LMFDB - Embedded newform 6930.2.a.bz.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{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +6.74456 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.74456 q^{17} +4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +8.74456 q^{23} +1.00000 q^{25} +6.74456 q^{26} +1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -6.74456 q^{34} -1.00000 q^{35} -6.74456 q^{37} +4.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} +8.74456 q^{46} +1.00000 q^{49} +1.00000 q^{50} +6.74456 q^{52} -2.00000 q^{53} +1.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -6.74456 q^{65} -0.744563 q^{67} -6.74456 q^{68} -1.00000 q^{70} +4.00000 q^{71} -2.00000 q^{73} -6.74456 q^{74} +4.00000 q^{76} -1.00000 q^{77} -1.00000 q^{80} -2.00000 q^{82} +8.00000 q^{83} +6.74456 q^{85} -1.00000 q^{88} -14.7446 q^{89} +6.74456 q^{91} +8.74456 q^{92} -4.00000 q^{95} -7.48913 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 8 q^{19} - 2 q^{20} - 2 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{35} - 2 q^{37} + 8 q^{38} - 2 q^{40} - 4 q^{41} - 2 q^{44} + 6 q^{46} + 2 q^{49} + 2 q^{50} + 2 q^{52} - 4 q^{53} + 2 q^{55} + 2 q^{56} - 4 q^{58} + 8 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{65} + 10 q^{67} - 2 q^{68} - 2 q^{70} + 8 q^{71} - 4 q^{73} - 2 q^{74} + 8 q^{76} - 2 q^{77} - 2 q^{80} - 4 q^{82} + 16 q^{83} + 2 q^{85} - 2 q^{88} - 18 q^{89} + 2 q^{91} + 6 q^{92} - 8 q^{95} + 8 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 + 2 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 8 * q^19 - 2 * q^20 - 2 * q^22 + 6 * q^23 + 2 * q^25 + 2 * q^26 + 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 2 * q^34 - 2 * q^35 - 2 * q^37 + 8 * q^38 - 2 * q^40 - 4 * q^41 - 2 * q^44 + 6 * q^46 + 2 * q^49 + 2 * q^50 + 2 * q^52 - 4 * q^53 + 2 * q^55 + 2 * q^56 - 4 * q^58 + 8 * q^59 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^65 + 10 * q^67 - 2 * q^68 - 2 * q^70 + 8 * q^71 - 4 * q^73 - 2 * q^74 + 8 * q^76 - 2 * q^77 - 2 * q^80 - 4 * q^82 + 16 * q^83 + 2 * q^85 - 2 * q^88 - 18 * q^89 + 2 * q^91 + 6 * q^92 - 8 * q^95 + 8 * 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
$13$	6.74456	1.87061	0.935303	−	0.353849i	$-0.115127\pi$
$13$	6.74456	1.87061	0.935303	+	0.353849i	$0.115127\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.74456	−1.63580	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−6.74456	−1.63580	−0.817898	+	0.575363i	$0.804861\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	8.74456	1.82337	0.911684	−	0.410893i	$-0.134783\pi$
$23$	8.74456	1.82337	0.911684	+	0.410893i	$0.134783\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.74456	1.32272
$27$	0	0
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.74456	−1.15668
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.74456	−1.10880	−0.554400	−	0.832251i	$-0.687052\pi$
$37$	−6.74456	−1.10880	−0.554400	+	0.832251i	$0.687052\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	8.74456	1.28932
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	6.74456	0.935303
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.74456	−0.836560
$66$	0	0
$67$	−0.744563	−0.0909628	−0.0454814	−	0.998965i	$-0.514482\pi$
$67$	−0.744563	−0.0909628	−0.0454814	+	0.998965i	$0.514482\pi$
$68$	−6.74456	−0.817898
$69$	0	0
$70$	−1.00000	−0.119523
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−6.74456	−0.784039
$75$	0	0
$76$	4.00000	0.458831
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−2.00000	−0.220863
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	6.74456	0.731551
$86$	0	0
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−14.7446	−1.56292	−0.781460	−	0.623955i	$-0.785525\pi$
$89$	−14.7446	−1.56292	−0.781460	+	0.623955i	$0.785525\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	8.74456	0.911684
$93$	0	0
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−7.48913	−0.760405	−0.380203	−	0.924903i	$-0.624146\pi$
$97$	−7.48913	−0.760405	−0.380203	+	0.924903i	$0.624146\pi$
$98$	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
$103$	13.4891	1.32912	0.664562	−	0.747234i	$-0.268618\pi$
$103$	13.4891	1.32912	0.664562	+	0.747234i	$0.268618\pi$
$104$	6.74456	0.661359
$105$	0	0
