LMFDB - Embedded newform 6825.2.a.bi.1.2

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:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 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			$0.785261$ 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)

$f(q)$	$=$	\(q-0.785261 q^{2} +1.00000 q^{3} -1.38336 q^{4} -0.785261 q^{6} +1.00000 q^{7} +2.65683 q^{8} +1.00000 q^{9} +4.44209 q^{11} -1.38336 q^{12} +1.00000 q^{13} -0.785261 q^{14} +0.680426 q^{16} +3.93029 q^{17} -0.785261 q^{18} -3.89010 q^{19} +1.00000 q^{21} -3.48820 q^{22} -5.54692 q^{23} +2.65683 q^{24} -0.785261 q^{26} +1.00000 q^{27} -1.38336 q^{28} -5.44716 q^{29} -0.343174 q^{31} -5.84796 q^{32} +4.44209 q^{33} -3.08630 q^{34} -1.38336 q^{36} -10.9118 q^{37} +3.05474 q^{38} +1.00000 q^{39} -11.0753 q^{41} -0.785261 q^{42} -6.46460 q^{43} -6.14503 q^{44} +4.35578 q^{46} -7.61469 q^{47} +0.680426 q^{48} +1.00000 q^{49} +3.93029 q^{51} -1.38336 q^{52} -11.8019 q^{53} -0.785261 q^{54} +2.65683 q^{56} -3.89010 q^{57} +4.27744 q^{58} -7.16754 q^{59} -1.42550 q^{61} +0.269482 q^{62} +1.00000 q^{63} +3.23133 q^{64} -3.48820 q^{66} +14.8732 q^{67} -5.43702 q^{68} -5.54692 q^{69} +13.6745 q^{71} +2.65683 q^{72} +3.08521 q^{73} +8.56858 q^{74} +5.38142 q^{76} +4.44209 q^{77} -0.785261 q^{78} -4.82147 q^{79} +1.00000 q^{81} +8.69702 q^{82} +11.9694 q^{83} -1.38336 q^{84} +5.07640 q^{86} -5.44716 q^{87} +11.8019 q^{88} -15.5997 q^{89} +1.00000 q^{91} +7.67342 q^{92} -0.343174 q^{93} +5.97952 q^{94} -5.84796 q^{96} +2.43304 q^{97} -0.785261 q^{98} +4.44209 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 4 q^{3} + q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + 4 q^{9} + 2 q^{11} + q^{12} + 4 q^{13} - q^{14} - q^{16} - 5 q^{17} - q^{18} - 15 q^{19} + 4 q^{21} - 9 q^{22} - 8 q^{23} - 3 q^{24} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - q^2 + 4 * q^3 + q^4 - q^6 + 4 * q^7 - 3 * q^8 + 4 * q^9 + 2 * q^11 + q^12 + 4 * q^13 - q^14 - q^16 - 5 * q^17 - q^18 - 15 * q^19 + 4 * q^21 - 9 * q^22 - 8 * q^23 - 3 * q^24 - q^26 + 4 * q^27 + q^28 - 5 * q^29 - 15 * q^31 - 2 * q^32 + 2 * q^33 - 3 * q^34 + q^36 - 12 * q^37 + 13 * q^38 + 4 * q^39 - 5 * q^41 - q^42 - 11 * q^43 - 6 * q^44 + 11 * q^46 + 4 * q^47 - q^48 + 4 * q^49 - 5 * q^51 + q^52 - 15 * q^53 - q^54 - 3 * q^56 - 15 * q^57 - 11 * q^59 - 20 * q^61 + 4 * q^62 + 4 * q^63 - 15 * q^64 - 9 * q^66 - 2 * q^67 - 7 * q^68 - 8 * q^69 + 20 * q^71 - 3 * q^72 - 6 * q^73 - 11 * q^74 - 26 * q^76 + 2 * q^77 - q^78 - 7 * q^79 + 4 * q^81 + q^82 - 2 * q^83 + q^84 - 10 * q^86 - 5 * q^87 + 15 * q^88 - 8 * q^89 + 4 * q^91 - 13 * q^92 - 15 * q^93 - 28 * q^94 - 2 * q^96 + 5 * 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 $

$2$	−0.785261	−0.555264	−0.277632	−	0.960688i	$-0.589550\pi$
$2$	−0.785261	−0.555264	−0.277632	+	0.960688i	$0.589550\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.38336	−0.691682
$5$	0	0
$5$	0	0
$6$	−0.785261	−0.320582
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.65683	0.939330
$9$	1.00000	0.333333
$10$	0	0
$11$	4.44209	1.33934	0.669670	−	0.742659i	$-0.266436\pi$
$11$	4.44209	1.33934	0.669670	+	0.742659i	$0.266436\pi$
$12$	−1.38336	−0.399343
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.785261	−0.209870
$15$	0	0
$16$	0.680426	0.170107
$17$	3.93029	0.953235	0.476617	−	0.879111i	$-0.341863\pi$
$17$	3.93029	0.953235	0.476617	+	0.879111i	$0.341863\pi$
$18$	−0.785261	−0.185088
$19$	−3.89010	−0.892449	−0.446225	−	0.894921i	$-0.647232\pi$
$19$	−3.89010	−0.892449	−0.446225	+	0.894921i	$0.647232\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−3.48820	−0.743687
$23$	−5.54692	−1.15661	−0.578307	−	0.815819i	$-0.696286\pi$
$23$	−5.54692	−1.15661	−0.578307	+	0.815819i	$0.696286\pi$
$24$	2.65683	0.542322
$25$	0	0
$26$	−0.785261	−0.154002
$27$	1.00000	0.192450
$28$	−1.38336	−0.261431
$29$	−5.44716	−1.01151	−0.505756	−	0.862677i	$-0.668786\pi$
$29$	−5.44716	−1.01151	−0.505756	+	0.862677i	$0.668786\pi$
$30$	0	0
$31$	−0.343174	−0.0616359	−0.0308180	−	0.999525i	$-0.509811\pi$
$31$	−0.343174	−0.0616359	−0.0308180	+	0.999525i	$0.509811\pi$
$32$	−5.84796	−1.03378
$33$	4.44209	0.773268
$34$	−3.08630	−0.529297
$35$	0	0
$36$	−1.38336	−0.230561
$37$	−10.9118	−1.79388	−0.896941	−	0.442151i	$-0.854216\pi$
$37$	−10.9118	−1.79388	−0.896941	+	0.442151i	$0.854216\pi$
$38$	3.05474	0.495545
$39$	1.00000	0.160128
$40$	0	0
$41$	−11.0753	−1.72967	−0.864837	−	0.502053i	$-0.832578\pi$
$41$	−11.0753	−1.72967	−0.864837	+	0.502053i	$0.832578\pi$
$42$	−0.785261	−0.121168
$43$	−6.46460	−0.985842	−0.492921	−	0.870074i	$-0.664071\pi$
$43$	−6.46460	−0.985842	−0.492921	+	0.870074i	$0.664071\pi$
$44$	−6.14503	−0.926397
$45$	0	0
$46$	4.35578	0.642225
$47$	−7.61469	−1.11072	−0.555359	−	0.831611i	$-0.687419\pi$
$47$	−7.61469	−1.11072	−0.555359	+	0.831611i	$0.687419\pi$
$48$	0.680426	0.0982111
$49$	1.00000	0.142857
$50$	0	0
$51$	3.93029	0.550350
$52$	−1.38336	−0.191838
$53$	−11.8019	−1.62111	−0.810554	−	0.585663i	$-0.800834\pi$
$53$	−11.8019	−1.62111	−0.810554	+	0.585663i	$0.800834\pi$
$54$	−0.785261	−0.106861
$55$	0	0
$56$	2.65683	0.355033
$57$	−3.89010	−0.515256
$58$	4.27744	0.561656
$59$	−7.16754	−0.933134	−0.466567	−	0.884486i	$-0.654509\pi$
$59$	−7.16754	−0.933134	−0.466567	+	0.884486i	$0.654509\pi$
$60$	0	0
$61$	−1.42550	−0.182516	−0.0912581	−	0.995827i	$-0.529089\pi$
