LMFDB - Embedded newform 6664.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6664,2,Mod(1,6664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6664, 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("6664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.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:	$53.2123079070$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{3} +0.381966 q^{5} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} +0.381966 q^{5} -2.61803 q^{9} -4.47214 q^{11} +4.47214 q^{13} -0.236068 q^{15} -1.00000 q^{17} -2.47214 q^{19} -9.23607 q^{23} -4.85410 q^{25} +3.47214 q^{27} +2.00000 q^{29} -2.09017 q^{31} +2.76393 q^{33} +4.00000 q^{37} -2.76393 q^{39} -1.09017 q^{41} +1.09017 q^{43} -1.00000 q^{45} +6.76393 q^{47} +0.618034 q^{51} +8.09017 q^{53} -1.70820 q^{55} +1.52786 q^{57} +4.00000 q^{59} -5.38197 q^{61} +1.70820 q^{65} -0.0901699 q^{67} +5.70820 q^{69} +11.5623 q^{73} +3.00000 q^{75} -14.1803 q^{79} +5.70820 q^{81} +6.76393 q^{83} -0.381966 q^{85} -1.23607 q^{87} +11.7082 q^{89} +1.29180 q^{93} -0.944272 q^{95} -15.0344 q^{97} +11.7082 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 3 q^{5} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 3 * q^5 - 3 * q^9	$ 2 q + q^{3} + 3 q^{5} - 3 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{19} - 14 q^{23} - 3 q^{25} - 2 q^{27} + 4 q^{29} + 7 q^{31} + 10 q^{33} + 8 q^{37} - 10 q^{39} + 9 q^{41} - 9 q^{43} - 2 q^{45} + 18 q^{47} - q^{51} + 5 q^{53} + 10 q^{55} + 12 q^{57} + 8 q^{59} - 13 q^{61} - 10 q^{65} + 11 q^{67} - 2 q^{69} + 3 q^{73} + 6 q^{75} - 6 q^{79} - 2 q^{81} + 18 q^{83} - 3 q^{85} + 2 q^{87} + 10 q^{89} + 16 q^{93} + 16 q^{95} - q^{97} + 10 q^{99}+O(q^{100}) $ 2 * q + q^3 + 3 * q^5 - 3 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^19 - 14 * q^23 - 3 * q^25 - 2 * q^27 + 4 * q^29 + 7 * q^31 + 10 * q^33 + 8 * q^37 - 10 * q^39 + 9 * q^41 - 9 * q^43 - 2 * q^45 + 18 * q^47 - q^51 + 5 * q^53 + 10 * q^55 + 12 * q^57 + 8 * q^59 - 13 * q^61 - 10 * q^65 + 11 * q^67 - 2 * q^69 + 3 * q^73 + 6 * q^75 - 6 * q^79 - 2 * q^81 + 18 * q^83 - 3 * q^85 + 2 * q^87 + 10 * q^89 + 16 * q^93 + 16 * q^95 - q^97 + 10 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	0.381966	0.170820	0.0854102	−	0.996346i	$-0.472780\pi$
$5$	0.381966	0.170820	0.0854102	+	0.996346i	$0.472780\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	−0.236068	−0.0609525
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.23607	−1.92585	−0.962927	−	0.269763i	$-0.913055\pi$
$23$	−9.23607	−1.92585	−0.962927	+	0.269763i	$0.913055\pi$
$24$	0	0
$25$	−4.85410	−0.970820
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−2.09017	−0.375406	−0.187703	−	0.982226i	$-0.560104\pi$
$31$	−2.09017	−0.375406	−0.187703	+	0.982226i	$0.560104\pi$
$32$	0	0
$33$	2.76393	0.481139
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−2.76393	−0.442583
$40$	0	0
$41$	−1.09017	−0.170256	−0.0851280	−	0.996370i	$-0.527130\pi$
$41$	−1.09017	−0.170256	−0.0851280	+	0.996370i	$0.527130\pi$
$42$	0	0
$43$	1.09017	0.166249	0.0831247	−	0.996539i	$-0.473510\pi$
$43$	1.09017	0.166249	0.0831247	+	0.996539i	$0.473510\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.76393	0.986621	0.493310	−	0.869853i	$-0.335787\pi$
$47$	6.76393	0.986621	0.493310	+	0.869853i	$0.335787\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.618034	0.0865421
$52$	0	0
$53$	8.09017	1.11127	0.555635	−	0.831426i	$-0.312475\pi$
$53$	8.09017	1.11127	0.555635	+	0.831426i	$0.312475\pi$
$54$	0	0
$55$	−1.70820	−0.230334
$56$	0	0
$57$	1.52786	0.202371
$58$	0	0
$59$	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
$61$	−5.38197	−0.689090	−0.344545	−	0.938770i	$-0.611967\pi$
$61$	−5.38197	−0.689090	−0.344545	+	0.938770i	$0.611967\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.70820	0.211877
$66$	0	0
$67$	−0.0901699	−0.0110160	−0.00550801	−	0.999985i	$-0.501753\pi$
$67$	−0.0901699	−0.0110160	−0.00550801	+	0.999985i	$0.501753\pi$
$68$	0	0
$69$	5.70820	0.687187
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	11.5623	1.35327	0.676633	−	0.736321i	$-0.263438\pi$
$73$	11.5623	1.35327	0.676633	+	0.736321i	$0.263438\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.1803	−1.59541	−0.797706	−	0.603046i	$-0.793954\pi$
$79$	−14.1803	−1.59541	−0.797706	+	0.603046i	$0.793954\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	6.76393	0.742438	0.371219	−	0.928545i	$-0.378940\pi$
