LMFDB - Embedded newform 6825.2.a.bl.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6825,2,Mod(1,6825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6825.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:	$54.4978993795$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.14013.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 6x + 3 $ x^4 - x^3 - 6x^2 + 6x + 3
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			1.4
Root			$2.20800$ of defining polynomial
Character	$\chi$	$=$	6825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.20800 q^{2} +1.00000 q^{3} +2.87525 q^{4} +2.20800 q^{6} -1.00000 q^{7} +1.93254 q^{8} +1.00000 q^{9} -4.47504 q^{11} +2.87525 q^{12} -1.00000 q^{13} -2.20800 q^{14} -1.48345 q^{16} +2.32433 q^{17} +2.20800 q^{18} -5.93254 q^{19} -1.00000 q^{21} -9.88086 q^{22} -4.86508 q^{23} +1.93254 q^{24} -2.20800 q^{26} +1.00000 q^{27} -2.87525 q^{28} -0.932540 q^{29} +5.09902 q^{31} -7.14054 q^{32} -4.47504 q^{33} +5.13212 q^{34} +2.87525 q^{36} -2.63415 q^{37} -13.0990 q^{38} -1.00000 q^{39} -6.50940 q^{41} -2.20800 q^{42} -1.35028 q^{43} -12.8668 q^{44} -10.7421 q^{46} +0.0259483 q^{47} -1.48345 q^{48} +1.00000 q^{49} +2.32433 q^{51} -2.87525 q^{52} +0.734710 q^{53} +2.20800 q^{54} -1.93254 q^{56} -5.93254 q^{57} -2.05904 q^{58} +13.5741 q^{59} +4.56543 q^{61} +11.2586 q^{62} -1.00000 q^{63} -12.7994 q^{64} -9.88086 q^{66} -13.2052 q^{67} +6.68303 q^{68} -4.86508 q^{69} -6.63240 q^{71} +1.93254 q^{72} -14.2238 q^{73} -5.81620 q^{74} -17.0575 q^{76} +4.47504 q^{77} -2.20800 q^{78} -0.618372 q^{79} +1.00000 q^{81} -14.3727 q^{82} +2.10919 q^{83} -2.87525 q^{84} -2.98142 q^{86} -0.932540 q^{87} -8.64819 q^{88} -10.7818 q^{89} +1.00000 q^{91} -13.9883 q^{92} +5.09902 q^{93} +0.0572938 q^{94} -7.14054 q^{96} +0.464870 q^{97} +2.20800 q^{98} -4.47504 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 4 q^{3} + 5 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} + 5 q^{12} - 4 q^{13} - q^{14} - q^{16} - 3 q^{17} + q^{18} - 13 q^{19} - 4 q^{21} + 7 q^{22} + 2 q^{23} - 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q + q^2 + 4 * q^3 + 5 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 + 5 * q^12 - 4 * q^13 - q^14 - q^16 - 3 * q^17 + q^18 - 13 * q^19 - 4 * q^21 + 7 * q^22 + 2 * q^23 - 3 * q^24 - q^26 + 4 * q^27 - 5 * q^28 + 7 * q^29 - 19 * q^31 - 10 * q^32 - 2 * q^33 - 9 * q^34 + 5 * q^36 - 2 * q^37 - 13 * q^38 - 4 * q^39 - 11 * q^41 - q^42 + 17 * q^43 - 34 * q^44 - 19 * q^46 - 10 * q^47 - q^48 + 4 * q^49 - 3 * q^51 - 5 * q^52 + 15 * q^53 + q^54 + 3 * q^56 - 13 * q^57 - 8 * q^58 - q^59 + 8 * q^61 + 26 * q^62 - 4 * q^63 - 23 * q^64 + 7 * q^66 + 6 * q^67 + 3 * q^68 + 2 * q^69 - 14 * q^71 - 3 * q^72 - 24 * q^73 - 17 * q^74 - 8 * q^76 + 2 * q^77 - q^78 - 19 * q^79 + 4 * q^81 - 17 * q^82 - 16 * q^83 - 5 * q^84 + 14 * q^86 + 7 * q^87 + 3 * q^88 - 36 * q^89 + 4 * q^91 + 19 * q^92 - 19 * q^93 - 4 * q^94 - 10 * q^96 - 29 * q^97 + q^98 - 2 * 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 $

$2$	2.20800	1.56129	0.780644	−	0.624975i	$-0.214891\pi$
$2$	2.20800	1.56129	0.780644	+	0.624975i	$0.214891\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.87525	1.43762
$5$	0	0
$5$	0	0
$6$	2.20800	0.901411
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.93254	0.683256
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.47504	−1.34927	−0.674637	−	0.738150i	$-0.735700\pi$
$11$	−4.47504	−1.34927	−0.674637	+	0.738150i	$0.735700\pi$
$12$	2.87525	0.830012
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.20800	−0.590112
$15$	0	0
$16$	−1.48345	−0.370863
$17$	2.32433	0.563734	0.281867	−	0.959454i	$-0.409046\pi$
$17$	2.32433	0.563734	0.281867	+	0.959454i	$0.409046\pi$
$18$	2.20800	0.520430
$19$	−5.93254	−1.36102	−0.680509	−	0.732740i	$-0.738241\pi$
$19$	−5.93254	−1.36102	−0.680509	+	0.732740i	$0.738241\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−9.88086	−2.10661
$23$	−4.86508	−1.01444	−0.507220	−	0.861817i	$-0.669327\pi$
$23$	−4.86508	−1.01444	−0.507220	+	0.861817i	$0.669327\pi$
$24$	1.93254	0.394478
$25$	0	0
$26$	−2.20800	−0.433024
$27$	1.00000	0.192450
$28$	−2.87525	−0.543370
$29$	−0.932540	−0.173168	−0.0865842	−	0.996245i	$-0.527595\pi$
$29$	−0.932540	−0.173168	−0.0865842	+	0.996245i	$0.527595\pi$
$30$	0	0
$31$	5.09902	0.915812	0.457906	−	0.889001i	$-0.348600\pi$
$31$	5.09902	0.915812	0.457906	+	0.889001i	$0.348600\pi$
$32$	−7.14054	−1.26228
$33$	−4.47504	−0.779004
$34$	5.13212	0.880151
$35$	0	0
$36$	2.87525	0.479208
$37$	−2.63415	−0.433052	−0.216526	−	0.976277i	$-0.569473\pi$
$37$	−2.63415	−0.433052	−0.216526	+	0.976277i	$0.569473\pi$
$38$	−13.0990	−2.12494
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−6.50940	−1.01660	−0.508299	−	0.861181i	$-0.669725\pi$
$41$	−6.50940	−1.01660	−0.508299	+	0.861181i	$0.669725\pi$
$42$	−2.20800	−0.340701
$43$	−1.35028	−0.205916	−0.102958	−	0.994686i	$-0.532831\pi$
$43$	−1.35028	−0.205916	−0.102958	+	0.994686i	$0.532831\pi$
$44$	−12.8668	−1.93975
$45$	0	0
$46$	−10.7421	−1.58383
$47$	0.0259483	0.00378495	0.00189247	−	0.999998i	$-0.499398\pi$
$47$	0.0259483	0.00378495	0.00189247	+	0.999998i	$0.499398\pi$
$48$	−1.48345	−0.214118
$49$	1.00000	0.142857
$50$	0	0
$51$	2.32433	0.325472
$52$	−2.87525	−0.398725
$53$	0.734710	0.100920	0.0504601	−	0.998726i	$-0.483931\pi$
$53$	0.734710	0.100920	0.0504601	+	0.998726i	$0.483931\pi$
