LMFDB - Embedded newform 945.2.a.c.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} -3.23607 q^{11} +2.47214 q^{13} -0.618034 q^{14} +1.85410 q^{16} +4.23607 q^{17} -7.47214 q^{19} -1.61803 q^{20} -2.00000 q^{22} -6.23607 q^{23} +1.00000 q^{25} +1.52786 q^{26} +1.61803 q^{28} -7.70820 q^{29} -9.00000 q^{31} +5.61803 q^{32} +2.61803 q^{34} -1.00000 q^{35} -9.70820 q^{37} -4.61803 q^{38} -2.23607 q^{40} +8.47214 q^{41} -8.47214 q^{43} +5.23607 q^{44} -3.85410 q^{46} -4.00000 q^{47} +1.00000 q^{49} +0.618034 q^{50} -4.00000 q^{52} +7.94427 q^{53} -3.23607 q^{55} +2.23607 q^{56} -4.76393 q^{58} +11.7082 q^{59} +0.708204 q^{61} -5.56231 q^{62} -0.236068 q^{64} +2.47214 q^{65} +10.9443 q^{67} -6.85410 q^{68} -0.618034 q^{70} +2.00000 q^{71} +0.472136 q^{73} -6.00000 q^{74} +12.0902 q^{76} +3.23607 q^{77} -17.1803 q^{79} +1.85410 q^{80} +5.23607 q^{82} -11.0000 q^{83} +4.23607 q^{85} -5.23607 q^{86} +7.23607 q^{88} -8.18034 q^{89} -2.47214 q^{91} +10.0902 q^{92} -2.47214 q^{94} -7.47214 q^{95} -5.23607 q^{97} +0.618034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{11} - 4 q^{13} + q^{14} - 3 q^{16} + 4 q^{17} - 6 q^{19} - q^{20} - 4 q^{22} - 8 q^{23} + 2 q^{25} + 12 q^{26} + q^{28} - 2 q^{29} - 18 q^{31} + 9 q^{32} + 3 q^{34} - 2 q^{35} - 6 q^{37} - 7 q^{38} + 8 q^{41} - 8 q^{43} + 6 q^{44} - q^{46} - 8 q^{47} + 2 q^{49} - q^{50} - 8 q^{52} - 2 q^{53} - 2 q^{55} - 14 q^{58} + 10 q^{59} - 12 q^{61} + 9 q^{62} + 4 q^{64} - 4 q^{65} + 4 q^{67} - 7 q^{68} + q^{70} + 4 q^{71} - 8 q^{73} - 12 q^{74} + 13 q^{76} + 2 q^{77} - 12 q^{79} - 3 q^{80} + 6 q^{82} - 22 q^{83} + 4 q^{85} - 6 q^{86} + 10 q^{88} + 6 q^{89} + 4 q^{91} + 9 q^{92} + 4 q^{94} - 6 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 - 2 * q^11 - 4 * q^13 + q^14 - 3 * q^16 + 4 * q^17 - 6 * q^19 - q^20 - 4 * q^22 - 8 * q^23 + 2 * q^25 + 12 * q^26 + q^28 - 2 * q^29 - 18 * q^31 + 9 * q^32 + 3 * q^34 - 2 * q^35 - 6 * q^37 - 7 * q^38 + 8 * q^41 - 8 * q^43 + 6 * q^44 - q^46 - 8 * q^47 + 2 * q^49 - q^50 - 8 * q^52 - 2 * q^53 - 2 * q^55 - 14 * q^58 + 10 * q^59 - 12 * q^61 + 9 * q^62 + 4 * q^64 - 4 * q^65 + 4 * q^67 - 7 * q^68 + q^70 + 4 * q^71 - 8 * q^73 - 12 * q^74 + 13 * q^76 + 2 * q^77 - 12 * q^79 - 3 * q^80 + 6 * q^82 - 22 * q^83 + 4 * q^85 - 6 * q^86 + 10 * q^88 + 6 * q^89 + 4 * q^91 + 9 * q^92 + 4 * q^94 - 6 * q^95 - 6 * 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.618034	0.195440
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	2.47214	0.685647	0.342824	−	0.939400i	$-0.388617\pi$
$13$	2.47214	0.685647	0.342824	+	0.939400i	$0.388617\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	4.23607	1.02740	0.513699	−	0.857971i	$-0.328275\pi$
$17$	4.23607	1.02740	0.513699	+	0.857971i	$0.328275\pi$
$18$	0	0
$19$	−7.47214	−1.71423	−0.857113	−	0.515129i	$-0.827744\pi$
$19$	−7.47214	−1.71423	−0.857113	+	0.515129i	$0.827744\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−6.23607	−1.30031	−0.650155	−	0.759802i	$-0.725296\pi$
$23$	−6.23607	−1.30031	−0.650155	+	0.759802i	$0.725296\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.52786	0.299639
$27$	0	0
$28$	1.61803	0.305780
$29$	−7.70820	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$29$	−7.70820	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	2.61803	0.448989
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.70820	−1.59602	−0.798009	−	0.602645i	$-0.794114\pi$
$37$	−9.70820	−1.59602	−0.798009	+	0.602645i	$0.794114\pi$
$38$	−4.61803	−0.749144
$39$	0	0
$40$	−2.23607	−0.353553
$41$	8.47214	1.32313	0.661563	−	0.749890i	$-0.269894\pi$
$41$	8.47214	1.32313	0.661563	+	0.749890i	$0.269894\pi$
$42$	0	0
$43$	−8.47214	−1.29199	−0.645994	−	0.763342i	$-0.723557\pi$
$43$	−8.47214	−1.29199	−0.645994	+	0.763342i	$0.723557\pi$
$44$	5.23607	0.789367
$45$	0	0
$46$	−3.85410	−0.568256
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.618034	0.0874032
$51$	0	0
$52$	−4.00000	−0.554700
$53$	7.94427	1.09123	0.545615	−	0.838036i	$-0.316296\pi$
$53$	7.94427	1.09123	0.545615	+	0.838036i	$0.316296\pi$
$54$	0	0
$55$	−3.23607	−0.436351
$56$	2.23607	0.298807
$57$	0	0
$58$	−4.76393	−0.625535
$59$	11.7082	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$59$	11.7082	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$60$	0	0
$61$	0.708204	0.0906762	0.0453381	−	0.998972i	$-0.485563\pi$
$61$	0.708204	0.0906762	0.0453381	+	0.998972i	$0.485563\pi$
$62$	−5.56231	−0.706414
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	2.47214	0.306631
$66$	0	0
$67$	10.9443	1.33706	0.668528	−	0.743687i	$-0.266925\pi$
$67$	10.9443	1.33706	0.668528	+	0.743687i	$0.266925\pi$
$68$	−6.85410	−0.831182
$69$	0	0
$70$	−0.618034	−0.0738692
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	0.472136	0.0552593	0.0276297	−	0.999618i	$-0.491204\pi$
$73$	0.472136	0.0552593	0.0276297	+	0.999618i	$0.491204\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	12.0902	1.38684
$77$	3.23607	0.368784
$78$	0	0
$79$	−17.1803	−1.93294	−0.966470	−	0.256781i	$-0.917338\pi$