$61$	−1.42550	−0.182516	−0.0912581	+	0.995827i	$0.529089\pi$
$62$	0.269482	0.0342242
$63$	1.00000	0.125988
$64$	3.23133	0.403916
$65$	0	0
$66$	−3.48820	−0.429368
$67$	14.8732	1.81705	0.908524	−	0.417832i	$-0.137210\pi$
$67$	14.8732	1.81705	0.908524	+	0.417832i	$0.137210\pi$
$68$	−5.43702	−0.659335
$69$	−5.54692	−0.667771
$70$	0	0
$71$	13.6745	1.62287	0.811433	−	0.584446i	$-0.198688\pi$
$71$	13.6745	1.62287	0.811433	+	0.584446i	$0.198688\pi$
$72$	2.65683	0.313110
$73$	3.08521	0.361097	0.180549	−	0.983566i	$-0.442213\pi$
$73$	3.08521	0.361097	0.180549	+	0.983566i	$0.442213\pi$
$74$	8.56858	0.996077
$75$	0	0
$76$	5.38142	0.617291
$77$	4.44209	0.506223
$78$	−0.785261	−0.0889134
$79$	−4.82147	−0.542458	−0.271229	−	0.962515i	$-0.587430\pi$
$79$	−4.82147	−0.542458	−0.271229	+	0.962515i	$0.587430\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	8.69702	0.960425
$83$	11.9694	1.31381	0.656906	−	0.753973i	$-0.271865\pi$
$83$	11.9694	1.31381	0.656906	+	0.753973i	$0.271865\pi$
$84$	−1.38336	−0.150937
$85$	0	0
$86$	5.07640	0.547402
$87$	−5.44716	−0.583996
$88$	11.8019	1.25808
$89$	−15.5997	−1.65357	−0.826784	−	0.562520i	$-0.809832\pi$
$89$	−15.5997	−1.65357	−0.826784	+	0.562520i	$0.809832\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	7.67342	0.800009
$93$	−0.343174	−0.0355855
$94$	5.97952	0.616741
$95$	0	0
$96$	−5.84796	−0.596855
$97$	2.43304	0.247038	0.123519	−	0.992342i	$-0.460582\pi$
$97$	2.43304	0.247038	0.123519	+	0.992342i	$0.460582\pi$
$98$	−0.785261	−0.0793234
$99$	4.44209	0.446447
$100$	0	0
$101$	8.77427	0.873073	0.436536	−	0.899687i	$-0.356205\pi$
$101$	8.77427	0.873073	0.436536	+	0.899687i	$0.356205\pi$
$102$	−3.08630	−0.305590
$103$	−11.6222	−1.14517	−0.572586	−	0.819844i	$-0.694060\pi$
$103$	−11.6222	−1.14517	−0.572586	+	0.819844i	$0.694060\pi$
$104$	2.65683	0.260523
$105$	0	0
$106$	9.26754	0.900143
$107$	−7.44209	−0.719454	−0.359727	−	0.933058i	$-0.617130\pi$
$107$	−7.44209	−0.719454	−0.359727	+	0.933058i	$0.617130\pi$
$108$	−1.38336	−0.133114
$109$	−8.00754	−0.766983	−0.383492	−	0.923544i	$-0.625279\pi$
$109$	−8.00754	−0.766983	−0.383492	+	0.923544i	$0.625279\pi$
$110$	0	0
$111$	−10.9118	−1.03570
$112$	0.680426	0.0642942
$113$	−7.68688	−0.723121	−0.361560	−	0.932349i	$-0.617756\pi$
$113$	−7.68688	−0.723121	−0.361560	+	0.932349i	$0.617756\pi$
$114$	3.05474	0.286103
$115$	0	0
$116$	7.53540	0.699644
$117$	1.00000	0.0924500
$118$	5.62839	0.518135
$119$	3.93029	0.360289
$120$	0	0
$121$	8.73214	0.793831
$122$	1.11939	0.101345
$123$	−11.0753	−0.998627
$124$	0.474735	0.0426325
$125$	0	0
$126$	−0.785261	−0.0699566
$127$	21.4327	1.90185	0.950923	−	0.309428i	$-0.100137\pi$
$127$	21.4327	1.90185	0.950923	+	0.309428i	$0.100137\pi$
$128$	9.15849	0.809504
$129$	−6.46460	−0.569176
$130$	0	0
$131$	4.26506	0.372640	0.186320	−	0.982489i	$-0.440344\pi$
$131$	4.26506	0.372640	0.186320	+	0.982489i	$0.440344\pi$
$132$	−6.14503	−0.534856
$133$	−3.89010	−0.337314
$134$	−11.6793	−1.00894
$135$	0	0
$136$	10.4421	0.895402
$137$	−7.44318	−0.635913	−0.317957	−	0.948105i	$-0.602997\pi$
$137$	−7.44318	−0.635913	−0.317957	+	0.948105i	$0.602997\pi$
$138$	4.35578	0.370789
$139$	−0.389287	−0.0330189	−0.0165094	−	0.999864i	$-0.505255\pi$
$139$	−0.389287	−0.0330189	−0.0165094	+	0.999864i	$0.505255\pi$
$140$	0	0
$141$	−7.61469	−0.641273
$142$	−10.7381	−0.901118
$143$	4.44209	0.371466
$144$	0.680426	0.0567022
$145$	0	0
$146$	−2.42270	−0.200504
$147$	1.00000	0.0824786
$148$	15.0949	1.24080
$149$	11.8966	0.974603	0.487302	−	0.873234i	$-0.337981\pi$
$149$	11.8966	0.974603	0.487302	+	0.873234i	$0.337981\pi$
$150$	0	0
$151$	−16.1627	−1.31530	−0.657651	−	0.753323i	$-0.728450\pi$
$151$	−16.1627	−1.31530	−0.657651	+	0.753323i	$0.728450\pi$
$152$	−10.3353	−0.838304
$153$	3.93029	0.317745
$154$	−3.48820	−0.281087
$155$	0	0
$156$	−1.38336	−0.110758
$157$	−12.1951	−0.973276	−0.486638	−	0.873604i	$-0.661777\pi$
$157$	−12.1951	−0.973276	−0.486638	+	0.873604i	$0.661777\pi$
$158$	3.78612	0.301207
$159$	−11.8019	−0.935948
$160$	0	0
$161$	−5.54692	−0.437159
$162$	−0.785261	−0.0616960
$163$	−6.59551	−0.516600	−0.258300	−	0.966065i	$-0.583162\pi$
$163$	−6.59551	−0.516600	−0.258300	+	0.966065i	$0.583162\pi$
$164$	15.3212	1.19638
$165$	0	0
$166$	−9.39910	−0.729512
$167$	−11.6734	−0.903316	−0.451658	−	0.892191i	$-0.649167\pi$
$167$	−11.6734	−0.903316	−0.451658	+	0.892191i	$0.649167\pi$
$168$	2.65683	0.204979
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.89010	−0.297483
$172$	8.94290	0.681889
$173$	18.0553	1.37272	0.686359	−	0.727263i	$-0.259208\pi$
$173$	18.0553	1.37272	0.686359	+	0.727263i	$0.259208\pi$
$174$	4.27744	0.324272
$175$	0	0
$176$	3.02251	0.227830
$177$	−7.16754	−0.538745
$178$	12.2499	0.918166
$179$	−7.27455	−0.543725	−0.271863	−	0.962336i	$-0.587640\pi$
$179$	−7.27455	−0.543725	−0.271863	+	0.962336i	$0.587640\pi$
$180$	0	0
$181$	−23.6986	−1.76151	−0.880753	−	0.473576i	$-0.842963\pi$
$181$	−23.6986	−1.76151	−0.880753	+	0.473576i	$0.842963\pi$
$182$	−0.785261	−0.0582075
$183$	−1.42550	−0.105376
$184$	−14.7372	−1.08644
$185$	0	0
$186$	0.269482	0.0197593
$187$	17.4587	1.27670
$188$	10.5339	0.768263
$189$	1.00000	0.0727393
$190$	0	0
$191$	0.907775	0.0656843	0.0328421	−	0.999461i	$-0.489544\pi$
$191$	0.907775	0.0656843	0.0328421	+	0.999461i	$0.489544\pi$
$192$	3.23133	0.233201
$193$	15.5936	1.12245	0.561225	−	0.827663i	$-0.310330\pi$
$193$	15.5936	1.12245	0.561225	+	0.827663i	$0.310330\pi$
