LMFDB - Embedded newform 3675.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +3.47214 q^{11} -1.85410 q^{12} +5.23607 q^{13} +3.14590 q^{16} +5.70820 q^{17} -0.381966 q^{18} -1.23607 q^{19} -1.32624 q^{22} -5.00000 q^{23} +1.47214 q^{24} -2.00000 q^{26} +1.00000 q^{27} -8.70820 q^{29} +4.47214 q^{31} -4.14590 q^{32} +3.47214 q^{33} -2.18034 q^{34} -1.85410 q^{36} +3.47214 q^{37} +0.472136 q^{38} +5.23607 q^{39} +8.00000 q^{41} -3.76393 q^{43} -6.43769 q^{44} +1.90983 q^{46} +2.76393 q^{47} +3.14590 q^{48} +5.70820 q^{51} -9.70820 q^{52} +8.47214 q^{53} -0.381966 q^{54} -1.23607 q^{57} +3.32624 q^{58} +5.23607 q^{59} -11.4164 q^{61} -1.70820 q^{62} -4.70820 q^{64} -1.32624 q^{66} -10.7082 q^{67} -10.5836 q^{68} -5.00000 q^{69} -9.47214 q^{71} +1.47214 q^{72} -3.23607 q^{73} -1.32624 q^{74} +2.29180 q^{76} -2.00000 q^{78} +6.23607 q^{79} +1.00000 q^{81} -3.05573 q^{82} +3.52786 q^{83} +1.43769 q^{86} -8.70820 q^{87} +5.11146 q^{88} -7.70820 q^{89} +9.27051 q^{92} +4.47214 q^{93} -1.05573 q^{94} -4.14590 q^{96} +3.52786 q^{97} +3.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{11} + 3 q^{12} + 6 q^{13} + 13 q^{16} - 2 q^{17} - 3 q^{18} + 2 q^{19} + 13 q^{22} - 10 q^{23} - 6 q^{24} - 4 q^{26} + 2 q^{27} - 4 q^{29} - 15 q^{32} - 2 q^{33} + 18 q^{34} + 3 q^{36} - 2 q^{37} - 8 q^{38} + 6 q^{39} + 16 q^{41} - 12 q^{43} - 33 q^{44} + 15 q^{46} + 10 q^{47} + 13 q^{48} - 2 q^{51} - 6 q^{52} + 8 q^{53} - 3 q^{54} + 2 q^{57} - 9 q^{58} + 6 q^{59} + 4 q^{61} + 10 q^{62} + 4 q^{64} + 13 q^{66} - 8 q^{67} - 48 q^{68} - 10 q^{69} - 10 q^{71} - 6 q^{72} - 2 q^{73} + 13 q^{74} + 18 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} - 24 q^{82} + 16 q^{83} + 23 q^{86} - 4 q^{87} + 46 q^{88} - 2 q^{89} - 15 q^{92} - 20 q^{94} - 15 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 - 2 * q^11 + 3 * q^12 + 6 * q^13 + 13 * q^16 - 2 * q^17 - 3 * q^18 + 2 * q^19 + 13 * q^22 - 10 * q^23 - 6 * q^24 - 4 * q^26 + 2 * q^27 - 4 * q^29 - 15 * q^32 - 2 * q^33 + 18 * q^34 + 3 * q^36 - 2 * q^37 - 8 * q^38 + 6 * q^39 + 16 * q^41 - 12 * q^43 - 33 * q^44 + 15 * q^46 + 10 * q^47 + 13 * q^48 - 2 * q^51 - 6 * q^52 + 8 * q^53 - 3 * q^54 + 2 * q^57 - 9 * q^58 + 6 * q^59 + 4 * q^61 + 10 * q^62 + 4 * q^64 + 13 * q^66 - 8 * q^67 - 48 * q^68 - 10 * q^69 - 10 * q^71 - 6 * q^72 - 2 * q^73 + 13 * q^74 + 18 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 - 24 * q^82 + 16 * q^83 + 23 * q^86 - 4 * q^87 + 46 * q^88 - 2 * q^89 - 15 * q^92 - 20 * q^94 - 15 * q^96 + 16 * q^97 - 2 * 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 $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.85410	−0.927051
$5$	0	0
$5$	0	0
$6$	−0.381966	−0.155937
$7$	0	0
$7$	0	0
$8$	1.47214	0.520479
$9$	1.00000	0.333333
$10$	0	0
$11$	3.47214	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$11$	3.47214	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$12$	−1.85410	−0.535233
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	0	0
$15$	0	0
$16$	3.14590	0.786475
$17$	5.70820	1.38444	0.692221	−	0.721685i	$-0.256632\pi$
$17$	5.70820	1.38444	0.692221	+	0.721685i	$0.256632\pi$
$18$	−0.381966	−0.0900303
$19$	−1.23607	−0.283573	−0.141787	−	0.989897i	$-0.545285\pi$
$19$	−1.23607	−0.283573	−0.141787	+	0.989897i	$0.545285\pi$
$20$	0	0
$21$	0	0
$22$	−1.32624	−0.282755
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	1.47214	0.300498
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.70820	−1.61707	−0.808536	−	0.588446i	$-0.799740\pi$
$29$	−8.70820	−1.61707	−0.808536	+	0.588446i	$0.799740\pi$
$30$	0	0
$31$	4.47214	0.803219	0.401610	−	0.915811i	$-0.368451\pi$
$31$	4.47214	0.803219	0.401610	+	0.915811i	$0.368451\pi$
$32$	−4.14590	−0.732898
$33$	3.47214	0.604421
$34$	−2.18034	−0.373925
$35$	0	0
$36$	−1.85410	−0.309017
$37$	3.47214	0.570816	0.285408	−	0.958406i	$-0.407871\pi$
$37$	3.47214	0.570816	0.285408	+	0.958406i	$0.407871\pi$
$38$	0.472136	0.0765906
$39$	5.23607	0.838442
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−3.76393	−0.573994	−0.286997	−	0.957931i	$-0.592657\pi$
$43$	−3.76393	−0.573994	−0.286997	+	0.957931i	$0.592657\pi$
$44$	−6.43769	−0.970519
$45$	0	0
$46$	1.90983	0.281589
$47$	2.76393	0.403161	0.201580	−	0.979472i	$-0.435392\pi$
$47$	2.76393	0.403161	0.201580	+	0.979472i	$0.435392\pi$
$48$	3.14590	0.454071
$49$	0	0
$50$	0	0
$51$	5.70820	0.799308
$52$	−9.70820	−1.34629
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	−0.381966	−0.0519790
$55$	0	0
$56$	0	0
$57$	−1.23607	−0.163721
$58$	3.32624	0.436756
$59$	5.23607	0.681678	0.340839	−	0.940122i	$-0.389289\pi$
$59$	5.23607	0.681678	0.340839	+	0.940122i	$0.389289\pi$
$60$	0	0
$61$	−11.4164	−1.46172	−0.730861	−	0.682527i	$-0.760881\pi$
$61$	−11.4164	−1.46172	−0.730861	+	0.682527i	$0.760881\pi$
$62$	−1.70820	−0.216942
$63$	0	0
$64$	−4.70820	−0.588525
$65$	0	0
$66$	−1.32624	−0.163249
$67$	−10.7082	−1.30822	−0.654108	−	0.756401i	$-0.726956\pi$
$67$	−10.7082	−1.30822	−0.654108	+	0.756401i	$0.726956\pi$
$68$	−10.5836	−1.28345
$69$	−5.00000	−0.601929
$70$	0	0
