LMFDB - Embedded newform 945.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, 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("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	\(q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} -1.61803 q^{10} +4.23607 q^{11} +1.38197 q^{13} +1.61803 q^{14} -4.85410 q^{16} +1.61803 q^{17} -7.09017 q^{19} +0.618034 q^{20} -6.85410 q^{22} +5.38197 q^{23} +1.00000 q^{25} -2.23607 q^{26} -0.618034 q^{28} -9.56231 q^{29} +6.70820 q^{31} +3.38197 q^{32} -2.61803 q^{34} -1.00000 q^{35} +6.70820 q^{37} +11.4721 q^{38} +2.23607 q^{40} +8.09017 q^{41} -9.94427 q^{43} +2.61803 q^{44} -8.70820 q^{46} +11.0000 q^{47} +1.00000 q^{49} -1.61803 q^{50} +0.854102 q^{52} -4.38197 q^{53} +4.23607 q^{55} -2.23607 q^{56} +15.4721 q^{58} +1.29180 q^{59} -13.8541 q^{61} -10.8541 q^{62} +4.23607 q^{64} +1.38197 q^{65} +11.3262 q^{67} +1.00000 q^{68} +1.61803 q^{70} +3.85410 q^{71} +13.9443 q^{73} -10.8541 q^{74} -4.38197 q^{76} -4.23607 q^{77} +5.61803 q^{79} -4.85410 q^{80} -13.0902 q^{82} +10.7082 q^{83} +1.61803 q^{85} +16.0902 q^{86} +9.47214 q^{88} +1.47214 q^{89} -1.38197 q^{91} +3.32624 q^{92} -17.7984 q^{94} -7.09017 q^{95} -3.32624 q^{97} -1.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} + 4 q^{11} + 5 q^{13} + q^{14} - 3 q^{16} + q^{17} - 3 q^{19} - q^{20} - 7 q^{22} + 13 q^{23} + 2 q^{25} + q^{28} + q^{29} + 9 q^{32} - 3 q^{34} - 2 q^{35} + 14 q^{38} + 5 q^{41} - 2 q^{43} + 3 q^{44} - 4 q^{46} + 22 q^{47} + 2 q^{49} - 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^{70} + q^{71} + 10 q^{73} - 15 q^{74} - 11 q^{76} - 4 q^{77} + 9 q^{79} - 3 q^{80} - 15 q^{82} + 8 q^{83} + q^{85} + 21 q^{86} + 10 q^{88} - 6 q^{89} - 5 q^{91} - 9 q^{92} - 11 q^{94} - 3 q^{95} + 9 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 + 4 * q^11 + 5 * q^13 + q^14 - 3 * q^16 + q^17 - 3 * q^19 - q^20 - 7 * q^22 + 13 * q^23 + 2 * q^25 + q^28 + q^29 + 9 * q^32 - 3 * q^34 - 2 * q^35 + 14 * q^38 + 5 * q^41 - 2 * q^43 + 3 * q^44 - 4 * q^46 + 22 * q^47 + 2 * q^49 - 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^70 + q^71 + 10 * q^73 - 15 * q^74 - 11 * q^76 - 4 * q^77 + 9 * q^79 - 3 * q^80 - 15 * q^82 + 8 * q^83 + q^85 + 21 * q^86 + 10 * q^88 - 6 * q^89 - 5 * q^91 - 9 * q^92 - 11 * q^94 - 3 * q^95 + 9 * q^97 - 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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	−1.61803	−0.511667
$11$	4.23607	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$11$	4.23607	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$12$	0	0
$13$	1.38197	0.383288	0.191644	−	0.981464i	$-0.438618\pi$
$13$	1.38197	0.383288	0.191644	+	0.981464i	$0.438618\pi$
$14$	1.61803	0.432438
$15$	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	$-0.802330\pi$
$19$	−7.09017	−1.62660	−0.813298	+	0.581847i	$0.802330\pi$
$20$	0.618034	0.138197
$21$	0	0
$22$	−6.85410	−1.46130
$23$	5.38197	1.12222	0.561109	−	0.827742i	$-0.310375\pi$
$23$	5.38197	1.12222	0.561109	+	0.827742i	$0.310375\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.23607	−0.438529
