LMFDB - Embedded newform 1275.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1275,2,Mod(1,1275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1275, 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("1275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1275.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:	$10.1809262577$
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:	no (minimal twist has level 255)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.61803 q^{12} -2.00000 q^{13} -0.618034 q^{14} +1.85410 q^{16} +1.00000 q^{17} -0.618034 q^{18} +0.236068 q^{19} -1.00000 q^{21} +1.85410 q^{22} +8.47214 q^{23} -2.23607 q^{24} +1.23607 q^{26} -1.00000 q^{27} -1.61803 q^{28} -1.76393 q^{29} -6.47214 q^{31} -5.61803 q^{32} +3.00000 q^{33} -0.618034 q^{34} -1.61803 q^{36} +8.70820 q^{37} -0.145898 q^{38} +2.00000 q^{39} -8.23607 q^{41} +0.618034 q^{42} +0.472136 q^{43} +4.85410 q^{44} -5.23607 q^{46} -0.708204 q^{47} -1.85410 q^{48} -6.00000 q^{49} -1.00000 q^{51} +3.23607 q^{52} +3.00000 q^{53} +0.618034 q^{54} +2.23607 q^{56} -0.236068 q^{57} +1.09017 q^{58} +8.94427 q^{59} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} -0.236068 q^{64} -1.85410 q^{66} -14.4721 q^{67} -1.61803 q^{68} -8.47214 q^{69} -4.94427 q^{71} +2.23607 q^{72} -6.23607 q^{73} -5.38197 q^{74} -0.381966 q^{76} -3.00000 q^{77} -1.23607 q^{78} +2.00000 q^{79} +1.00000 q^{81} +5.09017 q^{82} -12.9443 q^{83} +1.61803 q^{84} -0.291796 q^{86} +1.76393 q^{87} -6.70820 q^{88} -12.0000 q^{89} -2.00000 q^{91} -13.7082 q^{92} +6.47214 q^{93} +0.437694 q^{94} +5.61803 q^{96} -13.4164 q^{97} +3.70820 q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{7} + 2 q^{9} - 6 q^{11} + q^{12} - 4 q^{13} + q^{14} - 3 q^{16} + 2 q^{17} + q^{18} - 4 q^{19} - 2 q^{21} - 3 q^{22} + 8 q^{23} - 2 q^{26} - 2 q^{27} - q^{28}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 2 * q^7 + 2 * q^9 - 6 * q^11 + q^12 - 4 * q^13 + q^14 - 3 * q^16 + 2 * q^17 + q^18 - 4 * q^19 - 2 * q^21 - 3 * q^22 + 8 * q^23 - 2 * q^26 - 2 * q^27 - q^28 - 8 * q^29 - 4 * q^31 - 9 * q^32 + 6 * q^33 + q^34 - q^36 + 4 * q^37 - 7 * q^38 + 4 * q^39 - 12 * q^41 - q^42 - 8 * q^43 + 3 * q^44 - 6 * q^46 + 12 * q^47 + 3 * q^48 - 12 * q^49 - 2 * q^51 + 2 * q^52 + 6 * q^53 - q^54 + 4 * q^57 - 9 * q^58 - 20 * q^61 + 8 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^66 - 20 * q^67 - q^68 - 8 * q^69 + 8 * q^71 - 8 * q^73 - 13 * q^74 - 3 * q^76 - 6 * q^77 + 2 * q^78 + 4 * q^79 + 2 * q^81 - q^82 - 8 * q^83 + q^84 - 14 * q^86 + 8 * q^87 - 24 * q^89 - 4 * q^91 - 14 * q^92 + 4 * q^93 + 21 * q^94 + 9 * q^96 - 6 * q^98 - 6 * q^99

Coefficient data

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

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

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

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.61803	0.467086
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−0.618034	−0.145672
$19$	0.236068	0.0541577	0.0270789	−	0.999633i	$-0.491379\pi$
$19$	0.236068	0.0541577	0.0270789	+	0.999633i	$0.491379\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	1.85410	0.395296
$23$	8.47214	1.76656	0.883281	−	0.468844i	$-0.155329\pi$
$23$	8.47214	1.76656	0.883281	+	0.468844i	$0.155329\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	1.23607	0.242413
$27$	−1.00000	−0.192450
$28$	−1.61803	−0.305780
$29$	−1.76393	−0.327554	−0.163777	−	0.986497i	$-0.552368\pi$
$29$	−1.76393	−0.327554	−0.163777	+	0.986497i	$0.552368\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	−5.61803	−0.993137
$33$	3.00000	0.522233
$34$	−0.618034	−0.105992
$35$	0	0
$36$	−1.61803	−0.269672
$37$	8.70820	1.43162	0.715810	−	0.698295i	$-0.246058\pi$
$37$	8.70820	1.43162	0.715810	+	0.698295i	$0.246058\pi$
$38$	−0.145898	−0.0236678
$39$	2.00000	0.320256
$40$	0	0
$41$	−8.23607	−1.28626	−0.643129	−	0.765758i	$-0.722364\pi$
$41$	−8.23607	−1.28626	−0.643129	+	0.765758i	$0.722364\pi$
$42$	0.618034	0.0953647
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	4.85410	0.731783
$45$	0	0
$46$	−5.23607	−0.772016
$47$	−0.708204	−0.103302	−0.0516511	−	0.998665i	$-0.516448\pi$
$47$	−0.708204	−0.103302	−0.0516511	+	0.998665i	$0.516448\pi$
$48$	−1.85410	−0.267617
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	3.23607	0.448762
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	2.23607	0.298807
$57$	−0.236068	−0.0312680
$58$	1.09017	0.143146
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	−1.85410	−0.228224
$67$	−14.4721	−1.76805	−0.884026	−	0.467437i	$-0.845177\pi$
$67$	−14.4721	−1.76805	−0.884026	+	0.467437i	$0.845177\pi$
$68$	−1.61803	−0.196215
$69$	−8.47214	−1.01993
$70$	0	0
$71$	−4.94427	−0.586777	−0.293389	−	0.955993i	$-0.594783\pi$
$71$	−4.94427	−0.586777	−0.293389	+	0.955993i	$0.594783\pi$
$72$	2.23607	0.263523
$73$	−6.23607	−0.729877	−0.364938	−	0.931032i	$-0.618910\pi$
$73$	−6.23607	−0.729877	−0.364938	+	0.931032i	$0.618910\pi$
$74$	−5.38197	−0.625641
$75$	0	0
$76$	−0.381966	−0.0438145
$77$	−3.00000	−0.341882
$78$	−1.23607	−0.139957
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.09017	0.562115
$83$	−12.9443	−1.42082	−0.710409	−	0.703789i	$-0.751490\pi$
