LMFDB - Embedded newform 6930.2.a.cc.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +6.60555 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.60555 q^{17} +2.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +4.60555 q^{23} +1.00000 q^{25} +6.60555 q^{26} +1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +4.60555 q^{34} +1.00000 q^{35} +0.605551 q^{37} +2.00000 q^{38} +1.00000 q^{40} -3.21110 q^{41} -7.21110 q^{43} -1.00000 q^{44} +4.60555 q^{46} -3.21110 q^{47} +1.00000 q^{49} +1.00000 q^{50} +6.60555 q^{52} -3.21110 q^{53} -1.00000 q^{55} +1.00000 q^{56} +6.00000 q^{58} -13.2111 q^{61} -4.00000 q^{62} +1.00000 q^{64} +6.60555 q^{65} -11.8167 q^{67} +4.60555 q^{68} +1.00000 q^{70} +12.4222 q^{71} -10.0000 q^{73} +0.605551 q^{74} +2.00000 q^{76} -1.00000 q^{77} -1.21110 q^{79} +1.00000 q^{80} -3.21110 q^{82} -9.21110 q^{83} +4.60555 q^{85} -7.21110 q^{86} -1.00000 q^{88} +1.39445 q^{89} +6.60555 q^{91} +4.60555 q^{92} -3.21110 q^{94} +2.00000 q^{95} +4.78890 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 2 q^{10} - 2 q^{11} + 6 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 2 q^{20} - 2 q^{22} + 2 q^{23} + 2 q^{25} + 6 q^{26} + 2 q^{28} + 12 q^{29} - 8 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} - 6 q^{37} + 4 q^{38} + 2 q^{40} + 8 q^{41} - 2 q^{44} + 2 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} + 6 q^{52} + 8 q^{53} - 2 q^{55} + 2 q^{56} + 12 q^{58} - 12 q^{61} - 8 q^{62} + 2 q^{64} + 6 q^{65} - 2 q^{67} + 2 q^{68} + 2 q^{70} - 4 q^{71} - 20 q^{73} - 6 q^{74} + 4 q^{76} - 2 q^{77} + 12 q^{79} + 2 q^{80} + 8 q^{82} - 4 q^{83} + 2 q^{85} - 2 q^{88} + 10 q^{89} + 6 q^{91} + 2 q^{92} + 8 q^{94} + 4 q^{95} + 24 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8 + 2 * q^10 - 2 * q^11 + 6 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 4 * q^19 + 2 * q^20 - 2 * q^22 + 2 * q^23 + 2 * q^25 + 6 * q^26 + 2 * q^28 + 12 * q^29 - 8 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^35 - 6 * q^37 + 4 * q^38 + 2 * q^40 + 8 * q^41 - 2 * q^44 + 2 * q^46 + 8 * q^47 + 2 * q^49 + 2 * q^50 + 6 * q^52 + 8 * q^53 - 2 * q^55 + 2 * q^56 + 12 * q^58 - 12 * q^61 - 8 * q^62 + 2 * q^64 + 6 * q^65 - 2 * q^67 + 2 * q^68 + 2 * q^70 - 4 * q^71 - 20 * q^73 - 6 * q^74 + 4 * q^76 - 2 * q^77 + 12 * q^79 + 2 * q^80 + 8 * q^82 - 4 * q^83 + 2 * q^85 - 2 * q^88 + 10 * q^89 + 6 * q^91 + 2 * q^92 + 8 * q^94 + 4 * q^95 + 24 * q^97 + 2 * 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
$13$	6.60555	1.83205	0.916025	−	0.401121i	$-0.131379\pi$
$13$	6.60555	1.83205	0.916025	+	0.401121i	$0.131379\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.60555	1.11701	0.558505	−	0.829501i	$-0.311375\pi$
$17$	4.60555	1.11701	0.558505	+	0.829501i	$0.311375\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.60555	0.960324	0.480162	−	0.877180i	$-0.340578\pi$
$23$	4.60555	0.960324	0.480162	+	0.877180i	$0.340578\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.60555	1.29546
$27$	0	0
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.60555	0.789846
$35$	1.00000	0.169031
$36$	0	0
$37$	0.605551	0.0995520	0.0497760	−	0.998760i	$-0.484149\pi$
$37$	0.605551	0.0995520	0.0497760	+	0.998760i	$0.484149\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.21110	−0.501490	−0.250745	−	0.968053i	$-0.580676\pi$
$41$	−3.21110	−0.501490	−0.250745	+	0.968053i	$0.580676\pi$
$42$	0	0
$43$	−7.21110	−1.09968	−0.549841	−	0.835269i	$-0.685312\pi$
$43$	−7.21110	−1.09968	−0.549841	+	0.835269i	$0.685312\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.60555	0.679051
$47$	−3.21110	−0.468387	−0.234194	−	0.972190i	$-0.575245\pi$
$47$	−3.21110	−0.468387	−0.234194	+	0.972190i	$0.575245\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	6.60555	0.916025
$53$	−3.21110	−0.441079	−0.220539	−	0.975378i	$-0.570782\pi$
$53$	−3.21110	−0.441079	−0.220539	+	0.975378i	$0.570782\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−13.2111	−1.69151	−0.845754	−	0.533573i	$-0.820849\pi$
$61$	−13.2111	−1.69151	−0.845754	+	0.533573i	$0.820849\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.60555	0.819318
$66$	0	0
$67$	−11.8167	−1.44363	−0.721817	−	0.692084i	$-0.756693\pi$
$67$	−11.8167	−1.44363	−0.721817	+	0.692084i	$0.756693\pi$
$68$	4.60555	0.558505
$69$	0	0
$70$	1.00000	0.119523
$71$	12.4222	1.47424	0.737122	−	0.675759i	$-0.236184\pi$
$71$	12.4222	1.47424	0.737122	+	0.675759i	$0.236184\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0.605551	0.0703939
$75$	0	0
$76$	2.00000	0.229416
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−1.21110	−0.136260	−0.0681298	−	0.997676i	$-0.521703\pi$
$79$	−1.21110	−0.136260	−0.0681298	+	0.997676i	$0.521703\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−3.21110	−0.354607