$27$	0	0
$28$	−0.618034	−0.116797
$29$	−9.56231	−1.77568	−0.887838	−	0.460157i	$-0.847793\pi$
$29$	−9.56231	−1.77568	−0.887838	+	0.460157i	$0.847793\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−2.61803	−0.448989
$35$	−1.00000	−0.169031
$36$	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
$49$	1.00000	0.142857
$50$	−1.61803	−0.228825
$51$	0	0
$52$	0.854102	0.118443
$53$	−4.38197	−0.601909	−0.300955	−	0.953638i	$-0.597305\pi$
$53$	−4.38197	−0.601909	−0.300955	+	0.953638i	$0.597305\pi$
$54$	0	0
$55$	4.23607	0.571191
$56$	−2.23607	−0.298807
$57$	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	$-0.847158\pi$
$61$	−13.8541	−1.77384	−0.886918	+	0.461927i	$0.847158\pi$
$62$	−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
$70$	1.61803	0.193392
$71$	3.85410	0.457398	0.228699	−	0.973497i	$-0.426553\pi$
$71$	3.85410	0.457398	0.228699	+	0.973497i	$0.426553\pi$
$72$	0	0
$73$	13.9443	1.63205	0.816027	−	0.578014i	$-0.196172\pi$
$73$	13.9443	1.63205	0.816027	+	0.578014i	$0.196172\pi$
$74$	−10.8541	−1.26176
$75$	0	0
$76$	−4.38197	−0.502646
$77$	−4.23607	−0.482745
$78$	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
$91$	−1.38197	−0.144869
$92$	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	$-0.554010\pi$
$97$	−3.32624	−0.337728	−0.168864	+	0.985639i	$0.554010\pi$
$98$	−1.61803	−0.163446
$99$	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	$-0.324771\pi$
$103$	10.6180	1.04623	0.523113	+	0.852263i	$0.324771\pi$
$104$	3.09017	0.303016
$105$	0	0
$106$	7.09017	0.688658
$107$	−2.70820	−0.261812	−0.130906	−	0.991395i	$-0.541789\pi$
$107$	−2.70820	−0.261812	−0.130906	+	0.991395i	$0.541789\pi$
$108$	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
$112$	4.85410	0.458670
$113$	5.85410	0.550708	0.275354	−	0.961343i	$-0.411205\pi$
$113$	5.85410	0.550708	0.275354	+	0.961343i	$0.411205\pi$
$114$	0	0
$115$	5.38197	0.501871
$116$	−5.90983	−0.548714
$117$	0	0
$118$	−2.09017	−0.192416
$119$	−1.61803	−0.148325
$120$	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
$133$	7.09017	0.614796
$134$	−18.3262	−1.58315
$135$	0	0
$136$	3.61803	0.310244
$137$	13.8885	1.18658	0.593289	−	0.804989i	$-0.297829\pi$
$137$	13.8885	1.18658	0.593289	+	0.804989i	$0.297829\pi$
$138$	0	0
$139$	3.18034	0.269753	0.134876	−	0.990862i	$-0.456936\pi$
$139$	3.18034	0.269753	0.134876	+	0.990862i	$0.456936\pi$
$140$	−0.618034	−0.0522334
$141$	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	$-0.724029\pi$
$149$	−15.7984	−1.29425	−0.647127	+	0.762383i	$0.724029\pi$
$150$	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
$154$	6.85410	0.552319
$155$	6.70820	0.538816
$156$	0	0
$157$	−5.23607	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$157$	−5.23607	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$158$	−9.09017	−0.723175
$159$	0	0
$160$	3.38197	0.267368
$161$	−5.38197	−0.424158
$162$	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
$175$	−1.00000	−0.0755929
$176$	−20.5623	−1.54994
$177$	0	0
$178$	−2.38197	−0.178536
$179$	−7.00000	−0.523205	−0.261602	−	0.965176i	$-0.584251\pi$