$194$	−1.91057	−0.137171
$195$	0	0
$196$	−1.38336	−0.0988117
$197$	14.0844	1.00347	0.501735	−	0.865022i	$-0.332695\pi$
$197$	14.0844	1.00347	0.501735	+	0.865022i	$0.332695\pi$
$198$	−3.48820	−0.247896
$199$	18.6916	1.32501	0.662507	−	0.749056i	$-0.269492\pi$
$199$	18.6916	1.32501	0.662507	+	0.749056i	$0.269492\pi$
$200$	0	0
$201$	14.8732	1.04907
$202$	−6.89010	−0.484786
$203$	−5.44716	−0.382315
$204$	−5.43702	−0.380667
$205$	0	0
$206$	9.12649	0.635873
$207$	−5.54692	−0.385538
$208$	0.680426	0.0471791
$209$	−17.2801	−1.19529
$210$	0	0
$211$	−13.5997	−0.936244	−0.468122	−	0.883664i	$-0.655069\pi$
$211$	−13.5997	−0.936244	−0.468122	+	0.883664i	$0.655069\pi$
$212$	16.3263	1.12129
$213$	13.6745	0.936962
$214$	5.84398	0.399487
$215$	0	0
$216$	2.65683	0.180774
$217$	−0.343174	−0.0232962
$218$	6.28801	0.425878
$219$	3.08521	0.208479
$220$	0	0
$221$	3.93029	0.264380
$222$	8.56858	0.575085
$223$	−20.9809	−1.40499	−0.702493	−	0.711691i	$-0.747930\pi$
$223$	−20.9809	−1.40499	−0.702493	+	0.711691i	$0.747930\pi$
$224$	−5.84796	−0.390734
$225$	0	0
$226$	6.03621	0.401523
$227$	−1.44209	−0.0957147	−0.0478573	−	0.998854i	$-0.515239\pi$
$227$	−1.44209	−0.0957147	−0.0478573	+	0.998854i	$0.515239\pi$
$228$	5.38142	0.356393
$229$	−23.1118	−1.52727	−0.763634	−	0.645649i	$-0.776587\pi$
$229$	−23.1118	−1.52727	−0.763634	+	0.645649i	$0.776587\pi$
$230$	0	0
$231$	4.44209	0.292268
$232$	−14.4721	−0.950143
$233$	20.3713	1.33457	0.667284	−	0.744804i	$-0.267457\pi$
$233$	20.3713	1.33457	0.667284	+	0.744804i	$0.267457\pi$
$234$	−0.785261	−0.0513341
$235$	0	0
$236$	9.91532	0.645432
$237$	−4.82147	−0.313188
$238$	−3.08630	−0.200055
$239$	−27.0539	−1.74997	−0.874985	−	0.484150i	$-0.839129\pi$
$239$	−27.0539	−1.74997	−0.874985	+	0.484150i	$0.839129\pi$
$240$	0	0
$241$	−17.6600	−1.13758	−0.568789	−	0.822484i	$-0.692588\pi$
$241$	−17.6600	−1.13758	−0.568789	+	0.822484i	$0.692588\pi$
$242$	−6.85701	−0.440785
$243$	1.00000	0.0641500
$244$	1.97198	0.126243
$245$	0	0
$246$	8.69702	0.554501
$247$	−3.89010	−0.247521
$248$	−0.911754	−0.0578965
$249$	11.9694	0.758529
$250$	0	0
$251$	−0.0827403	−0.00522252	−0.00261126	−	0.999997i	$-0.500831\pi$
$251$	−0.0827403	−0.00522252	−0.00261126	+	0.999997i	$0.500831\pi$
$252$	−1.38336	−0.0871438
$253$	−24.6399	−1.54910
$254$	−16.8303	−1.05603
$255$	0	0
$256$	−13.6545	−0.853404
$257$	0.589706	0.0367849	0.0183924	−	0.999831i	$-0.494145\pi$
$257$	0.589706	0.0367849	0.0183924	+	0.999831i	$0.494145\pi$
$258$	5.07640	0.316043
$259$	−10.9118	−0.678023
$260$	0	0
$261$	−5.44716	−0.337170
$262$	−3.34919	−0.206914
$263$	8.88568	0.547914	0.273957	−	0.961742i	$-0.411667\pi$
$263$	8.88568	0.547914	0.273957	+	0.961742i	$0.411667\pi$
$264$	11.8019	0.726354
$265$	0	0
$266$	3.05474	0.187298
$267$	−15.5997	−0.954688
$268$	−20.5750	−1.25682
$269$	−19.1037	−1.16477	−0.582385	−	0.812913i	$-0.697880\pi$
$269$	−19.1037	−1.16477	−0.582385	+	0.812913i	$0.697880\pi$
$270$	0	0
$271$	9.56351	0.580942	0.290471	−	0.956884i	$-0.406188\pi$
$271$	9.56351	0.580942	0.290471	+	0.956884i	$0.406188\pi$
$272$	2.67427	0.162151
$273$	1.00000	0.0605228
$274$	5.84484	0.353100
$275$	0	0
$276$	7.67342	0.461885
$277$	−17.8265	−1.07109	−0.535546	−	0.844506i	$-0.679894\pi$
$277$	−17.8265	−1.07109	−0.535546	+	0.844506i	$0.679894\pi$
$278$	0.305692	0.0183342
$279$	−0.343174	−0.0205453
$280$	0	0
$281$	−6.92038	−0.412836	−0.206418	−	0.978464i	$-0.566181\pi$
$281$	−6.92038	−0.412836	−0.206418	+	0.978464i	$0.566181\pi$
$282$	5.97952	0.356076
$283$	18.7286	1.11330	0.556649	−	0.830748i	$-0.312087\pi$
$283$	18.7286	1.11330	0.556649	+	0.830748i	$0.312087\pi$
$284$	−18.9168	−1.12251
$285$	0	0
$286$	−3.48820	−0.206262
$287$	−11.0753	−0.653755
$288$	−5.84796	−0.344595
$289$	−1.55284	−0.0913438
$290$	0	0
$291$	2.43304	0.142627
$292$	−4.26798	−0.249764
$293$	6.06712	0.354445	0.177222	−	0.984171i	$-0.443289\pi$
$293$	6.06712	0.354445	0.177222	+	0.984171i	$0.443289\pi$
$294$	−0.785261	−0.0457974
$295$	0	0
$296$	−28.9906	−1.68505
$297$	4.44209	0.257756
$298$	−9.34190	−0.541162
$299$	−5.54692	−0.320787
$300$	0	0
$301$	−6.46460	−0.372613
$302$	12.6919	0.730340
$303$	8.77427	0.504069
$304$	−2.64692	−0.151811
$305$	0	0
$306$	−3.08630	−0.176432
$307$	12.8504	0.733413	0.366707	−	0.930337i	$-0.380485\pi$
$307$	12.8504	0.733413	0.366707	+	0.930337i	$0.380485\pi$
$308$	−6.14503	−0.350145
$309$	−11.6222	−0.661166
$310$	0	0
$311$	18.2771	1.03640	0.518200	−	0.855259i	$-0.326602\pi$
$311$	18.2771	1.03640	0.518200	+	0.855259i	$0.326602\pi$
$312$	2.65683	0.150413
$313$	27.7830	1.57039	0.785194	−	0.619250i	$-0.212563\pi$
$313$	27.7830	1.57039	0.785194	+	0.619250i	$0.212563\pi$
$314$	9.57636	0.540425
$315$	0	0
$316$	6.66985	0.375209
$317$	28.9555	1.62630	0.813152	−	0.582052i	$-0.197750\pi$
$317$	28.9555	1.62630	0.813152	+	0.582052i	$0.197750\pi$
$318$	9.26754	0.519698
$319$	−24.1967	−1.35476
$320$	0	0
$321$	−7.44209	−0.415377
$322$	4.35578	0.242738
$323$	−15.2892	−0.850714
$324$	−1.38336	−0.0768536
$325$	0	0
$326$	5.17920	0.286849
$327$	−8.00754	−0.442818
$328$	−29.4252	−1.62473
$329$	−7.61469	−0.419812
$330$	0	0
$331$	−18.0191	−0.990417	−0.495209	−	0.868774i	$-0.664908\pi$
$331$	−18.0191	−0.990417	−0.495209	+	0.868774i	$0.664908\pi$
$332$	−16.5580	−0.908740
$333$	−10.9118	−0.597960
$334$	9.16668	0.501579
$335$	0	0
$336$	0.680426	0.0371203
$337$	−11.1289	−0.606227	−0.303114	−	0.952954i	$-0.598026\pi$