$83$	−9.21110	−1.01105	−0.505525	−	0.862812i	$-0.668701\pi$
$83$	−9.21110	−1.01105	−0.505525	+	0.862812i	$0.668701\pi$
$84$	0	0
$85$	4.60555	0.499542
$86$	−7.21110	−0.777593
$87$	0	0
$88$	−1.00000	−0.106600
$89$	1.39445	0.147811	0.0739056	−	0.997265i	$-0.476454\pi$
$89$	1.39445	0.147811	0.0739056	+	0.997265i	$0.476454\pi$
$90$	0	0
$91$	6.60555	0.692450
$92$	4.60555	0.480162
$93$	0	0
$94$	−3.21110	−0.331200
$95$	2.00000	0.205196
$96$	0	0
$97$	4.78890	0.486239	0.243119	−	0.969996i	$-0.421829\pi$
$97$	4.78890	0.486239	0.243119	+	0.969996i	$0.421829\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.60555	0.647728
$105$	0	0
$106$	−3.21110	−0.311890
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	15.8167	1.51496	0.757480	−	0.652858i	$-0.226430\pi$
$109$	15.8167	1.51496	0.757480	+	0.652858i	$0.226430\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	1.39445	0.131179	0.0655894	−	0.997847i	$-0.479107\pi$
$113$	1.39445	0.131179	0.0655894	+	0.997847i	$0.479107\pi$
$114$	0	0
$115$	4.60555	0.429470
$116$	6.00000	0.557086
$117$	0	0
$118$	0	0
$119$	4.60555	0.422190
$120$	0	0
$121$	1.00000	0.0909091
$122$	−13.2111	−1.19608
$123$	0	0
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.21110	0.462411	0.231205	−	0.972905i	$-0.425733\pi$
$127$	5.21110	0.462411	0.231205	+	0.972905i	$0.425733\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.60555	0.579345
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	−11.8167	−1.02080
$135$	0	0
$136$	4.60555	0.394923
$137$	−10.6056	−0.906093	−0.453047	−	0.891487i	$-0.649663\pi$
$137$	−10.6056	−0.906093	−0.453047	+	0.891487i	$0.649663\pi$
$138$	0	0
$139$	−0.788897	−0.0669134	−0.0334567	−	0.999440i	$-0.510652\pi$
$139$	−0.788897	−0.0669134	−0.0334567	+	0.999440i	$0.510652\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	12.4222	1.04245
$143$	−6.60555	−0.552384
$144$	0	0
$145$	6.00000	0.498273
$146$	−10.0000	−0.827606
$147$	0	0
$148$	0.605551	0.0497760
$149$	12.4222	1.01767	0.508833	−	0.860865i	$-0.330077\pi$
$149$	12.4222	1.01767	0.508833	+	0.860865i	$0.330077\pi$
$150$	0	0
$151$	−6.78890	−0.552473	−0.276236	−	0.961090i	$-0.589087\pi$
$151$	−6.78890	−0.552473	−0.276236	+	0.961090i	$0.589087\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−4.00000	−0.321288
$156$	0	0
$157$	11.2111	0.894743	0.447372	−	0.894348i	$-0.352360\pi$
$157$	11.2111	0.894743	0.447372	+	0.894348i	$0.352360\pi$
$158$	−1.21110	−0.0963501
$159$	0	0
$160$	1.00000	0.0790569
$161$	4.60555	0.362968
$162$	0	0
$163$	−11.8167	−0.925552	−0.462776	−	0.886475i	$-0.653147\pi$
$163$	−11.8167	−0.925552	−0.462776	+	0.886475i	$0.653147\pi$
$164$	−3.21110	−0.250745
$165$	0	0
$166$	−9.21110	−0.714920
$167$	−7.81665	−0.604871	−0.302435	−	0.953170i	$-0.597800\pi$
$167$	−7.81665	−0.604871	−0.302435	+	0.953170i	$0.597800\pi$
$168$	0	0
$169$	30.6333	2.35641
$170$	4.60555	0.353230
$171$	0	0
$172$	−7.21110	−0.549841
$173$	21.6333	1.64475	0.822375	−	0.568946i	$-0.192649\pi$
$173$	21.6333	1.64475	0.822375	+	0.568946i	$0.192649\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	1.39445	0.104518
$179$	2.78890	0.208452	0.104226	−	0.994554i	$-0.466763\pi$
$179$	2.78890	0.208452	0.104226	+	0.994554i	$0.466763\pi$
$180$	0	0
$181$	24.6056	1.82892	0.914458	−	0.404681i	$-0.132617\pi$
$181$	24.6056	1.82892	0.914458	+	0.404681i	$0.132617\pi$
$182$	6.60555	0.489636
$183$	0	0
$184$	4.60555	0.339526
$185$	0.605551	0.0445210
$186$	0	0
$187$	−4.60555	−0.336791
$188$	−3.21110	−0.234194
$189$	0	0
$190$	2.00000	0.145095
$191$	8.78890	0.635942	0.317971	−	0.948100i	$-0.396998\pi$
$191$	8.78890	0.635942	0.317971	+	0.948100i	$0.396998\pi$
$192$	0	0
$193$	−7.21110	−0.519067	−0.259533	−	0.965734i	$-0.583569\pi$
$193$	−7.21110	−0.519067	−0.259533	+	0.965734i	$0.583569\pi$
$194$	4.78890	0.343823
$195$	0	0
$196$	1.00000	0.0714286
$197$	15.2111	1.08375	0.541873	−	0.840460i	$-0.317715\pi$
$197$	15.2111	1.08375	0.541873	+	0.840460i	$0.317715\pi$
$198$	0	0
$199$	−6.78890	−0.481252	−0.240626	−	0.970618i	$-0.577353\pi$
$199$	−6.78890	−0.481252	−0.240626	+	0.970618i	$0.577353\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	6.00000	0.421117
$204$	0	0
$205$	−3.21110	−0.224273
$206$	8.00000	0.557386
$207$	0	0
$208$	6.60555	0.458013
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−8.60555	−0.592431	−0.296215	−	0.955121i	$-0.595725\pi$
$211$	−8.60555	−0.592431	−0.296215	+	0.955121i	$0.595725\pi$
$212$	−3.21110	−0.220539
$213$	0	0
$214$	0	0
$215$	−7.21110	−0.491793
$216$	0	0
$217$	−4.00000	−0.271538
$218$	15.8167	1.07124
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	30.4222	2.04642
$222$	0	0
$223$	−22.4222	−1.50150	−0.750751	−	0.660585i	$-0.770308\pi$
