LMFDB - Embedded newform 6675.2.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6675,2,Mod(1,6675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6675 = 3 \cdot 5^{2} \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6675.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:	$53.3001433492$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.970031$ of defining polynomial
Character	$\chi$	$=$	6675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.970031 q^{2} +1.00000 q^{3} -1.05904 q^{4} -0.970031 q^{6} +4.84165 q^{7} +2.96736 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.970031 q^{2} +1.00000 q^{3} -1.05904 q^{4} -0.970031 q^{6} +4.84165 q^{7} +2.96736 q^{8} +1.00000 q^{9} +1.26097 q^{11} -1.05904 q^{12} +0.957447 q^{13} -4.69655 q^{14} -0.760356 q^{16} -2.50226 q^{17} -0.970031 q^{18} +7.26642 q^{19} +4.84165 q^{21} -1.22318 q^{22} +2.94416 q^{23} +2.96736 q^{24} -0.928753 q^{26} +1.00000 q^{27} -5.12750 q^{28} +2.51044 q^{29} -1.30054 q^{31} -5.19716 q^{32} +1.26097 q^{33} +2.42727 q^{34} -1.05904 q^{36} -10.6521 q^{37} -7.04865 q^{38} +0.957447 q^{39} +10.7945 q^{41} -4.69655 q^{42} -1.68646 q^{43} -1.33541 q^{44} -2.85593 q^{46} +7.95583 q^{47} -0.760356 q^{48} +16.4416 q^{49} -2.50226 q^{51} -1.01397 q^{52} +10.9614 q^{53} -0.970031 q^{54} +14.3669 q^{56} +7.26642 q^{57} -2.43520 q^{58} +5.13128 q^{59} -0.773449 q^{61} +1.26157 q^{62} +4.84165 q^{63} +6.56212 q^{64} -1.22318 q^{66} -8.58486 q^{67} +2.64999 q^{68} +2.94416 q^{69} +3.16973 q^{71} +2.96736 q^{72} +0.179310 q^{73} +10.3329 q^{74} -7.69543 q^{76} +6.10515 q^{77} -0.928753 q^{78} -11.8490 q^{79} +1.00000 q^{81} -10.4710 q^{82} +0.924640 q^{83} -5.12750 q^{84} +1.63592 q^{86} +2.51044 q^{87} +3.74174 q^{88} -1.00000 q^{89} +4.63562 q^{91} -3.11798 q^{92} -1.30054 q^{93} -7.71741 q^{94} -5.19716 q^{96} -11.3807 q^{97} -15.9488 q^{98} +1.26097 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 10 q^{3} + 13 q^{4} - q^{6} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - q^2 + 10 * q^3 + 13 * q^4 - q^6 + q^7 + 10 * q^9	$ 10 q - q^{2} + 10 q^{3} + 13 q^{4} - q^{6} + q^{7} + 10 q^{9} + 10 q^{11} + 13 q^{12} - 5 q^{13} + 13 q^{14} + 19 q^{16} - 9 q^{17} - q^{18} - 6 q^{19} + q^{21} + 3 q^{23} + 14 q^{26} + 10 q^{27} + 18 q^{28} + 38 q^{29} + 2 q^{31} - 16 q^{32} + 10 q^{33} - 8 q^{34} + 13 q^{36} - 9 q^{37} - 20 q^{38} - 5 q^{39} + 36 q^{41} + 13 q^{42} + 7 q^{43} + 16 q^{44} + 2 q^{46} + 23 q^{47} + 19 q^{48} + 25 q^{49} - 9 q^{51} - 13 q^{52} - 27 q^{53} - q^{54} + 41 q^{56} - 6 q^{57} + 32 q^{58} + 20 q^{59} + 30 q^{61} + 2 q^{62} + q^{63} - 2 q^{64} + 5 q^{67} + 10 q^{68} + 3 q^{69} + 24 q^{71} + 19 q^{73} + 42 q^{74} - 30 q^{76} - 18 q^{77} + 14 q^{78} + 12 q^{79} + 10 q^{81} - 29 q^{82} - 3 q^{83} + 18 q^{84} + 38 q^{86} + 38 q^{87} + 16 q^{88} - 10 q^{89} - 6 q^{91} - 32 q^{92} + 2 q^{93} + 17 q^{94} - 16 q^{96} - 3 q^{97} + 37 q^{98} + 10 q^{99}+O(q^{100}) $ 10 * q - q^2 + 10 * q^3 + 13 * q^4 - q^6 + q^7 + 10 * q^9 + 10 * q^11 + 13 * q^12 - 5 * q^13 + 13 * q^14 + 19 * q^16 - 9 * q^17 - q^18 - 6 * q^19 + q^21 + 3 * q^23 + 14 * q^26 + 10 * q^27 + 18 * q^28 + 38 * q^29 + 2 * q^31 - 16 * q^32 + 10 * q^33 - 8 * q^34 + 13 * q^36 - 9 * q^37 - 20 * q^38 - 5 * q^39 + 36 * q^41 + 13 * q^42 + 7 * q^43 + 16 * q^44 + 2 * q^46 + 23 * q^47 + 19 * q^48 + 25 * q^49 - 9 * q^51 - 13 * q^52 - 27 * q^53 - q^54 + 41 * q^56 - 6 * q^57 + 32 * q^58 + 20 * q^59 + 30 * q^61 + 2 * q^62 + q^63 - 2 * q^64 + 5 * q^67 + 10 * q^68 + 3 * q^69 + 24 * q^71 + 19 * q^73 + 42 * q^74 - 30 * q^76 - 18 * q^77 + 14 * q^78 + 12 * q^79 + 10 * q^81 - 29 * q^82 - 3 * q^83 + 18 * q^84 + 38 * q^86 + 38 * q^87 + 16 * q^88 - 10 * q^89 - 6 * q^91 - 32 * q^92 + 2 * q^93 + 17 * q^94 - 16 * q^96 - 3 * q^97 + 37 * q^98 + 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 $

$2$	−0.970031	−0.685916	−0.342958	−	0.939351i	$-0.611429\pi$
$2$	−0.970031	−0.685916	−0.342958	+	0.939351i	$0.611429\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.05904	−0.529520
$5$	0	0
$5$	0	0
$6$	−0.970031	−0.396014
$7$	4.84165	1.82997	0.914985	−	0.403487i	$-0.132202\pi$
$7$	4.84165	1.82997	0.914985	+	0.403487i	$0.132202\pi$
$8$	2.96736	1.04912
$9$	1.00000	0.333333
$10$	0	0
$11$	1.26097	0.380195	0.190098	−	0.981765i	$-0.439120\pi$
$11$	1.26097	0.380195	0.190098	+	0.981765i	$0.439120\pi$
$12$	−1.05904	−0.305718
$13$	0.957447	0.265548	0.132774	−	0.991146i	$-0.457612\pi$
$13$	0.957447	0.265548	0.132774	+	0.991146i	$0.457612\pi$
$14$	−4.69655	−1.25521
$15$	0	0
$16$	−0.760356	−0.190089
$17$	−2.50226	−0.606887	−0.303444	−	0.952849i	$-0.598136\pi$
$17$	−2.50226	−0.606887	−0.303444	+	0.952849i	$0.598136\pi$
$18$	−0.970031	−0.228639
$19$	7.26642	1.66703	0.833516	−	0.552496i	$-0.186325\pi$
$19$	7.26642	1.66703	0.833516	+	0.552496i	$0.186325\pi$
$20$	0	0
$21$	4.84165	1.05653
$22$	−1.22318	−0.260782
$23$	2.94416	0.613900	0.306950	−	0.951726i	$-0.400692\pi$
$23$	2.94416	0.613900	0.306950	+	0.951726i	$0.400692\pi$
$24$	2.96736	0.605711
$25$	0	0
$26$	−0.928753	−0.182144
$27$	1.00000	0.192450
$28$	−5.12750	−0.969006
$29$	2.51044	0.466176	0.233088	−	0.972456i	$-0.425117\pi$
$29$	2.51044	0.466176	0.233088	+	0.972456i	$0.425117\pi$
$30$	0	0
$31$	−1.30054	−0.233584	−0.116792	−	0.993156i	$-0.537261\pi$
$31$	−1.30054	−0.233584	−0.116792	+	0.993156i	$0.537261\pi$
$32$	−5.19716	−0.918736
$33$	1.26097	0.219506
$34$	2.42727	0.416274
$35$	0	0
$36$	−1.05904	−0.176507
$37$	−10.6521	−1.75120	−0.875598	−	0.483040i	$-0.839532\pi$
$37$	−10.6521	−1.75120	−0.875598	+	0.483040i	$0.839532\pi$
$38$	−7.04865	−1.14344
$39$	0.957447	0.153314
$40$	0	0
