176508
$221$	−2.23607	−0.150414
$222$	0	0
$223$	−7.47214	−0.500371	−0.250186	−	0.968198i	$-0.580492\pi$
$223$	−7.47214	−0.500371	−0.250186	+	0.968198i	$0.580492\pi$
$224$	0	0
$225$	0	0
$226$	−9.47214	−0.630077
$227$	7.43769	0.493657	0.246829	−	0.969059i	$-0.420612\pi$
$227$	7.43769	0.493657	0.246829	+	0.969059i	$0.420612\pi$
$228$	0	0
$229$	17.9443	1.18579	0.592895	−	0.805279i	$-0.297985\pi$
$229$	17.9443	1.18579	0.592895	+	0.805279i	$0.297985\pi$
$230$	−8.70820	−0.574202
$231$	0	0
$232$	−21.3820	−1.40379
$233$	−17.0902	−1.11961	−0.559807	−	0.828623i	$-0.689125\pi$
$233$	−17.0902	−1.11961	−0.559807	+	0.828623i	$0.689125\pi$
$234$	0	0
$235$	11.0000	0.717561
$236$	0.798374	0.0519697
$237$	0	0
$238$	0	0
$239$	−11.6525	−0.753736	−0.376868	−	0.926267i	$-0.622999\pi$
$239$	−11.6525	−0.753736	−0.376868	+	0.926267i	$0.622999\pi$
$240$	0	0
$241$	7.61803	0.490721	0.245360	−	0.969432i	$-0.421094\pi$
$241$	7.61803	0.490721	0.245360	+	0.969432i	$0.421094\pi$
$242$	11.2361	0.722282
$243$	0	0
$244$	8.56231	0.548145
$245$	0	0
$246$	0	0
$247$	−9.79837	−0.623456
$248$	15.0000	0.952501
$249$	0	0
$250$	1.61803	0.102333
$251$	1.23607	0.0780199	0.0390100	−	0.999239i	$-0.487580\pi$
$251$	1.23607	0.0780199	0.0390100	+	0.999239i	$0.487580\pi$
$252$	0	0
$253$	22.7984	1.43332
$254$	0.145898	0.00915446
$255$	0	0
$256$	13.5623	0.847644
$257$	−8.94427	−0.557928	−0.278964	−	0.960302i	$-0.589991\pi$
$257$	−8.94427	−0.557928	−0.278964	+	0.960302i	$0.589991\pi$
$258$	0	0
$259$	0	0
$260$	−0.854102	−0.0529692
$261$	0	0
$262$	7.09017	0.438032
$263$	30.7426	1.89567	0.947836	−	0.318757i	$-0.103265\pi$
$263$	30.7426	1.89567	0.947836	+	0.318757i	$0.103265\pi$
$264$	0	0
$265$	4.38197	0.269182
$266$	0	0
$267$	0	0
$268$	7.00000	0.427593
$269$	10.7082	0.652891	0.326445	−	0.945216i	$-0.394149\pi$
$269$	10.7082	0.652891	0.326445	+	0.945216i	$0.394149\pi$
$270$	0	0
$271$	24.5623	1.49205	0.746027	−	0.665916i	$-0.231959\pi$
$271$	24.5623	1.49205	0.746027	+	0.665916i	$0.231959\pi$
$272$	−7.85410	−0.476225
$273$	0	0
$274$	−22.4721	−1.35759
$275$	−4.23607	−0.255445
$276$	0	0
$277$	4.67376	0.280819	0.140410	−	0.990094i	$-0.455158\pi$
$277$	4.67376	0.280819	0.140410	+	0.990094i	$0.455158\pi$
$278$	−5.14590	−0.308630
$279$	0	0
$280$	0	0
$281$	13.6180	0.812384	0.406192	−	0.913788i	$-0.366856\pi$
$281$	13.6180	0.812384	0.406192	+	0.913788i	$0.366856\pi$
$282$	0	0
$283$	−21.1459	−1.25699	−0.628497	−	0.777812i	$-0.716329\pi$
$283$	−21.1459	−1.25699	−0.628497	+	0.777812i	$0.716329\pi$
$284$	−2.38197	−0.141344
$285$	0	0
$286$	9.47214	0.560099
$287$	0	0
$288$	0	0
$289$	−14.3820	−0.845998
$290$	15.4721	0.908555
$291$	0	0
$292$	−8.61803	−0.504332
$293$	−6.29180	−0.367571	−0.183785	−	0.982966i	$-0.558835\pi$
$293$	−6.29180	−0.367571	−0.183785	+	0.982966i	$0.558835\pi$
$294$	0	0
$295$	1.29180	0.0752113
$296$	−15.0000	−0.871857
$297$	0	0
$298$	25.5623	1.48078
$299$	7.43769	0.430133
$300$	0	0
$301$	0	0
$302$	−0.0901699	−0.00518870
$303$	0	0
$304$	−34.4164	−1.97392
$305$	13.8541	0.793284
$306$	0	0
$307$	−19.4164	−1.10815	−0.554076	−	0.832466i	$-0.686928\pi$
$307$	−19.4164	−1.10815	−0.554076	+	0.832466i	$0.686928\pi$
$308$	0	0
$309$	0	0
$310$	−10.8541	−0.616472
$311$	14.3262	0.812366	0.406183	−	0.913792i	$-0.366859\pi$
$311$	14.3262	0.812366	0.406183	+	0.913792i	$0.366859\pi$
$312$	0	0
$313$	23.1803	1.31023	0.655115	−	0.755529i	$-0.272620\pi$
$313$	23.1803	1.31023	0.655115	+	0.755529i	$0.272620\pi$
$314$	8.47214	0.478110
$315$	0	0
$316$	3.47214	0.195323
$317$	6.05573	0.340124	0.170062	−	0.985433i	$-0.445603\pi$
$317$	6.05573	0.340124	0.170062	+	0.985433i	$0.445603\pi$