$71$	−9.47214	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$71$	−9.47214	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$72$	1.47214	0.173493
$73$	−3.23607	−0.378753	−0.189377	−	0.981905i	$-0.560647\pi$
$73$	−3.23607	−0.378753	−0.189377	+	0.981905i	$0.560647\pi$
$74$	−1.32624	−0.154172
$75$	0	0
$76$	2.29180	0.262887
$77$	0	0
$78$	−2.00000	−0.226455
$79$	6.23607	0.701612	0.350806	−	0.936448i	$-0.385908\pi$
$79$	6.23607	0.701612	0.350806	+	0.936448i	$0.385908\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.05573	−0.337449
$83$	3.52786	0.387233	0.193617	−	0.981077i	$-0.437978\pi$
$83$	3.52786	0.387233	0.193617	+	0.981077i	$0.437978\pi$
$84$	0	0
$85$	0	0
$86$	1.43769	0.155031
$87$	−8.70820	−0.933617
$88$	5.11146	0.544883
$89$	−7.70820	−0.817068	−0.408534	−	0.912743i	$-0.633960\pi$
$89$	−7.70820	−0.817068	−0.408534	+	0.912743i	$0.633960\pi$
$90$	0	0
$91$	0	0
$92$	9.27051	0.966517
$93$	4.47214	0.463739
$94$	−1.05573	−0.108890
$95$	0	0
$96$	−4.14590	−0.423139
$97$	3.52786	0.358200	0.179100	−	0.983831i	$-0.442681\pi$
$97$	3.52786	0.358200	0.179100	+	0.983831i	$0.442681\pi$
$98$	0	0
$99$	3.47214	0.348963
$100$	0	0
$101$	17.7082	1.76203	0.881016	−	0.473086i	$-0.156860\pi$
$101$	17.7082	1.76203	0.881016	+	0.473086i	$0.156860\pi$
$102$	−2.18034	−0.215886
$103$	−3.70820	−0.365380	−0.182690	−	0.983171i	$-0.558480\pi$
$103$	−3.70820	−0.365380	−0.182690	+	0.983171i	$0.558480\pi$
$104$	7.70820	0.755852
$105$	0	0
$106$	−3.23607	−0.314315
$107$	−12.9443	−1.25137	−0.625685	−	0.780076i	$-0.715180\pi$
$107$	−12.9443	−1.25137	−0.625685	+	0.780076i	$0.715180\pi$
$108$	−1.85410	−0.178411
$109$	20.4164	1.95554	0.977769	−	0.209687i	$-0.0672444\pi$
$109$	20.4164	1.95554	0.977769	+	0.209687i	$0.0672444\pi$
$110$	0	0
$111$	3.47214	0.329561
$112$	0	0
$113$	11.7639	1.10666	0.553329	−	0.832963i	$-0.313357\pi$
$113$	11.7639	1.10666	0.553329	+	0.832963i	$0.313357\pi$
$114$	0.472136	0.0442196
$115$	0	0
$116$	16.1459	1.49911
$117$	5.23607	0.484075
$118$	−2.00000	−0.184115
$119$	0	0
$120$	0	0
$121$	1.05573	0.0959753
$122$	4.36068	0.394797
$123$	8.00000	0.721336
$124$	−8.29180	−0.744625
$125$	0	0
$126$	0	0
$127$	7.76393	0.688938	0.344469	−	0.938798i	$-0.388059\pi$
$127$	7.76393	0.688938	0.344469	+	0.938798i	$0.388059\pi$
$128$	10.0902	0.891853
$129$	−3.76393	−0.331396
$130$	0	0
$131$	17.7082	1.54717	0.773586	−	0.633691i	$-0.218461\pi$
$131$	17.7082	1.54717	0.773586	+	0.633691i	$0.218461\pi$
$132$	−6.43769	−0.560329
$133$	0	0
$134$	4.09017	0.353337
$135$	0	0
$136$	8.40325	0.720573
$137$	−8.47214	−0.723823	−0.361912	−	0.932212i	$-0.617876\pi$
$137$	−8.47214	−0.723823	−0.361912	+	0.932212i	$0.617876\pi$
$138$	1.90983	0.162576
$139$	9.70820	0.823439	0.411720	−	0.911311i	$-0.364928\pi$
$139$	9.70820	0.823439	0.411720	+	0.911311i	$0.364928\pi$
$140$	0	0
$141$	2.76393	0.232765
$142$	3.61803	0.303619
$143$	18.1803	1.52032
$144$	3.14590	0.262158
$145$	0	0
$146$	1.23607	0.102298
$147$	0	0
$148$	−6.43769	−0.529175
$149$	3.76393	0.308353	0.154177	−	0.988043i	$-0.450728\pi$
$149$	3.76393	0.308353	0.154177	+	0.988043i	$0.450728\pi$
$150$	0	0
$151$	14.7082	1.19694	0.598468	−	0.801146i	$-0.295776\pi$
$151$	14.7082	1.19694	0.598468	+	0.801146i	$0.295776\pi$
$152$	−1.81966	−0.147594
$153$	5.70820	0.461481
$154$	0	0
$155$	0	0
$156$	−9.70820	−0.777278
$157$	−0.944272	−0.0753611	−0.0376806	−	0.999290i	$-0.511997\pi$
$157$	−0.944272	−0.0753611	−0.0376806	+	0.999290i	$0.511997\pi$
$158$	−2.38197	−0.189499
$159$	8.47214	0.671884
$160$	0	0
$161$	0	0
$162$	−0.381966	−0.0300101
$163$	−9.52786	−0.746280	−0.373140	−	0.927775i	$-0.621719\pi$
$163$	−9.52786	−0.746280	−0.373140	+	0.927775i	$0.621719\pi$
$164$	−14.8328	−1.15825
$165$	0	0
$166$	−1.34752	−0.104588
$167$	−19.7082	−1.52507	−0.762533	−	0.646949i	$-0.776045\pi$
$167$	−19.7082	−1.52507	−0.762533	+	0.646949i	$0.776045\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	−1.23607	−0.0945245
$172$	6.97871	0.532122
$173$	−26.1803	−1.99045	−0.995227	−	0.0975850i	$-0.968888\pi$
$173$	−26.1803	−1.99045	−0.995227	+	0.0975850i	$0.968888\pi$
$174$	3.32624	0.252161
$175$	0	0
$176$	10.9230	0.823351
$177$	5.23607	0.393567
$178$	2.94427	0.220683
$179$	−12.9443	−0.967500	−0.483750	−	0.875206i	$-0.660726\pi$
$179$	−12.9443	−0.967500	−0.483750	+	0.875206i	$0.660726\pi$
$180$	0	0
$181$	26.6525	1.98106	0.990531	−	0.137286i	$-0.0438379\pi$
$181$	26.6525	1.98106	0.990531	+	0.137286i	$0.0438379\pi$
$182$	0	0
$183$	−11.4164	−0.843925
$184$	−7.36068	−0.542637
$185$	0	0
$186$	−1.70820	−0.125252
$187$	19.8197	1.44936
$188$	−5.12461	−0.373751
$189$	0	0
$190$	0	0
$191$	15.4164	1.11549	0.557746	−	0.830012i	$-0.311666\pi$
$191$	15.4164	1.11549	0.557746	+	0.830012i	$0.311666\pi$
$192$	−4.70820	−0.339785
$193$	26.4164	1.90149	0.950747	−	0.309967i	$-0.100318\pi$
$193$	26.4164	1.90149	0.950747	+	0.309967i	$0.100318\pi$
$194$	−1.34752	−0.0967466
$195$	0	0
$196$	0	0
$197$	10.2361	0.729290	0.364645	−	0.931147i	$-0.381190\pi$
$197$	10.2361	0.729290	0.364645	+	0.931147i	$0.381190\pi$
$198$	−1.32624	−0.0942516
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−10.7082	−0.755298
$202$	−6.76393	−0.475909
$203$	0	0
$204$	−10.5836	−0.741000