$83$	−12.9443	−1.42082	−0.710409	+	0.703789i	$0.751490\pi$
$84$	1.61803	0.176542
$85$	0	0
$86$	−0.291796	−0.0314652
$87$	1.76393	0.189113
$88$	−6.70820	−0.715097
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−13.7082	−1.42918
$93$	6.47214	0.671129
$94$	0.437694	0.0451447
$95$	0	0
$96$	5.61803	0.573388
$97$	−13.4164	−1.36223	−0.681115	−	0.732177i	$-0.738505\pi$
$97$	−13.4164	−1.36223	−0.681115	+	0.732177i	$0.738505\pi$
$98$	3.70820	0.374585
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−5.52786	−0.550043	−0.275022	−	0.961438i	$-0.588685\pi$
$101$	−5.52786	−0.550043	−0.275022	+	0.961438i	$0.588685\pi$
$102$	0.618034	0.0611945
$103$	−6.94427	−0.684239	−0.342120	−	0.939656i	$-0.611145\pi$
$103$	−6.94427	−0.684239	−0.342120	+	0.939656i	$0.611145\pi$
$104$	−4.47214	−0.438529
$105$	0	0
$106$	−1.85410	−0.180086
$107$	2.47214	0.238990	0.119495	−	0.992835i	$-0.461872\pi$
$107$	2.47214	0.238990	0.119495	+	0.992835i	$0.461872\pi$
$108$	1.61803	0.155695
$109$	−6.47214	−0.619918	−0.309959	−	0.950750i	$-0.600315\pi$
$109$	−6.47214	−0.619918	−0.309959	+	0.950750i	$0.600315\pi$
$110$	0	0
$111$	−8.70820	−0.826546
$112$	1.85410	0.175196
$113$	1.05573	0.0993145	0.0496573	−	0.998766i	$-0.484187\pi$
$113$	1.05573	0.0993145	0.0496573	+	0.998766i	$0.484187\pi$
$114$	0.145898	0.0136646
$115$	0	0
$116$	2.85410	0.264997
$117$	−2.00000	−0.184900
$118$	−5.52786	−0.508881
$119$	1.00000	0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	6.18034	0.559542
$123$	8.23607	0.742621
$124$	10.4721	0.940426
$125$	0	0
$126$	−0.618034	−0.0550588
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	11.3820	1.00603
$129$	−0.472136	−0.0415693
$130$	0	0
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	−4.85410	−0.422495
$133$	0.236068	0.0204697
$134$	8.94427	0.772667
$135$	0	0
$136$	2.23607	0.191741
$137$	16.4164	1.40255	0.701274	−	0.712892i	$-0.252615\pi$
$137$	16.4164	1.40255	0.701274	+	0.712892i	$0.252615\pi$
$138$	5.23607	0.445724
$139$	−12.9443	−1.09792	−0.548959	−	0.835849i	$-0.684976\pi$
$139$	−12.9443	−1.09792	−0.548959	+	0.835849i	$0.684976\pi$
$140$	0	0
$141$	0.708204	0.0596415
$142$	3.05573	0.256431
$143$	6.00000	0.501745
$144$	1.85410	0.154508
$145$	0	0
$146$	3.85410	0.318968
$147$	6.00000	0.494872
$148$	−14.0902	−1.15820
$149$	1.52786	0.125167	0.0625837	−	0.998040i	$-0.480066\pi$
$149$	1.52786	0.125167	0.0625837	+	0.998040i	$0.480066\pi$
$150$	0	0
$151$	13.7639	1.12009	0.560046	−	0.828461i	$-0.310783\pi$
$151$	13.7639	1.12009	0.560046	+	0.828461i	$0.310783\pi$
$152$	0.527864	0.0428154
$153$	1.00000	0.0808452
$154$	1.85410	0.149408
$155$	0	0
$156$	−3.23607	−0.259093
$157$	−19.4164	−1.54960	−0.774799	−	0.632208i	$-0.782149\pi$
$157$	−19.4164	−1.54960	−0.774799	+	0.632208i	$0.782149\pi$
$158$	−1.23607	−0.0983363
$159$	−3.00000	−0.237915
$160$	0	0
$161$	8.47214	0.667698
$162$	−0.618034	−0.0485573
$163$	−16.4164	−1.28583	−0.642916	−	0.765937i	$-0.722276\pi$
$163$	−16.4164	−1.28583	−0.642916	+	0.765937i	$0.722276\pi$
$164$	13.3262	1.04060
$165$	0	0
$166$	8.00000	0.620920
$167$	18.4721	1.42942	0.714708	−	0.699423i	$-0.246559\pi$
$167$	18.4721	1.42942	0.714708	+	0.699423i	$0.246559\pi$
$168$	−2.23607	−0.172516
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0.236068	0.0180526
$172$	−0.763932	−0.0582493
$173$	−20.3607	−1.54799	−0.773997	−	0.633189i	$-0.781745\pi$
$173$	−20.3607	−1.54799	−0.773997	+	0.633189i	$0.781745\pi$
$174$	−1.09017	−0.0826456
$175$	0	0
$176$	−5.56231	−0.419275
$177$	−8.94427	−0.672293
$178$	7.41641	0.555883
$179$	−1.52786	−0.114198	−0.0570990	−	0.998369i	$-0.518185\pi$
$179$	−1.52786	−0.114198	−0.0570990	+	0.998369i	$0.518185\pi$
$180$	0	0
$181$	13.8885	1.03233	0.516164	−	0.856490i	$-0.327360\pi$
$181$	13.8885	1.03233	0.516164	+	0.856490i	$0.327360\pi$
$182$	1.23607	0.0916235
$183$	10.0000	0.739221
$184$	18.9443	1.39659
$185$	0	0
$186$	−4.00000	−0.293294
$187$	−3.00000	−0.219382
$188$	1.14590	0.0835732
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	4.47214	0.323592	0.161796	−	0.986824i	$-0.448271\pi$
$191$	4.47214	0.323592	0.161796	+	0.986824i	$0.448271\pi$
$192$	0.236068	0.0170367
$193$	9.41641	0.677808	0.338904	−	0.940821i	$-0.389944\pi$
$193$	9.41641	0.677808	0.338904	+	0.940821i	$0.389944\pi$
$194$	8.29180	0.595316
$195$	0	0
$196$	9.70820	0.693443
$197$	22.4721	1.60107	0.800537	−	0.599284i	$-0.204548\pi$
$197$	22.4721	1.60107	0.800537	+	0.599284i	$0.204548\pi$
$198$	1.85410	0.131765
$199$	13.4164	0.951064	0.475532	−	0.879698i	$-0.342256\pi$
$199$	13.4164	0.951064	0.475532	+	0.879698i	$0.342256\pi$
$200$	0	0
$201$	14.4721	1.02079
$202$	3.41641	0.240378
$203$	−1.76393	−0.123804
$204$	1.61803	0.113285
$205$	0	0
$206$	4.29180	0.299024
$207$	8.47214	0.588854
$208$	−3.70820	−0.257118
$209$	−0.708204	−0.0489875
$210$	0	0
$211$	−18.3607	−1.26400	−0.632001	−	0.774968i	$-0.717766\pi$
$211$	−18.3607	−1.26400	−0.632001	+	0.774968i	$0.717766\pi$
$212$	−4.85410	−0.333381
$213$	4.94427	0.338776
$214$	−1.52786	−0.104443
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−6.47214	−0.439357
$218$	4.00000	0.270914
$219$	6.23607	0.421394
$220$	0	0
$221$	−2.00000	−0.134535
$222$	5.38197	0.361214
$223$	1.41641	0.0948497	0.0474248	−	0.998875i	$-0.484899\pi$