$337$	−11.1289	−0.606227	−0.303114	+	0.952954i	$0.598026\pi$
$338$	−0.785261	−0.0427126
$339$	−7.68688	−0.417494
$340$	0	0
$341$	−1.52441	−0.0825514
$342$	3.05474	0.165182
$343$	1.00000	0.0539949
$344$	−17.1753	−0.926031
$345$	0	0
$346$	−14.1781	−0.762220
$347$	1.87156	0.100471	0.0502354	−	0.998737i	$-0.484003\pi$
$347$	1.87156	0.100471	0.0502354	+	0.998737i	$0.484003\pi$
$348$	7.53540	0.403940
$349$	6.08232	0.325579	0.162790	−	0.986661i	$-0.447951\pi$
$349$	6.08232	0.325579	0.162790	+	0.986661i	$0.447951\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−25.9772	−1.38459
$353$	29.7860	1.58535	0.792675	−	0.609644i	$-0.208688\pi$
$353$	29.7860	1.58535	0.792675	+	0.609644i	$0.208688\pi$
$354$	5.62839	0.299146
$355$	0	0
$356$	21.5801	1.14374
$357$	3.93029	0.208013
$358$	5.71242	0.301911
$359$	−29.6380	−1.56423	−0.782116	−	0.623133i	$-0.785860\pi$
$359$	−29.6380	−1.56423	−0.782116	+	0.623133i	$0.785860\pi$
$360$	0	0
$361$	−3.86715	−0.203534
$362$	18.6096	0.978100
$363$	8.73214	0.458318
$364$	−1.38336	−0.0725080
$365$	0	0
$366$	1.11939	0.0585114
$367$	25.7726	1.34532	0.672658	−	0.739953i	$-0.265152\pi$
$367$	25.7726	1.34532	0.672658	+	0.739953i	$0.265152\pi$
$368$	−3.77427	−0.196747
$369$	−11.0753	−0.576558
$370$	0	0
$371$	−11.8019	−0.612722
$372$	0.474735	0.0246139
$373$	9.95322	0.515358	0.257679	−	0.966231i	$-0.417042\pi$
$373$	9.95322	0.515358	0.257679	+	0.966231i	$0.417042\pi$
$374$	−13.7096	−0.708908
$375$	0	0
$376$	−20.2309	−1.04333
$377$	−5.44716	−0.280543
$378$	−0.785261	−0.0403895
$379$	−12.9288	−0.664107	−0.332053	−	0.943261i	$-0.607741\pi$
$379$	−12.9288	−0.664107	−0.332053	+	0.943261i	$0.607741\pi$
$380$	0	0
$381$	21.4327	1.09803
$382$	−0.712840	−0.0364721
$383$	−6.27617	−0.320697	−0.160349	−	0.987060i	$-0.551262\pi$
$383$	−6.27617	−0.320697	−0.160349	+	0.987060i	$0.551262\pi$
$384$	9.15849	0.467367
$385$	0	0
$386$	−12.2450	−0.623255
$387$	−6.46460	−0.328614
$388$	−3.36578	−0.170872
$389$	29.6812	1.50490	0.752448	−	0.658652i	$-0.228873\pi$
$389$	29.6812	1.50490	0.752448	+	0.658652i	$0.228873\pi$
$390$	0	0
$391$	−21.8010	−1.10252
$392$	2.65683	0.134190
$393$	4.26506	0.215144
$394$	−11.0599	−0.557190
$395$	0	0
$396$	−6.14503	−0.308799
$397$	−34.1072	−1.71179	−0.855896	−	0.517148i	$-0.826994\pi$
$397$	−34.1072	−1.71179	−0.855896	+	0.517148i	$0.826994\pi$
$398$	−14.6778	−0.735732
$399$	−3.89010	−0.194748
$400$	0	0
$401$	12.9220	0.645294	0.322647	−	0.946519i	$-0.395427\pi$
$401$	12.9220	0.645294	0.322647	+	0.946519i	$0.395427\pi$
$402$	−11.6793	−0.582512
$403$	−0.343174	−0.0170947
$404$	−12.1380	−0.603889
$405$	0	0
$406$	4.27744	0.212286
$407$	−48.4710	−2.40262
$408$	10.4421	0.516960
$409$	26.3329	1.30208	0.651040	−	0.759043i	$-0.274333\pi$
$409$	26.3329	1.30208	0.651040	+	0.759043i	$0.274333\pi$
$410$	0	0
$411$	−7.44318	−0.367145
$412$	16.0778	0.792096
$413$	−7.16754	−0.352691
$414$	4.35578	0.214075
$415$	0	0
$416$	−5.84796	−0.286720
$417$	−0.389287	−0.0190635
$418$	13.5694	0.663703
$419$	−3.70639	−0.181069	−0.0905344	−	0.995893i	$-0.528858\pi$
$419$	−3.70639	−0.181069	−0.0905344	+	0.995893i	$0.528858\pi$
$420$	0	0
$421$	4.04635	0.197207	0.0986034	−	0.995127i	$-0.468562\pi$
$421$	4.04635	0.197207	0.0986034	+	0.995127i	$0.468562\pi$
$422$	10.6793	0.519862
$423$	−7.61469	−0.370239
$424$	−31.3555	−1.52276
$425$	0	0
$426$	−10.7381	−0.520261
$427$	−1.42550	−0.0689847
$428$	10.2951	0.497633
$429$	4.44209	0.214466
$430$	0	0
$431$	−28.5509	−1.37525	−0.687624	−	0.726067i	$-0.741346\pi$
$431$	−28.5509	−1.37525	−0.687624	+	0.726067i	$0.741346\pi$
$432$	0.680426	0.0327370
$433$	−2.84066	−0.136513	−0.0682566	−	0.997668i	$-0.521744\pi$
$433$	−2.84066	−0.136513	−0.0682566	+	0.997668i	$0.521744\pi$
$434$	0.269482	0.0129355
$435$	0	0
$436$	11.0773	0.530509
$437$	21.5781	1.03222
$438$	−2.42270	−0.115761
$439$	−10.8628	−0.518451	−0.259225	−	0.965817i	$-0.583467\pi$
$439$	−10.8628	−0.518451	−0.259225	+	0.965817i	$0.583467\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−3.08630	−0.146800
$443$	16.2620	0.772633	0.386316	−	0.922366i	$-0.373747\pi$
$443$	16.2620	0.772633	0.386316	+	0.922366i	$0.373747\pi$
$444$	15.0949	0.716374
$445$	0	0
$446$	16.4755	0.780138
$447$	11.8966	0.562688
$448$	3.23133	0.152666
$449$	−11.1922	−0.528194	−0.264097	−	0.964496i	$-0.585074\pi$
$449$	−11.1922	−0.528194	−0.264097	+	0.964496i	$0.585074\pi$
$450$	0	0
$451$	−49.1975	−2.31662
$452$	10.6338	0.500170
$453$	−16.1627	−0.759390
$454$	1.13242	0.0531469
$455$	0	0
$456$	−10.3353	−0.483995
$457$	32.8364	1.53602	0.768010	−	0.640438i	$-0.221247\pi$
$457$	32.8364	1.53602	0.768010	+	0.640438i	$0.221247\pi$
$458$	18.1488	0.848036
$459$	3.93029	0.183450
$460$	0	0
$461$	−22.8699	−1.06516	−0.532578	−	0.846381i	$-0.678777\pi$
$461$	−22.8699	−1.06516	−0.532578	+	0.846381i	$0.678777\pi$
$462$	−3.48820	−0.162286
$463$	38.4806	1.78835	0.894173	−	0.447721i	$-0.147764\pi$
$463$	38.4806	1.78835	0.894173	+	0.447721i	$0.147764\pi$
$464$	−3.70639	−0.172065
$465$	0	0
$466$	−15.9968	−0.741037
$467$	23.9851	1.10990	0.554950	−	0.831884i	$-0.312738\pi$
$467$	23.9851	1.10990	0.554950	+	0.831884i	$0.312738\pi$
$468$	−1.38336	−0.0639460
$469$	14.8732	0.686780
$470$	0	0
$471$	−12.1951	−0.561921
$472$	−19.0429	−0.876520
$473$	−28.7163	−1.32038
$474$	3.78612	0.173902
$475$	0	0
$476$	−5.43702	−0.249205
$477$	−11.8019	−0.540370
$478$	21.2444	0.971695