$54$	2.20800	0.300470
$55$	0	0
$56$	−1.93254	−0.258247
$57$	−5.93254	−0.785784
$58$	−2.05904	−0.270366
$59$	13.5741	1.76719	0.883596	−	0.468250i	$-0.155115\pi$
$59$	13.5741	1.76719	0.883596	+	0.468250i	$0.155115\pi$
$60$	0	0
$61$	4.56543	0.584543	0.292271	−	0.956335i	$-0.405589\pi$
$61$	4.56543	0.584543	0.292271	+	0.956335i	$0.405589\pi$
$62$	11.2586	1.42985
$63$	−1.00000	−0.125988
$64$	−12.7994	−1.59992
$65$	0	0
$66$	−9.88086	−1.21625
$67$	−13.2052	−1.61327	−0.806636	−	0.591049i	$-0.798714\pi$
$67$	−13.2052	−1.61327	−0.806636	+	0.591049i	$0.798714\pi$
$68$	6.68303	0.810437
$69$	−4.86508	−0.585687
$70$	0	0
$71$	−6.63240	−0.787122	−0.393561	−	0.919299i	$-0.628757\pi$
$71$	−6.63240	−0.787122	−0.393561	+	0.919299i	$0.628757\pi$
$72$	1.93254	0.227752
$73$	−14.2238	−1.66477	−0.832384	−	0.554200i	$-0.813024\pi$
$73$	−14.2238	−1.66477	−0.832384	+	0.554200i	$0.813024\pi$
$74$	−5.81620	−0.676120
$75$	0	0
$76$	−17.0575	−1.95663
$77$	4.47504	0.509978
$78$	−2.20800	−0.250006
$79$	−0.618372	−0.0695723	−0.0347862	−	0.999395i	$-0.511075\pi$
$79$	−0.618372	−0.0695723	−0.0347862	+	0.999395i	$0.511075\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−14.3727	−1.58720
$83$	2.10919	0.231514	0.115757	−	0.993278i	$-0.463071\pi$
$83$	2.10919	0.231514	0.115757	+	0.993278i	$0.463071\pi$
$84$	−2.87525	−0.313715
$85$	0	0
$86$	−2.98142	−0.321495
$87$	−0.932540	−0.0999788
$88$	−8.64819	−0.921900
$89$	−10.7818	−1.14287	−0.571436	−	0.820646i	$-0.693614\pi$
$89$	−10.7818	−1.14287	−0.571436	+	0.820646i	$0.693614\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−13.9883	−1.45838
$93$	5.09902	0.528744
$94$	0.0572938	0.00590940
$95$	0	0
$96$	−7.14054	−0.728778
$97$	0.464870	0.0472004	0.0236002	−	0.999721i	$-0.492487\pi$
$97$	0.464870	0.0472004	0.0236002	+	0.999721i	$0.492487\pi$
$98$	2.20800	0.223041
$99$	−4.47504	−0.449758
$100$	0	0
$101$	−17.0659	−1.69812	−0.849062	−	0.528294i	$-0.822832\pi$
$101$	−17.0659	−1.69812	−0.849062	+	0.528294i	$0.822832\pi$
$102$	5.13212	0.508156
$103$	−4.44067	−0.437552	−0.218776	−	0.975775i	$-0.570206\pi$
$103$	−4.44067	−0.437552	−0.218776	+	0.975775i	$0.570206\pi$
$104$	−1.93254	−0.189501
$105$	0	0
$106$	1.62224	0.157566
$107$	2.27546	0.219977	0.109988	−	0.993933i	$-0.464919\pi$
$107$	2.27546	0.219977	0.109988	+	0.993933i	$0.464919\pi$
$108$	2.87525	0.276671
$109$	3.93254	0.376669	0.188335	−	0.982105i	$-0.439691\pi$
$109$	3.93254	0.376669	0.188335	+	0.982105i	$0.439691\pi$
$110$	0	0
$111$	−2.63415	−0.250023
$112$	1.48345	0.140173
$113$	20.0214	1.88345	0.941727	−	0.336377i	$-0.109202\pi$
$113$	20.0214	1.88345	0.941727	+	0.336377i	$0.109202\pi$
$114$	−13.0990	−1.22684
$115$	0	0
$116$	−2.68128	−0.248951
$117$	−1.00000	−0.0924500
$118$	29.9715	2.75910
$119$	−2.32433	−0.213071
$120$	0	0
$121$	9.02595	0.820541
$122$	10.0804	0.912640
$123$	−6.50940	−0.586933
$124$	14.6609	1.31659
$125$	0	0
$126$	−2.20800	−0.196704
$127$	16.9170	1.50114	0.750569	−	0.660792i	$-0.229779\pi$
$127$	16.9170	1.50114	0.750569	+	0.660792i	$0.229779\pi$
$128$	−13.9799	−1.23566
$129$	−1.35028	−0.118886
$130$	0	0
$131$	−1.73471	−0.151562	−0.0757812	−	0.997124i	$-0.524145\pi$
$131$	−1.73471	−0.151562	−0.0757812	+	0.997124i	$0.524145\pi$
$132$	−12.8668	−1.11991
$133$	5.93254	0.514416
$134$	−29.1570	−2.51878
$135$	0	0
$136$	4.49187	0.385174
$137$	−3.94959	−0.337436	−0.168718	−	0.985664i	$-0.553963\pi$
$137$	−3.94959	−0.337436	−0.168718	+	0.985664i	$0.553963\pi$
$138$	−10.7421	−0.914426
$139$	5.22378	0.443075	0.221538	−	0.975152i	$-0.428892\pi$
$139$	5.22378	0.443075	0.221538	+	0.975152i	$0.428892\pi$
$140$	0	0
$141$	0.0259483	0.00218524
$142$	−14.6443	−1.22892
$143$	4.47504	0.374221
$144$	−1.48345	−0.123621
$145$	0	0
$146$	−31.4060	−2.59918
$147$	1.00000	0.0824786
$148$	−7.57384	−0.622566
$149$	7.60541	0.623059	0.311530	−	0.950236i	$-0.399159\pi$
$149$	7.60541	0.623059	0.311530	+	0.950236i	$0.399159\pi$
$150$	0	0
$151$	−1.43331	−0.116641	−0.0583204	−	0.998298i	$-0.518574\pi$
$151$	−1.43331	−0.116641	−0.0583204	+	0.998298i	$0.518574\pi$
$152$	−11.4649	−0.929924
$153$	2.32433	0.187911
$154$	9.88086	0.796223
$155$	0	0
$156$	−2.87525	−0.230204
$157$	−9.42616	−0.752289	−0.376145	−	0.926561i	$-0.622750\pi$
$157$	−9.42616	−0.752289	−0.376145	+	0.926561i	$0.622750\pi$
$158$	−1.36536	−0.108622
$159$	0.734710	0.0582663
$160$	0	0
$161$	4.86508	0.383422
$162$	2.20800	0.173477
$163$	14.6975	1.15120	0.575600	−	0.817731i	$-0.304768\pi$
$163$	14.6975	1.15120	0.575600	+	0.817731i	$0.304768\pi$
$164$	−18.7161	−1.46148
$165$	0	0
$166$	4.65708	0.361460
$167$	−21.4409	−1.65915	−0.829573	−	0.558398i	$-0.811416\pi$
$167$	−21.4409	−1.65915	−0.829573	+	0.558398i	$0.811416\pi$
$168$	−1.93254	−0.149099
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.93254	−0.453673
$172$	−3.88239	−0.296030
$173$	3.47329	0.264069	0.132035	−	0.991245i	$-0.457849\pi$
$173$	3.47329	0.264069	0.132035	+	0.991245i	$0.457849\pi$
$174$	−2.05904	−0.156096
$175$	0	0
$176$	6.63850	0.500396
$177$	13.5741	1.02029
$178$	−23.8063	−1.78435
$179$	8.86635	0.662702	0.331351	−	0.943508i	$-0.392496\pi$
$179$	8.86635	0.662702	0.331351	+	0.943508i	$0.392496\pi$
$180$	0	0
$181$	15.2912	1.13659	0.568294	−	0.822825i	$-0.307603\pi$
$181$	15.2912	1.13659	0.568294	+	0.822825i	$0.307603\pi$
$182$	2.20800	0.163668
$183$	4.56543	0.337486
$184$	−9.40196	−0.693122
$185$	0	0
$186$	11.2586	0.825522
$187$	−10.4015	−0.760631
$188$	0.0746078	0.00544133
