LMFDB - Embedded newform 2475.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
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 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +3.85410 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+0.618034 q^{2} -1.61803 q^{4} +3.85410 q^{7} -2.23607 q^{8} -1.00000 q^{11} +1.76393 q^{13} +2.38197 q^{14} +1.85410 q^{16} -1.61803 q^{17} +6.70820 q^{19} -0.618034 q^{22} -7.09017 q^{23} +1.09017 q^{26} -6.23607 q^{28} +3.61803 q^{29} -3.00000 q^{31} +5.61803 q^{32} -1.00000 q^{34} +5.76393 q^{37} +4.14590 q^{38} +3.00000 q^{41} -6.00000 q^{43} +1.61803 q^{44} -4.38197 q^{46} +5.94427 q^{47} +7.85410 q^{49} -2.85410 q^{52} +6.32624 q^{53} -8.61803 q^{56} +2.23607 q^{58} -9.47214 q^{59} -11.0902 q^{61} -1.85410 q^{62} -0.236068 q^{64} +8.00000 q^{67} +2.61803 q^{68} +14.1803 q^{71} +12.6180 q^{73} +3.56231 q^{74} -10.8541 q^{76} -3.85410 q^{77} -0.854102 q^{79} +1.85410 q^{82} +16.8541 q^{83} -3.70820 q^{86} +2.23607 q^{88} +18.0902 q^{89} +6.79837 q^{91} +11.4721 q^{92} +3.67376 q^{94} -0.618034 q^{97} +4.85410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + q^7	$ 2 q - q^{2} - q^{4} + q^{7} - 2 q^{11} + 8 q^{13} + 7 q^{14} - 3 q^{16} - q^{17} + q^{22} - 3 q^{23} - 9 q^{26} - 8 q^{28} + 5 q^{29} - 6 q^{31} + 9 q^{32} - 2 q^{34} + 16 q^{37} + 15 q^{38} + 6 q^{41} - 12 q^{43} + q^{44} - 11 q^{46} - 6 q^{47} + 9 q^{49} + q^{52} - 3 q^{53} - 15 q^{56} - 10 q^{59} - 11 q^{61} + 3 q^{62} + 4 q^{64} + 16 q^{67} + 3 q^{68} + 6 q^{71} + 23 q^{73} - 13 q^{74} - 15 q^{76} - q^{77} + 5 q^{79} - 3 q^{82} + 27 q^{83} + 6 q^{86} + 25 q^{89} - 11 q^{91} + 14 q^{92} + 23 q^{94} + q^{97} + 3 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + q^7 - 2 * q^11 + 8 * q^13 + 7 * q^14 - 3 * q^16 - q^17 + q^22 - 3 * q^23 - 9 * q^26 - 8 * q^28 + 5 * q^29 - 6 * q^31 + 9 * q^32 - 2 * q^34 + 16 * q^37 + 15 * q^38 + 6 * q^41 - 12 * q^43 + q^44 - 11 * q^46 - 6 * q^47 + 9 * q^49 + q^52 - 3 * q^53 - 15 * q^56 - 10 * q^59 - 11 * q^61 + 3 * q^62 + 4 * q^64 + 16 * q^67 + 3 * q^68 + 6 * q^71 + 23 * q^73 - 13 * q^74 - 15 * q^76 - q^77 + 5 * q^79 - 3 * q^82 + 27 * q^83 + 6 * q^86 + 25 * q^89 - 11 * q^91 + 14 * q^92 + 23 * q^94 + q^97 + 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.85410	1.45671	0.728357	−	0.685198i	$-0.240284\pi$
$7$	3.85410	1.45671	0.728357	+	0.685198i	$0.240284\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	2.38197	0.636607
$15$	0	0
$16$	1.85410	0.463525
$17$	−1.61803	−0.392431	−0.196215	−	0.980561i	$-0.562865\pi$
$17$	−1.61803	−0.392431	−0.196215	+	0.980561i	$0.562865\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	0	0
$22$	−0.618034	−0.131765
$23$	−7.09017	−1.47840	−0.739201	−	0.673485i	$-0.764797\pi$
$23$	−7.09017	−1.47840	−0.739201	+	0.673485i	$0.764797\pi$
$24$	0	0
$25$	0	0
$26$	1.09017	0.213800
$27$	0	0
$28$	−6.23607	−1.17851
$29$	3.61803	0.671852	0.335926	−	0.941888i	$-0.390951\pi$
$29$	3.61803	0.671852	0.335926	+	0.941888i	$0.390951\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	5.76393	0.947585	0.473792	−	0.880637i	$-0.342885\pi$
$37$	5.76393	0.947585	0.473792	+	0.880637i	$0.342885\pi$
$38$	4.14590	0.672553
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	1.61803	0.243928
$45$	0	0
$46$	−4.38197	−0.646086
$47$	5.94427	0.867061	0.433531	−	0.901139i	$-0.357268\pi$
$47$	5.94427	0.867061	0.433531	+	0.901139i	$0.357268\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	0	0
$51$	0	0
$52$	−2.85410	−0.395793
$53$	6.32624	0.868976	0.434488	−	0.900678i	$-0.356929\pi$
$53$	6.32624	0.868976	0.434488	+	0.900678i	$0.356929\pi$
$54$	0	0
$55$	0	0
$56$	−8.61803	−1.15163
$57$	0	0
$58$	2.23607	0.293610
$59$	−9.47214	−1.23317	−0.616584	−	0.787289i	$-0.711484\pi$
$59$	−9.47214	−1.23317	−0.616584	+	0.787289i	$0.711484\pi$
$60$	0	0
$61$	−11.0902	−1.41995	−0.709975	−	0.704226i	$-0.751294\pi$
$61$	−11.0902	−1.41995	−0.709975	+	0.704226i	$0.751294\pi$
$62$	−1.85410	−0.235471
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.61803	0.317483
$69$	0	0
$70$	0	0
$71$	14.1803	1.68290	0.841448	−	0.540338i	$-0.181703\pi$
$71$	14.1803	1.68290	0.841448	+	0.540338i	$0.181703\pi$
$72$	0	0
$73$	12.6180	1.47683	0.738415	−	0.674347i	$-0.235575\pi$
$73$	12.6180	1.47683	0.738415	+	0.674347i	$0.235575\pi$
$74$	3.56231	0.414110
$75$	0	0
$76$	−10.8541	−1.24505
$77$	−3.85410	−0.439216
$78$	0	0
$79$	−0.854102	−0.0960940	−0.0480470	−	0.998845i	$-0.515300\pi$
$79$	−0.854102	−0.0960940	−0.0480470	+	0.998845i	$0.515300\pi$
$80$	0	0
$81$	0	0
$82$	1.85410	0.204751
$83$	16.8541	1.84998	0.924989	−	0.379994i	$-0.124074\pi$
$83$	16.8541	1.84998	0.924989	+	0.379994i	$0.124074\pi$
$84$	0	0
$85$	0	0