$479$	41.6669	1.90381	0.951905	−	0.306392i	$-0.0991218\pi$
$479$	41.6669	1.90381	0.951905	+	0.306392i	$0.0991218\pi$
$480$	0	0
$481$	−10.9118	−0.497533
$482$	13.8677	0.631655
$483$	−5.54692	−0.252394
$484$	−12.0797	−0.549079
$485$	0	0
$486$	−0.785261	−0.0356202
$487$	−38.9230	−1.76377	−0.881884	−	0.471467i	$-0.843725\pi$
$487$	−38.9230	−1.76377	−0.881884	+	0.471467i	$0.843725\pi$
$488$	−3.78730	−0.171443
$489$	−6.59551	−0.298259
$490$	0	0
$491$	35.2153	1.58924	0.794621	−	0.607105i	$-0.207669\pi$
$491$	35.2153	1.58924	0.794621	+	0.607105i	$0.207669\pi$
$492$	15.3212	0.690733
$493$	−21.4089	−0.964208
$494$	3.05474	0.137439
$495$	0	0
$496$	−0.233505	−0.0104847
$497$	13.6745	0.613385
$498$	−9.39910	−0.421184
$499$	−0.390235	−0.0174693	−0.00873467	−	0.999962i	$-0.502780\pi$
$499$	−0.390235	−0.0174693	−0.00873467	+	0.999962i	$0.502780\pi$
$500$	0	0
$501$	−11.6734	−0.521530
$502$	0.0649728	0.00289988
$503$	−10.0548	−0.448323	−0.224162	−	0.974552i	$-0.571964\pi$
$503$	−10.0548	−0.448323	−0.224162	+	0.974552i	$0.571964\pi$
$504$	2.65683	0.118344
$505$	0	0
$506$	19.3488	0.860158
$507$	1.00000	0.0444116
$508$	−29.6493	−1.31547
$509$	30.0527	1.33206	0.666030	−	0.745925i	$-0.267992\pi$
$509$	30.0527	1.33206	0.666030	+	0.745925i	$0.267992\pi$
$510$	0	0
$511$	3.08521	0.136482
$512$	−7.59465	−0.335639
$513$	−3.89010	−0.171752
$514$	−0.463074	−0.0204253
$515$	0	0
$516$	8.94290	0.393689
$517$	−33.8251	−1.48763
$518$	8.56858	0.376482
$519$	18.0553	0.792539
$520$	0	0
$521$	−13.3168	−0.583419	−0.291709	−	0.956507i	$-0.594224\pi$
$521$	−13.3168	−0.583419	−0.291709	+	0.956507i	$0.594224\pi$
$522$	4.27744	0.187219
$523$	11.2662	0.492637	0.246319	−	0.969189i	$-0.420779\pi$
$523$	11.2662	0.492637	0.246319	+	0.969189i	$0.420779\pi$
$524$	−5.90014	−0.257749
$525$	0	0
$526$	−6.97758	−0.304237
$527$	−1.34877	−0.0587535
$528$	3.02251	0.131538
$529$	7.76835	0.337754
$530$	0	0
$531$	−7.16754	−0.311045
$532$	5.38142	0.233314
$533$	−11.0753	−0.479725
$534$	12.2499	0.530103
$535$	0	0
$536$	39.5155	1.70681
$537$	−7.27455	−0.313920
$538$	15.0014	0.646755
$539$	4.44209	0.191334
$540$	0	0
$541$	2.78815	0.119872	0.0599360	−	0.998202i	$-0.480910\pi$
$541$	2.78815	0.119872	0.0599360	+	0.998202i	$0.480910\pi$
$542$	−7.50986	−0.322576
$543$	−23.6986	−1.01701
$544$	−22.9842	−0.985438
$545$	0	0
$546$	−0.785261	−0.0336061
$547$	−0.906453	−0.0387571	−0.0193786	−	0.999812i	$-0.506169\pi$
$547$	−0.906453	−0.0387571	−0.0193786	+	0.999812i	$0.506169\pi$
$548$	10.2966	0.439850
$549$	−1.42550	−0.0608387
$550$	0	0
$551$	21.1900	0.902723
$552$	−14.7372	−0.627257
$553$	−4.82147	−0.205030
$554$	13.9985	0.594739
$555$	0	0
$556$	0.538526	0.0228386
$557$	20.2794	0.859265	0.429633	−	0.903004i	$-0.358643\pi$
$557$	20.2794	0.859265	0.429633	+	0.903004i	$0.358643\pi$
$558$	0.269482	0.0114081
$559$	−6.46460	−0.273423
$560$	0	0
$561$	17.4587	0.737106
$562$	5.43431	0.229233
$563$	−23.8201	−1.00390	−0.501948	−	0.864898i	$-0.667383\pi$
$563$	−23.8201	−1.00390	−0.501948	+	0.864898i	$0.667383\pi$
$564$	10.5339	0.443557
$565$	0	0
$566$	−14.7068	−0.618174
$567$	1.00000	0.0419961
$568$	36.3308	1.52441
$569$	−9.00375	−0.377457	−0.188728	−	0.982029i	$-0.560437\pi$
$569$	−9.00375	−0.377457	−0.188728	+	0.982029i	$0.560437\pi$
$570$	0	0
$571$	−8.33454	−0.348790	−0.174395	−	0.984676i	$-0.555797\pi$
$571$	−8.33454	−0.348790	−0.174395	+	0.984676i	$0.555797\pi$
$572$	−6.14503	−0.256936
$573$	0.907775	0.0379228
$574$	8.69702	0.363006
$575$	0	0
$576$	3.23133	0.134639
$577$	−14.3589	−0.597769	−0.298885	−	0.954289i	$-0.596615\pi$
$577$	−14.3589	−0.597769	−0.298885	+	0.954289i	$0.596615\pi$
$578$	1.21939	0.0507199
$579$	15.5936	0.648047
$580$	0	0
$581$	11.9694	0.496574
$582$	−1.91057	−0.0791958
$583$	−52.4249	−2.17122
$584$	8.19688	0.339189
$585$	0	0
$586$	−4.76427	−0.196810
$587$	7.83397	0.323342	0.161671	−	0.986845i	$-0.448312\pi$
$587$	7.83397	0.323342	0.161671	+	0.986845i	$0.448312\pi$
$588$	−1.38336	−0.0570490
$589$	1.33498	0.0550069
$590$	0	0
$591$	14.0844	0.579353
$592$	−7.42464	−0.305151
$593$	−11.4547	−0.470388	−0.235194	−	0.971948i	$-0.575573\pi$
$593$	−11.4547	−0.470388	−0.235194	+	0.971948i	$0.575573\pi$
$594$	−3.48820	−0.143123
$595$	0	0
$596$	−16.4573	−0.674116
$597$	18.6916	0.764997
$598$	4.35578	0.178121
$599$	5.93464	0.242483	0.121241	−	0.992623i	$-0.461312\pi$
$599$	5.93464	0.242483	0.121241	+	0.992623i	$0.461312\pi$
$600$	0	0
$601$	12.1173	0.494275	0.247138	−	0.968980i	$-0.420510\pi$
$601$	12.1173	0.494275	0.247138	+	0.968980i	$0.420510\pi$
$602$	5.07640	0.206899
$603$	14.8732	0.605683
$604$	22.3589	0.909771
$605$	0	0
$606$	−6.89010	−0.279891
$607$	−46.2065	−1.87547	−0.937733	−	0.347357i	$-0.887079\pi$
$607$	−46.2065	−1.87547	−0.937733	+	0.347357i	$0.887079\pi$
$608$	22.7491	0.922600
$609$	−5.44716	−0.220730
$610$	0	0
$611$	−7.61469	−0.308057
$612$	−5.43702	−0.219778
$613$	−35.9679	−1.45273	−0.726365	−	0.687310i	$-0.758792\pi$
$613$	−35.9679	−1.45273	−0.726365	+	0.687310i	$0.758792\pi$
$614$	−10.0910	−0.407238
$615$	0	0
$616$	11.8019	0.475510
$617$	−21.0454	−0.847255	−0.423627	−	0.905837i	$-0.639243\pi$
$617$	−21.0454	−0.847255	−0.423627	+	0.905837i	$0.639243\pi$
$618$	9.12649	0.367121
$619$	39.9069	1.60399	0.801997	−	0.597328i	$-0.203771\pi$
$619$	39.9069	1.60399	0.801997	+	0.597328i	$0.203771\pi$
$620$	0	0
$621$	−5.54692	−0.222590
$622$	−14.3523	−0.575475
$623$	−15.5997	−0.624990
$624$	0.680426	0.0272388