$83$	6.76393	0.742438	0.371219	+	0.928545i	$0.378940\pi$
$84$	0	0
$85$	−0.381966	−0.0414300
$86$	0	0
$87$	−1.23607	−0.132520
$88$	0	0
$89$	11.7082	1.24107	0.620534	−	0.784180i	$-0.286916\pi$
$89$	11.7082	1.24107	0.620534	+	0.784180i	$0.286916\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.29180	0.133953
$94$	0	0
$95$	−0.944272	−0.0968803
$96$	0	0
$97$	−15.0344	−1.52652	−0.763258	−	0.646094i	$-0.776402\pi$
$97$	−15.0344	−1.52652	−0.763258	+	0.646094i	$0.776402\pi$
$98$	0	0
$99$	11.7082	1.17672
$100$	0	0
$101$	−0.944272	−0.0939586	−0.0469793	−	0.998896i	$-0.514959\pi$
$101$	−0.944272	−0.0939586	−0.0469793	+	0.998896i	$0.514959\pi$
$102$	0	0
$103$	0.291796	0.0287515	0.0143758	−	0.999897i	$-0.495424\pi$
$103$	0.291796	0.0287515	0.0143758	+	0.999897i	$0.495424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.52786	−0.727746	−0.363873	−	0.931449i	$-0.618546\pi$
$107$	−7.52786	−0.727746	−0.363873	+	0.931449i	$0.618546\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.47214	−0.234645
$112$	0	0
$113$	7.70820	0.725127	0.362563	−	0.931959i	$-0.381902\pi$
$113$	7.70820	0.725127	0.362563	+	0.931959i	$0.381902\pi$
$114$	0	0
$115$	−3.52786	−0.328975
$116$	0	0
$117$	−11.7082	−1.08242
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0.673762	0.0607511
$124$	0	0
$125$	−3.76393	−0.336656
$126$	0	0
$127$	16.0902	1.42777	0.713886	−	0.700262i	$-0.246934\pi$
$127$	16.0902	1.42777	0.713886	+	0.700262i	$0.246934\pi$
$128$	0	0
$129$	−0.673762	−0.0593214
$130$	0	0
$131$	10.4721	0.914955	0.457477	−	0.889221i	$-0.348753\pi$
$131$	10.4721	0.914955	0.457477	+	0.889221i	$0.348753\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.32624	0.114144
$136$	0	0
$137$	−4.90983	−0.419475	−0.209738	−	0.977758i	$-0.567261\pi$
$137$	−4.90983	−0.419475	−0.209738	+	0.977758i	$0.567261\pi$
$138$	0	0
$139$	11.7984	1.00073	0.500363	−	0.865816i	$-0.333200\pi$
$139$	11.7984	1.00073	0.500363	+	0.865816i	$0.333200\pi$
$140$	0	0
$141$	−4.18034	−0.352048
$142$	0	0
$143$	−20.0000	−1.67248
$144$	0	0
$145$	0.763932	0.0634411
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.14590	0.175799	0.0878994	−	0.996129i	$-0.471985\pi$
$149$	2.14590	0.175799	0.0878994	+	0.996129i	$0.471985\pi$
$150$	0	0
$151$	−14.3262	−1.16585	−0.582926	−	0.812525i	$-0.698092\pi$
$151$	−14.3262	−1.16585	−0.582926	+	0.812525i	$0.698092\pi$
$152$	0	0
$153$	2.61803	0.211656
$154$	0	0
$155$	−0.798374	−0.0641269
$156$	0	0
$157$	−5.52786	−0.441172	−0.220586	−	0.975368i	$-0.570797\pi$
$157$	−5.52786	−0.441172	−0.220586	+	0.975368i	$0.570797\pi$
$158$	0	0
$159$	−5.00000	−0.396526
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.94427	0.387265	0.193633	−	0.981074i	$-0.437973\pi$
$163$	4.94427	0.387265	0.193633	+	0.981074i	$0.437973\pi$
$164$	0	0
$165$	1.05573	0.0821883
$166$	0	0
$167$	8.32624	0.644304	0.322152	−	0.946688i	$-0.395594\pi$
$167$	8.32624	0.644304	0.322152	+	0.946688i	$0.395594\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	6.47214	0.494937
$172$	0	0
$173$	16.1459	1.22755	0.613775	−	0.789481i	$-0.289650\pi$
$173$	16.1459	1.22755	0.613775	+	0.789481i	$0.289650\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.47214	−0.185817
$178$	0	0
$179$	15.2705	1.14137	0.570686	−	0.821169i	$-0.306677\pi$
$179$	15.2705	1.14137	0.570686	+	0.821169i	$0.306677\pi$
$180$	0	0
$181$	−15.8885	−1.18099	−0.590493	−	0.807043i	$-0.701067\pi$
$181$	−15.8885	−1.18099	−0.590493	+	0.807043i	$0.701067\pi$
$182$	0	0
$183$	3.32624	0.245883
$184$	0	0
$185$	1.52786	0.112331
$186$	0	0
$187$	4.47214	0.327035
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.32624	0.530108	0.265054	−	0.964234i	$-0.414610\pi$
$191$	7.32624	0.530108	0.265054	+	0.964234i	$0.414610\pi$
$192$	0	0
$193$	−12.4721	−0.897764	−0.448882	−	0.893591i	$-0.648178\pi$
$193$	−12.4721	−0.897764	−0.448882	+	0.893591i	$0.648178\pi$
$194$	0	0
$195$	−1.05573	−0.0756023
$196$	0	0
$197$	26.4721	1.88606	0.943031	−	0.332705i	$-0.107961\pi$
$197$	26.4721	1.88606	0.943031	+	0.332705i	$0.107961\pi$
$198$	0	0
$199$	−19.0902	−1.35327	−0.676633	−	0.736320i	$-0.736562\pi$
$199$	−19.0902	−1.35327	−0.676633	+	0.736320i	$0.736562\pi$
$200$	0	0
$201$	0.0557281	0.00393076
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.416408	−0.0290832
$206$	0	0
$207$	24.1803	1.68065
$208$	0	0
$209$	11.0557	0.764741
$210$	0	0
$211$	−15.1246	−1.04122	−0.520611	−	0.853794i	$-0.674296\pi$