$86$	−3.70820	−0.399866
$87$	0	0
$88$	2.23607	0.238366
$89$	18.0902	1.91755	0.958777	−	0.284159i	$-0.0917144\pi$
$89$	18.0902	1.91755	0.958777	+	0.284159i	$0.0917144\pi$
$90$	0	0
$91$	6.79837	0.712663
$92$	11.4721	1.19605
$93$	0	0
$94$	3.67376	0.378920
$95$	0	0
$96$	0	0
$97$	−0.618034	−0.0627518	−0.0313759	−	0.999508i	$-0.509989\pi$
$97$	−0.618034	−0.0627518	−0.0313759	+	0.999508i	$0.509989\pi$
$98$	4.85410	0.490338
$99$	0	0
$100$	0	0
$101$	−5.09017	−0.506491	−0.253245	−	0.967402i	$-0.581498\pi$
$101$	−5.09017	−0.506491	−0.253245	+	0.967402i	$0.581498\pi$
$102$	0	0
$103$	7.61803	0.750627	0.375314	−	0.926898i	$-0.377535\pi$
$103$	7.61803	0.750627	0.375314	+	0.926898i	$0.377535\pi$
$104$	−3.94427	−0.386768
$105$	0	0
$106$	3.90983	0.379756
$107$	−0.236068	−0.0228216	−0.0114108	−	0.999935i	$-0.503632\pi$
$107$	−0.236068	−0.0228216	−0.0114108	+	0.999935i	$0.503632\pi$
$108$	0	0
$109$	8.09017	0.774898	0.387449	−	0.921891i	$-0.373356\pi$
$109$	8.09017	0.774898	0.387449	+	0.921891i	$0.373356\pi$
$110$	0	0
$111$	0	0
$112$	7.14590	0.675224
$113$	−19.6525	−1.84875	−0.924375	−	0.381486i	$-0.875413\pi$
$113$	−19.6525	−1.84875	−0.924375	+	0.381486i	$0.875413\pi$
$114$	0	0
$115$	0	0
$116$	−5.85410	−0.543540
$117$	0	0
$118$	−5.85410	−0.538914
$119$	−6.23607	−0.571659
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.85410	−0.620541
$123$	0	0
$124$	4.85410	0.435911
$125$	0	0
$126$	0	0
$127$	1.61803	0.143577	0.0717886	−	0.997420i	$-0.477129\pi$
$127$	1.61803	0.143577	0.0717886	+	0.997420i	$0.477129\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	0	0
$131$	1.09017	0.0952486	0.0476243	−	0.998865i	$-0.484835\pi$
$131$	1.09017	0.0952486	0.0476243	+	0.998865i	$0.484835\pi$
$132$	0	0
$133$	25.8541	2.24183
$134$	4.94427	0.427120
$135$	0	0
$136$	3.61803	0.310244
$137$	14.5623	1.24414	0.622071	−	0.782961i	$-0.286292\pi$
$137$	14.5623	1.24414	0.622071	+	0.782961i	$0.286292\pi$
$138$	0	0
$139$	−16.7082	−1.41717	−0.708586	−	0.705625i	$-0.750666\pi$
$139$	−16.7082	−1.41717	−0.708586	+	0.705625i	$0.750666\pi$
$140$	0	0
$141$	0	0
$142$	8.76393	0.735453
$143$	−1.76393	−0.147507
$144$	0	0
$145$	0	0
$146$	7.79837	0.645398
$147$	0	0
$148$	−9.32624	−0.766612
$149$	−8.94427	−0.732743	−0.366372	−	0.930469i	$-0.619400\pi$
$149$	−8.94427	−0.732743	−0.366372	+	0.930469i	$0.619400\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	−15.0000	−1.21666
$153$	0	0
$154$	−2.38197	−0.191944
$155$	0	0
$156$	0	0
$157$	21.4164	1.70922	0.854608	−	0.519274i	$-0.173798\pi$
$157$	21.4164	1.70922	0.854608	+	0.519274i	$0.173798\pi$
$158$	−0.527864	−0.0419946
$159$	0	0
$160$	0	0
$161$	−27.3262	−2.15361
$162$	0	0
$163$	−0.145898	−0.0114276	−0.00571381	−	0.999984i	$-0.501819\pi$
$163$	−0.145898	−0.0114276	−0.00571381	+	0.999984i	$0.501819\pi$
$164$	−4.85410	−0.379042
$165$	0	0
$166$	10.4164	0.808470
$167$	18.7082	1.44768	0.723842	−	0.689966i	$-0.242374\pi$
$167$	18.7082	1.44768	0.723842	+	0.689966i	$0.242374\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	0	0
$172$	9.70820	0.740244
$173$	−3.47214	−0.263982	−0.131991	−	0.991251i	$-0.542137\pi$
$173$	−3.47214	−0.263982	−0.131991	+	0.991251i	$0.542137\pi$
$174$	0	0
$175$	0	0
$176$	−1.85410	−0.139758
$177$	0	0
$178$	11.1803	0.838002
$179$	11.3820	0.850728	0.425364	−	0.905022i	$-0.360146\pi$
$179$	11.3820	0.850728	0.425364	+	0.905022i	$0.360146\pi$
$180$	0	0
$181$	5.09017	0.378349	0.189175	−	0.981943i	$-0.439419\pi$
$181$	5.09017	0.378349	0.189175	+	0.981943i	$0.439419\pi$
$182$	4.20163	0.311445
$183$	0	0
$184$	15.8541	1.16878
$185$	0	0
$186$	0	0
$187$	1.61803	0.118322
$188$	−9.61803	−0.701467
$189$	0	0
$190$	0	0
$191$	9.90983	0.717050	0.358525	−	0.933520i	$-0.383280\pi$
$191$	9.90983	0.717050	0.358525	+	0.933520i	$0.383280\pi$
$192$	0	0
$193$	−4.94427	−0.355896	−0.177948	−	0.984040i	$-0.556946\pi$
$193$	−4.94427	−0.355896	−0.177948	+	0.984040i	$0.556946\pi$
$194$	−0.381966	−0.0274236
$195$	0	0
$196$	−12.7082	−0.907729
$197$	8.90983	0.634799	0.317400	−	0.948292i	$-0.397190\pi$
$197$	8.90983	0.634799	0.317400	+	0.948292i	$0.397190\pi$
$198$	0	0
$199$	8.09017	0.573497	0.286748	−	0.958006i	$-0.407426\pi$
$199$	8.09017	0.573497	0.286748	+	0.958006i	$0.407426\pi$
$200$	0	0
$201$	0	0
$202$	−3.14590	−0.221345
$203$	13.9443	0.978696
$204$	0	0
$205$	0	0
$206$	4.70820	0.328036
$207$	0	0
$208$	3.27051	0.226769
$209$	−6.70820	−0.464016
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	−10.2361	−0.703016
$213$	0	0
$214$	−0.145898	−0.00997338
$215$	0	0
$216$	0	0
$217$	−11.5623	−0.784900
$218$	5.00000	0.338643
$219$	0	0
$220$	0	0
$221$	−2.85410	−0.191988
$222$	0	0
$223$	−18.8885	−1.26487	−0.632435	−	0.774613i	$-0.717945\pi$
$223$	−18.8885	−1.26487	−0.632435	+	0.774613i	$0.717945\pi$