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−12.7994	−0.923715
$193$	−14.2846	−1.02823	−0.514113	−	0.857722i	$-0.671879\pi$
$193$	−14.2846	−1.02823	−0.514113	+	0.857722i	$0.671879\pi$
$194$	1.02643	0.0736934
$195$	0	0
$196$	2.87525	0.205375
$197$	−5.84650	−0.416546	−0.208273	−	0.978071i	$-0.566784\pi$
$197$	−5.84650	−0.416546	−0.208273	+	0.978071i	$0.566784\pi$
$198$	−9.88086	−0.702202
$199$	1.41726	0.100467	0.0502335	−	0.998738i	$-0.484003\pi$
$199$	1.41726	0.100467	0.0502335	+	0.998738i	$0.484003\pi$
$200$	0	0
$201$	−13.2052	−0.931423
$202$	−37.6815	−2.65126
$203$	0.932540	0.0654515
$204$	6.68303	0.467906
$205$	0	0
$206$	−9.80499	−0.683146
$207$	−4.86508	−0.338146
$208$	1.48345	0.102859
$209$	26.5483	1.83639
$210$	0	0
$211$	−23.0629	−1.58772	−0.793858	−	0.608103i	$-0.791931\pi$
$211$	−23.0629	−1.58772	−0.793858	+	0.608103i	$0.791931\pi$
$212$	2.11247	0.145085
$213$	−6.63240	−0.454445
$214$	5.02420	0.343447
$215$	0	0
$216$	1.93254	0.131493
$217$	−5.09902	−0.346144
$218$	8.68303	0.588089
$219$	−14.2238	−0.961154
$220$	0	0
$221$	−2.32433	−0.156352
$222$	−5.81620	−0.390358
$223$	22.5375	1.50922	0.754610	−	0.656173i	$-0.227826\pi$
$223$	22.5375	1.50922	0.754610	+	0.656173i	$0.227826\pi$
$224$	7.14054	0.477097
$225$	0	0
$226$	44.2072	2.94062
$227$	14.6568	0.972807	0.486404	−	0.873734i	$-0.338308\pi$
$227$	14.6568	0.972807	0.486404	+	0.873734i	$0.338308\pi$
$228$	−17.0575	−1.12966
$229$	0.814934	0.0538523	0.0269262	−	0.999637i	$-0.491428\pi$
$229$	0.814934	0.0538523	0.0269262	+	0.999637i	$0.491428\pi$
$230$	0	0
$231$	4.47504	0.294436
$232$	−1.80217	−0.118318
$233$	24.9241	1.63283	0.816417	−	0.577463i	$-0.195957\pi$
$233$	24.9241	1.63283	0.816417	+	0.577463i	$0.195957\pi$
$234$	−2.20800	−0.144341
$235$	0	0
$236$	39.0288	2.54056
$237$	−0.618372	−0.0401676
$238$	−5.13212	−0.332666
$239$	18.9213	1.22392	0.611959	−	0.790889i	$-0.290382\pi$
$239$	18.9213	1.22392	0.611959	+	0.790889i	$0.290382\pi$
$240$	0	0
$241$	−0.124754	−0.00803610	−0.00401805	−	0.999992i	$-0.501279\pi$
$241$	−0.124754	−0.00803610	−0.00401805	+	0.999992i	$0.501279\pi$
$242$	19.9293	1.28110
$243$	1.00000	0.0641500
$244$	13.1267	0.840352
$245$	0	0
$246$	−14.3727	−0.916372
$247$	5.93254	0.377478
$248$	9.85407	0.625734
$249$	2.10919	0.133665
$250$	0	0
$251$	−13.5985	−0.858331	−0.429166	−	0.903226i	$-0.641192\pi$
$251$	−13.5985	−0.858331	−0.429166	+	0.903226i	$0.641192\pi$
$252$	−2.87525	−0.181123
$253$	21.7714	1.36876
$254$	37.3526	2.34371
$255$	0	0
$256$	−5.26879	−0.329299
$257$	−17.5407	−1.09416	−0.547081	−	0.837080i	$-0.684261\pi$
$257$	−17.5407	−1.09416	−0.547081	+	0.837080i	$0.684261\pi$
$258$	−2.98142	−0.185615
$259$	2.63415	0.163678
$260$	0	0
$261$	−0.932540	−0.0577228
$262$	−3.83023	−0.236633
$263$	17.9055	1.10410	0.552051	−	0.833810i	$-0.313845\pi$
$263$	17.9055	1.10410	0.552051	+	0.833810i	$0.313845\pi$
$264$	−8.64819	−0.532259
$265$	0	0
$266$	13.0990	0.803153
$267$	−10.7818	−0.659838
$268$	−37.9682	−2.31928
$269$	30.7148	1.87272	0.936358	−	0.351048i	$-0.114174\pi$
$269$	30.7148	1.87272	0.936358	+	0.351048i	$0.114174\pi$
$270$	0	0
$271$	−20.1016	−1.22109	−0.610543	−	0.791983i	$-0.709049\pi$
$271$	−20.1016	−1.22109	−0.610543	+	0.791983i	$0.709049\pi$
$272$	−3.44804	−0.209068
$273$	1.00000	0.0605228
$274$	−8.72068	−0.526835
$275$	0	0
$276$	−13.9883	−0.841997
$277$	−27.6899	−1.66373	−0.831863	−	0.554981i	$-0.812725\pi$
$277$	−27.6899	−1.66373	−0.831863	+	0.554981i	$0.812725\pi$
$278$	11.5341	0.691768
$279$	5.09902	0.305271
$280$	0	0
$281$	−10.9929	−0.655779	−0.327889	−	0.944716i	$-0.606337\pi$
$281$	−10.9929	−0.655779	−0.327889	+	0.944716i	$0.606337\pi$
$282$	0.0572938	0.00341179
$283$	7.32687	0.435537	0.217769	−	0.976000i	$-0.430122\pi$
$283$	7.32687	0.435537	0.217769	+	0.976000i	$0.430122\pi$
$284$	−19.0698	−1.13158
$285$	0	0
$286$	9.88086	0.584268
$287$	6.50940	0.384238
$288$	−7.14054	−0.420760
$289$	−11.5975	−0.682204
$290$	0	0
$291$	0.464870	0.0272511
$292$	−40.8969	−2.39331
$293$	2.97427	0.173759	0.0868794	−	0.996219i	$-0.472311\pi$
$293$	2.97427	0.173759	0.0868794	+	0.996219i	$0.472311\pi$
$294$	2.20800	0.128773
$295$	0	0
$296$	−5.09061	−0.295886
$297$	−4.47504	−0.259668
$298$	16.7927	0.972775
$299$	4.86508	0.281355
$300$	0	0
$301$	1.35028	0.0778290
$302$	−3.16473	−0.182110
$303$	−17.0659	−0.980412
$304$	8.80064	0.504751
$305$	0	0
$306$	5.13212	0.293384
$307$	−20.6843	−1.18052	−0.590258	−	0.807215i	$-0.700974\pi$
$307$	−20.6843	−1.18052	−0.590258	+	0.807215i	$0.700974\pi$
$308$	12.8668	0.733156
$309$	−4.44067	−0.252621
$310$	0	0
$311$	−24.9399	−1.41421	−0.707106	−	0.707107i	$-0.750000\pi$
$311$	−24.9399	−1.41421	−0.707106	+	0.707107i	$0.750000\pi$
$312$	−1.93254	−0.109409
$313$	−13.1896	−0.745522	−0.372761	−	0.927927i	$-0.621589\pi$
$313$	−13.1896	−0.745522	−0.372761	+	0.927927i	$0.621589\pi$
$314$	−20.8129	−1.17454
$315$	0	0
$316$	−1.77797	−0.100019
$317$	6.24538	0.350775	0.175388	−	0.984499i	$-0.443882\pi$
$317$	6.24538	0.350775	0.175388	+	0.984499i	$0.443882\pi$
$318$	1.62224	0.0909706
$319$	4.17315	0.233652
$320$	0	0
$321$	2.27546	0.127004
$322$	10.7421	0.598633
$323$	−13.7892	−0.767252
$324$	2.87525	0.159736
$325$	0	0
$326$	32.4521	1.79736
$327$	3.93254	0.217470
$328$	−12.5797	−0.694596
$329$	−0.0259483	−0.00143058
$330$	0	0
$331$	−18.3322	−1.00763	−0.503814	−	0.863812i	$-0.668070\pi$
$331$	−18.3322	−1.00763	−0.503814	+	0.863812i	$0.668070\pi$