$205$	0	0
$206$	1.41641	0.0986858
$207$	−5.00000	−0.347524
$208$	16.4721	1.14214
$209$	−4.29180	−0.296870
$210$	0	0
$211$	−4.94427	−0.340378	−0.170189	−	0.985411i	$-0.554438\pi$
$211$	−4.94427	−0.340378	−0.170189	+	0.985411i	$0.554438\pi$
$212$	−15.7082	−1.07884
$213$	−9.47214	−0.649020
$214$	4.94427	0.337983
$215$	0	0
$216$	1.47214	0.100166
$217$	0	0
$218$	−7.79837	−0.528173
$219$	−3.23607	−0.218673
$220$	0	0
$221$	29.8885	2.01052
$222$	−1.32624	−0.0890113
$223$	3.23607	0.216703	0.108352	−	0.994113i	$-0.465443\pi$
$223$	3.23607	0.216703	0.108352	+	0.994113i	$0.465443\pi$
$224$	0	0
$225$	0	0
$226$	−4.49342	−0.298898
$227$	−9.23607	−0.613019	−0.306510	−	0.951868i	$-0.599161\pi$
$227$	−9.23607	−0.613019	−0.306510	+	0.951868i	$0.599161\pi$
$228$	2.29180	0.151778
$229$	22.3607	1.47764	0.738818	−	0.673905i	$-0.235384\pi$
$229$	22.3607	1.47764	0.738818	+	0.673905i	$0.235384\pi$
$230$	0	0
$231$	0	0
$232$	−12.8197	−0.841652
$233$	−20.7082	−1.35664	−0.678320	−	0.734767i	$-0.737292\pi$
$233$	−20.7082	−1.35664	−0.678320	+	0.734767i	$0.737292\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−9.70820	−0.631950
$237$	6.23607	0.405076
$238$	0	0
$239$	16.3607	1.05828	0.529142	−	0.848533i	$-0.322514\pi$
$239$	16.3607	1.05828	0.529142	+	0.848533i	$0.322514\pi$
$240$	0	0
$241$	−14.7639	−0.951028	−0.475514	−	0.879708i	$-0.657738\pi$
$241$	−14.7639	−0.951028	−0.475514	+	0.879708i	$0.657738\pi$
$242$	−0.403252	−0.0259220
$243$	1.00000	0.0641500
$244$	21.1672	1.35509
$245$	0	0
$246$	−3.05573	−0.194826
$247$	−6.47214	−0.411812
$248$	6.58359	0.418059
$249$	3.52786	0.223569
$250$	0	0
$251$	−7.23607	−0.456737	−0.228368	−	0.973575i	$-0.573339\pi$
$251$	−7.23607	−0.456737	−0.228368	+	0.973575i	$0.573339\pi$
$252$	0	0
$253$	−17.3607	−1.09146
$254$	−2.96556	−0.186076
$255$	0	0
$256$	5.56231	0.347644
$257$	0.472136	0.0294510	0.0147255	−	0.999892i	$-0.495313\pi$
$257$	0.472136	0.0294510	0.0147255	+	0.999892i	$0.495313\pi$
$258$	1.43769	0.0895069
$259$	0	0
$260$	0	0
$261$	−8.70820	−0.539024
$262$	−6.76393	−0.417877
$263$	−19.9443	−1.22982	−0.614908	−	0.788599i	$-0.710807\pi$
$263$	−19.9443	−1.22982	−0.614908	+	0.788599i	$0.710807\pi$
$264$	5.11146	0.314588
$265$	0	0
$266$	0	0
$267$	−7.70820	−0.471734
$268$	19.8541	1.21278
$269$	19.5279	1.19063	0.595317	−	0.803491i	$-0.297026\pi$
$269$	19.5279	1.19063	0.595317	+	0.803491i	$0.297026\pi$
$270$	0	0
$271$	−18.7639	−1.13983	−0.569914	−	0.821704i	$-0.693023\pi$
$271$	−18.7639	−1.13983	−0.569914	+	0.821704i	$0.693023\pi$
$272$	17.9574	1.08883
$273$	0	0
$274$	3.23607	0.195498
$275$	0	0
$276$	9.27051	0.558019
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	−3.70820	−0.222403
$279$	4.47214	0.267740
$280$	0	0
$281$	−17.6525	−1.05306	−0.526529	−	0.850157i	$-0.676507\pi$
$281$	−17.6525	−1.05306	−0.526529	+	0.850157i	$0.676507\pi$
$282$	−1.05573	−0.0628677
$283$	13.4164	0.797523	0.398761	−	0.917055i	$-0.369440\pi$
$283$	13.4164	0.797523	0.398761	+	0.917055i	$0.369440\pi$
$284$	17.5623	1.04213
$285$	0	0
$286$	−6.94427	−0.410623
$287$	0	0
$288$	−4.14590	−0.244299
$289$	15.5836	0.916682
$290$	0	0
$291$	3.52786	0.206807
$292$	6.00000	0.351123
$293$	4.65248	0.271801	0.135900	−	0.990723i	$-0.456607\pi$
$293$	4.65248	0.271801	0.135900	+	0.990723i	$0.456607\pi$
$294$	0	0
$295$	0	0
$296$	5.11146	0.297097
$297$	3.47214	0.201474
$298$	−1.43769	−0.0832834
$299$	−26.1803	−1.51405
$300$	0	0
$301$	0	0
$302$	−5.61803	−0.323282
$303$	17.7082	1.01731
$304$	−3.88854	−0.223023
$305$	0	0
$306$	−2.18034	−0.124642
$307$	−6.65248	−0.379677	−0.189838	−	0.981815i	$-0.560796\pi$
$307$	−6.65248	−0.379677	−0.189838	+	0.981815i	$0.560796\pi$
$308$	0	0
$309$	−3.70820	−0.210952
$310$	0	0
$311$	−8.76393	−0.496957	−0.248478	−	0.968637i	$-0.579931\pi$
$311$	−8.76393	−0.496957	−0.248478	+	0.968637i	$0.579931\pi$
$312$	7.70820	0.436391
$313$	7.70820	0.435693	0.217847	−	0.975983i	$-0.430097\pi$
$313$	7.70820	0.435693	0.217847	+	0.975983i	$0.430097\pi$
$314$	0.360680	0.0203543
$315$	0	0
$316$	−11.5623	−0.650431
$317$	13.6525	0.766799	0.383400	−	0.923583i	$-0.374753\pi$
$317$	13.6525	0.766799	0.383400	+	0.923583i	$0.374753\pi$
$318$	−3.23607	−0.181470
$319$	−30.2361	−1.69289
$320$	0	0
$321$	−12.9443	−0.722479
$322$	0	0
$323$	−7.05573	−0.392591
$324$	−1.85410	−0.103006
$325$	0	0
$326$	3.63932	0.201563
$327$	20.4164	1.12903
$328$	11.7771	0.650281
$329$	0	0
$330$	0	0
$331$	18.2361	1.00234	0.501172	−	0.865347i	$-0.332902\pi$
$331$	18.2361	1.00234	0.501172	+	0.865347i	$0.332902\pi$
$332$	−6.54102	−0.358985
$333$	3.47214	0.190272
$334$	7.52786	0.411906
$335$	0	0
$336$	0	0
$337$	19.5279	1.06375	0.531875	−	0.846823i	$-0.321488\pi$
$337$	19.5279	1.06375	0.531875	+	0.846823i	$0.321488\pi$
$338$	−5.50658	−0.299518
$339$	11.7639	0.638929
$340$	0	0
$341$	15.5279	0.840881
$342$	0.472136	0.0255302
$343$	0	0
$344$	−5.54102	−0.298752
$345$	0	0
$346$	10.0000	0.537603
$347$	13.0000	0.697877	0.348938	−	0.937146i	$-0.386542\pi$
$347$	13.0000	0.697877	0.348938	+	0.937146i	$0.386542\pi$
$348$	16.1459	0.865511
$349$	−5.23607	−0.280280	−0.140140	−	0.990132i	$-0.544755\pi$