$318$	0	0
$319$	−40.5066	−2.26793
$320$	4.23607	0.236803
$321$	0	0
$322$	0	0
$323$	11.4721	0.638327
$324$	0	0
$325$	−1.38197	−0.0766577
$326$	−4.76393	−0.263850
$327$	0	0
$328$	−18.0902	−0.998863
$329$	0	0
$330$	0	0
$331$	−31.8541	−1.75086	−0.875430	−	0.483345i	$-0.839422\pi$
$331$	−31.8541	−1.75086	−0.875430	+	0.483345i	$0.839422\pi$
$332$	6.61803	0.363212
$333$	0	0
$334$	−12.0902	−0.661545
$335$	11.3262	0.618818
$336$	0	0
$337$	−31.7426	−1.72913	−0.864566	−	0.502519i	$-0.832407\pi$
$337$	−31.7426	−1.72913	−0.864566	+	0.502519i	$0.832407\pi$
$338$	−17.9443	−0.976040
$339$	0	0
$340$	1.00000	0.0542326
$341$	28.4164	1.53883
$342$	0	0
$343$	0	0
$344$	22.2361	1.19889
$345$	0	0
$346$	18.5623	0.997916
$347$	12.7639	0.685204	0.342602	−	0.939481i	$-0.388692\pi$
$347$	12.7639	0.685204	0.342602	+	0.939481i	$0.388692\pi$
$348$	0	0
$349$	−31.3607	−1.67870	−0.839349	−	0.543592i	$-0.817064\pi$
$349$	−31.3607	−1.67870	−0.839349	+	0.543592i	$0.817064\pi$
$350$	0	0
$351$	0	0
$352$	14.3262	0.763591
$353$	11.6738	0.621332	0.310666	−	0.950519i	$-0.399448\pi$
$353$	11.6738	0.621332	0.310666	+	0.950519i	$0.399448\pi$
$354$	0	0
$355$	−3.85410	−0.204554
$356$	0.909830	0.0482209
$357$	0	0
$358$	11.3262	0.598610
$359$	−36.4164	−1.92198	−0.960992	−	0.276575i	$-0.910800\pi$
$359$	−36.4164	−1.92198	−0.960992	+	0.276575i	$0.910800\pi$
$360$	0	0
$361$	31.2705	1.64582
$362$	31.5066	1.65595
$363$	0	0
$364$	0	0
$365$	−13.9443	−0.729877
$366$	0	0
$367$	14.8541	0.775378	0.387689	−	0.921790i	$-0.373273\pi$
$367$	14.8541	0.775378	0.387689	+	0.921790i	$0.373273\pi$
$368$	26.1246	1.36184
$369$	0	0
$370$	10.8541	0.564278
$371$	0	0
$372$	0	0
$373$	−16.9787	−0.879124	−0.439562	−	0.898212i	$-0.644866\pi$
$373$	−16.9787	−0.879124	−0.439562	+	0.898212i	$0.644866\pi$
$374$	−11.0902	−0.573459
$375$	0	0
$376$	−24.5967	−1.26848
$377$	−13.2148	−0.680596
$378$	0	0
$379$	11.8885	0.610673	0.305337	−	0.952244i	$-0.401231\pi$
$379$	11.8885	0.610673	0.305337	+	0.952244i	$0.401231\pi$
$380$	4.38197	0.224790
$381$	0	0
$382$	−30.4164	−1.55624
$383$	−11.6525	−0.595414	−0.297707	−	0.954657i	$-0.596222\pi$
$383$	−11.6525	−0.595414	−0.297707	+	0.954657i	$0.596222\pi$
$384$	0	0
$385$	0	0
$386$	−1.38197	−0.0703402
$387$	0	0
$388$	2.05573	0.104364
$389$	−25.7984	−1.30803	−0.654015	−	0.756482i	$-0.726917\pi$
$389$	−25.7984	−1.30803	−0.654015	+	0.756482i	$0.726917\pi$
$390$	0	0
$391$	−8.70820	−0.440393
$392$	0	0
$393$	0	0
$394$	31.5066	1.58728
$395$	5.61803	0.282674
$396$	0	0
$397$	−9.05573	−0.454494	−0.227247	−	0.973837i	$-0.572972\pi$
$397$	−9.05573	−0.454494	−0.227247	+	0.973837i	$0.572972\pi$
$398$	−2.61803	−0.131230
$399$	0	0
$400$	−4.85410	−0.242705
$401$	17.3820	0.868014	0.434007	−	0.900910i	$-0.357099\pi$
$401$	17.3820	0.868014	0.434007	+	0.900910i	$0.357099\pi$
$402$	0	0
$403$	9.27051	0.461797
$404$	3.41641	0.169973
$405$	0	0
$406$	0	0
$407$	−28.4164	−1.40855
$408$	0	0
$409$	13.0557	0.645564	0.322782	−	0.946473i	$-0.395382\pi$
$409$	13.0557	0.645564	0.322782	+	0.946473i	$0.395382\pi$
$410$	13.0902	0.646477
$411$	0	0
$412$	−6.56231	−0.323302
$413$	0	0
$414$	0	0
$415$	10.7082	0.525645
$416$	4.67376	0.229150
$417$	0	0
$418$	−48.5967	−2.37694
$419$	7.47214	0.365038	0.182519	−	0.983202i	$-0.441575\pi$
$419$	7.47214	0.365038	0.182519	+	0.983202i	$0.441575\pi$
$420$	0	0
$421$	8.43769	0.411228	0.205614	−	0.978633i	$-0.434081\pi$
$421$	8.43769	0.411228	0.205614	+	0.978633i	$0.434081\pi$
$422$	−27.7082	−1.34881
$423$	0	0
$424$	−9.79837	−0.475851
$425$	1.61803	0.0784862
$426$	0	0
$427$	0	0
$428$	1.67376	0.0809043
$429$	0	0
$430$	−16.0902	−0.775937