$79$	−17.1803	−1.93294	−0.966470	+	0.256781i	$0.917338\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	5.23607	0.578227
$83$	−11.0000	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$84$	0	0
$85$	4.23607	0.459466
$86$	−5.23607	−0.564620
$87$	0	0
$88$	7.23607	0.771367
$89$	−8.18034	−0.867114	−0.433557	−	0.901126i	$-0.642742\pi$
$89$	−8.18034	−0.867114	−0.433557	+	0.901126i	$0.642742\pi$
$90$	0	0
$91$	−2.47214	−0.259150
$92$	10.0902	1.05197
$93$	0	0
$94$	−2.47214	−0.254981
$95$	−7.47214	−0.766625
$96$	0	0
$97$	−5.23607	−0.531642	−0.265821	−	0.964022i	$-0.585643\pi$
$97$	−5.23607	−0.531642	−0.265821	+	0.964022i	$0.585643\pi$
$98$	0.618034	0.0624309
$99$	0	0
$100$	−1.61803	−0.161803
$101$	6.18034	0.614967	0.307483	−	0.951553i	$-0.400513\pi$
$101$	6.18034	0.614967	0.307483	+	0.951553i	$0.400513\pi$
$102$	0	0
$103$	−10.7639	−1.06060	−0.530301	−	0.847810i	$-0.677921\pi$
$103$	−10.7639	−1.06060	−0.530301	+	0.847810i	$0.677921\pi$
$104$	−5.52786	−0.542052
$105$	0	0
$106$	4.90983	0.476885
$107$	11.4164	1.10367	0.551833	−	0.833955i	$-0.313929\pi$
$107$	11.4164	1.10367	0.551833	+	0.833955i	$0.313929\pi$
$108$	0	0
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	−1.85410	−0.175196
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−6.23607	−0.581516
$116$	12.4721	1.15801
$117$	0	0
$118$	7.23607	0.666134
$119$	−4.23607	−0.388320
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0.437694	0.0396270
$123$	0	0
$124$	14.5623	1.30773
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.1803	1.43577	0.717886	−	0.696160i	$-0.245110\pi$
$127$	16.1803	1.43577	0.717886	+	0.696160i	$0.245110\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	1.52786	0.134003
$131$	16.7639	1.46467	0.732336	−	0.680944i	$-0.238430\pi$
$131$	16.7639	1.46467	0.732336	+	0.680944i	$0.238430\pi$
$132$	0	0
$133$	7.47214	0.647916
$134$	6.76393	0.584315
$135$	0	0
$136$	−9.47214	−0.812229
$137$	1.94427	0.166110	0.0830552	−	0.996545i	$-0.473532\pi$
$137$	1.94427	0.166110	0.0830552	+	0.996545i	$0.473532\pi$
$138$	0	0
$139$	5.52786	0.468867	0.234434	−	0.972132i	$-0.424676\pi$
$139$	5.52786	0.468867	0.234434	+	0.972132i	$0.424676\pi$
$140$	1.61803	0.136749
$141$	0	0
$142$	1.23607	0.103729
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−7.70820	−0.640131
$146$	0.291796	0.0241492
$147$	0	0
$148$	15.7082	1.29121
$149$	6.94427	0.568897	0.284448	−	0.958691i	$-0.408190\pi$
$149$	6.94427	0.568897	0.284448	+	0.958691i	$0.408190\pi$
$150$	0	0
$151$	17.8885	1.45575	0.727875	−	0.685710i	$-0.240508\pi$
$151$	17.8885	1.45575	0.727875	+	0.685710i	$0.240508\pi$
$152$	16.7082	1.35521
$153$	0	0
$154$	2.00000	0.161165
$155$	−9.00000	−0.722897
$156$	0	0
$157$	−0.763932	−0.0609684	−0.0304842	−	0.999535i	$-0.509705\pi$
$157$	−0.763932	−0.0609684	−0.0304842	+	0.999535i	$0.509705\pi$
$158$	−10.6180	−0.844725
$159$	0	0
$160$	5.61803	0.444145
$161$	6.23607	0.491471
$162$	0	0
$163$	11.2361	0.880077	0.440038	−	0.897979i	$-0.354965\pi$
$163$	11.2361	0.880077	0.440038	+	0.897979i	$0.354965\pi$
$164$	−13.7082	−1.07043
$165$	0	0
$166$	−6.79837	−0.527656
$167$	−5.94427	−0.459982	−0.229991	−	0.973193i	$-0.573870\pi$
$167$	−5.94427	−0.459982	−0.229991	+	0.973193i	$0.573870\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	2.61803	0.200794
$171$	0	0
$172$	13.7082	1.04524
$173$	−13.1803	−1.00208	−0.501041	−	0.865423i	$-0.667050\pi$
$173$	−13.1803	−1.00208	−0.501041	+	0.865423i	$0.667050\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−6.00000	−0.452267
$177$	0	0
$178$	−5.05573	−0.378943
$179$	15.4164	1.15228	0.576138	−	0.817352i	$-0.304559\pi$
$179$	15.4164	1.15228	0.576138	+	0.817352i	$0.304559\pi$
$180$	0	0
$181$	3.76393	0.279771	0.139885	−	0.990168i	$-0.455327\pi$
$181$	3.76393	0.279771	0.139885	+	0.990168i	$0.455327\pi$
$182$	−1.52786	−0.113253
$183$	0	0
$184$	13.9443	1.02799
$185$	−9.70820	−0.713761
$186$	0	0
$187$	−13.7082	−1.00244
$188$	6.47214	0.472029
$189$	0	0
$190$	−4.61803	−0.335027
$191$	20.7639	1.50243	0.751213	−	0.660060i	$-0.229469\pi$
$191$	20.7639	1.50243	0.751213	+	0.660060i	$0.229469\pi$
$192$	0	0
$193$	−3.41641	−0.245918	−0.122959	−	0.992412i	$-0.539238\pi$
$193$	−3.41641	−0.245918	−0.122959	+	0.992412i	$0.539238\pi$
$194$	−3.23607	−0.232336
$195$	0	0
$196$	−1.61803	−0.115574
$197$	0.0557281	0.00397046	0.00198523	−	0.999998i	$-0.499368\pi$
$197$	0.0557281	0.00397046	0.00198523	+	0.999998i	$0.499368\pi$
$198$	0	0
$199$	−26.4721	−1.87656	−0.938280	−	0.345877i	$-0.887582\pi$
$199$	−26.4721	−1.87656	−0.938280	+	0.345877i	$0.887582\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	3.81966	0.268750
$203$	7.70820	0.541010
$204$	0	0
$205$	8.47214	0.591720
$206$	−6.65248	−0.463500
$207$	0	0
$208$	4.58359	0.317815
$209$	24.1803	1.67259
$210$	0	0
$211$	0.708204	0.0487548	0.0243774	−	0.999703i	$-0.492240\pi$
$211$	0.708204	0.0487548	0.0243774	+	0.999703i	$0.492240\pi$
$212$	−12.8541	−0.882823
$213$	0	0
$214$	7.05573	0.482320
$215$	−8.47214	−0.577795
$216$	0	0
$217$	9.00000	0.610960