$349$	−5.23607	−0.280280	−0.140140	+	0.990132i	$0.544755\pi$
$350$	0	0
$351$	5.23607	0.279481
$352$	−14.3951	−0.767263
$353$	18.9443	1.00830	0.504151	−	0.863616i	$-0.331806\pi$
$353$	18.9443	1.00830	0.504151	+	0.863616i	$0.331806\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	14.2918	0.757464
$357$	0	0
$358$	4.94427	0.261313
$359$	8.05573	0.425165	0.212583	−	0.977143i	$-0.431813\pi$
$359$	8.05573	0.425165	0.212583	+	0.977143i	$0.431813\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	−10.1803	−0.535067
$363$	1.05573	0.0554114
$364$	0	0
$365$	0	0
$366$	4.36068	0.227936
$367$	8.94427	0.466887	0.233444	−	0.972370i	$-0.425001\pi$
$367$	8.94427	0.466887	0.233444	+	0.972370i	$0.425001\pi$
$368$	−15.7295	−0.819956
$369$	8.00000	0.416463
$370$	0	0
$371$	0	0
$372$	−8.29180	−0.429910
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	−7.57044	−0.391458
$375$	0	0
$376$	4.06888	0.209837
$377$	−45.5967	−2.34835
$378$	0	0
$379$	−32.5967	−1.67438	−0.837191	−	0.546910i	$-0.815804\pi$
$379$	−32.5967	−1.67438	−0.837191	+	0.546910i	$0.815804\pi$
$380$	0	0
$381$	7.76393	0.397758
$382$	−5.88854	−0.301284
$383$	−15.1246	−0.772832	−0.386416	−	0.922325i	$-0.626287\pi$
$383$	−15.1246	−0.772832	−0.386416	+	0.922325i	$0.626287\pi$
$384$	10.0902	0.514912
$385$	0	0
$386$	−10.0902	−0.513576
$387$	−3.76393	−0.191331
$388$	−6.54102	−0.332070
$389$	−6.23607	−0.316181	−0.158091	−	0.987425i	$-0.550534\pi$
$389$	−6.23607	−0.316181	−0.158091	+	0.987425i	$0.550534\pi$
$390$	0	0
$391$	−28.5410	−1.44338
$392$	0	0
$393$	17.7082	0.893261
$394$	−3.90983	−0.196974
$395$	0	0
$396$	−6.43769	−0.323506
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−6.11146	−0.306340
$399$	0	0
$400$	0	0
$401$	−13.2918	−0.663761	−0.331880	−	0.943322i	$-0.607683\pi$
$401$	−13.2918	−0.663761	−0.331880	+	0.943322i	$0.607683\pi$
$402$	4.09017	0.203999
$403$	23.4164	1.16645
$404$	−32.8328	−1.63349
$405$	0	0
$406$	0	0
$407$	12.0557	0.597580
$408$	8.40325	0.416023
$409$	−14.1803	−0.701173	−0.350586	−	0.936530i	$-0.614018\pi$
$409$	−14.1803	−0.701173	−0.350586	+	0.936530i	$0.614018\pi$
$410$	0	0
$411$	−8.47214	−0.417900
$412$	6.87539	0.338726
$413$	0	0
$414$	1.90983	0.0938630
$415$	0	0
$416$	−21.7082	−1.06433
$417$	9.70820	0.475413
$418$	1.63932	0.0801818
$419$	−26.9443	−1.31631	−0.658157	−	0.752881i	$-0.728664\pi$
$419$	−26.9443	−1.31631	−0.658157	+	0.752881i	$0.728664\pi$
$420$	0	0
$421$	26.4164	1.28746	0.643728	−	0.765254i	$-0.277387\pi$
$421$	26.4164	1.28746	0.643728	+	0.765254i	$0.277387\pi$
$422$	1.88854	0.0919329
$423$	2.76393	0.134387
$424$	12.4721	0.605700
$425$	0	0
$426$	3.61803	0.175294
$427$	0	0
$428$	24.0000	1.16008
$429$	18.1803	0.877755
$430$	0	0
$431$	−1.88854	−0.0909680	−0.0454840	−	0.998965i	$-0.514483\pi$
$431$	−1.88854	−0.0909680	−0.0454840	+	0.998965i	$0.514483\pi$
$432$	3.14590	0.151357
$433$	−25.3050	−1.21608	−0.608039	−	0.793907i	$-0.708044\pi$
$433$	−25.3050	−1.21608	−0.608039	+	0.793907i	$0.708044\pi$
$434$	0	0
$435$	0	0
$436$	−37.8541	−1.81288
$437$	6.18034	0.295646
$438$	1.23607	0.0590616
$439$	6.76393	0.322825	0.161412	−	0.986887i	$-0.448395\pi$
$439$	6.76393	0.322825	0.161412	+	0.986887i	$0.448395\pi$
$440$	0	0
$441$	0	0
$442$	−11.4164	−0.543023
$443$	−39.4164	−1.87273	−0.936365	−	0.351028i	$-0.885832\pi$
$443$	−39.4164	−1.87273	−0.936365	+	0.351028i	$0.885832\pi$
$444$	−6.43769	−0.305519
$445$	0	0
$446$	−1.23607	−0.0585295
$447$	3.76393	0.178028
$448$	0	0
$449$	16.7082	0.788509	0.394254	−	0.919001i	$-0.371003\pi$
$449$	16.7082	0.788509	0.394254	+	0.919001i	$0.371003\pi$
$450$	0	0
$451$	27.7771	1.30797
$452$	−21.8115	−1.02593
$453$	14.7082	0.691052
$454$	3.52786	0.165571
$455$	0	0
$456$	−1.81966	−0.0852134
$457$	20.5279	0.960253	0.480126	−	0.877199i	$-0.340591\pi$
$457$	20.5279	0.960253	0.480126	+	0.877199i	$0.340591\pi$
$458$	−8.54102	−0.399096
$459$	5.70820	0.266436
$460$	0	0
$461$	20.6525	0.961882	0.480941	−	0.876753i	$-0.340295\pi$
$461$	20.6525	0.961882	0.480941	+	0.876753i	$0.340295\pi$
$462$	0	0
$463$	16.3607	0.760345	0.380173	−	0.924916i	$-0.375865\pi$
$463$	16.3607	0.760345	0.380173	+	0.924916i	$0.375865\pi$
$464$	−27.3951	−1.27179
$465$	0	0
$466$	7.90983	0.366416
$467$	−11.8885	−0.550136	−0.275068	−	0.961425i	$-0.588700\pi$
$467$	−11.8885	−0.550136	−0.275068	+	0.961425i	$0.588700\pi$
$468$	−9.70820	−0.448762
$469$	0	0
$470$	0	0
$471$	−0.944272	−0.0435098
$472$	7.70820	0.354799
$473$	−13.0689	−0.600908
$474$	−2.38197	−0.109407
$475$	0	0
$476$	0	0
$477$	8.47214	0.387912
$478$	−6.24922	−0.285833
$479$	18.7639	0.857346	0.428673	−	0.903460i	$-0.358981\pi$
$479$	18.7639	0.857346	0.428673	+	0.903460i	$0.358981\pi$
$480$	0	0
$481$	18.1803	0.828952
$482$	5.63932	0.256864
$483$	0	0
$484$	−1.95743	−0.0889740
$485$	0	0
$486$	−0.381966	−0.0173263
$487$	29.1803	1.32229	0.661144	−	0.750259i	$-0.270071\pi$
$487$	29.1803	1.32229	0.661144	+	0.750259i	$0.270071\pi$
$488$	−16.8065	−0.760795
$489$	−9.52786	−0.430865
$490$	0	0
$491$	−9.47214	−0.427472	−0.213736	−	0.976892i	$-0.568563\pi$
$491$	−9.47214	−0.427472	−0.213736	+	0.976892i	$0.568563\pi$
$492$	−14.8328	−0.668715