$431$	−30.4508	−1.46677	−0.733383	−	0.679816i	$-0.762060\pi$
$431$	−30.4508	−1.46677	−0.733383	+	0.679816i	$0.762060\pi$
$432$	0	0
$433$	21.2148	1.01952	0.509759	−	0.860317i	$-0.329735\pi$
$433$	21.2148	1.01952	0.509759	+	0.860317i	$0.329735\pi$
$434$	0	0
$435$	0	0
$436$	7.76393	0.371825
$437$	−38.1591	−1.82540
$438$	0	0
$439$	−11.2918	−0.538928	−0.269464	−	0.963010i	$-0.586847\pi$
$439$	−11.2918	−0.538928	−0.269464	+	0.963010i	$0.586847\pi$
$440$	9.47214	0.451566
$441$	0	0
$442$	−3.61803	−0.172092
$443$	40.6869	1.93309	0.966547	−	0.256490i	$-0.0825661\pi$
$443$	40.6869	1.93309	0.966547	+	0.256490i	$0.0825661\pi$
$444$	0	0
$445$	1.47214	0.0697859
$446$	−12.0902	−0.572486
$447$	0	0
$448$	0	0
$449$	−20.4721	−0.966140	−0.483070	−	0.875582i	$-0.660478\pi$
$449$	−20.4721	−0.966140	−0.483070	+	0.875582i	$0.660478\pi$
$450$	0	0
$451$	−34.2705	−1.61374
$452$	−3.61803	−0.170178
$453$	0	0
$454$	12.0344	0.564804
$455$	0	0
$456$	0	0
$457$	17.3262	0.810487	0.405244	−	0.914209i	$-0.367187\pi$
$457$	17.3262	0.810487	0.405244	+	0.914209i	$0.367187\pi$
$458$	29.0344	1.35669
$459$	0	0
$460$	−3.32624	−0.155087
$461$	−22.7984	−1.06183	−0.530913	−	0.847426i	$-0.678151\pi$
$461$	−22.7984	−1.06183	−0.530913	+	0.847426i	$0.678151\pi$
$462$	0	0
$463$	−0.618034	−0.0287225	−0.0143612	−	0.999897i	$-0.504571\pi$
$463$	−0.618034	−0.0287225	−0.0143612	+	0.999897i	$0.504571\pi$
$464$	−46.4164	−2.15483
$465$	0	0
$466$	−27.6525	−1.28098
$467$	16.9443	0.784087	0.392044	−	0.919947i	$-0.371768\pi$
$467$	16.9443	0.784087	0.392044	+	0.919947i	$0.371768\pi$
$468$	0	0
$469$	0	0
$470$	17.7984	0.820978
$471$	0	0
$472$	−2.88854	−0.132956
$473$	42.1246	1.93689
$474$	0	0
$475$	7.09017	0.325319
$476$	0	0
$477$	0	0
$478$	−18.8541	−0.862367
$479$	25.9098	1.18385	0.591925	−	0.805993i	$-0.298368\pi$
$479$	25.9098	1.18385	0.591925	+	0.805993i	$0.298368\pi$
$480$	0	0
$481$	−9.27051	−0.422699
$482$	12.3262	0.561445
$483$	0	0
$484$	4.29180	0.195082
$485$	3.32624	0.151037
$486$	0	0
$487$	−15.5279	−0.703635	−0.351817	−	0.936069i	$-0.614436\pi$
$487$	−15.5279	−0.703635	−0.351817	+	0.936069i	$0.614436\pi$
$488$	−30.9787	−1.40234
$489$	0	0
$490$	0	0
$491$	19.5066	0.880320	0.440160	−	0.897919i	$-0.354922\pi$
$491$	19.5066	0.880320	0.440160	+	0.897919i	$0.354922\pi$
$492$	0	0
$493$	15.4721	0.696830
$494$	−15.8541	−0.713310
$495$	0	0
$496$	32.5623	1.46209
$497$	0	0
$498$	0	0
$499$	3.36068	0.150445	0.0752223	−	0.997167i	$-0.476033\pi$
$499$	3.36068	0.150445	0.0752223	+	0.997167i	$0.476033\pi$
$500$	0.618034	0.0276393
$501$	0	0
$502$	2.00000	0.0892644
$503$	30.9098	1.37820	0.689101	−	0.724666i	$-0.258006\pi$
$503$	30.9098	1.37820	0.689101	+	0.724666i	$0.258006\pi$
$504$	0	0
$505$	5.52786	0.245987
$506$	36.8885	1.63990
$507$	0	0
$508$	0.0557281	0.00247253
$509$	32.3050	1.43189	0.715946	−	0.698156i	$-0.245996\pi$
$509$	32.3050	1.43189	0.715946	+	0.698156i	$0.245996\pi$
$510$	0	0
$511$	0	0
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−14.4721	−0.638339
$515$	−10.6180	−0.467886
$516$	0	0
$517$	−46.5967	−2.04932
$518$	0	0
$519$	0	0
$520$	3.09017	0.135513
$521$	−19.8328	−0.868891	−0.434446	−	0.900698i	$-0.643056\pi$
$521$	−19.8328	−0.868891	−0.434446	+	0.900698i	$0.643056\pi$
$522$	0	0
$523$	1.20163	0.0525434	0.0262717	−	0.999655i	$-0.491636\pi$
$523$	1.20163	0.0525434	0.0262717	+	0.999655i	$0.491636\pi$
$524$	2.70820	0.118308
$525$	0	0
$526$	49.7426	2.16888
$527$	−10.8541	−0.472812
$528$	0	0
$529$	5.96556	0.259372
$530$	7.09017	0.307977
$531$	0	0
$532$	0	0
$533$	−11.1803	−0.484274
$534$	0	0
$535$	2.70820	0.117086
$536$	−25.3262	−1.09393