$218$	0.618034	0.0418585
$219$	0	0
$220$	5.23607	0.353016
$221$	10.4721	0.704432
$222$	0	0
$223$	18.6525	1.24906	0.624531	−	0.781000i	$-0.285290\pi$
$223$	18.6525	1.24906	0.624531	+	0.781000i	$0.285290\pi$
$224$	−5.61803	−0.375371
$225$	0	0
$226$	−1.23607	−0.0822220
$227$	−11.0000	−0.730096	−0.365048	−	0.930989i	$-0.618947\pi$
$227$	−11.0000	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$228$	0	0
$229$	8.23607	0.544255	0.272127	−	0.962261i	$-0.412273\pi$
$229$	8.23607	0.544255	0.272127	+	0.962261i	$0.412273\pi$
$230$	−3.85410	−0.254132
$231$	0	0
$232$	17.2361	1.13160
$233$	−18.3607	−1.20285	−0.601424	−	0.798930i	$-0.705400\pi$
$233$	−18.3607	−1.20285	−0.601424	+	0.798930i	$0.705400\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	−18.9443	−1.23317
$237$	0	0
$238$	−2.61803	−0.169702
$239$	12.4721	0.806755	0.403378	−	0.915034i	$-0.367836\pi$
$239$	12.4721	0.806755	0.403378	+	0.915034i	$0.367836\pi$
$240$	0	0
$241$	−5.65248	−0.364108	−0.182054	−	0.983289i	$-0.558275\pi$
$241$	−5.65248	−0.364108	−0.182054	+	0.983289i	$0.558275\pi$
$242$	−0.326238	−0.0209714
$243$	0	0
$244$	−1.14590	−0.0733586
$245$	1.00000	0.0638877
$246$	0	0
$247$	−18.4721	−1.17535
$248$	20.1246	1.27791
$249$	0	0
$250$	0.618034	0.0390879
$251$	2.76393	0.174458	0.0872289	−	0.996188i	$-0.472199\pi$
$251$	2.76393	0.174458	0.0872289	+	0.996188i	$0.472199\pi$
$252$	0	0
$253$	20.1803	1.26873
$254$	10.0000	0.627456
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−24.5967	−1.53430	−0.767151	−	0.641466i	$-0.778327\pi$
$257$	−24.5967	−1.53430	−0.767151	+	0.641466i	$0.778327\pi$
$258$	0	0
$259$	9.70820	0.603238
$260$	−4.00000	−0.248069
$261$	0	0
$262$	10.3607	0.640085
$263$	−9.52786	−0.587513	−0.293757	−	0.955880i	$-0.594906\pi$
$263$	−9.52786	−0.587513	−0.293757	+	0.955880i	$0.594906\pi$
$264$	0	0
$265$	7.94427	0.488013
$266$	4.61803	0.283150
$267$	0	0
$268$	−17.7082	−1.08170
$269$	−5.70820	−0.348035	−0.174018	−	0.984743i	$-0.555675\pi$
$269$	−5.70820	−0.348035	−0.174018	+	0.984743i	$0.555675\pi$
$270$	0	0
$271$	6.41641	0.389769	0.194885	−	0.980826i	$-0.437567\pi$
$271$	6.41641	0.389769	0.194885	+	0.980826i	$0.437567\pi$
$272$	7.85410	0.476225
$273$	0	0
$274$	1.20163	0.0725929
$275$	−3.23607	−0.195142
$276$	0	0
$277$	−14.3607	−0.862850	−0.431425	−	0.902149i	$-0.641989\pi$
$277$	−14.3607	−0.862850	−0.431425	+	0.902149i	$0.641989\pi$
$278$	3.41641	0.204903
$279$	0	0
$280$	2.23607	0.133631
$281$	−20.6525	−1.23202	−0.616012	−	0.787737i	$-0.711253\pi$
$281$	−20.6525	−1.23202	−0.616012	+	0.787737i	$0.711253\pi$
$282$	0	0
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−3.23607	−0.192025
$285$	0	0
$286$	−4.94427	−0.292361
$287$	−8.47214	−0.500094
$288$	0	0
$289$	0.944272	0.0555454
$290$	−4.76393	−0.279748
$291$	0	0
$292$	−0.763932	−0.0447057
$293$	−22.7082	−1.32663	−0.663314	−	0.748341i	$-0.730850\pi$
$293$	−22.7082	−1.32663	−0.663314	+	0.748341i	$0.730850\pi$
$294$	0	0
$295$	11.7082	0.681678
$296$	21.7082	1.26176
$297$	0	0
$298$	4.29180	0.248617
$299$	−15.4164	−0.891554
$300$	0	0
$301$	8.47214	0.488326
$302$	11.0557	0.636186
$303$	0	0
$304$	−13.8541	−0.794587
$305$	0.708204	0.0405516
$306$	0	0
$307$	−17.1246	−0.977353	−0.488677	−	0.872465i	$-0.662520\pi$
$307$	−17.1246	−0.977353	−0.488677	+	0.872465i	$0.662520\pi$
$308$	−5.23607	−0.298353
$309$	0	0
$310$	−5.56231	−0.315918
$311$	16.9443	0.960822	0.480411	−	0.877044i	$-0.340488\pi$
$311$	16.9443	0.960822	0.480411	+	0.877044i	$0.340488\pi$
$312$	0	0
$313$	−1.52786	−0.0863600	−0.0431800	−	0.999067i	$-0.513749\pi$
$313$	−1.52786	−0.0863600	−0.0431800	+	0.999067i	$0.513749\pi$
$314$	−0.472136	−0.0266442
$315$	0	0
$316$	27.7984	1.56378
$317$	−5.94427	−0.333864	−0.166932	−	0.985968i	$-0.553386\pi$
$317$	−5.94427	−0.333864	−0.166932	+	0.985968i	$0.553386\pi$
$318$	0	0
$319$	24.9443	1.39661
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	3.85410	0.214781
$323$	−31.6525	−1.76119
$324$	0	0
$325$	2.47214	0.137129
$326$	6.94427	0.384608
$327$	0	0
$328$	−18.9443	−1.04602
$329$	4.00000	0.220527
$330$	0	0
$331$	7.41641	0.407643	0.203821	−	0.979008i	$-0.434664\pi$
$331$	7.41641	0.407643	0.203821	+	0.979008i	$0.434664\pi$
$332$	17.7984	0.976813
$333$	0	0
$334$	−3.67376	−0.201019
$335$	10.9443	0.597949
$336$	0	0
$337$	3.05573	0.166456	0.0832281	−	0.996531i	$-0.473477\pi$
$337$	3.05573	0.166456	0.0832281	+	0.996531i	$0.473477\pi$
$338$	−4.25735	−0.231570
$339$	0	0
$340$	−6.85410	−0.371716
$341$	29.1246	1.57719
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	18.9443	1.02141
$345$	0	0
$346$	−8.14590	−0.437926
$347$	3.05573	0.164040	0.0820200	−	0.996631i	$-0.473863\pi$
$347$	3.05573	0.164040	0.0820200	+	0.996631i	$0.473863\pi$
$348$	0	0
$349$	11.1803	0.598470	0.299235	−	0.954179i	$-0.403269\pi$
$349$	11.1803	0.598470	0.299235	+	0.954179i	$0.403269\pi$
$350$	−0.618034	−0.0330353
$351$	0	0
$352$	−18.1803	−0.969015
$353$	4.47214	0.238028	0.119014	−	0.992893i	$-0.462027\pi$