$224$	21.6525	1.44672
$225$	0	0
$226$	−12.1459	−0.807933
$227$	−5.03444	−0.334148	−0.167074	−	0.985944i	$-0.553432\pi$
$227$	−5.03444	−0.334148	−0.167074	+	0.985944i	$0.553432\pi$
$228$	0	0
$229$	−17.0344	−1.12567	−0.562834	−	0.826570i	$-0.690289\pi$
$229$	−17.0344	−1.12567	−0.562834	+	0.826570i	$0.690289\pi$
$230$	0	0
$231$	0	0
$232$	−8.09017	−0.531146
$233$	22.5066	1.47445	0.737227	−	0.675645i	$-0.236135\pi$
$233$	22.5066	1.47445	0.737227	+	0.675645i	$0.236135\pi$
$234$	0	0
$235$	0	0
$236$	15.3262	0.997653
$237$	0	0
$238$	−3.85410	−0.249824
$239$	−8.61803	−0.557454	−0.278727	−	0.960370i	$-0.589913\pi$
$239$	−8.61803	−0.557454	−0.278727	+	0.960370i	$0.589913\pi$
$240$	0	0
$241$	−12.2705	−0.790413	−0.395207	−	0.918592i	$-0.629327\pi$
$241$	−12.2705	−0.790413	−0.395207	+	0.918592i	$0.629327\pi$
$242$	0.618034	0.0397287
$243$	0	0
$244$	17.9443	1.14876
$245$	0	0
$246$	0	0
$247$	11.8328	0.752904
$248$	6.70820	0.425971
$249$	0	0
$250$	0	0
$251$	−6.27051	−0.395791	−0.197896	−	0.980223i	$-0.563411\pi$
$251$	−6.27051	−0.395791	−0.197896	+	0.980223i	$0.563411\pi$
$252$	0	0
$253$	7.09017	0.445755
$254$	1.00000	0.0627456
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−6.94427	−0.433172	−0.216586	−	0.976264i	$-0.569492\pi$
$257$	−6.94427	−0.433172	−0.216586	+	0.976264i	$0.569492\pi$
$258$	0	0
$259$	22.2148	1.38036
$260$	0	0
$261$	0	0
$262$	0.673762	0.0416252
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	0	0
$266$	15.9787	0.979718
$267$	0	0
$268$	−12.9443	−0.790697
$269$	−14.6738	−0.894675	−0.447338	−	0.894365i	$-0.647628\pi$
$269$	−14.6738	−0.894675	−0.447338	+	0.894365i	$0.647628\pi$
$270$	0	0
$271$	−9.18034	−0.557666	−0.278833	−	0.960340i	$-0.589948\pi$
$271$	−9.18034	−0.557666	−0.278833	+	0.960340i	$0.589948\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	9.00000	0.543710
$275$	0	0
$276$	0	0
$277$	−2.52786	−0.151885	−0.0759423	−	0.997112i	$-0.524196\pi$
$277$	−2.52786	−0.151885	−0.0759423	+	0.997112i	$0.524196\pi$
$278$	−10.3262	−0.619327
$279$	0	0
$280$	0	0
$281$	−19.3607	−1.15496	−0.577481	−	0.816404i	$-0.695964\pi$
$281$	−19.3607	−1.15496	−0.577481	+	0.816404i	$0.695964\pi$
$282$	0	0
$283$	−9.61803	−0.571733	−0.285866	−	0.958269i	$-0.592281\pi$
$283$	−9.61803	−0.571733	−0.285866	+	0.958269i	$0.592281\pi$
$284$	−22.9443	−1.36149
$285$	0	0
$286$	−1.09017	−0.0644631
$287$	11.5623	0.682501
$288$	0	0
$289$	−14.3820	−0.845998
$290$	0	0
$291$	0	0
$292$	−20.4164	−1.19478
$293$	−11.8885	−0.694536	−0.347268	−	0.937766i	$-0.612891\pi$
$293$	−11.8885	−0.694536	−0.347268	+	0.937766i	$0.612891\pi$
$294$	0	0
$295$	0	0
$296$	−12.8885	−0.749131
$297$	0	0
$298$	−5.52786	−0.320221
$299$	−12.5066	−0.723274
$300$	0	0
$301$	−23.1246	−1.33288
$302$	−1.85410	−0.106692
$303$	0	0
$304$	12.4377	0.713351
$305$	0	0
$306$	0	0
$307$	−22.4508	−1.28134	−0.640669	−	0.767817i	$-0.721343\pi$
$307$	−22.4508	−1.28134	−0.640669	+	0.767817i	$0.721343\pi$
$308$	6.23607	0.355333
$309$	0	0
$310$	0	0
$311$	−3.18034	−0.180341	−0.0901703	−	0.995926i	$-0.528741\pi$
$311$	−3.18034	−0.180341	−0.0901703	+	0.995926i	$0.528741\pi$
$312$	0	0
$313$	1.23607	0.0698667	0.0349333	−	0.999390i	$-0.488878\pi$
$313$	1.23607	0.0698667	0.0349333	+	0.999390i	$0.488878\pi$
$314$	13.2361	0.746955
$315$	0	0
$316$	1.38197	0.0777417
$317$	−14.3820	−0.807772	−0.403886	−	0.914809i	$-0.632341\pi$
$317$	−14.3820	−0.807772	−0.403886	+	0.914809i	$0.632341\pi$
$318$	0	0
$319$	−3.61803	−0.202571
$320$	0	0
$321$	0	0
$322$	−16.8885	−0.941162
$323$	−10.8541	−0.603938
$324$	0	0
$325$	0	0
$326$	−0.0901699	−0.00499405
$327$	0	0
$328$	−6.70820	−0.370399
$329$	22.9098	1.26306
$330$	0	0
$331$	3.18034	0.174807	0.0874036	−	0.996173i	$-0.472143\pi$
$331$	3.18034	0.174807	0.0874036	+	0.996173i	$0.472143\pi$
$332$	−27.2705	−1.49666
$333$	0	0
$334$	11.5623	0.632661
$335$	0	0
$336$	0	0
$337$	−25.4164	−1.38452	−0.692260	−	0.721648i	$-0.743385\pi$
$337$	−25.4164	−1.38452	−0.692260	+	0.721648i	$0.743385\pi$
$338$	−6.11146	−0.332419
$339$	0	0
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	3.29180	0.177740
$344$	13.4164	0.723364
$345$	0	0
$346$	−2.14590	−0.115364
$347$	−0.437694	−0.0234967	−0.0117483	−	0.999931i	$-0.503740\pi$
$347$	−0.437694	−0.0234967	−0.0117483	+	0.999931i	$0.503740\pi$
$348$	0	0
$349$	−31.8328	−1.70397	−0.851986	−	0.523565i	$-0.824602\pi$
$349$	−31.8328	−1.70397	−0.851986	+	0.523565i	$0.824602\pi$
$350$	0	0
$351$	0	0
$352$	−5.61803	−0.299442
$353$	23.3607	1.24336	0.621682	−	0.783270i	$-0.286450\pi$
$353$	23.3607	1.24336	0.621682	+	0.783270i	$0.286450\pi$
$354$	0	0
$355$	0	0
$356$	−29.2705	−1.55133
$357$	0	0
$358$	7.03444	0.371782
$359$	−4.47214	−0.236030	−0.118015	−	0.993012i	$-0.537653\pi$