$537$	0	0
$538$	17.3262	0.746987
$539$	0	0
$540$	0	0
$541$	−13.5066	−0.580693	−0.290347	−	0.956922i	$-0.593771\pi$
$541$	−13.5066	−0.580693	−0.290347	+	0.956922i	$0.593771\pi$
$542$	39.7426	1.70709
$543$	0	0
$544$	−5.47214	−0.234616
$545$	12.5623	0.538110
$546$	0	0
$547$	19.3607	0.827803	0.413901	−	0.910322i	$-0.364166\pi$
$547$	19.3607	0.827803	0.413901	+	0.910322i	$0.364166\pi$
$548$	−8.58359	−0.366673
$549$	0	0
$550$	−6.85410	−0.292260
$551$	67.7984	2.88831
$552$	0	0
$553$	0	0
$554$	7.56231	0.321292
$555$	0	0
$556$	−1.96556	−0.0833582
$557$	16.7984	0.711770	0.355885	−	0.934530i	$-0.384179\pi$
$557$	16.7984	0.711770	0.355885	+	0.934530i	$0.384179\pi$
$558$	0	0
$559$	13.7426	0.581252
$560$	0	0
$561$	0	0
$562$	22.0344	0.929467
$563$	21.6869	0.913995	0.456997	−	0.889468i	$-0.348925\pi$
$563$	21.6869	0.913995	0.456997	+	0.889468i	$0.348925\pi$
$564$	0	0
$565$	−5.85410	−0.246284
$566$	−34.2148	−1.43815
$567$	0	0
$568$	8.61803	0.361605
$569$	37.7639	1.58315	0.791573	−	0.611074i	$-0.209262\pi$
$569$	37.7639	1.58315	0.791573	+	0.611074i	$0.209262\pi$
$570$	0	0
$571$	1.25735	0.0526186	0.0263093	−	0.999654i	$-0.491625\pi$
$571$	1.25735	0.0526186	0.0263093	+	0.999654i	$0.491625\pi$
$572$	3.61803	0.151278
$573$	0	0
$574$	0	0
$575$	−5.38197	−0.224443
$576$	0	0
$577$	−22.1246	−0.921060	−0.460530	−	0.887644i	$-0.652341\pi$
$577$	−22.1246	−0.921060	−0.460530	+	0.887644i	$0.652341\pi$
$578$	−23.2705	−0.967926
$579$	0	0
$580$	5.90983	0.245392
$581$	0	0
$582$	0	0
$583$	−18.5623	−0.768772
$584$	31.1803	1.29025
$585$	0	0
$586$	−10.1803	−0.420546
$587$	−7.20163	−0.297243	−0.148621	−	0.988894i	$-0.547484\pi$
$587$	−7.20163	−0.297243	−0.148621	+	0.988894i	$0.547484\pi$
$588$	0	0
$589$	−47.5623	−1.95977
$590$	2.09017	0.0860509
$591$	0	0
$592$	−32.5623	−1.33830
$593$	−34.3050	−1.40874	−0.704368	−	0.709835i	$-0.748769\pi$
$593$	−34.3050	−1.40874	−0.704368	+	0.709835i	$0.748769\pi$
$594$	0	0
$595$	0	0
$596$	9.76393	0.399946
$597$	0	0
$598$	12.0344	0.492125
$599$	38.9443	1.59122	0.795610	−	0.605809i	$-0.207151\pi$
$599$	38.9443	1.59122	0.795610	+	0.605809i	$0.207151\pi$
$600$	0	0
$601$	−0.798374	−0.0325663	−0.0162832	−	0.999867i	$-0.505183\pi$
$601$	−0.798374	−0.0325663	−0.0162832	+	0.999867i	$0.505183\pi$
$602$	0	0
$603$	0	0
$604$	−0.0344419	−0.00140142
$605$	6.94427	0.282325
$606$	0	0
$607$	−24.4164	−0.991031	−0.495516	−	0.868599i	$-0.665021\pi$
$607$	−24.4164	−0.991031	−0.495516	+	0.868599i	$0.665021\pi$
$608$	−23.9787	−0.972465
$609$	0	0
$610$	22.4164	0.907614
$611$	−15.2016	−0.614992
$612$	0	0
$613$	−7.29180	−0.294513	−0.147256	−	0.989098i	$-0.547044\pi$
$613$	−7.29180	−0.294513	−0.147256	+	0.989098i	$0.547044\pi$
$614$	−31.4164	−1.26786
$615$	0	0
$616$	0	0
$617$	37.7984	1.52171	0.760853	−	0.648925i	$-0.224781\pi$
$617$	37.7984	1.52171	0.760853	+	0.648925i	$0.224781\pi$
$618$	0	0
$619$	47.5410	1.91083	0.955417	−	0.295258i	$-0.0954057\pi$
$619$	47.5410	1.91083	0.955417	+	0.295258i	$0.0954057\pi$
$620$	−4.14590	−0.166503
$621$	0	0
$622$	23.1803	0.929447
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	37.5066	1.49906
$627$	0	0
$628$	3.23607	0.129133
$629$	10.8541	0.432781
$630$	0	0
$631$	41.8885	1.66756	0.833778	−	0.552099i	$-0.186173\pi$
$631$	41.8885	1.66756	0.833778	+	0.552099i	$0.186173\pi$
$632$	−12.5623	−0.499702
$633$	0	0
$634$	9.79837	0.389143
$635$	0.0901699	0.00357829
$636$	0	0
$637$	0	0
$638$	−65.5410	−2.59479
$639$	0	0
$640$	13.6180	0.538300
$641$	0.729490	0.0288131	0.0144066	−	0.999896i	$-0.495414\pi$
$641$	0.729490	0.0288131	0.0144066	+	0.999896i	$0.495414\pi$
$642$	0	0