$493$	−49.7082	−2.23874
$494$	2.47214	0.111227
$495$	0	0
$496$	14.0689	0.631712
$497$	0	0
$498$	−1.34752	−0.0603840
$499$	39.7771	1.78067	0.890333	−	0.455309i	$-0.150471\pi$
$499$	39.7771	1.78067	0.890333	+	0.455309i	$0.150471\pi$
$500$	0	0
$501$	−19.7082	−0.880498
$502$	2.76393	0.123360
$503$	−9.41641	−0.419857	−0.209928	−	0.977717i	$-0.567323\pi$
$503$	−9.41641	−0.419857	−0.209928	+	0.977717i	$0.567323\pi$
$504$	0	0
$505$	0	0
$506$	6.63119	0.294792
$507$	14.4164	0.640255
$508$	−14.3951	−0.638680
$509$	−0.652476	−0.0289205	−0.0144602	−	0.999895i	$-0.504603\pi$
$509$	−0.652476	−0.0289205	−0.0144602	+	0.999895i	$0.504603\pi$
$510$	0	0
$511$	0	0
$512$	−22.3050	−0.985749
$513$	−1.23607	−0.0545737
$514$	−0.180340	−0.00795445
$515$	0	0
$516$	6.97871	0.307221
$517$	9.59675	0.422064
$518$	0	0
$519$	−26.1803	−1.14919
$520$	0	0
$521$	−33.7771	−1.47980	−0.739901	−	0.672716i	$-0.765127\pi$
$521$	−33.7771	−1.47980	−0.739901	+	0.672716i	$0.765127\pi$
$522$	3.32624	0.145585
$523$	35.7771	1.56442	0.782211	−	0.623013i	$-0.214092\pi$
$523$	35.7771	1.56442	0.782211	+	0.623013i	$0.214092\pi$
$524$	−32.8328	−1.43431
$525$	0	0
$526$	7.61803	0.332162
$527$	25.5279	1.11201
$528$	10.9230	0.475362
$529$	2.00000	0.0869565
$530$	0	0
$531$	5.23607	0.227226
$532$	0	0
$533$	41.8885	1.81439
$534$	2.94427	0.127411
$535$	0	0
$536$	−15.7639	−0.680898
$537$	−12.9443	−0.558587
$538$	−7.45898	−0.321579
$539$	0	0
$540$	0	0
$541$	−0.0557281	−0.00239594	−0.00119797	−	0.999999i	$-0.500381\pi$
$541$	−0.0557281	−0.00239594	−0.00119797	+	0.999999i	$0.500381\pi$
$542$	7.16718	0.307857
$543$	26.6525	1.14377
$544$	−23.6656	−1.01466
$545$	0	0
$546$	0	0
$547$	−23.0689	−0.986354	−0.493177	−	0.869929i	$-0.664165\pi$
$547$	−23.0689	−0.986354	−0.493177	+	0.869929i	$0.664165\pi$
$548$	15.7082	0.671021
$549$	−11.4164	−0.487240
$550$	0	0
$551$	10.7639	0.458559
$552$	−7.36068	−0.313291
$553$	0	0
$554$	7.59675	0.322755
$555$	0	0
$556$	−18.0000	−0.763370
$557$	−23.6525	−1.00219	−0.501094	−	0.865393i	$-0.667069\pi$
$557$	−23.6525	−1.00219	−0.501094	+	0.865393i	$0.667069\pi$
$558$	−1.70820	−0.0723140
$559$	−19.7082	−0.833568
$560$	0	0
$561$	19.8197	0.836787
$562$	6.74265	0.284421
$563$	−27.3050	−1.15077	−0.575383	−	0.817884i	$-0.695147\pi$
$563$	−27.3050	−1.15077	−0.575383	+	0.817884i	$0.695147\pi$
$564$	−5.12461	−0.215785
$565$	0	0
$566$	−5.12461	−0.215404
$567$	0	0
$568$	−13.9443	−0.585089
$569$	−7.76393	−0.325481	−0.162740	−	0.986669i	$-0.552033\pi$
$569$	−7.76393	−0.325481	−0.162740	+	0.986669i	$0.552033\pi$
$570$	0	0
$571$	−27.2918	−1.14213	−0.571063	−	0.820906i	$-0.693469\pi$
$571$	−27.2918	−1.14213	−0.571063	+	0.820906i	$0.693469\pi$
$572$	−33.7082	−1.40941
$573$	15.4164	0.644030
$574$	0	0
$575$	0	0
$576$	−4.70820	−0.196175
$577$	6.76393	0.281586	0.140793	−	0.990039i	$-0.455035\pi$
$577$	6.76393	0.281586	0.140793	+	0.990039i	$0.455035\pi$
$578$	−5.95240	−0.247587
$579$	26.4164	1.09783
$580$	0	0
$581$	0	0
$582$	−1.34752	−0.0558567
$583$	29.4164	1.21830
$584$	−4.76393	−0.197133
$585$	0	0
$586$	−1.77709	−0.0734108
$587$	33.1246	1.36720	0.683600	−	0.729857i	$-0.260414\pi$
$587$	33.1246	1.36720	0.683600	+	0.729857i	$0.260414\pi$
$588$	0	0
$589$	−5.52786	−0.227772
$590$	0	0
$591$	10.2361	0.421056
$592$	10.9230	0.448932
$593$	−37.3050	−1.53193	−0.765965	−	0.642882i	$-0.777739\pi$
$593$	−37.3050	−1.53193	−0.765965	+	0.642882i	$0.777739\pi$
$594$	−1.32624	−0.0544162
$595$	0	0
$596$	−6.97871	−0.285859
$597$	16.0000	0.654836
$598$	10.0000	0.408930
$599$	−28.0557	−1.14633	−0.573163	−	0.819441i	$-0.694284\pi$
$599$	−28.0557	−1.14633	−0.573163	+	0.819441i	$0.694284\pi$
$600$	0	0
$601$	−16.3607	−0.667366	−0.333683	−	0.942685i	$-0.608292\pi$
$601$	−16.3607	−0.667366	−0.333683	+	0.942685i	$0.608292\pi$
$602$	0	0
$603$	−10.7082	−0.436072
$604$	−27.2705	−1.10962
$605$	0	0
$606$	−6.76393	−0.274766
$607$	15.7082	0.637576	0.318788	−	0.947826i	$-0.396724\pi$
$607$	15.7082	0.637576	0.318788	+	0.947826i	$0.396724\pi$
$608$	5.12461	0.207830
$609$	0	0
$610$	0	0
$611$	14.4721	0.585480
$612$	−10.5836	−0.427816
$613$	−27.9443	−1.12866	−0.564329	−	0.825550i	$-0.690865\pi$
$613$	−27.9443	−1.12866	−0.564329	+	0.825550i	$0.690865\pi$
$614$	2.54102	0.102547
$615$	0	0
$616$	0	0
$617$	−9.65248	−0.388594	−0.194297	−	0.980943i	$-0.562243\pi$
$617$	−9.65248	−0.388594	−0.194297	+	0.980943i	$0.562243\pi$
$618$	1.41641	0.0569763
$619$	15.1246	0.607909	0.303955	−	0.952686i	$-0.401693\pi$
$619$	15.1246	0.607909	0.303955	+	0.952686i	$0.401693\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	3.34752	0.134223
$623$	0	0
$624$	16.4721	0.659413
$625$	0	0
$626$	−2.94427	−0.117677
$627$	−4.29180	−0.171398
$628$	1.75078	0.0698636
$629$	19.8197	0.790262
$630$	0	0
$631$	0.124612	0.00496072	0.00248036	−	0.999997i	$-0.499210\pi$
$631$	0.124612	0.00496072	0.00248036	+	0.999997i	$0.499210\pi$
$632$	9.18034	0.365174
$633$	−4.94427	−0.196517
$634$	−5.21478	−0.207105
$635$	0	0
$636$	−15.7082	−0.622871
$637$	0	0
$638$	11.5492	0.457235
$639$	−9.47214	−0.374712
$640$	0	0
$641$	−28.7082	−1.13391	−0.566953	−	0.823750i	$-0.691878\pi$
