LMFDB - Embedded newform 6930.2.a.ch.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, 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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -3.80642 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.05086 q^{17} -1.00000 q^{20} +1.00000 q^{22} +3.05086 q^{23} +1.00000 q^{25} +3.80642 q^{26} +1.00000 q^{28} -0.755569 q^{29} +7.61285 q^{31} -1.00000 q^{32} +5.05086 q^{34} -1.00000 q^{35} +6.56199 q^{37} +1.00000 q^{40} -6.85728 q^{41} +8.85728 q^{43} -1.00000 q^{44} -3.05086 q^{46} +7.61285 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.80642 q^{52} +10.8573 q^{53} +1.00000 q^{55} -1.00000 q^{56} +0.755569 q^{58} +4.85728 q^{59} -12.1017 q^{61} -7.61285 q^{62} +1.00000 q^{64} +3.80642 q^{65} +4.29529 q^{67} -5.05086 q^{68} +1.00000 q^{70} -1.24443 q^{71} -12.3684 q^{73} -6.56199 q^{74} -1.00000 q^{77} -1.51114 q^{79} -1.00000 q^{80} +6.85728 q^{82} -8.85728 q^{83} +5.05086 q^{85} -8.85728 q^{86} +1.00000 q^{88} -12.6637 q^{89} -3.80642 q^{91} +3.05086 q^{92} -7.61285 q^{94} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{10} - 3 q^{11} + 2 q^{13} - 3 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{20} + 3 q^{22} - 4 q^{23} + 3 q^{25} - 2 q^{26} + 3 q^{28} - 2 q^{29} - 4 q^{31} - 3 q^{32} + 2 q^{34} - 3 q^{35} + 6 q^{37} + 3 q^{40} + 6 q^{41} - 3 q^{44} + 4 q^{46} - 4 q^{47} + 3 q^{49} - 3 q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{55} - 3 q^{56} + 2 q^{58} - 12 q^{59} - 10 q^{61} + 4 q^{62} + 3 q^{64} - 2 q^{65} - 2 q^{68} + 3 q^{70} - 4 q^{71} - 10 q^{73} - 6 q^{74} - 3 q^{77} - 4 q^{79} - 3 q^{80} - 6 q^{82} + 2 q^{85} + 3 q^{88} + 2 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{94} - 6 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 - 3 * q^8 + 3 * q^10 - 3 * q^11 + 2 * q^13 - 3 * q^14 + 3 * q^16 - 2 * q^17 - 3 * q^20 + 3 * q^22 - 4 * q^23 + 3 * q^25 - 2 * q^26 + 3 * q^28 - 2 * q^29 - 4 * q^31 - 3 * q^32 + 2 * q^34 - 3 * q^35 + 6 * q^37 + 3 * q^40 + 6 * q^41 - 3 * q^44 + 4 * q^46 - 4 * q^47 + 3 * q^49 - 3 * q^50 + 2 * q^52 + 6 * q^53 + 3 * q^55 - 3 * q^56 + 2 * q^58 - 12 * q^59 - 10 * q^61 + 4 * q^62 + 3 * q^64 - 2 * q^65 - 2 * q^68 + 3 * q^70 - 4 * q^71 - 10 * q^73 - 6 * q^74 - 3 * q^77 - 4 * q^79 - 3 * q^80 - 6 * q^82 + 2 * q^85 + 3 * q^88 + 2 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^94 - 6 * q^97 - 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.80642	−1.05571	−0.527856	−	0.849334i	$-0.677004\pi$
$13$	−3.80642	−1.05571	−0.527856	+	0.849334i	$0.677004\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.05086	−1.22501	−0.612506	−	0.790466i	$-0.709839\pi$
$17$	−5.05086	−1.22501	−0.612506	+	0.790466i	$0.709839\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	3.05086	0.636147	0.318074	−	0.948066i	$-0.396964\pi$
$23$	3.05086	0.636147	0.318074	+	0.948066i	$0.396964\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.80642	0.746501
$27$	0	0
$28$	1.00000	0.188982
$29$	−0.755569	−0.140306	−0.0701528	−	0.997536i	$-0.522349\pi$
$29$	−0.755569	−0.140306	−0.0701528	+	0.997536i	$0.522349\pi$
$30$	0	0
$31$	7.61285	1.36731	0.683654	−	0.729806i	$-0.260390\pi$
$31$	7.61285	1.36731	0.683654	+	0.729806i	$0.260390\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.05086	0.866215
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.56199	1.07878	0.539392	−	0.842054i	$-0.318654\pi$
$37$	6.56199	1.07878	0.539392	+	0.842054i	$0.318654\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	0.158114
$41$	−6.85728	−1.07093	−0.535464	−	0.844558i	$-0.679863\pi$
$41$	−6.85728	−1.07093	−0.535464	+	0.844558i	$0.679863\pi$
$42$	0	0
$43$	8.85728	1.35072	0.675361	−	0.737487i	$-0.263988\pi$
$43$	8.85728	1.35072	0.675361	+	0.737487i	$0.263988\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−3.05086	−0.449824
$47$	7.61285	1.11045	0.555224	−	0.831701i	$-0.312633\pi$
$47$	7.61285	1.11045	0.555224	+	0.831701i	$0.312633\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.80642	−0.527856
$53$	10.8573	1.49136	0.745681	−	0.666303i	$-0.232124\pi$
$53$	10.8573	1.49136	0.745681	+	0.666303i	$0.232124\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	0.755569	0.0992110
$59$	4.85728	0.632364	0.316182	−	0.948699i	$-0.397599\pi$
$59$	4.85728	0.632364	0.316182	+	0.948699i	$0.397599\pi$
$60$	0	0
$61$	−12.1017	−1.54947	−0.774733	−	0.632289i	$-0.782116\pi$
$61$	−12.1017	−1.54947	−0.774733	+	0.632289i	$0.782116\pi$
$62$	−7.61285	−0.966833
$63$	0	0
$64$	1.00000	0.125000
$65$	3.80642	0.472129
$66$	0	0
$67$	4.29529	0.524753	0.262376	−	0.964966i	$-0.415494\pi$
$67$	4.29529	0.524753	0.262376	+	0.964966i	$0.415494\pi$
$68$	−5.05086	−0.612506
$69$	0	0
$70$	1.00000	0.119523
$71$	−1.24443	−0.147687	−0.0738434	−	0.997270i	$-0.523527\pi$
$71$	−1.24443	−0.147687	−0.0738434	+	0.997270i	$0.523527\pi$
$72$	0	0
$73$	−12.3684	−1.44761	−0.723807	−	0.690003i	$-0.757609\pi$