$211$	−15.1246	−1.04122	−0.520611	+	0.853794i	$0.674296\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.416408	0.0283988
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.14590	−0.482875
$220$	0	0
$221$	−4.47214	−0.300828
$222$	0	0
$223$	−4.94427	−0.331093	−0.165546	−	0.986202i	$-0.552939\pi$
$223$	−4.94427	−0.331093	−0.165546	+	0.986202i	$0.552939\pi$
$224$	0	0
$225$	12.7082	0.847214
$226$	0	0
$227$	6.85410	0.454923	0.227461	−	0.973787i	$-0.426957\pi$
$227$	6.85410	0.454923	0.227461	+	0.973787i	$0.426957\pi$
$228$	0	0
$229$	−28.4721	−1.88149	−0.940746	−	0.339112i	$-0.889873\pi$
$229$	−28.4721	−1.88149	−0.940746	+	0.339112i	$0.889873\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	2.58359	0.168535
$236$	0	0
$237$	8.76393	0.569279
$238$	0	0
$239$	−6.32624	−0.409210	−0.204605	−	0.978845i	$-0.565591\pi$
$239$	−6.32624	−0.409210	−0.204605	+	0.978845i	$0.565591\pi$
$240$	0	0
$241$	11.7984	0.760000	0.380000	−	0.924986i	$-0.375924\pi$
$241$	11.7984	0.760000	0.380000	+	0.924986i	$0.375924\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.0557	−0.703459
$248$	0	0
$249$	−4.18034	−0.264918
$250$	0	0
$251$	21.8885	1.38159	0.690796	−	0.723049i	$-0.257260\pi$
$251$	21.8885	1.38159	0.690796	+	0.723049i	$0.257260\pi$
$252$	0	0
$253$	41.3050	2.59682
$254$	0	0
$255$	0.236068	0.0147832
$256$	0	0
$257$	31.5967	1.97095	0.985475	−	0.169818i	$-0.0543179\pi$
$257$	31.5967	1.97095	0.985475	+	0.169818i	$0.0543179\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.23607	−0.324104
$262$	0	0
$263$	26.8328	1.65458	0.827291	−	0.561773i	$-0.189881\pi$
$263$	26.8328	1.65458	0.827291	+	0.561773i	$0.189881\pi$
$264$	0	0
$265$	3.09017	0.189828
$266$	0	0
$267$	−7.23607	−0.442840
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	5.23607	0.318068	0.159034	−	0.987273i	$-0.449162\pi$
$271$	5.23607	0.318068	0.159034	+	0.987273i	$0.449162\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.7082	1.30905
$276$	0	0
$277$	12.6525	0.760214	0.380107	−	0.924943i	$-0.375887\pi$
$277$	12.6525	0.760214	0.380107	+	0.924943i	$0.375887\pi$
$278$	0	0
$279$	5.47214	0.327608
$280$	0	0
$281$	4.85410	0.289571	0.144786	−	0.989463i	$-0.453751\pi$
$281$	4.85410	0.289571	0.144786	+	0.989463i	$0.453751\pi$
$282$	0	0
$283$	−28.7426	−1.70857	−0.854286	−	0.519802i	$-0.826006\pi$
$283$	−28.7426	−1.70857	−0.854286	+	0.519802i	$0.826006\pi$
$284$	0	0
$285$	0.583592	0.0345690
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	9.29180	0.544695
$292$	0	0
$293$	8.18034	0.477901	0.238950	−	0.971032i	$-0.423197\pi$
$293$	8.18034	0.477901	0.238950	+	0.971032i	$0.423197\pi$
$294$	0	0
$295$	1.52786	0.0889557
$296$	0	0
$297$	−15.5279	−0.901018
$298$	0	0
$299$	−41.3050	−2.38873
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.583592	0.0335265
$304$	0	0
$305$	−2.05573	−0.117711
$306$	0	0
$307$	19.2361	1.09786	0.548930	−	0.835868i	$-0.315035\pi$
$307$	19.2361	1.09786	0.548930	+	0.835868i	$0.315035\pi$
$308$	0	0
$309$	−0.180340	−0.0102592
$310$	0	0
$311$	5.50658	0.312249	0.156125	−	0.987737i	$-0.450100\pi$
$311$	5.50658	0.312249	0.156125	+	0.987737i	$0.450100\pi$
$312$	0	0
$313$	30.2705	1.71099	0.855495	−	0.517811i	$-0.173253\pi$
$313$	30.2705	1.71099	0.855495	+	0.517811i	$0.173253\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.12461	−0.512489	−0.256245	−	0.966612i	$-0.582485\pi$
$317$	−9.12461	−0.512489	−0.256245	+	0.966612i	$0.582485\pi$
$318$	0	0
$319$	−8.94427	−0.500783
$320$	0	0
$321$	4.65248	0.259676
$322$	0	0
$323$	2.47214	0.137553
$324$	0	0
$325$	−21.7082	−1.20415
$326$	0	0
$327$	−6.18034	−0.341774
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.8541	−0.706525	−0.353263	−	0.935524i	$-0.614928\pi$
$331$	−12.8541	−0.706525	−0.353263	+	0.935524i	$0.614928\pi$
$332$	0	0
$333$	−10.4721	−0.573870
$334$	0	0
$335$	−0.0344419	−0.00188176
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	0	0
$339$	−4.76393	−0.258741
$340$	0	0
$341$	9.34752	0.506197
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.18034	0.117386
$346$	0	0
$347$	18.3607	0.985653	0.492826	−	0.870128i	$-0.335964\pi$
$347$	18.3607	0.985653	0.492826	+	0.870128i	$0.335964\pi$
$348$	0	0
$349$	−4.18034	−0.223768	−0.111884	−	0.993721i	$-0.535689\pi$
$349$	−4.18034	−0.223768	−0.111884	+	0.993721i	$0.535689\pi$
$350$	0	0
$351$	15.5279	0.828816
$352$	0	0
$353$	33.4164	1.77858	0.889288	−	0.457348i	$-0.151201\pi$
$353$	33.4164	1.77858	0.889288	+	0.457348i	$0.151201\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.7984	−1.57270	−0.786349	−	0.617783i	$-0.788031\pi$