$353$	4.47214	0.238028	0.119014	+	0.992893i	$0.462027\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	13.2361	0.701510
$357$	0	0
$358$	9.52786	0.503563
$359$	9.41641	0.496979	0.248489	−	0.968635i	$-0.420066\pi$
$359$	9.41641	0.496979	0.248489	+	0.968635i	$0.420066\pi$
$360$	0	0
$361$	36.8328	1.93857
$362$	2.32624	0.122264
$363$	0	0
$364$	4.00000	0.209657
$365$	0.472136	0.0247127
$366$	0	0
$367$	−23.4164	−1.22233	−0.611163	−	0.791505i	$-0.709298\pi$
$367$	−23.4164	−1.22233	−0.611163	+	0.791505i	$0.709298\pi$
$368$	−11.5623	−0.602727
$369$	0	0
$370$	−6.00000	−0.311925
$371$	−7.94427	−0.412446
$372$	0	0
$373$	−11.4164	−0.591119	−0.295560	−	0.955324i	$-0.595506\pi$
$373$	−11.4164	−0.591119	−0.295560	+	0.955324i	$0.595506\pi$
$374$	−8.47214	−0.438084
$375$	0	0
$376$	8.94427	0.461266
$377$	−19.0557	−0.981420
$378$	0	0
$379$	−2.34752	−0.120584	−0.0602921	−	0.998181i	$-0.519203\pi$
$379$	−2.34752	−0.120584	−0.0602921	+	0.998181i	$0.519203\pi$
$380$	12.0902	0.620213
$381$	0	0
$382$	12.8328	0.656584
$383$	27.9443	1.42789	0.713943	−	0.700204i	$-0.246908\pi$
$383$	27.9443	1.42789	0.713943	+	0.700204i	$0.246908\pi$
$384$	0	0
$385$	3.23607	0.164925
$386$	−2.11146	−0.107470
$387$	0	0
$388$	8.47214	0.430108
$389$	−6.65248	−0.337294	−0.168647	−	0.985677i	$-0.553940\pi$
$389$	−6.65248	−0.337294	−0.168647	+	0.985677i	$0.553940\pi$
$390$	0	0
$391$	−26.4164	−1.33594
$392$	−2.23607	−0.112938
$393$	0	0
$394$	0.0344419	0.00173516
$395$	−17.1803	−0.864437
$396$	0	0
$397$	−6.76393	−0.339472	−0.169736	−	0.985490i	$-0.554292\pi$
$397$	−6.76393	−0.339472	−0.169736	+	0.985490i	$0.554292\pi$
$398$	−16.3607	−0.820087
$399$	0	0
$400$	1.85410	0.0927051
$401$	−24.4721	−1.22208	−0.611040	−	0.791600i	$-0.709249\pi$
$401$	−24.4721	−1.22208	−0.611040	+	0.791600i	$0.709249\pi$
$402$	0	0
$403$	−22.2492	−1.10831
$404$	−10.0000	−0.497519
$405$	0	0
$406$	4.76393	0.236430
$407$	31.4164	1.55725
$408$	0	0
$409$	−0.236068	−0.0116728	−0.00583641	−	0.999983i	$-0.501858\pi$
$409$	−0.236068	−0.0116728	−0.00583641	+	0.999983i	$0.501858\pi$
$410$	5.23607	0.258591
$411$	0	0
$412$	17.4164	0.858045
$413$	−11.7082	−0.576123
$414$	0	0
$415$	−11.0000	−0.539969
$416$	13.8885	0.680942
$417$	0	0
$418$	14.9443	0.730948
$419$	18.6525	0.911233	0.455617	−	0.890176i	$-0.349419\pi$
$419$	18.6525	0.911233	0.455617	+	0.890176i	$0.349419\pi$
$420$	0	0
$421$	−27.8328	−1.35649	−0.678244	−	0.734837i	$-0.737259\pi$
$421$	−27.8328	−1.35649	−0.678244	+	0.734837i	$0.737259\pi$
$422$	0.437694	0.0213066
$423$	0	0
$424$	−17.7639	−0.862693
$425$	4.23607	0.205479
$426$	0	0
$427$	−0.708204	−0.0342724
$428$	−18.4721	−0.892884
$429$	0	0
$430$	−5.23607	−0.252506
$431$	−18.4721	−0.889771	−0.444886	−	0.895587i	$-0.646756\pi$
$431$	−18.4721	−0.889771	−0.444886	+	0.895587i	$0.646756\pi$
$432$	0	0
$433$	5.23607	0.251629	0.125815	−	0.992054i	$-0.459846\pi$
$433$	5.23607	0.251629	0.125815	+	0.992054i	$0.459846\pi$
$434$	5.56231	0.266999
$435$	0	0
$436$	−1.61803	−0.0774898
$437$	46.5967	2.22902
$438$	0	0
$439$	−21.0000	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$440$	7.23607	0.344966
$441$	0	0
$442$	6.47214	0.307848
$443$	17.2918	0.821558	0.410779	−	0.911735i	$-0.365257\pi$
$443$	17.2918	0.821558	0.410779	+	0.911735i	$0.365257\pi$
$444$	0	0
$445$	−8.18034	−0.387785
$446$	11.5279	0.545860
$447$	0	0
$448$	0.236068	0.0111532
$449$	24.9443	1.17719	0.588596	−	0.808427i	$-0.299681\pi$
$449$	24.9443	1.17719	0.588596	+	0.808427i	$0.299681\pi$
$450$	0	0
$451$	−27.4164	−1.29099
$452$	3.23607	0.152212
$453$	0	0
$454$	−6.79837	−0.319063
$455$	−2.47214	−0.115896
$456$	0	0
$457$	40.0689	1.87434	0.937172	−	0.348869i	$-0.113434\pi$
$457$	40.0689	1.87434	0.937172	+	0.348869i	$0.113434\pi$
$458$	5.09017	0.237848
$459$	0	0
$460$	10.0902	0.470457
$461$	19.5279	0.909503	0.454752	−	0.890618i	$-0.349728\pi$
$461$	19.5279	0.909503	0.454752	+	0.890618i	$0.349728\pi$
$462$	0	0
$463$	−23.5279	−1.09343	−0.546716	−	0.837318i	$-0.684122\pi$
$463$	−23.5279	−1.09343	−0.546716	+	0.837318i	$0.684122\pi$
$464$	−14.2918	−0.663480
$465$	0	0
$466$	−11.3475	−0.525664
$467$	−32.5279	−1.50521	−0.752605	−	0.658472i	$-0.771203\pi$
$467$	−32.5279	−1.50521	−0.752605	+	0.658472i	$0.771203\pi$
$468$	0	0
$469$	−10.9443	−0.505360
$470$	−2.47214	−0.114031
$471$	0	0
$472$	−26.1803	−1.20505
$473$	27.4164	1.26061
$474$	0	0
$475$	−7.47214	−0.342845
$476$	6.85410	0.314157
$477$	0	0
$478$	7.70820	0.352565
$479$	−27.5967	−1.26093	−0.630464	−	0.776219i	$-0.717135\pi$
$479$	−27.5967	−1.26093	−0.630464	+	0.776219i	$0.717135\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	−3.49342	−0.159121
$483$	0	0
$484$	0.854102	0.0388228
$485$	−5.23607	−0.237758
$486$	0	0
$487$	3.23607	0.146640	0.0733201	−	0.997308i	$-0.476641\pi$
$487$	3.23607	0.146640	0.0733201	+	0.997308i	$0.476641\pi$
$488$	−1.58359	−0.0716858
$489$	0	0
$490$	0.618034	0.0279199
$491$	−31.8885	−1.43911	−0.719555	−	0.694436i	$-0.755654\pi$
$491$	−31.8885	−1.43911	−0.719555	+	0.694436i	$0.755654\pi$
$492$	0	0
$493$	−32.6525	−1.47059
$494$	−11.4164	−0.513648
$495$	0	0