$359$	−4.47214	−0.236030	−0.118015	+	0.993012i	$0.537653\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	3.14590	0.165345
$363$	0	0
$364$	−11.0000	−0.576557
$365$	0	0
$366$	0	0
$367$	13.8541	0.723178	0.361589	−	0.932338i	$-0.382234\pi$
$367$	13.8541	0.723178	0.361589	+	0.932338i	$0.382234\pi$
$368$	−13.1459	−0.685277
$369$	0	0
$370$	0	0
$371$	24.3820	1.26585
$372$	0	0
$373$	25.1803	1.30379	0.651894	−	0.758310i	$-0.273975\pi$
$373$	25.1803	1.30379	0.651894	+	0.758310i	$0.273975\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	−13.2918	−0.685472
$377$	6.38197	0.328688
$378$	0	0
$379$	−22.2361	−1.14219	−0.571095	−	0.820884i	$-0.693481\pi$
$379$	−22.2361	−1.14219	−0.571095	+	0.820884i	$0.693481\pi$
$380$	0	0
$381$	0	0
$382$	6.12461	0.313362
$383$	−22.9443	−1.17240	−0.586199	−	0.810167i	$-0.699376\pi$
$383$	−22.9443	−1.17240	−0.586199	+	0.810167i	$0.699376\pi$
$384$	0	0
$385$	0	0
$386$	−3.05573	−0.155532
$387$	0	0
$388$	1.00000	0.0507673
$389$	−14.4721	−0.733766	−0.366883	−	0.930267i	$-0.619575\pi$
$389$	−14.4721	−0.733766	−0.366883	+	0.930267i	$0.619575\pi$
$390$	0	0
$391$	11.4721	0.580171
$392$	−17.5623	−0.887030
$393$	0	0
$394$	5.50658	0.277417
$395$	0	0
$396$	0	0
$397$	26.0902	1.30943	0.654714	−	0.755877i	$-0.272789\pi$
$397$	26.0902	1.30943	0.654714	+	0.755877i	$0.272789\pi$
$398$	5.00000	0.250627
$399$	0	0
$400$	0	0
$401$	10.3607	0.517388	0.258694	−	0.965959i	$-0.416708\pi$
$401$	10.3607	0.517388	0.258694	+	0.965959i	$0.416708\pi$
$402$	0	0
$403$	−5.29180	−0.263603
$404$	8.23607	0.409760
$405$	0	0
$406$	8.61803	0.427706
$407$	−5.76393	−0.285708
$408$	0	0
$409$	−20.1246	−0.995098	−0.497549	−	0.867436i	$-0.665767\pi$
$409$	−20.1246	−0.995098	−0.497549	+	0.867436i	$0.665767\pi$
$410$	0	0
$411$	0	0
$412$	−12.3262	−0.607270
$413$	−36.5066	−1.79637
$414$	0	0
$415$	0	0
$416$	9.90983	0.485869
$417$	0	0
$418$	−4.14590	−0.202783
$419$	1.18034	0.0576634	0.0288317	−	0.999584i	$-0.490821\pi$
$419$	1.18034	0.0576634	0.0288317	+	0.999584i	$0.490821\pi$
$420$	0	0
$421$	21.2705	1.03666	0.518331	−	0.855180i	$-0.326554\pi$
$421$	21.2705	1.03666	0.518331	+	0.855180i	$0.326554\pi$
$422$	10.5066	0.511452
$423$	0	0
$424$	−14.1459	−0.686986
$425$	0	0
$426$	0	0
$427$	−42.7426	−2.06846
$428$	0.381966	0.0184630
$429$	0	0
$430$	0	0
$431$	−23.1803	−1.11656	−0.558279	−	0.829653i	$-0.688538\pi$
$431$	−23.1803	−1.11656	−0.558279	+	0.829653i	$0.688538\pi$
$432$	0	0
$433$	16.8885	0.811612	0.405806	−	0.913959i	$-0.366991\pi$
$433$	16.8885	0.811612	0.405806	+	0.913959i	$0.366991\pi$
$434$	−7.14590	−0.343014
$435$	0	0
$436$	−13.0902	−0.626905
$437$	−47.5623	−2.27521
$438$	0	0
$439$	−34.2705	−1.63564	−0.817821	−	0.575473i	$-0.804818\pi$
$439$	−34.2705	−1.63564	−0.817821	+	0.575473i	$0.804818\pi$
$440$	0	0
$441$	0	0
$442$	−1.76393	−0.0839017
$443$	5.34752	0.254069	0.127034	−	0.991898i	$-0.459454\pi$
$443$	5.34752	0.254069	0.127034	+	0.991898i	$0.459454\pi$
$444$	0	0
$445$	0	0
$446$	−11.6738	−0.552769
$447$	0	0
$448$	−0.909830	−0.0429854
$449$	−0.326238	−0.0153961	−0.00769806	−	0.999970i	$-0.502450\pi$
$449$	−0.326238	−0.0153961	−0.00769806	+	0.999970i	$0.502450\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	31.7984	1.49567
$453$	0	0
$454$	−3.11146	−0.146028
$455$	0	0
$456$	0	0
$457$	38.9787	1.82335	0.911674	−	0.410915i	$-0.134791\pi$
$457$	38.9787	1.82335	0.911674	+	0.410915i	$0.134791\pi$
$458$	−10.5279	−0.491935
$459$	0	0
$460$	0	0
$461$	−13.1803	−0.613870	−0.306935	−	0.951731i	$-0.599303\pi$
$461$	−13.1803	−0.613870	−0.306935	+	0.951731i	$0.599303\pi$
$462$	0	0
$463$	−33.3607	−1.55040	−0.775201	−	0.631714i	$-0.782352\pi$
$463$	−33.3607	−1.55040	−0.775201	+	0.631714i	$0.782352\pi$
$464$	6.70820	0.311421
$465$	0	0
$466$	13.9098	0.644360
$467$	−28.5279	−1.32011	−0.660056	−	0.751216i	$-0.729467\pi$
$467$	−28.5279	−1.32011	−0.660056	+	0.751216i	$0.729467\pi$
$468$	0	0
$469$	30.8328	1.42373
$470$	0	0
$471$	0	0
$472$	21.1803	0.974904
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	10.0902	0.462482
$477$	0	0
$478$	−5.32624	−0.243616
$479$	−1.58359	−0.0723562	−0.0361781	−	0.999345i	$-0.511518\pi$
$479$	−1.58359	−0.0723562	−0.0361781	+	0.999345i	$0.511518\pi$
$480$	0	0
$481$	10.1672	0.463584
$482$	−7.58359	−0.345423
$483$	0	0
$484$	−1.61803	−0.0735470
$485$	0	0
$486$	0	0
$487$	−3.58359	−0.162388	−0.0811940	−	0.996698i	$-0.525873\pi$
$487$	−3.58359	−0.162388	−0.0811940	+	0.996698i	$0.525873\pi$
$488$	24.7984	1.12257
$489$	0	0
$490$	0	0
$491$	29.1803	1.31689	0.658445	−	0.752629i	$-0.271214\pi$
$491$	29.1803	1.31689	0.658445	+	0.752629i	$0.271214\pi$
$492$	0	0
$493$	−5.85410	−0.263655
$494$	7.31308	0.329031
$495$	0	0
$496$	−5.56231	−0.249755