$359$	−29.7984	−1.57270	−0.786349	+	0.617783i	$0.788031\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	−5.56231	−0.291945
$364$	0	0
$365$	4.41641	0.231165
$366$	0	0
$367$	−7.67376	−0.400567	−0.200284	−	0.979738i	$-0.564186\pi$
$367$	−7.67376	−0.400567	−0.200284	+	0.979738i	$0.564186\pi$
$368$	0	0
$369$	2.85410	0.148579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−27.0902	−1.40268	−0.701338	−	0.712829i	$-0.747414\pi$
$373$	−27.0902	−1.40268	−0.701338	+	0.712829i	$0.747414\pi$
$374$	0	0
$375$	2.32624	0.120126
$376$	0	0
$377$	8.94427	0.460653
$378$	0	0
$379$	31.3050	1.60803	0.804014	−	0.594611i	$-0.202694\pi$
$379$	31.3050	1.60803	0.804014	+	0.594611i	$0.202694\pi$
$380$	0	0
$381$	−9.94427	−0.509460
$382$	0	0
$383$	30.3607	1.55136	0.775679	−	0.631127i	$-0.217408\pi$
$383$	30.3607	1.55136	0.775679	+	0.631127i	$0.217408\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.85410	−0.145082
$388$	0	0
$389$	−15.3262	−0.777071	−0.388536	−	0.921434i	$-0.627019\pi$
$389$	−15.3262	−0.777071	−0.388536	+	0.921434i	$0.627019\pi$
$390$	0	0
$391$	9.23607	0.467088
$392$	0	0
$393$	−6.47214	−0.326476
$394$	0	0
$395$	−5.41641	−0.272529
$396$	0	0
$397$	36.7426	1.84406	0.922030	−	0.387118i	$-0.126529\pi$
$397$	36.7426	1.84406	0.922030	+	0.387118i	$0.126529\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.9443	0.946032	0.473016	−	0.881054i	$-0.343165\pi$
$401$	18.9443	0.946032	0.473016	+	0.881054i	$0.343165\pi$
$402$	0	0
$403$	−9.34752	−0.465633
$404$	0	0
$405$	2.18034	0.108342
$406$	0	0
$407$	−17.8885	−0.886702
$408$	0	0
$409$	−20.0689	−0.992342	−0.496171	−	0.868225i	$-0.665261\pi$
$409$	−20.0689	−0.992342	−0.496171	+	0.868225i	$0.665261\pi$
$410$	0	0
$411$	3.03444	0.149678
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.58359	0.126824
$416$	0	0
$417$	−7.29180	−0.357081
$418$	0	0
$419$	−39.2148	−1.91577	−0.957884	−	0.287156i	$-0.907290\pi$
$419$	−39.2148	−1.91577	−0.957884	+	0.287156i	$0.907290\pi$
$420$	0	0
$421$	−0.0344419	−0.00167859	−0.000839297	−	1.00000i	$-0.500267\pi$
$421$	−0.0344419	−0.00167859	−0.000839297		1.00000i	$0.500267\pi$
$422$	0	0
$423$	−17.7082	−0.861002
$424$	0	0
$425$	4.85410	0.235459
$426$	0	0
$427$	0	0
$428$	0	0
$429$	12.3607	0.596779
$430$	0	0
$431$	−30.6525	−1.47648	−0.738239	−	0.674539i	$-0.764342\pi$
$431$	−30.6525	−1.47648	−0.738239	+	0.674539i	$0.764342\pi$
$432$	0	0
$433$	17.7082	0.851002	0.425501	−	0.904958i	$-0.360098\pi$
$433$	17.7082	0.851002	0.425501	+	0.904958i	$0.360098\pi$
$434$	0	0
$435$	−0.472136	−0.0226372
$436$	0	0
$437$	22.8328	1.09224
$438$	0	0
$439$	13.5623	0.647294	0.323647	−	0.946178i	$-0.395091\pi$
$439$	13.5623	0.647294	0.323647	+	0.946178i	$0.395091\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.4721	−1.82787	−0.913933	−	0.405865i	$-0.866970\pi$
$443$	−38.4721	−1.82787	−0.913933	+	0.405865i	$0.866970\pi$
$444$	0	0
$445$	4.47214	0.212000
$446$	0	0
$447$	−1.32624	−0.0627289
$448$	0	0
$449$	−9.52786	−0.449648	−0.224824	−	0.974399i	$-0.572181\pi$
$449$	−9.52786	−0.449648	−0.224824	+	0.974399i	$0.572181\pi$
$450$	0	0
$451$	4.87539	0.229573
$452$	0	0
$453$	8.85410	0.416002
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.6738	−0.592854	−0.296427	−	0.955056i	$-0.595795\pi$
$457$	−12.6738	−0.592854	−0.296427	+	0.955056i	$0.595795\pi$
$458$	0	0
$459$	−3.47214	−0.162065
$460$	0	0
$461$	−3.81966	−0.177899	−0.0889497	−	0.996036i	$-0.528351\pi$
$461$	−3.81966	−0.177899	−0.0889497	+	0.996036i	$0.528351\pi$
$462$	0	0
$463$	34.9787	1.62560	0.812799	−	0.582544i	$-0.197943\pi$
$463$	34.9787	1.62560	0.812799	+	0.582544i	$0.197943\pi$
$464$	0	0
$465$	0.493422	0.0228819
$466$	0	0
$467$	0.180340	0.00834513	0.00417257	−	0.999991i	$-0.498672\pi$
$467$	0.180340	0.00834513	0.00417257	+	0.999991i	$0.498672\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.41641	0.157420
$472$	0	0
$473$	−4.87539	−0.224171
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	−21.1803	−0.969781
$478$	0	0
$479$	−29.8541	−1.36407	−0.682034	−	0.731320i	$-0.738905\pi$
$479$	−29.8541	−1.36407	−0.682034	+	0.731320i	$0.738905\pi$
$480$	0	0
$481$	17.8885	0.815647
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.74265	−0.260760
$486$	0	0
$487$	−0.763932	−0.0346171	−0.0173085	−	0.999850i	$-0.505510\pi$
$487$	−0.763932	−0.0346171	−0.0173085	+	0.999850i	$0.505510\pi$
$488$	0	0
$489$	−3.05573	−0.138185
$490$	0	0