$41$	10.7945	1.68582	0.842909	−	0.538057i	$-0.180841\pi$
$41$	10.7945	1.68582	0.842909	+	0.538057i	$0.180841\pi$
$42$	−4.69655	−0.724693
$43$	−1.68646	−0.257183	−0.128591	−	0.991698i	$-0.541046\pi$
$43$	−1.68646	−0.257183	−0.128591	+	0.991698i	$0.541046\pi$
$44$	−1.33541	−0.201321
$45$	0	0
$46$	−2.85593	−0.421084
$47$	7.95583	1.16048	0.580239	−	0.814446i	$-0.302959\pi$
$47$	7.95583	1.16048	0.580239	+	0.814446i	$0.302959\pi$
$48$	−0.760356	−0.109748
$49$	16.4416	2.34879
$50$	0	0
$51$	−2.50226	−0.350387
$52$	−1.01397	−0.140613
$53$	10.9614	1.50566	0.752830	−	0.658214i	$-0.228688\pi$
$53$	10.9614	1.50566	0.752830	+	0.658214i	$0.228688\pi$
$54$	−0.970031	−0.132005
$55$	0	0
$56$	14.3669	1.91986
$57$	7.26642	0.962461
$58$	−2.43520	−0.319758
$59$	5.13128	0.668035	0.334018	−	0.942567i	$-0.391596\pi$
$59$	5.13128	0.668035	0.334018	+	0.942567i	$0.391596\pi$
$60$	0	0
$61$	−0.773449	−0.0990300	−0.0495150	−	0.998773i	$-0.515768\pi$
$61$	−0.773449	−0.0990300	−0.0495150	+	0.998773i	$0.515768\pi$
$62$	1.26157	0.160219
$63$	4.84165	0.609990
$64$	6.56212	0.820265
$65$	0	0
$66$	−1.22318	−0.150563
$67$	−8.58486	−1.04881	−0.524404	−	0.851470i	$-0.675712\pi$
$67$	−8.58486	−1.04881	−0.524404	+	0.851470i	$0.675712\pi$
$68$	2.64999	0.321359
$69$	2.94416	0.354435
$70$	0	0
$71$	3.16973	0.376178	0.188089	−	0.982152i	$-0.439771\pi$
$71$	3.16973	0.376178	0.188089	+	0.982152i	$0.439771\pi$
$72$	2.96736	0.349707
$73$	0.179310	0.0209867	0.0104933	−	0.999945i	$-0.496660\pi$
$73$	0.179310	0.0209867	0.0104933	+	0.999945i	$0.496660\pi$
$74$	10.3329	1.20117
$75$	0	0
$76$	−7.69543	−0.882726
$77$	6.10515	0.695746
$78$	−0.928753	−0.105161
$79$	−11.8490	−1.33311	−0.666557	−	0.745454i	$-0.732233\pi$
$79$	−11.8490	−1.33311	−0.666557	+	0.745454i	$0.732233\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−10.4710	−1.15633
$83$	0.924640	0.101492	0.0507462	−	0.998712i	$-0.483840\pi$
$83$	0.924640	0.101492	0.0507462	+	0.998712i	$0.483840\pi$
$84$	−5.12750	−0.559456
$85$	0	0
$86$	1.63592	0.176406
$87$	2.51044	0.269147
$88$	3.74174	0.398871
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	4.63562	0.485945
$92$	−3.11798	−0.325072
$93$	−1.30054	−0.134860
$94$	−7.71741	−0.795990
$95$	0	0
$96$	−5.19716	−0.530433
$97$	−11.3807	−1.15553	−0.577767	−	0.816201i	$-0.696076\pi$
$97$	−11.3807	−1.15553	−0.577767	+	0.816201i	$0.696076\pi$
$98$	−15.9488	−1.61107
$99$	1.26097	0.126732
$100$	0	0
$101$	−0.540046	−0.0537366	−0.0268683	−	0.999639i	$-0.508553\pi$
$101$	−0.540046	−0.0537366	−0.0268683	+	0.999639i	$0.508553\pi$
$102$	2.42727	0.240336
$103$	5.16142	0.508570	0.254285	−	0.967129i	$-0.418160\pi$
$103$	5.16142	0.508570	0.254285	+	0.967129i	$0.418160\pi$
$104$	2.84109	0.278592
$105$	0	0
$106$	−10.6329	−1.03276
$107$	−9.22424	−0.891741	−0.445871	−	0.895097i	$-0.647106\pi$
$107$	−9.22424	−0.891741	−0.445871	+	0.895097i	$0.647106\pi$
$108$	−1.05904	−0.101906
$109$	0.907932	0.0869641	0.0434821	−	0.999054i	$-0.486155\pi$
$109$	0.907932	0.0869641	0.0434821	+	0.999054i	$0.486155\pi$
$110$	0	0
$111$	−10.6521	−1.01105
$112$	−3.68138	−0.347858
$113$	−6.56595	−0.617673	−0.308836	−	0.951115i	$-0.599940\pi$
$113$	−6.56595	−0.617673	−0.308836	+	0.951115i	$0.599940\pi$
$114$	−7.04865	−0.660167
$115$	0	0
$116$	−2.65865	−0.246850
$117$	0.957447	0.0885160
$118$	−4.97750	−0.458216
$119$	−12.1151	−1.11059
$120$	0	0
$121$	−9.40997	−0.855452
$122$	0.750270	0.0679262
$123$	10.7945	0.973307
$124$	1.37733	0.123688
$125$	0	0
$126$	−4.69655	−0.418402
$127$	5.25118	0.465966	0.232983	−	0.972481i	$-0.425151\pi$
$127$	5.25118	0.465966	0.232983	+	0.972481i	$0.425151\pi$
$128$	4.02886	0.356104
$129$	−1.68646	−0.148485
$130$	0	0
$131$	10.8386	0.946972	0.473486	−	0.880801i	$-0.342996\pi$
$131$	10.8386	0.946972	0.473486	+	0.880801i	$0.342996\pi$
$132$	−1.33541	−0.116233
$133$	35.1814	3.05062
$134$	8.32758	0.719394
$135$	0	0
$136$	−7.42512	−0.636699
$137$	−3.29433	−0.281454	−0.140727	−	0.990048i	$-0.544944\pi$
$137$	−3.29433	−0.281454	−0.140727	+	0.990048i	$0.544944\pi$
$138$	−2.85593	−0.243113
$139$	8.14346	0.690719	0.345360	−	0.938470i	$-0.387757\pi$
$139$	8.14346	0.690719	0.345360	+	0.938470i	$0.387757\pi$
$140$	0	0
$141$	7.95583	0.670002
$142$	−3.07474	−0.258026
$143$	1.20731	0.100960
$144$	−0.760356	−0.0633630
$145$	0	0
$146$	−0.173936	−0.0143951
$147$	16.4416	1.35608
$148$	11.2810	0.927293
$149$	0.0801450	0.00656574	0.00328287	−	0.999995i	$-0.498955\pi$
$149$	0.0801450	0.00656574	0.00328287	+	0.999995i	$0.498955\pi$
$150$	0	0
$151$	8.88114	0.722737	0.361369	−	0.932423i	$-0.382310\pi$
$151$	8.88114	0.722737	0.361369	+	0.932423i	$0.382310\pi$
$152$	21.5621	1.74892
$153$	−2.50226	−0.202296
$154$	−5.92219	−0.477223
$155$	0	0
$156$	−1.01397	−0.0811829
$157$	−17.5406	−1.39989	−0.699946	−	0.714196i	$-0.746793\pi$
$157$	−17.5406	−1.39989	−0.699946	+	0.714196i	$0.746793\pi$
$158$	11.4939	0.914404
$159$	10.9614	0.869294
$160$	0	0
$161$	14.2546	1.12342
$162$	−0.970031	−0.0762128
$163$	−12.9358	−1.01321	−0.506605	−	0.862179i	$-0.669100\pi$
$163$	−12.9358	−1.01321	−0.506605	+	0.862179i	$0.669100\pi$
$164$	−11.4318	−0.892673
$165$	0	0
$166$	−0.896930	−0.0696153
$167$	−10.4471	−0.808423	−0.404211	−	0.914666i	$-0.632454\pi$
$167$	−10.4471	−0.808423	−0.404211	+	0.914666i	$0.632454\pi$
$168$	14.3669	1.10843
$169$	−12.0833	−0.929484
$170$	0	0
$171$	7.26642	0.555677
$172$	1.78603	0.136183
$173$	−16.5164	−1.25572	−0.627859	−	0.778327i	$-0.716069\pi$
$173$	−16.5164	−1.25572	−0.627859	+	0.778327i	$0.716069\pi$
$174$	−2.43520	−0.184612
$175$	0	0
$176$	−0.958783	−0.0722710
$177$	5.13128	0.385690
$178$	0.970031	0.0727069
$179$	0.591766	0.0442307	0.0221154	−	0.999755i	$-0.492960\pi$