$106$	−2.00000	−0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	5.25544	0.503380	0.251690	−	0.967808i	$-0.419014\pi$
$109$	5.25544	0.503380	0.251690	+	0.967808i	$0.419014\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	20.2337	1.90343	0.951713	−	0.306990i	$-0.0993219\pi$
$113$	20.2337	1.90343	0.951713	+	0.306990i	$0.0993219\pi$
$114$	0	0
$115$	−8.74456	−0.815435
$116$	−2.00000	−0.185695
$117$	0	0
$118$	4.00000	0.368230
$119$	−6.74456	−0.618273
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.74456	−0.591537
$131$	17.4891	1.52803	0.764016	−	0.645197i	$-0.223225\pi$
$131$	17.4891	1.52803	0.764016	+	0.645197i	$0.223225\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−0.744563	−0.0643204
$135$	0	0
$136$	−6.74456	−0.578341
$137$	−14.7446	−1.25971	−0.629857	−	0.776712i	$-0.716886\pi$
$137$	−14.7446	−1.25971	−0.629857	+	0.776712i	$0.716886\pi$
$138$	0	0
$139$	−5.48913	−0.465582	−0.232791	−	0.972527i	$-0.574786\pi$
$139$	−5.48913	−0.465582	−0.232791	+	0.972527i	$0.574786\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	4.00000	0.335673
$143$	−6.74456	−0.564009
$144$	0	0
$145$	2.00000	0.166091
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−6.74456	−0.554400
$149$	−19.4891	−1.59661	−0.798306	−	0.602252i	$-0.794270\pi$
$149$	−19.4891	−1.59661	−0.798306	+	0.602252i	$0.794270\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−4.00000	−0.321288
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	8.74456	0.689168
$162$	0	0
$163$	−8.74456	−0.684927	−0.342464	−	0.939531i	$-0.611261\pi$
$163$	−8.74456	−0.684927	−0.342464	+	0.939531i	$0.611261\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	8.00000	0.620920
$167$	20.7446	1.60526	0.802631	−	0.596476i	$-0.203433\pi$
$167$	20.7446	1.60526	0.802631	+	0.596476i	$0.203433\pi$
$168$	0	0
$169$	32.4891	2.49916
$170$	6.74456	0.517284
$171$	0	0
$172$	0	0
$173$	15.4891	1.17762	0.588808	−	0.808273i	$-0.299597\pi$
$173$	15.4891	1.17762	0.588808	+	0.808273i	$0.299597\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−14.7446	−1.10515
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	14.7446	1.09595	0.547977	−	0.836493i	$-0.315398\pi$
$181$	14.7446	1.09595	0.547977	+	0.836493i	$0.315398\pi$
$182$	6.74456	0.499940
$183$	0	0
$184$	8.74456	0.644658
$185$	6.74456	0.495870
$186$	0	0
$187$	6.74456	0.493211
$188$	0	0
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−13.4891	−0.976039	−0.488019	−	0.872833i	$-0.662280\pi$
$191$	−13.4891	−0.976039	−0.488019	+	0.872833i	$0.662280\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−7.48913	−0.537688
$195$	0	0
$196$	1.00000	0.0714286
$197$	15.4891	1.10355	0.551777	−	0.833992i	$-0.313950\pi$
$197$	15.4891	1.10355	0.551777	+	0.833992i	$0.313950\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	−2.00000	−0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	13.4891	0.939832
$207$	0	0
$208$	6.74456	0.467651
$209$	−4.00000	−0.276686
$210$	0	0
$211$	14.2337	0.979887	0.489944	−	0.871754i	$-0.337017\pi$
$211$	14.2337	0.979887	0.489944	+	0.871754i	$0.337017\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	5.25544	0.355943
$219$	0	0
$220$	1.00000	0.0674200
$221$	−45.4891	−3.05993
$222$	0	0
$223$	21.4891	1.43902	0.719509	−	0.694483i	$-0.244367\pi$
$223$	21.4891	1.43902	0.719509	+	0.694483i	$0.244367\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	20.2337	1.34592
$227$	−26.9783	−1.79061	−0.895305	−	0.445454i	$-0.853042\pi$
$227$	−26.9783	−1.79061	−0.895305	+	0.445454i	$0.853042\pi$
$228$	0	0
$229$	5.25544	0.347289	0.173645	−	0.984808i	$-0.444446\pi$
$229$	5.25544	0.347289	0.173645	+	0.984808i	$0.444446\pi$
$230$	−8.74456	−0.576599
$231$	0	0
$232$	−2.00000	−0.131306
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	−6.74456	−0.437185
$239$	−3.25544	−0.210577	−0.105288	−	0.994442i	$-0.533577\pi$
$239$	−3.25544	−0.210577	−0.105288	+	0.994442i	$0.533577\pi$
$240$	0	0
$241$	−20.9783	−1.35133	−0.675664	−	0.737210i	$-0.736143\pi$
$241$	−20.9783	−1.35133	−0.675664	+	0.737210i	$0.736143\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	26.9783	1.71658
$248$	4.00000	0.254000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−22.9783	−1.45037	−0.725187	−	0.688552i	$-0.758247\pi$
$251$	−22.9783	−1.45037	−0.725187	+	0.688552i	$0.758247\pi$
$252$	0	0