$179$	−7.00000	−0.523205	−0.261602	+	0.965176i	$0.584251\pi$
$180$	0	0
$181$	−19.4721	−1.44735	−0.723676	−	0.690140i	$-0.757549\pi$
$181$	−19.4721	−1.44735	−0.723676	+	0.690140i	$0.757549\pi$
$182$	2.23607	0.165748
$183$	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	$-0.261936\pi$
$191$	18.7984	1.36020	0.680101	+	0.733118i	$0.261936\pi$
$192$	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
$196$	0.618034	0.0441453
$197$	−19.4721	−1.38733	−0.693666	−	0.720297i	$-0.744006\pi$
$197$	−19.4721	−1.38733	−0.693666	+	0.720297i	$0.744006\pi$
$198$	0	0
$199$	1.61803	0.114699	0.0573497	−	0.998354i	$-0.481735\pi$
$199$	1.61803	0.114699	0.0573497	+	0.998354i	$0.481735\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	−8.94427	−0.629317
$203$	9.56231	0.671142
$204$	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
$217$	−6.70820	−0.455383
$218$	−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	$-0.419508\pi$
$223$	7.47214	0.500371	0.250186	+	0.968198i	$0.419508\pi$
$224$	−3.38197	−0.225967
$225$	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	$-0.702015\pi$
$229$	−17.9443	−1.18579	−0.592895	+	0.805279i	$0.702015\pi$
$230$	−8.70820	−0.574202
$231$	0	0
$232$	−21.3820	−1.40379
$233$	17.0902	1.11961	0.559807	−	0.828623i	$-0.310875\pi$
$233$	17.0902	1.11961	0.559807	+	0.828623i	$0.310875\pi$
$234$	0	0
$235$	11.0000	0.717561
$236$	0.798374	0.0519697
$237$	0	0
$238$	2.61803	0.169702
$239$	11.6525	0.753736	0.376868	−	0.926267i	$-0.377001\pi$
$239$	11.6525	0.753736	0.376868	+	0.926267i	$0.377001\pi$
$240$	0	0
$241$	−7.61803	−0.490721	−0.245360	−	0.969432i	$-0.578906\pi$
$241$	−7.61803	−0.490721	−0.245360	+	0.969432i	$0.578906\pi$
$242$	−11.2361	−0.722282
$243$	0	0
$244$	−8.56231	−0.548145
$245$	1.00000	0.0638877
$246$	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
$259$	−6.70820	−0.416828
$260$	0.854102	0.0529692
$261$	0	0
$262$	−7.09017	−0.438032
$263$	−30.7426	−1.89567	−0.947836	−	0.318757i	$-0.896735\pi$
$263$	−30.7426	−1.89567	−0.947836	+	0.318757i	$0.896735\pi$
$264$	0	0
$265$	−4.38197	−0.269182
$266$	−11.4721	−0.703402
$267$	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	$-0.768041\pi$
$271$	−24.5623	−1.49205	−0.746027	+	0.665916i	$0.768041\pi$
$272$	−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
$280$	−2.23607	−0.133631
$281$	−13.6180	−0.812384	−0.406192	−	0.913788i	$-0.633144\pi$
$281$	−13.6180	−0.812384	−0.406192	+	0.913788i	$0.633144\pi$
$282$	0	0
$283$	21.1459	1.25699	0.628497	−	0.777812i	$-0.283671\pi$
$283$	21.1459	1.25699	0.628497	+	0.777812i	$0.283671\pi$
$284$	2.38197	0.141344
$285$	0	0
$286$	−9.47214	−0.560099
$287$	−8.09017	−0.477548
$288$	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
$301$	9.94427	0.573178
$302$	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	$-0.313072\pi$
$307$	19.4164	1.10815	0.554076	+	0.832466i	$0.313072\pi$
$308$	−2.61803	−0.149176
$309$	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	$-0.727380\pi$
$313$	−23.1803	−1.31023	−0.655115	+	0.755529i	$0.727380\pi$
$314$	8.47214	0.478110
