LMFDB - Embedded newform 945.2.a.l.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			$-0.618034$ 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)

$f(q)$	$=$	$q+0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.47214 q^{8} +O(q^{10})$	\(q+0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.47214 q^{8} -0.381966 q^{10} +4.23607 q^{11} -2.61803 q^{13} +0.381966 q^{14} +3.14590 q^{16} -0.854102 q^{17} -1.38197 q^{19} +1.85410 q^{20} +1.61803 q^{22} +2.14590 q^{23} +1.00000 q^{25} -1.00000 q^{26} -1.85410 q^{28} +5.85410 q^{29} +1.47214 q^{31} +4.14590 q^{32} -0.326238 q^{34} -1.00000 q^{35} +7.94427 q^{37} -0.527864 q^{38} +1.47214 q^{40} +12.5623 q^{41} -2.23607 q^{43} -7.85410 q^{44} +0.819660 q^{46} +5.47214 q^{47} +1.00000 q^{49} +0.381966 q^{50} +4.85410 q^{52} +8.09017 q^{53} -4.23607 q^{55} -1.47214 q^{56} +2.23607 q^{58} +5.94427 q^{59} -0.145898 q^{61} +0.562306 q^{62} -4.70820 q^{64} +2.61803 q^{65} -5.14590 q^{67} +1.58359 q^{68} -0.381966 q^{70} +2.90983 q^{71} -7.94427 q^{73} +3.03444 q^{74} +2.56231 q^{76} +4.23607 q^{77} -9.79837 q^{79} -3.14590 q^{80} +4.79837 q^{82} +12.2361 q^{83} +0.854102 q^{85} -0.854102 q^{86} -6.23607 q^{88} -8.70820 q^{89} -2.61803 q^{91} -3.97871 q^{92} +2.09017 q^{94} +1.38197 q^{95} +8.09017 q^{97} +0.381966 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8	$ 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8} - 3 q^{10} + 4 q^{11} - 3 q^{13} + 3 q^{14} + 13 q^{16} + 5 q^{17} - 5 q^{19} - 3 q^{20} + q^{22} + 11 q^{23} + 2 q^{25} - 2 q^{26} + 3 q^{28} + 5 q^{29} - 6 q^{31} + 15 q^{32} + 15 q^{34} - 2 q^{35} - 2 q^{37} - 10 q^{38} - 6 q^{40} + 5 q^{41} - 9 q^{44} + 24 q^{46} + 2 q^{47} + 2 q^{49} + 3 q^{50} + 3 q^{52} + 5 q^{53} - 4 q^{55} + 6 q^{56} - 6 q^{59} - 7 q^{61} - 19 q^{62} + 4 q^{64} + 3 q^{65} - 17 q^{67} + 30 q^{68} - 3 q^{70} + 17 q^{71} + 2 q^{73} - 23 q^{74} - 15 q^{76} + 4 q^{77} + 5 q^{79} - 13 q^{80} - 15 q^{82} + 20 q^{83} - 5 q^{85} + 5 q^{86} - 8 q^{88} - 4 q^{89} - 3 q^{91} + 39 q^{92} - 7 q^{94} + 5 q^{95} + 5 q^{97} + 3 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 - 3 * q^10 + 4 * q^11 - 3 * q^13 + 3 * q^14 + 13 * q^16 + 5 * q^17 - 5 * q^19 - 3 * q^20 + q^22 + 11 * q^23 + 2 * q^25 - 2 * q^26 + 3 * q^28 + 5 * q^29 - 6 * q^31 + 15 * q^32 + 15 * q^34 - 2 * q^35 - 2 * q^37 - 10 * q^38 - 6 * q^40 + 5 * q^41 - 9 * q^44 + 24 * q^46 + 2 * q^47 + 2 * q^49 + 3 * q^50 + 3 * q^52 + 5 * q^53 - 4 * q^55 + 6 * q^56 - 6 * q^59 - 7 * q^61 - 19 * q^62 + 4 * q^64 + 3 * q^65 - 17 * q^67 + 30 * q^68 - 3 * q^70 + 17 * q^71 + 2 * q^73 - 23 * q^74 - 15 * q^76 + 4 * q^77 + 5 * q^79 - 13 * q^80 - 15 * q^82 + 20 * q^83 - 5 * q^85 + 5 * q^86 - 8 * q^88 - 4 * q^89 - 3 * q^91 + 39 * q^92 - 7 * q^94 + 5 * q^95 + 5 * 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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.47214	−0.520479
$9$	0	0
$10$	−0.381966	−0.120788
$11$	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
$13$	−2.61803	−0.726112	−0.363056	−	0.931767i	$-0.618267\pi$
$13$	−2.61803	−0.726112	−0.363056	+	0.931767i	$0.618267\pi$
$14$	0.381966	0.102085
$15$	0	0
$16$	3.14590	0.786475
$17$	−0.854102	−0.207150	−0.103575	−	0.994622i	$-0.533028\pi$
$17$	−0.854102	−0.207150	−0.103575	+	0.994622i	$0.533028\pi$
$18$	0	0
$19$	−1.38197	−0.317045	−0.158522	−	0.987355i	$-0.550673\pi$
$19$	−1.38197	−0.317045	−0.158522	+	0.987355i	$0.550673\pi$
$20$	1.85410	0.414590
$21$	0	0
$22$	1.61803	0.344966
$23$	2.14590	0.447451	0.223725	−	0.974652i	$-0.428178\pi$
$23$	2.14590	0.447451	0.223725	+	0.974652i	$0.428178\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−1.85410	−0.350392
$29$	5.85410	1.08708	0.543540	−	0.839383i	$-0.317084\pi$
$29$	5.85410	1.08708	0.543540	+	0.839383i	$0.317084\pi$
$30$	0	0
$31$	1.47214	0.264403	0.132202	−	0.991223i	$-0.457795\pi$
$31$	1.47214	0.264403	0.132202	+	0.991223i	$0.457795\pi$
$32$	4.14590	0.732898
$33$	0	0
$34$	−0.326238	−0.0559493
$35$	−1.00000	−0.169031
$36$	0	0
$37$	7.94427	1.30603	0.653015	−	0.757345i	$-0.273504\pi$
$37$	7.94427	1.30603	0.653015	+	0.757345i	$0.273504\pi$
$38$	−0.527864	−0.0856309
$39$	0	0
$40$	1.47214	0.232765
$41$	12.5623	1.96190	0.980951	−	0.194254i	$-0.0622286\pi$
$41$	12.5623	1.96190	0.980951	+	0.194254i	$0.0622286\pi$
$42$	0	0
$43$	−2.23607	−0.340997	−0.170499	−	0.985358i	$-0.554538\pi$
$43$	−2.23607	−0.340997	−0.170499	+	0.985358i	$0.554538\pi$
$44$	−7.85410	−1.18405
$45$	0	0
$46$	0.819660	0.120852
$47$	5.47214	0.798193	0.399097	−	0.916909i	$-0.369324\pi$
$47$	5.47214	0.798193	0.399097	+	0.916909i	$0.369324\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.381966	0.0540182
$51$	0	0
$52$	4.85410	0.673143
$53$	8.09017	1.11127	0.555635	−	0.831426i	$-0.312475\pi$
$53$	8.09017	1.11127	0.555635	+	0.831426i	$0.312475\pi$
$54$	0	0
$55$	−4.23607	−0.571191
$56$	−1.47214	−0.196722
$57$	0	0
$58$	2.23607	0.293610
$59$	5.94427	0.773878	0.386939	−	0.922105i	$-0.373532\pi$
$59$	5.94427	0.773878	0.386939	+	0.922105i	$0.373532\pi$
$60$	0	0
$61$	−0.145898	−0.0186803	−0.00934016	−	0.999956i	$-0.502973\pi$
$61$	−0.145898	−0.0186803	−0.00934016	+	0.999956i	$0.502973\pi$
$62$	0.562306	0.0714129