$223$	−22.4222	−1.50150	−0.750751	+	0.660585i	$0.770308\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	1.39445	0.0927573
$227$	6.42221	0.426257	0.213128	−	0.977024i	$-0.431635\pi$
$227$	6.42221	0.426257	0.213128	+	0.977024i	$0.431635\pi$
$228$	0	0
$229$	−20.6056	−1.36165	−0.680827	−	0.732445i	$-0.738379\pi$
$229$	−20.6056	−1.36165	−0.680827	+	0.732445i	$0.738379\pi$
$230$	4.60555	0.303681
$231$	0	0
$232$	6.00000	0.393919
$233$	−12.4222	−0.813806	−0.406903	−	0.913471i	$-0.633391\pi$
$233$	−12.4222	−0.813806	−0.406903	+	0.913471i	$0.633391\pi$
$234$	0	0
$235$	−3.21110	−0.209469
$236$	0	0
$237$	0	0
$238$	4.60555	0.298534
$239$	−19.8167	−1.28183	−0.640916	−	0.767611i	$-0.721445\pi$
$239$	−19.8167	−1.28183	−0.640916	+	0.767611i	$0.721445\pi$
$240$	0	0
$241$	−25.6333	−1.65119	−0.825593	−	0.564266i	$-0.809159\pi$
$241$	−25.6333	−1.65119	−0.825593	+	0.564266i	$0.809159\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−13.2111	−0.845754
$245$	1.00000	0.0638877
$246$	0	0
$247$	13.2111	0.840602
$248$	−4.00000	−0.254000
$249$	0	0
$250$	1.00000	0.0632456
$251$	27.6333	1.74420	0.872099	−	0.489329i	$-0.162758\pi$
$251$	27.6333	1.74420	0.872099	+	0.489329i	$0.162758\pi$
$252$	0	0
$253$	−4.60555	−0.289549
$254$	5.21110	0.326974
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.21110	0.574573	0.287286	−	0.957845i	$-0.407247\pi$
$257$	9.21110	0.574573	0.287286	+	0.957845i	$0.407247\pi$
$258$	0	0
$259$	0.605551	0.0376271
$260$	6.60555	0.409659
$261$	0	0
$262$	0	0
$263$	−2.78890	−0.171971	−0.0859854	−	0.996296i	$-0.527404\pi$
$263$	−2.78890	−0.171971	−0.0859854	+	0.996296i	$0.527404\pi$
$264$	0	0
$265$	−3.21110	−0.197256
$266$	2.00000	0.122628
$267$	0	0
$268$	−11.8167	−0.721817
$269$	−0.422205	−0.0257423	−0.0128711	−	0.999917i	$-0.504097\pi$
$269$	−0.422205	−0.0257423	−0.0128711	+	0.999917i	$0.504097\pi$
$270$	0	0
$271$	14.4222	0.876087	0.438043	−	0.898954i	$-0.355672\pi$
$271$	14.4222	0.876087	0.438043	+	0.898954i	$0.355672\pi$
$272$	4.60555	0.279253
$273$	0	0
$274$	−10.6056	−0.640705
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	2.42221	0.145536	0.0727681	−	0.997349i	$-0.476817\pi$
$277$	2.42221	0.145536	0.0727681	+	0.997349i	$0.476817\pi$
$278$	−0.788897	−0.0473149
$279$	0	0
$280$	1.00000	0.0597614
$281$	−11.0278	−0.657861	−0.328930	−	0.944354i	$-0.606688\pi$
$281$	−11.0278	−0.657861	−0.328930	+	0.944354i	$0.606688\pi$
$282$	0	0
$283$	12.6056	0.749322	0.374661	−	0.927162i	$-0.377759\pi$
$283$	12.6056	0.749322	0.374661	+	0.927162i	$0.377759\pi$
$284$	12.4222	0.737122
$285$	0	0
$286$	−6.60555	−0.390594
$287$	−3.21110	−0.189545
$288$	0	0
$289$	4.21110	0.247712
$290$	6.00000	0.352332
$291$	0	0
$292$	−10.0000	−0.585206
$293$	−12.4222	−0.725713	−0.362856	−	0.931845i	$-0.618198\pi$
$293$	−12.4222	−0.725713	−0.362856	+	0.931845i	$0.618198\pi$
$294$	0	0
$295$	0	0
$296$	0.605551	0.0351970
$297$	0	0
$298$	12.4222	0.719599
$299$	30.4222	1.75936
$300$	0	0
$301$	−7.21110	−0.415641
$302$	−6.78890	−0.390657
$303$	0	0
$304$	2.00000	0.114708
$305$	−13.2111	−0.756466
$306$	0	0
$307$	12.6056	0.719437	0.359718	−	0.933061i	$-0.382873\pi$
$307$	12.6056	0.719437	0.359718	+	0.933061i	$0.382873\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−1.81665	−0.103013	−0.0515065	−	0.998673i	$-0.516402\pi$
$311$	−1.81665	−0.103013	−0.0515065	+	0.998673i	$0.516402\pi$
$312$	0	0
$313$	26.8444	1.51734	0.758668	−	0.651478i	$-0.225851\pi$
$313$	26.8444	1.51734	0.758668	+	0.651478i	$0.225851\pi$
$314$	11.2111	0.632679
$315$	0	0
$316$	−1.21110	−0.0681298
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	1.00000	0.0559017
$321$	0	0
$322$	4.60555	0.256657
$323$	9.21110	0.512519
$324$	0	0
$325$	6.60555	0.366410
$326$	−11.8167	−0.654464
$327$	0	0
$328$	−3.21110	−0.177303
$329$	−3.21110	−0.177034
$330$	0	0
$331$	2.42221	0.133136	0.0665682	−	0.997782i	$-0.478795\pi$
$331$	2.42221	0.133136	0.0665682	+	0.997782i	$0.478795\pi$
$332$	−9.21110	−0.505525
$333$	0	0
$334$	−7.81665	−0.427708
$335$	−11.8167	−0.645613
$336$	0	0
$337$	−13.6333	−0.742654	−0.371327	−	0.928502i	$-0.621097\pi$
$337$	−13.6333	−0.742654	−0.371327	+	0.928502i	$0.621097\pi$
$338$	30.6333	1.66623
$339$	0	0
$340$	4.60555	0.249771
$341$	4.00000	0.216612
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.21110	−0.388797
$345$	0	0
$346$	21.6333	1.16301
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−1.21110	−0.0648288	−0.0324144	−	0.999475i	$-0.510320\pi$
$349$	−1.21110	−0.0648288	−0.0324144	+	0.999475i	$0.510320\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	27.6333	1.47077	0.735386	−	0.677648i	$-0.237001\pi$
$353$	27.6333	1.47077	0.735386	+	0.677648i	$0.237001\pi$
$354$	0	0
$355$	12.4222	0.659302
$356$	1.39445	0.0739056