$496$	−16.6869	−0.749265
$497$	−2.00000	−0.0897123
$498$	0	0
$499$	5.76393	0.258029	0.129015	−	0.991643i	$-0.458819\pi$
$499$	5.76393	0.258029	0.129015	+	0.991643i	$0.458819\pi$
$500$	−1.61803	−0.0723607
$501$	0	0
$502$	1.70820	0.0762409
$503$	7.94427	0.354218	0.177109	−	0.984191i	$-0.443326\pi$
$503$	7.94427	0.354218	0.177109	+	0.984191i	$0.443326\pi$
$504$	0	0
$505$	6.18034	0.275022
$506$	12.4721	0.554454
$507$	0	0
$508$	−26.1803	−1.16156
$509$	8.36068	0.370581	0.185290	−	0.982684i	$-0.440677\pi$
$509$	8.36068	0.370581	0.185290	+	0.982684i	$0.440677\pi$
$510$	0	0
$511$	−0.472136	−0.0208861
$512$	18.7082	0.826794
$513$	0	0
$514$	−15.2016	−0.670515
$515$	−10.7639	−0.474316
$516$	0	0
$517$	12.9443	0.569288
$518$	6.00000	0.263625
$519$	0	0
$520$	−5.52786	−0.242413
$521$	−41.7082	−1.82727	−0.913635	−	0.406536i	$-0.866737\pi$
$521$	−41.7082	−1.82727	−0.913635	+	0.406536i	$0.866737\pi$
$522$	0	0
$523$	−20.0689	−0.877551	−0.438776	−	0.898597i	$-0.644588\pi$
$523$	−20.0689	−0.877551	−0.438776	+	0.898597i	$0.644588\pi$
$524$	−27.1246	−1.18494
$525$	0	0
$526$	−5.88854	−0.256753
$527$	−38.1246	−1.66073
$528$	0	0
$529$	15.8885	0.690806
$530$	4.90983	0.213269
$531$	0	0
$532$	−12.0902	−0.524175
$533$	20.9443	0.907197
$534$	0	0
$535$	11.4164	0.493574
$536$	−24.4721	−1.05704
$537$	0	0
$538$	−3.52786	−0.152097
$539$	−3.23607	−0.139387
$540$	0	0
$541$	−0.472136	−0.0202987	−0.0101494	−	0.999948i	$-0.503231\pi$
$541$	−0.472136	−0.0202987	−0.0101494	+	0.999948i	$0.503231\pi$
$542$	3.96556	0.170335
$543$	0	0
$544$	23.7984	1.02035
$545$	1.00000	0.0428353
$546$	0	0
$547$	−11.2361	−0.480420	−0.240210	−	0.970721i	$-0.577216\pi$
$547$	−11.2361	−0.480420	−0.240210	+	0.970721i	$0.577216\pi$
$548$	−3.14590	−0.134386
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	57.5967	2.45370
$552$	0	0
$553$	17.1803	0.730582
$554$	−8.87539	−0.377079
$555$	0	0
$556$	−8.94427	−0.379322
$557$	2.94427	0.124753	0.0623764	−	0.998053i	$-0.480132\pi$
$557$	2.94427	0.124753	0.0623764	+	0.998053i	$0.480132\pi$
$558$	0	0
$559$	−20.9443	−0.885848
$560$	−1.85410	−0.0783501
$561$	0	0
$562$	−12.7639	−0.538414
$563$	0.583592	0.0245955	0.0122977	−	0.999924i	$-0.496085\pi$
$563$	0.583592	0.0245955	0.0122977	+	0.999924i	$0.496085\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	−13.5967	−0.571514
$567$	0	0
$568$	−4.47214	−0.187647
$569$	42.4721	1.78052	0.890262	−	0.455448i	$-0.150521\pi$
$569$	42.4721	1.78052	0.890262	+	0.455448i	$0.150521\pi$
$570$	0	0
$571$	17.1803	0.718975	0.359487	−	0.933150i	$-0.382952\pi$
$571$	17.1803	0.718975	0.359487	+	0.933150i	$0.382952\pi$
$572$	12.9443	0.541227
$573$	0	0
$574$	−5.23607	−0.218549
$575$	−6.23607	−0.260062
$576$	0	0
$577$	−28.5410	−1.18818	−0.594089	−	0.804399i	$-0.702487\pi$
$577$	−28.5410	−1.18818	−0.594089	+	0.804399i	$0.702487\pi$
$578$	0.583592	0.0242742
$579$	0	0
$580$	12.4721	0.517877
$581$	11.0000	0.456357
$582$	0	0
$583$	−25.7082	−1.06473
$584$	−1.05573	−0.0436863
$585$	0	0
$586$	−14.0344	−0.579757
$587$	16.3050	0.672977	0.336489	−	0.941688i	$-0.390761\pi$
$587$	16.3050	0.672977	0.336489	+	0.941688i	$0.390761\pi$
$588$	0	0
$589$	67.2492	2.77096
$590$	7.23607	0.297904
$591$	0	0
$592$	−18.0000	−0.739795
$593$	−0.819660	−0.0336594	−0.0168297	−	0.999858i	$-0.505357\pi$
$593$	−0.819660	−0.0336594	−0.0168297	+	0.999858i	$0.505357\pi$
$594$	0	0
$595$	−4.23607	−0.173662
$596$	−11.2361	−0.460247
$597$	0	0
$598$	−9.52786	−0.389623
$599$	6.11146	0.249707	0.124854	−	0.992175i	$-0.460154\pi$
$599$	6.11146	0.249707	0.124854	+	0.992175i	$0.460154\pi$
$600$	0	0
$601$	−0.236068	−0.00962941	−0.00481471	−	0.999988i	$-0.501533\pi$
$601$	−0.236068	−0.00962941	−0.00481471	+	0.999988i	$0.501533\pi$
$602$	5.23607	0.213406
$603$	0	0
$604$	−28.9443	−1.17773
$605$	−0.527864	−0.0214607
$606$	0	0
$607$	−29.4164	−1.19398	−0.596988	−	0.802250i	$-0.703636\pi$
$607$	−29.4164	−1.19398	−0.596988	+	0.802250i	$0.703636\pi$
$608$	−41.9787	−1.70246
$609$	0	0
$610$	0.437694	0.0177217
$611$	−9.88854	−0.400048
$612$	0	0
$613$	12.2918	0.496461	0.248230	−	0.968701i	$-0.420151\pi$
$613$	12.2918	0.496461	0.248230	+	0.968701i	$0.420151\pi$
$614$	−10.5836	−0.427119
$615$	0	0
$616$	−7.23607	−0.291549
$617$	−26.8885	−1.08249	−0.541246	−	0.840864i	$-0.682047\pi$
$617$	−26.8885	−1.08249	−0.541246	+	0.840864i	$0.682047\pi$
$618$	0	0
$619$	42.8328	1.72160	0.860798	−	0.508947i	$-0.169965\pi$
$619$	42.8328	1.72160	0.860798	+	0.508947i	$0.169965\pi$
$620$	14.5623	0.584836
$621$	0	0
$622$	10.4721	0.419894
$623$	8.18034	0.327738
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.944272	−0.0377407
$627$	0	0
$628$	1.23607	0.0493245
$629$	−41.1246	−1.63975
$630$	0	0
$631$	−5.18034	−0.206226	−0.103113	−	0.994670i	$-0.532880\pi$
$631$	−5.18034	−0.206226	−0.103113	+	0.994670i	$0.532880\pi$
$632$	38.4164	1.52812
$633$	0	0
$634$	−3.67376	−0.145904
$635$	16.1803	0.642097
$636$	0	0
$637$	2.47214	0.0979496
$638$	15.4164	0.610341
$639$	0	0
$640$	−11.3820	−0.449912
$641$	−27.1246	−1.07136	−0.535679	−	0.844422i	$-0.679944\pi$