$63$	0	0
$64$	−4.70820	−0.588525
$65$	2.61803	0.324727
$66$	0	0
$67$	−5.14590	−0.628672	−0.314336	−	0.949312i	$-0.601782\pi$
$67$	−5.14590	−0.628672	−0.314336	+	0.949312i	$0.601782\pi$
$68$	1.58359	0.192039
$69$	0	0
$70$	−0.381966	−0.0456537
$71$	2.90983	0.345333	0.172667	−	0.984980i	$-0.444762\pi$
$71$	2.90983	0.345333	0.172667	+	0.984980i	$0.444762\pi$
$72$	0	0
$73$	−7.94427	−0.929807	−0.464903	−	0.885361i	$-0.653911\pi$
$73$	−7.94427	−0.929807	−0.464903	+	0.885361i	$0.653911\pi$
$74$	3.03444	0.352747
$75$	0	0
$76$	2.56231	0.293917
$77$	4.23607	0.482745
$78$	0	0
$79$	−9.79837	−1.10240	−0.551202	−	0.834372i	$-0.685830\pi$
$79$	−9.79837	−1.10240	−0.551202	+	0.834372i	$0.685830\pi$
$80$	−3.14590	−0.351722
$81$	0	0
$82$	4.79837	0.529892
$83$	12.2361	1.34308	0.671541	−	0.740967i	$-0.265633\pi$
$83$	12.2361	1.34308	0.671541	+	0.740967i	$0.265633\pi$
$84$	0	0
$85$	0.854102	0.0926404
$86$	−0.854102	−0.0921002
$87$	0	0
$88$	−6.23607	−0.664767
$89$	−8.70820	−0.923068	−0.461534	−	0.887123i	$-0.652701\pi$
$89$	−8.70820	−0.923068	−0.461534	+	0.887123i	$0.652701\pi$
$90$	0	0
$91$	−2.61803	−0.274445
$92$	−3.97871	−0.414810
$93$	0	0
$94$	2.09017	0.215585
$95$	1.38197	0.141787
$96$	0	0
$97$	8.09017	0.821432	0.410716	−	0.911763i	$-0.365279\pi$
$97$	8.09017	0.821432	0.410716	+	0.911763i	$0.365279\pi$
$98$	0.381966	0.0385844
$99$	0	0
$100$	−1.85410	−0.185410
$101$	−15.4164	−1.53399	−0.766995	−	0.641653i	$-0.778249\pi$
$101$	−15.4164	−1.53399	−0.766995	+	0.641653i	$0.778249\pi$
$102$	0	0
$103$	−19.8541	−1.95628	−0.978141	−	0.207941i	$-0.933324\pi$
$103$	−19.8541	−1.95628	−0.978141	+	0.207941i	$0.933324\pi$
$104$	3.85410	0.377926
$105$	0	0
$106$	3.09017	0.300144
$107$	−17.4721	−1.68910	−0.844548	−	0.535481i	$-0.820131\pi$
$107$	−17.4721	−1.68910	−0.844548	+	0.535481i	$0.820131\pi$
$108$	0	0
$109$	12.5623	1.20325	0.601625	−	0.798778i	$-0.294520\pi$
$109$	12.5623	1.20325	0.601625	+	0.798778i	$0.294520\pi$
$110$	−1.61803	−0.154273
$111$	0	0
$112$	3.14590	0.297259
$113$	9.09017	0.855131	0.427566	−	0.903984i	$-0.359371\pi$
$113$	9.09017	0.855131	0.427566	+	0.903984i	$0.359371\pi$
$114$	0	0
$115$	−2.14590	−0.200106
$116$	−10.8541	−1.00778
$117$	0	0
$118$	2.27051	0.209017
$119$	−0.854102	−0.0782954
$120$	0	0
$121$	6.94427	0.631297
$122$	−0.0557281	−0.00504538
$123$	0	0
$124$	−2.72949	−0.245115
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.09017	0.185473	0.0927363	−	0.995691i	$-0.470439\pi$
$127$	2.09017	0.185473	0.0927363	+	0.995691i	$0.470439\pi$
$128$	−10.0902	−0.891853
$129$	0	0
$130$	1.00000	0.0877058
$131$	−10.5623	−0.922833	−0.461416	−	0.887184i	$-0.652659\pi$
$131$	−10.5623	−0.922833	−0.461416	+	0.887184i	$0.652659\pi$
$132$	0	0
$133$	−1.38197	−0.119832
$134$	−1.96556	−0.169798
$135$	0	0
$136$	1.25735	0.107817
$137$	6.47214	0.552952	0.276476	−	0.961021i	$-0.410833\pi$
$137$	6.47214	0.552952	0.276476	+	0.961021i	$0.410833\pi$
$138$	0	0
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	1.85410	0.156700
$141$	0	0
$142$	1.11146	0.0932713
$143$	−11.0902	−0.927407
$144$	0	0
$145$	−5.85410	−0.486157
$146$	−3.03444	−0.251132
$147$	0	0
$148$	−14.7295	−1.21076
$149$	−9.32624	−0.764035	−0.382018	−	0.924155i	$-0.624771\pi$
$149$	−9.32624	−0.764035	−0.382018	+	0.924155i	$0.624771\pi$
$150$	0	0
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	2.03444	0.165015
$153$	0	0
$154$	1.61803	0.130385
$155$	−1.47214	−0.118245
$156$	0	0
$157$	−8.65248	−0.690543	−0.345271	−	0.938503i	$-0.612213\pi$
$157$	−8.65248	−0.690543	−0.345271	+	0.938503i	$0.612213\pi$
$158$	−3.74265	−0.297749
$159$	0	0
$160$	−4.14590	−0.327762
$161$	2.14590	0.169120
$162$	0	0
$163$	−5.05573	−0.395995	−0.197998	−	0.980203i	$-0.563444\pi$
$163$	−5.05573	−0.395995	−0.197998	+	0.980203i	$0.563444\pi$
$164$	−23.2918	−1.81878
$165$	0	0
$166$	4.67376	0.362754
$167$	14.4164	1.11558	0.557788	−	0.829984i	$-0.311650\pi$
$167$	14.4164	1.11558	0.557788	+	0.829984i	$0.311650\pi$
$168$	0	0
$169$	−6.14590	−0.472761
$170$	0.326238	0.0250213
$171$	0	0
$172$	4.14590	0.316122
$173$	−13.4721	−1.02427	−0.512134	−	0.858906i	$-0.671145\pi$
$173$	−13.4721	−1.02427	−0.512134	+	0.858906i	$0.671145\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	13.3262	1.00450
$177$	0	0
$178$	−3.32624	−0.249312
$179$	13.9443	1.04224	0.521122	−	0.853482i	$-0.325514\pi$
$179$	13.9443	1.04224	0.521122	+	0.853482i	$0.325514\pi$
$180$	0	0
$181$	0.236068	0.0175468	0.00877340	−	0.999962i	$-0.497207\pi$
$181$	0.236068	0.0175468	0.00877340	+	0.999962i	$0.497207\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−3.15905	−0.232889
$185$	−7.94427	−0.584074
$186$	0	0
$187$	−3.61803	−0.264577
$188$	−10.1459	−0.739966
$189$	0	0
$190$	0.527864	0.0382953
$191$	21.8541	1.58131	0.790654	−	0.612264i	$-0.209741\pi$
$191$	21.8541	1.58131	0.790654	+	0.612264i	$0.209741\pi$
$192$	0	0
$193$	−14.2705	−1.02721	−0.513607	−	0.858026i	$-0.671691\pi$
$193$	−14.2705	−1.02721	−0.513607	+	0.858026i	$0.671691\pi$
$194$	3.09017	0.221861
$195$	0	0
$196$	−1.85410	−0.132436
$197$	−6.23607	−0.444301	−0.222151	−	0.975012i	$-0.571308\pi$
$197$	−6.23607	−0.444301	−0.222151	+	0.975012i	$0.571308\pi$
$198$	0	0