$253$	−8.74456	−0.549766
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.4891	1.46521	0.732606	−	0.680653i	$-0.238304\pi$
$257$	23.4891	1.46521	0.732606	+	0.680653i	$0.238304\pi$
$258$	0	0
$259$	−6.74456	−0.419087
$260$	−6.74456	−0.418280
$261$	0	0
$262$	17.4891	1.08048
$263$	−25.4891	−1.57173	−0.785863	−	0.618400i	$-0.787781\pi$
$263$	−25.4891	−1.57173	−0.785863	+	0.618400i	$0.787781\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	4.00000	0.245256
$267$	0	0
$268$	−0.744563	−0.0454814
$269$	3.48913	0.212736	0.106368	−	0.994327i	$-0.466078\pi$
$269$	3.48913	0.212736	0.106368	+	0.994327i	$0.466078\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−6.74456	−0.408949
$273$	0	0
$274$	−14.7446	−0.890752
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−15.4891	−0.930651	−0.465326	−	0.885140i	$-0.654063\pi$
$277$	−15.4891	−0.930651	−0.465326	+	0.885140i	$0.654063\pi$
$278$	−5.48913	−0.329216
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−2.74456	−0.163727	−0.0818634	−	0.996644i	$-0.526087\pi$
$281$	−2.74456	−0.163727	−0.0818634	+	0.996644i	$0.526087\pi$
$282$	0	0
$283$	−0.744563	−0.0442597	−0.0221298	−	0.999755i	$-0.507045\pi$
$283$	−0.744563	−0.0442597	−0.0221298	+	0.999755i	$0.507045\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−6.74456	−0.398814
$287$	−2.00000	−0.118056
$288$	0	0
$289$	28.4891	1.67583
$290$	2.00000	0.117444
$291$	0	0
$292$	−2.00000	−0.117041
$293$	16.9783	0.991880	0.495940	−	0.868357i	$-0.334824\pi$
$293$	16.9783	0.991880	0.495940	+	0.868357i	$0.334824\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−6.74456	−0.392020
$297$	0	0
$298$	−19.4891	−1.12897
$299$	58.9783	3.41080
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	4.00000	0.229416
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−8.74456	−0.499079	−0.249539	−	0.968365i	$-0.580279\pi$
$307$	−8.74456	−0.499079	−0.249539	+	0.968365i	$0.580279\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−4.00000	−0.227185
$311$	30.2337	1.71440	0.857198	−	0.514988i	$-0.172204\pi$
$311$	30.2337	1.71440	0.857198	+	0.514988i	$0.172204\pi$
$312$	0	0
$313$	−32.9783	−1.86404	−0.932020	−	0.362406i	$-0.881956\pi$
$313$	−32.9783	−1.86404	−0.932020	+	0.362406i	$0.881956\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	8.74456	0.487315
$323$	−26.9783	−1.50111
$324$	0	0
$325$	6.74456	0.374121
$326$	−8.74456	−0.484317
$327$	0	0
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	−22.9783	−1.26300	−0.631499	−	0.775376i	$-0.717560\pi$
$331$	−22.9783	−1.26300	−0.631499	+	0.775376i	$0.717560\pi$
$332$	8.00000	0.439057
$333$	0	0
$334$	20.7446	1.13509
$335$	0.744563	0.0406798
$336$	0	0
$337$	−4.51087	−0.245723	−0.122862	−	0.992424i	$-0.539207\pi$
$337$	−4.51087	−0.245723	−0.122862	+	0.992424i	$0.539207\pi$
$338$	32.4891	1.76718
$339$	0	0
$340$	6.74456	0.365775
$341$	−4.00000	−0.216612
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	15.4891	0.832701
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−27.4891	−1.46310	−0.731549	−	0.681789i	$-0.761202\pi$
$353$	−27.4891	−1.46310	−0.731549	+	0.681789i	$0.761202\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	−14.7446	−0.781460
$357$	0	0
$358$	−12.0000	−0.634220
$359$	14.2337	0.751225	0.375613	−	0.926777i	$-0.377432\pi$
$359$	14.2337	0.751225	0.375613	+	0.926777i	$0.377432\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	14.7446	0.774957
$363$	0	0
$364$	6.74456	0.353511
$365$	2.00000	0.104685
$366$	0	0
$367$	5.48913	0.286530	0.143265	−	0.989684i	$-0.454240\pi$
$367$	5.48913	0.286530	0.143265	+	0.989684i	$0.454240\pi$
$368$	8.74456	0.455842
$369$	0	0
$370$	6.74456	0.350633
$371$	−2.00000	−0.103835
$372$	0	0
$373$	12.9783	0.671988	0.335994	−	0.941864i	$-0.390928\pi$
$373$	12.9783	0.671988	0.335994	+	0.941864i	$0.390928\pi$
$374$	6.74456	0.348753
$375$	0	0
$376$	0	0
$377$	−13.4891	−0.694725
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−13.4891	−0.690164
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	10.0000	0.508987
$387$	0	0
$388$	−7.48913	−0.380203
$389$	7.48913	0.379714	0.189857	−	0.981812i	$-0.439198\pi$
$389$	7.48913	0.379714	0.189857	+	0.981812i	$0.439198\pi$
$390$	0	0
$391$	−58.9783	−2.98266
$392$	1.00000	0.0505076
$393$	0	0
$394$	15.4891	0.780331
$395$	0	0
$396$	0	0