$315$	0	0
$316$	3.47214	0.195323
$317$	−6.05573	−0.340124	−0.170062	−	0.985433i	$-0.554397\pi$
$317$	−6.05573	−0.340124	−0.170062	+	0.985433i	$0.554397\pi$
$318$	0	0
$319$	−40.5066	−2.26793
$320$	4.23607	0.236803
$321$	0	0
$322$	8.70820	0.485289
$323$	−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
$329$	−11.0000	−0.606450
$330$	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
$343$	−1.00000	−0.0539949
$344$	−22.2361	−1.19889
$345$	0	0
$346$	−18.5623	−0.997916
$347$	−12.7639	−0.685204	−0.342602	−	0.939481i	$-0.611308\pi$
$347$	−12.7639	−0.685204	−0.342602	+	0.939481i	$0.611308\pi$
$348$	0	0
$349$	31.3607	1.67870	0.839349	−	0.543592i	$-0.182936\pi$
$349$	31.3607	1.67870	0.839349	+	0.543592i	$0.182936\pi$
$350$	1.61803	0.0864876
$351$	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	$-0.0891995\pi$
$359$	36.4164	1.92198	0.960992	+	0.276575i	$0.0891995\pi$
$360$	0	0
$361$	31.2705	1.64582
$362$	31.5066	1.65595
$363$	0	0
$364$	−0.854102	−0.0447671
$365$	13.9443	0.729877
$366$	0	0
$367$	−14.8541	−0.775378	−0.387689	−	0.921790i	$-0.626727\pi$
$367$	−14.8541	−0.775378	−0.387689	+	0.921790i	$0.626727\pi$
$368$	−26.1246	−1.36184
$369$	0	0
$370$	−10.8541	−0.564278
$371$	4.38197	0.227500
$372$	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
$385$	−4.23607	−0.215890
$386$	1.38197	0.0703402
$387$	0	0
$388$	−2.05573	−0.104364
$389$	25.7984	1.30803	0.654015	−	0.756482i	$-0.273083\pi$
$389$	25.7984	1.30803	0.654015	+	0.756482i	$0.273083\pi$
$390$	0	0
$391$	8.70820	0.440393
$392$	2.23607	0.112938
$393$	0	0
$394$	31.5066	1.58728
$395$	5.61803	0.282674
$396$	0	0
$397$	9.05573	0.454494	0.227247	−	0.973837i	$-0.427028\pi$
$397$	9.05573	0.454494	0.227247	+	0.973837i	$0.427028\pi$
$398$	−2.61803	−0.131230
$399$	0	0
$400$	−4.85410	−0.242705
$401$	−17.3820	−0.868014	−0.434007	−	0.900910i	$-0.642901\pi$
$401$	−17.3820	−0.868014	−0.434007	+	0.900910i	$0.642901\pi$
$402$	0	0
$403$	9.27051	0.461797
$404$	3.41641	0.169973
$405$	0	0
$406$	−15.4721	−0.767869
$407$	28.4164	1.40855
$408$	0	0
$409$	−13.0557	−0.645564	−0.322782	−	0.946473i	$-0.604618\pi$
$409$	−13.0557	−0.645564	−0.322782	+	0.946473i	$0.604618\pi$
$410$	−13.0902	−0.646477
$411$	0	0
$412$	6.56231	0.323302
$413$	−1.29180	−0.0635651
$414$	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
$427$	13.8541	0.670447
$428$	−1.67376	−0.0809043
$429$	0	0
$430$	16.0902	0.775937
$431$	30.4508	1.46677	0.733383	−	0.679816i	$-0.237940\pi$
$431$	30.4508	1.46677	0.733383	+	0.679816i	$0.237940\pi$
$432$	0	0
$433$	−21.2148	−1.01952	−0.509759	−	0.860317i	$-0.670265\pi$
$433$	−21.2148	−1.01952	−0.509759	+	0.860317i	$0.670265\pi$
$434$	10.8541	0.521014
$435$	0	0
$436$	7.76393	0.371825
$437$	−38.1591	−1.82540
$438$	0	0
$439$	11.2918	0.538928	0.269464	−	0.963010i	$-0.413153\pi$
$439$	11.2918	0.538928	0.269464	+	0.963010i	$0.413153\pi$
$440$	9.47214	0.451566
$441$	0	0
$442$	−3.61803	−0.172092
$443$	−40.6869	−1.93309	−0.966547	−	0.256490i	$-0.917434\pi$
$443$	−40.6869	−1.93309	−0.966547	+	0.256490i	$0.917434\pi$
$444$	0	0
$445$	1.47214	0.0697859
$446$	−12.0902	−0.572486