$179$	0.591766	0.0442307	0.0221154	+	0.999755i	$0.492960\pi$
$180$	0	0
$181$	6.57081	0.488404	0.244202	−	0.969724i	$-0.421474\pi$
$181$	6.57081	0.488404	0.244202	+	0.969724i	$0.421474\pi$
$182$	−4.49670	−0.333317
$183$	−0.773449	−0.0571750
$184$	8.73640	0.644056
$185$	0	0
$186$	1.26157	0.0925026
$187$	−3.15526	−0.230736
$188$	−8.42554	−0.614496
$189$	4.84165	0.352178
$190$	0	0
$191$	−17.5759	−1.27175	−0.635876	−	0.771792i	$-0.719361\pi$
$191$	−17.5759	−1.27175	−0.635876	+	0.771792i	$0.719361\pi$
$192$	6.56212	0.473580
$193$	22.1071	1.59131	0.795653	−	0.605753i	$-0.207128\pi$
$193$	22.1071	1.59131	0.795653	+	0.605753i	$0.207128\pi$
$194$	11.0396	0.792600
$195$	0	0
$196$	−17.4123	−1.24373
$197$	−23.5198	−1.67572	−0.837860	−	0.545886i	$-0.816193\pi$
$197$	−23.5198	−1.67572	−0.837860	+	0.545886i	$0.816193\pi$
$198$	−1.22318	−0.0869273
$199$	0.406042	0.0287835	0.0143918	−	0.999896i	$-0.495419\pi$
$199$	0.406042	0.0287835	0.0143918	+	0.999896i	$0.495419\pi$
$200$	0	0
$201$	−8.58486	−0.605529
$202$	0.523861	0.0368588
$203$	12.1546	0.853089
$204$	2.64999	0.185537
$205$	0	0
$206$	−5.00674	−0.348836
$207$	2.94416	0.204633
$208$	−0.728001	−0.0504778
$209$	9.16270	0.633797
$210$	0	0
$211$	−8.95696	−0.616623	−0.308311	−	0.951285i	$-0.599764\pi$
$211$	−8.95696	−0.616623	−0.308311	+	0.951285i	$0.599764\pi$
$212$	−11.6085	−0.797277
$213$	3.16973	0.217186
$214$	8.94781	0.611659
$215$	0	0
$216$	2.96736	0.201904
$217$	−6.29677	−0.427453
$218$	−0.880722	−0.0596500
$219$	0.179310	0.0121167
$220$	0	0
$221$	−2.39578	−0.161158
$222$	10.3329	0.693497
$223$	13.0602	0.874578	0.437289	−	0.899321i	$-0.355939\pi$
$223$	13.0602	0.874578	0.437289	+	0.899321i	$0.355939\pi$
$224$	−25.1628	−1.68126
$225$	0	0
$226$	6.36918	0.423672
$227$	2.54072	0.168634	0.0843169	−	0.996439i	$-0.473129\pi$
$227$	2.54072	0.168634	0.0843169	+	0.996439i	$0.473129\pi$
$228$	−7.69543	−0.509642
$229$	3.14557	0.207865	0.103932	−	0.994584i	$-0.466857\pi$
$229$	3.14557	0.207865	0.103932	+	0.994584i	$0.466857\pi$
$230$	0	0
$231$	6.10515	0.401689
$232$	7.44938	0.489076
$233$	19.5193	1.27875	0.639377	−	0.768893i	$-0.279192\pi$
$233$	19.5193	1.27875	0.639377	+	0.768893i	$0.279192\pi$
$234$	−0.928753	−0.0607145
$235$	0	0
$236$	−5.43422	−0.353738
$237$	−11.8490	−0.769674
$238$	11.7520	0.761768
$239$	14.4309	0.933459	0.466730	−	0.884400i	$-0.345432\pi$
$239$	14.4309	0.933459	0.466730	+	0.884400i	$0.345432\pi$
$240$	0	0
$241$	−27.1333	−1.74781	−0.873905	−	0.486097i	$-0.838420\pi$
$241$	−27.1333	−1.74781	−0.873905	+	0.486097i	$0.838420\pi$
$242$	9.12796	0.586768
$243$	1.00000	0.0641500
$244$	0.819113	0.0524383
$245$	0	0
$246$	−10.4710	−0.667606
$247$	6.95721	0.442677
$248$	−3.85918	−0.245058
$249$	0.924640	0.0585967
$250$	0	0
$251$	17.1350	1.08155	0.540777	−	0.841166i	$-0.318130\pi$
$251$	17.1350	1.08155	0.540777	+	0.841166i	$0.318130\pi$
$252$	−5.12750	−0.323002
$253$	3.71248	0.233402
$254$	−5.09380	−0.319614
$255$	0	0
$256$	−17.0324	−1.06452
$257$	−21.1235	−1.31765	−0.658824	−	0.752297i	$-0.728946\pi$
$257$	−21.1235	−1.31765	−0.658824	+	0.752297i	$0.728946\pi$
$258$	1.63592	0.101848
$259$	−51.5738	−3.20464
$260$	0	0
$261$	2.51044	0.155392
$262$	−10.5138	−0.649543
$263$	20.0742	1.23783	0.618913	−	0.785459i	$-0.287573\pi$
$263$	20.0742	1.23783	0.618913	+	0.785459i	$0.287573\pi$
$264$	3.74174	0.230288
$265$	0	0
$266$	−34.1271	−2.09247
$267$	−1.00000	−0.0611990
$268$	9.09171	0.555364
$269$	−10.9077	−0.665052	−0.332526	−	0.943094i	$-0.607901\pi$
$269$	−10.9077	−0.665052	−0.332526	+	0.943094i	$0.607901\pi$
$270$	0	0
$271$	−20.4886	−1.24459	−0.622297	−	0.782782i	$-0.713800\pi$
$271$	−20.4886	−1.24459	−0.622297	+	0.782782i	$0.713800\pi$
$272$	1.90261	0.115363
$273$	4.63562	0.280560
$274$	3.19560	0.193053
$275$	0	0
$276$	−3.11798	−0.187680
$277$	−27.0530	−1.62546	−0.812728	−	0.582643i	$-0.802018\pi$
$277$	−27.0530	−1.62546	−0.812728	+	0.582643i	$0.802018\pi$
$278$	−7.89941	−0.473775
$279$	−1.30054	−0.0778614
$280$	0	0
$281$	16.3438	0.974990	0.487495	−	0.873126i	$-0.337911\pi$
$281$	16.3438	0.974990	0.487495	+	0.873126i	$0.337911\pi$
$282$	−7.71741	−0.459565
$283$	15.5600	0.924946	0.462473	−	0.886633i	$-0.346962\pi$
$283$	15.5600	0.924946	0.462473	+	0.886633i	$0.346962\pi$
$284$	−3.35687	−0.199194
$285$	0	0
$286$	−1.17113	−0.0692501
$287$	52.2631	3.08500
$288$	−5.19716	−0.306245
$289$	−10.7387	−0.631688
$290$	0	0
$291$	−11.3807	−0.667148
$292$	−0.189896	−0.0111128
$293$	17.7412	1.03645	0.518226	−	0.855244i	$-0.326593\pi$
$293$	17.7412	1.03645	0.518226	+	0.855244i	$0.326593\pi$
$294$	−15.9488	−0.930154
$295$	0	0
$296$	−31.6087	−1.83722
$297$	1.26097	0.0731686
$298$	−0.0777432	−0.00450354
$299$	2.81888	0.163020
$300$	0	0
$301$	−8.16525	−0.470637
$302$	−8.61499	−0.495737
$303$	−0.540046	−0.0310248
$304$	−5.52507	−0.316884
$305$	0	0
$306$	2.42727	0.138758
$307$	14.5519	0.830522	0.415261	−	0.909702i	$-0.363690\pi$
$307$	14.5519	0.830522	0.415261	+	0.909702i	$0.363690\pi$
$308$	−6.46559	−0.368411
$309$	5.16142	0.293623
$310$	0	0
$311$	−11.2035	−0.635292	−0.317646	−	0.948209i	$-0.602892\pi$
$311$	−11.2035	−0.635292	−0.317646	+	0.948209i	$0.602892\pi$
$312$	2.84109	0.160845
$313$	−21.4370	−1.21169	−0.605846	−	0.795582i	$-0.707165\pi$
$313$	−21.4370	−1.21169	−0.605846	+	0.795582i	$0.707165\pi$
$314$	17.0149	0.960208
$315$	0	0
$316$	12.5485	0.705910
$317$	21.0534	1.18248	0.591239	−	0.806496i	$-0.298639\pi$
$317$	21.0534	1.18248	0.591239	+	0.806496i	$0.298639\pi$
$318$	−10.6329	−0.596262
$319$	3.16557	0.177238
$320$	0	0
$321$	−9.22424	−0.514847
$322$	−13.8274	−0.770571
$323$	−18.1825	−1.01170
$324$	−1.05904	−0.0588355
$325$	0	0
$326$	12.5481	0.694976
$327$	0.907932	0.0502087