$491$	−27.5066	−1.24135	−0.620677	−	0.784066i	$-0.713142\pi$
$491$	−27.5066	−1.24135	−0.620677	+	0.784066i	$0.713142\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	4.47214	0.201008
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.1803	−0.545267	−0.272633	−	0.962118i	$-0.587895\pi$
$499$	−12.1803	−0.545267	−0.272633	+	0.962118i	$0.587895\pi$
$500$	0	0
$501$	−5.14590	−0.229902
$502$	0	0
$503$	20.7426	0.924869	0.462434	−	0.886653i	$-0.346976\pi$
$503$	20.7426	0.924869	0.462434	+	0.886653i	$0.346976\pi$
$504$	0	0
$505$	−0.360680	−0.0160500
$506$	0	0
$507$	−4.32624	−0.192135
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.58359	−0.378975
$514$	0	0
$515$	0.111456	0.00491135
$516$	0	0
$517$	−30.2492	−1.33036
$518$	0	0
$519$	−9.97871	−0.438017
$520$	0	0
$521$	12.1459	0.532121	0.266061	−	0.963956i	$-0.414278\pi$
$521$	12.1459	0.532121	0.266061	+	0.963956i	$0.414278\pi$
$522$	0	0
$523$	−4.29180	−0.187667	−0.0938336	−	0.995588i	$-0.529912\pi$
$523$	−4.29180	−0.187667	−0.0938336	+	0.995588i	$0.529912\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.09017	0.0910492
$528$	0	0
$529$	62.3050	2.70891
$530$	0	0
$531$	−10.4721	−0.454452
$532$	0	0
$533$	−4.87539	−0.211177
$534$	0	0
$535$	−2.87539	−0.124314
$536$	0	0
$537$	−9.43769	−0.407267
$538$	0	0
$539$	0	0
$540$	0	0
$541$	43.7082	1.87916	0.939581	−	0.342326i	$-0.111215\pi$
$541$	43.7082	1.87916	0.939581	+	0.342326i	$0.111215\pi$
$542$	0	0
$543$	9.81966	0.421402
$544$	0	0
$545$	3.81966	0.163616
$546$	0	0
$547$	−14.8328	−0.634205	−0.317103	−	0.948391i	$-0.602710\pi$
$547$	−14.8328	−0.634205	−0.317103	+	0.948391i	$0.602710\pi$
$548$	0	0
$549$	14.0902	0.601354
$550$	0	0
$551$	−4.94427	−0.210633
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.944272	−0.0400821
$556$	0	0
$557$	−9.41641	−0.398986	−0.199493	−	0.979899i	$-0.563930\pi$
$557$	−9.41641	−0.398986	−0.199493	+	0.979899i	$0.563930\pi$
$558$	0	0
$559$	4.87539	0.206207
$560$	0	0
$561$	−2.76393	−0.116693
$562$	0	0
$563$	38.3607	1.61671	0.808355	−	0.588695i	$-0.200358\pi$
$563$	38.3607	1.61671	0.808355	+	0.588695i	$0.200358\pi$
$564$	0	0
$565$	2.94427	0.123866
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.9230	1.58981	0.794907	−	0.606731i	$-0.207520\pi$
$569$	37.9230	1.58981	0.794907	+	0.606731i	$0.207520\pi$
$570$	0	0
$571$	−1.52786	−0.0639391	−0.0319696	−	0.999489i	$-0.510178\pi$
$571$	−1.52786	−0.0639391	−0.0319696	+	0.999489i	$0.510178\pi$
$572$	0	0
$573$	−4.52786	−0.189154
$574$	0	0
$575$	44.8328	1.86966
$576$	0	0
$577$	20.9443	0.871921	0.435961	−	0.899966i	$-0.356409\pi$
$577$	20.9443	0.871921	0.435961	+	0.899966i	$0.356409\pi$
$578$	0	0
$579$	7.70820	0.320342
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.1803	−1.49844
$584$	0	0
$585$	−4.47214	−0.184900
$586$	0	0
$587$	1.70820	0.0705051	0.0352526	−	0.999378i	$-0.488776\pi$
$587$	1.70820	0.0705051	0.0352526	+	0.999378i	$0.488776\pi$
$588$	0	0
$589$	5.16718	0.212910
$590$	0	0
$591$	−16.3607	−0.672988
$592$	0	0
$593$	31.4164	1.29012	0.645059	−	0.764133i	$-0.276833\pi$
$593$	31.4164	1.29012	0.645059	+	0.764133i	$0.276833\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.7984	0.482875
$598$	0	0
$599$	−10.9787	−0.448578	−0.224289	−	0.974523i	$-0.572006\pi$
$599$	−10.9787	−0.448578	−0.224289	+	0.974523i	$0.572006\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	0.236068	0.00961343
$604$	0	0
$605$	3.43769	0.139762
$606$	0	0
$607$	−13.3262	−0.540895	−0.270448	−	0.962735i	$-0.587172\pi$
$607$	−13.3262	−0.540895	−0.270448	+	0.962735i	$0.587172\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.2492	1.22375
$612$	0	0
$613$	−2.96556	−0.119778	−0.0598889	−	0.998205i	$-0.519075\pi$
$613$	−2.96556	−0.119778	−0.0598889	+	0.998205i	$0.519075\pi$
$614$	0	0
$615$	0.257354	0.0103775
$616$	0	0
$617$	−40.8328	−1.64387	−0.821934	−	0.569583i	$-0.807105\pi$
$617$	−40.8328	−1.64387	−0.821934	+	0.569583i	$0.807105\pi$
$618$	0	0
$619$	−21.5279	−0.865278	−0.432639	−	0.901567i	$-0.642418\pi$
$619$	−21.5279	−0.865278	−0.432639	+	0.901567i	$0.642418\pi$
$620$	0	0
$621$	−32.0689	−1.28688
$622$	0	0
$623$	0	0
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	−6.83282	−0.272876
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	3.90983	0.155648	0.0778239	−	0.996967i	$-0.475203\pi$
$631$	3.90983	0.155648	0.0778239	+	0.996967i	$0.475203\pi$
$632$	0	0
$633$	9.34752	0.371531
$634$	0	0
$635$	6.14590	0.243893