$332$	6.06444	0.332829
$333$	−2.63415	−0.144351
$334$	−47.3414	−2.59041
$335$	0	0
$336$	1.48345	0.0809289
$337$	22.4391	1.22234	0.611169	−	0.791500i	$-0.290700\pi$
$337$	22.4391	1.22234	0.611169	+	0.791500i	$0.290700\pi$
$338$	2.20800	0.120099
$339$	20.0214	1.08741
$340$	0	0
$341$	−22.8183	−1.23568
$342$	−13.0990	−0.708314
$343$	−1.00000	−0.0539949
$344$	−2.60947	−0.140693
$345$	0	0
$346$	7.66900	0.412288
$347$	−3.56410	−0.191331	−0.0956654	−	0.995414i	$-0.530498\pi$
$347$	−3.56410	−0.191331	−0.0956654	+	0.995414i	$0.530498\pi$
$348$	−2.68128	−0.143732
$349$	16.2281	0.868672	0.434336	−	0.900751i	$-0.356983\pi$
$349$	16.2281	0.868672	0.434336	+	0.900751i	$0.356983\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	31.9542	1.70316
$353$	12.4631	0.663345	0.331672	−	0.943395i	$-0.392387\pi$
$353$	12.4631	0.663345	0.331672	+	0.943395i	$0.392387\pi$
$354$	29.9715	1.59297
$355$	0	0
$356$	−31.0004	−1.64302
$357$	−2.32433	−0.123017
$358$	19.5769	1.03467
$359$	−32.8365	−1.73305	−0.866523	−	0.499136i	$-0.833651\pi$
$359$	−32.8365	−1.73305	−0.866523	+	0.499136i	$0.833651\pi$
$360$	0	0
$361$	16.1950	0.852370
$362$	33.7630	1.77454
$363$	9.02595	0.473739
$364$	2.87525	0.150704
$365$	0	0
$366$	10.0804	0.526913
$367$	−8.56494	−0.447086	−0.223543	−	0.974694i	$-0.571762\pi$
$367$	−8.56494	−0.447086	−0.223543	+	0.974694i	$0.571762\pi$
$368$	7.21711	0.376218
$369$	−6.50940	−0.338866
$370$	0	0
$371$	−0.734710	−0.0381443
$372$	14.6609	0.760135
$373$	12.9626	0.671177	0.335588	−	0.942009i	$-0.391065\pi$
$373$	12.9626	0.671177	0.335588	+	0.942009i	$0.391065\pi$
$374$	−22.9664	−1.18757
$375$	0	0
$376$	0.0501461	0.00258609
$377$	0.932540	0.0480282
$378$	−2.20800	−0.113567
$379$	−9.69636	−0.498069	−0.249034	−	0.968495i	$-0.580113\pi$
$379$	−9.69636	−0.498069	−0.249034	+	0.968495i	$0.580113\pi$
$380$	0	0
$381$	16.9170	0.866683
$382$	−6.62399	−0.338913
$383$	11.0048	0.562317	0.281159	−	0.959661i	$-0.409281\pi$
$383$	11.0048	0.562317	0.281159	+	0.959661i	$0.409281\pi$
$384$	−13.9799	−0.713408
$385$	0	0
$386$	−31.5403	−1.60536
$387$	−1.35028	−0.0686387
$388$	1.33661	0.0678563
$389$	24.5226	1.24335	0.621673	−	0.783277i	$-0.286453\pi$
$389$	24.5226	1.24335	0.621673	+	0.783277i	$0.286453\pi$
$390$	0	0
$391$	−11.3081	−0.571874
$392$	1.93254	0.0976080
$393$	−1.73471	−0.0875046
$394$	−12.9090	−0.650348
$395$	0	0
$396$	−12.8668	−0.646583
$397$	−19.9950	−1.00352	−0.501759	−	0.865007i	$-0.667314\pi$
$397$	−19.9950	−1.00352	−0.501759	+	0.865007i	$0.667314\pi$
$398$	3.12930	0.156858
$399$	5.93254	0.296998
$400$	0	0
$401$	29.4083	1.46858	0.734290	−	0.678836i	$-0.237515\pi$
$401$	29.4083	1.46858	0.734290	+	0.678836i	$0.237515\pi$
$402$	−29.1570	−1.45422
$403$	−5.09902	−0.254000
$404$	−49.0687	−2.44126
$405$	0	0
$406$	2.05904	0.102189
$407$	11.7879	0.584306
$408$	4.49187	0.222381
$409$	10.3062	0.509607	0.254804	−	0.966993i	$-0.417989\pi$
$409$	10.3062	0.509607	0.254804	+	0.966993i	$0.417989\pi$
$410$	0	0
$411$	−3.94959	−0.194819
$412$	−12.7680	−0.629035
$413$	−13.5741	−0.667936
$414$	−10.7421	−0.527944
$415$	0	0
$416$	7.14054	0.350094
$417$	5.22378	0.255810
$418$	58.6186	2.86713
$419$	−1.99698	−0.0975589	−0.0487795	−	0.998810i	$-0.515533\pi$
$419$	−1.99698	−0.0975589	−0.0487795	+	0.998810i	$0.515533\pi$
$420$	0	0
$421$	32.1241	1.56563	0.782817	−	0.622251i	$-0.213782\pi$
$421$	32.1241	1.56563	0.782817	+	0.622251i	$0.213782\pi$
$422$	−50.9228	−2.47888
$423$	0.0259483	0.00126165
$424$	1.41986	0.0689543
$425$	0	0
$426$	−14.6443	−0.709520
$427$	−4.56543	−0.220936
$428$	6.54250	0.316243
$429$	4.47504	0.216057
$430$	0	0
$431$	0.0501461	0.00241545	0.00120773	−	0.999999i	$-0.499616\pi$
$431$	0.0501461	0.00241545	0.00120773	+	0.999999i	$0.499616\pi$
$432$	−1.48345	−0.0713726
$433$	−1.14579	−0.0550630	−0.0275315	−	0.999621i	$-0.508765\pi$
$433$	−1.14579	−0.0550630	−0.0275315	+	0.999621i	$0.508765\pi$
$434$	−11.2586	−0.540431
$435$	0	0
$436$	11.3070	0.541508
$437$	28.8623	1.38067
$438$	−31.4060	−1.50064
$439$	−20.0300	−0.955981	−0.477991	−	0.878365i	$-0.658635\pi$
$439$	−20.0300	−0.955981	−0.477991	+	0.878365i	$0.658635\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−5.13212	−0.244110
$443$	−35.5998	−1.69140	−0.845699	−	0.533660i	$-0.820816\pi$
$443$	−35.5998	−1.69140	−0.845699	+	0.533660i	$0.820816\pi$
$444$	−7.57384	−0.359439
$445$	0	0
$446$	49.7626	2.35633
$447$	7.60541	0.359723
$448$	12.7994	0.604713
$449$	24.0649	1.13569	0.567846	−	0.823135i	$-0.307777\pi$
$449$	24.0649	1.13569	0.567846	+	0.823135i	$0.307777\pi$
$450$	0	0
$451$	29.1298	1.37167
$452$	57.5664	2.70770
$453$	−1.43331	−0.0673426
$454$	32.3622	1.51883
$455$	0	0
$456$	−11.4649	−0.536892
$457$	19.8273	0.927481	0.463741	−	0.885971i	$-0.346507\pi$
$457$	19.8273	0.927481	0.463741	+	0.885971i	$0.346507\pi$
$458$	1.79937	0.0840790
$459$	2.32433	0.108491
$460$	0	0
$461$	1.92062	0.0894523	0.0447262	−	0.998999i	$-0.485758\pi$
$461$	1.92062	0.0894523	0.0447262	+	0.998999i	$0.485758\pi$
$462$	9.88086	0.459699
$463$	2.24109	0.104152	0.0520762	−	0.998643i	$-0.483416\pi$
$463$	2.24109	0.104152	0.0520762	+	0.998643i	$0.483416\pi$
$464$	1.38338	0.0642217
$465$	0	0
$466$	55.0324	2.54933
$467$	26.6543	1.23341	0.616707	−	0.787193i	$-0.288467\pi$
$467$	26.6543	1.23341	0.616707	+	0.787193i	$0.288467\pi$
$468$	−2.87525	−0.132908
$469$	13.2052	0.609759
$470$	0	0
$471$	−9.42616	−0.434334
$472$	26.2324	1.20744
$473$	6.04256	0.277837