$223$	1.41641	0.0948497	0.0474248	+	0.998875i	$0.484899\pi$
$224$	−5.61803	−0.375371
$225$	0	0
$226$	−0.652476	−0.0434020
$227$	3.05573	0.202816	0.101408	−	0.994845i	$-0.467665\pi$
$227$	3.05573	0.202816	0.101408	+	0.994845i	$0.467665\pi$
$228$	0.381966	0.0252963
$229$	−16.4164	−1.08483	−0.542413	−	0.840112i	$-0.682489\pi$
$229$	−16.4164	−1.08483	−0.542413	+	0.840112i	$0.682489\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	−3.94427	−0.258954
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	1.23607	0.0808043
$235$	0	0
$236$	−14.4721	−0.942056
$237$	−2.00000	−0.129914
$238$	−0.618034	−0.0400612
$239$	0.472136	0.0305399	0.0152700	−	0.999883i	$-0.495139\pi$
$239$	0.472136	0.0305399	0.0152700	+	0.999883i	$0.495139\pi$
$240$	0	0
$241$	4.94427	0.318489	0.159244	−	0.987239i	$-0.449094\pi$
$241$	4.94427	0.318489	0.159244	+	0.987239i	$0.449094\pi$
$242$	1.23607	0.0794575
$243$	−1.00000	−0.0641500
$244$	16.1803	1.03584
$245$	0	0
$246$	−5.09017	−0.324537
$247$	−0.472136	−0.0300413
$248$	−14.4721	−0.918982
$249$	12.9443	0.820310
$250$	0	0
$251$	−11.8885	−0.750398	−0.375199	−	0.926944i	$-0.622426\pi$
$251$	−11.8885	−0.750398	−0.375199	+	0.926944i	$0.622426\pi$
$252$	−1.61803	−0.101927
$253$	−25.4164	−1.59792
$254$	3.70820	0.232673
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−13.0557	−0.814394	−0.407197	−	0.913340i	$-0.633494\pi$
$257$	−13.0557	−0.814394	−0.407197	+	0.913340i	$0.633494\pi$
$258$	0.291796	0.0181664
$259$	8.70820	0.541101
$260$	0	0
$261$	−1.76393	−0.109185
$262$	−10.4721	−0.646971
$263$	21.6525	1.33515	0.667574	−	0.744543i	$-0.267333\pi$
$263$	21.6525	1.33515	0.667574	+	0.744543i	$0.267333\pi$
$264$	6.70820	0.412861
$265$	0	0
$266$	−0.145898	−0.00894558
$267$	12.0000	0.734388
$268$	23.4164	1.43038
$269$	−7.18034	−0.437793	−0.218897	−	0.975748i	$-0.570246\pi$
$269$	−7.18034	−0.437793	−0.218897	+	0.975748i	$0.570246\pi$
$270$	0	0
$271$	21.8885	1.32963	0.664817	−	0.747006i	$-0.268509\pi$
$271$	21.8885	1.32963	0.664817	+	0.747006i	$0.268509\pi$
$272$	1.85410	0.112421
$273$	2.00000	0.121046
$274$	−10.1459	−0.612936
$275$	0	0
$276$	13.7082	0.825137
$277$	−29.4164	−1.76746	−0.883730	−	0.467996i	$-0.844976\pi$
$277$	−29.4164	−1.76746	−0.883730	+	0.467996i	$0.844976\pi$
$278$	8.00000	0.479808
$279$	−6.47214	−0.387477
$280$	0	0
$281$	−14.3607	−0.856686	−0.428343	−	0.903616i	$-0.640903\pi$
$281$	−14.3607	−0.856686	−0.428343	+	0.903616i	$0.640903\pi$
$282$	−0.437694	−0.0260643
$283$	−17.4721	−1.03861	−0.519305	−	0.854589i	$-0.673809\pi$
$283$	−17.4721	−1.03861	−0.519305	+	0.854589i	$0.673809\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−3.70820	−0.219271
$287$	−8.23607	−0.486160
$288$	−5.61803	−0.331046
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.4164	0.786484
$292$	10.0902	0.590483
$293$	8.88854	0.519274	0.259637	−	0.965706i	$-0.416397\pi$
$293$	8.88854	0.519274	0.259637	+	0.965706i	$0.416397\pi$
$294$	−3.70820	−0.216267
$295$	0	0
$296$	19.4721	1.13179
$297$	3.00000	0.174078
$298$	−0.944272	−0.0547002
$299$	−16.9443	−0.979913
$300$	0	0
$301$	0.472136	0.0272135
$302$	−8.50658	−0.489499
$303$	5.52786	0.317567
$304$	0.437694	0.0251035
$305$	0	0
$306$	−0.618034	−0.0353307
$307$	34.8328	1.98801	0.994007	−	0.109317i	$-0.0348665\pi$
$307$	34.8328	1.98801	0.994007	+	0.109317i	$0.0348665\pi$
$308$	4.85410	0.276588
$309$	6.94427	0.395046
$310$	0	0
$311$	10.5279	0.596980	0.298490	−	0.954413i	$-0.403517\pi$
$311$	10.5279	0.596980	0.298490	+	0.954413i	$0.403517\pi$
$312$	4.47214	0.253185
$313$	6.70820	0.379170	0.189585	−	0.981864i	$-0.439286\pi$
$313$	6.70820	0.379170	0.189585	+	0.981864i	$0.439286\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	−3.23607	−0.182043
$317$	−11.5279	−0.647469	−0.323735	−	0.946148i	$-0.604939\pi$
$317$	−11.5279	−0.647469	−0.323735	+	0.946148i	$0.604939\pi$
$318$	1.85410	0.103973
$319$	5.29180	0.296284
$320$	0	0
$321$	−2.47214	−0.137981
$322$	−5.23607	−0.291795
$323$	0.236068	0.0131352
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	10.1459	0.561929
$327$	6.47214	0.357910
$328$	−18.4164	−1.01688
$329$	−0.708204	−0.0390445
$330$	0	0
$331$	10.1246	0.556499	0.278249	−	0.960509i	$-0.410246\pi$
$331$	10.1246	0.556499	0.278249	+	0.960509i	$0.410246\pi$
$332$	20.9443	1.14947
$333$	8.70820	0.477207
$334$	−11.4164	−0.624678
$335$	0	0
$336$	−1.85410	−0.101150
$337$	31.6525	1.72422	0.862110	−	0.506721i	$-0.169143\pi$
$337$	31.6525	1.72422	0.862110	+	0.506721i	$0.169143\pi$
$338$	5.56231	0.302550
$339$	−1.05573	−0.0573393
$340$	0	0
$341$	19.4164	1.05146
$342$	−0.145898	−0.00788926
$343$	−13.0000	−0.701934
$344$	1.05573	0.0569210
$345$	0	0
$346$	12.5836	0.676498
$347$	−10.4721	−0.562174	−0.281087	−	0.959682i	$-0.590695\pi$
$347$	−10.4721	−0.562174	−0.281087	+	0.959682i	$0.590695\pi$
$348$	−2.85410	−0.152996
$349$	−32.4164	−1.73521	−0.867605	−	0.497254i	$-0.834342\pi$
$349$	−32.4164	−1.73521	−0.867605	+	0.497254i	$0.834342\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	16.8541	0.898327
$353$	−16.5279	−0.879689	−0.439845	−	0.898074i	$-0.644967\pi$
$353$	−16.5279	−0.879689	−0.439845	+	0.898074i	$0.644967\pi$
$354$	5.52786	0.293803
$355$	0	0
$356$	19.4164	1.02907