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.2361	0.601789	0.300894	−	0.953658i	$-0.402715\pi$
$641$	15.2361	0.601789	0.300894	+	0.953658i	$0.402715\pi$
$642$	0	0
$643$	1.56231	0.0616113	0.0308057	−	0.999525i	$-0.490193\pi$
$643$	1.56231	0.0616113	0.0308057	+	0.999525i	$0.490193\pi$
$644$	0	0
$645$	−0.257354	−0.0101333
$646$	0	0
$647$	49.4164	1.94276	0.971380	−	0.237532i	$-0.0763384\pi$
$647$	49.4164	1.94276	0.971380	+	0.237532i	$0.0763384\pi$
$648$	0	0
$649$	−17.8885	−0.702187
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.8885	0.621767	0.310883	−	0.950448i	$-0.399375\pi$
$653$	15.8885	0.621767	0.310883	+	0.950448i	$0.399375\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−30.2705	−1.18097
$658$	0	0
$659$	15.3262	0.597025	0.298513	−	0.954406i	$-0.403509\pi$
$659$	15.3262	0.597025	0.298513	+	0.954406i	$0.403509\pi$
$660$	0	0
$661$	31.3050	1.21762	0.608811	−	0.793315i	$-0.291647\pi$
$661$	31.3050	1.21762	0.608811	+	0.793315i	$0.291647\pi$
$662$	0	0
$663$	2.76393	0.107342
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.4721	−0.715244
$668$	0	0
$669$	3.05573	0.118141
$670$	0	0
$671$	24.0689	0.929169
$672$	0	0
$673$	18.2918	0.705097	0.352548	−	0.935794i	$-0.385315\pi$
$673$	18.2918	0.705097	0.352548	+	0.935794i	$0.385315\pi$
$674$	0	0
$675$	−16.8541	−0.648715
$676$	0	0
$677$	46.3607	1.78179	0.890893	−	0.454214i	$-0.150080\pi$
$677$	46.3607	1.78179	0.890893	+	0.454214i	$0.150080\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.23607	−0.162326
$682$	0	0
$683$	0.944272	0.0361316	0.0180658	−	0.999837i	$-0.494249\pi$
$683$	0.944272	0.0361316	0.0180658	+	0.999837i	$0.494249\pi$
$684$	0	0
$685$	−1.87539	−0.0716549
$686$	0	0
$687$	17.5967	0.671358
$688$	0	0
$689$	36.1803	1.37836
$690$	0	0
$691$	1.85410	0.0705334	0.0352667	−	0.999378i	$-0.488772\pi$
$691$	1.85410	0.0705334	0.0352667	+	0.999378i	$0.488772\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.50658	0.170944
$696$	0	0
$697$	1.09017	0.0412931
$698$	0	0
$699$	−3.70820	−0.140257
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	−9.88854	−0.372953
$704$	0	0
$705$	−1.59675	−0.0601370
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.58359	0.247252	0.123626	−	0.992329i	$-0.460548\pi$
$709$	6.58359	0.247252	0.123626	+	0.992329i	$0.460548\pi$
$710$	0	0
$711$	37.1246	1.39228
$712$	0	0
$713$	19.3050	0.722976
$714$	0	0
$715$	−7.63932	−0.285694
$716$	0	0
$717$	3.90983	0.146015
$718$	0	0
$719$	4.32624	0.161341	0.0806707	−	0.996741i	$-0.474294\pi$
$719$	4.32624	0.161341	0.0806707	+	0.996741i	$0.474294\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7.29180	−0.271185
$724$	0	0
$725$	−9.70820	−0.360554
$726$	0	0
$727$	36.1803	1.34185	0.670927	−	0.741523i	$-0.265896\pi$
$727$	36.1803	1.34185	0.670927	+	0.741523i	$0.265896\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	−1.09017	−0.0403214
$732$	0	0
$733$	−51.5967	−1.90577	−0.952885	−	0.303333i	$-0.901901\pi$
$733$	−51.5967	−1.90577	−0.952885	+	0.303333i	$0.901901\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.403252	0.0148540
$738$	0	0
$739$	40.7984	1.50079	0.750396	−	0.660988i	$-0.229863\pi$
$739$	40.7984	1.50079	0.750396	+	0.660988i	$0.229863\pi$
$740$	0	0
$741$	6.83282	0.251010
$742$	0	0
$743$	11.2361	0.412211	0.206106	−	0.978530i	$-0.433921\pi$
$743$	11.2361	0.412211	0.206106	+	0.978530i	$0.433921\pi$
$744$	0	0
$745$	0.819660	0.0300300
$746$	0	0
$747$	−17.7082	−0.647909
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	−13.5279	−0.492983
$754$	0	0
$755$	−5.47214	−0.199151
$756$	0	0
$757$	20.3262	0.738770	0.369385	−	0.929277i	$-0.379568\pi$
$757$	20.3262	0.738770	0.369385	+	0.929277i	$0.379568\pi$
$758$	0	0
$759$	−25.5279	−0.926603
$760$	0	0
$761$	−39.9574	−1.44846	−0.724228	−	0.689561i	$-0.757804\pi$
$761$	−39.9574	−1.44846	−0.724228	+	0.689561i	$0.757804\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	17.8885	0.645918
$768$	0	0
$769$	26.2492	0.946571	0.473286	−	0.880909i	$-0.343068\pi$
$769$	26.2492	0.946571	0.473286	+	0.880909i	$0.343068\pi$
$770$	0	0
$771$	−19.5279	−0.703279
$772$	0	0
$773$	−50.1803	−1.80486	−0.902431	−	0.430835i	$-0.858219\pi$
$773$	−50.1803	−1.80486	−0.902431	+	0.430835i	$0.858219\pi$
$774$	0	0
$775$	10.1459	0.364451
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.69505	0.0965601
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.94427	0.248168
$784$	0	0
$785$	−2.11146	−0.0753611
$786$	0	0
$787$	13.5279	0.482216	0.241108	−	0.970498i	$-0.422489\pi$