$328$	32.0312	1.76863
$329$	38.5193	2.12364
$330$	0	0
$331$	29.5188	1.62250	0.811249	−	0.584700i	$-0.198788\pi$
$331$	29.5188	1.62250	0.811249	+	0.584700i	$0.198788\pi$
$332$	−0.979231	−0.0537423
$333$	−10.6521	−0.583732
$334$	10.1340	0.554510
$335$	0	0
$336$	−3.68138	−0.200836
$337$	−29.4025	−1.60165	−0.800827	−	0.598895i	$-0.795606\pi$
$337$	−29.4025	−1.60165	−0.800827	+	0.598895i	$0.795606\pi$
$338$	11.7212	0.637548
$339$	−6.56595	−0.356614
$340$	0	0
$341$	−1.63994	−0.0888077
$342$	−7.04865	−0.381148
$343$	45.7127	2.46825
$344$	−5.00434	−0.269816
$345$	0	0
$346$	16.0214	0.861317
$347$	16.5560	0.888774	0.444387	−	0.895835i	$-0.353421\pi$
$347$	16.5560	0.888774	0.444387	+	0.895835i	$0.353421\pi$
$348$	−2.65865	−0.142519
$349$	15.2845	0.818160	0.409080	−	0.912499i	$-0.365850\pi$
$349$	15.2845	0.818160	0.409080	+	0.912499i	$0.365850\pi$
$350$	0	0
$351$	0.957447	0.0511047
$352$	−6.55344	−0.349299
$353$	−27.6140	−1.46975	−0.734873	−	0.678204i	$-0.762758\pi$
$353$	−27.6140	−1.46975	−0.734873	+	0.678204i	$0.762758\pi$
$354$	−4.97750	−0.264551
$355$	0	0
$356$	1.05904	0.0561290
$357$	−12.1151	−0.641197
$358$	−0.574032	−0.0303385
$359$	23.9146	1.26216	0.631082	−	0.775716i	$-0.282611\pi$
$359$	23.9146	1.26216	0.631082	+	0.775716i	$0.282611\pi$
$360$	0	0
$361$	33.8009	1.77899
$362$	−6.37389	−0.335004
$363$	−9.40997	−0.493895
$364$	−4.90930	−0.257317
$365$	0	0
$366$	0.750270	0.0392172
$367$	23.5377	1.22866	0.614329	−	0.789050i	$-0.289427\pi$
$367$	23.5377	1.22866	0.614329	+	0.789050i	$0.289427\pi$
$368$	−2.23861	−0.116696
$369$	10.7945	0.561939
$370$	0	0
$371$	53.0711	2.75532
$372$	1.37733	0.0714110
$373$	−5.05202	−0.261583	−0.130792	−	0.991410i	$-0.541752\pi$
$373$	−5.05202	−0.261583	−0.130792	+	0.991410i	$0.541752\pi$
$374$	3.06070	0.158265
$375$	0	0
$376$	23.6078	1.21748
$377$	2.40361	0.123792
$378$	−4.69655	−0.241564
$379$	17.2544	0.886297	0.443149	−	0.896448i	$-0.353861\pi$
$379$	17.2544	0.886297	0.443149	+	0.896448i	$0.353861\pi$
$380$	0	0
$381$	5.25118	0.269026
$382$	17.0492	0.872314
$383$	−27.2695	−1.39340	−0.696702	−	0.717361i	$-0.745350\pi$
$383$	−27.2695	−1.39340	−0.696702	+	0.717361i	$0.745350\pi$
$384$	4.02886	0.205597
$385$	0	0
$386$	−21.4446	−1.09150
$387$	−1.68646	−0.0857276
$388$	12.0526	0.611879
$389$	24.7299	1.25386	0.626928	−	0.779077i	$-0.284312\pi$
$389$	24.7299	1.25386	0.626928	+	0.779077i	$0.284312\pi$
$390$	0	0
$391$	−7.36706	−0.372568
$392$	48.7881	2.46417
$393$	10.8386	0.546734
$394$	22.8150	1.14940
$395$	0	0
$396$	−1.33541	−0.0671070
$397$	2.48853	0.124896	0.0624480	−	0.998048i	$-0.480109\pi$
$397$	2.48853	0.124896	0.0624480	+	0.998048i	$0.480109\pi$
$398$	−0.393873	−0.0197431
$399$	35.1814	1.76128
$400$	0	0
$401$	−12.0280	−0.600648	−0.300324	−	0.953837i	$-0.597095\pi$
$401$	−12.0280	−0.600648	−0.300324	+	0.953837i	$0.597095\pi$
$402$	8.32758	0.415342
$403$	−1.24520	−0.0620278
$404$	0.571930	0.0284546
$405$	0	0
$406$	−11.7904	−0.585147
$407$	−13.4319	−0.665797
$408$	−7.42512	−0.367598
$409$	−3.76565	−0.186199	−0.0930997	−	0.995657i	$-0.529678\pi$
$409$	−3.76565	−0.186199	−0.0930997	+	0.995657i	$0.529678\pi$
$410$	0	0
$411$	−3.29433	−0.162497
$412$	−5.46615	−0.269298
$413$	24.8438	1.22248
$414$	−2.85593	−0.140361
$415$	0	0
$416$	−4.97600	−0.243969
$417$	8.14346	0.398787
$418$	−8.88811	−0.434732
$419$	−7.01225	−0.342571	−0.171285	−	0.985221i	$-0.554792\pi$
$419$	−7.01225	−0.342571	−0.171285	+	0.985221i	$0.554792\pi$
$420$	0	0
$421$	14.2678	0.695371	0.347686	−	0.937611i	$-0.386968\pi$
$421$	14.2678	0.695371	0.347686	+	0.937611i	$0.386968\pi$
$422$	8.68854	0.422951
$423$	7.95583	0.386826
$424$	32.5264	1.57962
$425$	0	0
$426$	−3.07474	−0.148972
$427$	−3.74477	−0.181222
$428$	9.76884	0.472195
$429$	1.20731	0.0582893
$430$	0	0
$431$	18.8215	0.906600	0.453300	−	0.891358i	$-0.350247\pi$
$431$	18.8215	0.906600	0.453300	+	0.891358i	$0.350247\pi$
$432$	−0.760356	−0.0365827
$433$	−24.4512	−1.17505	−0.587524	−	0.809207i	$-0.699897\pi$
$433$	−24.4512	−1.17505	−0.587524	+	0.809207i	$0.699897\pi$
$434$	6.10806	0.293196
$435$	0	0
$436$	−0.961536	−0.0460492
$437$	21.3935	1.02339
$438$	−0.173936	−0.00831100
$439$	−38.8547	−1.85443	−0.927217	−	0.374524i	$-0.877806\pi$
$439$	−38.8547	−1.85443	−0.927217	+	0.374524i	$0.877806\pi$
$440$	0	0
$441$	16.4416	0.782931
$442$	2.32398	0.110541
$443$	−31.5057	−1.49688	−0.748440	−	0.663202i	$-0.769197\pi$
$443$	−31.5057	−1.49688	−0.748440	+	0.663202i	$0.769197\pi$
$444$	11.2810	0.535373
$445$	0	0
$446$	−12.6688	−0.599887
$447$	0.0801450	0.00379073
$448$	31.7715	1.50106
$449$	0.0428453	0.00202199	0.00101100	−	0.999999i	$-0.499678\pi$
$449$	0.0428453	0.00202199	0.00101100	+	0.999999i	$0.499678\pi$
$450$	0	0
$451$	13.6115	0.640940
$452$	6.95360	0.327070
$453$	8.88114	0.417273
$454$	−2.46458	−0.115669
$455$	0	0
$456$	21.5621	1.00974
$457$	6.01306	0.281279	0.140640	−	0.990061i	$-0.455084\pi$
$457$	6.01306	0.281279	0.140640	+	0.990061i	$0.455084\pi$
$458$	−3.05130	−0.142578
$459$	−2.50226	−0.116796
$460$	0	0
$461$	14.1798	0.660421	0.330211	−	0.943907i	$-0.392880\pi$
$461$	14.1798	0.660421	0.330211	+	0.943907i	$0.392880\pi$
$462$	−5.92219	−0.275525
$463$	−17.0659	−0.793120	−0.396560	−	0.918009i	$-0.629796\pi$
$463$	−17.0659	−0.793120	−0.396560	+	0.918009i	$0.629796\pi$
$464$	−1.90883	−0.0886150
$465$	0	0
$466$	−18.9344	−0.877118
$467$	16.7557	0.775362	0.387681	−	0.921794i	$-0.373276\pi$
$467$	16.7557	0.775362	0.387681	+	0.921794i	$0.373276\pi$
$468$	−1.01397	−0.0468710
$469$	−41.5649	−1.91929
$470$	0	0
$471$	−17.5406	−0.808228
$472$	15.2264	0.700850
$473$	−2.12657	−0.0977797
$474$	11.4939	0.527931
$475$	0	0
$476$	12.8303	0.588077
$477$	10.9614	0.501887