$643$	−41.7426	−1.64617	−0.823085	−	0.567919i	$-0.807749\pi$
$643$	−41.7426	−1.64617	−0.823085	+	0.567919i	$0.807749\pi$
$644$	0	0
$645$	0	0
$646$	18.5623	0.730324
$647$	−13.0557	−0.513274	−0.256637	−	0.966508i	$-0.582614\pi$
$647$	−13.0557	−0.513274	−0.256637	+	0.966508i	$0.582614\pi$
$648$	0	0
$649$	−5.47214	−0.214800
$650$	−2.23607	−0.0877058
$651$	0	0
$652$	−1.81966	−0.0712634
$653$	−6.03444	−0.236146	−0.118073	−	0.993005i	$-0.537672\pi$
$653$	−6.03444	−0.236146	−0.118073	+	0.993005i	$0.537672\pi$
$654$	0	0
$655$	4.38197	0.171218
$656$	−39.2705	−1.53326
$657$	0	0
$658$	0	0
$659$	0.763932	0.0297586	0.0148793	−	0.999889i	$-0.495264\pi$
$659$	0.763932	0.0297586	0.0148793	+	0.999889i	$0.495264\pi$
$660$	0	0
$661$	16.2148	0.630682	0.315341	−	0.948978i	$-0.397881\pi$
$661$	16.2148	0.630682	0.315341	+	0.948978i	$0.397881\pi$
$662$	−51.5410	−2.00320
$663$	0	0
$664$	−23.9443	−0.929218
$665$	0	0
$666$	0	0
$667$	−51.4640	−1.99269
$668$	−4.61803	−0.178677
$669$	0	0
$670$	18.3262	0.708004
$671$	−58.6869	−2.26558
$672$	0	0
$673$	−46.4721	−1.79137	−0.895685	−	0.444690i	$-0.853314\pi$
$673$	−46.4721	−1.79137	−0.895685	+	0.444690i	$0.853314\pi$
$674$	−51.3607	−1.97834
$675$	0	0
$676$	−6.85410	−0.263619
$677$	−17.6525	−0.678440	−0.339220	−	0.940707i	$-0.610163\pi$
$677$	−17.6525	−0.678440	−0.339220	+	0.940707i	$0.610163\pi$
$678$	0	0
$679$	0	0
$680$	−3.61803	−0.138745
$681$	0	0
$682$	45.9787	1.76062
$683$	42.9787	1.64453	0.822267	−	0.569101i	$-0.192709\pi$
$683$	42.9787	1.64453	0.822267	+	0.569101i	$0.192709\pi$
$684$	0	0
$685$	−13.8885	−0.530654
$686$	0	0
$687$	0	0
$688$	48.2705	1.84030
$689$	−6.05573	−0.230705
$690$	0	0
$691$	30.5279	1.16133	0.580667	−	0.814141i	$-0.302792\pi$
$691$	30.5279	1.16133	0.580667	+	0.814141i	$0.302792\pi$
$692$	7.09017	0.269528
$693$	0	0
$694$	20.6525	0.783957
$695$	−3.18034	−0.120637
$696$	0	0
$697$	13.0902	0.495826
$698$	−50.7426	−1.92064
$699$	0	0
$700$	0	0
$701$	−6.88854	−0.260177	−0.130088	−	0.991502i	$-0.541526\pi$
$701$	−6.88854	−0.260177	−0.130088	+	0.991502i	$0.541526\pi$
$702$	0	0
$703$	47.5623	1.79385
$704$	−17.9443	−0.676300
$705$	0	0
$706$	18.8885	0.710880
$707$	0	0
$708$	0	0
$709$	−9.59675	−0.360413	−0.180207	−	0.983629i	$-0.557677\pi$
$709$	−9.59675	−0.360413	−0.180207	+	0.983629i	$0.557677\pi$
$710$	−6.23607	−0.234035
$711$	0	0
$712$	−3.29180	−0.123365
$713$	36.1033	1.35208
$714$	0	0
$715$	5.85410	0.218931
$716$	4.32624	0.161679
$717$	0	0
$718$	−58.9230	−2.19899
$719$	12.9656	0.483534	0.241767	−	0.970334i	$-0.422273\pi$
$719$	12.9656	0.483534	0.241767	+	0.970334i	$0.422273\pi$
$720$	0	0
$721$	0	0
$722$	50.5967	1.88302
$723$	0	0
$724$	12.0344	0.447257
$725$	9.56231	0.355135
$726$	0	0
$727$	11.4508	0.424689	0.212344	−	0.977195i	$-0.431890\pi$
$727$	11.4508	0.424689	0.212344	+	0.977195i	$0.431890\pi$
$728$	0	0
$729$	0	0
$730$	−22.5623	−0.835068
$731$	−16.0902	−0.595116
$732$	0	0
$733$	−10.5623	−0.390128	−0.195064	−	0.980791i	$-0.562491\pi$
$733$	−10.5623	−0.390128	−0.195064	+	0.980791i	$0.562491\pi$
$734$	24.0344	0.887127
$735$	0	0
$736$	18.2016	0.670921
$737$	−47.9787	−1.76732
$738$	0	0
$739$	−20.6525	−0.759714	−0.379857	−	0.925045i	$-0.624027\pi$
$739$	−20.6525	−0.759714	−0.379857	+	0.925045i	$0.624027\pi$
$740$	4.14590	0.152406
$741$	0	0
$742$	0	0
$743$	3.32624	0.122028	0.0610139	−	0.998137i	$-0.480567\pi$
$743$	3.32624	0.122028	0.0610139	+	0.998137i	$0.480567\pi$
$744$	0	0
$745$	15.7984	0.578808
$746$	−27.4721	−1.00583
$747$	0	0
$748$	−4.23607	−0.154886
$749$	0	0
$750$	0	0
$751$	−47.8885	−1.74748	−0.873739	−	0.486395i	$-0.838312\pi$
$751$	−47.8885	−1.74748	−0.873739	+	0.486395i	$0.838312\pi$