$397$	35.4891	1.78115	0.890574	−	0.454838i	$-0.150303\pi$
$397$	35.4891	1.78115	0.890574	+	0.454838i	$0.150303\pi$
$398$	12.0000	0.601506
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	26.9783	1.34388
$404$	6.00000	0.298511
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	6.74456	0.334316
$408$	0	0
$409$	−8.51087	−0.420836	−0.210418	−	0.977612i	$-0.567482\pi$
$409$	−8.51087	−0.420836	−0.210418	+	0.977612i	$0.567482\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	13.4891	0.664562
$413$	4.00000	0.196827
$414$	0	0
$415$	−8.00000	−0.392705
$416$	6.74456	0.330679
$417$	0	0
$418$	−4.00000	−0.195646
$419$	−29.4891	−1.44064	−0.720319	−	0.693643i	$-0.756005\pi$
$419$	−29.4891	−1.44064	−0.720319	+	0.693643i	$0.756005\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	14.2337	0.692885
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	−6.74456	−0.327159
$426$	0	0
$427$	2.00000	0.0967868
$428$	4.00000	0.193347
$429$	0	0
$430$	0	0
$431$	31.7228	1.52803	0.764017	−	0.645196i	$-0.223224\pi$
$431$	31.7228	1.52803	0.764017	+	0.645196i	$0.223224\pi$
$432$	0	0
$433$	27.4891	1.32104	0.660522	−	0.750807i	$-0.270335\pi$
$433$	27.4891	1.32104	0.660522	+	0.750807i	$0.270335\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	5.25544	0.251690
$437$	34.9783	1.67324
$438$	0	0
$439$	18.9783	0.905782	0.452891	−	0.891566i	$-0.350393\pi$
$439$	18.9783	0.905782	0.452891	+	0.891566i	$0.350393\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−45.4891	−2.16370
$443$	38.9783	1.85191	0.925956	−	0.377631i	$-0.123261\pi$
$443$	38.9783	1.85191	0.925956	+	0.377631i	$0.123261\pi$
$444$	0	0
$445$	14.7446	0.698959
$446$	21.4891	1.01754
$447$	0	0
$448$	1.00000	0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	20.2337	0.951713
$453$	0	0
$454$	−26.9783	−1.26615
$455$	−6.74456	−0.316190
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	5.25544	0.245570
$459$	0	0
$460$	−8.74456	−0.407717
$461$	4.51087	0.210092	0.105046	−	0.994467i	$-0.466501\pi$
$461$	4.51087	0.210092	0.105046	+	0.994467i	$0.466501\pi$
$462$	0	0
$463$	−33.4891	−1.55637	−0.778186	−	0.628034i	$-0.783860\pi$
$463$	−33.4891	−1.55637	−0.778186	+	0.628034i	$0.783860\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	−6.97825	−0.322915	−0.161457	−	0.986880i	$-0.551619\pi$
$467$	−6.97825	−0.322915	−0.161457	+	0.986880i	$0.551619\pi$
$468$	0	0
$469$	−0.744563	−0.0343807
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	−6.74456	−0.309137
$477$	0	0
$478$	−3.25544	−0.148900
$479$	−26.9783	−1.23267	−0.616334	−	0.787485i	$-0.711383\pi$
$479$	−26.9783	−1.23267	−0.616334	+	0.787485i	$0.711383\pi$
$480$	0	0
$481$	−45.4891	−2.07413
$482$	−20.9783	−0.955533
$483$	0	0
$484$	1.00000	0.0454545
$485$	7.48913	0.340064
$486$	0	0
$487$	−25.4891	−1.15502	−0.577511	−	0.816383i	$-0.695976\pi$
$487$	−25.4891	−1.15502	−0.577511	+	0.816383i	$0.695976\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	13.4891	0.607520
$494$	26.9783	1.21381
$495$	0	0
$496$	4.00000	0.179605
$497$	4.00000	0.179425
$498$	0	0
$499$	−32.4674	−1.45344	−0.726720	−	0.686934i	$-0.758956\pi$
$499$	−32.4674	−1.45344	−0.726720	+	0.686934i	$0.758956\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−22.9783	−1.02557
$503$	−14.2337	−0.634649	−0.317324	−	0.948317i	$-0.602784\pi$
$503$	−14.2337	−0.634649	−0.317324	+	0.948317i	$0.602784\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	−8.74456	−0.388743
$507$	0	0
$508$	0	0
$509$	0.510875	0.0226441	0.0113221	−	0.999936i	$-0.496396\pi$
$509$	0.510875	0.0226441	0.0113221	+	0.999936i	$0.496396\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	23.4891	1.03606
$515$	−13.4891	−0.594402
$516$	0	0
$517$	0	0
$518$	−6.74456	−0.296339
$519$	0	0
$520$	−6.74456	−0.295769
$521$	36.2337	1.58743	0.793713	−	0.608292i	$-0.208145\pi$
$521$	36.2337	1.58743	0.793713	+	0.608292i	$0.208145\pi$
$522$	0	0
$523$	−10.2337	−0.447488	−0.223744	−	0.974648i	$-0.571828\pi$
$523$	−10.2337	−0.447488	−0.223744	+	0.974648i	$0.571828\pi$
$524$	17.4891	0.764016
$525$	0	0
$526$	−25.4891	−1.11138
$527$	−26.9783	−1.17519
$528$	0	0
$529$	53.4674	2.32467
$530$	2.00000	0.0868744
$531$	0	0
$532$	4.00000	0.173422
$533$	−13.4891	−0.584279
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−0.744563	−0.0321602
$537$	0	0