$73$	−12.3684	−1.44761	−0.723807	+	0.690003i	$0.757609\pi$
$74$	−6.56199	−0.762816
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−1.51114	−0.170016	−0.0850081	−	0.996380i	$-0.527092\pi$
$79$	−1.51114	−0.170016	−0.0850081	+	0.996380i	$0.527092\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.85728	0.757260
$83$	−8.85728	−0.972213	−0.486106	−	0.873900i	$-0.661583\pi$
$83$	−8.85728	−0.972213	−0.486106	+	0.873900i	$0.661583\pi$
$84$	0	0
$85$	5.05086	0.547842
$86$	−8.85728	−0.955105
$87$	0	0
$88$	1.00000	0.106600
$89$	−12.6637	−1.34235	−0.671175	−	0.741299i	$-0.734210\pi$
$89$	−12.6637	−1.34235	−0.671175	+	0.741299i	$0.734210\pi$
$90$	0	0
$91$	−3.80642	−0.399022
$92$	3.05086	0.318074
$93$	0	0
$94$	−7.61285	−0.785205
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−5.24443	−0.516749	−0.258375	−	0.966045i	$-0.583187\pi$
$103$	−5.24443	−0.516749	−0.258375	+	0.966045i	$0.583187\pi$
$104$	3.80642	0.373251
$105$	0	0
$106$	−10.8573	−1.05455
$107$	7.61285	0.735962	0.367981	−	0.929833i	$-0.380049\pi$
$107$	7.61285	0.735962	0.367981	+	0.929833i	$0.380049\pi$
$108$	0	0
$109$	−6.29529	−0.602979	−0.301490	−	0.953469i	$-0.597484\pi$
$109$	−6.29529	−0.602979	−0.301490	+	0.953469i	$0.597484\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	−1.05086	−0.0988561	−0.0494281	−	0.998778i	$-0.515740\pi$
$113$	−1.05086	−0.0988561	−0.0494281	+	0.998778i	$0.515740\pi$
$114$	0	0
$115$	−3.05086	−0.284494
$116$	−0.755569	−0.0701528
$117$	0	0
$118$	−4.85728	−0.447149
$119$	−5.05086	−0.463011
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.1017	1.09564
$123$	0	0
$124$	7.61285	0.683654
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.10171	−0.541439	−0.270720	−	0.962658i	$-0.587262\pi$
$127$	−6.10171	−0.541439	−0.270720	+	0.962658i	$0.587262\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.80642	−0.333845
$131$	12.8573	1.12335	0.561673	−	0.827359i	$-0.310158\pi$
$131$	12.8573	1.12335	0.561673	+	0.827359i	$0.310158\pi$
$132$	0	0
$133$	0	0
$134$	−4.29529	−0.371056
$135$	0	0
$136$	5.05086	0.433107
$137$	−6.56199	−0.560629	−0.280314	−	0.959908i	$-0.590439\pi$
$137$	−6.56199	−0.560629	−0.280314	+	0.959908i	$0.590439\pi$
$138$	0	0
$139$	−14.1017	−1.19609	−0.598046	−	0.801462i	$-0.704056\pi$
$139$	−14.1017	−1.19609	−0.598046	+	0.801462i	$0.704056\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	1.24443	0.104430
$143$	3.80642	0.318309
$144$	0	0
$145$	0.755569	0.0627466
$146$	12.3684	1.02362
$147$	0	0
$148$	6.56199	0.539392
$149$	−14.8573	−1.21716	−0.608578	−	0.793494i	$-0.708260\pi$
$149$	−14.8573	−1.21716	−0.608578	+	0.793494i	$0.708260\pi$
$150$	0	0
$151$	−9.51114	−0.774005	−0.387003	−	0.922079i	$-0.626490\pi$
$151$	−9.51114	−0.774005	−0.387003	+	0.922079i	$0.626490\pi$
$152$	0	0
$153$	0	0
$154$	1.00000	0.0805823
$155$	−7.61285	−0.611479
$156$	0	0
$157$	−6.85728	−0.547270	−0.273635	−	0.961834i	$-0.588226\pi$
$157$	−6.85728	−0.547270	−0.273635	+	0.961834i	$0.588226\pi$
$158$	1.51114	0.120220
$159$	0	0
$160$	1.00000	0.0790569
$161$	3.05086	0.240441
$162$	0	0
$163$	1.80642	0.141490	0.0707450	−	0.997494i	$-0.477462\pi$
$163$	1.80642	0.141490	0.0707450	+	0.997494i	$0.477462\pi$
$164$	−6.85728	−0.535464
$165$	0	0
$166$	8.85728	0.687458
$167$	13.1526	1.01778	0.508888	−	0.860833i	$-0.330057\pi$
$167$	13.1526	1.01778	0.508888	+	0.860833i	$0.330057\pi$
$168$	0	0
$169$	1.48886	0.114528
$170$	−5.05086	−0.387383
$171$	0	0
$172$	8.85728	0.675361
$173$	1.14272	0.0868795	0.0434397	−	0.999056i	$-0.486168\pi$
$173$	1.14272	0.0868795	0.0434397	+	0.999056i	$0.486168\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	12.6637	0.949185
$179$	−15.6128	−1.16696	−0.583480	−	0.812128i	$-0.698309\pi$
$179$	−15.6128	−1.16696	−0.583480	+	0.812128i	$0.698309\pi$
$180$	0	0
$181$	−15.8064	−1.17488	−0.587441	−	0.809267i	$-0.699865\pi$
$181$	−15.8064	−1.17488	−0.587441	+	0.809267i	$0.699865\pi$
$182$	3.80642	0.282151
$183$	0	0
$184$	−3.05086	−0.224912
$185$	−6.56199	−0.482447
$186$	0	0
$187$	5.05086	0.369355
$188$	7.61285	0.555224
$189$	0	0
$190$	0	0
$191$	−18.3684	−1.32909	−0.664546	−	0.747247i	$-0.731375\pi$
$191$	−18.3684	−1.32909	−0.664546	+	0.747247i	$0.731375\pi$
$192$	0	0
$193$	−1.14272	−0.0822549	−0.0411274	−	0.999154i	$-0.513095\pi$
$193$	−1.14272	−0.0822549	−0.0411274	+	0.999154i	$0.513095\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	−7.71456	−0.549639	−0.274820	−	0.961496i	$-0.588618\pi$
$197$	−7.71456	−0.549639	−0.274820	+	0.961496i	$0.588618\pi$
$198$	0	0
$199$	25.3274	1.79541	0.897706	−	0.440595i	$-0.145232\pi$
$199$	25.3274	1.79541	0.897706	+	0.440595i	$0.145232\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	2.00000	0.140720
$203$	−0.755569	−0.0530305
$204$	0	0
$205$	6.85728	0.478933
$206$	5.24443	0.365397
$207$	0	0
$208$	−3.80642	−0.263928
$209$	0	0
$210$	0	0
$211$	−4.29529	−0.295700	−0.147850	−	0.989010i	$-0.547235\pi$