$357$	−1.00000	−0.0529256
$358$	0.944272	0.0499063
$359$	−11.8885	−0.627453	−0.313727	−	0.949513i	$-0.601578\pi$
$359$	−11.8885	−0.627453	−0.313727	+	0.949513i	$0.601578\pi$
$360$	0	0
$361$	−18.9443	−0.997067
$362$	−8.58359	−0.451144
$363$	2.00000	0.104973
$364$	3.23607	0.169616
$365$	0	0
$366$	−6.18034	−0.323052
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	15.7082	0.818847
$369$	−8.23607	−0.428753
$370$	0	0
$371$	3.00000	0.155752
$372$	−10.4721	−0.542955
$373$	0.472136	0.0244463	0.0122231	−	0.999925i	$-0.496109\pi$
$373$	0.472136	0.0244463	0.0122231	+	0.999925i	$0.496109\pi$
$374$	1.85410	0.0958733
$375$	0	0
$376$	−1.58359	−0.0816675
$377$	3.52786	0.181694
$378$	0.618034	0.0317882
$379$	36.3607	1.86772	0.933861	−	0.357635i	$-0.116417\pi$
$379$	36.3607	1.86772	0.933861	+	0.357635i	$0.116417\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	−2.76393	−0.141415
$383$	5.65248	0.288828	0.144414	−	0.989517i	$-0.453870\pi$
$383$	5.65248	0.288828	0.144414	+	0.989517i	$0.453870\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	−5.81966	−0.296213
$387$	0.472136	0.0240000
$388$	21.7082	1.10207
$389$	−28.3607	−1.43794	−0.718972	−	0.695039i	$-0.755387\pi$
$389$	−28.3607	−1.43794	−0.718972	+	0.695039i	$0.755387\pi$
$390$	0	0
$391$	8.47214	0.428454
$392$	−13.4164	−0.677631
$393$	−16.9443	−0.854725
$394$	−13.8885	−0.699695
$395$	0	0
$396$	4.85410	0.243928
$397$	−16.2361	−0.814865	−0.407432	−	0.913235i	$-0.633576\pi$
$397$	−16.2361	−0.814865	−0.407432	+	0.913235i	$0.633576\pi$
$398$	−8.29180	−0.415630
$399$	−0.236068	−0.0118182
$400$	0	0
$401$	11.2918	0.563885	0.281943	−	0.959431i	$-0.409021\pi$
$401$	11.2918	0.563885	0.281943	+	0.959431i	$0.409021\pi$
$402$	−8.94427	−0.446100
$403$	12.9443	0.644800
$404$	8.94427	0.444994
$405$	0	0
$406$	1.09017	0.0541042
$407$	−26.1246	−1.29495
$408$	−2.23607	−0.110702
$409$	35.8328	1.77182	0.885909	−	0.463858i	$-0.153535\pi$
$409$	35.8328	1.77182	0.885909	+	0.463858i	$0.153535\pi$
$410$	0	0
$411$	−16.4164	−0.809762
$412$	11.2361	0.553561
$413$	8.94427	0.440119
$414$	−5.23607	−0.257339
$415$	0	0
$416$	11.2361	0.550894
$417$	12.9443	0.633884
$418$	0.437694	0.0214083
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	4.41641	0.215243	0.107621	−	0.994192i	$-0.465677\pi$
$421$	4.41641	0.215243	0.107621	+	0.994192i	$0.465677\pi$
$422$	11.3475	0.552389
$423$	−0.708204	−0.0344341
$424$	6.70820	0.325779
$425$	0	0
$426$	−3.05573	−0.148051
$427$	−10.0000	−0.483934
$428$	−4.00000	−0.193347
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−27.4721	−1.32329	−0.661643	−	0.749819i	$-0.730141\pi$
$431$	−27.4721	−1.32329	−0.661643	+	0.749819i	$0.730141\pi$
$432$	−1.85410	−0.0892055
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	10.4721	0.501524
$437$	2.00000	0.0956730
$438$	−3.85410	−0.184156
$439$	1.41641	0.0676015	0.0338007	−	0.999429i	$-0.489239\pi$
$439$	1.41641	0.0676015	0.0338007	+	0.999429i	$0.489239\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	1.23607	0.0587938
$443$	−4.94427	−0.234909	−0.117455	−	0.993078i	$-0.537474\pi$
$443$	−4.94427	−0.234909	−0.117455	+	0.993078i	$0.537474\pi$
$444$	14.0902	0.668690
$445$	0	0
$446$	−0.875388	−0.0414508
$447$	−1.52786	−0.0722655
$448$	−0.236068	−0.0111532
$449$	1.41641	0.0668444	0.0334222	−	0.999441i	$-0.489359\pi$
$449$	1.41641	0.0668444	0.0334222	+	0.999441i	$0.489359\pi$
$450$	0	0
$451$	24.7082	1.16346
$452$	−1.70820	−0.0803472
$453$	−13.7639	−0.646686
$454$	−1.88854	−0.0886338
$455$	0	0
$456$	−0.527864	−0.0247195
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	10.1459	0.474087
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	2.94427	0.137128	0.0685642	−	0.997647i	$-0.478158\pi$
$461$	2.94427	0.137128	0.0685642	+	0.997647i	$0.478158\pi$
$462$	−1.85410	−0.0862606
$463$	−3.41641	−0.158774	−0.0793870	−	0.996844i	$-0.525296\pi$
$463$	−3.41641	−0.158774	−0.0793870	+	0.996844i	$0.525296\pi$
$464$	−3.27051	−0.151830
$465$	0	0
$466$	−7.41641	−0.343558
$467$	−5.76393	−0.266723	−0.133361	−	0.991067i	$-0.542577\pi$
$467$	−5.76393	−0.266723	−0.133361	+	0.991067i	$0.542577\pi$
$468$	3.23607	0.149587
$469$	−14.4721	−0.668261
$470$	0	0
$471$	19.4164	0.894661
$472$	20.0000	0.920575
$473$	−1.41641	−0.0651265
$474$	1.23607	0.0567745
$475$	0	0
$476$	−1.61803	−0.0741625
$477$	3.00000	0.137361
$478$	−0.291796	−0.0133464
$479$	41.8885	1.91394	0.956968	−	0.290193i	$-0.0937194\pi$
$479$	41.8885	1.91394	0.956968	+	0.290193i	$0.0937194\pi$
$480$	0	0
$481$	−17.4164	−0.794120
$482$	−3.05573	−0.139185
$483$	−8.47214	−0.385496
$484$	3.23607	0.147094
$485$	0	0
$486$	0.618034	0.0280346
$487$	38.8328	1.75968	0.879841	−	0.475267i	$-0.157649\pi$
$487$	38.8328	1.75968	0.879841	+	0.475267i	$0.157649\pi$
$488$	−22.3607	−1.01222
$489$	16.4164	0.742376
$490$	0	0
$491$	−32.8328	−1.48172	−0.740862	−	0.671657i	$-0.765583\pi$
$491$	−32.8328	−1.48172	−0.740862	+	0.671657i	$0.765583\pi$
$492$	−13.3262	−0.600793
$493$	−1.76393	−0.0794435
$494$	0.291796	0.0131285
$495$	0	0
$496$	−12.0000	−0.538816
$497$	−4.94427	−0.221781
$498$	−8.00000	−0.358489
$499$	−38.3607	−1.71726	−0.858630	−	0.512596i	$-0.828684\pi$
$499$	−38.3607	−1.71726	−0.858630	+	0.512596i	$0.828684\pi$
$500$	0	0