$641$	−28.7082	−1.13391	−0.566953	+	0.823750i	$0.691878\pi$
$642$	4.94427	0.195135
$643$	44.7214	1.76364	0.881819	−	0.471588i	$-0.156319\pi$
$643$	44.7214	1.76364	0.881819	+	0.471588i	$0.156319\pi$
$644$	0	0
$645$	0	0
$646$	2.69505	0.106035
$647$	−18.9443	−0.744776	−0.372388	−	0.928077i	$-0.621461\pi$
$647$	−18.9443	−0.744776	−0.372388	+	0.928077i	$0.621461\pi$
$648$	1.47214	0.0578310
$649$	18.1803	0.713641
$650$	0	0
$651$	0	0
$652$	17.6656	0.691840
$653$	1.41641	0.0554283	0.0277142	−	0.999616i	$-0.491177\pi$
$653$	1.41641	0.0554283	0.0277142	+	0.999616i	$0.491177\pi$
$654$	−7.79837	−0.304941
$655$	0	0
$656$	25.1672	0.982613
$657$	−3.23607	−0.126251
$658$	0	0
$659$	23.0557	0.898124	0.449062	−	0.893501i	$-0.351758\pi$
$659$	23.0557	0.898124	0.449062	+	0.893501i	$0.351758\pi$
$660$	0	0
$661$	14.1803	0.551551	0.275776	−	0.961222i	$-0.411065\pi$
$661$	14.1803	0.551551	0.275776	+	0.961222i	$0.411065\pi$
$662$	−6.96556	−0.270724
$663$	29.8885	1.16077
$664$	5.19350	0.201547
$665$	0	0
$666$	−1.32624	−0.0513907
$667$	43.5410	1.68592
$668$	36.5410	1.41381
$669$	3.23607	0.125114
$670$	0	0
$671$	−39.6393	−1.53026
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	−7.45898	−0.287309
$675$	0	0
$676$	−26.7295	−1.02806
$677$	−19.3050	−0.741950	−0.370975	−	0.928643i	$-0.620976\pi$
$677$	−19.3050	−0.741950	−0.370975	+	0.928643i	$0.620976\pi$
$678$	−4.49342	−0.172569
$679$	0	0
$680$	0	0
$681$	−9.23607	−0.353927
$682$	−5.93112	−0.227114
$683$	−40.8885	−1.56456	−0.782278	−	0.622929i	$-0.785943\pi$
$683$	−40.8885	−1.56456	−0.782278	+	0.622929i	$0.785943\pi$
$684$	2.29180	0.0876290
$685$	0	0
$686$	0	0
$687$	22.3607	0.853113
$688$	−11.8409	−0.451432
$689$	44.3607	1.69001
$690$	0	0
$691$	−32.3607	−1.23106	−0.615529	−	0.788114i	$-0.711058\pi$
$691$	−32.3607	−1.23106	−0.615529	+	0.788114i	$0.711058\pi$
$692$	48.5410	1.84525
$693$	0	0
$694$	−4.96556	−0.188490
$695$	0	0
$696$	−12.8197	−0.485928
$697$	45.6656	1.72971
$698$	2.00000	0.0757011
$699$	−20.7082	−0.783256
$700$	0	0
$701$	6.58359	0.248659	0.124329	−	0.992241i	$-0.460322\pi$
$701$	6.58359	0.248659	0.124329	+	0.992241i	$0.460322\pi$
$702$	−2.00000	−0.0754851
$703$	−4.29180	−0.161868
$704$	−16.3475	−0.616120
$705$	0	0
$706$	−7.23607	−0.272333
$707$	0	0
$708$	−9.70820	−0.364857
$709$	8.11146	0.304632	0.152316	−	0.988332i	$-0.451327\pi$
$709$	8.11146	0.304632	0.152316	+	0.988332i	$0.451327\pi$
$710$	0	0
$711$	6.23607	0.233871
$712$	−11.3475	−0.425266
$713$	−22.3607	−0.837414
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	16.3607	0.611001
$718$	−3.07701	−0.114833
$719$	38.2492	1.42646	0.713228	−	0.700932i	$-0.247233\pi$
$719$	38.2492	1.42646	0.713228	+	0.700932i	$0.247233\pi$
$720$	0	0
$721$	0	0
$722$	6.67376	0.248372
$723$	−14.7639	−0.549077
$724$	−49.4164	−1.83655
$725$	0	0
$726$	−0.403252	−0.0149661
$727$	48.5410	1.80029	0.900143	−	0.435594i	$-0.143462\pi$
$727$	48.5410	1.80029	0.900143	+	0.435594i	$0.143462\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.4853	−0.794662
$732$	21.1672	0.782362
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	−3.41641	−0.126102
$735$	0	0
$736$	20.7295	0.764099
$737$	−37.1803	−1.36956
$738$	−3.05573	−0.112483
$739$	8.23607	0.302969	0.151484	−	0.988460i	$-0.451595\pi$
$739$	8.23607	0.302969	0.151484	+	0.988460i	$0.451595\pi$
$740$	0	0
$741$	−6.47214	−0.237760
$742$	0	0
$743$	−50.2492	−1.84347	−0.921733	−	0.387826i	$-0.873226\pi$
$743$	−50.2492	−1.84347	−0.921733	+	0.387826i	$0.873226\pi$
$744$	6.58359	0.241366
$745$	0	0
$746$	−5.72949	−0.209772
$747$	3.52786	0.129078
$748$	−36.7477	−1.34363
$749$	0	0
$750$	0	0
$751$	−4.36068	−0.159123	−0.0795617	−	0.996830i	$-0.525352\pi$
$751$	−4.36068	−0.159123	−0.0795617	+	0.996830i	$0.525352\pi$
$752$	8.69505	0.317076
$753$	−7.23607	−0.263697
$754$	17.4164	0.634268
$755$	0	0
$756$	0	0
$757$	−3.94427	−0.143357	−0.0716785	−	0.997428i	$-0.522836\pi$
$757$	−3.94427	−0.143357	−0.0716785	+	0.997428i	$0.522836\pi$
$758$	12.4508	0.452235
$759$	−17.3607	−0.630153
$760$	0	0
$761$	53.3050	1.93230	0.966151	−	0.257975i	$-0.0830553\pi$
$761$	53.3050	1.93230	0.966151	+	0.257975i	$0.0830553\pi$
$762$	−2.96556	−0.107431
$763$	0	0
$764$	−28.5836	−1.03412
$765$	0	0
$766$	5.77709	0.208735
$767$	27.4164	0.989949
$768$	5.56231	0.200712
$769$	−8.58359	−0.309532	−0.154766	−	0.987951i	$-0.549462\pi$
$769$	−8.58359	−0.309532	−0.154766	+	0.987951i	$0.549462\pi$
$770$	0	0
$771$	0.472136	0.0170036
$772$	−48.9787	−1.76278
$773$	−17.0557	−0.613452	−0.306726	−	0.951798i	$-0.599234\pi$
$773$	−17.0557	−0.613452	−0.306726	+	0.951798i	$0.599234\pi$
$774$	1.43769	0.0516768
$775$	0	0
$776$	5.19350	0.186436
$777$	0	0
$778$	2.38197	0.0853976
$779$	−9.88854	−0.354294
$780$	0	0
$781$	−32.8885	−1.17684
$782$	10.9017	0.389844
$783$	−8.70820	−0.311206
$784$	0	0
$785$	0	0
$786$	−6.76393	−0.241261
$787$	43.2361	1.54120	0.770600	−	0.637319i	$-0.219957\pi$
$787$	43.2361	1.54120	0.770600	+	0.637319i	$0.219957\pi$
$788$	−18.9787	−0.676089
$789$	−19.9443	−0.710035
$790$	0	0
$791$	0	0
$792$	5.11146	0.181628
$793$	−59.7771	−2.12275
$794$	6.87539	0.243998
$795$	0	0
$796$	−29.6656	−1.05147