$211$	−4.29529	−0.295700	−0.147850	+	0.989010i	$0.547235\pi$
$212$	10.8573	0.745681
$213$	0	0
$214$	−7.61285	−0.520404
$215$	−8.85728	−0.604061
$216$	0	0
$217$	7.61285	0.516794
$218$	6.29529	0.426371
$219$	0	0
$220$	1.00000	0.0674200
$221$	19.2257	1.29326
$222$	0	0
$223$	8.85728	0.593127	0.296564	−	0.955013i	$-0.404159\pi$
$223$	8.85728	0.593127	0.296564	+	0.955013i	$0.404159\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	1.05086	0.0699018
$227$	−11.3461	−0.753070	−0.376535	−	0.926402i	$-0.622885\pi$
$227$	−11.3461	−0.753070	−0.376535	+	0.926402i	$0.622885\pi$
$228$	0	0
$229$	−24.9304	−1.64745	−0.823724	−	0.566991i	$-0.808107\pi$
$229$	−24.9304	−1.64745	−0.823724	+	0.566991i	$0.808107\pi$
$230$	3.05086	0.201167
$231$	0	0
$232$	0.755569	0.0496055
$233$	15.7146	1.02949	0.514747	−	0.857342i	$-0.327886\pi$
$233$	15.7146	1.02949	0.514747	+	0.857342i	$0.327886\pi$
$234$	0	0
$235$	−7.61285	−0.496607
$236$	4.85728	0.316182
$237$	0	0
$238$	5.05086	0.327398
$239$	18.2766	1.18221	0.591106	−	0.806594i	$-0.298692\pi$
$239$	18.2766	1.18221	0.591106	+	0.806594i	$0.298692\pi$
$240$	0	0
$241$	−2.38715	−0.153770	−0.0768850	−	0.997040i	$-0.524497\pi$
$241$	−2.38715	−0.153770	−0.0768850	+	0.997040i	$0.524497\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−12.1017	−0.774733
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	−7.61285	−0.483416
$249$	0	0
$250$	1.00000	0.0632456
$251$	3.14272	0.198367	0.0991834	−	0.995069i	$-0.468377\pi$
$251$	3.14272	0.198367	0.0991834	+	0.995069i	$0.468377\pi$
$252$	0	0
$253$	−3.05086	−0.191806
$254$	6.10171	0.382855
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.1017	−0.754884	−0.377442	−	0.926033i	$-0.623196\pi$
$257$	−12.1017	−0.754884	−0.377442	+	0.926033i	$0.623196\pi$
$258$	0	0
$259$	6.56199	0.407742
$260$	3.80642	0.236064
$261$	0	0
$262$	−12.8573	−0.794325
$263$	11.6128	0.716079	0.358039	−	0.933706i	$-0.383445\pi$
$263$	11.6128	0.716079	0.358039	+	0.933706i	$0.383445\pi$
$264$	0	0
$265$	−10.8573	−0.666957
$266$	0	0
$267$	0	0
$268$	4.29529	0.262376
$269$	15.3274	0.934528	0.467264	−	0.884118i	$-0.345240\pi$
$269$	15.3274	0.934528	0.467264	+	0.884118i	$0.345240\pi$
$270$	0	0
$271$	1.71456	0.104152	0.0520760	−	0.998643i	$-0.483416\pi$
$271$	1.71456	0.104152	0.0520760	+	0.998643i	$0.483416\pi$
$272$	−5.05086	−0.306253
$273$	0	0
$274$	6.56199	0.396424
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	14.1017	0.845764
$279$	0	0
$280$	1.00000	0.0597614
$281$	9.90813	0.591070	0.295535	−	0.955332i	$-0.404502\pi$
$281$	9.90813	0.591070	0.295535	+	0.955332i	$0.404502\pi$
$282$	0	0
$283$	−31.2543	−1.85787	−0.928937	−	0.370238i	$-0.879276\pi$
$283$	−31.2543	−1.85787	−0.928937	+	0.370238i	$0.879276\pi$
$284$	−1.24443	−0.0738434
$285$	0	0
$286$	−3.80642	−0.225079
$287$	−6.85728	−0.404772
$288$	0	0
$289$	8.51114	0.500655
$290$	−0.755569	−0.0443685
$291$	0	0
$292$	−12.3684	−0.723807
$293$	15.2444	0.890589	0.445295	−	0.895384i	$-0.353099\pi$
$293$	15.2444	0.890589	0.445295	+	0.895384i	$0.353099\pi$
$294$	0	0
$295$	−4.85728	−0.282802
$296$	−6.56199	−0.381408
$297$	0	0
$298$	14.8573	0.860659
$299$	−11.6128	−0.671588
$300$	0	0
$301$	8.85728	0.510525
$302$	9.51114	0.547304
$303$	0	0
$304$	0	0
$305$	12.1017	0.692942
$306$	0	0
$307$	−0.561993	−0.0320746	−0.0160373	−	0.999871i	$-0.505105\pi$
$307$	−0.561993	−0.0320746	−0.0160373	+	0.999871i	$0.505105\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	7.61285	0.432381
$311$	−9.80642	−0.556071	−0.278036	−	0.960571i	$-0.589683\pi$
$311$	−9.80642	−0.556071	−0.278036	+	0.960571i	$0.589683\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	6.85728	0.386979
$315$	0	0
$316$	−1.51114	−0.0850081
$317$	8.36842	0.470017	0.235009	−	0.971993i	$-0.424488\pi$
$317$	8.36842	0.470017	0.235009	+	0.971993i	$0.424488\pi$
$318$	0	0
$319$	0.755569	0.0423037
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−3.05086	−0.170018
$323$	0	0
$324$	0	0
$325$	−3.80642	−0.211142
$326$	−1.80642	−0.100049
$327$	0	0
$328$	6.85728	0.378630
$329$	7.61285	0.419710
$330$	0	0
$331$	−28.9403	−1.59070	−0.795350	−	0.606150i	$-0.792713\pi$
$331$	−28.9403	−1.59070	−0.795350	+	0.606150i	$0.792713\pi$
$332$	−8.85728	−0.486106
$333$	0	0
$334$	−13.1526	−0.719676
$335$	−4.29529	−0.234677
$336$	0	0
$337$	1.34614	0.0733290	0.0366645	−	0.999328i	$-0.488327\pi$
$337$	1.34614	0.0733290	0.0366645	+	0.999328i	$0.488327\pi$
$338$	−1.48886	−0.0809834
$339$	0	0
$340$	5.05086	0.273921
$341$	−7.61285	−0.412259
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.85728	−0.477552
$345$	0	0
$346$	−1.14272	−0.0614331
$347$	−23.6128	−1.26760	−0.633802	−	0.773495i	$-0.718507\pi$
$347$	−23.6128	−1.26760	−0.633802	+	0.773495i	$0.718507\pi$
$348$	0	0
$349$	15.3274	0.820457	0.410229	−	0.911983i	$-0.365449\pi$
$349$	15.3274	0.820457	0.410229	+	0.911983i	$0.365449\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	−19.3274	−1.02869	−0.514347	−	0.857582i	$-0.671966\pi$