$447$	0	0
$448$	−4.23607	−0.200135
$449$	20.4721	0.966140	0.483070	−	0.875582i	$-0.339522\pi$
$449$	20.4721	0.966140	0.483070	+	0.875582i	$0.339522\pi$
$450$	0	0
$451$	34.2705	1.61374
$452$	3.61803	0.170178
$453$	0	0
$454$	−12.0344	−0.564804
$455$	−1.38197	−0.0647876
$456$	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
$469$	−11.3262	−0.522997
$470$	−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
$476$	−1.00000	−0.0458349
$477$	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
$490$	−1.61803	−0.0730953
$491$	−19.5066	−0.880320	−0.440160	−	0.897919i	$-0.645078\pi$
$491$	−19.5066	−0.880320	−0.440160	+	0.897919i	$0.645078\pi$
$492$	0	0
$493$	−15.4721	−0.696830
$494$	15.8541	0.713310
$495$	0	0
$496$	−32.5623	−1.46209
$497$	−3.85410	−0.172880
$498$	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
$511$	−13.9443	−0.616858
$512$	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
$518$	10.8541	0.476902
$519$	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	$-0.508364\pi$
$523$	−1.20163	−0.0525434	−0.0262717	+	0.999655i	$0.508364\pi$
$524$	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
$532$	4.38197	0.189982
$533$	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
$539$	4.23607	0.182460
$540$	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
$553$	−5.61803	−0.238903
$554$	−7.56231	−0.321292
$555$	0	0
$556$	1.96556	0.0833582
$557$	−16.7984	−0.711770	−0.355885	−	0.934530i	$-0.615821\pi$
$557$	−16.7984	−0.711770	−0.355885	+	0.934530i	$0.615821\pi$
$558$	0	0
$559$	−13.7426	−0.581252
$560$	4.85410	0.205123
$561$	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	$-0.790738\pi$
$569$	−37.7639	−1.58315	−0.791573	+	0.611074i	$0.790738\pi$
$570$	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
$574$	13.0902	0.546373
$575$	5.38197	0.224443
$576$	0	0
$577$	22.1246	0.921060	0.460530	−	0.887644i	$-0.347659\pi$
$577$	22.1246	0.921060	0.460530	+	0.887644i	$0.347659\pi$
$578$	23.2705	0.967926
$579$	0	0
$580$	−5.90983	−0.245392
$581$	−10.7082	−0.444251
$582$	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
$595$	−1.61803	−0.0663329
$596$	−9.76393	−0.399946
$597$	0	0
$598$	−12.0344	−0.492125
$599$	−38.9443	−1.59122	−0.795610	−	0.605809i	$-0.792849\pi$
$599$	−38.9443	−1.59122	−0.795610	+	0.605809i	$0.792849\pi$
$600$	0	0
$601$	0.798374	0.0325663	0.0162832	−	0.999867i	$-0.494817\pi$
$601$	0.798374	0.0325663	0.0162832	+	0.999867i	$0.494817\pi$
$602$	−16.0902	−0.655786
$603$	0	0
$604$	−0.0344419	−0.00140142
$605$	6.94427	0.282325
$606$	0	0
$607$	24.4164	0.991031	0.495516	−	0.868599i	$-0.334979\pi$
$607$	24.4164	0.991031	0.495516	+	0.868599i	$0.334979\pi$
$608$	−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
$616$	−9.47214	−0.381643
$617$	−37.7984	−1.52171	−0.760853	−	0.648925i	$-0.775219\pi$
$617$	−37.7984	−1.52171	−0.760853	+	0.648925i	$0.775219\pi$
$618$	0	0
$619$	−47.5410	−1.91083	−0.955417	−	0.295258i	$-0.904594\pi$
$619$	−47.5410	−1.91083	−0.955417	+	0.295258i	$0.904594\pi$
$620$	4.14590	0.166503
$621$	0	0
$622$	−23.1803	−0.929447