$752$	−53.3951	−1.94712
$753$	0	0
$754$	−21.3820	−0.778685
$755$	−0.0557281	−0.00202815
$756$	0	0
$757$	6.27051	0.227906	0.113953	−	0.993486i	$-0.463649\pi$
$757$	6.27051	0.227906	0.113953	+	0.993486i	$0.463649\pi$
$758$	19.2361	0.698685
$759$	0	0
$760$	−15.8541	−0.575089
$761$	−9.96556	−0.361251	−0.180626	−	0.983552i	$-0.557812\pi$
$761$	−9.96556	−0.361251	−0.180626	+	0.983552i	$0.557812\pi$
$762$	0	0
$763$	0	0
$764$	−11.6180	−0.420326
$765$	0	0
$766$	−18.8541	−0.681226
$767$	−1.78522	−0.0644605
$768$	0	0
$769$	−22.5967	−0.814860	−0.407430	−	0.913237i	$-0.633575\pi$
$769$	−22.5967	−0.814860	−0.407430	+	0.913237i	$0.633575\pi$
$770$	0	0
$771$	0	0
$772$	−0.527864	−0.0189982
$773$	−17.6738	−0.635681	−0.317841	−	0.948144i	$-0.602958\pi$
$773$	−17.6738	−0.635681	−0.317841	+	0.948144i	$0.602958\pi$
$774$	0	0
$775$	−6.70820	−0.240966
$776$	−7.43769	−0.266998
$777$	0	0
$778$	−41.7426	−1.49655
$779$	57.3607	2.05516
$780$	0	0
$781$	16.3262	0.584199
$782$	−14.0902	−0.503863
$783$	0	0
$784$	0	0
$785$	5.23607	0.186883
$786$	0	0
$787$	3.23607	0.115353	0.0576767	−	0.998335i	$-0.481631\pi$
$787$	3.23607	0.115353	0.0576767	+	0.998335i	$0.481631\pi$
$788$	12.0344	0.428709
$789$	0	0
$790$	9.09017	0.323414
$791$	0	0
$792$	0	0
$793$	−19.1459	−0.679891
$794$	−14.6525	−0.519997
$795$	0	0
$796$	−1.00000	−0.0354441
$797$	23.6869	0.839034	0.419517	−	0.907748i	$-0.362200\pi$
$797$	23.6869	0.839034	0.419517	+	0.907748i	$0.362200\pi$
$798$	0	0
$799$	17.7984	0.629661
$800$	−3.38197	−0.119571
$801$	0	0
$802$	28.1246	0.993115
$803$	59.0689	2.08450
$804$	0	0
$805$	0	0
$806$	15.0000	0.528352
$807$	0	0
$808$	−12.3607	−0.434847
$809$	6.63932	0.233426	0.116713	−	0.993166i	$-0.462764\pi$
$809$	6.63932	0.233426	0.116713	+	0.993166i	$0.462764\pi$
$810$	0	0
$811$	−8.32624	−0.292374	−0.146187	−	0.989257i	$-0.546700\pi$
$811$	−8.32624	−0.292374	−0.146187	+	0.989257i	$0.546700\pi$
$812$	0	0
$813$	0	0
$814$	−45.9787	−1.61155
$815$	−2.94427	−0.103133
$816$	0	0
$817$	−70.5066	−2.46671
$818$	21.1246	0.738605
$819$	0	0
$820$	5.00000	0.174608
$821$	29.5836	1.03247	0.516237	−	0.856446i	$-0.327332\pi$
$821$	29.5836	1.03247	0.516237	+	0.856446i	$0.327332\pi$
$822$	0	0
$823$	−37.9443	−1.32265	−0.661327	−	0.750098i	$-0.730006\pi$
$823$	−37.9443	−1.32265	−0.661327	+	0.750098i	$0.730006\pi$
$824$	23.7426	0.827114
$825$	0	0
$826$	0	0
$827$	11.5410	0.401321	0.200660	−	0.979661i	$-0.435691\pi$
$827$	11.5410	0.401321	0.200660	+	0.979661i	$0.435691\pi$
$828$	0	0
$829$	13.3607	0.464036	0.232018	−	0.972712i	$-0.425467\pi$
$829$	13.3607	0.464036	0.232018	+	0.972712i	$0.425467\pi$
$830$	17.3262	0.601402
$831$	0	0
$832$	−5.85410	−0.202954
$833$	0	0
$834$	0	0
$835$	−7.47214	−0.258584
$836$	−18.5623	−0.641991
$837$	0	0
$838$	12.0902	0.417648
$839$	0.381966	0.0131869	0.00659347	−	0.999978i	$-0.497901\pi$
$839$	0.381966	0.0131869	0.00659347	+	0.999978i	$0.497901\pi$
$840$	0	0
$841$	62.4377	2.15302
$842$	13.6525	0.470495
$843$	0	0
$844$	−10.5836	−0.364302
$845$	−11.0902	−0.381513
$846$	0	0
$847$	0	0
$848$	−21.2705	−0.730432
$849$	0	0
$850$	2.61803	0.0897978
$851$	−36.1033	−1.23761
$852$	0	0
$853$	2.16718	0.0742030	0.0371015	−	0.999312i	$-0.488188\pi$
$853$	2.16718	0.0742030	0.0371015	+	0.999312i	$0.488188\pi$
$854$	0	0
$855$	0	0
$856$	−6.05573	−0.206981
$857$	−26.2361	−0.896207	−0.448104	−	0.893982i	$-0.647900\pi$
$857$	−26.2361	−0.896207	−0.448104	+	0.893982i	$0.647900\pi$
$858$	0	0
$859$	−21.5967	−0.736872	−0.368436	−	0.929653i	$-0.620107\pi$
$859$	−21.5967	−0.736872	−0.368436	+	0.929653i	$0.620107\pi$
$860$	−6.14590	−0.209573