$501$	−18.4721	−0.825274
$502$	7.34752	0.327936
$503$	33.8885	1.51102	0.755508	−	0.655140i	$-0.227390\pi$
$503$	33.8885	1.51102	0.755508	+	0.655140i	$0.227390\pi$
$504$	2.23607	0.0996024
$505$	0	0
$506$	15.7082	0.698315
$507$	9.00000	0.399704
$508$	9.70820	0.430732
$509$	−19.0557	−0.844630	−0.422315	−	0.906449i	$-0.638782\pi$
$509$	−19.0557	−0.844630	−0.422315	+	0.906449i	$0.638782\pi$
$510$	0	0
$511$	−6.23607	−0.275867
$512$	−18.7082	−0.826794
$513$	−0.236068	−0.0104227
$514$	8.06888	0.355903
$515$	0	0
$516$	0.763932	0.0336302
$517$	2.12461	0.0934403
$518$	−5.38197	−0.236470
$519$	20.3607	0.893735
$520$	0	0
$521$	−15.7639	−0.690630	−0.345315	−	0.938487i	$-0.612228\pi$
$521$	−15.7639	−0.690630	−0.345315	+	0.938487i	$0.612228\pi$
$522$	1.09017	0.0477154
$523$	−12.8328	−0.561140	−0.280570	−	0.959834i	$-0.590523\pi$
$523$	−12.8328	−0.561140	−0.280570	+	0.959834i	$0.590523\pi$
$524$	−27.4164	−1.19769
$525$	0	0
$526$	−13.3820	−0.583481
$527$	−6.47214	−0.281931
$528$	5.56231	0.242068
$529$	48.7771	2.12074
$530$	0	0
$531$	8.94427	0.388148
$532$	−0.381966	−0.0165603
$533$	16.4721	0.713487
$534$	−7.41641	−0.320939
$535$	0	0
$536$	−32.3607	−1.39777
$537$	1.52786	0.0659322
$538$	4.43769	0.191323
$539$	18.0000	0.775315
$540$	0	0
$541$	−3.41641	−0.146883	−0.0734414	−	0.997300i	$-0.523398\pi$
$541$	−3.41641	−0.146883	−0.0734414	+	0.997300i	$0.523398\pi$
$542$	−13.5279	−0.581072
$543$	−13.8885	−0.596014
$544$	−5.61803	−0.240871
$545$	0	0
$546$	−1.23607	−0.0528988
$547$	−26.4164	−1.12948	−0.564742	−	0.825268i	$-0.691024\pi$
$547$	−26.4164	−1.12948	−0.564742	+	0.825268i	$0.691024\pi$
$548$	−26.5623	−1.13469
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−0.416408	−0.0177396
$552$	−18.9443	−0.806322
$553$	2.00000	0.0850487
$554$	18.1803	0.772409
$555$	0	0
$556$	20.9443	0.888235
$557$	43.8885	1.85962	0.929809	−	0.368043i	$-0.119972\pi$
$557$	43.8885	1.85962	0.929809	+	0.368043i	$0.119972\pi$
$558$	4.00000	0.169334
$559$	−0.944272	−0.0399384
$560$	0	0
$561$	3.00000	0.126660
$562$	8.87539	0.374386
$563$	−20.1246	−0.848151	−0.424076	−	0.905627i	$-0.639401\pi$
$563$	−20.1246	−0.848151	−0.424076	+	0.905627i	$0.639401\pi$
$564$	−1.14590	−0.0482510
$565$	0	0
$566$	10.7984	0.453890
$567$	1.00000	0.0419961
$568$	−11.0557	−0.463888
$569$	43.8885	1.83990	0.919952	−	0.392032i	$-0.128228\pi$
$569$	43.8885	1.83990	0.919952	+	0.392032i	$0.128228\pi$
$570$	0	0
$571$	−12.8328	−0.537037	−0.268518	−	0.963275i	$-0.586534\pi$
$571$	−12.8328	−0.537037	−0.268518	+	0.963275i	$0.586534\pi$
$572$	−9.70820	−0.405920
$573$	−4.47214	−0.186826
$574$	5.09017	0.212460
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	29.3050	1.21998	0.609991	−	0.792409i	$-0.291173\pi$
$577$	29.3050	1.21998	0.609991	+	0.792409i	$0.291173\pi$
$578$	−0.618034	−0.0257068
$579$	−9.41641	−0.391333
$580$	0	0
$581$	−12.9443	−0.537019
$582$	−8.29180	−0.343706
$583$	−9.00000	−0.372742
$584$	−13.9443	−0.577018
$585$	0	0
$586$	−5.49342	−0.226931
$587$	2.23607	0.0922924	0.0461462	−	0.998935i	$-0.485306\pi$
$587$	2.23607	0.0922924	0.0461462	+	0.998935i	$0.485306\pi$
$588$	−9.70820	−0.400360
$589$	−1.52786	−0.0629545
$590$	0	0
$591$	−22.4721	−0.924380
$592$	16.1459	0.663592
$593$	−8.52786	−0.350197	−0.175099	−	0.984551i	$-0.556024\pi$
$593$	−8.52786	−0.350197	−0.175099	+	0.984551i	$0.556024\pi$
$594$	−1.85410	−0.0760747
$595$	0	0
$596$	−2.47214	−0.101263
$597$	−13.4164	−0.549097
$598$	10.4721	0.428237
$599$	−4.47214	−0.182727	−0.0913633	−	0.995818i	$-0.529122\pi$
$599$	−4.47214	−0.182727	−0.0913633	+	0.995818i	$0.529122\pi$
$600$	0	0
$601$	−27.4164	−1.11834	−0.559169	−	0.829053i	$-0.688880\pi$
$601$	−27.4164	−1.11834	−0.559169	+	0.829053i	$0.688880\pi$
$602$	−0.291796	−0.0118927
$603$	−14.4721	−0.589351
$604$	−22.2705	−0.906174
$605$	0	0
$606$	−3.41641	−0.138782
$607$	11.1115	0.451000	0.225500	−	0.974243i	$-0.427598\pi$
$607$	11.1115	0.451000	0.225500	+	0.974243i	$0.427598\pi$
$608$	−1.32624	−0.0537861
$609$	1.76393	0.0714781
$610$	0	0
$611$	1.41641	0.0573017
$612$	−1.61803	−0.0654051
$613$	25.0557	1.01199	0.505996	−	0.862536i	$-0.331125\pi$
$613$	25.0557	1.01199	0.505996	+	0.862536i	$0.331125\pi$
$614$	−21.5279	−0.868794
$615$	0	0
$616$	−6.70820	−0.270281
$617$	−23.8885	−0.961717	−0.480858	−	0.876798i	$-0.659675\pi$
$617$	−23.8885	−0.961717	−0.480858	+	0.876798i	$0.659675\pi$
$618$	−4.29180	−0.172641
$619$	27.8885	1.12094	0.560468	−	0.828176i	$-0.310621\pi$
$619$	27.8885	1.12094	0.560468	+	0.828176i	$0.310621\pi$
$620$	0	0
$621$	−8.47214	−0.339975
$622$	−6.50658	−0.260890
$623$	−12.0000	−0.480770
$624$	3.70820	0.148447
$625$	0	0
$626$	−4.14590	−0.165703
$627$	0.708204	0.0282829
$628$	31.4164	1.25365
$629$	8.70820	0.347219
$630$	0	0
$631$	−32.1246	−1.27886	−0.639430	−	0.768849i	$-0.720830\pi$
$631$	−32.1246	−1.27886	−0.639430	+	0.768849i	$0.720830\pi$
$632$	4.47214	0.177892
$633$	18.3607	0.729772
$634$	7.12461	0.282954
$635$	0	0
$636$	4.85410	0.192478
$637$	12.0000	0.475457
$638$	−3.27051	−0.129481
$639$	−4.94427	−0.195592
$640$	0	0
$641$	−36.7082	−1.44989	−0.724943	−	0.688808i	$-0.758134\pi$
$641$	−36.7082	−1.44989	−0.724943	+	0.688808i	$0.758134\pi$
$642$	1.52786	0.0603000