$623$	−1.47214	−0.0589799
$624$	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
$637$	1.38197	0.0547555
$638$	65.5410	2.59479
$639$	0	0
$640$	−13.6180	−0.538300
$641$	−0.729490	−0.0288131	−0.0144066	−	0.999896i	$-0.504586\pi$
$641$	−0.729490	−0.0288131	−0.0144066	+	0.999896i	$0.504586\pi$
$642$	0	0
$643$	41.7426	1.64617	0.823085	−	0.567919i	$-0.192251\pi$
$643$	41.7426	1.64617	0.823085	+	0.567919i	$0.192251\pi$
$644$	−3.32624	−0.131072
$645$	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	$-0.462328\pi$
$653$	6.03444	0.236146	0.118073	+	0.993005i	$0.462328\pi$
$654$	0	0
$655$	4.38197	0.171218
$656$	−39.2705	−1.53326
$657$	0	0
$658$	17.7984	0.693853
$659$	−0.763932	−0.0297586	−0.0148793	−	0.999889i	$-0.504736\pi$
$659$	−0.763932	−0.0297586	−0.0148793	+	0.999889i	$0.504736\pi$
$660$	0	0
$661$	−16.2148	−0.630682	−0.315341	−	0.948978i	$-0.602119\pi$
$661$	−16.2148	−0.630682	−0.315341	+	0.948978i	$0.602119\pi$
$662$	51.5410	2.00320
$663$	0	0
$664$	23.9443	0.929218
$665$	7.09017	0.274945
$666$	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
$679$	3.32624	0.127649
$680$	3.61803	0.138745
$681$	0	0
$682$	−45.9787	−1.76062
$683$	−42.9787	−1.64453	−0.822267	−	0.569101i	$-0.807291\pi$
$683$	−42.9787	−1.64453	−0.822267	+	0.569101i	$0.807291\pi$
$684$	0	0
$685$	13.8885	0.530654
$686$	1.61803	0.0617768
$687$	0	0
$688$	48.2705	1.84030
$689$	−6.05573	−0.230705
$690$	0	0
$691$	−30.5279	−1.16133	−0.580667	−	0.814141i	$-0.697208\pi$
$691$	−30.5279	−1.16133	−0.580667	+	0.814141i	$0.697208\pi$
$692$	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
$700$	−0.618034	−0.0233595
$701$	6.88854	0.260177	0.130088	−	0.991502i	$-0.458474\pi$
$701$	6.88854	0.260177	0.130088	+	0.991502i	$0.458474\pi$
$702$	0	0
$703$	−47.5623	−1.79385
$704$	17.9443	0.676300
$705$	0	0
$706$	−18.8885	−0.710880
$707$	−5.52786	−0.207897
$708$	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
$721$	−10.6180	−0.395436
$722$	−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	$-0.568110\pi$
$727$	−11.4508	−0.424689	−0.212344	+	0.977195i	$0.568110\pi$
$728$	−3.09017	−0.114529
$729$	0	0
$730$	−22.5623	−0.835068
$731$	−16.0902	−0.595116
$732$	0	0
$733$	10.5623	0.390128	0.195064	−	0.980791i	$-0.437509\pi$
$733$	10.5623	0.390128	0.195064	+	0.980791i	$0.437509\pi$
$734$	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
$742$	−7.09017	−0.260288
$743$	−3.32624	−0.122028	−0.0610139	−	0.998137i	$-0.519433\pi$
$743$	−3.32624	−0.122028	−0.0610139	+	0.998137i	$0.519433\pi$
$744$	0	0
$745$	−15.7984	−0.578808
$746$	27.4721	1.00583
$747$	0	0
$748$	4.23607	0.154886
$749$	2.70820	0.0989556
$750$	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
$763$	−12.5623	−0.454786
$764$	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	$-0.366425\pi$
$769$	22.5967	0.814860	0.407430	+	0.913237i	$0.366425\pi$
$770$	6.85410	0.247005
$771$	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
$784$	−4.85410	−0.173361
$785$	−5.23607	−0.186883
$786$	0	0
$787$	−3.23607	−0.115353	−0.0576767	−	0.998335i	$-0.518369\pi$
$787$	−3.23607	−0.115353	−0.0576767	+	0.998335i	$0.518369\pi$