$861$	0	0
$862$	−49.2705	−1.67816
$863$	51.7771	1.76251	0.881256	−	0.472639i	$-0.156698\pi$
$863$	51.7771	1.76251	0.881256	+	0.472639i	$0.156698\pi$
$864$	0	0
$865$	11.4721	0.390064
$866$	34.3262	1.16645
$867$	0	0
$868$	0	0
$869$	−23.7984	−0.807305
$870$	0	0
$871$	−15.6525	−0.530364
$872$	−28.0902	−0.951253
$873$	0	0
$874$	−61.7426	−2.08848
$875$	0	0
$876$	0	0
$877$	27.9443	0.943611	0.471806	−	0.881703i	$-0.343602\pi$
$877$	27.9443	0.943611	0.471806	+	0.881703i	$0.343602\pi$
$878$	−18.2705	−0.616600
$879$	0	0
$880$	20.5623	0.693155
$881$	55.8885	1.88293	0.941466	−	0.337107i	$-0.109448\pi$
$881$	55.8885	1.88293	0.941466	+	0.337107i	$0.109448\pi$
$882$	0	0
$883$	−19.6525	−0.661358	−0.330679	−	0.943743i	$-0.607278\pi$
$883$	−19.6525	−0.661358	−0.330679	+	0.943743i	$0.607278\pi$
$884$	−1.38197	−0.0464805
$885$	0	0
$886$	65.8328	2.21170
$887$	−14.7639	−0.495724	−0.247862	−	0.968795i	$-0.579728\pi$
$887$	−14.7639	−0.495724	−0.247862	+	0.968795i	$0.579728\pi$
$888$	0	0
$889$	0	0
$890$	2.38197	0.0798437
$891$	0	0
$892$	−4.61803	−0.154623
$893$	77.9919	2.60990
$894$	0	0
$895$	7.00000	0.233984
$896$	0	0
$897$	0	0
$898$	−33.1246	−1.10538
$899$	−64.1459	−2.13939
$900$	0	0
$901$	7.09017	0.236208
$902$	−55.4508	−1.84631
$903$	0	0
$904$	13.0902	0.435373
$905$	19.4721	0.647276
$906$	0	0
$907$	8.14590	0.270480	0.135240	−	0.990813i	$-0.456819\pi$
$907$	8.14590	0.270480	0.135240	+	0.990813i	$0.456819\pi$
$908$	4.59675	0.152548
$909$	0	0
$910$	0	0
$911$	−7.63932	−0.253102	−0.126551	−	0.991960i	$-0.540391\pi$
$911$	−7.63932	−0.253102	−0.126551	+	0.991960i	$0.540391\pi$
$912$	0	0
$913$	−45.3607	−1.50122
$914$	28.0344	0.927297
$915$	0	0
$916$	11.0902	0.366430
$917$	0	0
$918$	0	0
$919$	42.1591	1.39070	0.695349	−	0.718672i	$-0.255250\pi$
$919$	42.1591	1.39070	0.695349	+	0.718672i	$0.255250\pi$
$920$	12.0344	0.396764
$921$	0	0
$922$	−36.8885	−1.21486
$923$	5.32624	0.175315
$924$	0	0
$925$	6.70820	0.220564
$926$	−1.00000	−0.0328620
$927$	0	0
$928$	−32.3394	−1.06159
$929$	−57.1033	−1.87350	−0.936750	−	0.350000i	$-0.886182\pi$
$929$	−57.1033	−1.87350	−0.936750	+	0.350000i	$0.886182\pi$
$930$	0	0
$931$	0	0
$932$	−10.5623	−0.345980
$933$	0	0
$934$	27.4164	0.897092
$935$	−6.85410	−0.224153
$936$	0	0
$937$	−26.9443	−0.880231	−0.440115	−	0.897941i	$-0.645063\pi$
$937$	−26.9443	−0.880231	−0.440115	+	0.897941i	$0.645063\pi$
$938$	0	0
$939$	0	0
$940$	6.79837	0.221739
$941$	−45.7639	−1.49186	−0.745931	−	0.666023i	$-0.767995\pi$
$941$	−45.7639	−1.49186	−0.745931	+	0.666023i	$0.767995\pi$
$942$	0	0
$943$	−43.5410	−1.41789
$944$	−6.27051	−0.204088
$945$	0	0
$946$	68.1591	2.21604
$947$	5.43769	0.176701	0.0883507	−	0.996089i	$-0.471840\pi$
$947$	5.43769	0.176701	0.0883507	+	0.996089i	$0.471840\pi$
$948$	0	0
$949$	19.2705	0.625547
$950$	11.4721	0.372205
$951$	0	0
$952$	0	0
$953$	35.6525	1.15490	0.577448	−	0.816427i	$-0.304048\pi$
$953$	35.6525	1.15490	0.577448	+	0.816427i	$0.304048\pi$
$954$	0	0
$955$	−18.7984	−0.608301
$956$	−7.20163	−0.232917
$957$	0	0
$958$	41.9230	1.35447
$959$	0	0
$960$	0	0
$961$	14.0000	0.451613
$962$	−15.0000	−0.483619
$963$	0	0
$964$	4.70820	0.151641
$965$	−0.854102	−0.0274945
$966$	0	0
$967$	−8.52786	−0.274238	−0.137119	−	0.990555i	$-0.543784\pi$
$967$	−8.52786	−0.274238	−0.137119	+	0.990555i	$0.543784\pi$
$968$	−15.5279	−0.499084
$969$	0	0
$970$	5.38197	0.172805
$971$	22.2492	0.714012	0.357006	−	0.934102i	$-0.383798\pi$
$971$	22.2492	0.714012	0.357006	+	0.934102i	$0.383798\pi$
$972$	0	0
$973$	0	0
$974$	−25.1246	−0.805044
$975$	0	0
$976$	−67.2492	−2.15260
$977$	−14.3607	−0.459439	−0.229719	−	0.973257i	$-0.573781\pi$