$787$	13.5279	0.482216	0.241108	+	0.970498i	$0.422489\pi$
$788$	0	0
$789$	−16.5836	−0.590392
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.0689	−0.854711
$794$	0	0
$795$	−1.90983	−0.0677347
$796$	0	0
$797$	−31.1246	−1.10249	−0.551245	−	0.834343i	$-0.685847\pi$
$797$	−31.1246	−1.10249	−0.551245	+	0.834343i	$0.685847\pi$
$798$	0	0
$799$	−6.76393	−0.239291
$800$	0	0
$801$	−30.6525	−1.08305
$802$	0	0
$803$	−51.7082	−1.82474
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.70820	0.130535
$808$	0	0
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	5.20163	0.182654	0.0913269	−	0.995821i	$-0.470889\pi$
$811$	5.20163	0.182654	0.0913269	+	0.995821i	$0.470889\pi$
$812$	0	0
$813$	−3.23607	−0.113494
$814$	0	0
$815$	1.88854	0.0661528
$816$	0	0
$817$	−2.69505	−0.0942878
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.583592	−0.0203675	−0.0101838	−	0.999948i	$-0.503242\pi$
$821$	−0.583592	−0.0203675	−0.0101838	+	0.999948i	$0.503242\pi$
$822$	0	0
$823$	−51.5967	−1.79855	−0.899275	−	0.437384i	$-0.855905\pi$
$823$	−51.5967	−1.79855	−0.899275	+	0.437384i	$0.855905\pi$
$824$	0	0
$825$	−13.4164	−0.467099
$826$	0	0
$827$	−11.2361	−0.390716	−0.195358	−	0.980732i	$-0.562587\pi$
$827$	−11.2361	−0.390716	−0.195358	+	0.980732i	$0.562587\pi$
$828$	0	0
$829$	25.7771	0.895275	0.447638	−	0.894215i	$-0.352265\pi$
$829$	25.7771	0.895275	0.447638	+	0.894215i	$0.352265\pi$
$830$	0	0
$831$	−7.81966	−0.271261
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.18034	0.110060
$836$	0	0
$837$	−7.25735	−0.250851
$838$	0	0
$839$	9.52786	0.328938	0.164469	−	0.986382i	$-0.447409\pi$
$839$	9.52786	0.328938	0.164469	+	0.986382i	$0.447409\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−3.00000	−0.103325
$844$	0	0
$845$	2.67376	0.0919802
$846$	0	0
$847$	0	0
$848$	0	0
$849$	17.7639	0.609657
$850$	0	0
$851$	−36.9443	−1.26643
$852$	0	0
$853$	−33.7771	−1.15651	−0.578253	−	0.815858i	$-0.696265\pi$
$853$	−33.7771	−1.15651	−0.578253	+	0.815858i	$0.696265\pi$
$854$	0	0
$855$	2.47214	0.0845453
$856$	0	0
$857$	−30.2705	−1.03402	−0.517010	−	0.855979i	$-0.672955\pi$
$857$	−30.2705	−1.03402	−0.517010	+	0.855979i	$0.672955\pi$
$858$	0	0
$859$	35.7082	1.21835	0.609174	−	0.793037i	$-0.291501\pi$
$859$	35.7082	1.21835	0.609174	+	0.793037i	$0.291501\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.2705	−0.553855	−0.276927	−	0.960891i	$-0.589316\pi$
$863$	−16.2705	−0.553855	−0.276927	+	0.960891i	$0.589316\pi$
$864$	0	0
$865$	6.16718	0.209691
$866$	0	0
$867$	−0.618034	−0.0209895
$868$	0	0
$869$	63.4164	2.15125
$870$	0	0
$871$	−0.403252	−0.0136637
$872$	0	0
$873$	39.3607	1.33216
$874$	0	0
$875$	0	0
$876$	0	0
$877$	55.4164	1.87128	0.935640	−	0.352957i	$-0.114824\pi$
$877$	55.4164	1.87128	0.935640	+	0.352957i	$0.114824\pi$
$878$	0	0
$879$	−5.05573	−0.170525
$880$	0	0
$881$	−18.4934	−0.623059	−0.311530	−	0.950236i	$-0.600841\pi$
$881$	−18.4934	−0.623059	−0.311530	+	0.950236i	$0.600841\pi$
$882$	0	0
$883$	−36.6312	−1.23274	−0.616369	−	0.787458i	$-0.711397\pi$
$883$	−36.6312	−1.23274	−0.616369	+	0.787458i	$0.711397\pi$
$884$	0	0
$885$	−0.944272	−0.0317414
$886$	0	0
$887$	−31.9787	−1.07374	−0.536870	−	0.843665i	$-0.680393\pi$
$887$	−31.9787	−1.07374	−0.536870	+	0.843665i	$0.680393\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−25.5279	−0.855216
$892$	0	0
$893$	−16.7214	−0.559559
$894$	0	0
$895$	5.83282	0.194970
$896$	0	0
$897$	25.5279	0.852351
$898$	0	0
$899$	−4.18034	−0.139422
$900$	0	0
$901$	−8.09017	−0.269523
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.06888	−0.201737
$906$	0	0
$907$	23.3050	0.773828	0.386914	−	0.922116i	$-0.373541\pi$
$907$	23.3050	0.773828	0.386914	+	0.922116i	$0.373541\pi$
$908$	0	0
$909$	2.47214	0.0819956
$910$	0	0
$911$	−12.4721	−0.413220	−0.206610	−	0.978423i	$-0.566243\pi$
$911$	−12.4721	−0.413220	−0.206610	+	0.978423i	$0.566243\pi$
$912$	0	0
$913$	−30.2492	−1.00110
$914$	0	0
$915$	1.27051	0.0420018
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.9230	−0.657199	−0.328599	−	0.944469i	$-0.606577\pi$
$919$	−19.9230	−0.657199	−0.328599	+	0.944469i	$0.606577\pi$
$920$	0	0
$921$	−11.8885	−0.391741
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−19.4164	−0.638408
$926$	0	0
$927$	−0.763932	−0.0250908
$928$	0	0
$929$	−49.1033	−1.61103	−0.805514	−	0.592577i	$-0.798111\pi$
$929$	−49.1033	−1.61103	−0.805514	+	0.592577i	$0.798111\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.40325	−0.111417
$934$	0	0