$478$	−13.9984	−0.640274
$479$	17.5127	0.800176	0.400088	−	0.916477i	$-0.368980\pi$
$479$	17.5127	0.800176	0.400088	+	0.916477i	$0.368980\pi$
$480$	0	0
$481$	−10.1988	−0.465027
$482$	26.3202	1.19885
$483$	14.2546	0.648606
$484$	9.96553	0.452978
$485$	0	0
$486$	−0.970031	−0.0440015
$487$	40.4312	1.83211	0.916056	−	0.401050i	$-0.131355\pi$
$487$	40.4312	1.83211	0.916056	+	0.401050i	$0.131355\pi$
$488$	−2.29511	−0.103895
$489$	−12.9358	−0.584977
$490$	0	0
$491$	42.9414	1.93792	0.968959	−	0.247220i	$-0.0795171\pi$
$491$	42.9414	1.93792	0.968959	+	0.247220i	$0.0795171\pi$
$492$	−11.4318	−0.515385
$493$	−6.28177	−0.282917
$494$	−6.74871	−0.303639
$495$	0	0
$496$	0.988876	0.0444018
$497$	15.3467	0.688394
$498$	−0.896930	−0.0401924
$499$	35.4613	1.58747	0.793733	−	0.608266i	$-0.208135\pi$
$499$	35.4613	1.58747	0.793733	+	0.608266i	$0.208135\pi$
$500$	0	0
$501$	−10.4471	−0.466743
$502$	−16.6215	−0.741855
$503$	−10.6586	−0.475241	−0.237621	−	0.971358i	$-0.576368\pi$
$503$	−10.6586	−0.475241	−0.237621	+	0.971358i	$0.576368\pi$
$504$	14.3669	0.639954
$505$	0	0
$506$	−3.60123	−0.160094
$507$	−12.0833	−0.536638
$508$	−5.56120	−0.246738
$509$	−15.3253	−0.679282	−0.339641	−	0.940555i	$-0.610306\pi$
$509$	−15.3253	−0.679282	−0.339641	+	0.940555i	$0.610306\pi$
$510$	0	0
$511$	0.868156	0.0384050
$512$	8.46420	0.374068
$513$	7.26642	0.320820
$514$	20.4905	0.903795
$515$	0	0
$516$	1.78603	0.0786255
$517$	10.0320	0.441208
$518$	50.0282	2.19811
$519$	−16.5164	−0.724989
$520$	0	0
$521$	−1.42539	−0.0624475	−0.0312237	−	0.999512i	$-0.509940\pi$
$521$	−1.42539	−0.0624475	−0.0312237	+	0.999512i	$0.509940\pi$
$522$	−2.43520	−0.106586
$523$	22.8141	0.997590	0.498795	−	0.866720i	$-0.333776\pi$
$523$	22.8141	0.997590	0.498795	+	0.866720i	$0.333776\pi$
$524$	−11.4785	−0.501440
$525$	0	0
$526$	−19.4726	−0.849044
$527$	3.25430	0.141759
$528$	−0.958783	−0.0417257
$529$	−14.3319	−0.623127
$530$	0	0
$531$	5.13128	0.222678
$532$	−37.2585	−1.61536
$533$	10.3352	0.447665
$534$	0.970031	0.0419774
$535$	0	0
$536$	−25.4744	−1.10033
$537$	0.591766	0.0255366
$538$	10.5808	0.456170
$539$	20.7322	0.893000
$540$	0	0
$541$	−13.4937	−0.580141	−0.290070	−	0.957005i	$-0.593679\pi$
$541$	−13.4937	−0.580141	−0.290070	+	0.957005i	$0.593679\pi$
$542$	19.8746	0.853686
$543$	6.57081	0.281980
$544$	13.0046	0.557570
$545$	0	0
$546$	−4.49670	−0.192441
$547$	28.2383	1.20738	0.603691	−	0.797219i	$-0.293696\pi$
$547$	28.2383	1.20738	0.603691	+	0.797219i	$0.293696\pi$
$548$	3.48883	0.149035
$549$	−0.773449	−0.0330100
$550$	0	0
$551$	18.2419	0.777130
$552$	8.73640	0.371846
$553$	−57.3686	−2.43956
$554$	26.2422	1.11493
$555$	0	0
$556$	−8.62425	−0.365749
$557$	−33.2231	−1.40771	−0.703855	−	0.710344i	$-0.748539\pi$
$557$	−33.2231	−1.40771	−0.703855	+	0.710344i	$0.748539\pi$
$558$	1.26157	0.0534064
$559$	−1.61470	−0.0682944
$560$	0	0
$561$	−3.15526	−0.133215
$562$	−15.8540	−0.668761
$563$	28.9628	1.22064	0.610319	−	0.792156i	$-0.291041\pi$
$563$	28.9628	1.22064	0.610319	+	0.792156i	$0.291041\pi$
$564$	−8.42554	−0.354779
$565$	0	0
$566$	−15.0937	−0.634435
$567$	4.84165	0.203330
$568$	9.40574	0.394656
$569$	−12.0048	−0.503268	−0.251634	−	0.967822i	$-0.580968\pi$
$569$	−12.0048	−0.503268	−0.251634	+	0.967822i	$0.580968\pi$
$570$	0	0
$571$	−11.7190	−0.490426	−0.245213	−	0.969469i	$-0.578858\pi$
$571$	−11.7190	−0.490426	−0.245213	+	0.969469i	$0.578858\pi$
$572$	−1.27859	−0.0534604
$573$	−17.5759	−0.734246
$574$	−50.6969	−2.11605
$575$	0	0
$576$	6.56212	0.273422
$577$	28.8315	1.20027	0.600135	−	0.799899i	$-0.295114\pi$
$577$	28.8315	1.20027	0.600135	+	0.799899i	$0.295114\pi$
$578$	10.4169	0.433285
$579$	22.1071	0.918741
$580$	0	0
$581$	4.47678	0.185728
$582$	11.0396	0.457608
$583$	13.8219	0.572445
$584$	0.532078	0.0220176
$585$	0	0
$586$	−17.2095	−0.710918
$587$	−47.1515	−1.94615	−0.973075	−	0.230488i	$-0.925968\pi$
$587$	−47.1515	−1.94615	−0.973075	+	0.230488i	$0.925968\pi$
$588$	−17.4123	−0.718069
$589$	−9.45029	−0.389392
$590$	0	0
$591$	−23.5198	−0.967477
$592$	8.09940	0.332883
$593$	−12.5235	−0.514280	−0.257140	−	0.966374i	$-0.582780\pi$
$593$	−12.5235	−0.514280	−0.257140	+	0.966374i	$0.582780\pi$
$594$	−1.22318	−0.0501875
$595$	0	0
$596$	−0.0848768	−0.00347669
$597$	0.406042	0.0166182
$598$	−2.73440	−0.111818
$599$	42.1633	1.72275	0.861374	−	0.507972i	$-0.169605\pi$
$599$	42.1633	1.72275	0.861374	+	0.507972i	$0.169605\pi$
$600$	0	0
$601$	−19.9645	−0.814369	−0.407185	−	0.913346i	$-0.633489\pi$
$601$	−19.9645	−0.814369	−0.407185	+	0.913346i	$0.633489\pi$
$602$	7.92055	0.322817
$603$	−8.58486	−0.349603
$604$	−9.40548	−0.382704
$605$	0	0
$606$	0.523861	0.0212804
$607$	−26.8828	−1.09114	−0.545570	−	0.838065i	$-0.683687\pi$
$607$	−26.8828	−1.09114	−0.545570	+	0.838065i	$0.683687\pi$
$608$	−37.7647	−1.53156
$609$	12.1546	0.492531
$610$	0	0
$611$	7.61729	0.308162
$612$	2.64999	0.107120
$613$	−28.8740	−1.16621	−0.583105	−	0.812397i	$-0.698162\pi$
$613$	−28.8740	−1.16621	−0.583105	+	0.812397i	$0.698162\pi$
$614$	−14.1158	−0.569668
$615$	0	0
$616$	18.1162	0.729922
$617$	27.1937	1.09478	0.547388	−	0.836879i	$-0.315622\pi$
$617$	27.1937	1.09478	0.547388	+	0.836879i	$0.315622\pi$
$618$	−5.00674	−0.201401
$619$	6.00427	0.241332	0.120666	−	0.992693i	$-0.461497\pi$
$619$	6.00427	0.241332	0.120666	+	0.992693i	$0.461497\pi$
$620$	0	0
$621$	2.94416	0.118145
$622$	10.8677	0.435757
$623$	−4.84165	−0.193977
$624$	−0.728001	−0.0291434
$625$	0	0
$626$	20.7946	0.831119
$627$	9.16270	0.365923
$628$	18.5762	0.741271
$629$	26.6544	1.06278
$630$	0	0
$631$	27.2201	1.08362	0.541808	−	0.840502i	$-0.317740\pi$
$631$	27.2201	1.08362	0.541808	+	0.840502i	$0.317740\pi$