$977$	−14.3607	−0.459439	−0.229719	+	0.973257i	$0.573781\pi$
$978$	0	0
$979$	−6.23607	−0.199306
$980$	0	0
$981$	0	0
$982$	31.5623	1.00719
$983$	46.9787	1.49839	0.749194	−	0.662350i	$-0.230441\pi$
$983$	46.9787	1.49839	0.749194	+	0.662350i	$0.230441\pi$
$984$	0	0
$985$	19.4721	0.620434
$986$	25.0344	0.797259
$987$	0	0
$988$	−6.05573	−0.192658
$989$	53.5197	1.70183
$990$	0	0
$991$	−42.8885	−1.36240	−0.681200	−	0.732098i	$-0.738541\pi$
$991$	−42.8885	−1.36240	−0.681200	+	0.732098i	$0.738541\pi$
$992$	22.6869	0.720310
$993$	0	0
$994$	0	0
$995$	−1.61803	−0.0512951
$996$	0	0
$997$	−14.7082	−0.465813	−0.232907	−	0.972499i	$-0.574824\pi$
$997$	−14.7082	−0.465813	−0.232907	+	0.972499i	$0.574824\pi$
$998$	5.43769	0.172127
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6615.2.a.r.1.2		2
3.2	odd	2		6615.2.a.o.1.1		2
7.6	odd	2		945.2.a.g.1.2	yes	2
21.20	even	2		945.2.a.e.1.1	✓	2
35.34	odd	2		4725.2.a.w.1.1		2
105.104	even	2		4725.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.e.1.1	✓	2	21.20	even	2
945.2.a.g.1.2	yes	2	7.6	odd	2
4725.2.a.w.1.1		2	35.34	odd	2
4725.2.a.be.1.2		2	105.104	even	2
6615.2.a.o.1.1		2	3.2	odd	2
6615.2.a.r.1.2		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:	\( N \)	\(=\)	\( 6615 = 3^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6615.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -2.23607 q^{8} +1.61803 q^{10} -4.23607 q^{11} -1.38197 q^{13} -4.85410 q^{16} +1.61803 q^{17} +7.09017 q^{19} +0.618034 q^{20} -6.85410 q^{22} -5.38197 q^{23} +1.00000 q^{25} -2.23607 q^{26} +9.56231 q^{29} -6.70820 q^{31} -3.38197 q^{32} +2.61803 q^{34} +6.70820 q^{37} +11.4721 q^{38} -2.23607 q^{40} +8.09017 q^{41} -9.94427 q^{43} -2.61803 q^{44} -8.70820 q^{46} +11.0000 q^{47} +1.61803 q^{50} -0.854102 q^{52} +4.38197 q^{53} -4.23607 q^{55} +15.4721 q^{58} +1.29180 q^{59} +13.8541 q^{61} -10.8541 q^{62} +4.23607 q^{64} -1.38197 q^{65} +11.3262 q^{67} +1.00000 q^{68} -3.85410 q^{71} -13.9443 q^{73} +10.8541 q^{74} +4.38197 q^{76} +5.61803 q^{79} -4.85410 q^{80} +13.0902 q^{82} +10.7082 q^{83} +1.61803 q^{85} -16.0902 q^{86} +9.47214 q^{88} +1.47214 q^{89} -3.32624 q^{92} +17.7984 q^{94} +7.09017 q^{95} +3.32624 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5}+O(q^{10}) \) 2 * q + q^2 - q^4 + 2 * q^5	\( 2 q + q^{2} - q^{4} + 2 q^{5} + q^{10} - 4 q^{11} - 5 q^{13} - 3 q^{16} + q^{17} + 3 q^{19} - q^{20} - 7 q^{22} - 13 q^{23} + 2 q^{25} - q^{29} - 9 q^{32} + 3 q^{34} + 14 q^{38} + 5 q^{41} - 2 q^{43} - 3 q^{44} - 4 q^{46} + 22 q^{47} + q^{50} + 5 q^{52} + 11 q^{53} - 4 q^{55} + 22 q^{58} + 16 q^{59} + 21 q^{61} - 15 q^{62} + 4 q^{64} - 5 q^{65} + 7 q^{67} + 2 q^{68} - q^{71} - 10 q^{73} + 15 q^{74} + 11 q^{76} + 9 q^{79} - 3 q^{80} + 15 q^{82} + 8 q^{83} + q^{85} - 21 q^{86} + 10 q^{88} - 6 q^{89} + 9 q^{92} + 11 q^{94} + 3 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q + q^2 - q^4 + 2 * q^5 + q^10 - 4 * q^11 - 5 * q^13 - 3 * q^16 + q^17 + 3 * q^19 - q^20 - 7 * q^22 - 13 * q^23 + 2 * q^25 - q^29 - 9 * q^32 + 3 * q^34 + 14 * q^38 + 5 * q^41 - 2 * q^43 - 3 * q^44 - 4 * q^46 + 22 * q^47 + q^50 + 5 * q^52 + 11 * q^53 - 4 * q^55 + 22 * q^58 + 16 * q^59 + 21 * q^61 - 15 * q^62 + 4 * q^64 - 5 * q^65 + 7 * q^67 + 2 * q^68 - q^71 - 10 * q^73 + 15 * q^74 + 11 * q^76 + 9 * q^79 - 3 * q^80 + 15 * q^82 + 8 * q^83 + q^85 - 21 * q^86 + 10 * q^88 - 6 * q^89 + 9 * q^92 + 11 * q^94 + 3 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

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Database

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