$199$	16.5623	1.17407	0.587035	−	0.809561i	$-0.300295\pi$
$199$	16.5623	1.17407	0.587035	+	0.809561i	$0.300295\pi$
$200$	−1.47214	−0.104096
$201$	0	0
$202$	−5.88854	−0.414316
$203$	5.85410	0.410877
$204$	0	0
$205$	−12.5623	−0.877390
$206$	−7.58359	−0.528374
$207$	0	0
$208$	−8.23607	−0.571069
$209$	−5.85410	−0.404937
$210$	0	0
$211$	22.6525	1.55946	0.779730	−	0.626115i	$-0.215356\pi$
$211$	22.6525	1.55946	0.779730	+	0.626115i	$0.215356\pi$
$212$	−15.0000	−1.03020
$213$	0	0
$214$	−6.67376	−0.456209
$215$	2.23607	0.152499
$216$	0	0
$217$	1.47214	0.0999351
$218$	4.79837	0.324987
$219$	0	0
$220$	7.85410	0.529523
$221$	2.23607	0.150414
$222$	0	0
$223$	6.52786	0.437138	0.218569	−	0.975821i	$-0.429861\pi$
$223$	6.52786	0.437138	0.218569	+	0.975821i	$0.429861\pi$
$224$	4.14590	0.277009
$225$	0	0
$226$	3.47214	0.230963
$227$	−13.5066	−0.896463	−0.448232	−	0.893918i	$-0.647946\pi$
$227$	−13.5066	−0.896463	−0.448232	+	0.893918i	$0.647946\pi$
$228$	0	0
$229$	27.6525	1.82733	0.913664	−	0.406471i	$-0.133241\pi$
$229$	27.6525	1.82733	0.913664	+	0.406471i	$0.133241\pi$
$230$	−0.819660	−0.0540468
$231$	0	0
$232$	−8.61803	−0.565802
$233$	−13.5623	−0.888496	−0.444248	−	0.895904i	$-0.646529\pi$
$233$	−13.5623	−0.888496	−0.444248	+	0.895904i	$0.646529\pi$
$234$	0	0
$235$	−5.47214	−0.356963
$236$	−11.0213	−0.717425
$237$	0	0
$238$	−0.326238	−0.0211469
$239$	5.18034	0.335088	0.167544	−	0.985865i	$-0.446416\pi$
$239$	5.18034	0.335088	0.167544	+	0.985865i	$0.446416\pi$
$240$	0	0
$241$	14.3820	0.926424	0.463212	−	0.886248i	$-0.346697\pi$
$241$	14.3820	0.926424	0.463212	+	0.886248i	$0.346697\pi$
$242$	2.65248	0.170508
$243$	0	0
$244$	0.270510	0.0173176
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	3.61803	0.230210
$248$	−2.16718	−0.137616
$249$	0	0
$250$	−0.381966	−0.0241577
$251$	−4.65248	−0.293662	−0.146831	−	0.989162i	$-0.546907\pi$
$251$	−4.65248	−0.293662	−0.146831	+	0.989162i	$0.546907\pi$
$252$	0	0
$253$	9.09017	0.571494
$254$	0.798374	0.0500944
$255$	0	0
$256$	5.56231	0.347644
$257$	−9.52786	−0.594332	−0.297166	−	0.954826i	$-0.596041\pi$
$257$	−9.52786	−0.594332	−0.297166	+	0.954826i	$0.596041\pi$
$258$	0	0
$259$	7.94427	0.493633
$260$	−4.85410	−0.301039
$261$	0	0
$262$	−4.03444	−0.249249
$263$	26.0902	1.60879	0.804394	−	0.594096i	$-0.202490\pi$
$263$	26.0902	1.60879	0.804394	+	0.594096i	$0.202490\pi$
$264$	0	0
$265$	−8.09017	−0.496975
$266$	−0.527864	−0.0323654
$267$	0	0
$268$	9.54102	0.582811
$269$	−7.47214	−0.455584	−0.227792	−	0.973710i	$-0.573151\pi$
$269$	−7.47214	−0.455584	−0.227792	+	0.973710i	$0.573151\pi$
$270$	0	0
$271$	−17.6180	−1.07022	−0.535110	−	0.844783i	$-0.679730\pi$
$271$	−17.6180	−1.07022	−0.535110	+	0.844783i	$0.679730\pi$
$272$	−2.68692	−0.162918
$273$	0	0
$274$	2.47214	0.149347
$275$	4.23607	0.255445
$276$	0	0
$277$	−3.79837	−0.228222	−0.114111	−	0.993468i	$-0.536402\pi$
$277$	−3.79837	−0.228222	−0.114111	+	0.993468i	$0.536402\pi$
$278$	3.43769	0.206179
$279$	0	0
$280$	1.47214	0.0879770
$281$	17.7984	1.06176	0.530881	−	0.847446i	$-0.321861\pi$
$281$	17.7984	1.06176	0.530881	+	0.847446i	$0.321861\pi$
$282$	0	0
$283$	−18.2705	−1.08607	−0.543035	−	0.839710i	$-0.682725\pi$
$283$	−18.2705	−1.08607	−0.543035	+	0.839710i	$0.682725\pi$
$284$	−5.39512	−0.320142
$285$	0	0
$286$	−4.23607	−0.250484
$287$	12.5623	0.741529
$288$	0	0
$289$	−16.2705	−0.957089
$290$	−2.23607	−0.131306
$291$	0	0
$292$	14.7295	0.861978
$293$	−24.1803	−1.41263	−0.706315	−	0.707897i	$-0.749644\pi$
$293$	−24.1803	−1.41263	−0.706315	+	0.707897i	$0.749644\pi$
$294$	0	0
$295$	−5.94427	−0.346089
$296$	−11.6950	−0.679761
$297$	0	0
$298$	−3.56231	−0.206359
$299$	−5.61803	−0.324899
$300$	0	0
$301$	−2.23607	−0.128885
$302$	8.78522	0.505533
$303$	0	0
$304$	−4.34752	−0.249348
$305$	0.145898	0.00835410
$306$	0	0
$307$	−8.94427	−0.510477	−0.255238	−	0.966878i	$-0.582154\pi$
$307$	−8.94427	−0.510477	−0.255238	+	0.966878i	$0.582154\pi$
$308$	−7.85410	−0.447529
$309$	0	0
$310$	−0.562306	−0.0319368
$311$	−11.7426	−0.665864	−0.332932	−	0.942951i	$-0.608038\pi$
$311$	−11.7426	−0.665864	−0.332932	+	0.942951i	$0.608038\pi$
$312$	0	0
$313$	7.29180	0.412157	0.206078	−	0.978535i	$-0.433930\pi$
$313$	7.29180	0.412157	0.206078	+	0.978535i	$0.433930\pi$
$314$	−3.30495	−0.186509
$315$	0	0
$316$	18.1672	1.02198
$317$	22.5967	1.26916	0.634580	−	0.772857i	$-0.281173\pi$
$317$	22.5967	1.26916	0.634580	+	0.772857i	$0.281173\pi$
$318$	0	0
$319$	24.7984	1.38844
$320$	4.70820	0.263197
$321$	0	0
$322$	0.819660	0.0456779
$323$	1.18034	0.0656759
$324$	0	0
$325$	−2.61803	−0.145222
$326$	−1.93112	−0.106955
$327$	0	0
$328$	−18.4934	−1.02113
$329$	5.47214	0.301689
$330$	0	0
$331$	14.0344	0.771403	0.385701	−	0.922624i	$-0.373960\pi$
$331$	14.0344	0.771403	0.385701	+	0.922624i	$0.373960\pi$
$332$	−22.6869	−1.24511
$333$	0	0
$334$	5.50658	0.301307
$335$	5.14590	0.281150
$336$	0	0
$337$	−25.0902	−1.36675	−0.683374	−	0.730068i	$-0.739488\pi$
$337$	−25.0902	−1.36675	−0.683374	+	0.730068i	$0.739488\pi$
$338$	−2.34752	−0.127688
$339$	0	0
$340$	−1.58359	−0.0858823
$341$	6.23607	0.337702
$342$	0	0
$343$	1.00000	0.0539949
$344$	3.29180	0.177482
$345$	0	0
$346$	−5.14590	−0.276645