$632$	−35.1602	−1.39860
$633$	−8.95696	−0.356007
$634$	−20.4225	−0.811080
$635$	0	0
$636$	−11.6085	−0.460308
$637$	15.7419	0.623717
$638$	−3.07070	−0.121570
$639$	3.16973	0.125393
$640$	0	0
$641$	3.63150	0.143436	0.0717179	−	0.997425i	$-0.477152\pi$
$641$	3.63150	0.143436	0.0717179	+	0.997425i	$0.477152\pi$
$642$	8.94781	0.353142
$643$	−32.5013	−1.28172	−0.640862	−	0.767656i	$-0.721423\pi$
$643$	−32.5013	−1.28172	−0.640862	+	0.767656i	$0.721423\pi$
$644$	−15.0962	−0.594872
$645$	0	0
$646$	17.6376	0.693941
$647$	34.7958	1.36796	0.683982	−	0.729499i	$-0.260247\pi$
$647$	34.7958	1.36796	0.683982	+	0.729499i	$0.260247\pi$
$648$	2.96736	0.116569
$649$	6.47036	0.253984
$650$	0	0
$651$	−6.29677	−0.246790
$652$	13.6995	0.536514
$653$	−8.75257	−0.342514	−0.171257	−	0.985226i	$-0.554783\pi$
$653$	−8.75257	−0.342514	−0.171257	+	0.985226i	$0.554783\pi$
$654$	−0.880722	−0.0344390
$655$	0	0
$656$	−8.20767	−0.320455
$657$	0.179310	0.00699555
$658$	−37.3650	−1.45664
$659$	−33.5109	−1.30540	−0.652701	−	0.757616i	$-0.726364\pi$
$659$	−33.5109	−1.30540	−0.652701	+	0.757616i	$0.726364\pi$
$660$	0	0
$661$	4.44926	0.173056	0.0865280	−	0.996249i	$-0.472423\pi$
$661$	4.44926	0.173056	0.0865280	+	0.996249i	$0.472423\pi$
$662$	−28.6341	−1.11290
$663$	−2.39578	−0.0930444
$664$	2.74374	0.106478
$665$	0	0
$666$	10.3329	0.400391
$667$	7.39113	0.286186
$668$	11.0639	0.428076
$669$	13.0602	0.504938
$670$	0	0
$671$	−0.975293	−0.0376508
$672$	−25.1628	−0.970676
$673$	14.9337	0.575650	0.287825	−	0.957683i	$-0.407068\pi$
$673$	14.9337	0.575650	0.287825	+	0.957683i	$0.407068\pi$
$674$	28.5213	1.09860
$675$	0	0
$676$	12.7967	0.492180
$677$	17.3442	0.666590	0.333295	−	0.942823i	$-0.391839\pi$
$677$	17.3442	0.666590	0.333295	+	0.942823i	$0.391839\pi$
$678$	6.36918	0.244607
$679$	−55.1013	−2.11460
$680$	0	0
$681$	2.54072	0.0973607
$682$	1.59079	0.0609146
$683$	36.2406	1.38671	0.693353	−	0.720598i	$-0.256133\pi$
$683$	36.2406	1.38671	0.693353	+	0.720598i	$0.256133\pi$
$684$	−7.69543	−0.294242
$685$	0	0
$686$	−44.3427	−1.69301
$687$	3.14557	0.120011
$688$	1.28231	0.0488877
$689$	10.4949	0.399825
$690$	0	0
$691$	−47.9692	−1.82483	−0.912417	−	0.409262i	$-0.865786\pi$
$691$	−47.9692	−1.82483	−0.912417	+	0.409262i	$0.865786\pi$
$692$	17.4915	0.664927
$693$	6.10515	0.231915
$694$	−16.0599	−0.609624
$695$	0	0
$696$	7.44938	0.282368
$697$	−27.0106	−1.02310
$698$	−14.8264	−0.561188
$699$	19.5193	0.738289
$700$	0	0
$701$	−22.7966	−0.861016	−0.430508	−	0.902587i	$-0.641665\pi$
$701$	−22.7966	−0.861016	−0.430508	+	0.902587i	$0.641665\pi$
$702$	−0.928753	−0.0350535
$703$	−77.4027	−2.91930
$704$	8.27460	0.311861
$705$	0	0
$706$	26.7865	1.00812
$707$	−2.61471	−0.0983364
$708$	−5.43422	−0.204231
$709$	29.4952	1.10772	0.553858	−	0.832611i	$-0.313155\pi$
$709$	29.4952	1.10772	0.553858	+	0.832611i	$0.313155\pi$
$710$	0	0
$711$	−11.8490	−0.444371
$712$	−2.96736	−0.111207
$713$	−3.82901	−0.143397
$714$	11.7520	0.439807
$715$	0	0
$716$	−0.626704	−0.0234210
$717$	14.4309	0.538933
$718$	−23.1979	−0.865738
$719$	15.7542	0.587534	0.293767	−	0.955877i	$-0.405091\pi$
$719$	15.7542	0.587534	0.293767	+	0.955877i	$0.405091\pi$
$720$	0	0
$721$	24.9898	0.930669
$722$	−32.7879	−1.22024
$723$	−27.1333	−1.00910
$724$	−6.95875	−0.258620
$725$	0	0
$726$	9.12796	0.338770
$727$	−16.2559	−0.602899	−0.301449	−	0.953482i	$-0.597470\pi$
$727$	−16.2559	−0.602899	−0.301449	+	0.953482i	$0.597470\pi$
$728$	13.7556	0.509815
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.21996	0.156081
$732$	0.819113	0.0302753
$733$	−14.7171	−0.543590	−0.271795	−	0.962355i	$-0.587617\pi$
$733$	−14.7171	−0.543590	−0.271795	+	0.962355i	$0.587617\pi$
$734$	−22.8323	−0.842755
$735$	0	0
$736$	−15.3013	−0.564012
$737$	−10.8252	−0.398752
$738$	−10.4710	−0.385443
$739$	17.7624	0.653400	0.326700	−	0.945128i	$-0.394063\pi$
$739$	17.7624	0.653400	0.326700	+	0.945128i	$0.394063\pi$
$740$	0	0
$741$	6.95721	0.255580
$742$	−51.4806	−1.88991
$743$	5.90489	0.216629	0.108315	−	0.994117i	$-0.465455\pi$
$743$	5.90489	0.216629	0.108315	+	0.994117i	$0.465455\pi$
$744$	−3.85918	−0.141485
$745$	0	0
$746$	4.90061	0.179424
$747$	0.924640	0.0338308
$748$	3.34155	0.122179
$749$	−44.6605	−1.63186
$750$	0	0
$751$	15.4581	0.564075	0.282037	−	0.959403i	$-0.408990\pi$
$751$	15.4581	0.564075	0.282037	+	0.959403i	$0.408990\pi$
$752$	−6.04927	−0.220594
$753$	17.1350	0.624435
$754$	−2.33158	−0.0849110
$755$	0	0
$756$	−5.12750	−0.186485
$757$	17.8323	0.648125	0.324062	−	0.946036i	$-0.394951\pi$
$757$	17.8323	0.648125	0.324062	+	0.946036i	$0.394951\pi$
$758$	−16.7373	−0.607925
$759$	3.71248	0.134755
$760$	0	0
$761$	2.06887	0.0749965	0.0374983	−	0.999297i	$-0.488061\pi$
$761$	2.06887	0.0749965	0.0374983	+	0.999297i	$0.488061\pi$
$762$	−5.09380	−0.184529
$763$	4.39589	0.159142
$764$	18.6136	0.673417
$765$	0	0
$766$	26.4522	0.955757
$767$	4.91292	0.177395
$768$	−17.0324	−0.614602
$769$	0.182522	0.00658191	0.00329096	−	0.999995i	$-0.498952\pi$
$769$	0.182522	0.00658191	0.00329096	+	0.999995i	$0.498952\pi$
$770$	0	0
$771$	−21.1235	−0.760744
$772$	−23.4123	−0.842628
$773$	22.5668	0.811672	0.405836	−	0.913946i	$-0.366980\pi$
$773$	22.5668	0.811672	0.405836	+	0.913946i	$0.366980\pi$
$774$	1.63592	0.0588019
$775$	0	0
$776$	−33.7707	−1.21230
$777$	−51.5738	−1.85020
$778$	−23.9888	−0.860040
$779$	78.4374	2.81031
$780$	0	0
$781$	3.99692	0.143021
$782$	7.14628	0.255550
$783$	2.51044	0.0897157
$784$	−12.5014	−0.446480
$785$	0	0
$786$	−10.5138	−0.375014
$787$	3.57295	0.127362	0.0636810	−	0.997970i	$-0.479716\pi$
$787$	3.57295	0.127362	0.0636810	+	0.997970i	$0.479716\pi$
$788$	24.9084	0.887326
$789$	20.0742	0.714659