$474$	−1.36536	−0.0627132
$475$	0	0
$476$	−6.68303	−0.306316
$477$	0.734710	0.0336401
$478$	41.7782	1.91089
$479$	−38.5562	−1.76168	−0.880838	−	0.473417i	$-0.843020\pi$
$479$	−38.5562	−1.76168	−0.880838	+	0.473417i	$0.843020\pi$
$480$	0	0
$481$	2.63415	0.120107
$482$	−0.275456	−0.0125467
$483$	4.86508	0.221369
$484$	25.9518	1.17963
$485$	0	0
$486$	2.20800	0.100157
$487$	−26.9961	−1.22331	−0.611656	−	0.791124i	$-0.709496\pi$
$487$	−26.9961	−1.22331	−0.611656	+	0.791124i	$0.709496\pi$
$488$	8.82287	0.399392
$489$	14.6975	0.664646
$490$	0	0
$491$	−15.0002	−0.676950	−0.338475	−	0.940975i	$-0.609911\pi$
$491$	−15.0002	−0.676950	−0.338475	+	0.940975i	$0.609911\pi$
$492$	−18.7161	−0.843788
$493$	−2.16753	−0.0976208
$494$	13.0990	0.589353
$495$	0	0
$496$	−7.56416	−0.339641
$497$	6.63240	0.297504
$498$	4.65708	0.208689
$499$	6.47868	0.290026	0.145013	−	0.989430i	$-0.453678\pi$
$499$	6.47868	0.290026	0.145013	+	0.989430i	$0.453678\pi$
$500$	0	0
$501$	−21.4409	−0.957908
$502$	−30.0255	−1.34010
$503$	−27.9939	−1.24819	−0.624094	−	0.781350i	$-0.714532\pi$
$503$	−27.9939	−1.24819	−0.624094	+	0.781350i	$0.714532\pi$
$504$	−1.93254	−0.0860822
$505$	0	0
$506$	48.0712	2.13702
$507$	1.00000	0.0444116
$508$	48.6405	2.15807
$509$	−29.3427	−1.30059	−0.650296	−	0.759681i	$-0.725355\pi$
$509$	−29.3427	−1.30059	−0.650296	+	0.759681i	$0.725355\pi$
$510$	0	0
$511$	14.2238	0.629223
$512$	16.3263	0.721527
$513$	−5.93254	−0.261928
$514$	−38.7299	−1.70830
$515$	0	0
$516$	−3.88239	−0.170913
$517$	−0.116120	−0.00510693
$518$	5.81620	0.255549
$519$	3.47329	0.152460
$520$	0	0
$521$	−18.6876	−0.818719	−0.409359	−	0.912373i	$-0.634248\pi$
$521$	−18.6876	−0.818719	−0.409359	+	0.912373i	$0.634248\pi$
$522$	−2.05904	−0.0901219
$523$	28.2173	1.23386	0.616929	−	0.787019i	$-0.288377\pi$
$523$	28.2173	1.23386	0.616929	+	0.787019i	$0.288377\pi$
$524$	−4.98772	−0.217890
$525$	0	0
$526$	39.5354	1.72382
$527$	11.8518	0.516274
$528$	6.63850	0.288904
$529$	0.669001	0.0290870
$530$	0	0
$531$	13.5741	0.589064
$532$	17.0575	0.739537
$533$	6.50940	0.281953
$534$	−23.8063	−1.03020
$535$	0	0
$536$	−25.5196	−1.10228
$537$	8.86635	0.382611
$538$	67.8182	2.92385
$539$	−4.47504	−0.192753
$540$	0	0
$541$	−28.2108	−1.21288	−0.606439	−	0.795130i	$-0.707402\pi$
$541$	−28.2108	−1.21288	−0.606439	+	0.795130i	$0.707402\pi$
$542$	−44.3843	−1.90647
$543$	15.2912	0.656210
$544$	−16.5970	−0.711590
$545$	0	0
$546$	2.20800	0.0944935
$547$	−13.9735	−0.597463	−0.298732	−	0.954337i	$-0.596564\pi$
$547$	−13.9735	−0.597463	−0.298732	+	0.954337i	$0.596564\pi$
$548$	−11.3560	−0.485106
$549$	4.56543	0.194848
$550$	0	0
$551$	5.53233	0.235685
$552$	−9.40196	−0.400174
$553$	0.618372	0.0262959
$554$	−61.1392	−2.59756
$555$	0	0
$556$	15.0196	0.636975
$557$	12.0510	0.510617	0.255308	−	0.966860i	$-0.417823\pi$
$557$	12.0510	0.510617	0.255308	+	0.966860i	$0.417823\pi$
$558$	11.2586	0.476616
$559$	1.35028	0.0571109
$560$	0	0
$561$	−10.4015	−0.439151
$562$	−24.2722	−1.02386
$563$	25.2839	1.06559	0.532794	−	0.846245i	$-0.321142\pi$
$563$	25.2839	1.06559	0.532794	+	0.846245i	$0.321142\pi$
$564$	0.0746078	0.00314155
$565$	0	0
$566$	16.1777	0.679999
$567$	−1.00000	−0.0419961
$568$	−12.8174	−0.537806
$569$	−16.4488	−0.689571	−0.344785	−	0.938682i	$-0.612048\pi$
$569$	−16.4488	−0.689571	−0.344785	+	0.938682i	$0.612048\pi$
$570$	0	0
$571$	−34.6755	−1.45113	−0.725563	−	0.688156i	$-0.758421\pi$
$571$	−34.6755	−1.45113	−0.725563	+	0.688156i	$0.758421\pi$
$572$	12.8668	0.537989
$573$	−3.00000	−0.125327
$574$	14.3727	0.599906
$575$	0	0
$576$	−12.7994	−0.533307
$577$	38.8294	1.61649	0.808245	−	0.588847i	$-0.200418\pi$
$577$	38.8294	1.61649	0.808245	+	0.588847i	$0.200418\pi$
$578$	−25.6072	−1.06512
$579$	−14.2846	−0.593647
$580$	0	0
$581$	−2.10919	−0.0875040
$582$	1.02643	0.0425469
$583$	−3.28786	−0.136169
$584$	−27.4880	−1.13746
$585$	0	0
$586$	6.56718	0.271288
$587$	−7.64762	−0.315651	−0.157825	−	0.987467i	$-0.550448\pi$
$587$	−7.64762	−0.315651	−0.157825	+	0.987467i	$0.550448\pi$
$588$	2.87525	0.118573
$589$	−30.2502	−1.24644
$590$	0	0
$591$	−5.84650	−0.240493
$592$	3.90764	0.160603
$593$	−16.7025	−0.685891	−0.342945	−	0.939355i	$-0.611425\pi$
$593$	−16.7025	−0.685891	−0.342945	+	0.939355i	$0.611425\pi$
$594$	−9.88086	−0.405417
$595$	0	0
$596$	21.8674	0.895724
$597$	1.41726	0.0580046
$598$	10.7421	0.439276
$599$	−4.20625	−0.171863	−0.0859313	−	0.996301i	$-0.527387\pi$
$599$	−4.20625	−0.171863	−0.0859313	+	0.996301i	$0.527387\pi$
$600$	0	0
$601$	−15.1025	−0.616045	−0.308022	−	0.951379i	$-0.599667\pi$
$601$	−15.1025	−0.616045	−0.308022	+	0.951379i	$0.599667\pi$
$602$	2.98142	0.121514
$603$	−13.2052	−0.537757
$604$	−4.12111	−0.167685
$605$	0	0
$606$	−37.6815	−1.53071
$607$	27.7546	1.12652	0.563262	−	0.826279i	$-0.309546\pi$
$607$	27.7546	1.12652	0.563262	+	0.826279i	$0.309546\pi$
$608$	42.3615	1.71799
$609$	0.932540	0.0377884
$610$	0	0
$611$	−0.0259483	−0.00104976
$612$	6.68303	0.270146
$613$	−8.43029	−0.340496	−0.170248	−	0.985401i	$-0.554457\pi$
$613$	−8.43029	−0.340496	−0.170248	+	0.985401i	$0.554457\pi$
$614$	−45.6709	−1.84313
$615$	0	0
$616$	8.64819	0.348445
$617$	17.7233	0.713512	0.356756	−	0.934198i	$-0.383883\pi$
$617$	17.7233	0.713512	0.356756	+	0.934198i	$0.383883\pi$
$618$	−9.80499	−0.394414
$619$	−39.1390	−1.57313	−0.786565	−	0.617508i	$-0.788142\pi$
$619$	−39.1390	−1.57313	−0.786565	+	0.617508i	$0.788142\pi$