$347$	−13.1246	−0.704566	−0.352283	−	0.935894i	$-0.614595\pi$
$347$	−13.1246	−0.704566	−0.352283	+	0.935894i	$0.614595\pi$
$348$	0	0
$349$	−29.0689	−1.55602	−0.778011	−	0.628251i	$-0.783771\pi$
$349$	−29.0689	−1.55602	−0.778011	+	0.628251i	$0.783771\pi$
$350$	0.381966	0.0204169
$351$	0	0
$352$	17.5623	0.936074
$353$	−28.3262	−1.50765	−0.753827	−	0.657073i	$-0.771794\pi$
$353$	−28.3262	−1.50765	−0.753827	+	0.657073i	$0.771794\pi$
$354$	0	0
$355$	−2.90983	−0.154438
$356$	16.1459	0.855731
$357$	0	0
$358$	5.32624	0.281500
$359$	−12.8885	−0.680231	−0.340116	−	0.940384i	$-0.610466\pi$
$359$	−12.8885	−0.680231	−0.340116	+	0.940384i	$0.610466\pi$
$360$	0	0
$361$	−17.0902	−0.899483
$362$	0.0901699	0.00473923
$363$	0	0
$364$	4.85410	0.254424
$365$	7.94427	0.415822
$366$	0	0
$367$	−1.90983	−0.0996923	−0.0498462	−	0.998757i	$-0.515873\pi$
$367$	−1.90983	−0.0996923	−0.0498462	+	0.998757i	$0.515873\pi$
$368$	6.75078	0.351909
$369$	0	0
$370$	−3.03444	−0.157753
$371$	8.09017	0.420021
$372$	0	0
$373$	11.5623	0.598674	0.299337	−	0.954148i	$-0.403235\pi$
$373$	11.5623	0.598674	0.299337	+	0.954148i	$0.403235\pi$
$374$	−1.38197	−0.0714598
$375$	0	0
$376$	−8.05573	−0.415442
$377$	−15.3262	−0.789341
$378$	0	0
$379$	−18.3607	−0.943125	−0.471562	−	0.881833i	$-0.656310\pi$
$379$	−18.3607	−0.943125	−0.471562	+	0.881833i	$0.656310\pi$
$380$	−2.56231	−0.131444
$381$	0	0
$382$	8.34752	0.427096
$383$	16.7082	0.853749	0.426875	−	0.904311i	$-0.359615\pi$
$383$	16.7082	0.853749	0.426875	+	0.904311i	$0.359615\pi$
$384$	0	0
$385$	−4.23607	−0.215890
$386$	−5.45085	−0.277441
$387$	0	0
$388$	−15.0000	−0.761510
$389$	−2.20163	−0.111627	−0.0558134	−	0.998441i	$-0.517775\pi$
$389$	−2.20163	−0.111627	−0.0558134	+	0.998441i	$0.517775\pi$
$390$	0	0
$391$	−1.83282	−0.0926895
$392$	−1.47214	−0.0743541
$393$	0	0
$394$	−2.38197	−0.120002
$395$	9.79837	0.493010
$396$	0	0
$397$	13.4164	0.673350	0.336675	−	0.941621i	$-0.390698\pi$
$397$	13.4164	0.673350	0.336675	+	0.941621i	$0.390698\pi$
$398$	6.32624	0.317106
$399$	0	0
$400$	3.14590	0.157295
$401$	−35.2705	−1.76133	−0.880663	−	0.473744i	$-0.842902\pi$
$401$	−35.2705	−1.76133	−0.880663	+	0.473744i	$0.842902\pi$
$402$	0	0
$403$	−3.85410	−0.191986
$404$	28.5836	1.42209
$405$	0	0
$406$	2.23607	0.110974
$407$	33.6525	1.66809
$408$	0	0
$409$	22.3607	1.10566	0.552832	−	0.833293i	$-0.313547\pi$
$409$	22.3607	1.10566	0.552832	+	0.833293i	$0.313547\pi$
$410$	−4.79837	−0.236975
$411$	0	0
$412$	36.8115	1.81357
$413$	5.94427	0.292498
$414$	0	0
$415$	−12.2361	−0.600645
$416$	−10.8541	−0.532166
$417$	0	0
$418$	−2.23607	−0.109370
$419$	3.76393	0.183880	0.0919401	−	0.995765i	$-0.470693\pi$
$419$	3.76393	0.183880	0.0919401	+	0.995765i	$0.470693\pi$
$420$	0	0
$421$	31.2705	1.52403	0.762016	−	0.647559i	$-0.224210\pi$
$421$	31.2705	1.52403	0.762016	+	0.647559i	$0.224210\pi$
$422$	8.65248	0.421196
$423$	0	0
$424$	−11.9098	−0.578392
$425$	−0.854102	−0.0414300
$426$	0	0
$427$	−0.145898	−0.00706050
$428$	32.3951	1.56588
$429$	0	0
$430$	0.854102	0.0411885
$431$	27.6180	1.33031	0.665157	−	0.746704i	$-0.268365\pi$
$431$	27.6180	1.33031	0.665157	+	0.746704i	$0.268365\pi$
$432$	0	0
$433$	16.0902	0.773244	0.386622	−	0.922238i	$-0.373642\pi$
$433$	16.0902	0.773244	0.386622	+	0.922238i	$0.373642\pi$
$434$	0.562306	0.0269915
$435$	0	0
$436$	−23.2918	−1.11547
$437$	−2.96556	−0.141862
$438$	0	0
$439$	13.4721	0.642990	0.321495	−	0.946911i	$-0.395815\pi$
$439$	13.4721	0.642990	0.321495	+	0.946911i	$0.395815\pi$
$440$	6.23607	0.297293
$441$	0	0
$442$	0.854102	0.0406255
$443$	35.8541	1.70348	0.851740	−	0.523965i	$-0.175548\pi$
$443$	35.8541	1.70348	0.851740	+	0.523965i	$0.175548\pi$
$444$	0	0
$445$	8.70820	0.412808
$446$	2.49342	0.118067
$447$	0	0
$448$	−4.70820	−0.222442
$449$	41.4164	1.95456	0.977281	−	0.211950i	$-0.0679813\pi$
$449$	41.4164	1.95456	0.977281	+	0.211950i	$0.0679813\pi$
$450$	0	0
$451$	53.2148	2.50579
$452$	−16.8541	−0.792750
$453$	0	0
$454$	−5.15905	−0.242126
$455$	2.61803	0.122735
$456$	0	0
$457$	−16.6738	−0.779966	−0.389983	−	0.920822i	$-0.627519\pi$
$457$	−16.6738	−0.779966	−0.389983	+	0.920822i	$0.627519\pi$
$458$	10.5623	0.493544
$459$	0	0
$460$	3.97871	0.185508
$461$	1.38197	0.0643646	0.0321823	−	0.999482i	$-0.489754\pi$
$461$	1.38197	0.0643646	0.0321823	+	0.999482i	$0.489754\pi$
$462$	0	0
$463$	−28.7984	−1.33837	−0.669187	−	0.743094i	$-0.733357\pi$
$463$	−28.7984	−1.33837	−0.669187	+	0.743094i	$0.733357\pi$
$464$	18.4164	0.854960
$465$	0	0
$466$	−5.18034	−0.239975
$467$	−3.05573	−0.141402	−0.0707011	−	0.997498i	$-0.522524\pi$
$467$	−3.05573	−0.141402	−0.0707011	+	0.997498i	$0.522524\pi$
$468$	0	0
$469$	−5.14590	−0.237615
$470$	−2.09017	−0.0964124
$471$	0	0
$472$	−8.75078	−0.402787
$473$	−9.47214	−0.435529
$474$	0	0
$475$	−1.38197	−0.0634089
$476$	1.58359	0.0725838
$477$	0	0
$478$	1.97871	0.0905043
$479$	−4.09017	−0.186885	−0.0934423	−	0.995625i	$-0.529787\pi$
$479$	−4.09017	−0.186885	−0.0934423	+	0.995625i	$0.529787\pi$
$480$	0	0
$481$	−20.7984	−0.948324
$482$	5.49342	0.250219
$483$	0	0
$484$	−12.8754	−0.585245
$485$	−8.09017	−0.367356
$486$	0	0
$487$	−11.8885	−0.538721	−0.269361	−	0.963039i	$-0.586812\pi$