$790$	0	0
$791$	−31.7900	−1.13032
$792$	3.74174	0.132957
$793$	−0.740536	−0.0262972
$794$	−2.41396	−0.0856681
$795$	0	0
$796$	−0.430014	−0.0152415
$797$	38.9586	1.37998	0.689992	−	0.723817i	$-0.257614\pi$
$797$	38.9586	1.37998	0.689992	+	0.723817i	$0.257614\pi$
$798$	−34.1271	−1.20809
$799$	−19.9076	−0.704279
$800$	0	0
$801$	−1.00000	−0.0353333
$802$	11.6675	0.411994
$803$	0.226104	0.00797903
$804$	9.09171	0.320640
$805$	0	0
$806$	1.20788	0.0425459
$807$	−10.9077	−0.383968
$808$	−1.60251	−0.0563762
$809$	21.2303	0.746417	0.373208	−	0.927748i	$-0.378258\pi$
$809$	21.2303	0.746417	0.373208	+	0.927748i	$0.378258\pi$
$810$	0	0
$811$	−12.4400	−0.436827	−0.218414	−	0.975856i	$-0.570088\pi$
$811$	−12.4400	−0.436827	−0.218414	+	0.975856i	$0.570088\pi$
$812$	−12.8723	−0.451727
$813$	−20.4886	−0.718566
$814$	13.0294	0.456680
$815$	0	0
$816$	1.90261	0.0666047
$817$	−12.2545	−0.428732
$818$	3.65280	0.127717
$819$	4.63562	0.161982
$820$	0	0
$821$	−7.62904	−0.266255	−0.133128	−	0.991099i	$-0.542502\pi$
$821$	−7.62904	−0.266255	−0.133128	+	0.991099i	$0.542502\pi$
$822$	3.19560	0.111459
$823$	50.8434	1.77229	0.886144	−	0.463410i	$-0.153374\pi$
$823$	50.8434	1.77229	0.886144	+	0.463410i	$0.153374\pi$
$824$	15.3158	0.533552
$825$	0	0
$826$	−24.0993	−0.838522
$827$	−25.6950	−0.893502	−0.446751	−	0.894658i	$-0.647419\pi$
$827$	−25.6950	−0.893502	−0.446751	+	0.894658i	$0.647419\pi$
$828$	−3.11798	−0.108357
$829$	28.1275	0.976909	0.488454	−	0.872589i	$-0.337561\pi$
$829$	28.1275	0.976909	0.488454	+	0.872589i	$0.337561\pi$
$830$	0	0
$831$	−27.0530	−0.938457
$832$	6.28288	0.217820
$833$	−41.1410	−1.42545
$834$	−7.89941	−0.273534
$835$	0	0
$836$	−9.70367	−0.335608
$837$	−1.30054	−0.0449533
$838$	6.80210	0.234975
$839$	−28.4960	−0.983790	−0.491895	−	0.870654i	$-0.663696\pi$
$839$	−28.4960	−0.983790	−0.491895	+	0.870654i	$0.663696\pi$
$840$	0	0
$841$	−22.6977	−0.782680
$842$	−13.8402	−0.476966
$843$	16.3438	0.562911
$844$	9.48578	0.326514
$845$	0	0
$846$	−7.71741	−0.265330
$847$	−45.5597	−1.56545
$848$	−8.33455	−0.286210
$849$	15.5600	0.534018
$850$	0	0
$851$	−31.3615	−1.07506
$852$	−3.35687	−0.115004
$853$	6.37854	0.218397	0.109198	−	0.994020i	$-0.465172\pi$
$853$	6.37854	0.218397	0.109198	+	0.994020i	$0.465172\pi$
$854$	3.63254	0.124303
$855$	0	0
$856$	−27.3717	−0.935545
$857$	2.79032	0.0953157	0.0476578	−	0.998864i	$-0.484824\pi$
$857$	2.79032	0.0953157	0.0476578	+	0.998864i	$0.484824\pi$
$858$	−1.17113	−0.0399816
$859$	−52.1048	−1.77779	−0.888897	−	0.458108i	$-0.848527\pi$
$859$	−52.1048	−1.77779	−0.888897	+	0.458108i	$0.848527\pi$
$860$	0	0
$861$	52.2631	1.78112
$862$	−18.2574	−0.621851
$863$	−11.9685	−0.407413	−0.203707	−	0.979032i	$-0.565299\pi$
$863$	−11.9685	−0.407413	−0.203707	+	0.979032i	$0.565299\pi$
$864$	−5.19716	−0.176811
$865$	0	0
$866$	23.7184	0.805984
$867$	−10.7387	−0.364705
$868$	6.66853	0.226345
$869$	−14.9411	−0.506844
$870$	0	0
$871$	−8.21955	−0.278509
$872$	2.69416	0.0912359
$873$	−11.3807	−0.385178
$874$	−20.7524	−0.701959
$875$	0	0
$876$	−0.189896	−0.00641601
$877$	7.55377	0.255073	0.127536	−	0.991834i	$-0.459293\pi$
$877$	7.55377	0.255073	0.127536	+	0.991834i	$0.459293\pi$
$878$	37.6903	1.27199
$879$	17.7412	0.598395
$880$	0	0
$881$	42.2046	1.42191	0.710955	−	0.703238i	$-0.248263\pi$
$881$	42.2046	1.42191	0.710955	+	0.703238i	$0.248263\pi$
$882$	−15.9488	−0.537025
$883$	48.5407	1.63352	0.816762	−	0.576974i	$-0.195767\pi$
$883$	48.5407	1.63352	0.816762	+	0.576974i	$0.195767\pi$
$884$	2.53723	0.0853362
$885$	0	0
$886$	30.5615	1.02673
$887$	−31.4482	−1.05593	−0.527963	−	0.849267i	$-0.677044\pi$
$887$	−31.4482	−1.05593	−0.527963	+	0.849267i	$0.677044\pi$
$888$	−31.6087	−1.06072
$889$	25.4243	0.852705
$890$	0	0
$891$	1.26097	0.0422439
$892$	−13.8313	−0.463106
$893$	57.8104	1.93455
$894$	−0.0777432	−0.00260012
$895$	0	0
$896$	19.5063	0.651660
$897$	2.81888	0.0941196
$898$	−0.0415613	−0.00138692
$899$	−3.26493	−0.108891
$900$	0	0
$901$	−27.4282	−0.913767
$902$	−13.2036	−0.439631
$903$	−8.16525	−0.271722
$904$	−19.4836	−0.648014
$905$	0	0
$906$	−8.61499	−0.286214
$907$	29.8819	0.992212	0.496106	−	0.868262i	$-0.334763\pi$
$907$	29.8819	0.992212	0.496106	+	0.868262i	$0.334763\pi$
$908$	−2.69073	−0.0892949
$909$	−0.540046	−0.0179122
$910$	0	0
$911$	−4.70778	−0.155976	−0.0779879	−	0.996954i	$-0.524850\pi$
$911$	−4.70778	−0.155976	−0.0779879	+	0.996954i	$0.524850\pi$
$912$	−5.52507	−0.182953
$913$	1.16594	0.0385870
$914$	−5.83286	−0.192934
$915$	0	0
$916$	−3.33128	−0.110069
$917$	52.4766	1.73293
$918$	2.42727	0.0801119
$919$	49.0844	1.61914	0.809572	−	0.587021i	$-0.199699\pi$
$919$	49.0844	1.61914	0.809572	+	0.587021i	$0.199699\pi$
$920$	0	0
$921$	14.5519	0.479502
$922$	−13.7549	−0.452993
$923$	3.03485	0.0998933
$924$	−6.46559	−0.212702
$925$	0	0
$926$	16.5545	0.544013
$927$	5.16142	0.169523
$928$	−13.0471	−0.428293
$929$	−40.3235	−1.32297	−0.661485	−	0.749958i	$-0.730074\pi$
$929$	−40.3235	−1.32297	−0.661485	+	0.749958i	$0.730074\pi$
$930$	0	0
$931$	119.471	3.91551
$932$	−20.6718	−0.677126
$933$	−11.2035	−0.366786
$934$	−16.2536	−0.531833
$935$	0	0
$936$	2.84109	0.0928640
$937$	11.9997	0.392013	0.196006	−	0.980603i	$-0.437203\pi$
$937$	11.9997	0.392013	0.196006	+	0.980603i	$0.437203\pi$
$938$	40.3192	1.31647
$939$	−21.4370	−0.699571
$940$	0	0
$941$	−50.4797	−1.64559	−0.822795	−	0.568338i	$-0.807587\pi$
$941$	−50.4797	−1.64559	−0.822795	+	0.568338i	$0.807587\pi$
$942$	17.0149	0.554376
$943$	31.7807	1.03492
$944$	−3.90160	−0.126986
$945$	0	0
$946$	2.06284	0.0670686
$947$	50.8704	1.65307	0.826533	−	0.562889i	$-0.190310\pi$
$947$	50.8704	1.65307	0.826533	+	0.562889i	$0.190310\pi$