$788$	−12.0344	−0.428709
$789$	0	0
$790$	−9.09017	−0.323414
$791$	−5.85410	−0.208148
$792$	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
$805$	−5.38197	−0.189689
$806$	−15.0000	−0.528352
$807$	0	0
$808$	12.3607	0.434847
$809$	−6.63932	−0.233426	−0.116713	−	0.993166i	$-0.537236\pi$
$809$	−6.63932	−0.233426	−0.116713	+	0.993166i	$0.537236\pi$
$810$	0	0
$811$	8.32624	0.292374	0.146187	−	0.989257i	$-0.453300\pi$
$811$	8.32624	0.292374	0.146187	+	0.989257i	$0.453300\pi$
$812$	5.90983	0.207394
$813$	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	$-0.672668\pi$
$821$	−29.5836	−1.03247	−0.516237	+	0.856446i	$0.672668\pi$
$822$	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
$826$	2.09017	0.0727263
$827$	−11.5410	−0.401321	−0.200660	−	0.979661i	$-0.564309\pi$
$827$	−11.5410	−0.401321	−0.200660	+	0.979661i	$0.564309\pi$
$828$	0	0
$829$	−13.3607	−0.464036	−0.232018	−	0.972712i	$-0.574533\pi$
$829$	−13.3607	−0.464036	−0.232018	+	0.972712i	$0.574533\pi$
$830$	−17.3262	−0.601402
$831$	0	0
$832$	5.85410	0.202954
$833$	1.61803	0.0560616
$834$	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
$847$	−6.94427	−0.238608
$848$	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	$-0.511812\pi$
$853$	−2.16718	−0.0742030	−0.0371015	+	0.999312i	$0.511812\pi$
$854$	−22.4164	−0.767074
$855$	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	$-0.379893\pi$
$859$	21.5967	0.736872	0.368436	+	0.929653i	$0.379893\pi$
$860$	−6.14590	−0.209573
$861$	0	0
$862$	−49.2705	−1.67816
$863$	−51.7771	−1.76251	−0.881256	−	0.472639i	$-0.843302\pi$
$863$	−51.7771	−1.76251	−0.881256	+	0.472639i	$0.843302\pi$
$864$	0	0
$865$	11.4721	0.390064
$866$	34.3262	1.16645
$867$	0	0
$868$	−4.14590	−0.140721
$869$	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
$875$	−1.00000	−0.0338062
$876$	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
$889$	−0.0901699	−0.00302420
$890$	−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
$896$	13.6180	0.454947
$897$	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
$910$	2.23607	0.0741249
$911$	7.63932	0.253102	0.126551	−	0.991960i	$-0.459609\pi$
$911$	7.63932	0.253102	0.126551	+	0.991960i	$0.459609\pi$
$912$	0	0
$913$	45.3607	1.50122
$914$	−28.0344	−0.927297
$915$	0	0
$916$	−11.0902	−0.366430
$917$	−4.38197	−0.144705
$918$	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
$931$	−7.09017	−0.232371
$932$	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	$-0.354937\pi$
$937$	26.9443	0.880231	0.440115	+	0.897941i	$0.354937\pi$
$938$	18.3262	0.598373
$939$	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	$-0.528160\pi$
$947$	−5.43769	−0.176701	−0.0883507	+	0.996089i	$0.528160\pi$
$948$	0	0
$949$	19.2705	0.625547
$950$	11.4721	0.372205
$951$	0	0
$952$	−3.61803	−0.117261
$953$	−35.6525	−1.15490	−0.577448	−	0.816427i	$-0.695952\pi$
$953$	−35.6525	−1.15490	−0.577448	+	0.816427i	$0.695952\pi$
$954$	0	0
$955$	18.7984	0.608301
$956$	7.20163	0.232917
$957$	0	0
$958$	−41.9230	−1.35447
$959$	−13.8885	−0.448484