$487$	−11.8885	−0.538721	−0.269361	+	0.963039i	$0.586812\pi$
$488$	0.214782	0.00972271
$489$	0	0
$490$	−0.381966	−0.0172555
$491$	−16.6738	−0.752476	−0.376238	−	0.926523i	$-0.622783\pi$
$491$	−16.6738	−0.752476	−0.376238	+	0.926523i	$0.622783\pi$
$492$	0	0
$493$	−5.00000	−0.225189
$494$	1.38197	0.0621776
$495$	0	0
$496$	4.63119	0.207947
$497$	2.90983	0.130524
$498$	0	0
$499$	17.4721	0.782160	0.391080	−	0.920357i	$-0.372102\pi$
$499$	17.4721	0.782160	0.391080	+	0.920357i	$0.372102\pi$
$500$	1.85410	0.0829180
$501$	0	0
$502$	−1.77709	−0.0793153
$503$	23.2705	1.03758	0.518790	−	0.854901i	$-0.326383\pi$
$503$	23.2705	1.03758	0.518790	+	0.854901i	$0.326383\pi$
$504$	0	0
$505$	15.4164	0.686021
$506$	3.47214	0.154355
$507$	0	0
$508$	−3.87539	−0.171943
$509$	−38.2361	−1.69478	−0.847392	−	0.530968i	$-0.821828\pi$
$509$	−38.2361	−1.69478	−0.847392	+	0.530968i	$0.821828\pi$
$510$	0	0
$511$	−7.94427	−0.351434
$512$	22.3050	0.985749
$513$	0	0
$514$	−3.63932	−0.160524
$515$	19.8541	0.874876
$516$	0	0
$517$	23.1803	1.01947
$518$	3.03444	0.133326
$519$	0	0
$520$	−3.85410	−0.169014
$521$	40.5967	1.77858	0.889288	−	0.457348i	$-0.151201\pi$
$521$	40.5967	1.77858	0.889288	+	0.457348i	$0.151201\pi$
$522$	0	0
$523$	−16.0344	−0.701137	−0.350569	−	0.936537i	$-0.614012\pi$
$523$	−16.0344	−0.701137	−0.350569	+	0.936537i	$0.614012\pi$
$524$	19.5836	0.855513
$525$	0	0
$526$	9.96556	0.434519
$527$	−1.25735	−0.0547712
$528$	0	0
$529$	−18.3951	−0.799788
$530$	−3.09017	−0.134228
$531$	0	0
$532$	2.56231	0.111090
$533$	−32.8885	−1.42456
$534$	0	0
$535$	17.4721	0.755386
$536$	7.57546	0.327210
$537$	0	0
$538$	−2.85410	−0.123049
$539$	4.23607	0.182460
$540$	0	0
$541$	−10.0902	−0.433810	−0.216905	−	0.976193i	$-0.569596\pi$
$541$	−10.0902	−0.433810	−0.216905	+	0.976193i	$0.569596\pi$
$542$	−6.72949	−0.289056
$543$	0	0
$544$	−3.54102	−0.151820
$545$	−12.5623	−0.538110
$546$	0	0
$547$	−42.5967	−1.82131	−0.910653	−	0.413173i	$-0.864421\pi$
$547$	−42.5967	−1.82131	−0.910653	+	0.413173i	$0.864421\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	1.61803	0.0689932
$551$	−8.09017	−0.344653
$552$	0	0
$553$	−9.79837	−0.416669
$554$	−1.45085	−0.0616407
$555$	0	0
$556$	−16.6869	−0.707683
$557$	34.2148	1.44973	0.724863	−	0.688893i	$-0.241903\pi$
$557$	34.2148	1.44973	0.724863	+	0.688893i	$0.241903\pi$
$558$	0	0
$559$	5.85410	0.247602
$560$	−3.14590	−0.132938
$561$	0	0
$562$	6.79837	0.286772
$563$	−27.6180	−1.16396	−0.581981	−	0.813203i	$-0.697722\pi$
$563$	−27.6180	−1.16396	−0.581981	+	0.813203i	$0.697722\pi$
$564$	0	0
$565$	−9.09017	−0.382426
$566$	−6.97871	−0.293337
$567$	0	0
$568$	−4.28367	−0.179739
$569$	31.1803	1.30715	0.653574	−	0.756863i	$-0.273269\pi$
$569$	31.1803	1.30715	0.653574	+	0.756863i	$0.273269\pi$
$570$	0	0
$571$	−20.8541	−0.872717	−0.436359	−	0.899773i	$-0.643732\pi$
$571$	−20.8541	−0.872717	−0.436359	+	0.899773i	$0.643732\pi$
$572$	20.5623	0.859753
$573$	0	0
$574$	4.79837	0.200280
$575$	2.14590	0.0894901
$576$	0	0
$577$	−6.81966	−0.283906	−0.141953	−	0.989873i	$-0.545338\pi$
$577$	−6.81966	−0.283906	−0.141953	+	0.989873i	$0.545338\pi$
$578$	−6.21478	−0.258501
$579$	0	0
$580$	10.8541	0.450692
$581$	12.2361	0.507638
$582$	0	0
$583$	34.2705	1.41934
$584$	11.6950	0.483945
$585$	0	0
$586$	−9.23607	−0.381538
$587$	23.2705	0.960477	0.480238	−	0.877138i	$-0.340550\pi$
$587$	23.2705	0.960477	0.480238	+	0.877138i	$0.340550\pi$
$588$	0	0
$589$	−2.03444	−0.0838277
$590$	−2.27051	−0.0934754
$591$	0	0
$592$	24.9919	1.02716
$593$	7.36068	0.302267	0.151133	−	0.988513i	$-0.451708\pi$
$593$	7.36068	0.302267	0.151133	+	0.988513i	$0.451708\pi$
$594$	0	0
$595$	0.854102	0.0350148
$596$	17.2918	0.708300
$597$	0	0
$598$	−2.14590	−0.0877523
$599$	24.4721	0.999904	0.499952	−	0.866053i	$-0.333351\pi$
$599$	24.4721	0.999904	0.499952	+	0.866053i	$0.333351\pi$
$600$	0	0
$601$	−3.38197	−0.137953	−0.0689766	−	0.997618i	$-0.521973\pi$
$601$	−3.38197	−0.137953	−0.0689766	+	0.997618i	$0.521973\pi$
$602$	−0.854102	−0.0348106
$603$	0	0
$604$	−42.6443	−1.73517
$605$	−6.94427	−0.282325
$606$	0	0
$607$	16.4164	0.666321	0.333161	−	0.942870i	$-0.391885\pi$
$607$	16.4164	0.666321	0.333161	+	0.942870i	$0.391885\pi$
$608$	−5.72949	−0.232362
$609$	0	0
$610$	0.0557281	0.00225636
$611$	−14.3262	−0.579578
$612$	0	0
$613$	25.9443	1.04788	0.523940	−	0.851755i	$-0.324462\pi$
$613$	25.9443	1.04788	0.523940	+	0.851755i	$0.324462\pi$
$614$	−3.41641	−0.137875
$615$	0	0
$616$	−6.23607	−0.251258
$617$	−37.9098	−1.52619	−0.763096	−	0.646285i	$-0.776322\pi$
$617$	−37.9098	−1.52619	−0.763096	+	0.646285i	$0.776322\pi$
$618$	0	0
$619$	−27.8328	−1.11870	−0.559348	−	0.828933i	$-0.688948\pi$
$619$	−27.8328	−1.11870	−0.559348	+	0.828933i	$0.688948\pi$
$620$	2.72949	0.109619
$621$	0	0
$622$	−4.48529	−0.179844
$623$	−8.70820	−0.348887
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.78522	0.111320
$627$	0	0
$628$	16.0426	0.640168
$629$	−6.78522	−0.270544
$630$	0	0
$631$	30.8328	1.22744	0.613718	−	0.789526i	$-0.289673\pi$
$631$	30.8328	1.22744	0.613718	+	0.789526i	$0.289673\pi$
$632$	14.4245	0.573777
$633$	0	0
$634$	8.63119	0.342788
$635$	−2.09017	−0.0829459
$636$	0	0
$637$	−2.61803	−0.103730
$638$	9.47214	0.375005