$948$	12.5485	0.407557
$949$	0.171680	0.00557296
$950$	0	0
$951$	21.0534	0.682704
$952$	−35.9498	−1.16514
$953$	32.0577	1.03845	0.519225	−	0.854637i	$-0.326221\pi$
$953$	32.0577	1.03845	0.519225	+	0.854637i	$0.326221\pi$
$954$	−10.6329	−0.344252
$955$	0	0
$956$	−15.2829	−0.494285
$957$	3.16557	0.102328
$958$	−16.9879	−0.548853
$959$	−15.9500	−0.515052
$960$	0	0
$961$	−29.3086	−0.945438
$962$	9.89318	0.318969
$963$	−9.22424	−0.297247
$964$	28.7352	0.925500
$965$	0	0
$966$	−13.8274	−0.444889
$967$	−28.7578	−0.924789	−0.462394	−	0.886674i	$-0.653010\pi$
$967$	−28.7578	−0.924789	−0.462394	+	0.886674i	$0.653010\pi$
$968$	−27.9228	−0.897473
$969$	−18.1825	−0.584105
$970$	0	0
$971$	15.8385	0.508280	0.254140	−	0.967167i	$-0.418208\pi$
$971$	15.8385	0.508280	0.254140	+	0.967167i	$0.418208\pi$
$972$	−1.05904	−0.0339687
$973$	39.4278	1.26400
$974$	−39.2195	−1.25667
$975$	0	0
$976$	0.588097	0.0188245
$977$	4.99133	0.159687	0.0798434	−	0.996807i	$-0.474558\pi$
$977$	4.99133	0.159687	0.0798434	+	0.996807i	$0.474558\pi$
$978$	12.5481	0.401245
$979$	−1.26097	−0.0403006
$980$	0	0
$981$	0.907932	0.0289880
$982$	−41.6545	−1.32925
$983$	−41.2405	−1.31537	−0.657683	−	0.753295i	$-0.728463\pi$
$983$	−41.2405	−1.31537	−0.657683	+	0.753295i	$0.728463\pi$
$984$	32.0312	1.02112
$985$	0	0
$986$	6.09351	0.194057
$987$	38.5193	1.22608
$988$	−7.36796	−0.234406
$989$	−4.96521	−0.157885
$990$	0	0
$991$	29.2806	0.930128	0.465064	−	0.885277i	$-0.346031\pi$
$991$	29.2806	0.930128	0.465064	+	0.885277i	$0.346031\pi$
$992$	6.75913	0.214602
$993$	29.5188	0.936750
$994$	−14.8868	−0.472181
$995$	0	0
$996$	−0.979231	−0.0310281
$997$	−25.6186	−0.811348	−0.405674	−	0.914018i	$-0.632963\pi$
$997$	−25.6186	−0.811348	−0.405674	+	0.914018i	$0.632963\pi$
$998$	−34.3986	−1.08887
$999$	−10.6521	−0.337018

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6675.2.a.z.1.4		10
5.4	even	2		1335.2.a.j.1.7	✓	10
15.14	odd	2		4005.2.a.s.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.7	✓	10	5.4	even	2
4005.2.a.s.1.4		10	15.14	odd	2
6675.2.a.z.1.4		10	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:	\( N \)	\(=\)	\( 6675 = 3 \cdot 5^{2} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6675.a (trivial)

\(f(q)\)	\(=\)	\(q-0.970031 q^{2} +1.00000 q^{3} -1.05904 q^{4} -0.970031 q^{6} +4.84165 q^{7} +2.96736 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.970031 q^{2} +1.00000 q^{3} -1.05904 q^{4} -0.970031 q^{6} +4.84165 q^{7} +2.96736 q^{8} +1.00000 q^{9} +1.26097 q^{11} -1.05904 q^{12} +0.957447 q^{13} -4.69655 q^{14} -0.760356 q^{16} -2.50226 q^{17} -0.970031 q^{18} +7.26642 q^{19} +4.84165 q^{21} -1.22318 q^{22} +2.94416 q^{23} +2.96736 q^{24} -0.928753 q^{26} +1.00000 q^{27} -5.12750 q^{28} +2.51044 q^{29} -1.30054 q^{31} -5.19716 q^{32} +1.26097 q^{33} +2.42727 q^{34} -1.05904 q^{36} -10.6521 q^{37} -7.04865 q^{38} +0.957447 q^{39} +10.7945 q^{41} -4.69655 q^{42} -1.68646 q^{43} -1.33541 q^{44} -2.85593 q^{46} +7.95583 q^{47} -0.760356 q^{48} +16.4416 q^{49} -2.50226 q^{51} -1.01397 q^{52} +10.9614 q^{53} -0.970031 q^{54} +14.3669 q^{56} +7.26642 q^{57} -2.43520 q^{58} +5.13128 q^{59} -0.773449 q^{61} +1.26157 q^{62} +4.84165 q^{63} +6.56212 q^{64} -1.22318 q^{66} -8.58486 q^{67} +2.64999 q^{68} +2.94416 q^{69} +3.16973 q^{71} +2.96736 q^{72} +0.179310 q^{73} +10.3329 q^{74} -7.69543 q^{76} +6.10515 q^{77} -0.928753 q^{78} -11.8490 q^{79} +1.00000 q^{81} -10.4710 q^{82} +0.924640 q^{83} -5.12750 q^{84} +1.63592 q^{86} +2.51044 q^{87} +3.74174 q^{88} -1.00000 q^{89} +4.63562 q^{91} -3.11798 q^{92} -1.30054 q^{93} -7.71741 q^{94} -5.19716 q^{96} -11.3807 q^{97} -15.9488 q^{98} +1.26097 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 10 q^{3} + 13 q^{4} - q^{6} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - q^2 + 10 * q^3 + 13 * q^4 - q^6 + q^7 + 10 * q^9	\( 10 q - q^{2} + 10 q^{3} + 13 q^{4} - q^{6} + q^{7} + 10 q^{9} + 10 q^{11} + 13 q^{12} - 5 q^{13} + 13 q^{14} + 19 q^{16} - 9 q^{17} - q^{18} - 6 q^{19} + q^{21} + 3 q^{23} + 14 q^{26} + 10 q^{27} + 18 q^{28} + 38 q^{29} + 2 q^{31} - 16 q^{32} + 10 q^{33} - 8 q^{34} + 13 q^{36} - 9 q^{37} - 20 q^{38} - 5 q^{39} + 36 q^{41} + 13 q^{42} + 7 q^{43} + 16 q^{44} + 2 q^{46} + 23 q^{47} + 19 q^{48} + 25 q^{49} - 9 q^{51} - 13 q^{52} - 27 q^{53} - q^{54} + 41 q^{56} - 6 q^{57} + 32 q^{58} + 20 q^{59} + 30 q^{61} + 2 q^{62} + q^{63} - 2 q^{64} + 5 q^{67} + 10 q^{68} + 3 q^{69} + 24 q^{71} + 19 q^{73} + 42 q^{74} - 30 q^{76} - 18 q^{77} + 14 q^{78} + 12 q^{79} + 10 q^{81} - 29 q^{82} - 3 q^{83} + 18 q^{84} + 38 q^{86} + 38 q^{87} + 16 q^{88} - 10 q^{89} - 6 q^{91} - 32 q^{92} + 2 q^{93} + 17 q^{94} - 16 q^{96} - 3 q^{97} + 37 q^{98} + 10 q^{99}+O(q^{100}) \) 10 * q - q^2 + 10 * q^3 + 13 * q^4 - q^6 + q^7 + 10 * q^9 + 10 * q^11 + 13 * q^12 - 5 * q^13 + 13 * q^14 + 19 * q^16 - 9 * q^17 - q^18 - 6 * q^19 + q^21 + 3 * q^23 + 14 * q^26 + 10 * q^27 + 18 * q^28 + 38 * q^29 + 2 * q^31 - 16 * q^32 + 10 * q^33 - 8 * q^34 + 13 * q^36 - 9 * q^37 - 20 * q^38 - 5 * q^39 + 36 * q^41 + 13 * q^42 + 7 * q^43 + 16 * q^44 + 2 * q^46 + 23 * q^47 + 19 * q^48 + 25 * q^49 - 9 * q^51 - 13 * q^52 - 27 * q^53 - q^54 + 41 * q^56 - 6 * q^57 + 32 * q^58 + 20 * q^59 + 30 * q^61 + 2 * q^62 + q^63 - 2 * q^64 + 5 * q^67 + 10 * q^68 + 3 * q^69 + 24 * q^71 + 19 * q^73 + 42 * q^74 - 30 * q^76 - 18 * q^77 + 14 * q^78 + 12 * q^79 + 10 * q^81 - 29 * q^82 - 3 * q^83 + 18 * q^84 + 38 * q^86 + 38 * q^87 + 16 * q^88 - 10 * q^89 - 6 * q^91 - 32 * q^92 + 2 * q^93 + 17 * q^94 - 16 * q^96 - 3 * q^97 + 37 * q^98 + 10 * q^99

Introduction

L-functions

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