$538$	3.48913	0.150427
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−36.2337	−1.55781	−0.778904	−	0.627143i	$-0.784224\pi$
$541$	−36.2337	−1.55781	−0.778904	+	0.627143i	$0.784224\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	−6.74456	−0.289171
$545$	−5.25544	−0.225118
$546$	0	0
$547$	1.48913	0.0636704	0.0318352	−	0.999493i	$-0.489865\pi$
$547$	1.48913	0.0636704	0.0318352	+	0.999493i	$0.489865\pi$
$548$	−14.7446	−0.629857
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	−15.4891	−0.658070
$555$	0	0
$556$	−5.48913	−0.232791
$557$	−27.4891	−1.16475	−0.582376	−	0.812920i	$-0.697877\pi$
$557$	−27.4891	−1.16475	−0.582376	+	0.812920i	$0.697877\pi$
$558$	0	0
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−2.74456	−0.115772
$563$	−28.4674	−1.19976	−0.599878	−	0.800091i	$-0.704784\pi$
$563$	−28.4674	−1.19976	−0.599878	+	0.800091i	$0.704784\pi$
$564$	0	0
$565$	−20.2337	−0.851238
$566$	−0.744563	−0.0312963
$567$	0	0
$568$	4.00000	0.167836
$569$	−20.2337	−0.848240	−0.424120	−	0.905606i	$-0.639417\pi$
$569$	−20.2337	−0.848240	−0.424120	+	0.905606i	$0.639417\pi$
$570$	0	0
$571$	−30.2337	−1.26524	−0.632620	−	0.774462i	$-0.718021\pi$
$571$	−30.2337	−1.26524	−0.632620	+	0.774462i	$0.718021\pi$
$572$	−6.74456	−0.282004
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	8.74456	0.364673
$576$	0	0
$577$	−24.9783	−1.03986	−0.519929	−	0.854209i	$-0.674042\pi$
$577$	−24.9783	−1.03986	−0.519929	+	0.854209i	$0.674042\pi$
$578$	28.4891	1.18499
$579$	0	0
$580$	2.00000	0.0830455
$581$	8.00000	0.331896
$582$	0	0
$583$	2.00000	0.0828315
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	16.9783	0.701365
$587$	38.9783	1.60880	0.804402	−	0.594085i	$-0.202486\pi$
$587$	38.9783	1.60880	0.804402	+	0.594085i	$0.202486\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−4.00000	−0.164677
$591$	0	0
$592$	−6.74456	−0.277200
$593$	−8.23369	−0.338117	−0.169059	−	0.985606i	$-0.554073\pi$
$593$	−8.23369	−0.338117	−0.169059	+	0.985606i	$0.554073\pi$
$594$	0	0
$595$	6.74456	0.276500
$596$	−19.4891	−0.798306
$597$	0	0
$598$	58.9783	2.41180
$599$	5.48913	0.224280	0.112140	−	0.993692i	$-0.464230\pi$
$599$	5.48913	0.224280	0.112140	+	0.993692i	$0.464230\pi$
$600$	0	0
$601$	−8.51087	−0.347166	−0.173583	−	0.984819i	$-0.555534\pi$
$601$	−8.51087	−0.347166	−0.173583	+	0.984819i	$0.555534\pi$
$602$	0	0
$603$	0	0
$604$	16.0000	0.651031
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	0	0
$613$	27.4891	1.11028	0.555138	−	0.831758i	$-0.312666\pi$
$613$	27.4891	1.11028	0.555138	+	0.831758i	$0.312666\pi$
$614$	−8.74456	−0.352902
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−38.7446	−1.55980	−0.779899	−	0.625906i	$-0.784729\pi$
$617$	−38.7446	−1.55980	−0.779899	+	0.625906i	$0.784729\pi$
$618$	0	0
$619$	−0.744563	−0.0299265	−0.0149632	−	0.999888i	$-0.504763\pi$
$619$	−0.744563	−0.0299265	−0.0149632	+	0.999888i	$0.504763\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	30.2337	1.21226
$623$	−14.7446	−0.590728
$624$	0	0
$625$	1.00000	0.0400000
$626$	−32.9783	−1.31808
$627$	0	0
$628$	10.0000	0.399043
$629$	45.4891	1.81377
$630$	0	0
$631$	−42.9783	−1.71094	−0.855469	−	0.517855i	$-0.826731\pi$
$631$	−42.9783	−1.71094	−0.855469	+	0.517855i	$0.826731\pi$
$632$	0	0
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	6.74456	0.267229
$638$	2.00000	0.0791808
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	−30.9783	−1.22166	−0.610831	−	0.791761i	$-0.709165\pi$
$643$	−30.9783	−1.22166	−0.610831	+	0.791761i	$0.709165\pi$
$644$	8.74456	0.344584
$645$	0	0
$646$	−26.9783	−1.06145
$647$	18.9783	0.746112	0.373056	−	0.927809i	$-0.378310\pi$
$647$	18.9783	0.746112	0.373056	+	0.927809i	$0.378310\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	6.74456	0.264544
$651$	0	0
$652$	−8.74456	−0.342464
$653$	−3.48913	−0.136540	−0.0682700	−	0.997667i	$-0.521748\pi$
$653$	−3.48913	−0.136540	−0.0682700	+	0.997667i	$0.521748\pi$
$654$	0	0
$655$	−17.4891	−0.683357
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	0	0
$659$	22.9783	0.895106	0.447553	−	0.894258i	$-0.352296\pi$
$659$	22.9783	0.895106	0.447553	+	0.894258i	$0.352296\pi$
$660$	0	0
$661$	33.7228	1.31167	0.655833	−	0.754906i	$-0.272318\pi$
$661$	33.7228	1.31167	0.655833	+	0.754906i	$0.272318\pi$