$639$	0	0
$640$	10.0902	0.398849
$641$	−42.3951	−1.67451	−0.837253	−	0.546815i	$-0.815840\pi$
$641$	−42.3951	−1.67451	−0.837253	+	0.546815i	$0.815840\pi$
$642$	0	0
$643$	10.9098	0.430242	0.215121	−	0.976587i	$-0.430985\pi$
$643$	10.9098	0.430242	0.215121	+	0.976587i	$0.430985\pi$
$644$	−3.97871	−0.156783
$645$	0	0
$646$	0.450850	0.0177384
$647$	25.0557	0.985042	0.492521	−	0.870300i	$-0.336075\pi$
$647$	25.0557	0.985042	0.492521	+	0.870300i	$0.336075\pi$
$648$	0	0
$649$	25.1803	0.988415
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	9.37384	0.367108
$653$	6.43769	0.251926	0.125963	−	0.992035i	$-0.459798\pi$
$653$	6.43769	0.251926	0.125963	+	0.992035i	$0.459798\pi$
$654$	0	0
$655$	10.5623	0.412703
$656$	39.5197	1.54299
$657$	0	0
$658$	2.09017	0.0814833
$659$	7.23607	0.281877	0.140939	−	0.990018i	$-0.454988\pi$
$659$	7.23607	0.281877	0.140939	+	0.990018i	$0.454988\pi$
$660$	0	0
$661$	−33.9230	−1.31945	−0.659726	−	0.751507i	$-0.729327\pi$
$661$	−33.9230	−1.31945	−0.659726	+	0.751507i	$0.729327\pi$
$662$	5.36068	0.208349
$663$	0	0
$664$	−18.0132	−0.699046
$665$	1.38197	0.0535903
$666$	0	0
$667$	12.5623	0.486414
$668$	−26.7295	−1.03420
$669$	0	0
$670$	1.96556	0.0759361
$671$	−0.618034	−0.0238589
$672$	0	0
$673$	46.2492	1.78278	0.891388	−	0.453240i	$-0.149732\pi$
$673$	46.2492	1.78278	0.891388	+	0.453240i	$0.149732\pi$
$674$	−9.58359	−0.369146
$675$	0	0
$676$	11.3951	0.438274
$677$	−50.0132	−1.92216	−0.961081	−	0.276267i	$-0.910903\pi$
$677$	−50.0132	−1.92216	−0.961081	+	0.276267i	$0.910903\pi$
$678$	0	0
$679$	8.09017	0.310472
$680$	−1.25735	−0.0482173
$681$	0	0
$682$	2.38197	0.0912102
$683$	12.6180	0.482816	0.241408	−	0.970424i	$-0.422391\pi$
$683$	12.6180	0.482816	0.241408	+	0.970424i	$0.422391\pi$
$684$	0	0
$685$	−6.47214	−0.247288
$686$	0.381966	0.0145835
$687$	0	0
$688$	−7.03444	−0.268186
$689$	−21.1803	−0.806907
$690$	0	0
$691$	−13.0689	−0.497164	−0.248582	−	0.968611i	$-0.579965\pi$
$691$	−13.0689	−0.497164	−0.248582	+	0.968611i	$0.579965\pi$
$692$	24.9787	0.949548
$693$	0	0
$694$	−5.01316	−0.190297
$695$	−9.00000	−0.341389
$696$	0	0
$697$	−10.7295	−0.406408
$698$	−11.1033	−0.420267
$699$	0	0
$700$	−1.85410	−0.0700785
$701$	−9.83282	−0.371380	−0.185690	−	0.982608i	$-0.559452\pi$
$701$	−9.83282	−0.371380	−0.185690	+	0.982608i	$0.559452\pi$
$702$	0	0
$703$	−10.9787	−0.414070
$704$	−19.9443	−0.751678
$705$	0	0
$706$	−10.8197	−0.407203
$707$	−15.4164	−0.579794
$708$	0	0
$709$	5.59675	0.210190	0.105095	−	0.994462i	$-0.466485\pi$
$709$	5.59675	0.210190	0.105095	+	0.994462i	$0.466485\pi$
$710$	−1.11146	−0.0417122
$711$	0	0
$712$	12.8197	0.480437
$713$	3.15905	0.118307
$714$	0	0
$715$	11.0902	0.414749
$716$	−25.8541	−0.966213
$717$	0	0
$718$	−4.92299	−0.183724
$719$	0.854102	0.0318526	0.0159263	−	0.999873i	$-0.494930\pi$
$719$	0.854102	0.0318526	0.0159263	+	0.999873i	$0.494930\pi$
$720$	0	0
$721$	−19.8541	−0.739405
$722$	−6.52786	−0.242942
$723$	0	0
$724$	−0.437694	−0.0162668
$725$	5.85410	0.217416
$726$	0	0
$727$	40.6869	1.50899	0.754497	−	0.656303i	$-0.227881\pi$
$727$	40.6869	1.50899	0.754497	+	0.656303i	$0.227881\pi$
$728$	3.85410	0.142843
$729$	0	0
$730$	3.03444	0.112310
$731$	1.90983	0.0706376
$732$	0	0
$733$	7.14590	0.263940	0.131970	−	0.991254i	$-0.457870\pi$
$733$	7.14590	0.263940	0.131970	+	0.991254i	$0.457870\pi$
$734$	−0.729490	−0.0269260
$735$	0	0
$736$	8.89667	0.327936
$737$	−21.7984	−0.802953
$738$	0	0
$739$	6.76393	0.248815	0.124408	−	0.992231i	$-0.460297\pi$
$739$	6.76393	0.248815	0.124408	+	0.992231i	$0.460297\pi$
$740$	14.7295	0.541467
$741$	0	0
$742$	3.09017	0.113444
$743$	−49.1033	−1.80143	−0.900713	−	0.434414i	$-0.856955\pi$
$743$	−49.1033	−1.80143	−0.900713	+	0.434414i	$0.856955\pi$
$744$	0	0
$745$	9.32624	0.341687
$746$	4.41641	0.161696
$747$	0	0
$748$	6.70820	0.245276
$749$	−17.4721	−0.638418
$750$	0	0
$751$	−35.8885	−1.30959	−0.654796	−	0.755806i	$-0.727245\pi$
$751$	−35.8885	−1.30959	−0.654796	+	0.755806i	$0.727245\pi$
$752$	17.2148	0.627758
$753$	0	0
$754$	−5.85410	−0.213194
$755$	−23.0000	−0.837056
$756$	0	0
$757$	−48.4508	−1.76098	−0.880488	−	0.474068i	$-0.842785\pi$
$757$	−48.4508	−1.76098	−0.880488	+	0.474068i	$0.842785\pi$
$758$	−7.01316	−0.254729
$759$	0	0
$760$	−2.03444	−0.0737970
$761$	−8.25735	−0.299329	−0.149664	−	0.988737i	$-0.547819\pi$
$761$	−8.25735	−0.299329	−0.149664	+	0.988737i	$0.547819\pi$
$762$	0	0
$763$	12.5623	0.454786
$764$	−40.5197	−1.46595
$765$	0	0
$766$	6.38197	0.230590
$767$	−15.5623	−0.561922
$768$	0	0
$769$	44.4164	1.60170	0.800848	−	0.598867i	$-0.204382\pi$
$769$	44.4164	1.60170	0.800848	+	0.598867i	$0.204382\pi$
$770$	−1.61803	−0.0583099
$771$	0	0
$772$	26.4590	0.952280
$773$	24.7984	0.891936	0.445968	−	0.895049i	$-0.352860\pi$
$773$	24.7984	0.891936	0.445968	+	0.895049i	$0.352860\pi$
$774$	0	0
$775$	1.47214	0.0528807
$776$	−11.9098	−0.427538
$777$	0	0
$778$	−0.840946	−0.0301494
$779$	−17.3607	−0.622011
$780$	0	0
$781$	12.3262	0.441067
$782$	−0.700073	−0.0250346
$783$	0	0
$784$	3.14590	0.112354
$785$	8.65248	0.308820
$786$	0	0
$787$	−28.1803	−1.00452	−0.502260	−	0.864716i	$-0.667498\pi$
$787$	−28.1803	−1.00452	−0.502260	+	0.864716i	$0.667498\pi$