$797$	18.0689	0.640033	0.320016	−	0.947412i	$-0.396312\pi$
$797$	18.0689	0.640033	0.320016	+	0.947412i	$0.396312\pi$
$798$	0	0
$799$	15.7771	0.558153
$800$	0	0
$801$	−7.70820	−0.272356
$802$	5.07701	0.179276
$803$	−11.2361	−0.396512
$804$	19.8541	0.700200
$805$	0	0
$806$	−8.94427	−0.315049
$807$	19.5279	0.687413
$808$	26.0689	0.917100
$809$	−29.1803	−1.02593	−0.512963	−	0.858411i	$-0.671452\pi$
$809$	−29.1803	−1.02593	−0.512963	+	0.858411i	$0.671452\pi$
$810$	0	0
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	−18.7639	−0.658080
$814$	−4.60488	−0.161401
$815$	0	0
$816$	17.9574	0.628636
$817$	4.65248	0.162770
$818$	5.41641	0.189380
$819$	0	0
$820$	0	0
$821$	6.94427	0.242357	0.121178	−	0.992631i	$-0.461333\pi$
$821$	6.94427	0.242357	0.121178	+	0.992631i	$0.461333\pi$
$822$	3.23607	0.112871
$823$	−32.0132	−1.11591	−0.557954	−	0.829872i	$-0.688414\pi$
$823$	−32.0132	−1.11591	−0.557954	+	0.829872i	$0.688414\pi$
$824$	−5.45898	−0.190173
$825$	0	0
$826$	0	0
$827$	−14.0557	−0.488766	−0.244383	−	0.969679i	$-0.578585\pi$
$827$	−14.0557	−0.488766	−0.244383	+	0.969679i	$0.578585\pi$
$828$	9.27051	0.322172
$829$	−31.2361	−1.08487	−0.542437	−	0.840097i	$-0.682498\pi$
$829$	−31.2361	−1.08487	−0.542437	+	0.840097i	$0.682498\pi$
$830$	0	0
$831$	−19.8885	−0.689926
$832$	−24.6525	−0.854671
$833$	0	0
$834$	−3.70820	−0.128405
$835$	0	0
$836$	7.95743	0.275213
$837$	4.47214	0.154580
$838$	10.2918	0.355524
$839$	13.1246	0.453112	0.226556	−	0.973998i	$-0.427253\pi$
$839$	13.1246	0.453112	0.226556	+	0.973998i	$0.427253\pi$
$840$	0	0
$841$	46.8328	1.61492
$842$	−10.0902	−0.347730
$843$	−17.6525	−0.607984
$844$	9.16718	0.315547
$845$	0	0
$846$	−1.05573	−0.0362967
$847$	0	0
$848$	26.6525	0.915250
$849$	13.4164	0.460450
$850$	0	0
$851$	−17.3607	−0.595116
$852$	17.5623	0.601675
$853$	7.63932	0.261565	0.130783	−	0.991411i	$-0.458251\pi$
$853$	7.63932	0.261565	0.130783	+	0.991411i	$0.458251\pi$
$854$	0	0
$855$	0	0
$856$	−19.0557	−0.651311
$857$	12.5836	0.429847	0.214924	−	0.976631i	$-0.431050\pi$
$857$	12.5836	0.429847	0.214924	+	0.976631i	$0.431050\pi$
$858$	−6.94427	−0.237074
$859$	−8.47214	−0.289066	−0.144533	−	0.989500i	$-0.546168\pi$
$859$	−8.47214	−0.289066	−0.144533	+	0.989500i	$0.546168\pi$
$860$	0	0
$861$	0	0
$862$	0.721360	0.0245696
$863$	−19.9443	−0.678911	−0.339455	−	0.940622i	$-0.610243\pi$
$863$	−19.9443	−0.678911	−0.339455	+	0.940622i	$0.610243\pi$
$864$	−4.14590	−0.141046
$865$	0	0
$866$	9.66563	0.328452
$867$	15.5836	0.529247
$868$	0	0
$869$	21.6525	0.734510
$870$	0	0
$871$	−56.0689	−1.89982
$872$	30.0557	1.01782
$873$	3.52786	0.119400
$874$	−2.36068	−0.0798512
$875$	0	0
$876$	6.00000	0.202721
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	−2.58359	−0.0871920
$879$	4.65248	0.156924
$880$	0	0
$881$	−6.36068	−0.214297	−0.107148	−	0.994243i	$-0.534172\pi$
$881$	−6.36068	−0.214297	−0.107148	+	0.994243i	$0.534172\pi$
$882$	0	0
$883$	−29.7639	−1.00164	−0.500818	−	0.865553i	$-0.666967\pi$
$883$	−29.7639	−1.00164	−0.500818	+	0.865553i	$0.666967\pi$
$884$	−55.4164	−1.86386
$885$	0	0
$886$	15.0557	0.505807
$887$	11.4164	0.383325	0.191663	−	0.981461i	$-0.438612\pi$
$887$	11.4164	0.383325	0.191663	+	0.981461i	$0.438612\pi$
$888$	5.11146	0.171529
$889$	0	0
$890$	0	0
$891$	3.47214	0.116321
$892$	−6.00000	−0.200895
$893$	−3.41641	−0.114326
$894$	−1.43769	−0.0480837
$895$	0	0
$896$	0	0
$897$	−26.1803	−0.874136
$898$	−6.38197	−0.212969
$899$	−38.9443	−1.29886
$900$	0	0
$901$	48.3607	1.61113
$902$	−10.6099	−0.353271
$903$	0	0
$904$	17.3181	0.575992
$905$	0	0
$906$	−5.61803	−0.186647
$907$	−28.3607	−0.941701	−0.470850	−	0.882213i	$-0.656053\pi$
$907$	−28.3607	−0.941701	−0.470850	+	0.882213i	$0.656053\pi$
$908$	17.1246	0.568300
$909$	17.7082	0.587344
$910$	0	0
$911$	34.4164	1.14027	0.570133	−	0.821552i	$-0.306892\pi$
$911$	34.4164	1.14027	0.570133	+	0.821552i	$0.306892\pi$
$912$	−3.88854	−0.128763
$913$	12.2492	0.405390
$914$	−7.84095	−0.259355
$915$	0	0
$916$	−41.4590	−1.36984
$917$	0	0
$918$	−2.18034	−0.0719619
$919$	−40.0132	−1.31991	−0.659956	−	0.751304i	$-0.729425\pi$
$919$	−40.0132	−1.31991	−0.659956	+	0.751304i	$0.729425\pi$
$920$	0	0
$921$	−6.65248	−0.219207
$922$	−7.88854	−0.259795
$923$	−49.5967	−1.63250
$924$	0	0
$925$	0	0
$926$	−6.24922	−0.205362
$927$	−3.70820	−0.121793
$928$	36.1033	1.18515
$929$	7.81966	0.256555	0.128277	−	0.991738i	$-0.459055\pi$
$929$	7.81966	0.256555	0.128277	+	0.991738i	$0.459055\pi$
$930$	0	0
$931$	0	0
$932$	38.3951	1.25767
$933$	−8.76393	−0.286918
$934$	4.54102	0.148587
$935$	0	0
$936$	7.70820	0.251951
$937$	−40.3607	−1.31853	−0.659263	−	0.751912i	$-0.729132\pi$
$937$	−40.3607	−1.31853	−0.659263	+	0.751912i	$0.729132\pi$
$938$	0	0
$939$	7.70820	0.251548
$940$	0	0
$941$	8.83282	0.287942	0.143971	−	0.989582i	$-0.454013\pi$
$941$	8.83282	0.287942	0.143971	+	0.989582i	$0.454013\pi$
$942$	0.360680	0.0117516
$943$	−40.0000	−1.30258
$944$	16.4721	0.536122
$945$	0	0
$946$	4.99187	0.162300
$947$	−15.0557	−0.489245	−0.244623	−	0.969618i	$-0.578664\pi$
$947$	−15.0557	−0.489245	−0.244623	+	0.969618i	$0.578664\pi$
$948$	−11.5623	−0.375526