$662$	−22.9783	−0.893075
$663$	0	0
$664$	8.00000	0.310460
$665$	−4.00000	−0.155113
$666$	0	0
$667$	−17.4891	−0.677182
$668$	20.7446	0.802631
$669$	0	0
$670$	0.744563	0.0287650
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−42.4674	−1.63700	−0.818499	−	0.574509i	$-0.805193\pi$
$673$	−42.4674	−1.63700	−0.818499	+	0.574509i	$0.805193\pi$
$674$	−4.51087	−0.173752
$675$	0	0
$676$	32.4891	1.24958
$677$	−43.4891	−1.67142	−0.835711	−	0.549169i	$-0.814944\pi$
$677$	−43.4891	−1.67142	−0.835711	+	0.549169i	$0.814944\pi$
$678$	0	0
$679$	−7.48913	−0.287406
$680$	6.74456	0.258642
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	14.7446	0.563361
$686$	1.00000	0.0381802
$687$	0	0
$688$	0	0
$689$	−13.4891	−0.513895
$690$	0	0
$691$	37.2119	1.41561	0.707804	−	0.706408i	$-0.249686\pi$
$691$	37.2119	1.41561	0.707804	+	0.706408i	$0.249686\pi$
$692$	15.4891	0.588808
$693$	0	0
$694$	−20.0000	−0.759190
$695$	5.48913	0.208214
$696$	0	0
$697$	13.4891	0.510937
$698$	34.0000	1.28692
$699$	0	0
$700$	1.00000	0.0377964
$701$	−39.9565	−1.50914	−0.754568	−	0.656222i	$-0.772154\pi$
$701$	−39.9565	−1.50914	−0.754568	+	0.656222i	$0.772154\pi$
$702$	0	0
$703$	−26.9783	−1.01750
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−27.4891	−1.03457
$707$	6.00000	0.225653
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	−4.00000	−0.150117
$711$	0	0
$712$	−14.7446	−0.552576
$713$	34.9783	1.30995
$714$	0	0
$715$	6.74456	0.252232
$716$	−12.0000	−0.448461
$717$	0	0
$718$	14.2337	0.531197
$719$	36.7446	1.37034	0.685170	−	0.728383i	$-0.259728\pi$
$719$	36.7446	1.37034	0.685170	+	0.728383i	$0.259728\pi$
$720$	0	0
$721$	13.4891	0.502361
$722$	−3.00000	−0.111648
$723$	0	0
$724$	14.7446	0.547977
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−2.51087	−0.0931232	−0.0465616	−	0.998915i	$-0.514826\pi$
$727$	−2.51087	−0.0931232	−0.0465616	+	0.998915i	$0.514826\pi$
$728$	6.74456	0.249970
$729$	0	0
$730$	2.00000	0.0740233
$731$	0	0
$732$	0	0
$733$	−25.2554	−0.932831	−0.466415	−	0.884566i	$-0.654455\pi$
$733$	−25.2554	−0.932831	−0.466415	+	0.884566i	$0.654455\pi$
$734$	5.48913	0.202607
$735$	0	0
$736$	8.74456	0.322329
$737$	0.744563	0.0274263
$738$	0	0
$739$	23.7228	0.872658	0.436329	−	0.899787i	$-0.356278\pi$
$739$	23.7228	0.872658	0.436329	+	0.899787i	$0.356278\pi$
$740$	6.74456	0.247935
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−5.02175	−0.184230	−0.0921151	−	0.995748i	$-0.529363\pi$
$743$	−5.02175	−0.184230	−0.0921151	+	0.995748i	$0.529363\pi$
$744$	0	0
$745$	19.4891	0.714026
$746$	12.9783	0.475168
$747$	0	0
$748$	6.74456	0.246606
$749$	4.00000	0.146157
$750$	0	0
$751$	18.9783	0.692526	0.346263	−	0.938137i	$-0.387450\pi$
$751$	18.9783	0.692526	0.346263	+	0.938137i	$0.387450\pi$
$752$	0	0
$753$	0	0
$754$	−13.4891	−0.491245
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−25.7228	−0.934912	−0.467456	−	0.884016i	$-0.654829\pi$
$757$	−25.7228	−0.934912	−0.467456	+	0.884016i	$0.654829\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−4.97825	−0.180461	−0.0902307	−	0.995921i	$-0.528760\pi$
$761$	−4.97825	−0.180461	−0.0902307	+	0.995921i	$0.528760\pi$
$762$	0	0
$763$	5.25544	0.190260
$764$	−13.4891	−0.488019
$765$	0	0
$766$	24.0000	0.867155
$767$	26.9783	0.974128
$768$	0	0
$769$	−8.51087	−0.306910	−0.153455	−	0.988156i	$-0.549040\pi$
$769$	−8.51087	−0.306910	−0.153455	+	0.988156i	$0.549040\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	10.0000	0.359908
$773$	3.48913	0.125495	0.0627476	−	0.998029i	$-0.480014\pi$
$773$	3.48913	0.125495	0.0627476	+	0.998029i	$0.480014\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	−7.48913	−0.268844
$777$	0	0
$778$	7.48913	0.268498
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−58.9783	−2.10906
$783$	0	0
$784$	1.00000	0.0357143
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−0.744563	−0.0265408	−0.0132704	−	0.999912i	$-0.504224\pi$
$787$	−0.744563	−0.0265408	−0.0132704	+	0.999912i	$0.504224\pi$
$788$	15.4891	0.551777
$789$	0	0
$790$	0	0
$791$	20.2337	0.719427
$792$	0	0
$793$	13.4891	0.479013
$794$	35.4891	1.25946
$795$	0	0
$796$	12.0000	0.425329
$797$	3.48913	0.123591	0.0617956	−	0.998089i	$-0.480317\pi$
$797$	3.48913	0.123591	0.0617956	+	0.998089i	$0.480317\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	14.0000	0.494357