$620$	0	0
$621$	−4.86508	−0.195229
$622$	−55.0672	−2.20799
$623$	10.7818	0.431965
$624$	1.48345	0.0593856
$625$	0	0
$626$	−29.1227	−1.16398
$627$	26.5483	1.06024
$628$	−27.1025	−1.08151
$629$	−6.12265	−0.244126
$630$	0	0
$631$	13.1016	0.521565	0.260782	−	0.965398i	$-0.416019\pi$
$631$	13.1016	0.521565	0.260782	+	0.965398i	$0.416019\pi$
$632$	−1.19503	−0.0475357
$633$	−23.0629	−0.916668
$634$	13.7898	0.547662
$635$	0	0
$636$	2.11247	0.0837650
$637$	−1.00000	−0.0396214
$638$	9.21430	0.364798
$639$	−6.63240	−0.262374
$640$	0	0
$641$	27.2594	1.07668	0.538341	−	0.842727i	$-0.319051\pi$
$641$	27.2594	1.07668	0.538341	+	0.842727i	$0.319051\pi$
$642$	5.02420	0.198289
$643$	−5.59789	−0.220759	−0.110380	−	0.993889i	$-0.535207\pi$
$643$	−5.59789	−0.220759	−0.110380	+	0.993889i	$0.535207\pi$
$644$	13.9883	0.551216
$645$	0	0
$646$	−30.4465	−1.19790
$647$	0.537931	0.0211482	0.0105741	−	0.999944i	$-0.496634\pi$
$647$	0.537931	0.0211482	0.0105741	+	0.999944i	$0.496634\pi$
$648$	1.93254	0.0759173
$649$	−60.7444	−2.38443
$650$	0	0
$651$	−5.09902	−0.199847
$652$	42.2591	1.65499
$653$	27.5821	1.07937	0.539686	−	0.841866i	$-0.318543\pi$
$653$	27.5821	1.07937	0.539686	+	0.841866i	$0.318543\pi$
$654$	8.68303	0.339533
$655$	0	0
$656$	9.65638	0.377018
$657$	−14.2238	−0.554922
$658$	−0.0572938	−0.00223354
$659$	−15.4625	−0.602333	−0.301167	−	0.953572i	$-0.597376\pi$
$659$	−15.4625	−0.602333	−0.301167	+	0.953572i	$0.597376\pi$
$660$	0	0
$661$	27.1392	1.05559	0.527797	−	0.849371i	$-0.323018\pi$
$661$	27.1392	1.05559	0.527797	+	0.849371i	$0.323018\pi$
$662$	−40.4774	−1.57320
$663$	−2.32433	−0.0902696
$664$	4.07609	0.158183
$665$	0	0
$666$	−5.81620	−0.225373
$667$	4.53688	0.175669
$668$	−61.6478	−2.38523
$669$	22.5375	0.871349
$670$	0	0
$671$	−20.4304	−0.788709
$672$	7.14054	0.275452
$673$	44.3959	1.71134	0.855668	−	0.517526i	$-0.173147\pi$
$673$	44.3959	1.71134	0.855668	+	0.517526i	$0.173147\pi$
$674$	49.5455	1.90842
$675$	0	0
$676$	2.87525	0.110586
$677$	9.24999	0.355506	0.177753	−	0.984075i	$-0.443117\pi$
$677$	9.24999	0.355506	0.177753	+	0.984075i	$0.443117\pi$
$678$	44.2072	1.69777
$679$	−0.464870	−0.0178401
$680$	0	0
$681$	14.6568	0.561651
$682$	−50.3828	−1.92926
$683$	−0.278739	−0.0106656	−0.00533282	−	0.999986i	$-0.501697\pi$
$683$	−0.278739	−0.0106656	−0.00533282	+	0.999986i	$0.501697\pi$
$684$	−17.0575	−0.652210
$685$	0	0
$686$	−2.20800	−0.0843017
$687$	0.814934	0.0310917
$688$	2.00308	0.0763667
$689$	−0.734710	−0.0279902
$690$	0	0
$691$	−46.9622	−1.78653	−0.893264	−	0.449533i	$-0.851590\pi$
$691$	−46.9622	−1.78653	−0.893264	+	0.449533i	$0.851590\pi$
$692$	9.98655	0.379632
$693$	4.47504	0.169993
$694$	−7.86951	−0.298723
$695$	0	0
$696$	−1.80217	−0.0683111
$697$	−15.1300	−0.573090
$698$	35.8316	1.35625
$699$	24.9241	0.942717
$700$	0	0
$701$	−25.6464	−0.968652	−0.484326	−	0.874887i	$-0.660935\pi$
$701$	−25.6464	−0.968652	−0.484326	+	0.874887i	$0.660935\pi$
$702$	−2.20800	−0.0833354
$703$	15.6272	0.589392
$704$	57.2776	2.15873
$705$	0	0
$706$	27.5185	1.03567
$707$	17.0659	0.641830
$708$	39.0288	1.46679
$709$	24.4080	0.916662	0.458331	−	0.888782i	$-0.348447\pi$
$709$	24.4080	0.916662	0.458331	+	0.888782i	$0.348447\pi$
$710$	0	0
$711$	−0.618372	−0.0231908
$712$	−20.8363	−0.780875
$713$	−24.8072	−0.929035
$714$	−5.13212	−0.192065
$715$	0	0
$716$	25.4929	0.952716
$717$	18.9213	0.706630
$718$	−72.5030	−2.70579
$719$	−10.8760	−0.405608	−0.202804	−	0.979219i	$-0.565005\pi$
$719$	−10.8760	−0.405608	−0.202804	+	0.979219i	$0.565005\pi$
$720$	0	0
$721$	4.44067	0.165379
$722$	35.7586	1.33080
$723$	−0.124754	−0.00463965
$724$	43.9661	1.63399
$725$	0	0
$726$	19.9293	0.739644
$727$	−4.82399	−0.178912	−0.0894559	−	0.995991i	$-0.528513\pi$
$727$	−4.82399	−0.178912	−0.0894559	+	0.995991i	$0.528513\pi$
$728$	1.93254	0.0716247
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.13851	−0.116082
$732$	13.1267	0.485178
$733$	−21.6571	−0.799923	−0.399961	−	0.916532i	$-0.630976\pi$
$733$	−21.6571	−0.799923	−0.399961	+	0.916532i	$0.630976\pi$
$734$	−18.9114	−0.698031
$735$	0	0
$736$	34.7393	1.28051
$737$	59.0937	2.17675
$738$	−14.3727	−0.529067
$739$	−27.9263	−1.02729	−0.513643	−	0.858004i	$-0.671704\pi$
$739$	−27.9263	−1.02729	−0.513643	+	0.858004i	$0.671704\pi$
$740$	0	0
$741$	5.93254	0.217937
$742$	−1.62224	−0.0595542
$743$	42.0836	1.54390	0.771949	−	0.635685i	$-0.219282\pi$
$743$	42.0836	1.54390	0.771949	+	0.635685i	$0.219282\pi$
$744$	9.85407	0.361268
$745$	0	0
$746$	28.6213	1.04790
$747$	2.10919	0.0771712
$748$	−29.9068	−1.09350
$749$	−2.27546	−0.0831434
$750$	0	0
$751$	−3.24214	−0.118307	−0.0591537	−	0.998249i	$-0.518840\pi$
$751$	−3.24214	−0.118307	−0.0591537	+	0.998249i	$0.518840\pi$
$752$	−0.0384931	−0.00140370
$753$	−13.5985	−0.495558
$754$	2.05904	0.0749860
$755$	0	0
$756$	−2.87525	−0.104572
$757$	32.1576	1.16879	0.584394	−	0.811470i	$-0.301332\pi$
$757$	32.1576	1.16879	0.584394	+	0.811470i	$0.301332\pi$
$758$	−21.4095	−0.777629
$759$	21.7714	0.790252
$760$	0	0
$761$	−42.5786	−1.54347	−0.771737	−	0.635942i	$-0.780612\pi$
$761$	−42.5786	−1.54347	−0.771737	+	0.635942i	$0.780612\pi$
$762$	37.3526	1.35314
$763$	−3.93254	−0.142368
$764$	−8.62574	−0.312068
$765$	0	0
$766$	24.2985	0.877940
$767$	−13.5741	−0.490131
$768$	−5.26879	−0.190121
$769$	6.99263	0.252161	0.126080	−	0.992020i	$-0.459760\pi$
$769$	6.99263	0.252161	0.126080	+	0.992020i	$0.459760\pi$
$770$	0	0