$497$	54.6525	2.45150
$498$	0	0
$499$	5.20163	0.232857	0.116428	−	0.993199i	$-0.462855\pi$
$499$	5.20163	0.232857	0.116428	+	0.993199i	$0.462855\pi$
$500$	0	0
$501$	0	0
$502$	−3.87539	−0.172967
$503$	−24.6525	−1.09920	−0.549600	−	0.835428i	$-0.685220\pi$
$503$	−24.6525	−1.09920	−0.549600	+	0.835428i	$0.685220\pi$
$504$	0	0
$505$	0	0
$506$	4.38197	0.194802
$507$	0	0
$508$	−2.61803	−0.116156
$509$	−18.6180	−0.825230	−0.412615	−	0.910906i	$-0.635384\pi$
$509$	−18.6180	−0.825230	−0.412615	+	0.910906i	$0.635384\pi$
$510$	0	0
$511$	48.6312	2.15132
$512$	18.7082	0.826794
$513$	0	0
$514$	−4.29180	−0.189303
$515$	0	0
$516$	0	0
$517$	−5.94427	−0.261429
$518$	13.7295	0.603239
$519$	0	0
$520$	0	0
$521$	24.1803	1.05936	0.529680	−	0.848198i	$-0.322312\pi$
$521$	24.1803	1.05936	0.529680	+	0.848198i	$0.322312\pi$
$522$	0	0
$523$	5.05573	0.221072	0.110536	−	0.993872i	$-0.464743\pi$
$523$	5.05573	0.221072	0.110536	+	0.993872i	$0.464743\pi$
$524$	−1.76393	−0.0770577
$525$	0	0
$526$	12.9787	0.565899
$527$	4.85410	0.211448
$528$	0	0
$529$	27.2705	1.18567
$530$	0	0
$531$	0	0
$532$	−41.8328	−1.81368
$533$	5.29180	0.229213
$534$	0	0
$535$	0	0
$536$	−17.8885	−0.772667
$537$	0	0
$538$	−9.06888	−0.390987
$539$	−7.85410	−0.338300
$540$	0	0
$541$	−8.72949	−0.375310	−0.187655	−	0.982235i	$-0.560089\pi$
$541$	−8.72949	−0.375310	−0.187655	+	0.982235i	$0.560089\pi$
$542$	−5.67376	−0.243709
$543$	0	0
$544$	−9.09017	−0.389738
$545$	0	0
$546$	0	0
$547$	−27.8541	−1.19096	−0.595478	−	0.803372i	$-0.703037\pi$
$547$	−27.8541	−1.19096	−0.595478	+	0.803372i	$0.703037\pi$
$548$	−23.5623	−1.00653
$549$	0	0
$550$	0	0
$551$	24.2705	1.03396
$552$	0	0
$553$	−3.29180	−0.139981
$554$	−1.56231	−0.0663760
$555$	0	0
$556$	27.0344	1.14652
$557$	14.2361	0.603202	0.301601	−	0.953434i	$-0.402479\pi$
$557$	14.2361	0.603202	0.301601	+	0.953434i	$0.402479\pi$
$558$	0	0
$559$	−10.5836	−0.447638
$560$	0	0
$561$	0	0
$562$	−11.9656	−0.504737
$563$	−16.0344	−0.675771	−0.337886	−	0.941187i	$-0.609712\pi$
$563$	−16.0344	−0.675771	−0.337886	+	0.941187i	$0.609712\pi$
$564$	0	0
$565$	0	0
$566$	−5.94427	−0.249856
$567$	0	0
$568$	−31.7082	−1.33045
$569$	−28.6180	−1.19973	−0.599865	−	0.800101i	$-0.704779\pi$
$569$	−28.6180	−1.19973	−0.599865	+	0.800101i	$0.704779\pi$
$570$	0	0
$571$	2.72949	0.114226	0.0571128	−	0.998368i	$-0.481811\pi$
$571$	2.72949	0.114226	0.0571128	+	0.998368i	$0.481811\pi$
$572$	2.85410	0.119336
$573$	0	0
$574$	7.14590	0.298264
$575$	0	0
$576$	0	0
$577$	−4.56231	−0.189931	−0.0949656	−	0.995481i	$-0.530274\pi$
$577$	−4.56231	−0.189931	−0.0949656	+	0.995481i	$0.530274\pi$
$578$	−8.88854	−0.369715
$579$	0	0
$580$	0	0
$581$	64.9574	2.69489
$582$	0	0
$583$	−6.32624	−0.262006
$584$	−28.2148	−1.16754
$585$	0	0
$586$	−7.34752	−0.303523
$587$	−25.0344	−1.03328	−0.516641	−	0.856202i	$-0.672818\pi$
$587$	−25.0344	−1.03328	−0.516641	+	0.856202i	$0.672818\pi$
$588$	0	0
$589$	−20.1246	−0.829220
$590$	0	0
$591$	0	0
$592$	10.6869	0.439230
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	0	0
$595$	0	0
$596$	14.4721	0.592802
$597$	0	0
$598$	−7.72949	−0.316082
$599$	15.3262	0.626213	0.313107	−	0.949718i	$-0.398630\pi$
$599$	15.3262	0.626213	0.313107	+	0.949718i	$0.398630\pi$
$600$	0	0
$601$	11.2705	0.459734	0.229867	−	0.973222i	$-0.426171\pi$
$601$	11.2705	0.459734	0.229867	+	0.973222i	$0.426171\pi$
$602$	−14.2918	−0.582490
$603$	0	0
$604$	4.85410	0.197511
$605$	0	0
$606$	0	0
$607$	12.4721	0.506228	0.253114	−	0.967436i	$-0.418545\pi$
$607$	12.4721	0.506228	0.253114	+	0.967436i	$0.418545\pi$
$608$	37.6869	1.52841
$609$	0	0
$610$	0	0
$611$	10.4853	0.424189
$612$	0	0
$613$	26.5623	1.07284	0.536421	−	0.843951i	$-0.319776\pi$
$613$	26.5623	1.07284	0.536421	+	0.843951i	$0.319776\pi$
$614$	−13.8754	−0.559965
$615$	0	0
$616$	8.61803	0.347230
$617$	15.8197	0.636876	0.318438	−	0.947944i	$-0.396842\pi$
$617$	15.8197	0.636876	0.318438	+	0.947944i	$0.396842\pi$
$618$	0	0
$619$	12.8885	0.518034	0.259017	−	0.965873i	$-0.416601\pi$
$619$	12.8885	0.518034	0.259017	+	0.965873i	$0.416601\pi$
$620$	0	0
$621$	0	0
$622$	−1.96556	−0.0788117
$623$	69.7214	2.79333
$624$	0	0
$625$	0	0
$626$	0.763932	0.0305329
$627$	0	0
$628$	−34.6525	−1.38278
$629$	−9.32624	−0.371861
$630$	0	0
$631$	2.72949	0.108659	0.0543296	−	0.998523i	$-0.482698\pi$
$631$	2.72949	0.108659	0.0543296	+	0.998523i	$0.482698\pi$
$632$	1.90983	0.0759690
$633$	0	0
$634$	−8.88854	−0.353009
$635$	0	0
$636$	0	0
$637$	13.8541	0.548920
$638$	−2.23607	−0.0885268
$639$	0	0
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	37.4164	1.47556	0.737780	−	0.675042i	$-0.235874\pi$
$643$	37.4164	1.47556	0.737780	+	0.675042i	$0.235874\pi$
$644$	44.2148	1.74231
$645$	0	0