$803$	2.00000	0.0705785
$804$	0	0
$805$	−8.74456	−0.308205
$806$	26.9783	0.950268
$807$	0	0
$808$	6.00000	0.211079
$809$	−9.25544	−0.325404	−0.162702	−	0.986675i	$-0.552021\pi$
$809$	−9.25544	−0.325404	−0.162702	+	0.986675i	$0.552021\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	6.74456	0.236397
$815$	8.74456	0.306309
$816$	0	0
$817$	0	0
$818$	−8.51087	−0.297576
$819$	0	0
$820$	2.00000	0.0698430
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	13.4891	0.469916
$825$	0	0
$826$	4.00000	0.139178
$827$	−46.9783	−1.63359	−0.816797	−	0.576925i	$-0.804252\pi$
$827$	−46.9783	−1.63359	−0.816797	+	0.576925i	$0.804252\pi$
$828$	0	0
$829$	−4.23369	−0.147042	−0.0735210	−	0.997294i	$-0.523424\pi$
$829$	−4.23369	−0.147042	−0.0735210	+	0.997294i	$0.523424\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	6.74456	0.233826
$833$	−6.74456	−0.233685
$834$	0	0
$835$	−20.7446	−0.717895
$836$	−4.00000	−0.138343
$837$	0	0
$838$	−29.4891	−1.01868
$839$	−15.7228	−0.542812	−0.271406	−	0.962465i	$-0.587489\pi$
$839$	−15.7228	−0.542812	−0.271406	+	0.962465i	$0.587489\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	0	0
$844$	14.2337	0.489944
$845$	−32.4891	−1.11766
$846$	0	0
$847$	1.00000	0.0343604
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−6.74456	−0.231337
$851$	−58.9783	−2.02175
$852$	0	0
$853$	33.7228	1.15465	0.577324	−	0.816515i	$-0.304097\pi$
$853$	33.7228	1.15465	0.577324	+	0.816515i	$0.304097\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	4.00000	0.136717
$857$	2.74456	0.0937525	0.0468762	−	0.998901i	$-0.485073\pi$
$857$	2.74456	0.0937525	0.0468762	+	0.998901i	$0.485073\pi$
$858$	0	0
$859$	−23.2554	−0.793465	−0.396733	−	0.917934i	$-0.629856\pi$
$859$	−23.2554	−0.793465	−0.396733	+	0.917934i	$0.629856\pi$
$860$	0	0
$861$	0	0
$862$	31.7228	1.08048
$863$	8.74456	0.297668	0.148834	−	0.988862i	$-0.452448\pi$
$863$	8.74456	0.297668	0.148834	+	0.988862i	$0.452448\pi$
$864$	0	0
$865$	−15.4891	−0.526646
$866$	27.4891	0.934119
$867$	0	0
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	−5.02175	−0.170155
$872$	5.25544	0.177972
$873$	0	0
$874$	34.9783	1.18316
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−23.4891	−0.793171	−0.396586	−	0.917998i	$-0.629805\pi$
$877$	−23.4891	−0.793171	−0.396586	+	0.917998i	$0.629805\pi$
$878$	18.9783	0.640485
$879$	0	0
$880$	1.00000	0.0337100
$881$	31.2119	1.05156	0.525778	−	0.850622i	$-0.323774\pi$
$881$	31.2119	1.05156	0.525778	+	0.850622i	$0.323774\pi$
$882$	0	0
$883$	−24.7446	−0.832721	−0.416360	−	0.909200i	$-0.636695\pi$
$883$	−24.7446	−0.832721	−0.416360	+	0.909200i	$0.636695\pi$
$884$	−45.4891	−1.52996
$885$	0	0
$886$	38.9783	1.30950
$887$	49.2119	1.65238	0.826188	−	0.563395i	$-0.190505\pi$
$887$	49.2119	1.65238	0.826188	+	0.563395i	$0.190505\pi$
$888$	0	0
$889$	0	0
$890$	14.7446	0.494239
$891$	0	0
$892$	21.4891	0.719509
$893$	0	0
$894$	0	0
$895$	12.0000	0.401116
$896$	1.00000	0.0334077
$897$	0	0
$898$	−10.0000	−0.333704
$899$	−8.00000	−0.266815
$900$	0	0
$901$	13.4891	0.449388
$902$	2.00000	0.0665927
$903$	0	0
$904$	20.2337	0.672962
$905$	−14.7446	−0.490126
$906$	0	0
$907$	24.7446	0.821630	0.410815	−	0.911719i	$-0.365244\pi$
$907$	24.7446	0.821630	0.410815	+	0.911719i	$0.365244\pi$
$908$	−26.9783	−0.895305
$909$	0	0
$910$	−6.74456	−0.223580
$911$	38.9783	1.29141	0.645704	−	0.763588i	$-0.276564\pi$
$911$	38.9783	1.29141	0.645704	+	0.763588i	$0.276564\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	10.0000	0.330771
$915$	0	0
$916$	5.25544	0.173645
$917$	17.4891	0.577542
$918$	0	0
$919$	22.5109	0.742565	0.371283	−	0.928520i	$-0.378918\pi$
$919$	22.5109	0.742565	0.371283	+	0.928520i	$0.378918\pi$
$920$	−8.74456	−0.288300
$921$	0	0
$922$	4.51087	0.148558
$923$	26.9783	0.888000
$924$	0	0
$925$	−6.74456	−0.221760
$926$	−33.4891	−1.10052
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	50.7446	1.66488	0.832438	−	0.554119i	$-0.186945\pi$
$929$	50.7446	1.66488	0.832438	+	0.554119i	$0.186945\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	2.00000	0.0655122
$933$	0	0
$934$	−6.97825	−0.228335
$935$	−6.74456	−0.220571
$936$	0	0
$937$	15.4891	0.506008	0.253004	−	0.967465i	$-0.418581\pi$
$937$	15.4891	0.506008	0.253004	+	0.967465i	$0.418581\pi$