$641$	−27.1246	−1.07136	−0.535679	+	0.844422i	$0.679944\pi$
$642$	0	0
$643$	16.6525	0.656710	0.328355	−	0.944554i	$-0.393506\pi$
$643$	16.6525	0.656710	0.328355	+	0.944554i	$0.393506\pi$
$644$	−10.0902	−0.397608
$645$	0	0
$646$	−19.5623	−0.769669
$647$	−2.52786	−0.0993806	−0.0496903	−	0.998765i	$-0.515823\pi$
$647$	−2.52786	−0.0993806	−0.0496903	+	0.998765i	$0.515823\pi$
$648$	0	0
$649$	−37.8885	−1.48726
$650$	1.52786	0.0599278
$651$	0	0
$652$	−18.1803	−0.711997
$653$	−30.8885	−1.20876	−0.604381	−	0.796695i	$-0.706580\pi$
$653$	−30.8885	−1.20876	−0.604381	+	0.796695i	$0.706580\pi$
$654$	0	0
$655$	16.7639	0.655021
$656$	15.7082	0.613302
$657$	0	0
$658$	2.47214	0.0963739
$659$	22.4721	0.875390	0.437695	−	0.899123i	$-0.355795\pi$
$659$	22.4721	0.875390	0.437695	+	0.899123i	$0.355795\pi$
$660$	0	0
$661$	18.3607	0.714148	0.357074	−	0.934076i	$-0.383774\pi$
$661$	18.3607	0.714148	0.357074	+	0.934076i	$0.383774\pi$
$662$	4.58359	0.178146
$663$	0	0
$664$	24.5967	0.954539
$665$	7.47214	0.289757
$666$	0	0
$667$	48.0689	1.86123
$668$	9.61803	0.372133
$669$	0	0
$670$	6.76393	0.261313
$671$	−2.29180	−0.0884738
$672$	0	0
$673$	30.7639	1.18586	0.592931	−	0.805253i	$-0.297971\pi$
$673$	30.7639	1.18586	0.592931	+	0.805253i	$0.297971\pi$
$674$	1.88854	0.0727440
$675$	0	0
$676$	11.1459	0.428688
$677$	−24.4721	−0.940541	−0.470270	−	0.882522i	$-0.655844\pi$
$677$	−24.4721	−0.940541	−0.470270	+	0.882522i	$0.655844\pi$
$678$	0	0
$679$	5.23607	0.200942
$680$	−9.47214	−0.363240
$681$	0	0
$682$	18.0000	0.689256
$683$	−2.12461	−0.0812960	−0.0406480	−	0.999174i	$-0.512942\pi$
$683$	−2.12461	−0.0812960	−0.0406480	+	0.999174i	$0.512942\pi$
$684$	0	0
$685$	1.94427	0.0742868
$686$	−0.618034	−0.0235966
$687$	0	0
$688$	−15.7082	−0.598870
$689$	19.6393	0.748199
$690$	0	0
$691$	17.3607	0.660431	0.330216	−	0.943906i	$-0.392879\pi$
$691$	17.3607	0.660431	0.330216	+	0.943906i	$0.392879\pi$
$692$	21.3262	0.810702
$693$	0	0
$694$	1.88854	0.0716881
$695$	5.52786	0.209684
$696$	0	0
$697$	35.8885	1.35938
$698$	6.90983	0.261541
$699$	0	0
$700$	1.61803	0.0611559
$701$	−29.0557	−1.09742	−0.548710	−	0.836013i	$-0.684881\pi$
$701$	−29.0557	−1.09742	−0.548710	+	0.836013i	$0.684881\pi$
$702$	0	0
$703$	72.5410	2.73594
$704$	0.763932	0.0287918
$705$	0	0
$706$	2.76393	0.104022
$707$	−6.18034	−0.232436
$708$	0	0
$709$	21.0557	0.790764	0.395382	−	0.918517i	$-0.370612\pi$
$709$	21.0557	0.790764	0.395382	+	0.918517i	$0.370612\pi$
$710$	1.23607	0.0463888
$711$	0	0
$712$	18.2918	0.685514
$713$	56.1246	2.10188
$714$	0	0
$715$	−8.00000	−0.299183
$716$	−24.9443	−0.932211
$717$	0	0
$718$	5.81966	0.217188
$719$	−33.7771	−1.25967	−0.629836	−	0.776728i	$-0.716878\pi$
$719$	−33.7771	−1.25967	−0.629836	+	0.776728i	$0.716878\pi$
$720$	0	0
$721$	10.7639	0.400870
$722$	22.7639	0.847186
$723$	0	0
$724$	−6.09017	−0.226339
$725$	−7.70820	−0.286276
$726$	0	0
$727$	−32.9443	−1.22184	−0.610918	−	0.791694i	$-0.709199\pi$
$727$	−32.9443	−1.22184	−0.610918	+	0.791694i	$0.709199\pi$
$728$	5.52786	0.204876
$729$	0	0
$730$	0.291796	0.0107999
$731$	−35.8885	−1.32739
$732$	0	0
$733$	−15.1246	−0.558640	−0.279320	−	0.960198i	$-0.590109\pi$
$733$	−15.1246	−0.558640	−0.279320	+	0.960198i	$0.590109\pi$
$734$	−14.4721	−0.534176
$735$	0	0
$736$	−35.0344	−1.29139
$737$	−35.4164	−1.30458
$738$	0	0
$739$	−5.76393	−0.212030	−0.106015	−	0.994365i	$-0.533809\pi$
$739$	−5.76393	−0.212030	−0.106015	+	0.994365i	$0.533809\pi$
$740$	15.7082	0.577445
$741$	0	0
$742$	−4.90983	−0.180246
$743$	41.8885	1.53674	0.768371	−	0.640005i	$-0.221068\pi$
$743$	41.8885	1.53674	0.768371	+	0.640005i	$0.221068\pi$
$744$	0	0
$745$	6.94427	0.254418
$746$	−7.05573	−0.258329
$747$	0	0
$748$	22.1803	0.810994
$749$	−11.4164	−0.417146
$750$	0	0
$751$	−39.6525	−1.44694	−0.723470	−	0.690356i	$-0.757454\pi$
$751$	−39.6525	−1.44694	−0.723470	+	0.690356i	$0.757454\pi$
$752$	−7.41641	−0.270449
$753$	0	0
$754$	−11.7771	−0.428896
$755$	17.8885	0.651031
$756$	0	0
$757$	−25.4164	−0.923775	−0.461888	−	0.886939i	$-0.652828\pi$
$757$	−25.4164	−0.923775	−0.461888	+	0.886939i	$0.652828\pi$
$758$	−1.45085	−0.0526972
$759$	0	0
$760$	16.7082	0.606070
$761$	−6.47214	−0.234615	−0.117307	−	0.993096i	$-0.537426\pi$
$761$	−6.47214	−0.234615	−0.117307	+	0.993096i	$0.537426\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	−33.5967	−1.21549
$765$	0	0
$766$	17.2705	0.624009
$767$	28.9443	1.04512
$768$	0	0
$769$	−20.5967	−0.742738	−0.371369	−	0.928485i	$-0.621111\pi$
$769$	−20.5967	−0.742738	−0.371369	+	0.928485i	$0.621111\pi$
$770$	2.00000	0.0720750
$771$	0	0
$772$	5.52786	0.198952
$773$	24.8197	0.892701	0.446351	−	0.894858i	$-0.352723\pi$
$773$	24.8197	0.892701	0.446351	+	0.894858i	$0.352723\pi$
$774$	0	0
$775$	−9.00000	−0.323290
$776$	11.7082	0.420300
$777$	0	0
$778$	−4.11146	−0.147403
$779$	−63.3050	−2.26814
$780$	0	0
$781$	−6.47214	−0.231591
$782$	−16.3262	−0.583825
$783$	0	0
$784$	1.85410	0.0662179
$785$	−0.763932	−0.0272659
$786$	0	0
$787$	11.8197	0.421325	0.210663	−	0.977559i	$-0.432438\pi$