$353$	−19.3274	−1.02869	−0.514347	+	0.857582i	$0.671966\pi$
$354$	0	0
$355$	1.24443	0.0660476
$356$	−12.6637	−0.671175
$357$	0	0
$358$	15.6128	0.825165
$359$	−8.56199	−0.451885	−0.225942	−	0.974141i	$-0.572546\pi$
$359$	−8.56199	−0.451885	−0.225942	+	0.974141i	$0.572546\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	15.8064	0.830767
$363$	0	0
$364$	−3.80642	−0.199511
$365$	12.3684	0.647393
$366$	0	0
$367$	18.7556	0.979033	0.489516	−	0.871994i	$-0.337173\pi$
$367$	18.7556	0.979033	0.489516	+	0.871994i	$0.337173\pi$
$368$	3.05086	0.159037
$369$	0	0
$370$	6.56199	0.341142
$371$	10.8573	0.563682
$372$	0	0
$373$	−21.6128	−1.11907	−0.559535	−	0.828806i	$-0.689020\pi$
$373$	−21.6128	−1.11907	−0.559535	+	0.828806i	$0.689020\pi$
$374$	−5.05086	−0.261174
$375$	0	0
$376$	−7.61285	−0.392603
$377$	2.87601	0.148122
$378$	0	0
$379$	29.7146	1.52633	0.763167	−	0.646201i	$-0.223643\pi$
$379$	29.7146	1.52633	0.763167	+	0.646201i	$0.223643\pi$
$380$	0	0
$381$	0	0
$382$	18.3684	0.939810
$383$	0.387152	0.0197826	0.00989128	−	0.999951i	$-0.496851\pi$
$383$	0.387152	0.0197826	0.00989128	+	0.999951i	$0.496851\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	1.14272	0.0581630
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−27.4479	−1.39166	−0.695831	−	0.718206i	$-0.744964\pi$
$389$	−27.4479	−1.39166	−0.695831	+	0.718206i	$0.744964\pi$
$390$	0	0
$391$	−15.4094	−0.779288
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	7.71456	0.388654
$395$	1.51114	0.0760336
$396$	0	0
$397$	−20.9590	−1.05190	−0.525951	−	0.850515i	$-0.676290\pi$
$397$	−20.9590	−1.05190	−0.525951	+	0.850515i	$0.676290\pi$
$398$	−25.3274	−1.26955
$399$	0	0
$400$	1.00000	0.0500000
$401$	−30.2034	−1.50829	−0.754143	−	0.656710i	$-0.771948\pi$
$401$	−30.2034	−1.50829	−0.754143	+	0.656710i	$0.771948\pi$
$402$	0	0
$403$	−28.9777	−1.44348
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0.755569	0.0374982
$407$	−6.56199	−0.325266
$408$	0	0
$409$	−13.8163	−0.683170	−0.341585	−	0.939851i	$-0.610964\pi$
$409$	−13.8163	−0.683170	−0.341585	+	0.939851i	$0.610964\pi$
$410$	−6.85728	−0.338657
$411$	0	0
$412$	−5.24443	−0.258375
$413$	4.85728	0.239011
$414$	0	0
$415$	8.85728	0.434787
$416$	3.80642	0.186625
$417$	0	0
$418$	0	0
$419$	31.6958	1.54844	0.774221	−	0.632915i	$-0.218142\pi$
$419$	31.6958	1.54844	0.774221	+	0.632915i	$0.218142\pi$
$420$	0	0
$421$	13.2257	0.644581	0.322290	−	0.946641i	$-0.395547\pi$
$421$	13.2257	0.644581	0.322290	+	0.946641i	$0.395547\pi$
$422$	4.29529	0.209091
$423$	0	0
$424$	−10.8573	−0.527276
$425$	−5.05086	−0.245002
$426$	0	0
$427$	−12.1017	−0.585643
$428$	7.61285	0.367981
$429$	0	0
$430$	8.85728	0.427136
$431$	−14.6637	−0.706326	−0.353163	−	0.935562i	$-0.614894\pi$
$431$	−14.6637	−0.706326	−0.353163	+	0.935562i	$0.614894\pi$
$432$	0	0
$433$	−17.2257	−0.827814	−0.413907	−	0.910319i	$-0.635836\pi$
$433$	−17.2257	−0.827814	−0.413907	+	0.910319i	$0.635836\pi$
$434$	−7.61285	−0.365428
$435$	0	0
$436$	−6.29529	−0.301490
$437$	0	0
$438$	0	0
$439$	−35.6128	−1.69971	−0.849854	−	0.527018i	$-0.823310\pi$
$439$	−35.6128	−1.69971	−0.849854	+	0.527018i	$0.823310\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−19.2257	−0.914473
$443$	3.87955	0.184323	0.0921616	−	0.995744i	$-0.470622\pi$
$443$	3.87955	0.184323	0.0921616	+	0.995744i	$0.470622\pi$
$444$	0	0
$445$	12.6637	0.600317
$446$	−8.85728	−0.419404
$447$	0	0
$448$	1.00000	0.0472456
$449$	−22.7368	−1.07302	−0.536509	−	0.843895i	$-0.680257\pi$
$449$	−22.7368	−1.07302	−0.536509	+	0.843895i	$0.680257\pi$
$450$	0	0
$451$	6.85728	0.322897
$452$	−1.05086	−0.0494281
$453$	0	0
$454$	11.3461	0.532501
$455$	3.80642	0.178448
$456$	0	0
$457$	22.8573	1.06922	0.534609	−	0.845099i	$-0.320459\pi$
$457$	22.8573	1.06922	0.534609	+	0.845099i	$0.320459\pi$
$458$	24.9304	1.16492
$459$	0	0
$460$	−3.05086	−0.142247
$461$	26.2034	1.22041	0.610207	−	0.792242i	$-0.291086\pi$
$461$	26.2034	1.22041	0.610207	+	0.792242i	$0.291086\pi$
$462$	0	0
$463$	−3.87955	−0.180298	−0.0901491	−	0.995928i	$-0.528734\pi$
$463$	−3.87955	−0.180298	−0.0901491	+	0.995928i	$0.528734\pi$
$464$	−0.755569	−0.0350764
$465$	0	0
$466$	−15.7146	−0.727963
$467$	4.59057	0.212426	0.106213	−	0.994343i	$-0.466127\pi$
$467$	4.59057	0.212426	0.106213	+	0.994343i	$0.466127\pi$
$468$	0	0
$469$	4.29529	0.198338
$470$	7.61285	0.351154
$471$	0	0
$472$	−4.85728	−0.223574
$473$	−8.85728	−0.407258
$474$	0	0
$475$	0	0
$476$	−5.05086	−0.231506
$477$	0	0
$478$	−18.2766	−0.835950
$479$	−21.5111	−0.982869	−0.491434	−	0.870915i	$-0.663527\pi$
$479$	−21.5111	−0.982869	−0.491434	+	0.870915i	$0.663527\pi$
$480$	0	0
$481$	−24.9777	−1.13889
$482$	2.38715	0.108732
$483$	0	0
$484$	1.00000	0.0454545
$485$	2.00000	0.0908153
$486$	0	0
$487$	1.63158	0.0739341	0.0369671	−	0.999316i	$-0.488230\pi$
$487$	1.63158	0.0739341	0.0369671	+	0.999316i	$0.488230\pi$
$488$	12.1017	0.547819
$489$	0	0
$490$	1.00000	0.0451754