$771$	−17.5407	−0.631715
$772$	−41.0717	−1.47820
$773$	−30.6787	−1.10344	−0.551718	−	0.834031i	$-0.686028\pi$
$773$	−30.6787	−1.10344	−0.551718	+	0.834031i	$0.686028\pi$
$774$	−2.98142	−0.107165
$775$	0	0
$776$	0.898379	0.0322499
$777$	2.63415	0.0944998
$778$	54.1458	1.94122
$779$	38.6173	1.38361
$780$	0	0
$781$	29.6802	1.06204
$782$	−24.9682	−0.892860
$783$	−0.932540	−0.0333263
$784$	−1.48345	−0.0529804
$785$	0	0
$786$	−3.83023	−0.136620
$787$	−42.3381	−1.50919	−0.754595	−	0.656191i	$-0.772167\pi$
$787$	−42.3381	−1.50919	−0.754595	+	0.656191i	$0.772167\pi$
$788$	−16.8101	−0.598836
$789$	17.9055	0.637454
$790$	0	0
$791$	−20.0214	−0.711879
$792$	−8.64819	−0.307300
$793$	−4.56543	−0.162123
$794$	−44.1488	−1.56678
$795$	0	0
$796$	4.07497	0.144434
$797$	−19.8744	−0.703988	−0.351994	−	0.936002i	$-0.614496\pi$
$797$	−19.8744	−0.703988	−0.351994	+	0.936002i	$0.614496\pi$
$798$	13.0990	0.463700
$799$	0.0603125	0.00213370
$800$	0	0
$801$	−10.7818	−0.380958
$802$	64.9334	2.29288
$803$	63.6519	2.24623
$804$	−37.9682	−1.33903
$805$	0	0
$806$	−11.2586	−0.396568
$807$	30.7148	1.08121
$808$	−32.9806	−1.16025
$809$	29.7135	1.04467	0.522336	−	0.852740i	$-0.325061\pi$
$809$	29.7135	1.04467	0.522336	+	0.852740i	$0.325061\pi$
$810$	0	0
$811$	−41.0407	−1.44113	−0.720567	−	0.693386i	$-0.756118\pi$
$811$	−41.0407	−1.44113	−0.720567	+	0.693386i	$0.756118\pi$
$812$	2.68128	0.0940945
$813$	−20.1016	−0.704995
$814$	26.0277	0.912271
$815$	0	0
$816$	−3.44804	−0.120705
$817$	8.01060	0.280256
$818$	22.7560	0.795645
$819$	1.00000	0.0349428
$820$	0	0
$821$	41.5476	1.45002	0.725011	−	0.688737i	$-0.241835\pi$
$821$	41.5476	1.45002	0.725011	+	0.688737i	$0.241835\pi$
$822$	−8.72068	−0.304169
$823$	39.1253	1.36382	0.681912	−	0.731434i	$-0.261149\pi$
$823$	39.1253	1.36382	0.681912	+	0.731434i	$0.261149\pi$
$824$	−8.58178	−0.298960
$825$	0	0
$826$	−29.9715	−1.04284
$827$	11.3450	0.394505	0.197253	−	0.980353i	$-0.436798\pi$
$827$	11.3450	0.394505	0.197253	+	0.980353i	$0.436798\pi$
$828$	−13.9883	−0.486127
$829$	9.29054	0.322674	0.161337	−	0.986899i	$-0.448419\pi$
$829$	9.29054	0.322674	0.161337	+	0.986899i	$0.448419\pi$
$830$	0	0
$831$	−27.6899	−0.960553
$832$	12.7994	0.443738
$833$	2.32433	0.0805334
$834$	11.5341	0.399393
$835$	0	0
$836$	76.3330	2.64003
$837$	5.09902	0.176248
$838$	−4.40933	−0.152318
$839$	−3.90963	−0.134975	−0.0674876	−	0.997720i	$-0.521498\pi$
$839$	−3.90963	−0.134975	−0.0674876	+	0.997720i	$0.521498\pi$
$840$	0	0
$841$	−28.1304	−0.970013
$842$	70.9300	2.44441
$843$	−10.9929	−0.378614
$844$	−66.3115	−2.28254
$845$	0	0
$846$	0.0572938	0.00196980
$847$	−9.02595	−0.310135
$848$	−1.08991	−0.0374276
$849$	7.32687	0.251458
$850$	0	0
$851$	12.8154	0.439305
$852$	−19.0698	−0.653320
$853$	−38.9862	−1.33486	−0.667431	−	0.744672i	$-0.732606\pi$
$853$	−38.9862	−1.33486	−0.667431	+	0.744672i	$0.732606\pi$
$854$	−10.0804	−0.344946
$855$	0	0
$856$	4.39741	0.150300
$857$	27.4938	0.939172	0.469586	−	0.882887i	$-0.344403\pi$
$857$	27.4938	0.939172	0.469586	+	0.882887i	$0.344403\pi$
$858$	9.88086	0.337327
$859$	8.76978	0.299221	0.149610	−	0.988745i	$-0.452198\pi$
$859$	8.76978	0.299221	0.149610	+	0.988745i	$0.452198\pi$
$860$	0	0
$861$	6.50940	0.221840
$862$	0.110722	0.00377122
$863$	−33.7732	−1.14965	−0.574826	−	0.818276i	$-0.694930\pi$
$863$	−33.7732	−1.14965	−0.574826	+	0.818276i	$0.694930\pi$
$864$	−7.14054	−0.242926
$865$	0	0
$866$	−2.52989	−0.0859693
$867$	−11.5975	−0.393871
$868$	−14.6609	−0.497625
$869$	2.76724	0.0938721
$870$	0	0
$871$	13.2052	0.447441
$872$	7.59979	0.257361
$873$	0.464870	0.0157335
$874$	63.7278	2.15562
$875$	0	0
$876$	−40.8969	−1.38178
$877$	53.2805	1.79915	0.899577	−	0.436763i	$-0.143875\pi$
$877$	53.2805	1.79915	0.899577	+	0.436763i	$0.143875\pi$
$878$	−44.2262	−1.49256
$879$	2.97427	0.100320
$880$	0	0
$881$	−4.18625	−0.141038	−0.0705192	−	0.997510i	$-0.522466\pi$
$881$	−4.18625	−0.141038	−0.0705192	+	0.997510i	$0.522466\pi$
$882$	2.20800	0.0743471
$883$	−21.3360	−0.718013	−0.359007	−	0.933335i	$-0.616884\pi$
$883$	−21.3360	−0.718013	−0.359007	+	0.933335i	$0.616884\pi$
$884$	−6.68303	−0.224775
$885$	0	0
$886$	−78.6042	−2.64076
$887$	−30.1961	−1.01389	−0.506943	−	0.861980i	$-0.669224\pi$
$887$	−30.1961	−1.01389	−0.506943	+	0.861980i	$0.669224\pi$
$888$	−5.09061	−0.170830
$889$	−16.9170	−0.567377
$890$	0	0
$891$	−4.47504	−0.149919
$892$	64.8008	2.16969
$893$	−0.153939	−0.00515138
$894$	16.7927	0.561632
$895$	0	0
$896$	13.9799	0.467035
$897$	4.86508	0.162440
$898$	53.1352	1.77314
$899$	−4.75504	−0.158590
$900$	0	0
$901$	1.70771	0.0568921
$902$	64.3185	2.14157
$903$	1.35028	0.0449346
$904$	38.6921	1.28688
$905$	0	0
$906$	−3.16473	−0.105141
$907$	19.5703	0.649822	0.324911	−	0.945745i	$-0.394666\pi$
$907$	19.5703	0.649822	0.324911	+	0.945745i	$0.394666\pi$
$908$	42.1420	1.39853
$909$	−17.0659	−0.566041
$910$	0	0
$911$	−33.2612	−1.10199	−0.550997	−	0.834507i	$-0.685752\pi$
$911$	−33.2612	−1.10199	−0.550997	+	0.834507i	$0.685752\pi$
$912$	8.80064	0.291418
$913$	−9.43870	−0.312375
$914$	43.7786	1.44807
$915$	0	0
$916$	2.34313	0.0774193
$917$	1.73471	0.0572852
$918$	5.13212	0.169385
$919$	−14.5704	−0.480634	−0.240317	−	0.970695i	$-0.577251\pi$
$919$	−14.5704	−0.480634	−0.240317	+	0.970695i	$0.577251\pi$
$920$	0	0
$921$	−20.6843	−0.681571
$922$	4.24073	0.139661
$923$	6.63240	0.218308
$924$	12.8668	0.423288
$925$	0	0
$926$	4.94832	0.162612