$646$	−6.70820	−0.263931
$647$	−43.6525	−1.71616	−0.858078	−	0.513519i	$-0.828341\pi$
$647$	−43.6525	−1.71616	−0.858078	+	0.513519i	$0.828341\pi$
$648$	0	0
$649$	9.47214	0.371814
$650$	0	0
$651$	0	0
$652$	0.236068	0.00924514
$653$	34.7426	1.35958	0.679792	−	0.733405i	$-0.262070\pi$
$653$	34.7426	1.35958	0.679792	+	0.733405i	$0.262070\pi$
$654$	0	0
$655$	0	0
$656$	5.56231	0.217172
$657$	0	0
$658$	14.1591	0.551977
$659$	24.2705	0.945445	0.472722	−	0.881211i	$-0.343271\pi$
$659$	24.2705	0.945445	0.472722	+	0.881211i	$0.343271\pi$
$660$	0	0
$661$	−26.8197	−1.04316	−0.521582	−	0.853201i	$-0.674658\pi$
$661$	−26.8197	−1.04316	−0.521582	+	0.853201i	$0.674658\pi$
$662$	1.96556	0.0763936
$663$	0	0
$664$	−37.6869	−1.46254
$665$	0	0
$666$	0	0
$667$	−25.6525	−0.993268
$668$	−30.2705	−1.17120
$669$	0	0
$670$	0	0
$671$	11.0902	0.428131
$672$	0	0
$673$	−24.4164	−0.941183	−0.470592	−	0.882351i	$-0.655960\pi$
$673$	−24.4164	−0.941183	−0.470592	+	0.882351i	$0.655960\pi$
$674$	−15.7082	−0.605057
$675$	0	0
$676$	16.0000	0.615385
$677$	2.65248	0.101943	0.0509715	−	0.998700i	$-0.483768\pi$
$677$	2.65248	0.101943	0.0509715	+	0.998700i	$0.483768\pi$
$678$	0	0
$679$	−2.38197	−0.0914115
$680$	0	0
$681$	0	0
$682$	1.85410	0.0709972
$683$	−1.36068	−0.0520650	−0.0260325	−	0.999661i	$-0.508287\pi$
$683$	−1.36068	−0.0520650	−0.0260325	+	0.999661i	$0.508287\pi$
$684$	0	0
$685$	0	0
$686$	2.03444	0.0776754
$687$	0	0
$688$	−11.1246	−0.424122
$689$	11.1591	0.425126
$690$	0	0
$691$	18.9098	0.719364	0.359682	−	0.933075i	$-0.382885\pi$
$691$	18.9098	0.719364	0.359682	+	0.933075i	$0.382885\pi$
$692$	5.61803	0.213566
$693$	0	0
$694$	−0.270510	−0.0102684
$695$	0	0
$696$	0	0
$697$	−4.85410	−0.183862
$698$	−19.6738	−0.744663
$699$	0	0
$700$	0	0
$701$	−44.3607	−1.67548	−0.837740	−	0.546070i	$-0.816123\pi$
$701$	−44.3607	−1.67548	−0.837740	+	0.546070i	$0.816123\pi$
$702$	0	0
$703$	38.6656	1.45830
$704$	0.236068	0.00889715
$705$	0	0
$706$	14.4377	0.543370
$707$	−19.6180	−0.737812
$708$	0	0
$709$	21.3050	0.800124	0.400062	−	0.916488i	$-0.368989\pi$
$709$	21.3050	0.800124	0.400062	+	0.916488i	$0.368989\pi$
$710$	0	0
$711$	0	0
$712$	−40.4508	−1.51596
$713$	21.2705	0.796587
$714$	0	0
$715$	0	0
$716$	−18.4164	−0.688253
$717$	0	0
$718$	−2.76393	−0.103149
$719$	−15.6525	−0.583739	−0.291869	−	0.956458i	$-0.594277\pi$
$719$	−15.6525	−0.583739	−0.291869	+	0.956458i	$0.594277\pi$
$720$	0	0
$721$	29.3607	1.09345
$722$	16.0689	0.598022
$723$	0	0
$724$	−8.23607	−0.306091
$725$	0	0
$726$	0	0
$727$	−7.72949	−0.286671	−0.143335	−	0.989674i	$-0.545783\pi$
$727$	−7.72949	−0.286671	−0.143335	+	0.989674i	$0.545783\pi$
$728$	−15.2016	−0.563410
$729$	0	0
$730$	0	0
$731$	9.70820	0.359071
$732$	0	0
$733$	−7.83282	−0.289312	−0.144656	−	0.989482i	$-0.546207\pi$
$733$	−7.83282	−0.289312	−0.144656	+	0.989482i	$0.546207\pi$
$734$	8.56231	0.316040
$735$	0	0
$736$	−39.8328	−1.46826
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−35.8541	−1.31891	−0.659457	−	0.751742i	$-0.729214\pi$
$739$	−35.8541	−1.31891	−0.659457	+	0.751742i	$0.729214\pi$
$740$	0	0
$741$	0	0
$742$	15.0689	0.553196
$743$	−34.3262	−1.25931	−0.629654	−	0.776876i	$-0.716803\pi$
$743$	−34.3262	−1.25931	−0.629654	+	0.776876i	$0.716803\pi$
$744$	0	0
$745$	0	0
$746$	15.5623	0.569777
$747$	0	0
$748$	−2.61803	−0.0957248
$749$	−0.909830	−0.0332445
$750$	0	0
$751$	−22.2705	−0.812662	−0.406331	−	0.913726i	$-0.633192\pi$
$751$	−22.2705	−0.812662	−0.406331	+	0.913726i	$0.633192\pi$
$752$	11.0213	0.401905
$753$	0	0
$754$	3.94427	0.143642
$755$	0	0
$756$	0	0
$757$	−12.5279	−0.455333	−0.227666	−	0.973739i	$-0.573110\pi$
$757$	−12.5279	−0.455333	−0.227666	+	0.973739i	$0.573110\pi$
$758$	−13.7426	−0.499155
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	31.1803	1.12880
$764$	−16.0344	−0.580106
$765$	0	0
$766$	−14.1803	−0.512357
$767$	−16.7082	−0.603298
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	8.00000	0.287926
$773$	−9.20163	−0.330959	−0.165480	−	0.986213i	$-0.552917\pi$
$773$	−9.20163	−0.330959	−0.165480	+	0.986213i	$0.552917\pi$
$774$	0	0
$775$	0	0
$776$	1.38197	0.0496097
$777$	0	0
$778$	−8.94427	−0.320668
$779$	20.1246	0.721039
$780$	0	0
$781$	−14.1803	−0.507412
$782$	7.09017	0.253544
$783$	0	0
$784$	14.5623	0.520082
$785$	0	0
$786$	0	0
$787$	34.7082	1.23721	0.618607	−	0.785701i	$-0.287697\pi$
$787$	34.7082	1.23721	0.618607	+	0.785701i	$0.287697\pi$
$788$	−14.4164	−0.513563
$789$	0	0
$790$	0	0
$791$	−75.7426	−2.69310
$792$	0	0
$793$	−19.5623	−0.694678
$794$	16.1246	0.572241
$795$	0	0
$796$	−13.0902	−0.463969
$797$	−27.7984	−0.984669	−0.492334	−	0.870406i	$-0.663856\pi$
$797$	−27.7984	−0.984669	−0.492334	+	0.870406i	$0.663856\pi$
$798$	0	0