$491$	19.2257	0.867643	0.433822	−	0.900999i	$-0.357165\pi$
$491$	19.2257	0.867643	0.433822	+	0.900999i	$0.357165\pi$
$492$	0	0
$493$	3.81627	0.171876
$494$	0	0
$495$	0	0
$496$	7.61285	0.341827
$497$	−1.24443	−0.0558204
$498$	0	0
$499$	15.6128	0.698927	0.349464	−	0.936950i	$-0.386364\pi$
$499$	15.6128	0.698927	0.349464	+	0.936950i	$0.386364\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−3.14272	−0.140267
$503$	8.94914	0.399023	0.199511	−	0.979896i	$-0.436065\pi$
$503$	8.94914	0.399023	0.199511	+	0.979896i	$0.436065\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	3.05086	0.135627
$507$	0	0
$508$	−6.10171	−0.270720
$509$	15.3274	0.679375	0.339688	−	0.940538i	$-0.389679\pi$
$509$	15.3274	0.679375	0.339688	+	0.940538i	$0.389679\pi$
$510$	0	0
$511$	−12.3684	−0.547147
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.1017	0.533784
$515$	5.24443	0.231097
$516$	0	0
$517$	−7.61285	−0.334813
$518$	−6.56199	−0.288317
$519$	0	0
$520$	−3.80642	−0.166923
$521$	−3.72345	−0.163127	−0.0815636	−	0.996668i	$-0.525991\pi$
$521$	−3.72345	−0.163127	−0.0815636	+	0.996668i	$0.525991\pi$
$522$	0	0
$523$	−10.4603	−0.457396	−0.228698	−	0.973497i	$-0.573447\pi$
$523$	−10.4603	−0.457396	−0.228698	+	0.973497i	$0.573447\pi$
$524$	12.8573	0.561673
$525$	0	0
$526$	−11.6128	−0.506344
$527$	−38.4514	−1.67497
$528$	0	0
$529$	−13.6923	−0.595317
$530$	10.8573	0.471610
$531$	0	0
$532$	0	0
$533$	26.1017	1.13059
$534$	0	0
$535$	−7.61285	−0.329132
$536$	−4.29529	−0.185528
$537$	0	0
$538$	−15.3274	−0.660811
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	40.7467	1.75184	0.875918	−	0.482460i	$-0.160256\pi$
$541$	40.7467	1.75184	0.875918	+	0.482460i	$0.160256\pi$
$542$	−1.71456	−0.0736466
$543$	0	0
$544$	5.05086	0.216554
$545$	6.29529	0.269660
$546$	0	0
$547$	0.266706	0.0114035	0.00570177	−	0.999984i	$-0.498185\pi$
$547$	0.266706	0.0114035	0.00570177	+	0.999984i	$0.498185\pi$
$548$	−6.56199	−0.280314
$549$	0	0
$550$	1.00000	0.0426401
$551$	0	0
$552$	0	0
$553$	−1.51114	−0.0642601
$554$	2.00000	0.0849719
$555$	0	0
$556$	−14.1017	−0.598046
$557$	42.9403	1.81944	0.909718	−	0.415226i	$-0.136297\pi$
$557$	42.9403	1.81944	0.909718	+	0.415226i	$0.136297\pi$
$558$	0	0
$559$	−33.7146	−1.42597
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−9.90813	−0.417949
$563$	−25.9813	−1.09498	−0.547490	−	0.836812i	$-0.684417\pi$
$563$	−25.9813	−1.09498	−0.547490	+	0.836812i	$0.684417\pi$
$564$	0	0
$565$	1.05086	0.0442098
$566$	31.2543	1.31372
$567$	0	0
$568$	1.24443	0.0522152
$569$	−12.1936	−0.511181	−0.255591	−	0.966785i	$-0.582270\pi$
$569$	−12.1936	−0.511181	−0.255591	+	0.966785i	$0.582270\pi$
$570$	0	0
$571$	29.2355	1.22347	0.611735	−	0.791063i	$-0.290472\pi$
$571$	29.2355	1.22347	0.611735	+	0.791063i	$0.290472\pi$
$572$	3.80642	0.159155
$573$	0	0
$574$	6.85728	0.286217
$575$	3.05086	0.127229
$576$	0	0
$577$	−5.02227	−0.209080	−0.104540	−	0.994521i	$-0.533337\pi$
$577$	−5.02227	−0.209080	−0.104540	+	0.994521i	$0.533337\pi$
$578$	−8.51114	−0.354017
$579$	0	0
$580$	0.755569	0.0313733
$581$	−8.85728	−0.367462
$582$	0	0
$583$	−10.8573	−0.449663
$584$	12.3684	0.511809
$585$	0	0
$586$	−15.2444	−0.629742
$587$	−20.5906	−0.849864	−0.424932	−	0.905225i	$-0.639702\pi$
$587$	−20.5906	−0.849864	−0.424932	+	0.905225i	$0.639702\pi$
$588$	0	0
$589$	0	0
$590$	4.85728	0.199971
$591$	0	0
$592$	6.56199	0.269696
$593$	33.0509	1.35724	0.678618	−	0.734491i	$-0.262579\pi$
$593$	33.0509	1.35724	0.678618	+	0.734491i	$0.262579\pi$
$594$	0	0
$595$	5.05086	0.207065
$596$	−14.8573	−0.608578
$597$	0	0
$598$	11.6128	0.474885
$599$	−44.8573	−1.83282	−0.916409	−	0.400242i	$-0.868926\pi$
$599$	−44.8573	−1.83282	−0.916409	+	0.400242i	$0.868926\pi$
$600$	0	0
$601$	−33.6128	−1.37110	−0.685548	−	0.728027i	$-0.740437\pi$
$601$	−33.6128	−1.37110	−0.685548	+	0.728027i	$0.740437\pi$
$602$	−8.85728	−0.360996
$603$	0	0
$604$	−9.51114	−0.387003
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	27.4291	1.11331	0.556657	−	0.830743i	$-0.312084\pi$
$607$	27.4291	1.11331	0.556657	+	0.830743i	$0.312084\pi$
$608$	0	0
$609$	0	0
$610$	−12.1017	−0.489984
$611$	−28.9777	−1.17231
$612$	0	0
$613$	−38.7368	−1.56457	−0.782283	−	0.622923i	$-0.785945\pi$
$613$	−38.7368	−1.56457	−0.782283	+	0.622923i	$0.785945\pi$
$614$	0.561993	0.0226802
$615$	0	0
$616$	1.00000	0.0402911
$617$	43.1526	1.73726	0.868628	−	0.495464i	$-0.165002\pi$
$617$	43.1526	1.73726	0.868628	+	0.495464i	$0.165002\pi$
$618$	0	0
$619$	20.6824	0.831297	0.415649	−	0.909525i	$-0.363555\pi$
$619$	20.6824	0.831297	0.415649	+	0.909525i	$0.363555\pi$
$620$	−7.61285	−0.305739
$621$	0	0
$622$	9.80642	0.393202
$623$	−12.6637	−0.507361
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	−6.85728	−0.273635
$629$	−33.1437	−1.32152
$630$	0	0
$631$	27.6128	1.09925	0.549625	−	0.835411i	$-0.314771\pi$
$631$	27.6128	1.09925	0.549625	+	0.835411i	$0.314771\pi$
$632$	1.51114	0.0601098
$633$	0	0