$357$	0	0
$358$	2.78890	0.147398
$359$	26.2389	1.38483	0.692417	−	0.721498i	$-0.256546\pi$
$359$	26.2389	1.38483	0.692417	+	0.721498i	$0.256546\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	24.6056	1.29324
$363$	0	0
$364$	6.60555	0.346225
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−10.4222	−0.544035	−0.272017	−	0.962292i	$-0.587691\pi$
$367$	−10.4222	−0.544035	−0.272017	+	0.962292i	$0.587691\pi$
$368$	4.60555	0.240081
$369$	0	0
$370$	0.605551	0.0314811
$371$	−3.21110	−0.166712
$372$	0	0
$373$	−18.7889	−0.972852	−0.486426	−	0.873722i	$-0.661700\pi$
$373$	−18.7889	−0.972852	−0.486426	+	0.873722i	$0.661700\pi$
$374$	−4.60555	−0.238147
$375$	0	0
$376$	−3.21110	−0.165600
$377$	39.6333	2.04122
$378$	0	0
$379$	−28.8444	−1.48164	−0.740819	−	0.671705i	$-0.765562\pi$
$379$	−28.8444	−1.48164	−0.740819	+	0.671705i	$0.765562\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	8.78890	0.449679
$383$	−21.6333	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$383$	−21.6333	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−7.21110	−0.367035
$387$	0	0
$388$	4.78890	0.243119
$389$	−12.4222	−0.629831	−0.314915	−	0.949120i	$-0.601976\pi$
$389$	−12.4222	−0.629831	−0.314915	+	0.949120i	$0.601976\pi$
$390$	0	0
$391$	21.2111	1.07269
$392$	1.00000	0.0505076
$393$	0	0
$394$	15.2111	0.766324
$395$	−1.21110	−0.0609372
$396$	0	0
$397$	−28.4222	−1.42647	−0.713235	−	0.700925i	$-0.752771\pi$
$397$	−28.4222	−1.42647	−0.713235	+	0.700925i	$0.752771\pi$
$398$	−6.78890	−0.340297
$399$	0	0
$400$	1.00000	0.0500000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−26.4222	−1.31618
$404$	6.00000	0.298511
$405$	0	0
$406$	6.00000	0.297775
$407$	−0.605551	−0.0300161
$408$	0	0
$409$	−37.6333	−1.86085	−0.930424	−	0.366486i	$-0.880561\pi$
$409$	−37.6333	−1.86085	−0.930424	+	0.366486i	$0.880561\pi$
$410$	−3.21110	−0.158585
$411$	0	0
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	−9.21110	−0.452155
$416$	6.60555	0.323864
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	−9.21110	−0.449992	−0.224996	−	0.974360i	$-0.572237\pi$
$419$	−9.21110	−0.449992	−0.224996	+	0.974360i	$0.572237\pi$
$420$	0	0
$421$	8.42221	0.410473	0.205237	−	0.978712i	$-0.434204\pi$
$421$	8.42221	0.410473	0.205237	+	0.978712i	$0.434204\pi$
$422$	−8.60555	−0.418912
$423$	0	0
$424$	−3.21110	−0.155945
$425$	4.60555	0.223402
$426$	0	0
$427$	−13.2111	−0.639330
$428$	0	0
$429$	0	0
$430$	−7.21110	−0.347750
$431$	29.0278	1.39822	0.699109	−	0.715015i	$-0.253580\pi$
$431$	29.0278	1.39822	0.699109	+	0.715015i	$0.253580\pi$
$432$	0	0
$433$	36.0555	1.73272	0.866359	−	0.499422i	$-0.166454\pi$
$433$	36.0555	1.73272	0.866359	+	0.499422i	$0.166454\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	15.8167	0.757480
$437$	9.21110	0.440627
$438$	0	0
$439$	−25.2111	−1.20326	−0.601630	−	0.798775i	$-0.705482\pi$
$439$	−25.2111	−1.20326	−0.601630	+	0.798775i	$0.705482\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	30.4222	1.44704
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	1.39445	0.0661032
$446$	−22.4222	−1.06172
$447$	0	0
$448$	1.00000	0.0472456
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	3.21110	0.151205
$452$	1.39445	0.0655894
$453$	0	0
$454$	6.42221	0.301409
$455$	6.60555	0.309673
$456$	0	0
$457$	4.78890	0.224015	0.112008	−	0.993707i	$-0.464272\pi$
$457$	4.78890	0.224015	0.112008	+	0.993707i	$0.464272\pi$
$458$	−20.6056	−0.962834
$459$	0	0
$460$	4.60555	0.214735
$461$	8.78890	0.409340	0.204670	−	0.978831i	$-0.434388\pi$
$461$	8.78890	0.409340	0.204670	+	0.978831i	$0.434388\pi$
$462$	0	0
$463$	14.4222	0.670257	0.335128	−	0.942172i	$-0.391220\pi$
$463$	14.4222	0.670257	0.335128	+	0.942172i	$0.391220\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−12.4222	−0.575448
$467$	−15.6333	−0.723423	−0.361712	−	0.932290i	$-0.617807\pi$
$467$	−15.6333	−0.723423	−0.361712	+	0.932290i	$0.617807\pi$
$468$	0	0
$469$	−11.8167	−0.545642
$470$	−3.21110	−0.148117
$471$	0	0
$472$	0	0
$473$	7.21110	0.331567
$474$	0	0
$475$	2.00000	0.0917663
$476$	4.60555	0.211095
$477$	0	0
$478$	−19.8167	−0.906393
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−25.6333	−1.16757
$483$	0	0
$484$	1.00000	0.0454545
$485$	4.78890	0.217453
$486$	0	0
$487$	−6.78890	−0.307634	−0.153817	−	0.988099i	$-0.549157\pi$
$487$	−6.78890	−0.307634	−0.153817	+	0.988099i	$0.549157\pi$
$488$	−13.2111	−0.598039
$489$	0	0
$490$	1.00000	0.0451754
$491$	−33.2111	−1.49880	−0.749398	−	0.662120i	$-0.769657\pi$
$491$	−33.2111	−1.49880	−0.749398	+	0.662120i	$0.769657\pi$
$492$	0	0
$493$	27.6333	1.24454
$494$	13.2111	0.594396
$495$	0	0
$496$	−4.00000	−0.179605
$497$	12.4222	0.557212
$498$	0	0
$499$	−25.2111	−1.12860	−0.564302	−	0.825568i	$-0.690855\pi$