$787$	11.8197	0.421325	0.210663	+	0.977559i	$0.432438\pi$
$788$	−0.0901699	−0.00321217
$789$	0	0
$790$	−10.6180	−0.377773
$791$	2.00000	0.0711118
$792$	0	0
$793$	1.75078	0.0621719
$794$	−4.18034	−0.148355
$795$	0	0
$796$	42.8328	1.51817
$797$	−38.7082	−1.37111	−0.685557	−	0.728019i	$-0.740441\pi$
$797$	−38.7082	−1.37111	−0.685557	+	0.728019i	$0.740441\pi$
$798$	0	0
$799$	−16.9443	−0.599445
$800$	5.61803	0.198627
$801$	0	0
$802$	−15.1246	−0.534069
$803$	−1.52786	−0.0539172
$804$	0	0
$805$	6.23607	0.219793
$806$	−13.7508	−0.484350
$807$	0	0
$808$	−13.8197	−0.486174
$809$	−46.9443	−1.65047	−0.825236	−	0.564788i	$-0.808958\pi$
$809$	−46.9443	−1.65047	−0.825236	+	0.564788i	$0.808958\pi$
$810$	0	0
$811$	52.9443	1.85912	0.929562	−	0.368665i	$-0.120185\pi$
$811$	52.9443	1.85912	0.929562	+	0.368665i	$0.120185\pi$
$812$	−12.4721	−0.437686
$813$	0	0
$814$	19.4164	0.680545
$815$	11.2361	0.393582
$816$	0	0
$817$	63.3050	2.21476
$818$	−0.145898	−0.00510121
$819$	0	0
$820$	−13.7082	−0.478711
$821$	21.4164	0.747438	0.373719	−	0.927542i	$-0.378082\pi$
$821$	21.4164	0.747438	0.373719	+	0.927542i	$0.378082\pi$
$822$	0	0
$823$	−39.3050	−1.37008	−0.685042	−	0.728503i	$-0.740216\pi$
$823$	−39.3050	−1.37008	−0.685042	+	0.728503i	$0.740216\pi$
$824$	24.0689	0.838479
$825$	0	0
$826$	−7.23607	−0.251775
$827$	34.7082	1.20692	0.603461	−	0.797392i	$-0.293788\pi$
$827$	34.7082	1.20692	0.603461	+	0.797392i	$0.293788\pi$
$828$	0	0
$829$	19.5279	0.678231	0.339115	−	0.940745i	$-0.389872\pi$
$829$	19.5279	0.678231	0.339115	+	0.940745i	$0.389872\pi$
$830$	−6.79837	−0.235975
$831$	0	0
$832$	−0.583592	−0.0202324
$833$	4.23607	0.146771
$834$	0	0
$835$	−5.94427	−0.205710
$836$	−39.1246	−1.35315
$837$	0	0
$838$	11.5279	0.398223
$839$	26.1803	0.903846	0.451923	−	0.892057i	$-0.350738\pi$
$839$	26.1803	0.903846	0.451923	+	0.892057i	$0.350738\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	−17.2016	−0.592807
$843$	0	0
$844$	−1.14590	−0.0394434
$845$	−6.88854	−0.236973
$846$	0	0
$847$	0.527864	0.0181376
$848$	14.7295	0.505813
$849$	0	0
$850$	2.61803	0.0897978
$851$	60.5410	2.07532
$852$	0	0
$853$	38.2492	1.30963	0.654814	−	0.755790i	$-0.272747\pi$
$853$	38.2492	1.30963	0.654814	+	0.755790i	$0.272747\pi$
$854$	−0.437694	−0.0149776
$855$	0	0
$856$	−25.5279	−0.872524
$857$	−23.1803	−0.791825	−0.395913	−	0.918288i	$-0.629572\pi$
$857$	−23.1803	−0.791825	−0.395913	+	0.918288i	$0.629572\pi$
$858$	0	0
$859$	5.94427	0.202816	0.101408	−	0.994845i	$-0.467665\pi$
$859$	5.94427	0.202816	0.101408	+	0.994845i	$0.467665\pi$
$860$	13.7082	0.467446
$861$	0	0
$862$	−11.4164	−0.388844
$863$	−27.1803	−0.925230	−0.462615	−	0.886559i	$-0.653089\pi$
$863$	−27.1803	−0.925230	−0.462615	+	0.886559i	$0.653089\pi$
$864$	0	0
$865$	−13.1803	−0.448145
$866$	3.23607	0.109966
$867$	0	0
$868$	−14.5623	−0.494277
$869$	55.5967	1.88599
$870$	0	0
$871$	27.0557	0.916748
$872$	−2.23607	−0.0757228
$873$	0	0
$874$	28.7984	0.974120
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	21.3475	0.720855	0.360427	−	0.932787i	$-0.382631\pi$
$877$	21.3475	0.720855	0.360427	+	0.932787i	$0.382631\pi$
$878$	−12.9787	−0.438010
$879$	0	0
$880$	−6.00000	−0.202260
$881$	47.4853	1.59982	0.799910	−	0.600120i	$-0.204880\pi$
$881$	47.4853	1.59982	0.799910	+	0.600120i	$0.204880\pi$
$882$	0	0
$883$	−3.34752	−0.112653	−0.0563266	−	0.998412i	$-0.517939\pi$
$883$	−3.34752	−0.112653	−0.0563266	+	0.998412i	$0.517939\pi$
$884$	−16.9443	−0.569898
$885$	0	0
$886$	10.6869	0.359034
$887$	51.7214	1.73663	0.868317	−	0.496010i	$-0.165202\pi$
$887$	51.7214	1.73663	0.868317	+	0.496010i	$0.165202\pi$
$888$	0	0
$889$	−16.1803	−0.542671
$890$	−5.05573	−0.169468
$891$	0	0
$892$	−30.1803	−1.01051
$893$	29.8885	1.00018
$894$	0	0
$895$	15.4164	0.515314
$896$	11.3820	0.380245
$897$	0	0
$898$	15.4164	0.514452
$899$	69.3738	2.31375
$900$	0	0
$901$	33.6525	1.12113
$902$	−16.9443	−0.564183
$903$	0	0
$904$	4.47214	0.148741
$905$	3.76393	0.125117
$906$	0	0
$907$	27.1246	0.900658	0.450329	−	0.892863i	$-0.351307\pi$
$907$	27.1246	0.900658	0.450329	+	0.892863i	$0.351307\pi$
$908$	17.7984	0.590660
$909$	0	0
$910$	−1.52786	−0.0506482
$911$	−54.7639	−1.81441	−0.907205	−	0.420689i	$-0.861788\pi$
$911$	−54.7639	−1.81441	−0.907205	+	0.420689i	$0.861788\pi$
$912$	0	0
$913$	35.5967	1.17808
$914$	24.7639	0.819118
$915$	0	0
$916$	−13.3262	−0.440311
$917$	−16.7639	−0.553594
$918$	0	0
$919$	−49.3050	−1.62642	−0.813210	−	0.581970i	$-0.802282\pi$
$919$	−49.3050	−1.62642	−0.813210	+	0.581970i	$0.802282\pi$
$920$	13.9443	0.459729
$921$	0	0
$922$	12.0689	0.397468
$923$	4.94427	0.162743
$924$	0	0
$925$	−9.70820	−0.319204
$926$	−14.5410	−0.477848
$927$	0	0
$928$	−43.3050	−1.42155
$929$	5.12461	0.168133	0.0840665	−	0.996460i	$-0.473209\pi$
$929$	5.12461	0.168133	0.0840665	+	0.996460i	$0.473209\pi$
$930$	0	0
$931$	−7.47214	−0.244889
$932$	29.7082	0.973125
$933$	0	0
$934$	−20.1033	−0.657801
$935$	−13.7082	−0.448306
$936$	0	0
$937$	22.4721	0.734133	0.367066	−	0.930195i	$-0.380362\pi$