$634$	−8.36842	−0.332352
$635$	6.10171	0.242139
$636$	0	0
$637$	−3.80642	−0.150816
$638$	−0.755569	−0.0299133
$639$	0	0
$640$	1.00000	0.0395285
$641$	−20.4889	−0.809261	−0.404631	−	0.914480i	$-0.632600\pi$
$641$	−20.4889	−0.809261	−0.404631	+	0.914480i	$0.632600\pi$
$642$	0	0
$643$	26.4514	1.04314	0.521571	−	0.853208i	$-0.325346\pi$
$643$	26.4514	1.04314	0.521571	+	0.853208i	$0.325346\pi$
$644$	3.05086	0.120221
$645$	0	0
$646$	0	0
$647$	48.5531	1.90882	0.954410	−	0.298500i	$-0.0964862\pi$
$647$	48.5531	1.90882	0.954410	+	0.298500i	$0.0964862\pi$
$648$	0	0
$649$	−4.85728	−0.190665
$650$	3.80642	0.149300
$651$	0	0
$652$	1.80642	0.0707450
$653$	−6.26671	−0.245235	−0.122618	−	0.992454i	$-0.539129\pi$
$653$	−6.26671	−0.245235	−0.122618	+	0.992454i	$0.539129\pi$
$654$	0	0
$655$	−12.8573	−0.502375
$656$	−6.85728	−0.267732
$657$	0	0
$658$	−7.61285	−0.296780
$659$	−1.51114	−0.0588656	−0.0294328	−	0.999567i	$-0.509370\pi$
$659$	−1.51114	−0.0588656	−0.0294328	+	0.999567i	$0.509370\pi$
$660$	0	0
$661$	−28.5433	−1.11020	−0.555102	−	0.831782i	$-0.687321\pi$
$661$	−28.5433	−1.11020	−0.555102	+	0.831782i	$0.687321\pi$
$662$	28.9403	1.12479
$663$	0	0
$664$	8.85728	0.343729
$665$	0	0
$666$	0	0
$667$	−2.30513	−0.0892550
$668$	13.1526	0.508888
$669$	0	0
$670$	4.29529	0.165941
$671$	12.1017	0.467181
$672$	0	0
$673$	−10.8573	−0.418517	−0.209259	−	0.977860i	$-0.567105\pi$
$673$	−10.8573	−0.418517	−0.209259	+	0.977860i	$0.567105\pi$
$674$	−1.34614	−0.0518514
$675$	0	0
$676$	1.48886	0.0572639
$677$	15.0607	0.578830	0.289415	−	0.957204i	$-0.406539\pi$
$677$	15.0607	0.578830	0.289415	+	0.957204i	$0.406539\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	−5.05086	−0.193691
$681$	0	0
$682$	7.61285	0.291511
$683$	−18.5718	−0.710632	−0.355316	−	0.934746i	$-0.615627\pi$
$683$	−18.5718	−0.710632	−0.355316	+	0.934746i	$0.615627\pi$
$684$	0	0
$685$	6.56199	0.250721
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	8.85728	0.337681
$689$	−41.3274	−1.57445
$690$	0	0
$691$	−24.2953	−0.924236	−0.462118	−	0.886818i	$-0.652910\pi$
$691$	−24.2953	−0.924236	−0.462118	+	0.886818i	$0.652910\pi$
$692$	1.14272	0.0434397
$693$	0	0
$694$	23.6128	0.896331
$695$	14.1017	0.534908
$696$	0	0
$697$	34.6351	1.31190
$698$	−15.3274	−0.580151
$699$	0	0
$700$	1.00000	0.0377964
$701$	−12.1847	−0.460209	−0.230105	−	0.973166i	$-0.573907\pi$
$701$	−12.1847	−0.460209	−0.230105	+	0.973166i	$0.573907\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	19.3274	0.727397
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	1.79658	0.0674720	0.0337360	−	0.999431i	$-0.489259\pi$
$709$	1.79658	0.0674720	0.0337360	+	0.999431i	$0.489259\pi$
$710$	−1.24443	−0.0467027
$711$	0	0
$712$	12.6637	0.474592
$713$	23.2257	0.869809
$714$	0	0
$715$	−3.80642	−0.142352
$716$	−15.6128	−0.583480
$717$	0	0
$718$	8.56199	0.319531
$719$	−31.9081	−1.18997	−0.594986	−	0.803736i	$-0.702843\pi$
$719$	−31.9081	−1.18997	−0.594986	+	0.803736i	$0.702843\pi$
$720$	0	0
$721$	−5.24443	−0.195313
$722$	19.0000	0.707107
$723$	0	0
$724$	−15.8064	−0.587441
$725$	−0.755569	−0.0280611
$726$	0	0
$727$	14.3684	0.532895	0.266448	−	0.963849i	$-0.414150\pi$
$727$	14.3684	0.532895	0.266448	+	0.963849i	$0.414150\pi$
$728$	3.80642	0.141075
$729$	0	0
$730$	−12.3684	−0.457776
$731$	−44.7368	−1.65465
$732$	0	0
$733$	−43.8064	−1.61803	−0.809014	−	0.587790i	$-0.799998\pi$
$733$	−43.8064	−1.61803	−0.809014	+	0.587790i	$0.799998\pi$
$734$	−18.7556	−0.692281
$735$	0	0
$736$	−3.05086	−0.112456
$737$	−4.29529	−0.158219
$738$	0	0
$739$	14.7841	0.543844	0.271922	−	0.962319i	$-0.412341\pi$
$739$	14.7841	0.543844	0.271922	+	0.962319i	$0.412341\pi$
$740$	−6.56199	−0.241224
$741$	0	0
$742$	−10.8573	−0.398583
$743$	−20.9777	−0.769598	−0.384799	−	0.923000i	$-0.625729\pi$
$743$	−20.9777	−0.769598	−0.384799	+	0.923000i	$0.625729\pi$
$744$	0	0
$745$	14.8573	0.544329
$746$	21.6128	0.791303
$747$	0	0
$748$	5.05086	0.184678
$749$	7.61285	0.278167
$750$	0	0
$751$	−41.5308	−1.51548	−0.757741	−	0.652556i	$-0.773697\pi$
$751$	−41.5308	−1.51548	−0.757741	+	0.652556i	$0.773697\pi$
$752$	7.61285	0.277612
$753$	0	0
$754$	−2.87601	−0.104738
$755$	9.51114	0.346146
$756$	0	0
$757$	13.2543	0.481735	0.240867	−	0.970558i	$-0.422568\pi$
$757$	13.2543	0.481735	0.240867	+	0.970558i	$0.422568\pi$
$758$	−29.7146	−1.07928
$759$	0	0
$760$	0	0
$761$	15.5941	0.565286	0.282643	−	0.959225i	$-0.408789\pi$
$761$	15.5941	0.565286	0.282643	+	0.959225i	$0.408789\pi$
$762$	0	0
$763$	−6.29529	−0.227905
$764$	−18.3684	−0.664546
$765$	0	0
$766$	−0.387152	−0.0139884
$767$	−18.4889	−0.667594
$768$	0	0
$769$	25.0420	0.903036	0.451518	−	0.892262i	$-0.350883\pi$
$769$	25.0420	0.903036	0.451518	+	0.892262i	$0.350883\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	−1.14272	−0.0411274
$773$	1.46659	0.0527495	0.0263747	−	0.999652i	$-0.491604\pi$
$773$	1.46659	0.0527495	0.0263747	+	0.999652i	$0.491604\pi$
$774$	0	0