$625$	0	0
$626$	−21.8169	−0.871979
$627$	−17.2801	−0.690103
$628$	16.8703	0.673198
$629$	−42.8863	−1.70999
$630$	0	0
$631$	21.5073	0.856191	0.428095	−	0.903734i	$-0.359185\pi$
$631$	21.5073	0.856191	0.428095	+	0.903734i	$0.359185\pi$
$632$	−12.8098	−0.509547
$633$	−13.5997	−0.540540
$634$	−22.7376	−0.903027
$635$	0	0
$636$	16.3263	0.647378
$637$	1.00000	0.0396214
$638$	19.0008	0.752247
$639$	13.6745	0.540955
$640$	0	0
$641$	31.9699	1.26273	0.631367	−	0.775484i	$-0.282494\pi$
$641$	31.9699	1.26273	0.631367	+	0.775484i	$0.282494\pi$
$642$	5.84398	0.230644
$643$	17.9515	0.707939	0.353970	−	0.935257i	$-0.384832\pi$
$643$	17.9515	0.707939	0.353970	+	0.935257i	$0.384832\pi$
$644$	7.67342	0.302375
$645$	0	0
$646$	12.0060	0.472370
$647$	29.9900	1.17903	0.589513	−	0.807759i	$-0.299320\pi$
$647$	29.9900	1.17903	0.589513	+	0.807759i	$0.299320\pi$
$648$	2.65683	0.104370
$649$	−31.8388	−1.24978
$650$	0	0
$651$	−0.343174	−0.0134501
$652$	9.12399	0.357323
$653$	−18.4148	−0.720627	−0.360313	−	0.932831i	$-0.617330\pi$
$653$	−18.4148	−0.720627	−0.360313	+	0.932831i	$0.617330\pi$
$654$	6.28801	0.245881
$655$	0	0
$656$	−7.53593	−0.294229
$657$	3.08521	0.120366
$658$	5.97952	0.233106
$659$	−11.7754	−0.458703	−0.229351	−	0.973344i	$-0.573661\pi$
$659$	−11.7754	−0.458703	−0.229351	+	0.973344i	$0.573661\pi$
$660$	0	0
$661$	4.74843	0.184693	0.0923463	−	0.995727i	$-0.470563\pi$
$661$	4.74843	0.184693	0.0923463	+	0.995727i	$0.470563\pi$
$662$	14.1497	0.549943
$663$	3.93029	0.152640
$664$	31.8006	1.23410
$665$	0	0
$666$	8.56858	0.332026
$667$	30.2149	1.16993
$668$	16.1486	0.624808
$669$	−20.9809	−0.811169
$670$	0	0
$671$	−6.33218	−0.244451
$672$	−5.84796	−0.225590
$673$	−6.91346	−0.266494	−0.133247	−	0.991083i	$-0.542540\pi$
$673$	−6.91346	−0.266494	−0.133247	+	0.991083i	$0.542540\pi$
$674$	8.73906	0.336616
$675$	0	0
$676$	−1.38336	−0.0532063
$677$	12.1493	0.466937	0.233468	−	0.972364i	$-0.424992\pi$
$677$	12.1493	0.466937	0.233468	+	0.972364i	$0.424992\pi$
$678$	6.03621	0.231819
$679$	2.43304	0.0933715
$680$	0	0
$681$	−1.44209	−0.0552609
$682$	1.19706	0.0458378
$683$	16.4342	0.628838	0.314419	−	0.949284i	$-0.398190\pi$
$683$	16.4342	0.628838	0.314419	+	0.949284i	$0.398190\pi$
$684$	5.38142	0.205764
$685$	0	0
$686$	−0.785261	−0.0299814
$687$	−23.1118	−0.881769
$688$	−4.39868	−0.167698
$689$	−11.8019	−0.449615
$690$	0	0
$691$	−47.8805	−1.82146	−0.910730	−	0.413001i	$-0.864481\pi$
$691$	−47.8805	−1.82146	−0.910730	+	0.413001i	$0.864481\pi$
$692$	−24.9770	−0.949484
$693$	4.44209	0.168741
$694$	−1.46967	−0.0557878
$695$	0	0
$696$	−14.4721	−0.548565
$697$	−43.5292	−1.64878
$698$	−4.77621	−0.180782
$699$	20.3713	0.770513
$700$	0	0
$701$	−4.77760	−0.180447	−0.0902237	−	0.995922i	$-0.528758\pi$
$701$	−4.77760	−0.180447	−0.0902237	+	0.995922i	$0.528758\pi$
$702$	−0.785261	−0.0296378
$703$	42.4478	1.60095
$704$	14.3538	0.540981
$705$	0	0
$706$	−23.3898	−0.880287
$707$	8.77427	0.329990
$708$	9.91532	0.372640
$709$	35.6958	1.34058	0.670292	−	0.742097i	$-0.266169\pi$
$709$	35.6958	1.34058	0.670292	+	0.742097i	$0.266169\pi$
$710$	0	0
$711$	−4.82147	−0.180819
$712$	−41.4457	−1.55324
$713$	1.90356	0.0712889
$714$	−3.08630	−0.115502
$715$	0	0
$716$	10.0634	0.376085
$717$	−27.0539	−1.01035
$718$	23.2736	0.868561
$719$	7.69636	0.287026	0.143513	−	0.989648i	$-0.454160\pi$
$719$	7.69636	0.287026	0.143513	+	0.989648i	$0.454160\pi$
$720$	0	0
$721$	−11.6222	−0.432835
$722$	3.03672	0.113015
$723$	−17.6600	−0.656781
$724$	32.7838	1.21840
$725$	0	0
$726$	−6.85701	−0.254488
$727$	−50.9619	−1.89007	−0.945036	−	0.326967i	$-0.893973\pi$
$727$	−50.9619	−1.89007	−0.945036	+	0.326967i	$0.893973\pi$
$728$	2.65683	0.0984685
$729$	1.00000	0.0370370
$730$	0	0
$731$	−25.4077	−0.939739
$732$	1.97198	0.0728866
$733$	−35.7998	−1.32230	−0.661148	−	0.750256i	$-0.729930\pi$
$733$	−35.7998	−1.32230	−0.661148	+	0.750256i	$0.729930\pi$
$734$	−20.2382	−0.747006
$735$	0	0
$736$	32.4382	1.19569
$737$	66.0680	2.43365
$738$	8.69702	0.320142
$739$	−4.62881	−0.170273	−0.0851367	−	0.996369i	$-0.527133\pi$
$739$	−4.62881	−0.170273	−0.0851367	+	0.996369i	$0.527133\pi$
$740$	0	0
$741$	−3.89010	−0.142906
$742$	9.26754	0.340222
$743$	36.7084	1.34670	0.673351	−	0.739323i	$-0.264854\pi$
$743$	36.7084	1.34670	0.673351	+	0.739323i	$0.264854\pi$
$744$	−0.911754	−0.0334265
$745$	0	0
$746$	−7.81588	−0.286160
$747$	11.9694	0.437937
$748$	−24.1517	−0.883074
$749$	−7.44209	−0.271928
$750$	0	0
$751$	−22.0166	−0.803397	−0.401698	−	0.915772i	$-0.631580\pi$
$751$	−22.0166	−0.803397	−0.401698	+	0.915772i	$0.631580\pi$
$752$	−5.18124	−0.188940
$753$	−0.0827403	−0.00301523
$754$	4.27744	0.155775
$755$	0	0
$756$	−1.38336	−0.0503125
$757$	33.7558	1.22688	0.613438	−	0.789743i	$-0.289786\pi$
$757$	33.7558	1.22688	0.613438	+	0.789743i	$0.289786\pi$
$758$	10.1525	0.368754
$759$	−24.6399	−0.894372
$760$	0	0
$761$	−12.2115	−0.442666	−0.221333	−	0.975198i	$-0.571041\pi$
$761$	−12.2115	−0.442666	−0.221333	+	0.975198i	$0.571041\pi$
$762$	−16.8303	−0.609697
$763$	−8.00754	−0.289892
$764$	−1.25578	−0.0454326
$765$	0	0
$766$	4.92843	0.178072
$767$	−7.16754	−0.258805
$768$	−13.6545	−0.492713
$769$	47.6084	1.71680	0.858402	−	0.512977i	$-0.171457\pi$
$769$	47.6084	1.71680	0.858402	+	0.512977i	$0.171457\pi$
$770$	0	0
$771$	0.589706	0.0212378
$772$	−21.5716	−0.776378
$773$	18.4610	0.663997	0.331999	−	0.943280i	$-0.392277\pi$