$499$	−25.2111	−1.12860	−0.564302	+	0.825568i	$0.690855\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	27.6333	1.23333
$503$	16.1833	0.721580	0.360790	−	0.932647i	$-0.382507\pi$
$503$	16.1833	0.721580	0.360790	+	0.932647i	$0.382507\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−4.60555	−0.204742
$507$	0	0
$508$	5.21110	0.231205
$509$	9.63331	0.426989	0.213494	−	0.976944i	$-0.431515\pi$
$509$	9.63331	0.426989	0.213494	+	0.976944i	$0.431515\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.21110	0.406284
$515$	8.00000	0.352522
$516$	0	0
$517$	3.21110	0.141224
$518$	0.605551	0.0266064
$519$	0	0
$520$	6.60555	0.289673
$521$	1.39445	0.0610919	0.0305460	−	0.999533i	$-0.490275\pi$
$521$	1.39445	0.0610919	0.0305460	+	0.999533i	$0.490275\pi$
$522$	0	0
$523$	−8.60555	−0.376294	−0.188147	−	0.982141i	$-0.560248\pi$
$523$	−8.60555	−0.376294	−0.188147	+	0.982141i	$0.560248\pi$
$524$	0	0
$525$	0	0
$526$	−2.78890	−0.121602
$527$	−18.4222	−0.802484
$528$	0	0
$529$	−1.78890	−0.0777781
$530$	−3.21110	−0.139481
$531$	0	0
$532$	2.00000	0.0867110
$533$	−21.2111	−0.918755
$534$	0	0
$535$	0	0
$536$	−11.8167	−0.510402
$537$	0	0
$538$	−0.422205	−0.0182026
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−5.39445	−0.231925	−0.115963	−	0.993254i	$-0.536995\pi$
$541$	−5.39445	−0.231925	−0.115963	+	0.993254i	$0.536995\pi$
$542$	14.4222	0.619487
$543$	0	0
$544$	4.60555	0.197461
$545$	15.8167	0.677511
$546$	0	0
$547$	20.4222	0.873190	0.436595	−	0.899658i	$-0.356184\pi$
$547$	20.4222	0.873190	0.436595	+	0.899658i	$0.356184\pi$
$548$	−10.6056	−0.453047
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	12.0000	0.511217
$552$	0	0
$553$	−1.21110	−0.0515013
$554$	2.42221	0.102910
$555$	0	0
$556$	−0.788897	−0.0334567
$557$	−28.0555	−1.18875	−0.594375	−	0.804188i	$-0.702601\pi$
$557$	−28.0555	−1.18875	−0.594375	+	0.804188i	$0.702601\pi$
$558$	0	0
$559$	−47.6333	−2.01467
$560$	1.00000	0.0422577
$561$	0	0
$562$	−11.0278	−0.465178
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	1.39445	0.0586649
$566$	12.6056	0.529851
$567$	0	0
$568$	12.4222	0.521224
$569$	−44.2389	−1.85459	−0.927295	−	0.374332i	$-0.877872\pi$
$569$	−44.2389	−1.85459	−0.927295	+	0.374332i	$0.877872\pi$
$570$	0	0
$571$	−24.2389	−1.01436	−0.507182	−	0.861839i	$-0.669313\pi$
$571$	−24.2389	−1.01436	−0.507182	+	0.861839i	$0.669313\pi$
$572$	−6.60555	−0.276192
$573$	0	0
$574$	−3.21110	−0.134029
$575$	4.60555	0.192065
$576$	0	0
$577$	−4.42221	−0.184099	−0.0920494	−	0.995754i	$-0.529342\pi$
$577$	−4.42221	−0.184099	−0.0920494	+	0.995754i	$0.529342\pi$
$578$	4.21110	0.175159
$579$	0	0
$580$	6.00000	0.249136
$581$	−9.21110	−0.382141
$582$	0	0
$583$	3.21110	0.132990
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−12.4222	−0.513157
$587$	3.63331	0.149963	0.0749813	−	0.997185i	$-0.476110\pi$
$587$	3.63331	0.149963	0.0749813	+	0.997185i	$0.476110\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	0.605551	0.0248880
$593$	19.3944	0.796435	0.398217	−	0.917291i	$-0.369629\pi$
$593$	19.3944	0.796435	0.398217	+	0.917291i	$0.369629\pi$
$594$	0	0
$595$	4.60555	0.188809
$596$	12.4222	0.508833
$597$	0	0
$598$	30.4222	1.24406
$599$	21.6333	0.883913	0.441956	−	0.897036i	$-0.354285\pi$
$599$	21.6333	0.883913	0.441956	+	0.897036i	$0.354285\pi$
$600$	0	0
$601$	−13.6333	−0.556114	−0.278057	−	0.960565i	$-0.589690\pi$
$601$	−13.6333	−0.556114	−0.278057	+	0.960565i	$0.589690\pi$
$602$	−7.21110	−0.293903
$603$	0	0
$604$	−6.78890	−0.276236
$605$	1.00000	0.0406558
$606$	0	0
$607$	35.6333	1.44631	0.723156	−	0.690685i	$-0.242691\pi$
$607$	35.6333	1.44631	0.723156	+	0.690685i	$0.242691\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−13.2111	−0.534902
$611$	−21.2111	−0.858109
$612$	0	0
$613$	−22.4222	−0.905624	−0.452812	−	0.891606i	$-0.649579\pi$
$613$	−22.4222	−0.905624	−0.452812	+	0.891606i	$0.649579\pi$
$614$	12.6056	0.508719
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	14.2389	0.573235	0.286617	−	0.958045i	$-0.407469\pi$
$617$	14.2389	0.573235	0.286617	+	0.958045i	$0.407469\pi$
$618$	0	0
$619$	−17.3944	−0.699142	−0.349571	−	0.936910i	$-0.613673\pi$
$619$	−17.3944	−0.699142	−0.349571	+	0.936910i	$0.613673\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−1.81665	−0.0728412
$623$	1.39445	0.0558674
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.8444	1.07292
$627$	0	0
$628$	11.2111	0.447372
$629$	2.78890	0.111201
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−1.21110	−0.0481751
$633$	0	0
$634$	−6.00000	−0.238290
$635$	5.21110	0.206796
$636$	0	0
$637$	6.60555	0.261721
$638$	−6.00000	−0.237542
$639$	0	0
$640$	1.00000	0.0395285
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	−10.4222	−0.411012	−0.205506	−	0.978656i	$-0.565884\pi$