$775$	7.61285	0.273462
$776$	2.00000	0.0717958
$777$	0	0
$778$	27.4479	0.984053
$779$	0	0
$780$	0	0
$781$	1.24443	0.0445293
$782$	15.4094	0.551040
$783$	0	0
$784$	1.00000	0.0357143
$785$	6.85728	0.244747
$786$	0	0
$787$	33.6860	1.20078	0.600388	−	0.799709i	$-0.295013\pi$
$787$	33.6860	1.20078	0.600388	+	0.799709i	$0.295013\pi$
$788$	−7.71456	−0.274820
$789$	0	0
$790$	−1.51114	−0.0537639
$791$	−1.05086	−0.0373641
$792$	0	0
$793$	46.0642	1.63579
$794$	20.9590	0.743807
$795$	0	0
$796$	25.3274	0.897706
$797$	36.2480	1.28397	0.641984	−	0.766718i	$-0.278111\pi$
$797$	36.2480	1.28397	0.641984	+	0.766718i	$0.278111\pi$
$798$	0	0
$799$	−38.4514	−1.36031
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	30.2034	1.06652
$803$	12.3684	0.436472
$804$	0	0
$805$	−3.05086	−0.107529
$806$	28.9777	1.02070
$807$	0	0
$808$	2.00000	0.0703598
$809$	−30.6824	−1.07874	−0.539369	−	0.842070i	$-0.681337\pi$
$809$	−30.6824	−1.07874	−0.539369	+	0.842070i	$0.681337\pi$
$810$	0	0
$811$	−45.9180	−1.61240	−0.806199	−	0.591644i	$-0.798479\pi$
$811$	−45.9180	−1.61240	−0.806199	+	0.591644i	$0.798479\pi$
$812$	−0.755569	−0.0265153
$813$	0	0
$814$	6.56199	0.229998
$815$	−1.80642	−0.0632763
$816$	0	0
$817$	0	0
$818$	13.8163	0.483074
$819$	0	0
$820$	6.85728	0.239467
$821$	−45.4924	−1.58770	−0.793848	−	0.608116i	$-0.791925\pi$
$821$	−45.4924	−1.58770	−0.793848	+	0.608116i	$0.791925\pi$
$822$	0	0
$823$	−54.9590	−1.91575	−0.957875	−	0.287186i	$-0.907280\pi$
$823$	−54.9590	−1.91575	−0.957875	+	0.287186i	$0.907280\pi$
$824$	5.24443	0.182698
$825$	0	0
$826$	−4.85728	−0.169006
$827$	26.1017	0.907645	0.453823	−	0.891092i	$-0.350060\pi$
$827$	26.1017	0.907645	0.453823	+	0.891092i	$0.350060\pi$
$828$	0	0
$829$	32.1936	1.11813	0.559065	−	0.829124i	$-0.311160\pi$
$829$	32.1936	1.11813	0.559065	+	0.829124i	$0.311160\pi$
$830$	−8.85728	−0.307441
$831$	0	0
$832$	−3.80642	−0.131964
$833$	−5.05086	−0.175002
$834$	0	0
$835$	−13.1526	−0.455163
$836$	0	0
$837$	0	0
$838$	−31.6958	−1.09491
$839$	20.1116	0.694328	0.347164	−	0.937804i	$-0.387145\pi$
$839$	20.1116	0.694328	0.347164	+	0.937804i	$0.387145\pi$
$840$	0	0
$841$	−28.4291	−0.980314
$842$	−13.2257	−0.455788
$843$	0	0
$844$	−4.29529	−0.147850
$845$	−1.48886	−0.0512184
$846$	0	0
$847$	1.00000	0.0343604
$848$	10.8573	0.372840
$849$	0	0
$850$	5.05086	0.173243
$851$	20.0197	0.686266
$852$	0	0
$853$	−19.2730	−0.659895	−0.329948	−	0.943999i	$-0.607031\pi$
$853$	−19.2730	−0.659895	−0.329948	+	0.943999i	$0.607031\pi$
$854$	12.1017	0.414112
$855$	0	0
$856$	−7.61285	−0.260202
$857$	9.05086	0.309171	0.154586	−	0.987979i	$-0.450596\pi$
$857$	9.05086	0.309171	0.154586	+	0.987979i	$0.450596\pi$
$858$	0	0
$859$	−2.19358	−0.0748439	−0.0374219	−	0.999300i	$-0.511915\pi$
$859$	−2.19358	−0.0748439	−0.0374219	+	0.999300i	$0.511915\pi$
$860$	−8.85728	−0.302031
$861$	0	0
$862$	14.6637	0.499448
$863$	−15.4380	−0.525516	−0.262758	−	0.964862i	$-0.584632\pi$
$863$	−15.4380	−0.525516	−0.262758	+	0.964862i	$0.584632\pi$
$864$	0	0
$865$	−1.14272	−0.0388537
$866$	17.2257	0.585353
$867$	0	0
$868$	7.61285	0.258397
$869$	1.51114	0.0512618
$870$	0	0
$871$	−16.3497	−0.553988
$872$	6.29529	0.213185
$873$	0	0
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	23.1811	0.782772	0.391386	−	0.920227i	$-0.371996\pi$
$877$	23.1811	0.782772	0.391386	+	0.920227i	$0.371996\pi$
$878$	35.6128	1.20188
$879$	0	0
$880$	1.00000	0.0337100
$881$	−14.3783	−0.484416	−0.242208	−	0.970224i	$-0.577872\pi$
$881$	−14.3783	−0.484416	−0.242208	+	0.970224i	$0.577872\pi$
$882$	0	0
$883$	54.5433	1.83553	0.917763	−	0.397128i	$-0.129993\pi$
$883$	54.5433	1.83553	0.917763	+	0.397128i	$0.129993\pi$
$884$	19.2257	0.646630
$885$	0	0
$886$	−3.87955	−0.130336
$887$	−41.8894	−1.40651	−0.703254	−	0.710939i	$-0.748270\pi$
$887$	−41.8894	−1.40651	−0.703254	+	0.710939i	$0.748270\pi$
$888$	0	0
$889$	−6.10171	−0.204645
$890$	−12.6637	−0.424488
$891$	0	0
$892$	8.85728	0.296564
$893$	0	0
$894$	0	0
$895$	15.6128	0.521880
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	22.7368	0.758738
$899$	−5.75203	−0.191841
$900$	0	0
$901$	−54.8385	−1.82694
$902$	−6.85728	−0.228322
$903$	0	0
$904$	1.05086	0.0349509
$905$	15.8064	0.525423
$906$	0	0
$907$	8.68244	0.288296	0.144148	−	0.989556i	$-0.453956\pi$
$907$	8.68244	0.288296	0.144148	+	0.989556i	$0.453956\pi$
$908$	−11.3461	−0.376535
$909$	0	0
$910$	−3.80642	−0.126182
$911$	−32.4701	−1.07578	−0.537892	−	0.843014i	$-0.680779\pi$
$911$	−32.4701	−1.07578	−0.537892	+	0.843014i	$0.680779\pi$
$912$	0	0
$913$	8.85728	0.293133
$914$	−22.8573	−0.756052
$915$	0	0
$916$	−24.9304	−0.823724
$917$	12.8573	0.424585
$918$	0	0
$919$	2.87601	0.0948710	0.0474355	−	0.998874i	$-0.484895\pi$
$919$	2.87601	0.0948710	0.0474355	+	0.998874i	$0.484895\pi$
$920$	3.05086	0.100584
$921$	0	0
$922$	−26.2034	−0.862964
$923$	4.73683	0.155915
$924$	0	0
$925$	6.56199	0.215757
$926$	3.87955	0.127490
$927$	0	0