$927$	−4.44067	−0.145851
$928$	6.65883	0.218587
$929$	−50.6493	−1.66175	−0.830875	−	0.556459i	$-0.812160\pi$
$929$	−50.6493	−1.66175	−0.830875	+	0.556459i	$0.812160\pi$
$930$	0	0
$931$	−5.93254	−0.194431
$932$	71.6630	2.34740
$933$	−24.9399	−0.816496
$934$	58.8525	1.92571
$935$	0	0
$936$	−1.93254	−0.0631670
$937$	11.9733	0.391150	0.195575	−	0.980689i	$-0.437343\pi$
$937$	11.9733	0.391150	0.195575	+	0.980689i	$0.437343\pi$
$938$	29.1570	0.952010
$939$	−13.1896	−0.430427
$940$	0	0
$941$	5.56823	0.181519	0.0907595	−	0.995873i	$-0.471071\pi$
$941$	5.56823	0.181519	0.0907595	+	0.995873i	$0.471071\pi$
$942$	−20.8129	−0.678122
$943$	31.6688	1.03128
$944$	−20.1365	−0.655386
$945$	0	0
$946$	13.3420	0.433784
$947$	55.2586	1.79566	0.897832	−	0.440338i	$-0.145141\pi$
$947$	55.2586	1.79566	0.897832	+	0.440338i	$0.145141\pi$
$948$	−1.77797	−0.0577459
$949$	14.2238	0.461723
$950$	0	0
$951$	6.24538	0.202520
$952$	−4.49187	−0.145582
$953$	−46.0046	−1.49023	−0.745117	−	0.666934i	$-0.767606\pi$
$953$	−46.0046	−1.49023	−0.745117	+	0.666934i	$0.767606\pi$
$954$	1.62224	0.0525219
$955$	0	0
$956$	54.4035	1.75953
$957$	4.17315	0.134899
$958$	−85.1319	−2.75049
$959$	3.94959	0.127539
$960$	0	0
$961$	−4.99995	−0.161289
$962$	5.81620	0.187522
$963$	2.27546	0.0733255
$964$	−0.358698	−0.0115529
$965$	0	0
$966$	10.7421	0.345621
$967$	−47.5913	−1.53043	−0.765217	−	0.643772i	$-0.777368\pi$
$967$	−47.5913	−1.53043	−0.765217	+	0.643772i	$0.777368\pi$
$968$	17.4430	0.560639
$969$	−13.7892	−0.442973
$970$	0	0
$971$	−32.4394	−1.04103	−0.520514	−	0.853853i	$-0.674260\pi$
$971$	−32.4394	−1.04103	−0.520514	+	0.853853i	$0.674260\pi$
$972$	2.87525	0.0922236
$973$	−5.22378	−0.167467
$974$	−59.6074	−1.90994
$975$	0	0
$976$	−6.77259	−0.216785
$977$	48.4164	1.54898	0.774489	−	0.632587i	$-0.218007\pi$
$977$	48.4164	1.54898	0.774489	+	0.632587i	$0.218007\pi$
$978$	32.4521	1.03770
$979$	48.2491	1.54205
$980$	0	0
$981$	3.93254	0.125556
$982$	−33.1204	−1.05692
$983$	1.54054	0.0491357	0.0245678	−	0.999698i	$-0.492179\pi$
$983$	1.54054	0.0491357	0.0245678	+	0.999698i	$0.492179\pi$
$984$	−12.5797	−0.401025
$985$	0	0
$986$	−4.78591	−0.152414
$987$	−0.0259483	−0.000825944 0
$988$	17.0575	0.542672
$989$	6.56923	0.208889
$990$	0	0
$991$	−46.5319	−1.47813	−0.739067	−	0.673632i	$-0.764733\pi$
$991$	−46.5319	−1.47813	−0.739067	+	0.673632i	$0.764733\pi$
$992$	−36.4098	−1.15601
$993$	−18.3322	−0.581754
$994$	14.6443	0.464490
$995$	0	0
$996$	6.06444	0.192159
$997$	23.8997	0.756910	0.378455	−	0.925620i	$-0.376455\pi$
$997$	23.8997	0.756910	0.378455	+	0.925620i	$0.376455\pi$
$998$	14.3049	0.452814
$999$	−2.63415	−0.0833410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bl.1.4	yes	4
5.4	even	2		6825.2.a.bf.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6825.2.a.bf.1.1	✓	4	5.4	even	2
6825.2.a.bl.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20800 q^{2} +1.00000 q^{3} +2.87525 q^{4} +2.20800 q^{6} -1.00000 q^{7} +1.93254 q^{8} +1.00000 q^{9} -4.47504 q^{11} +2.87525 q^{12} -1.00000 q^{13} -2.20800 q^{14} -1.48345 q^{16} +2.32433 q^{17} +2.20800 q^{18} -5.93254 q^{19} -1.00000 q^{21} -9.88086 q^{22} -4.86508 q^{23} +1.93254 q^{24} -2.20800 q^{26} +1.00000 q^{27} -2.87525 q^{28} -0.932540 q^{29} +5.09902 q^{31} -7.14054 q^{32} -4.47504 q^{33} +5.13212 q^{34} +2.87525 q^{36} -2.63415 q^{37} -13.0990 q^{38} -1.00000 q^{39} -6.50940 q^{41} -2.20800 q^{42} -1.35028 q^{43} -12.8668 q^{44} -10.7421 q^{46} +0.0259483 q^{47} -1.48345 q^{48} +1.00000 q^{49} +2.32433 q^{51} -2.87525 q^{52} +0.734710 q^{53} +2.20800 q^{54} -1.93254 q^{56} -5.93254 q^{57} -2.05904 q^{58} +13.5741 q^{59} +4.56543 q^{61} +11.2586 q^{62} -1.00000 q^{63} -12.7994 q^{64} -9.88086 q^{66} -13.2052 q^{67} +6.68303 q^{68} -4.86508 q^{69} -6.63240 q^{71} +1.93254 q^{72} -14.2238 q^{73} -5.81620 q^{74} -17.0575 q^{76} +4.47504 q^{77} -2.20800 q^{78} -0.618372 q^{79} +1.00000 q^{81} -14.3727 q^{82} +2.10919 q^{83} -2.87525 q^{84} -2.98142 q^{86} -0.932540 q^{87} -8.64819 q^{88} -10.7818 q^{89} +1.00000 q^{91} -13.9883 q^{92} +5.09902 q^{93} +0.0572938 q^{94} -7.14054 q^{96} +0.464870 q^{97} +2.20800 q^{98} -4.47504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 4 q^{3} + 5 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} + 5 q^{12} - 4 q^{13} - q^{14} - q^{16} - 3 q^{17} + q^{18} - 13 q^{19} - 4 q^{21} + 7 q^{22} + 2 q^{23} - 3 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q + q^2 + 4 * q^3 + 5 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 4 * q^9 - 2 * q^11 + 5 * q^12 - 4 * q^13 - q^14 - q^16 - 3 * q^17 + q^18 - 13 * q^19 - 4 * q^21 + 7 * q^22 + 2 * q^23 - 3 * q^24 - q^26 + 4 * q^27 - 5 * q^28 + 7 * q^29 - 19 * q^31 - 10 * q^32 - 2 * q^33 - 9 * q^34 + 5 * q^36 - 2 * q^37 - 13 * q^38 - 4 * q^39 - 11 * q^41 - q^42 + 17 * q^43 - 34 * q^44 - 19 * q^46 - 10 * q^47 - q^48 + 4 * q^49 - 3 * q^51 - 5 * q^52 + 15 * q^53 + q^54 + 3 * q^56 - 13 * q^57 - 8 * q^58 - q^59 + 8 * q^61 + 26 * q^62 - 4 * q^63 - 23 * q^64 + 7 * q^66 + 6 * q^67 + 3 * q^68 + 2 * q^69 - 14 * q^71 - 3 * q^72 - 24 * q^73 - 17 * q^74 - 8 * q^76 + 2 * q^77 - q^78 - 19 * q^79 + 4 * q^81 - 17 * q^82 - 16 * q^83 - 5 * q^84 + 14 * q^86 + 7 * q^87 + 3 * q^88 - 36 * q^89 + 4 * q^91 + 19 * q^92 - 19 * q^93 - 4 * q^94 - 10 * q^96 - 29 * q^97 + q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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