$643$	−10.4222	−0.411012	−0.205506	+	0.978656i	$0.565884\pi$
$644$	4.60555	0.181484
$645$	0	0
$646$	9.21110	0.362406
$647$	9.63331	0.378724	0.189362	−	0.981907i	$-0.439358\pi$
$647$	9.63331	0.378724	0.189362	+	0.981907i	$0.439358\pi$
$648$	0	0
$649$	0	0
$650$	6.60555	0.259091
$651$	0	0
$652$	−11.8167	−0.462776
$653$	−15.2111	−0.595256	−0.297628	−	0.954682i	$-0.596196\pi$
$653$	−15.2111	−0.595256	−0.297628	+	0.954682i	$0.596196\pi$
$654$	0	0
$655$	0	0
$656$	−3.21110	−0.125372
$657$	0	0
$658$	−3.21110	−0.125182
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−23.3944	−0.909939	−0.454969	−	0.890507i	$-0.650350\pi$
$661$	−23.3944	−0.909939	−0.454969	+	0.890507i	$0.650350\pi$
$662$	2.42221	0.0941417
$663$	0	0
$664$	−9.21110	−0.357460
$665$	2.00000	0.0775567
$666$	0	0
$667$	27.6333	1.06997
$668$	−7.81665	−0.302435
$669$	0	0
$670$	−11.8167	−0.456517
$671$	13.2111	0.510009
$672$	0	0
$673$	28.7889	1.10973	0.554865	−	0.831940i	$-0.312770\pi$
$673$	28.7889	1.10973	0.554865	+	0.831940i	$0.312770\pi$
$674$	−13.6333	−0.525135
$675$	0	0
$676$	30.6333	1.17820
$677$	−0.422205	−0.0162267	−0.00811333	−	0.999967i	$-0.502583\pi$
$677$	−0.422205	−0.0162267	−0.00811333	+	0.999967i	$0.502583\pi$
$678$	0	0
$679$	4.78890	0.183781
$680$	4.60555	0.176615
$681$	0	0
$682$	4.00000	0.153168
$683$	51.6333	1.97569	0.987847	−	0.155431i	$-0.0496765\pi$
$683$	51.6333	1.97569	0.987847	+	0.155431i	$0.0496765\pi$
$684$	0	0
$685$	−10.6056	−0.405217
$686$	1.00000	0.0381802
$687$	0	0
$688$	−7.21110	−0.274921
$689$	−21.2111	−0.808079
$690$	0	0
$691$	−17.3944	−0.661716	−0.330858	−	0.943681i	$-0.607338\pi$
$691$	−17.3944	−0.661716	−0.330858	+	0.943681i	$0.607338\pi$
$692$	21.6333	0.822375
$693$	0	0
$694$	−24.0000	−0.911028
$695$	−0.788897	−0.0299246
$696$	0	0
$697$	−14.7889	−0.560169
$698$	−1.21110	−0.0458409
$699$	0	0
$700$	1.00000	0.0377964
$701$	48.4222	1.82888	0.914441	−	0.404720i	$-0.132631\pi$
$701$	48.4222	1.82888	0.914441	+	0.404720i	$0.132631\pi$
$702$	0	0
$703$	1.21110	0.0456776
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	27.6333	1.03999
$707$	6.00000	0.225653
$708$	0	0
$709$	−16.4222	−0.616749	−0.308374	−	0.951265i	$-0.599785\pi$
$709$	−16.4222	−0.616749	−0.308374	+	0.951265i	$0.599785\pi$
$710$	12.4222	0.466197
$711$	0	0
$712$	1.39445	0.0522592
$713$	−18.4222	−0.689917
$714$	0	0
$715$	−6.60555	−0.247034
$716$	2.78890	0.104226
$717$	0	0
$718$	26.2389	0.979226
$719$	49.8167	1.85785	0.928924	−	0.370271i	$-0.120735\pi$
$719$	49.8167	1.85785	0.928924	+	0.370271i	$0.120735\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	−15.0000	−0.558242
$723$	0	0
$724$	24.6056	0.914458
$725$	6.00000	0.222834
$726$	0	0
$727$	32.8444	1.21813	0.609066	−	0.793120i	$-0.291544\pi$
$727$	32.8444	1.21813	0.609066	+	0.793120i	$0.291544\pi$
$728$	6.60555	0.244818
$729$	0	0
$730$	−10.0000	−0.370117
$731$	−33.2111	−1.22836
$732$	0	0
$733$	6.60555	0.243982	0.121991	−	0.992531i	$-0.461072\pi$
$733$	6.60555	0.243982	0.121991	+	0.992531i	$0.461072\pi$
$734$	−10.4222	−0.384691
$735$	0	0
$736$	4.60555	0.169763
$737$	11.8167	0.435272
$738$	0	0
$739$	−23.3944	−0.860579	−0.430289	−	0.902691i	$-0.641588\pi$
$739$	−23.3944	−0.860579	−0.430289	+	0.902691i	$0.641588\pi$
$740$	0.605551	0.0222605
$741$	0	0
$742$	−3.21110	−0.117883
$743$	−42.4222	−1.55632	−0.778160	−	0.628066i	$-0.783847\pi$
$743$	−42.4222	−1.55632	−0.778160	+	0.628066i	$0.783847\pi$
$744$	0	0
$745$	12.4222	0.455114
$746$	−18.7889	−0.687910
$747$	0	0
$748$	−4.60555	−0.168396
$749$	0	0
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−3.21110	−0.117097
$753$	0	0
$754$	39.6333	1.44336
$755$	−6.78890	−0.247073
$756$	0	0
$757$	6.18335	0.224738	0.112369	−	0.993667i	$-0.464156\pi$
$757$	6.18335	0.224738	0.112369	+	0.993667i	$0.464156\pi$
$758$	−28.8444	−1.04768
$759$	0	0
$760$	2.00000	0.0725476
$761$	−9.63331	−0.349207	−0.174604	−	0.984639i	$-0.555864\pi$
$761$	−9.63331	−0.349207	−0.174604	+	0.984639i	$0.555864\pi$
$762$	0	0
$763$	15.8167	0.572601
$764$	8.78890	0.317971
$765$	0	0
$766$	−21.6333	−0.781643
$767$	0	0
$768$	0	0
$769$	24.0555	0.867464	0.433732	−	0.901042i	$-0.357196\pi$
$769$	24.0555	0.867464	0.433732	+	0.901042i	$0.357196\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	−7.21110	−0.259533
$773$	0.422205	0.0151857	0.00759283	−	0.999971i	$-0.497583\pi$
$773$	0.422205	0.0151857	0.00759283	+	0.999971i	$0.497583\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	4.78890	0.171911
$777$	0	0
$778$	−12.4222	−0.445358
$779$	−6.42221	−0.230099
$780$	0	0
$781$	−12.4222	−0.444501
$782$	21.2111	0.758507
$783$	0	0
$784$	1.00000	0.0357143
$785$	11.2111	0.400141
$786$	0	0
$787$	−24.2389	−0.864022	−0.432011	−	0.901868i	$-0.642196\pi$