$788$	11.5623	0.411890
$789$	0	0
$790$	3.74265	0.133157
$791$	9.09017	0.323209
$792$	0	0
$793$	0.381966	0.0135640
$794$	5.12461	0.181866
$795$	0	0
$796$	−30.7082	−1.08842
$797$	−29.0344	−1.02845	−0.514226	−	0.857655i	$-0.671921\pi$
$797$	−29.0344	−1.02845	−0.514226	+	0.857655i	$0.671921\pi$
$798$	0	0
$799$	−4.67376	−0.165346
$800$	4.14590	0.146580
$801$	0	0
$802$	−13.4721	−0.475718
$803$	−33.6525	−1.18757
$804$	0	0
$805$	−2.14590	−0.0756330
$806$	−1.47214	−0.0518538
$807$	0	0
$808$	22.6950	0.798409
$809$	7.47214	0.262706	0.131353	−	0.991336i	$-0.458068\pi$
$809$	7.47214	0.262706	0.131353	+	0.991336i	$0.458068\pi$
$810$	0	0
$811$	−46.6869	−1.63940	−0.819700	−	0.572793i	$-0.805860\pi$
$811$	−46.6869	−1.63940	−0.819700	+	0.572793i	$0.805860\pi$
$812$	−10.8541	−0.380904
$813$	0	0
$814$	12.8541	0.450536
$815$	5.05573	0.177094
$816$	0	0
$817$	3.09017	0.108111
$818$	8.54102	0.298630
$819$	0	0
$820$	23.2918	0.813385
$821$	5.47214	0.190979	0.0954894	−	0.995430i	$-0.469558\pi$
$821$	5.47214	0.190979	0.0954894	+	0.995430i	$0.469558\pi$
$822$	0	0
$823$	−50.2361	−1.75112	−0.875560	−	0.483110i	$-0.839507\pi$
$823$	−50.2361	−1.75112	−0.875560	+	0.483110i	$0.839507\pi$
$824$	29.2279	1.01820
$825$	0	0
$826$	2.27051	0.0790011
$827$	0.888544	0.0308977	0.0154488	−	0.999881i	$-0.495082\pi$
$827$	0.888544	0.0308977	0.0154488	+	0.999881i	$0.495082\pi$
$828$	0	0
$829$	−33.0689	−1.14853	−0.574265	−	0.818670i	$-0.694712\pi$
$829$	−33.0689	−1.14853	−0.574265	+	0.818670i	$0.694712\pi$
$830$	−4.67376	−0.162229
$831$	0	0
$832$	12.3262	0.427335
$833$	−0.854102	−0.0295929
$834$	0	0
$835$	−14.4164	−0.498900
$836$	10.8541	0.375397
$837$	0	0
$838$	1.43769	0.0496643
$839$	32.8541	1.13425	0.567125	−	0.823632i	$-0.308056\pi$
$839$	32.8541	1.13425	0.567125	+	0.823632i	$0.308056\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	11.9443	0.411627
$843$	0	0
$844$	−42.0000	−1.44570
$845$	6.14590	0.211425
$846$	0	0
$847$	6.94427	0.238608
$848$	25.4508	0.873986
$849$	0	0
$850$	−0.326238	−0.0111899
$851$	17.0476	0.584384
$852$	0	0
$853$	18.0557	0.618216	0.309108	−	0.951027i	$-0.399969\pi$
$853$	18.0557	0.618216	0.309108	+	0.951027i	$0.399969\pi$
$854$	−0.0557281	−0.00190698
$855$	0	0
$856$	25.7214	0.879138
$857$	−44.7082	−1.52720	−0.763602	−	0.645688i	$-0.776571\pi$
$857$	−44.7082	−1.52720	−0.763602	+	0.645688i	$0.776571\pi$
$858$	0	0
$859$	28.6525	0.977610	0.488805	−	0.872393i	$-0.337433\pi$
$859$	28.6525	0.977610	0.488805	+	0.872393i	$0.337433\pi$
$860$	−4.14590	−0.141374
$861$	0	0
$862$	10.5492	0.359305
$863$	−23.0557	−0.784826	−0.392413	−	0.919789i	$-0.628360\pi$
$863$	−23.0557	−0.784826	−0.392413	+	0.919789i	$0.628360\pi$
$864$	0	0
$865$	13.4721	0.458066
$866$	6.14590	0.208846
$867$	0	0
$868$	−2.72949	−0.0926449
$869$	−41.5066	−1.40801
$870$	0	0
$871$	13.4721	0.456486
$872$	−18.4934	−0.626266
$873$	0	0
$874$	−1.13274	−0.0383156
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−5.29180	−0.178691	−0.0893456	−	0.996001i	$-0.528478\pi$
$877$	−5.29180	−0.178691	−0.0893456	+	0.996001i	$0.528478\pi$
$878$	5.14590	0.173666
$879$	0	0
$880$	−13.3262	−0.449227
$881$	−18.9443	−0.638249	−0.319124	−	0.947713i	$-0.603389\pi$
$881$	−18.9443	−0.638249	−0.319124	+	0.947713i	$0.603389\pi$
$882$	0	0
$883$	30.8885	1.03948	0.519741	−	0.854324i	$-0.326028\pi$
$883$	30.8885	1.03948	0.519741	+	0.854324i	$0.326028\pi$
$884$	−4.14590	−0.139442
$885$	0	0
$886$	13.6950	0.460094
$887$	−27.1246	−0.910755	−0.455378	−	0.890298i	$-0.650496\pi$
$887$	−27.1246	−0.910755	−0.455378	+	0.890298i	$0.650496\pi$
$888$	0	0
$889$	2.09017	0.0701020
$890$	3.32624	0.111496
$891$	0	0
$892$	−12.1033	−0.405249
$893$	−7.56231	−0.253063
$894$	0	0
$895$	−13.9443	−0.466106
$896$	−10.0902	−0.337089
$897$	0	0
$898$	15.8197	0.527909
$899$	8.61803	0.287428
$900$	0	0
$901$	−6.90983	−0.230200
$902$	20.3262	0.676790
$903$	0	0
$904$	−13.3820	−0.445078
$905$	−0.236068	−0.00784717
$906$	0	0
$907$	−25.2016	−0.836806	−0.418403	−	0.908261i	$-0.637410\pi$
$907$	−25.2016	−0.836806	−0.418403	+	0.908261i	$0.637410\pi$
$908$	25.0426	0.831067
$909$	0	0
$910$	1.00000	0.0331497
$911$	−19.4164	−0.643294	−0.321647	−	0.946860i	$-0.604236\pi$
$911$	−19.4164	−0.643294	−0.321647	+	0.946860i	$0.604236\pi$
$912$	0	0
$913$	51.8328	1.71542
$914$	−6.36881	−0.210662
$915$	0	0
$916$	−51.2705	−1.69403
$917$	−10.5623	−0.348798
$918$	0	0
$919$	38.7426	1.27800	0.639001	−	0.769206i	$-0.279348\pi$
$919$	38.7426	1.27800	0.639001	+	0.769206i	$0.279348\pi$
$920$	3.15905	0.104151
$921$	0	0
$922$	0.527864	0.0173843
$923$	−7.61803	−0.250751
$924$	0	0
$925$	7.94427	0.261206
$926$	−11.0000	−0.361482
$927$	0	0
$928$	24.2705	0.796719
$929$	−23.9098	−0.784456	−0.392228	−	0.919868i	$-0.628296\pi$
$929$	−23.9098	−0.784456	−0.392228	+	0.919868i	$0.628296\pi$
$930$	0	0
$931$	−1.38197	−0.0452921
$932$	25.1459	0.823681
$933$	0	0
$934$	−1.16718	−0.0381914
$935$	3.61803	0.118322
$936$	0	0
$937$	−49.1935	−1.60708	−0.803541	−	0.595250i	$-0.797053\pi$
$937$	−49.1935	−1.60708	−0.803541	+	0.595250i	$0.797053\pi$
$938$	−1.96556	−0.0641777
$939$	0	0
$940$	10.1459	0.330923
$941$	4.41641	0.143971	0.0719854	−	0.997406i	$-0.477067\pi$