$949$	−16.9443	−0.550034
$950$	0	0
$951$	13.6525	0.442712
$952$	0	0
$953$	24.1246	0.781473	0.390736	−	0.920503i	$-0.372220\pi$
$953$	24.1246	0.781473	0.390736	+	0.920503i	$0.372220\pi$
$954$	−3.23607	−0.104772
$955$	0	0
$956$	−30.3344	−0.981084
$957$	−30.2361	−0.977393
$958$	−7.16718	−0.231561
$959$	0	0
$960$	0	0
$961$	−11.0000	−0.354839
$962$	−6.94427	−0.223892
$963$	−12.9443	−0.417123
$964$	27.3738	0.881652
$965$	0	0
$966$	0	0
$967$	37.8885	1.21841	0.609207	−	0.793011i	$-0.291488\pi$
$967$	37.8885	1.21841	0.609207	+	0.793011i	$0.291488\pi$
$968$	1.55418	0.0499531
$969$	−7.05573	−0.226663
$970$	0	0
$971$	37.5967	1.20654	0.603269	−	0.797538i	$-0.293865\pi$
$971$	37.5967	1.20654	0.603269	+	0.797538i	$0.293865\pi$
$972$	−1.85410	−0.0594703
$973$	0	0
$974$	−11.1459	−0.357138
$975$	0	0
$976$	−35.9149	−1.14961
$977$	−16.7082	−0.534543	−0.267271	−	0.963621i	$-0.586122\pi$
$977$	−16.7082	−0.534543	−0.267271	+	0.963621i	$0.586122\pi$
$978$	3.63932	0.116373
$979$	−26.7639	−0.855379
$980$	0	0
$981$	20.4164	0.651846
$982$	3.61803	0.115456
$983$	−29.4164	−0.938238	−0.469119	−	0.883135i	$-0.655428\pi$
$983$	−29.4164	−0.938238	−0.469119	+	0.883135i	$0.655428\pi$
$984$	11.7771	0.375440
$985$	0	0
$986$	18.9868	0.604664
$987$	0	0
$988$	12.0000	0.381771
$989$	18.8197	0.598430
$990$	0	0
$991$	−40.5967	−1.28960	−0.644799	−	0.764352i	$-0.723059\pi$
$991$	−40.5967	−1.28960	−0.644799	+	0.764352i	$0.723059\pi$
$992$	−18.5410	−0.588678
$993$	18.2361	0.578704
$994$	0	0
$995$	0	0
$996$	−6.54102	−0.207260
$997$	−14.5410	−0.460519	−0.230259	−	0.973129i	$-0.573957\pi$
$997$	−14.5410	−0.460519	−0.230259	+	0.973129i	$0.573957\pi$
$998$	−15.1935	−0.480942
$999$	3.47214	0.109854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.r.1.2		2
5.4	even	2		3675.2.a.bh.1.1		2
7.6	odd	2		525.2.a.e.1.2	✓	2
21.20	even	2		1575.2.a.v.1.1		2
28.27	even	2		8400.2.a.da.1.1		2
35.13	even	4		525.2.d.e.274.3		4
35.27	even	4		525.2.d.e.274.2		4
35.34	odd	2		525.2.a.i.1.1	yes	2
105.62	odd	4		1575.2.d.f.1324.3		4
105.83	odd	4		1575.2.d.f.1324.2		4
105.104	even	2		1575.2.a.l.1.2		2
140.139	even	2		8400.2.a.cy.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.e.1.2	✓	2	7.6	odd	2
525.2.a.i.1.1	yes	2	35.34	odd	2
525.2.d.e.274.2		4	35.27	even	4
525.2.d.e.274.3		4	35.13	even	4
1575.2.a.l.1.2		2	105.104	even	2
1575.2.a.v.1.1		2	21.20	even	2
1575.2.d.f.1324.2		4	105.83	odd	4
1575.2.d.f.1324.3		4	105.62	odd	4
3675.2.a.r.1.2		2	1.1	even	1	trivial
3675.2.a.bh.1.1		2	5.4	even	2
8400.2.a.cy.1.1		2	140.139	even	2
8400.2.a.da.1.1		2	28.27	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +3.47214 q^{11} -1.85410 q^{12} +5.23607 q^{13} +3.14590 q^{16} +5.70820 q^{17} -0.381966 q^{18} -1.23607 q^{19} -1.32624 q^{22} -5.00000 q^{23} +1.47214 q^{24} -2.00000 q^{26} +1.00000 q^{27} -8.70820 q^{29} +4.47214 q^{31} -4.14590 q^{32} +3.47214 q^{33} -2.18034 q^{34} -1.85410 q^{36} +3.47214 q^{37} +0.472136 q^{38} +5.23607 q^{39} +8.00000 q^{41} -3.76393 q^{43} -6.43769 q^{44} +1.90983 q^{46} +2.76393 q^{47} +3.14590 q^{48} +5.70820 q^{51} -9.70820 q^{52} +8.47214 q^{53} -0.381966 q^{54} -1.23607 q^{57} +3.32624 q^{58} +5.23607 q^{59} -11.4164 q^{61} -1.70820 q^{62} -4.70820 q^{64} -1.32624 q^{66} -10.7082 q^{67} -10.5836 q^{68} -5.00000 q^{69} -9.47214 q^{71} +1.47214 q^{72} -3.23607 q^{73} -1.32624 q^{74} +2.29180 q^{76} -2.00000 q^{78} +6.23607 q^{79} +1.00000 q^{81} -3.05573 q^{82} +3.52786 q^{83} +1.43769 q^{86} -8.70820 q^{87} +5.11146 q^{88} -7.70820 q^{89} +9.27051 q^{92} +4.47214 q^{93} -1.05573 q^{94} -4.14590 q^{96} +3.52786 q^{97} +3.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{11} + 3 q^{12} + 6 q^{13} + 13 q^{16} - 2 q^{17} - 3 q^{18} + 2 q^{19} + 13 q^{22} - 10 q^{23} - 6 q^{24} - 4 q^{26} + 2 q^{27} - 4 q^{29} - 15 q^{32} - 2 q^{33} + 18 q^{34} + 3 q^{36} - 2 q^{37} - 8 q^{38} + 6 q^{39} + 16 q^{41} - 12 q^{43} - 33 q^{44} + 15 q^{46} + 10 q^{47} + 13 q^{48} - 2 q^{51} - 6 q^{52} + 8 q^{53} - 3 q^{54} + 2 q^{57} - 9 q^{58} + 6 q^{59} + 4 q^{61} + 10 q^{62} + 4 q^{64} + 13 q^{66} - 8 q^{67} - 48 q^{68} - 10 q^{69} - 10 q^{71} - 6 q^{72} - 2 q^{73} + 13 q^{74} + 18 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} - 24 q^{82} + 16 q^{83} + 23 q^{86} - 4 q^{87} + 46 q^{88} - 2 q^{89} - 15 q^{92} - 20 q^{94} - 15 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 - 2 * q^11 + 3 * q^12 + 6 * q^13 + 13 * q^16 - 2 * q^17 - 3 * q^18 + 2 * q^19 + 13 * q^22 - 10 * q^23 - 6 * q^24 - 4 * q^26 + 2 * q^27 - 4 * q^29 - 15 * q^32 - 2 * q^33 + 18 * q^34 + 3 * q^36 - 2 * q^37 - 8 * q^38 + 6 * q^39 + 16 * q^41 - 12 * q^43 - 33 * q^44 + 15 * q^46 + 10 * q^47 + 13 * q^48 - 2 * q^51 - 6 * q^52 + 8 * q^53 - 3 * q^54 + 2 * q^57 - 9 * q^58 + 6 * q^59 + 4 * q^61 + 10 * q^62 + 4 * q^64 + 13 * q^66 - 8 * q^67 - 48 * q^68 - 10 * q^69 - 10 * q^71 - 6 * q^72 - 2 * q^73 + 13 * q^74 + 18 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 - 24 * q^82 + 16 * q^83 + 23 * q^86 - 4 * q^87 + 46 * q^88 - 2 * q^89 - 15 * q^92 - 20 * q^94 - 15 * q^96 + 16 * q^97 - 2 * q^99

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