$787$	−24.2389	−0.864022	−0.432011	+	0.901868i	$0.642196\pi$
$788$	15.2111	0.541873
$789$	0	0
$790$	−1.21110	−0.0430891
$791$	1.39445	0.0495809
$792$	0	0
$793$	−87.2666	−3.09893
$794$	−28.4222	−1.00867
$795$	0	0
$796$	−6.78890	−0.240626
$797$	30.8444	1.09257	0.546283	−	0.837601i	$-0.316042\pi$
$797$	30.8444	1.09257	0.546283	+	0.837601i	$0.316042\pi$
$798$	0	0
$799$	−14.7889	−0.523194
$800$	1.00000	0.0353553
$801$	0	0
$802$	24.0000	0.847469
$803$	10.0000	0.352892
$804$	0	0
$805$	4.60555	0.162324
$806$	−26.4222	−0.930682
$807$	0	0
$808$	6.00000	0.211079
$809$	−49.8167	−1.75146	−0.875730	−	0.482801i	$-0.839619\pi$
$809$	−49.8167	−1.75146	−0.875730	+	0.482801i	$0.839619\pi$
$810$	0	0
$811$	−10.0000	−0.351147	−0.175574	−	0.984466i	$-0.556178\pi$
$811$	−10.0000	−0.351147	−0.175574	+	0.984466i	$0.556178\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	−0.605551	−0.0212246
$815$	−11.8167	−0.413919
$816$	0	0
$817$	−14.4222	−0.504569
$818$	−37.6333	−1.31582
$819$	0	0
$820$	−3.21110	−0.112137
$821$	9.63331	0.336205	0.168102	−	0.985770i	$-0.446236\pi$
$821$	9.63331	0.336205	0.168102	+	0.985770i	$0.446236\pi$
$822$	0	0
$823$	−9.57779	−0.333861	−0.166930	−	0.985969i	$-0.553386\pi$
$823$	−9.57779	−0.333861	−0.166930	+	0.985969i	$0.553386\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	−18.4222	−0.640603	−0.320301	−	0.947316i	$-0.603784\pi$
$827$	−18.4222	−0.640603	−0.320301	+	0.947316i	$0.603784\pi$
$828$	0	0
$829$	12.6056	0.437809	0.218904	−	0.975746i	$-0.429752\pi$
$829$	12.6056	0.437809	0.218904	+	0.975746i	$0.429752\pi$
$830$	−9.21110	−0.319722
$831$	0	0
$832$	6.60555	0.229006
$833$	4.60555	0.159573
$834$	0	0
$835$	−7.81665	−0.270506
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	−9.21110	−0.318192
$839$	41.4500	1.43101	0.715506	−	0.698607i	$-0.246196\pi$
$839$	41.4500	1.43101	0.715506	+	0.698607i	$0.246196\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	8.42221	0.290248
$843$	0	0
$844$	−8.60555	−0.296215
$845$	30.6333	1.05382
$846$	0	0
$847$	1.00000	0.0343604
$848$	−3.21110	−0.110270
$849$	0	0
$850$	4.60555	0.157969
$851$	2.78890	0.0956022
$852$	0	0
$853$	30.6056	1.04791	0.523957	−	0.851745i	$-0.324455\pi$
$853$	30.6056	1.04791	0.523957	+	0.851745i	$0.324455\pi$
$854$	−13.2111	−0.452075
$855$	0	0
$856$	0	0
$857$	−29.4500	−1.00599	−0.502996	−	0.864289i	$-0.667769\pi$
$857$	−29.4500	−1.00599	−0.502996	+	0.864289i	$0.667769\pi$
$858$	0	0
$859$	−32.1833	−1.09808	−0.549041	−	0.835796i	$-0.685007\pi$
$859$	−32.1833	−1.09808	−0.549041	+	0.835796i	$0.685007\pi$
$860$	−7.21110	−0.245897
$861$	0	0
$862$	29.0278	0.988689
$863$	11.0278	0.375389	0.187695	−	0.982227i	$-0.439898\pi$
$863$	11.0278	0.375389	0.187695	+	0.982227i	$0.439898\pi$
$864$	0	0
$865$	21.6333	0.735555
$866$	36.0555	1.22522
$867$	0	0
$868$	−4.00000	−0.135769
$869$	1.21110	0.0410838
$870$	0	0
$871$	−78.0555	−2.64481
$872$	15.8167	0.535619
$873$	0	0
$874$	9.21110	0.311570
$875$	1.00000	0.0338062
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−25.2111	−0.850833
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	13.3944	0.451270	0.225635	−	0.974212i	$-0.427554\pi$
$881$	13.3944	0.451270	0.225635	+	0.974212i	$0.427554\pi$
$882$	0	0
$883$	−41.3944	−1.39303	−0.696517	−	0.717540i	$-0.745268\pi$
$883$	−41.3944	−1.39303	−0.696517	+	0.717540i	$0.745268\pi$
$884$	30.4222	1.02321
$885$	0	0
$886$	0	0
$887$	19.8167	0.665378	0.332689	−	0.943037i	$-0.392044\pi$
$887$	19.8167	0.665378	0.332689	+	0.943037i	$0.392044\pi$
$888$	0	0
$889$	5.21110	0.174775
$890$	1.39445	0.0467420
$891$	0	0
$892$	−22.4222	−0.750751
$893$	−6.42221	−0.214911
$894$	0	0
$895$	2.78890	0.0932226
$896$	1.00000	0.0334077
$897$	0	0
$898$	−36.0000	−1.20134
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−14.7889	−0.492690
$902$	3.21110	0.106918
$903$	0	0
$904$	1.39445	0.0463787
$905$	24.6056	0.817916
$906$	0	0
$907$	−21.0278	−0.698215	−0.349108	−	0.937083i	$-0.613515\pi$
$907$	−21.0278	−0.698215	−0.349108	+	0.937083i	$0.613515\pi$
$908$	6.42221	0.213128
$909$	0	0
$910$	6.60555	0.218972
$911$	−42.8444	−1.41950	−0.709749	−	0.704454i	$-0.751192\pi$
$911$	−42.8444	−1.41950	−0.709749	+	0.704454i	$0.751192\pi$
$912$	0	0
$913$	9.21110	0.304843
$914$	4.78890	0.158403
$915$	0	0
$916$	−20.6056	−0.680827
$917$	0	0
$918$	0	0
$919$	17.2111	0.567742	0.283871	−	0.958862i	$-0.408381\pi$
$919$	17.2111	0.567742	0.283871	+	0.958862i	$0.408381\pi$
$920$	4.60555	0.151841
$921$	0	0
$922$	8.78890	0.289447
$923$	82.0555	2.70089
$924$	0	0
$925$	0.605551	0.0199104
$926$	14.4222	0.473943
$927$	0	0
$928$	6.00000	0.196960
$929$	−43.8167	−1.43758	−0.718789	−	0.695228i	$-0.755303\pi$
$929$	−43.8167	−1.43758	−0.718789	+	0.695228i	$0.755303\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−12.4222	−0.406903