$937$	22.4721	0.734133	0.367066	+	0.930195i	$0.380362\pi$
$938$	−6.76393	−0.220850
$939$	0	0
$940$	6.47214	0.211098
$941$	−35.2361	−1.14866	−0.574331	−	0.818623i	$-0.694738\pi$
$941$	−35.2361	−1.14866	−0.574331	+	0.818623i	$0.694738\pi$
$942$	0	0
$943$	−52.8328	−1.72047
$944$	21.7082	0.706542
$945$	0	0
$946$	16.9443	0.550906
$947$	−41.5410	−1.34990	−0.674951	−	0.737863i	$-0.735835\pi$
$947$	−41.5410	−1.34990	−0.674951	+	0.737863i	$0.735835\pi$
$948$	0	0
$949$	1.16718	0.0378884
$950$	−4.61803	−0.149829
$951$	0	0
$952$	9.47214	0.306994
$953$	57.7771	1.87158	0.935792	−	0.352553i	$-0.114686\pi$
$953$	57.7771	1.87158	0.935792	+	0.352553i	$0.114686\pi$
$954$	0	0
$955$	20.7639	0.671905
$956$	−20.1803	−0.652679
$957$	0	0
$958$	−17.0557	−0.551046
$959$	−1.94427	−0.0627838
$960$	0	0
$961$	50.0000	1.61290
$962$	−14.8328	−0.478229
$963$	0	0
$964$	9.14590	0.294570
$965$	−3.41641	−0.109978
$966$	0	0
$967$	−19.5967	−0.630189	−0.315094	−	0.949060i	$-0.602036\pi$
$967$	−19.5967	−0.630189	−0.315094	+	0.949060i	$0.602036\pi$
$968$	1.18034	0.0379376
$969$	0	0
$970$	−3.23607	−0.103904
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	0	0
$973$	−5.52786	−0.177215
$974$	2.00000	0.0640841
$975$	0	0
$976$	1.31308	0.0420307
$977$	−20.1115	−0.643422	−0.321711	−	0.946838i	$-0.604258\pi$
$977$	−20.1115	−0.643422	−0.321711	+	0.946838i	$0.604258\pi$
$978$	0	0
$979$	26.4721	0.846053
$980$	−1.61803	−0.0516862
$981$	0	0
$982$	−19.7082	−0.628914
$983$	−41.0000	−1.30770	−0.653848	−	0.756626i	$-0.726847\pi$
$983$	−41.0000	−1.30770	−0.653848	+	0.756626i	$0.726847\pi$
$984$	0	0
$985$	0.0557281	0.00177564
$986$	−20.1803	−0.642673
$987$	0	0
$988$	29.8885	0.950881
$989$	52.8328	1.67999
$990$	0	0
$991$	34.0132	1.08046	0.540232	−	0.841516i	$-0.318337\pi$
$991$	34.0132	1.08046	0.540232	+	0.841516i	$0.318337\pi$
$992$	−50.5623	−1.60535
$993$	0	0
$994$	−1.23607	−0.0392057
$995$	−26.4721	−0.839223
$996$	0	0
$997$	6.58359	0.208504	0.104252	−	0.994551i	$-0.466755\pi$
$997$	6.58359	0.208504	0.104252	+	0.994551i	$0.466755\pi$
$998$	3.56231	0.112763
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.c.1.2	✓	2
3.2	odd	2		945.2.a.h.1.1	yes	2
5.4	even	2		4725.2.a.bd.1.1		2
7.6	odd	2		6615.2.a.n.1.2		2
15.14	odd	2		4725.2.a.y.1.2		2
21.20	even	2		6615.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.c.1.2	✓	2	1.1	even	1	trivial
945.2.a.h.1.1	yes	2	3.2	odd	2
4725.2.a.y.1.2		2	15.14	odd	2
4725.2.a.bd.1.1		2	5.4	even	2
6615.2.a.n.1.2		2	7.6	odd	2
6615.2.a.t.1.1		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)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} -3.23607 q^{11} +2.47214 q^{13} -0.618034 q^{14} +1.85410 q^{16} +4.23607 q^{17} -7.47214 q^{19} -1.61803 q^{20} -2.00000 q^{22} -6.23607 q^{23} +1.00000 q^{25} +1.52786 q^{26} +1.61803 q^{28} -7.70820 q^{29} -9.00000 q^{31} +5.61803 q^{32} +2.61803 q^{34} -1.00000 q^{35} -9.70820 q^{37} -4.61803 q^{38} -2.23607 q^{40} +8.47214 q^{41} -8.47214 q^{43} +5.23607 q^{44} -3.85410 q^{46} -4.00000 q^{47} +1.00000 q^{49} +0.618034 q^{50} -4.00000 q^{52} +7.94427 q^{53} -3.23607 q^{55} +2.23607 q^{56} -4.76393 q^{58} +11.7082 q^{59} +0.708204 q^{61} -5.56231 q^{62} -0.236068 q^{64} +2.47214 q^{65} +10.9443 q^{67} -6.85410 q^{68} -0.618034 q^{70} +2.00000 q^{71} +0.472136 q^{73} -6.00000 q^{74} +12.0902 q^{76} +3.23607 q^{77} -17.1803 q^{79} +1.85410 q^{80} +5.23607 q^{82} -11.0000 q^{83} +4.23607 q^{85} -5.23607 q^{86} +7.23607 q^{88} -8.18034 q^{89} -2.47214 q^{91} +10.0902 q^{92} -2.47214 q^{94} -7.47214 q^{95} -5.23607 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - q^{10} - 2 q^{11} - 4 q^{13} + q^{14} - 3 q^{16} + 4 q^{17} - 6 q^{19} - q^{20} - 4 q^{22} - 8 q^{23} + 2 q^{25} + 12 q^{26} + q^{28} - 2 q^{29} - 18 q^{31} + 9 q^{32} + 3 q^{34} - 2 q^{35} - 6 q^{37} - 7 q^{38} + 8 q^{41} - 8 q^{43} + 6 q^{44} - q^{46} - 8 q^{47} + 2 q^{49} - q^{50} - 8 q^{52} - 2 q^{53} - 2 q^{55} - 14 q^{58} + 10 q^{59} - 12 q^{61} + 9 q^{62} + 4 q^{64} - 4 q^{65} + 4 q^{67} - 7 q^{68} + q^{70} + 4 q^{71} - 8 q^{73} - 12 q^{74} + 13 q^{76} + 2 q^{77} - 12 q^{79} - 3 q^{80} + 6 q^{82} - 22 q^{83} + 4 q^{85} - 6 q^{86} + 10 q^{88} + 6 q^{89} + 4 q^{91} + 9 q^{92} + 4 q^{94} - 6 q^{95} - 6 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 2 * q^7 - q^10 - 2 * q^11 - 4 * q^13 + q^14 - 3 * q^16 + 4 * q^17 - 6 * q^19 - q^20 - 4 * q^22 - 8 * q^23 + 2 * q^25 + 12 * q^26 + q^28 - 2 * q^29 - 18 * q^31 + 9 * q^32 + 3 * q^34 - 2 * q^35 - 6 * q^37 - 7 * q^38 + 8 * q^41 - 8 * q^43 + 6 * q^44 - q^46 - 8 * q^47 + 2 * q^49 - q^50 - 8 * q^52 - 2 * q^53 - 2 * q^55 - 14 * q^58 + 10 * q^59 - 12 * q^61 + 9 * q^62 + 4 * q^64 - 4 * q^65 + 4 * q^67 - 7 * q^68 + q^70 + 4 * q^71 - 8 * q^73 - 12 * q^74 + 13 * q^76 + 2 * q^77 - 12 * q^79 - 3 * q^80 + 6 * q^82 - 22 * q^83 + 4 * q^85 - 6 * q^86 + 10 * q^88 + 6 * q^89 + 4 * q^91 + 9 * q^92 + 4 * q^94 - 6 * q^95 - 6 * q^97 - q^98

Introduction

L-functions

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