$773$	18.4610	0.663997	0.331999	+	0.943280i	$0.392277\pi$
$774$	5.07640	0.182467
$775$	0	0
$776$	6.46416	0.232050
$777$	−10.9118	−0.391457
$778$	−23.3075	−0.835614
$779$	43.0840	1.54365
$780$	0	0
$781$	60.7433	2.17357
$782$	17.1195	0.612191
$783$	−5.44716	−0.194665
$784$	0.680426	0.0243009
$785$	0	0
$786$	−3.34919	−0.119462
$787$	43.8059	1.56151	0.780757	−	0.624835i	$-0.214834\pi$
$787$	43.8059	1.56151	0.780757	+	0.624835i	$0.214834\pi$
$788$	−19.4838	−0.694082
$789$	8.88568	0.316339
$790$	0	0
$791$	−7.68688	−0.273314
$792$	11.8019	0.419361
$793$	−1.42550	−0.0506209
$794$	26.7831	0.950496
$795$	0	0
$796$	−25.8573	−0.916489
$797$	−35.9518	−1.27348	−0.636738	−	0.771080i	$-0.719717\pi$
$797$	−35.9518	−1.27348	−0.636738	+	0.771080i	$0.719717\pi$
$798$	3.05474	0.108137
$799$	−29.9279	−1.05877
$800$	0	0
$801$	−15.5997	−0.551189
$802$	−10.1472	−0.358308
$803$	13.7048	0.483632
$804$	−20.5750	−0.725625
$805$	0	0
$806$	0.269482	0.00949208
$807$	−19.1037	−0.672481
$808$	23.3117	0.820103
$809$	18.7954	0.660810	0.330405	−	0.943839i	$-0.392815\pi$
$809$	18.7954	0.660810	0.330405	+	0.943839i	$0.392815\pi$
$810$	0	0
$811$	30.2850	1.06345	0.531726	−	0.846917i	$-0.321544\pi$
$811$	30.2850	1.06345	0.531726	+	0.846917i	$0.321544\pi$
$812$	7.53540	0.264441
$813$	9.56351	0.335407
$814$	38.0624	1.33409
$815$	0	0
$816$	2.67427	0.0936182
$817$	25.1479	0.879814
$818$	−20.6782	−0.722998
$819$	1.00000	0.0349428
$820$	0	0
$821$	29.5739	1.03213	0.516067	−	0.856548i	$-0.327395\pi$
$821$	29.5739	1.03213	0.516067	+	0.856548i	$0.327395\pi$
$822$	5.84484	0.203862
$823$	−12.3767	−0.431426	−0.215713	−	0.976457i	$-0.569207\pi$
$823$	−12.3767	−0.431426	−0.215713	+	0.976457i	$0.569207\pi$
$824$	−30.8783	−1.07569
$825$	0	0
$826$	5.62839	0.195837
$827$	13.8824	0.482739	0.241369	−	0.970433i	$-0.422403\pi$
$827$	13.8824	0.482739	0.241369	+	0.970433i	$0.422403\pi$
$828$	7.67342	0.266670
$829$	−21.2005	−0.736323	−0.368162	−	0.929762i	$-0.620013\pi$
$829$	−21.2005	−0.736323	−0.368162	+	0.929762i	$0.620013\pi$
$830$	0	0
$831$	−17.8265	−0.618396
$832$	3.23133	0.112026
$833$	3.93029	0.136176
$834$	0.305692	0.0105853
$835$	0	0
$836$	23.9047	0.826763
$837$	−0.343174	−0.0118618
$838$	2.91048	0.100541
$839$	−12.6352	−0.436214	−0.218107	−	0.975925i	$-0.569988\pi$
$839$	−12.6352	−0.436214	−0.218107	+	0.975925i	$0.569988\pi$
$840$	0	0
$841$	0.671498	0.0231551
$842$	−3.17744	−0.109502
$843$	−6.92038	−0.238351
$844$	18.8134	0.647583
$845$	0	0
$846$	5.97952	0.205580
$847$	8.73214	0.300040
$848$	−8.03029	−0.275761
$849$	18.7286	0.642763
$850$	0	0
$851$	60.5267	2.07483
$852$	−18.9168	−0.648080
$853$	−33.6330	−1.15157	−0.575787	−	0.817600i	$-0.695304\pi$
$853$	−33.6330	−1.15157	−0.575787	+	0.817600i	$0.695304\pi$
$854$	1.11939	0.0383047
$855$	0	0
$856$	−19.7723	−0.675804
$857$	−21.1767	−0.723382	−0.361691	−	0.932298i	$-0.617801\pi$
$857$	−21.1767	−0.723382	−0.361691	+	0.932298i	$0.617801\pi$
$858$	−3.48820	−0.119085
$859$	9.19482	0.313723	0.156862	−	0.987621i	$-0.449862\pi$
$859$	9.19482	0.313723	0.156862	+	0.987621i	$0.449862\pi$
$860$	0	0
$861$	−11.0753	−0.377446
$862$	22.4199	0.763626
$863$	25.7046	0.874993	0.437497	−	0.899220i	$-0.355865\pi$
$863$	25.7046	0.874993	0.437497	+	0.899220i	$0.355865\pi$
$864$	−5.84796	−0.198952
$865$	0	0
$866$	2.23066	0.0758008
$867$	−1.55284	−0.0527374
$868$	0.474735	0.0161136
$869$	−21.4174	−0.726535
$870$	0	0
$871$	14.8732	0.503959
$872$	−21.2746	−0.720450
$873$	2.43304	0.0823459
$874$	−16.9444	−0.573154
$875$	0	0
$876$	−4.26798	−0.144202
$877$	34.3037	1.15836	0.579178	−	0.815201i	$-0.303374\pi$
$877$	34.3037	1.15836	0.579178	+	0.815201i	$0.303374\pi$
$878$	8.53010	0.287877
$879$	6.06712	0.204639
$880$	0	0
$881$	27.9014	0.940023	0.470012	−	0.882660i	$-0.344250\pi$
$881$	27.9014	0.940023	0.470012	+	0.882660i	$0.344250\pi$
$882$	−0.785261	−0.0264411
$883$	−28.4426	−0.957170	−0.478585	−	0.878041i	$-0.658850\pi$
$883$	−28.4426	−0.957170	−0.478585	+	0.878041i	$0.658850\pi$
$884$	−5.43702	−0.182867
$885$	0	0
$886$	−12.7699	−0.429015
$887$	33.4374	1.12272	0.561360	−	0.827572i	$-0.310278\pi$
$887$	33.4374	1.12272	0.561360	+	0.827572i	$0.310278\pi$
$888$	−28.9906	−0.972862
$889$	21.4327	0.718830
$890$	0	0
$891$	4.44209	0.148816
$892$	29.0242	0.971804
$893$	29.6219	0.991259
$894$	−9.34190	−0.312440
$895$	0	0
$896$	9.15849	0.305964
$897$	−5.54692	−0.185206
$898$	8.78882	0.293287
$899$	1.86932	0.0623454
$900$	0	0
$901$	−46.3847	−1.54530
$902$	38.6329	1.28633
$903$	−6.46460	−0.215128
$904$	−20.4227	−0.679249
$905$	0	0
$906$	12.6919	0.421662
$907$	54.9066	1.82314	0.911572	−	0.411141i	$-0.134870\pi$
$907$	54.9066	1.82314	0.911572	+	0.411141i	$0.134870\pi$
$908$	1.99493	0.0662041
$909$	8.77427	0.291024
$910$	0	0
$911$	−41.0350	−1.35955	−0.679776	−	0.733420i	$-0.737923\pi$
$911$	−41.0350	−1.35955	−0.679776	+	0.733420i	$0.737923\pi$
$912$	−2.64692	−0.0876484
$913$	53.1691	1.75964
$914$	−25.7851	−0.852896
$915$	0	0
$916$	31.9720	1.05638
$917$	4.26506	0.140845
$918$	−3.08630	−0.101863
$919$	−28.8154	−0.950533	−0.475266	−	0.879842i	$-0.657648\pi$
$919$	−28.8154	−0.950533	−0.475266	+	0.879842i	$0.657648\pi$
$920$	0	0
$921$	12.8504	0.423436
$922$	17.9588	0.591442
$923$	13.6745	0.450102
$924$	−6.14503	−0.202156
$925$	0	0
$926$	−30.2174	−0.993004
$927$	−11.6222	−0.381724
$928$	31.8548	1.04568
$929$	−36.7756	−1.20657	−0.603284	−	0.797527i	$-0.706141\pi$