$643$	31.2492	1.23235	0.616175	−	0.787610i	$-0.288682\pi$
$643$	31.2492	1.23235	0.616175	+	0.787610i	$0.288682\pi$
$644$	−13.7082	−0.540179
$645$	0	0
$646$	−0.145898	−0.00574028
$647$	19.0557	0.749158	0.374579	−	0.927195i	$-0.377787\pi$
$647$	19.0557	0.749158	0.374579	+	0.927195i	$0.377787\pi$
$648$	2.23607	0.0878410
$649$	−26.8328	−1.05328
$650$	0	0
$651$	6.47214	0.253663
$652$	26.5623	1.04026
$653$	−29.3050	−1.14679	−0.573396	−	0.819279i	$-0.694374\pi$
$653$	−29.3050	−1.14679	−0.573396	+	0.819279i	$0.694374\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	−15.2705	−0.596213
$657$	−6.23607	−0.243292
$658$	0.437694	0.0170631
$659$	−26.8328	−1.04526	−0.522629	−	0.852560i	$-0.675049\pi$
$659$	−26.8328	−1.04526	−0.522629	+	0.852560i	$0.675049\pi$
$660$	0	0
$661$	−28.4164	−1.10527	−0.552635	−	0.833423i	$-0.686378\pi$
$661$	−28.4164	−1.10527	−0.552635	+	0.833423i	$0.686378\pi$
$662$	−6.25735	−0.243199
$663$	2.00000	0.0776736
$664$	−28.9443	−1.12326
$665$	0	0
$666$	−5.38197	−0.208547
$667$	−14.9443	−0.578645
$668$	−29.8885	−1.15642
$669$	−1.41641	−0.0547615
$670$	0	0
$671$	30.0000	1.15814
$672$	5.61803	0.216720
$673$	12.4721	0.480766	0.240383	−	0.970678i	$-0.422727\pi$
$673$	12.4721	0.480766	0.240383	+	0.970678i	$0.422727\pi$
$674$	−19.5623	−0.753512
$675$	0	0
$676$	14.5623	0.560089
$677$	30.9443	1.18928	0.594642	−	0.803990i	$-0.297294\pi$
$677$	30.9443	1.18928	0.594642	+	0.803990i	$0.297294\pi$
$678$	0.652476	0.0250582
$679$	−13.4164	−0.514874
$680$	0	0
$681$	−3.05573	−0.117096
$682$	−12.0000	−0.459504
$683$	41.7771	1.59856	0.799278	−	0.600962i	$-0.205216\pi$
$683$	41.7771	1.59856	0.799278	+	0.600962i	$0.205216\pi$
$684$	−0.381966	−0.0146048
$685$	0	0
$686$	8.03444	0.306756
$687$	16.4164	0.626325
$688$	0.875388	0.0333739
$689$	−6.00000	−0.228582
$690$	0	0
$691$	34.2492	1.30290	0.651451	−	0.758691i	$-0.274161\pi$
$691$	34.2492	1.30290	0.651451	+	0.758691i	$0.274161\pi$
$692$	32.9443	1.25235
$693$	−3.00000	−0.113961
$694$	6.47214	0.245679
$695$	0	0
$696$	3.94427	0.149507
$697$	−8.23607	−0.311963
$698$	20.0344	0.758315
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−1.23607	−0.0466524
$703$	2.05573	0.0775333
$704$	0.708204	0.0266914
$705$	0	0
$706$	10.2148	0.384438
$707$	−5.52786	−0.207897
$708$	14.4721	0.543896
$709$	−10.5836	−0.397475	−0.198738	−	0.980053i	$-0.563684\pi$
$709$	−10.5836	−0.397475	−0.198738	+	0.980053i	$0.563684\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	−26.8328	−1.00560
$713$	−54.8328	−2.05351
$714$	0.618034	0.0231293
$715$	0	0
$716$	2.47214	0.0923881
$717$	−0.472136	−0.0176322
$718$	7.34752	0.274207
$719$	−34.3050	−1.27936	−0.639679	−	0.768642i	$-0.720933\pi$
$719$	−34.3050	−1.27936	−0.639679	+	0.768642i	$0.720933\pi$
$720$	0	0
$721$	−6.94427	−0.258618
$722$	11.7082	0.435734
$723$	−4.94427	−0.183879
$724$	−22.4721	−0.835170
$725$	0	0
$726$	−1.23607	−0.0458748
$727$	35.3050	1.30939	0.654694	−	0.755894i	$-0.272797\pi$
$727$	35.3050	1.30939	0.654694	+	0.755894i	$0.272797\pi$
$728$	−4.47214	−0.165748
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.472136	0.0174626
$732$	−16.1803	−0.598043
$733$	5.52786	0.204176	0.102088	−	0.994775i	$-0.467448\pi$
$733$	5.52786	0.204176	0.102088	+	0.994775i	$0.467448\pi$
$734$	7.41641	0.273745
$735$	0	0
$736$	−47.5967	−1.75444
$737$	43.4164	1.59926
$738$	5.09017	0.187372
$739$	−38.8328	−1.42849	−0.714244	−	0.699897i	$-0.753229\pi$
$739$	−38.8328	−1.42849	−0.714244	+	0.699897i	$0.753229\pi$
$740$	0	0
$741$	0.472136	0.0173443
$742$	−1.85410	−0.0680662
$743$	15.8885	0.582894	0.291447	−	0.956587i	$-0.405863\pi$
$743$	15.8885	0.582894	0.291447	+	0.956587i	$0.405863\pi$
$744$	14.4721	0.530574
$745$	0	0
$746$	−0.291796	−0.0106834
$747$	−12.9443	−0.473606
$748$	4.85410	0.177484
$749$	2.47214	0.0903299
$750$	0	0
$751$	39.8885	1.45555	0.727777	−	0.685814i	$-0.240554\pi$
$751$	39.8885	1.45555	0.727777	+	0.685814i	$0.240554\pi$
$752$	−1.31308	−0.0478832
$753$	11.8885	0.433243
$754$	−2.18034	−0.0794033
$755$	0	0
$756$	1.61803	0.0588473
$757$	−31.4164	−1.14185	−0.570924	−	0.821003i	$-0.693415\pi$
$757$	−31.4164	−1.14185	−0.570924	+	0.821003i	$0.693415\pi$
$758$	−22.4721	−0.816225
$759$	25.4164	0.922557
$760$	0	0
$761$	−22.9443	−0.831729	−0.415865	−	0.909427i	$-0.636521\pi$
$761$	−22.9443	−0.831729	−0.415865	+	0.909427i	$0.636521\pi$
$762$	−3.70820	−0.134334
$763$	−6.47214	−0.234307
$764$	−7.23607	−0.261792
$765$	0	0
$766$	−3.49342	−0.126222
$767$	−17.8885	−0.645918
$768$	6.56231	0.236797
$769$	31.9443	1.15194	0.575970	−	0.817471i	$-0.304625\pi$
$769$	31.9443	1.15194	0.575970	+	0.817471i	$0.304625\pi$
$770$	0	0
$771$	13.0557	0.470191
$772$	−15.2361	−0.548358
$773$	−50.7771	−1.82632	−0.913162	−	0.407596i	$-0.866367\pi$
$773$	−50.7771	−1.82632	−0.913162	+	0.407596i	$0.866367\pi$
$774$	−0.291796	−0.0104884
$775$	0	0
$776$	−30.0000	−1.07694
$777$	−8.70820	−0.312405
$778$	17.5279	0.628404
$779$	−1.94427	−0.0696608
$780$	0	0
$781$	14.8328	0.530760
$782$	−5.23607	−0.187241
$783$	1.76393	0.0630378
$784$	−11.1246	−0.397308
$785$	0	0
$786$	10.4721	0.373529
$787$	−46.4164	−1.65457	−0.827283	−	0.561785i	$-0.810115\pi$
$787$	−46.4164	−1.65457	−0.827283	+	0.561785i	$0.810115\pi$
$788$	−36.3607	−1.29530