$935$	1.70820	0.0558642
$936$	0	0
$937$	−48.0000	−1.56809	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−48.0000	−1.56809	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	−18.7082	−0.610519
$940$	0	0
$941$	−21.7426	−0.708790	−0.354395	−	0.935096i	$-0.615313\pi$
$941$	−21.7426	−0.708790	−0.354395	+	0.935096i	$0.615313\pi$
$942$	0	0
$943$	10.0689	0.327888
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.16718	0.102920	0.0514598	−	0.998675i	$-0.483613\pi$
$947$	3.16718	0.102920	0.0514598	+	0.998675i	$0.483613\pi$
$948$	0	0
$949$	51.7082	1.67852
$950$	0	0
$951$	5.63932	0.182867
$952$	0	0
$953$	2.56231	0.0830012	0.0415006	−	0.999138i	$-0.486786\pi$
$953$	2.56231	0.0830012	0.0415006	+	0.999138i	$0.486786\pi$
$954$	0	0
$955$	2.79837	0.0905533
$956$	0	0
$957$	5.52786	0.178690
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.6312	−0.859071
$962$	0	0
$963$	19.7082	0.635088
$964$	0	0
$965$	−4.76393	−0.153356
$966$	0	0
$967$	−16.4377	−0.528601	−0.264300	−	0.964440i	$-0.585141\pi$
$967$	−16.4377	−0.528601	−0.264300	+	0.964440i	$0.585141\pi$
$968$	0	0
$969$	−1.52786	−0.0490821
$970$	0	0
$971$	9.41641	0.302187	0.151093	−	0.988519i	$-0.451721\pi$
$971$	9.41641	0.302187	0.151093	+	0.988519i	$0.451721\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	13.4164	0.429669
$976$	0	0
$977$	−35.5066	−1.13596	−0.567978	−	0.823044i	$-0.692274\pi$
$977$	−35.5066	−1.13596	−0.567978	+	0.823044i	$0.692274\pi$
$978$	0	0
$979$	−52.3607	−1.67345
$980$	0	0
$981$	−26.1803	−0.835874
$982$	0	0
$983$	−9.50658	−0.303213	−0.151606	−	0.988441i	$-0.548445\pi$
$983$	−9.50658	−0.303213	−0.151606	+	0.988441i	$0.548445\pi$
$984$	0	0
$985$	10.1115	0.322178
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.0689	−0.320172
$990$	0	0
$991$	45.4164	1.44270	0.721350	−	0.692571i	$-0.243522\pi$
$991$	45.4164	1.44270	0.721350	+	0.692571i	$0.243522\pi$
$992$	0	0
$993$	7.94427	0.252104
$994$	0	0
$995$	−7.29180	−0.231165
$996$	0	0
$997$	55.5623	1.75968	0.879838	−	0.475274i	$-0.157651\pi$
$997$	55.5623	1.75968	0.879838	+	0.475274i	$0.157651\pi$
$998$	0	0
$999$	13.8885	0.439414

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.h.1.1		2
7.6	odd	2		952.2.a.a.1.2	✓	2
21.20	even	2		8568.2.a.v.1.1		2
28.27	even	2		1904.2.a.i.1.1		2
56.13	odd	2		7616.2.a.w.1.1		2
56.27	even	2		7616.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.a.1.2	✓	2	7.6	odd	2
1904.2.a.i.1.1		2	28.27	even	2
6664.2.a.h.1.1		2	1.1	even	1	trivial
7616.2.a.r.1.2		2	56.27	even	2
7616.2.a.w.1.1		2	56.13	odd	2
8568.2.a.v.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} +0.381966 q^{5} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} +0.381966 q^{5} -2.61803 q^{9} -4.47214 q^{11} +4.47214 q^{13} -0.236068 q^{15} -1.00000 q^{17} -2.47214 q^{19} -9.23607 q^{23} -4.85410 q^{25} +3.47214 q^{27} +2.00000 q^{29} -2.09017 q^{31} +2.76393 q^{33} +4.00000 q^{37} -2.76393 q^{39} -1.09017 q^{41} +1.09017 q^{43} -1.00000 q^{45} +6.76393 q^{47} +0.618034 q^{51} +8.09017 q^{53} -1.70820 q^{55} +1.52786 q^{57} +4.00000 q^{59} -5.38197 q^{61} +1.70820 q^{65} -0.0901699 q^{67} +5.70820 q^{69} +11.5623 q^{73} +3.00000 q^{75} -14.1803 q^{79} +5.70820 q^{81} +6.76393 q^{83} -0.381966 q^{85} -1.23607 q^{87} +11.7082 q^{89} +1.29180 q^{93} -0.944272 q^{95} -15.0344 q^{97} +11.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 3 q^{5} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 3 * q^5 - 3 * q^9	\( 2 q + q^{3} + 3 q^{5} - 3 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{19} - 14 q^{23} - 3 q^{25} - 2 q^{27} + 4 q^{29} + 7 q^{31} + 10 q^{33} + 8 q^{37} - 10 q^{39} + 9 q^{41} - 9 q^{43} - 2 q^{45} + 18 q^{47} - q^{51} + 5 q^{53} + 10 q^{55} + 12 q^{57} + 8 q^{59} - 13 q^{61} - 10 q^{65} + 11 q^{67} - 2 q^{69} + 3 q^{73} + 6 q^{75} - 6 q^{79} - 2 q^{81} + 18 q^{83} - 3 q^{85} + 2 q^{87} + 10 q^{89} + 16 q^{93} + 16 q^{95} - q^{97} + 10 q^{99}+O(q^{100}) \) 2 * q + q^3 + 3 * q^5 - 3 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^19 - 14 * q^23 - 3 * q^25 - 2 * q^27 + 4 * q^29 + 7 * q^31 + 10 * q^33 + 8 * q^37 - 10 * q^39 + 9 * q^41 - 9 * q^43 - 2 * q^45 + 18 * q^47 - q^51 + 5 * q^53 + 10 * q^55 + 12 * q^57 + 8 * q^59 - 13 * q^61 - 10 * q^65 + 11 * q^67 - 2 * q^69 + 3 * q^73 + 6 * q^75 - 6 * q^79 - 2 * q^81 + 18 * q^83 - 3 * q^85 + 2 * q^87 + 10 * q^89 + 16 * q^93 + 16 * q^95 - q^97 + 10 * q^99

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