$941$	4.41641	0.143971	0.0719854	+	0.997406i	$0.477067\pi$
$942$	0	0
$943$	26.9574	0.877855
$944$	18.7001	0.608636
$945$	0	0
$946$	−3.61803	−0.117632
$947$	23.6180	0.767483	0.383741	−	0.923441i	$-0.374635\pi$
$947$	23.6180	0.767483	0.383741	+	0.923441i	$0.374635\pi$
$948$	0	0
$949$	20.7984	0.675144
$950$	−0.527864	−0.0171262
$951$	0	0
$952$	1.25735	0.0407511
$953$	−30.4164	−0.985284	−0.492642	−	0.870232i	$-0.663969\pi$
$953$	−30.4164	−0.985284	−0.492642	+	0.870232i	$0.663969\pi$
$954$	0	0
$955$	−21.8541	−0.707182
$956$	−9.60488	−0.310644
$957$	0	0
$958$	−1.56231	−0.0504758
$959$	6.47214	0.208996
$960$	0	0
$961$	−28.8328	−0.930091
$962$	−7.94427	−0.256134
$963$	0	0
$964$	−26.6656	−0.858842
$965$	14.2705	0.459384
$966$	0	0
$967$	−33.1803	−1.06701	−0.533504	−	0.845798i	$-0.679125\pi$
$967$	−33.1803	−1.06701	−0.533504	+	0.845798i	$0.679125\pi$
$968$	−10.2229	−0.328577
$969$	0	0
$970$	−3.09017	−0.0992194
$971$	−36.9443	−1.18560	−0.592799	−	0.805350i	$-0.701977\pi$
$971$	−36.9443	−1.18560	−0.592799	+	0.805350i	$0.701977\pi$
$972$	0	0
$973$	9.00000	0.288527
$974$	−4.54102	−0.145504
$975$	0	0
$976$	−0.458980	−0.0146916
$977$	−2.58359	−0.0826564	−0.0413282	−	0.999146i	$-0.513159\pi$
$977$	−2.58359	−0.0826564	−0.0413282	+	0.999146i	$0.513159\pi$
$978$	0	0
$979$	−36.8885	−1.17896
$980$	1.85410	0.0592271
$981$	0	0
$982$	−6.36881	−0.203237
$983$	58.9787	1.88113	0.940564	−	0.339615i	$-0.110297\pi$
$983$	58.9787	1.88113	0.940564	+	0.339615i	$0.110297\pi$
$984$	0	0
$985$	6.23607	0.198698
$986$	−1.90983	−0.0608214
$987$	0	0
$988$	−6.70820	−0.213416
$989$	−4.79837	−0.152579
$990$	0	0
$991$	9.47214	0.300892	0.150446	−	0.988618i	$-0.451929\pi$
$991$	9.47214	0.300892	0.150446	+	0.988618i	$0.451929\pi$
$992$	6.10333	0.193781
$993$	0	0
$994$	1.11146	0.0352532
$995$	−16.5623	−0.525060
$996$	0	0
$997$	−45.2918	−1.43441	−0.717203	−	0.696865i	$-0.754578\pi$
$997$	−45.2918	−1.43441	−0.717203	+	0.696865i	$0.754578\pi$
$998$	6.67376	0.211254
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.l.1.1	yes	2
3.2	odd	2		945.2.a.a.1.2	✓	2
5.4	even	2		4725.2.a.u.1.2		2
7.6	odd	2		6615.2.a.x.1.1		2
15.14	odd	2		4725.2.a.bh.1.1		2
21.20	even	2		6615.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.a.1.2	✓	2	3.2	odd	2
945.2.a.l.1.1	yes	2	1.1	even	1	trivial
4725.2.a.u.1.2		2	5.4	even	2
4725.2.a.bh.1.1		2	15.14	odd	2
6615.2.a.k.1.2		2	21.20	even	2
6615.2.a.x.1.1		2	7.6	odd	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)

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.47214 q^{8} +O(q^{10})\)	\(q+0.381966 q^{2} -1.85410 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.47214 q^{8} -0.381966 q^{10} +4.23607 q^{11} -2.61803 q^{13} +0.381966 q^{14} +3.14590 q^{16} -0.854102 q^{17} -1.38197 q^{19} +1.85410 q^{20} +1.61803 q^{22} +2.14590 q^{23} +1.00000 q^{25} -1.00000 q^{26} -1.85410 q^{28} +5.85410 q^{29} +1.47214 q^{31} +4.14590 q^{32} -0.326238 q^{34} -1.00000 q^{35} +7.94427 q^{37} -0.527864 q^{38} +1.47214 q^{40} +12.5623 q^{41} -2.23607 q^{43} -7.85410 q^{44} +0.819660 q^{46} +5.47214 q^{47} +1.00000 q^{49} +0.381966 q^{50} +4.85410 q^{52} +8.09017 q^{53} -4.23607 q^{55} -1.47214 q^{56} +2.23607 q^{58} +5.94427 q^{59} -0.145898 q^{61} +0.562306 q^{62} -4.70820 q^{64} +2.61803 q^{65} -5.14590 q^{67} +1.58359 q^{68} -0.381966 q^{70} +2.90983 q^{71} -7.94427 q^{73} +3.03444 q^{74} +2.56231 q^{76} +4.23607 q^{77} -9.79837 q^{79} -3.14590 q^{80} +4.79837 q^{82} +12.2361 q^{83} +0.854102 q^{85} -0.854102 q^{86} -6.23607 q^{88} -8.70820 q^{89} -2.61803 q^{91} -3.97871 q^{92} +2.09017 q^{94} +1.38197 q^{95} +8.09017 q^{97} +0.381966 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8	\( 2 q + 3 q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8} - 3 q^{10} + 4 q^{11} - 3 q^{13} + 3 q^{14} + 13 q^{16} + 5 q^{17} - 5 q^{19} - 3 q^{20} + q^{22} + 11 q^{23} + 2 q^{25} - 2 q^{26} + 3 q^{28} + 5 q^{29} - 6 q^{31} + 15 q^{32} + 15 q^{34} - 2 q^{35} - 2 q^{37} - 10 q^{38} - 6 q^{40} + 5 q^{41} - 9 q^{44} + 24 q^{46} + 2 q^{47} + 2 q^{49} + 3 q^{50} + 3 q^{52} + 5 q^{53} - 4 q^{55} + 6 q^{56} - 6 q^{59} - 7 q^{61} - 19 q^{62} + 4 q^{64} + 3 q^{65} - 17 q^{67} + 30 q^{68} - 3 q^{70} + 17 q^{71} + 2 q^{73} - 23 q^{74} - 15 q^{76} + 4 q^{77} + 5 q^{79} - 13 q^{80} - 15 q^{82} + 20 q^{83} - 5 q^{85} + 5 q^{86} - 8 q^{88} - 4 q^{89} - 3 q^{91} + 39 q^{92} - 7 q^{94} + 5 q^{95} + 5 q^{97} + 3 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 - 3 * q^10 + 4 * q^11 - 3 * q^13 + 3 * q^14 + 13 * q^16 + 5 * q^17 - 5 * q^19 - 3 * q^20 + q^22 + 11 * q^23 + 2 * q^25 - 2 * q^26 + 3 * q^28 + 5 * q^29 - 6 * q^31 + 15 * q^32 + 15 * q^34 - 2 * q^35 - 2 * q^37 - 10 * q^38 - 6 * q^40 + 5 * q^41 - 9 * q^44 + 24 * q^46 + 2 * q^47 + 2 * q^49 + 3 * q^50 + 3 * q^52 + 5 * q^53 - 4 * q^55 + 6 * q^56 - 6 * q^59 - 7 * q^61 - 19 * q^62 + 4 * q^64 + 3 * q^65 - 17 * q^67 + 30 * q^68 - 3 * q^70 + 17 * q^71 + 2 * q^73 - 23 * q^74 - 15 * q^76 + 4 * q^77 + 5 * q^79 - 13 * q^80 - 15 * q^82 + 20 * q^83 - 5 * q^85 + 5 * q^86 - 8 * q^88 - 4 * q^89 - 3 * q^91 + 39 * q^92 - 7 * q^94 + 5 * q^95 + 5 * q^97 + 3 * q^98

Introduction

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