$938$	−0.744563	−0.0243108
$939$	0	0
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	−17.4891	−0.569524
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	−18.5109	−0.601523	−0.300761	−	0.953699i	$-0.597241\pi$
$947$	−18.5109	−0.601523	−0.300761	+	0.953699i	$0.597241\pi$
$948$	0	0
$949$	−13.4891	−0.437876
$950$	4.00000	0.129777
$951$	0	0
$952$	−6.74456	−0.218593
$953$	−23.4891	−0.760887	−0.380444	−	0.924804i	$-0.624229\pi$
$953$	−23.4891	−0.760887	−0.380444	+	0.924804i	$0.624229\pi$
$954$	0	0
$955$	13.4891	0.436498
$956$	−3.25544	−0.105288
$957$	0	0
$958$	−26.9783	−0.871628
$959$	−14.7446	−0.476127
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−45.4891	−1.46663
$963$	0	0
$964$	−20.9783	−0.675664
$965$	−10.0000	−0.321911
$966$	0	0
$967$	−13.0217	−0.418751	−0.209376	−	0.977835i	$-0.567143\pi$
$967$	−13.0217	−0.418751	−0.209376	+	0.977835i	$0.567143\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	7.48913	0.240461
$971$	−29.4891	−0.946351	−0.473176	−	0.880968i	$-0.656892\pi$
$971$	−29.4891	−0.946351	−0.473176	+	0.880968i	$0.656892\pi$
$972$	0	0
$973$	−5.48913	−0.175973
$974$	−25.4891	−0.816724
$975$	0	0
$976$	2.00000	0.0640184
$977$	−32.2337	−1.03125	−0.515624	−	0.856815i	$-0.672440\pi$
$977$	−32.2337	−1.03125	−0.515624	+	0.856815i	$0.672440\pi$
$978$	0	0
$979$	14.7446	0.471238
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	28.0000	0.893516
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	0	0
$985$	−15.4891	−0.493525
$986$	13.4891	0.429581
$987$	0	0
$988$	26.9783	0.858292
$989$	0	0
$990$	0	0
$991$	26.9783	0.856992	0.428496	−	0.903544i	$-0.359044\pi$
$991$	26.9783	0.856992	0.428496	+	0.903544i	$0.359044\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	4.00000	0.126872
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−7.76631	−0.245962	−0.122981	−	0.992409i	$-0.539245\pi$
$997$	−7.76631	−0.245962	−0.122981	+	0.992409i	$0.539245\pi$
$998$	−32.4674	−1.02774
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.bz.1.2		2
3.2	odd	2		2310.2.a.y.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.y.1.2	✓	2	3.2	odd	2
6930.2.a.bz.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +6.74456 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.74456 q^{17} +4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +8.74456 q^{23} +1.00000 q^{25} +6.74456 q^{26} +1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -6.74456 q^{34} -1.00000 q^{35} -6.74456 q^{37} +4.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} +8.74456 q^{46} +1.00000 q^{49} +1.00000 q^{50} +6.74456 q^{52} -2.00000 q^{53} +1.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -6.74456 q^{65} -0.744563 q^{67} -6.74456 q^{68} -1.00000 q^{70} +4.00000 q^{71} -2.00000 q^{73} -6.74456 q^{74} +4.00000 q^{76} -1.00000 q^{77} -1.00000 q^{80} -2.00000 q^{82} +8.00000 q^{83} +6.74456 q^{85} -1.00000 q^{88} -14.7446 q^{89} +6.74456 q^{91} +8.74456 q^{92} -4.00000 q^{95} -7.48913 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 8 q^{19} - 2 q^{20} - 2 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{35} - 2 q^{37} + 8 q^{38} - 2 q^{40} - 4 q^{41} - 2 q^{44} + 6 q^{46} + 2 q^{49} + 2 q^{50} + 2 q^{52} - 4 q^{53} + 2 q^{55} + 2 q^{56} - 4 q^{58} + 8 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 2 q^{65} + 10 q^{67} - 2 q^{68} - 2 q^{70} + 8 q^{71} - 4 q^{73} - 2 q^{74} + 8 q^{76} - 2 q^{77} - 2 q^{80} - 4 q^{82} + 16 q^{83} + 2 q^{85} - 2 q^{88} - 18 q^{89} + 2 q^{91} + 6 q^{92} - 8 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 + 2 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 8 * q^19 - 2 * q^20 - 2 * q^22 + 6 * q^23 + 2 * q^25 + 2 * q^26 + 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 2 * q^34 - 2 * q^35 - 2 * q^37 + 8 * q^38 - 2 * q^40 - 4 * q^41 - 2 * q^44 + 6 * q^46 + 2 * q^49 + 2 * q^50 + 2 * q^52 - 4 * q^53 + 2 * q^55 + 2 * q^56 - 4 * q^58 + 8 * q^59 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 2 * q^65 + 10 * q^67 - 2 * q^68 - 2 * q^70 + 8 * q^71 - 4 * q^73 - 2 * q^74 + 8 * q^76 - 2 * q^77 - 2 * q^80 - 4 * q^82 + 16 * q^83 + 2 * q^85 - 2 * q^88 - 18 * q^89 + 2 * q^91 + 6 * q^92 - 8 * q^95 + 8 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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