$789$	−21.6525	−0.770849
$790$	0	0
$791$	1.05573	0.0375374
$792$	−6.70820	−0.238366
$793$	20.0000	0.710221
$794$	10.0344	0.356109
$795$	0	0
$796$	−21.7082	−0.769427
$797$	−39.0000	−1.38145	−0.690725	−	0.723117i	$-0.742709\pi$
$797$	−39.0000	−1.38145	−0.690725	+	0.723117i	$0.742709\pi$
$798$	0.145898	0.00516473
$799$	−0.708204	−0.0250545
$800$	0	0
$801$	−12.0000	−0.423999
$802$	−6.97871	−0.246427
$803$	18.7082	0.660198
$804$	−23.4164	−0.825833
$805$	0	0
$806$	−8.00000	−0.281788
$807$	7.18034	0.252760
$808$	−12.3607	−0.434847
$809$	16.4721	0.579129	0.289565	−	0.957158i	$-0.406489\pi$
$809$	16.4721	0.579129	0.289565	+	0.957158i	$0.406489\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	2.85410	0.100159
$813$	−21.8885	−0.767665
$814$	16.1459	0.565913
$815$	0	0
$816$	−1.85410	−0.0649066
$817$	0.111456	0.00389936
$818$	−22.1459	−0.774313
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	16.4721	0.574882	0.287441	−	0.957798i	$-0.407196\pi$
$821$	16.4721	0.574882	0.287441	+	0.957798i	$0.407196\pi$
$822$	10.1459	0.353879
$823$	7.00000	0.244005	0.122002	−	0.992530i	$-0.461068\pi$
$823$	7.00000	0.244005	0.122002	+	0.992530i	$0.461068\pi$
$824$	−15.5279	−0.540939
$825$	0	0
$826$	−5.52786	−0.192339
$827$	35.8885	1.24797	0.623983	−	0.781438i	$-0.285513\pi$
$827$	35.8885	1.24797	0.623983	+	0.781438i	$0.285513\pi$
$828$	−13.7082	−0.476393
$829$	−23.3607	−0.811350	−0.405675	−	0.914017i	$-0.632964\pi$
$829$	−23.3607	−0.811350	−0.405675	+	0.914017i	$0.632964\pi$
$830$	0	0
$831$	29.4164	1.02044
$832$	0.472136	0.0163684
$833$	−6.00000	−0.207888
$834$	−8.00000	−0.277017
$835$	0	0
$836$	1.14590	0.0396317
$837$	6.47214	0.223710
$838$	12.9787	0.448342
$839$	10.4164	0.359614	0.179807	−	0.983702i	$-0.442453\pi$
$839$	10.4164	0.359614	0.179807	+	0.983702i	$0.442453\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	−2.72949	−0.0940644
$843$	14.3607	0.494608
$844$	29.7082	1.02260
$845$	0	0
$846$	0.437694	0.0150482
$847$	−2.00000	−0.0687208
$848$	5.56231	0.191010
$849$	17.4721	0.599642
$850$	0	0
$851$	73.7771	2.52905
$852$	−8.00000	−0.274075
$853$	56.2492	1.92594	0.962968	−	0.269614i	$-0.0868962\pi$
$853$	56.2492	1.92594	0.962968	+	0.269614i	$0.0868962\pi$
$854$	6.18034	0.211487
$855$	0	0
$856$	5.52786	0.188939
$857$	−9.63932	−0.329273	−0.164636	−	0.986354i	$-0.552645\pi$
$857$	−9.63932	−0.329273	−0.164636	+	0.986354i	$0.552645\pi$
$858$	3.70820	0.126596
$859$	14.8197	0.505640	0.252820	−	0.967513i	$-0.418642\pi$
$859$	14.8197	0.505640	0.252820	+	0.967513i	$0.418642\pi$
$860$	0	0
$861$	8.23607	0.280684
$862$	16.9787	0.578297
$863$	27.7639	0.945095	0.472548	−	0.881305i	$-0.343334\pi$
$863$	27.7639	0.945095	0.472548	+	0.881305i	$0.343334\pi$
$864$	5.61803	0.191129
$865$	0	0
$866$	−3.70820	−0.126010
$867$	−1.00000	−0.0339618
$868$	10.4721	0.355447
$869$	−6.00000	−0.203536
$870$	0	0
$871$	28.9443	0.980739
$872$	−14.4721	−0.490088
$873$	−13.4164	−0.454077
$874$	−1.23607	−0.0418106
$875$	0	0
$876$	−10.0902	−0.340915
$877$	13.8754	0.468539	0.234269	−	0.972172i	$-0.424730\pi$
$877$	13.8754	0.468539	0.234269	+	0.972172i	$0.424730\pi$
$878$	−0.875388	−0.0295429
$879$	−8.88854	−0.299803
$880$	0	0
$881$	−51.0689	−1.72055	−0.860277	−	0.509827i	$-0.829710\pi$
$881$	−51.0689	−1.72055	−0.860277	+	0.509827i	$0.829710\pi$
$882$	3.70820	0.124862
$883$	−3.16718	−0.106584	−0.0532921	−	0.998579i	$-0.516971\pi$
$883$	−3.16718	−0.106584	−0.0532921	+	0.998579i	$0.516971\pi$
$884$	3.23607	0.108841
$885$	0	0
$886$	3.05573	0.102659
$887$	−31.3050	−1.05112	−0.525559	−	0.850757i	$-0.676144\pi$
$887$	−31.3050	−1.05112	−0.525559	+	0.850757i	$0.676144\pi$
$888$	−19.4721	−0.653442
$889$	−6.00000	−0.201234
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−2.29180	−0.0767350
$893$	−0.167184	−0.00559461
$894$	0.944272	0.0315812
$895$	0	0
$896$	11.3820	0.380245
$897$	16.9443	0.565753
$898$	−0.875388	−0.0292121
$899$	11.4164	0.380759
$900$	0	0
$901$	3.00000	0.0999445
$902$	−15.2705	−0.508452
$903$	−0.472136	−0.0157117
$904$	2.36068	0.0785150
$905$	0	0
$906$	8.50658	0.282612
$907$	40.4164	1.34200	0.671002	−	0.741455i	$-0.265864\pi$
$907$	40.4164	1.34200	0.671002	+	0.741455i	$0.265864\pi$
$908$	−4.94427	−0.164081
$909$	−5.52786	−0.183348
$910$	0	0
$911$	31.2492	1.03533	0.517666	−	0.855582i	$-0.326801\pi$
$911$	31.2492	1.03533	0.517666	+	0.855582i	$0.326801\pi$
$912$	−0.437694	−0.0144935
$913$	38.8328	1.28518
$914$	21.0132	0.695053
$915$	0	0
$916$	26.5623	0.877643
$917$	16.9443	0.559549
$918$	0.618034	0.0203982
$919$	−5.29180	−0.174560	−0.0872801	−	0.996184i	$-0.527818\pi$
$919$	−5.29180	−0.174560	−0.0872801	+	0.996184i	$0.527818\pi$
$920$	0	0
$921$	−34.8328	−1.14778
$922$	−1.81966	−0.0599273
$923$	9.88854	0.325485
$924$	−4.85410	−0.159688
$925$	0	0
$926$	2.11146	0.0693868
$927$	−6.94427	−0.228080
$928$	9.90983	0.325306
$929$	52.4853	1.72199	0.860993	−	0.508616i	$-0.169843\pi$
$929$	52.4853	1.72199	0.860993	+	0.508616i	$0.169843\pi$
$930$	0	0
$931$	−1.41641	−0.0464209
$932$	−19.4164	−0.636006
$933$	−10.5279	−0.344667
$934$	3.56231	0.116562
$935$	0	0
$936$	−4.47214	−0.146176
$937$	9.88854	0.323045	0.161522	−	0.986869i	$-0.448360\pi$
$937$	9.88854	0.323045	0.161522	+	0.986869i	$0.448360\pi$