$929$	−36.7756	−1.20657	−0.603284	+	0.797527i	$0.706141\pi$
$930$	0	0
$931$	−3.89010	−0.127493
$932$	−28.1809	−0.923096
$933$	18.2771	0.598366
$934$	−18.8346	−0.616287
$935$	0	0
$936$	2.65683	0.0868411
$937$	−17.3914	−0.568151	−0.284076	−	0.958802i	$-0.591687\pi$
$937$	−17.3914	−0.568151	−0.284076	+	0.958802i	$0.591687\pi$
$938$	−11.6793	−0.381344
$939$	27.7830	0.906664
$940$	0	0
$941$	−29.2628	−0.953940	−0.476970	−	0.878920i	$-0.658265\pi$
$941$	−29.2628	−0.953940	−0.476970	+	0.878920i	$0.658265\pi$
$942$	9.57636	0.312015
$943$	61.4339	2.00056
$944$	−4.87698	−0.158732
$945$	0	0
$946$	22.5498	0.733158
$947$	14.0943	0.458004	0.229002	−	0.973426i	$-0.426454\pi$
$947$	14.0943	0.458004	0.229002	+	0.973426i	$0.426454\pi$
$948$	6.66985	0.216627
$949$	3.08521	0.100150
$950$	0	0
$951$	28.9555	0.938947
$952$	10.4421	0.338430
$953$	−10.4133	−0.337320	−0.168660	−	0.985674i	$-0.553944\pi$
$953$	−10.4133	−0.337320	−0.168660	+	0.985674i	$0.553944\pi$
$954$	9.26754	0.300048
$955$	0	0
$956$	37.4254	1.21042
$957$	−24.1967	−0.782169
$958$	−32.7194	−1.05712
$959$	−7.44318	−0.240353
$960$	0	0
$961$	−30.8822	−0.996201
$962$	8.56858	0.276262
$963$	−7.44209	−0.239818
$964$	24.4301	0.786842
$965$	0	0
$966$	4.35578	0.140145
$967$	−19.2259	−0.618264	−0.309132	−	0.951019i	$-0.600039\pi$
$967$	−19.2259	−0.618264	−0.309132	+	0.951019i	$0.600039\pi$
$968$	23.1998	0.745669
$969$	−15.2892	−0.491160
$970$	0	0
$971$	−61.0257	−1.95841	−0.979203	−	0.202881i	$-0.934969\pi$
$971$	−61.0257	−1.95841	−0.979203	+	0.202881i	$0.934969\pi$
$972$	−1.38336	−0.0443714
$973$	−0.389287	−0.0124800
$974$	30.5647	0.979356
$975$	0	0
$976$	−0.969946	−0.0310472
$977$	−33.5580	−1.07362	−0.536808	−	0.843704i	$-0.680370\pi$
$977$	−33.5580	−1.07362	−0.536808	+	0.843704i	$0.680370\pi$
$978$	5.17920	0.165613
$979$	−69.2953	−2.21469
$980$	0	0
$981$	−8.00754	−0.255661
$982$	−27.6532	−0.882449
$983$	−27.0187	−0.861764	−0.430882	−	0.902408i	$-0.641797\pi$
$983$	−27.0187	−0.861764	−0.430882	+	0.902408i	$0.641797\pi$
$984$	−29.4252	−0.938040
$985$	0	0
$986$	16.8116	0.535389
$987$	−7.61469	−0.242378
$988$	5.38142	0.171206
$989$	35.8586	1.14024
$990$	0	0
$991$	−35.7526	−1.13572	−0.567859	−	0.823126i	$-0.692228\pi$
$991$	−35.7526	−1.13572	−0.567859	+	0.823126i	$0.692228\pi$
$992$	2.00687	0.0637182
$993$	−18.0191	−0.571818
$994$	−10.7381	−0.340591
$995$	0	0
$996$	−16.5580	−0.524661
$997$	−55.6051	−1.76103	−0.880515	−	0.474018i	$-0.842803\pi$
$997$	−55.6051	−1.76103	−0.880515	+	0.474018i	$0.842803\pi$
$998$	0.306437	0.00970009
$999$	−10.9118	−0.345233

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.bi.1.2	✓	4
5.4	even	2		6825.2.a.bj.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6825.2.a.bi.1.2	✓	4	1.1	even	1	trivial
6825.2.a.bj.1.3	yes	4	5.4	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:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q-0.785261 q^{2} +1.00000 q^{3} -1.38336 q^{4} -0.785261 q^{6} +1.00000 q^{7} +2.65683 q^{8} +1.00000 q^{9} +4.44209 q^{11} -1.38336 q^{12} +1.00000 q^{13} -0.785261 q^{14} +0.680426 q^{16} +3.93029 q^{17} -0.785261 q^{18} -3.89010 q^{19} +1.00000 q^{21} -3.48820 q^{22} -5.54692 q^{23} +2.65683 q^{24} -0.785261 q^{26} +1.00000 q^{27} -1.38336 q^{28} -5.44716 q^{29} -0.343174 q^{31} -5.84796 q^{32} +4.44209 q^{33} -3.08630 q^{34} -1.38336 q^{36} -10.9118 q^{37} +3.05474 q^{38} +1.00000 q^{39} -11.0753 q^{41} -0.785261 q^{42} -6.46460 q^{43} -6.14503 q^{44} +4.35578 q^{46} -7.61469 q^{47} +0.680426 q^{48} +1.00000 q^{49} +3.93029 q^{51} -1.38336 q^{52} -11.8019 q^{53} -0.785261 q^{54} +2.65683 q^{56} -3.89010 q^{57} +4.27744 q^{58} -7.16754 q^{59} -1.42550 q^{61} +0.269482 q^{62} +1.00000 q^{63} +3.23133 q^{64} -3.48820 q^{66} +14.8732 q^{67} -5.43702 q^{68} -5.54692 q^{69} +13.6745 q^{71} +2.65683 q^{72} +3.08521 q^{73} +8.56858 q^{74} +5.38142 q^{76} +4.44209 q^{77} -0.785261 q^{78} -4.82147 q^{79} +1.00000 q^{81} +8.69702 q^{82} +11.9694 q^{83} -1.38336 q^{84} +5.07640 q^{86} -5.44716 q^{87} +11.8019 q^{88} -15.5997 q^{89} +1.00000 q^{91} +7.67342 q^{92} -0.343174 q^{93} +5.97952 q^{94} -5.84796 q^{96} +2.43304 q^{97} -0.785261 q^{98} +4.44209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 4 q^{3} + q^{4} - q^{6} + 4 q^{7} - 3 q^{8} + 4 q^{9} + 2 q^{11} + q^{12} + 4 q^{13} - q^{14} - q^{16} - 5 q^{17} - q^{18} - 15 q^{19} + 4 q^{21} - 9 q^{22} - 8 q^{23} - 3 q^{24} - q^{26}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - q^2 + 4 * q^3 + q^4 - q^6 + 4 * q^7 - 3 * q^8 + 4 * q^9 + 2 * q^11 + q^12 + 4 * q^13 - q^14 - q^16 - 5 * q^17 - q^18 - 15 * q^19 + 4 * q^21 - 9 * q^22 - 8 * q^23 - 3 * q^24 - q^26 + 4 * q^27 + q^28 - 5 * q^29 - 15 * q^31 - 2 * q^32 + 2 * q^33 - 3 * q^34 + q^36 - 12 * q^37 + 13 * q^38 + 4 * q^39 - 5 * q^41 - q^42 - 11 * q^43 - 6 * q^44 + 11 * q^46 + 4 * q^47 - q^48 + 4 * q^49 - 5 * q^51 + q^52 - 15 * q^53 - q^54 - 3 * q^56 - 15 * q^57 - 11 * q^59 - 20 * q^61 + 4 * q^62 + 4 * q^63 - 15 * q^64 - 9 * q^66 - 2 * q^67 - 7 * q^68 - 8 * q^69 + 20 * q^71 - 3 * q^72 - 6 * q^73 - 11 * q^74 - 26 * q^76 + 2 * q^77 - q^78 - 7 * q^79 + 4 * q^81 + q^82 - 2 * q^83 + q^84 - 10 * q^86 - 5 * q^87 + 15 * q^88 - 8 * q^89 + 4 * q^91 - 13 * q^92 - 15 * q^93 - 28 * q^94 - 2 * q^96 + 5 * q^97 - q^98 + 2 * q^99

Introduction

L-functions

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Database

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