$933$	0	0
$934$	−15.6333	−0.511537
$935$	−4.60555	−0.150618
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−11.8167	−0.385827
$939$	0	0
$940$	−3.21110	−0.104735
$941$	42.8444	1.39669	0.698344	−	0.715762i	$-0.253921\pi$
$941$	42.8444	1.39669	0.698344	+	0.715762i	$0.253921\pi$
$942$	0	0
$943$	−14.7889	−0.481593
$944$	0	0
$945$	0	0
$946$	7.21110	0.234453
$947$	−42.4222	−1.37854	−0.689268	−	0.724506i	$-0.742068\pi$
$947$	−42.4222	−1.37854	−0.689268	+	0.724506i	$0.742068\pi$
$948$	0	0
$949$	−66.0555	−2.14425
$950$	2.00000	0.0648886
$951$	0	0
$952$	4.60555	0.149267
$953$	9.63331	0.312053	0.156027	−	0.987753i	$-0.450131\pi$
$953$	9.63331	0.312053	0.156027	+	0.987753i	$0.450131\pi$
$954$	0	0
$955$	8.78890	0.284402
$956$	−19.8167	−0.640916
$957$	0	0
$958$	12.0000	0.387702
$959$	−10.6056	−0.342471
$960$	0	0
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	0	0
$964$	−25.6333	−0.825593
$965$	−7.21110	−0.232134
$966$	0	0
$967$	47.6333	1.53178	0.765892	−	0.642969i	$-0.222298\pi$
$967$	47.6333	1.53178	0.765892	+	0.642969i	$0.222298\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	4.78890	0.153762
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	−0.788897	−0.0252909
$974$	−6.78890	−0.217530
$975$	0	0
$976$	−13.2111	−0.422877
$977$	−37.3944	−1.19635	−0.598177	−	0.801364i	$-0.704108\pi$
$977$	−37.3944	−1.19635	−0.598177	+	0.801364i	$0.704108\pi$
$978$	0	0
$979$	−1.39445	−0.0445668
$980$	1.00000	0.0319438
$981$	0	0
$982$	−33.2111	−1.05981
$983$	−45.6333	−1.45548	−0.727738	−	0.685855i	$-0.759428\pi$
$983$	−45.6333	−1.45548	−0.727738	+	0.685855i	$0.759428\pi$
$984$	0	0
$985$	15.2111	0.484666
$986$	27.6333	0.880024
$987$	0	0
$988$	13.2111	0.420301
$989$	−33.2111	−1.05605
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	12.4222	0.394008
$995$	−6.78890	−0.215223
$996$	0	0
$997$	39.8167	1.26101	0.630503	−	0.776187i	$-0.282849\pi$
$997$	39.8167	1.26101	0.630503	+	0.776187i	$0.282849\pi$
$998$	−25.2111	−0.798044
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.cc.1.2	yes	2
3.2	odd	2		6930.2.a.bp.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6930.2.a.bp.1.2	✓	2	3.2	odd	2
6930.2.a.cc.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +6.60555 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.60555 q^{17} +2.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +4.60555 q^{23} +1.00000 q^{25} +6.60555 q^{26} +1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +4.60555 q^{34} +1.00000 q^{35} +0.605551 q^{37} +2.00000 q^{38} +1.00000 q^{40} -3.21110 q^{41} -7.21110 q^{43} -1.00000 q^{44} +4.60555 q^{46} -3.21110 q^{47} +1.00000 q^{49} +1.00000 q^{50} +6.60555 q^{52} -3.21110 q^{53} -1.00000 q^{55} +1.00000 q^{56} +6.00000 q^{58} -13.2111 q^{61} -4.00000 q^{62} +1.00000 q^{64} +6.60555 q^{65} -11.8167 q^{67} +4.60555 q^{68} +1.00000 q^{70} +12.4222 q^{71} -10.0000 q^{73} +0.605551 q^{74} +2.00000 q^{76} -1.00000 q^{77} -1.21110 q^{79} +1.00000 q^{80} -3.21110 q^{82} -9.21110 q^{83} +4.60555 q^{85} -7.21110 q^{86} -1.00000 q^{88} +1.39445 q^{89} +6.60555 q^{91} +4.60555 q^{92} -3.21110 q^{94} +2.00000 q^{95} +4.78890 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} + 2 q^{8} + 2 q^{10} - 2 q^{11} + 6 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 2 q^{20} - 2 q^{22} + 2 q^{23} + 2 q^{25} + 6 q^{26} + 2 q^{28} + 12 q^{29} - 8 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} - 6 q^{37} + 4 q^{38} + 2 q^{40} + 8 q^{41} - 2 q^{44} + 2 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} + 6 q^{52} + 8 q^{53} - 2 q^{55} + 2 q^{56} + 12 q^{58} - 12 q^{61} - 8 q^{62} + 2 q^{64} + 6 q^{65} - 2 q^{67} + 2 q^{68} + 2 q^{70} - 4 q^{71} - 20 q^{73} - 6 q^{74} + 4 q^{76} - 2 q^{77} + 12 q^{79} + 2 q^{80} + 8 q^{82} - 4 q^{83} + 2 q^{85} - 2 q^{88} + 10 q^{89} + 6 q^{91} + 2 q^{92} + 8 q^{94} + 4 q^{95} + 24 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 + 2 * q^8 + 2 * q^10 - 2 * q^11 + 6 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 + 4 * q^19 + 2 * q^20 - 2 * q^22 + 2 * q^23 + 2 * q^25 + 6 * q^26 + 2 * q^28 + 12 * q^29 - 8 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^35 - 6 * q^37 + 4 * q^38 + 2 * q^40 + 8 * q^41 - 2 * q^44 + 2 * q^46 + 8 * q^47 + 2 * q^49 + 2 * q^50 + 6 * q^52 + 8 * q^53 - 2 * q^55 + 2 * q^56 + 12 * q^58 - 12 * q^61 - 8 * q^62 + 2 * q^64 + 6 * q^65 - 2 * q^67 + 2 * q^68 + 2 * q^70 - 4 * q^71 - 20 * q^73 - 6 * q^74 + 4 * q^76 - 2 * q^77 + 12 * q^79 + 2 * q^80 + 8 * q^82 - 4 * q^83 + 2 * q^85 - 2 * q^88 + 10 * q^89 + 6 * q^91 + 2 * q^92 + 8 * q^94 + 4 * q^95 + 24 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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