$960$	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
$973$	−3.18034	−0.101957
$974$	25.1246	0.805044
$975$	0	0
$976$	67.2492	2.15260
$977$	14.3607	0.459439	0.229719	−	0.973257i	$-0.426219\pi$
$977$	14.3607	0.459439	0.229719	+	0.973257i	$0.426219\pi$
$978$	0	0
$979$	6.23607	0.199306
$980$	0.618034	0.0197424
$981$	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
$994$	6.23607	0.197796
$995$	1.61803	0.0512951
$996$	0	0
$997$	14.7082	0.465813	0.232907	−	0.972499i	$-0.425176\pi$
$997$	14.7082	0.465813	0.232907	+	0.972499i	$0.425176\pi$
$998$	−5.43769	−0.172127
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.e.1.1	✓	2	1.1	even	1	trivial
945.2.a.g.1.2	yes	2	3.2	odd	2
4725.2.a.w.1.1		2	15.14	odd	2
4725.2.a.be.1.2		2	5.4	even	2
6615.2.a.o.1.1		2	7.6	odd	2
6615.2.a.r.1.2		2	21.20	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} -1.61803 q^{10} +4.23607 q^{11} +1.38197 q^{13} +1.61803 q^{14} -4.85410 q^{16} +1.61803 q^{17} -7.09017 q^{19} +0.618034 q^{20} -6.85410 q^{22} +5.38197 q^{23} +1.00000 q^{25} -2.23607 q^{26} -0.618034 q^{28} -9.56231 q^{29} +6.70820 q^{31} +3.38197 q^{32} -2.61803 q^{34} -1.00000 q^{35} +6.70820 q^{37} +11.4721 q^{38} +2.23607 q^{40} +8.09017 q^{41} -9.94427 q^{43} +2.61803 q^{44} -8.70820 q^{46} +11.0000 q^{47} +1.00000 q^{49} -1.61803 q^{50} +0.854102 q^{52} -4.38197 q^{53} +4.23607 q^{55} -2.23607 q^{56} +15.4721 q^{58} +1.29180 q^{59} -13.8541 q^{61} -10.8541 q^{62} +4.23607 q^{64} +1.38197 q^{65} +11.3262 q^{67} +1.00000 q^{68} +1.61803 q^{70} +3.85410 q^{71} +13.9443 q^{73} -10.8541 q^{74} -4.38197 q^{76} -4.23607 q^{77} +5.61803 q^{79} -4.85410 q^{80} -13.0902 q^{82} +10.7082 q^{83} +1.61803 q^{85} +16.0902 q^{86} +9.47214 q^{88} +1.47214 q^{89} -1.38197 q^{91} +3.32624 q^{92} -17.7984 q^{94} -7.09017 q^{95} -3.32624 q^{97} -1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} + 4 q^{11} + 5 q^{13} + q^{14} - 3 q^{16} + q^{17} - 3 q^{19} - q^{20} - 7 q^{22} + 13 q^{23} + 2 q^{25} + q^{28} + q^{29} + 9 q^{32} - 3 q^{34} - 2 q^{35} + 14 q^{38} + 5 q^{41} - 2 q^{43} + 3 q^{44} - 4 q^{46} + 22 q^{47} + 2 q^{49} - q^{50} - 5 q^{52} - 11 q^{53} + 4 q^{55} + 22 q^{58} + 16 q^{59} - 21 q^{61} - 15 q^{62} + 4 q^{64} + 5 q^{65} + 7 q^{67} + 2 q^{68} + q^{70} + q^{71} + 10 q^{73} - 15 q^{74} - 11 q^{76} - 4 q^{77} + 9 q^{79} - 3 q^{80} - 15 q^{82} + 8 q^{83} + q^{85} + 21 q^{86} + 10 q^{88} - 6 q^{89} - 5 q^{91} - 9 q^{92} - 11 q^{94} - 3 q^{95} + 9 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 + 4 * q^11 + 5 * q^13 + q^14 - 3 * q^16 + q^17 - 3 * q^19 - q^20 - 7 * q^22 + 13 * q^23 + 2 * q^25 + q^28 + q^29 + 9 * q^32 - 3 * q^34 - 2 * q^35 + 14 * q^38 + 5 * q^41 - 2 * q^43 + 3 * q^44 - 4 * q^46 + 22 * q^47 + 2 * q^49 - 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^70 + q^71 + 10 * q^73 - 15 * q^74 - 11 * q^76 - 4 * q^77 + 9 * q^79 - 3 * q^80 - 15 * q^82 + 8 * q^83 + q^85 + 21 * q^86 + 10 * q^88 - 6 * q^89 - 5 * q^91 - 9 * q^92 - 11 * q^94 - 3 * q^95 + 9 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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