$799$	−9.61803	−0.340262
$800$	0	0
$801$	0	0
$802$	6.40325	0.226107
$803$	−12.6180	−0.445281
$804$	0	0
$805$	0	0
$806$	−3.27051	−0.115199
$807$	0	0
$808$	11.3820	0.400416
$809$	43.4164	1.52644	0.763220	−	0.646139i	$-0.223617\pi$
$809$	43.4164	1.52644	0.763220	+	0.646139i	$0.223617\pi$
$810$	0	0
$811$	17.0000	0.596951	0.298475	−	0.954417i	$-0.403522\pi$
$811$	17.0000	0.596951	0.298475	+	0.954417i	$0.403522\pi$
$812$	−22.5623	−0.791782
$813$	0	0
$814$	−3.56231	−0.124859
$815$	0	0
$816$	0	0
$817$	−40.2492	−1.40814
$818$	−12.4377	−0.434874
$819$	0	0
$820$	0	0
$821$	40.3607	1.40860	0.704299	−	0.709904i	$-0.251262\pi$
$821$	40.3607	1.40860	0.704299	+	0.709904i	$0.251262\pi$
$822$	0	0
$823$	0.583592	0.0203427	0.0101714	−	0.999948i	$-0.496762\pi$
$823$	0.583592	0.0203427	0.0101714	+	0.999948i	$0.496762\pi$
$824$	−17.0344	−0.593423
$825$	0	0
$826$	−22.5623	−0.785043
$827$	26.0689	0.906504	0.453252	−	0.891382i	$-0.350264\pi$
$827$	26.0689	0.906504	0.453252	+	0.891382i	$0.350264\pi$
$828$	0	0
$829$	51.1033	1.77489	0.887446	−	0.460912i	$-0.152478\pi$
$829$	51.1033	1.77489	0.887446	+	0.460912i	$0.152478\pi$
$830$	0	0
$831$	0	0
$832$	−0.416408	−0.0144363
$833$	−12.7082	−0.440313
$834$	0	0
$835$	0	0
$836$	10.8541	0.375397
$837$	0	0
$838$	0.729490	0.0251998
$839$	−43.3394	−1.49624	−0.748121	−	0.663562i	$-0.769044\pi$
$839$	−43.3394	−1.49624	−0.748121	+	0.663562i	$0.769044\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	13.1459	0.453038
$843$	0	0
$844$	−27.5066	−0.946815
$845$	0	0
$846$	0	0
$847$	3.85410	0.132429
$848$	11.7295	0.402792
$849$	0	0
$850$	0	0
$851$	−40.8673	−1.40091
$852$	0	0
$853$	43.1459	1.47729	0.738644	−	0.674096i	$-0.235467\pi$
$853$	43.1459	1.47729	0.738644	+	0.674096i	$0.235467\pi$
$854$	−26.4164	−0.903951
$855$	0	0
$856$	0.527864	0.0180420
$857$	30.8197	1.05278	0.526390	−	0.850243i	$-0.323545\pi$
$857$	30.8197	1.05278	0.526390	+	0.850243i	$0.323545\pi$
$858$	0	0
$859$	−6.18034	−0.210870	−0.105435	−	0.994426i	$-0.533624\pi$
$859$	−6.18034	−0.210870	−0.105435	+	0.994426i	$0.533624\pi$
$860$	0	0
$861$	0	0
$862$	−14.3262	−0.487954
$863$	−48.5967	−1.65425	−0.827126	−	0.562016i	$-0.810026\pi$
$863$	−48.5967	−1.65425	−0.827126	+	0.562016i	$0.810026\pi$
$864$	0	0
$865$	0	0
$866$	10.4377	0.354687
$867$	0	0
$868$	18.7082	0.634998
$869$	0.854102	0.0289734
$870$	0	0
$871$	14.1115	0.478148
$872$	−18.0902	−0.612610
$873$	0	0
$874$	−29.3951	−0.994305
$875$	0	0
$876$	0	0
$877$	31.4164	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$877$	31.4164	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$878$	−21.1803	−0.714802
$879$	0	0
$880$	0	0
$881$	1.09017	0.0367288	0.0183644	−	0.999831i	$-0.494154\pi$
$881$	1.09017	0.0367288	0.0183644	+	0.999831i	$0.494154\pi$
$882$	0	0
$883$	−0.347524	−0.0116951	−0.00584756	−	0.999983i	$-0.501861\pi$
$883$	−0.347524	−0.0116951	−0.00584756	+	0.999983i	$0.501861\pi$
$884$	4.61803	0.155321
$885$	0	0
$886$	3.30495	0.111032
$887$	42.7771	1.43631	0.718157	−	0.695881i	$-0.244986\pi$
$887$	42.7771	1.43631	0.718157	+	0.695881i	$0.244986\pi$
$888$	0	0
$889$	6.23607	0.209151
$890$	0	0
$891$	0	0
$892$	30.5623	1.02330
$893$	39.8754	1.33438
$894$	0	0
$895$	0	0
$896$	−43.8673	−1.46550
$897$	0	0
$898$	−0.201626	−0.00672835
$899$	−10.8541	−0.362005
$900$	0	0
$901$	−10.2361	−0.341013
$902$	−1.85410	−0.0617348
$903$	0	0
$904$	43.9443	1.46156
$905$	0	0
$906$	0	0
$907$	−38.8328	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$907$	−38.8328	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$908$	8.14590	0.270331
$909$	0	0
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	−16.8541	−0.557789
$914$	24.0902	0.796832
$915$	0	0
$916$	27.5623	0.910684
$917$	4.20163	0.138750
$918$	0	0
$919$	−3.41641	−0.112697	−0.0563484	−	0.998411i	$-0.517946\pi$
$919$	−3.41641	−0.112697	−0.0563484	+	0.998411i	$0.517946\pi$
$920$	0	0
$921$	0	0
$922$	−8.14590	−0.268271
$923$	25.0132	0.823318
$924$	0	0
$925$	0	0
$926$	−20.6180	−0.677551
$927$	0	0
$928$	20.3262	0.667241
$929$	−5.40325	−0.177275	−0.0886375	−	0.996064i	$-0.528251\pi$
$929$	−5.40325	−0.177275	−0.0886375	+	0.996064i	$0.528251\pi$
$930$	0	0
$931$	52.6869	1.72674
$932$	−36.4164	−1.19286
$933$	0	0
$934$	−17.6312	−0.576910
$935$	0	0
$936$	0	0
$937$	14.8328	0.484567	0.242283	−	0.970206i	$-0.422104\pi$
$937$	14.8328	0.484567	0.242283	+	0.970206i	$0.422104\pi$
$938$	19.0557	0.622192
$939$	0	0
$940$	0	0
$941$	47.7214	1.55567	0.777836	−	0.628467i	$-0.216317\pi$
$941$	47.7214	1.55567	0.777836	+	0.628467i	$0.216317\pi$
$942$	0	0
$943$	−21.2705	−0.692663
$944$	−17.5623	−0.571604
$945$	0	0
$946$	3.70820	0.120564
$947$	−5.36068	−0.174199	−0.0870993	−	0.996200i	$-0.527760\pi$