$938$	8.94427	0.292041
$939$	−6.70820	−0.218914
$940$	0	0
$941$	−3.52786	−0.115005	−0.0575025	−	0.998345i	$-0.518314\pi$
$941$	−3.52786	−0.115005	−0.0575025	+	0.998345i	$0.518314\pi$
$942$	−12.0000	−0.390981
$943$	−69.7771	−2.27225
$944$	16.5836	0.539750
$945$	0	0
$946$	0.875388	0.0284613
$947$	19.5279	0.634570	0.317285	−	0.948330i	$-0.397229\pi$
$947$	19.5279	0.634570	0.317285	+	0.948330i	$0.397229\pi$
$948$	3.23607	0.105103
$949$	12.4721	0.404863
$950$	0	0
$951$	11.5279	0.373817
$952$	2.23607	0.0724714
$953$	5.36068	0.173649	0.0868247	−	0.996224i	$-0.472328\pi$
$953$	5.36068	0.173649	0.0868247	+	0.996224i	$0.472328\pi$
$954$	−1.85410	−0.0600288
$955$	0	0
$956$	−0.763932	−0.0247073
$957$	−5.29180	−0.171059
$958$	−25.8885	−0.836421
$959$	16.4164	0.530113
$960$	0	0
$961$	10.8885	0.351243
$962$	10.7639	0.347043
$963$	2.47214	0.0796635
$964$	−8.00000	−0.257663
$965$	0	0
$966$	5.23607	0.168468
$967$	−46.0000	−1.47926	−0.739630	−	0.673014i	$-0.765000\pi$
$967$	−46.0000	−1.47926	−0.739630	+	0.673014i	$0.765000\pi$
$968$	−4.47214	−0.143740
$969$	−0.236068	−0.00758360
$970$	0	0
$971$	41.3050	1.32554	0.662769	−	0.748823i	$-0.269381\pi$
$971$	41.3050	1.32554	0.662769	+	0.748823i	$0.269381\pi$
$972$	1.61803	0.0518985
$973$	−12.9443	−0.414974
$974$	−24.0000	−0.769010
$975$	0	0
$976$	−18.5410	−0.593484
$977$	−30.9443	−0.989995	−0.494997	−	0.868894i	$-0.664831\pi$
$977$	−30.9443	−0.989995	−0.494997	+	0.868894i	$0.664831\pi$
$978$	−10.1459	−0.324430
$979$	36.0000	1.15056
$980$	0	0
$981$	−6.47214	−0.206639
$982$	20.2918	0.647537
$983$	42.4721	1.35465	0.677325	−	0.735684i	$-0.263139\pi$
$983$	42.4721	1.35465	0.677325	+	0.735684i	$0.263139\pi$
$984$	18.4164	0.587094
$985$	0	0
$986$	1.09017	0.0347181
$987$	0.708204	0.0225424
$988$	0.763932	0.0243039
$989$	4.00000	0.127193
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	36.3607	1.15445
$993$	−10.1246	−0.321295
$994$	3.05573	0.0969218
$995$	0	0
$996$	−20.9443	−0.663645
$997$	−14.5967	−0.462284	−0.231142	−	0.972920i	$-0.574246\pi$
$997$	−14.5967	−0.462284	−0.231142	+	0.972920i	$0.574246\pi$
$998$	23.7082	0.750470
$999$	−8.70820	−0.275515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.a.m.1.1		2
3.2	odd	2		3825.2.a.q.1.2		2
5.2	odd	4		255.2.b.b.154.2	✓	4
5.3	odd	4		255.2.b.b.154.3	yes	4
5.4	even	2		1275.2.a.i.1.2		2
15.2	even	4		765.2.b.b.154.3		4
15.8	even	4		765.2.b.b.154.2		4
15.14	odd	2		3825.2.a.w.1.1		2
20.3	even	4		4080.2.m.n.2449.4		4
20.7	even	4		4080.2.m.n.2449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.b.b.154.2	✓	4	5.2	odd	4
255.2.b.b.154.3	yes	4	5.3	odd	4
765.2.b.b.154.2		4	15.8	even	4
765.2.b.b.154.3		4	15.2	even	4
1275.2.a.i.1.2		2	5.4	even	2
1275.2.a.m.1.1		2	1.1	even	1	trivial
3825.2.a.q.1.2		2	3.2	odd	2
3825.2.a.w.1.1		2	15.14	odd	2
4080.2.m.n.2449.2		4	20.7	even	4
4080.2.m.n.2449.4		4	20.3	even	4

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:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.61803 q^{12} -2.00000 q^{13} -0.618034 q^{14} +1.85410 q^{16} +1.00000 q^{17} -0.618034 q^{18} +0.236068 q^{19} -1.00000 q^{21} +1.85410 q^{22} +8.47214 q^{23} -2.23607 q^{24} +1.23607 q^{26} -1.00000 q^{27} -1.61803 q^{28} -1.76393 q^{29} -6.47214 q^{31} -5.61803 q^{32} +3.00000 q^{33} -0.618034 q^{34} -1.61803 q^{36} +8.70820 q^{37} -0.145898 q^{38} +2.00000 q^{39} -8.23607 q^{41} +0.618034 q^{42} +0.472136 q^{43} +4.85410 q^{44} -5.23607 q^{46} -0.708204 q^{47} -1.85410 q^{48} -6.00000 q^{49} -1.00000 q^{51} +3.23607 q^{52} +3.00000 q^{53} +0.618034 q^{54} +2.23607 q^{56} -0.236068 q^{57} +1.09017 q^{58} +8.94427 q^{59} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} -0.236068 q^{64} -1.85410 q^{66} -14.4721 q^{67} -1.61803 q^{68} -8.47214 q^{69} -4.94427 q^{71} +2.23607 q^{72} -6.23607 q^{73} -5.38197 q^{74} -0.381966 q^{76} -3.00000 q^{77} -1.23607 q^{78} +2.00000 q^{79} +1.00000 q^{81} +5.09017 q^{82} -12.9443 q^{83} +1.61803 q^{84} -0.291796 q^{86} +1.76393 q^{87} -6.70820 q^{88} -12.0000 q^{89} -2.00000 q^{91} -13.7082 q^{92} +6.47214 q^{93} +0.437694 q^{94} +5.61803 q^{96} -13.4164 q^{97} +3.70820 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{7} + 2 q^{9} - 6 q^{11} + q^{12} - 4 q^{13} + q^{14} - 3 q^{16} + 2 q^{17} + q^{18} - 4 q^{19} - 2 q^{21} - 3 q^{22} + 8 q^{23} - 2 q^{26} - 2 q^{27} - q^{28}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 2 * q^7 + 2 * q^9 - 6 * q^11 + q^12 - 4 * q^13 + q^14 - 3 * q^16 + 2 * q^17 + q^18 - 4 * q^19 - 2 * q^21 - 3 * q^22 + 8 * q^23 - 2 * q^26 - 2 * q^27 - q^28 - 8 * q^29 - 4 * q^31 - 9 * q^32 + 6 * q^33 + q^34 - q^36 + 4 * q^37 - 7 * q^38 + 4 * q^39 - 12 * q^41 - q^42 - 8 * q^43 + 3 * q^44 - 6 * q^46 + 12 * q^47 + 3 * q^48 - 12 * q^49 - 2 * q^51 + 2 * q^52 + 6 * q^53 - q^54 + 4 * q^57 - 9 * q^58 - 20 * q^61 + 8 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^66 - 20 * q^67 - q^68 - 8 * q^69 + 8 * q^71 - 8 * q^73 - 13 * q^74 - 3 * q^76 - 6 * q^77 + 2 * q^78 + 4 * q^79 + 2 * q^81 - q^82 - 8 * q^83 + q^84 - 14 * q^86 + 8 * q^87 - 24 * q^89 - 4 * q^91 - 14 * q^92 + 4 * q^93 + 21 * q^94 + 9 * q^96 - 6 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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