$928$	0.755569	0.0248028
$929$	−17.0509	−0.559420	−0.279710	−	0.960084i	$-0.590238\pi$
$929$	−17.0509	−0.559420	−0.279710	+	0.960084i	$0.590238\pi$
$930$	0	0
$931$	0	0
$932$	15.7146	0.514747
$933$	0	0
$934$	−4.59057	−0.150208
$935$	−5.05086	−0.165181
$936$	0	0
$937$	−11.2444	−0.367340	−0.183670	−	0.982988i	$-0.558798\pi$
$937$	−11.2444	−0.367340	−0.183670	+	0.982988i	$0.558798\pi$
$938$	−4.29529	−0.140246
$939$	0	0
$940$	−7.61285	−0.248304
$941$	−30.2034	−0.984603	−0.492302	−	0.870425i	$-0.663844\pi$
$941$	−30.2034	−0.984603	−0.492302	+	0.870425i	$0.663844\pi$
$942$	0	0
$943$	−20.9206	−0.681267
$944$	4.85728	0.158091
$945$	0	0
$946$	8.85728	0.287975
$947$	−38.1847	−1.24084	−0.620418	−	0.784272i	$-0.713037\pi$
$947$	−38.1847	−1.24084	−0.620418	+	0.784272i	$0.713037\pi$
$948$	0	0
$949$	47.0794	1.52826
$950$	0	0
$951$	0	0
$952$	5.05086	0.163699
$953$	31.8983	1.03329	0.516643	−	0.856201i	$-0.327181\pi$
$953$	31.8983	1.03329	0.516643	+	0.856201i	$0.327181\pi$
$954$	0	0
$955$	18.3684	0.594388
$956$	18.2766	0.591106
$957$	0	0
$958$	21.5111	0.694993
$959$	−6.56199	−0.211898
$960$	0	0
$961$	26.9555	0.869531
$962$	24.9777	0.805314
$963$	0	0
$964$	−2.38715	−0.0768850
$965$	1.14272	0.0367855
$966$	0	0
$967$	−34.3051	−1.10318	−0.551589	−	0.834116i	$-0.685978\pi$
$967$	−34.3051	−1.10318	−0.551589	+	0.834116i	$0.685978\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−29.4479	−0.945027	−0.472513	−	0.881324i	$-0.656653\pi$
$971$	−29.4479	−0.945027	−0.472513	+	0.881324i	$0.656653\pi$
$972$	0	0
$973$	−14.1017	−0.452080
$974$	−1.63158	−0.0522793
$975$	0	0
$976$	−12.1017	−0.387366
$977$	−43.1151	−1.37937	−0.689687	−	0.724108i	$-0.742252\pi$
$977$	−43.1151	−1.37937	−0.689687	+	0.724108i	$0.742252\pi$
$978$	0	0
$979$	12.6637	0.404734
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−19.2257	−0.613517
$983$	−36.5906	−1.16706	−0.583529	−	0.812092i	$-0.698329\pi$
$983$	−36.5906	−1.16706	−0.583529	+	0.812092i	$0.698329\pi$
$984$	0	0
$985$	7.71456	0.245806
$986$	−3.81627	−0.121535
$987$	0	0
$988$	0	0
$989$	27.0223	0.859258
$990$	0	0
$991$	43.3720	1.37776	0.688878	−	0.724878i	$-0.258104\pi$
$991$	43.3720	1.37776	0.688878	+	0.724878i	$0.258104\pi$
$992$	−7.61285	−0.241708
$993$	0	0
$994$	1.24443	0.0394710
$995$	−25.3274	−0.802933
$996$	0	0
$997$	50.0544	1.58524	0.792619	−	0.609717i	$-0.208717\pi$
$997$	50.0544	1.58524	0.792619	+	0.609717i	$0.208717\pi$
$998$	−15.6128	−0.494216
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.ch.1.1		3
3.2	odd	2		2310.2.a.bd.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.bd.1.1	✓	3	3.2	odd	2
6930.2.a.ch.1.1		3	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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -3.80642 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.05086 q^{17} -1.00000 q^{20} +1.00000 q^{22} +3.05086 q^{23} +1.00000 q^{25} +3.80642 q^{26} +1.00000 q^{28} -0.755569 q^{29} +7.61285 q^{31} -1.00000 q^{32} +5.05086 q^{34} -1.00000 q^{35} +6.56199 q^{37} +1.00000 q^{40} -6.85728 q^{41} +8.85728 q^{43} -1.00000 q^{44} -3.05086 q^{46} +7.61285 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.80642 q^{52} +10.8573 q^{53} +1.00000 q^{55} -1.00000 q^{56} +0.755569 q^{58} +4.85728 q^{59} -12.1017 q^{61} -7.61285 q^{62} +1.00000 q^{64} +3.80642 q^{65} +4.29529 q^{67} -5.05086 q^{68} +1.00000 q^{70} -1.24443 q^{71} -12.3684 q^{73} -6.56199 q^{74} -1.00000 q^{77} -1.51114 q^{79} -1.00000 q^{80} +6.85728 q^{82} -8.85728 q^{83} +5.05086 q^{85} -8.85728 q^{86} +1.00000 q^{88} -12.6637 q^{89} -3.80642 q^{91} +3.05086 q^{92} -7.61285 q^{94} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{10} - 3 q^{11} + 2 q^{13} - 3 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{20} + 3 q^{22} - 4 q^{23} + 3 q^{25} - 2 q^{26} + 3 q^{28} - 2 q^{29} - 4 q^{31} - 3 q^{32} + 2 q^{34} - 3 q^{35} + 6 q^{37} + 3 q^{40} + 6 q^{41} - 3 q^{44} + 4 q^{46} - 4 q^{47} + 3 q^{49} - 3 q^{50} + 2 q^{52} + 6 q^{53} + 3 q^{55} - 3 q^{56} + 2 q^{58} - 12 q^{59} - 10 q^{61} + 4 q^{62} + 3 q^{64} - 2 q^{65} - 2 q^{68} + 3 q^{70} - 4 q^{71} - 10 q^{73} - 6 q^{74} - 3 q^{77} - 4 q^{79} - 3 q^{80} - 6 q^{82} + 2 q^{85} + 3 q^{88} + 2 q^{89} + 2 q^{91} - 4 q^{92} + 4 q^{94} - 6 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 - 3 * q^8 + 3 * q^10 - 3 * q^11 + 2 * q^13 - 3 * q^14 + 3 * q^16 - 2 * q^17 - 3 * q^20 + 3 * q^22 - 4 * q^23 + 3 * q^25 - 2 * q^26 + 3 * q^28 - 2 * q^29 - 4 * q^31 - 3 * q^32 + 2 * q^34 - 3 * q^35 + 6 * q^37 + 3 * q^40 + 6 * q^41 - 3 * q^44 + 4 * q^46 - 4 * q^47 + 3 * q^49 - 3 * q^50 + 2 * q^52 + 6 * q^53 + 3 * q^55 - 3 * q^56 + 2 * q^58 - 12 * q^59 - 10 * q^61 + 4 * q^62 + 3 * q^64 - 2 * q^65 - 2 * q^68 + 3 * q^70 - 4 * q^71 - 10 * q^73 - 6 * q^74 - 3 * q^77 - 4 * q^79 - 3 * q^80 - 6 * q^82 + 2 * q^85 + 3 * q^88 + 2 * q^89 + 2 * q^91 - 4 * q^92 + 4 * q^94 - 6 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more