$947$	−5.36068	−0.174199	−0.0870993	+	0.996200i	$0.527760\pi$
$948$	0	0
$949$	22.2574	0.722504
$950$	0	0
$951$	0	0
$952$	13.9443	0.451936
$953$	0.472136	0.0152940	0.00764699	−	0.999971i	$-0.497566\pi$
$953$	0.472136	0.0152940	0.00764699	+	0.999971i	$0.497566\pi$
$954$	0	0
$955$	0	0
$956$	13.9443	0.450990
$957$	0	0
$958$	−0.978714	−0.0316208
$959$	56.1246	1.81236
$960$	0	0
$961$	−22.0000	−0.709677
$962$	6.28367	0.202594
$963$	0	0
$964$	19.8541	0.639458
$965$	0	0
$966$	0	0
$967$	3.72949	0.119932	0.0599662	−	0.998200i	$-0.480901\pi$
$967$	3.72949	0.119932	0.0599662	+	0.998200i	$0.480901\pi$
$968$	−2.23607	−0.0718699
$969$	0	0
$970$	0	0
$971$	−35.0902	−1.12610	−0.563049	−	0.826424i	$-0.690372\pi$
$971$	−35.0902	−1.12610	−0.563049	+	0.826424i	$0.690372\pi$
$972$	0	0
$973$	−64.3951	−2.06441
$974$	−2.21478	−0.0709662
$975$	0	0
$976$	−20.5623	−0.658183
$977$	1.06888	0.0341966	0.0170983	−	0.999854i	$-0.494557\pi$
$977$	1.06888	0.0341966	0.0170983	+	0.999854i	$0.494557\pi$
$978$	0	0
$979$	−18.0902	−0.578164
$980$	0	0
$981$	0	0
$982$	18.0344	0.575502
$983$	10.4721	0.334009	0.167005	−	0.985956i	$-0.446591\pi$
$983$	10.4721	0.334009	0.167005	+	0.985956i	$0.446591\pi$
$984$	0	0
$985$	0	0
$986$	−3.61803	−0.115222
$987$	0	0
$988$	−19.1459	−0.609112
$989$	42.5410	1.35273
$990$	0	0
$991$	−12.2705	−0.389786	−0.194893	−	0.980825i	$-0.562436\pi$
$991$	−12.2705	−0.389786	−0.194893	+	0.980825i	$0.562436\pi$
$992$	−16.8541	−0.535118
$993$	0	0
$994$	33.7771	1.07134
$995$	0	0
$996$	0	0
$997$	38.8541	1.23052	0.615261	−	0.788324i	$-0.289051\pi$
$997$	38.8541	1.23052	0.615261	+	0.788324i	$0.289051\pi$
$998$	3.21478	0.101762
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.n.1.2		2
3.2	odd	2		275.2.a.g.1.1	yes	2
5.2	odd	4		2475.2.c.p.199.3		4
5.3	odd	4		2475.2.c.p.199.2		4
5.4	even	2		2475.2.a.s.1.1		2
12.11	even	2		4400.2.a.bg.1.2		2
15.2	even	4		275.2.b.e.199.2		4
15.8	even	4		275.2.b.e.199.3		4
15.14	odd	2		275.2.a.d.1.2	✓	2
33.32	even	2		3025.2.a.i.1.2		2
60.23	odd	4		4400.2.b.x.4049.2		4
60.47	odd	4		4400.2.b.x.4049.3		4
60.59	even	2		4400.2.a.bv.1.1		2
165.164	even	2		3025.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.d.1.2	✓	2	15.14	odd	2
275.2.a.g.1.1	yes	2	3.2	odd	2
275.2.b.e.199.2		4	15.2	even	4
275.2.b.e.199.3		4	15.8	even	4
2475.2.a.n.1.2		2	1.1	even	1	trivial
2475.2.a.s.1.1		2	5.4	even	2
2475.2.c.p.199.2		4	5.3	odd	4
2475.2.c.p.199.3		4	5.2	odd	4
3025.2.a.i.1.2		2	33.32	even	2
3025.2.a.m.1.1		2	165.164	even	2
4400.2.a.bg.1.2		2	12.11	even	2
4400.2.a.bv.1.1		2	60.59	even	2
4400.2.b.x.4049.2		4	60.23	odd	4
4400.2.b.x.4049.3		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +3.85410 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +3.85410 q^{7} -2.23607 q^{8} -1.00000 q^{11} +1.76393 q^{13} +2.38197 q^{14} +1.85410 q^{16} -1.61803 q^{17} +6.70820 q^{19} -0.618034 q^{22} -7.09017 q^{23} +1.09017 q^{26} -6.23607 q^{28} +3.61803 q^{29} -3.00000 q^{31} +5.61803 q^{32} -1.00000 q^{34} +5.76393 q^{37} +4.14590 q^{38} +3.00000 q^{41} -6.00000 q^{43} +1.61803 q^{44} -4.38197 q^{46} +5.94427 q^{47} +7.85410 q^{49} -2.85410 q^{52} +6.32624 q^{53} -8.61803 q^{56} +2.23607 q^{58} -9.47214 q^{59} -11.0902 q^{61} -1.85410 q^{62} -0.236068 q^{64} +8.00000 q^{67} +2.61803 q^{68} +14.1803 q^{71} +12.6180 q^{73} +3.56231 q^{74} -10.8541 q^{76} -3.85410 q^{77} -0.854102 q^{79} +1.85410 q^{82} +16.8541 q^{83} -3.70820 q^{86} +2.23607 q^{88} +18.0902 q^{89} +6.79837 q^{91} +11.4721 q^{92} +3.67376 q^{94} -0.618034 q^{97} +4.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + q^7	\( 2 q - q^{2} - q^{4} + q^{7} - 2 q^{11} + 8 q^{13} + 7 q^{14} - 3 q^{16} - q^{17} + q^{22} - 3 q^{23} - 9 q^{26} - 8 q^{28} + 5 q^{29} - 6 q^{31} + 9 q^{32} - 2 q^{34} + 16 q^{37} + 15 q^{38} + 6 q^{41} - 12 q^{43} + q^{44} - 11 q^{46} - 6 q^{47} + 9 q^{49} + q^{52} - 3 q^{53} - 15 q^{56} - 10 q^{59} - 11 q^{61} + 3 q^{62} + 4 q^{64} + 16 q^{67} + 3 q^{68} + 6 q^{71} + 23 q^{73} - 13 q^{74} - 15 q^{76} - q^{77} + 5 q^{79} - 3 q^{82} + 27 q^{83} + 6 q^{86} + 25 q^{89} - 11 q^{91} + 14 q^{92} + 23 q^{94} + q^{97} + 3 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + q^7 - 2 * q^11 + 8 * q^13 + 7 * q^14 - 3 * q^16 - q^17 + q^22 - 3 * q^23 - 9 * q^26 - 8 * q^28 + 5 * q^29 - 6 * q^31 + 9 * q^32 - 2 * q^34 + 16 * q^37 + 15 * q^38 + 6 * q^41 - 12 * q^43 + q^44 - 11 * q^46 - 6 * q^47 + 9 * q^49 + q^52 - 3 * q^53 - 15 * q^56 - 10 * q^59 - 11 * q^61 + 3 * q^62 + 4 * q^64 + 16 * q^67 + 3 * q^68 + 6 * q^71 + 23 * q^73 - 13 * q^74 - 15 * q^76 - q^77 + 5 * q^79 - 3 * q^82 + 27 * q^83 + 6 * q^86 + 25 * q^89 - 11 * q^91 + 14 * q^92 + 23 * q^94 + q^97 + 3 * q^98

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