LMFDB - Embedded newform 2475.2.a.n.1.1

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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} -2.85410 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-1.61803 q^{2} +0.618034 q^{4} -2.85410 q^{7} +2.23607 q^{8} -1.00000 q^{11} +6.23607 q^{13} +4.61803 q^{14} -4.85410 q^{16} +0.618034 q^{17} -6.70820 q^{19} +1.61803 q^{22} +4.09017 q^{23} -10.0902 q^{26} -1.76393 q^{28} +1.38197 q^{29} -3.00000 q^{31} +3.38197 q^{32} -1.00000 q^{34} +10.2361 q^{37} +10.8541 q^{38} +3.00000 q^{41} -6.00000 q^{43} -0.618034 q^{44} -6.61803 q^{46} -11.9443 q^{47} +1.14590 q^{49} +3.85410 q^{52} -9.32624 q^{53} -6.38197 q^{56} -2.23607 q^{58} -0.527864 q^{59} +0.0901699 q^{61} +4.85410 q^{62} +4.23607 q^{64} +8.00000 q^{67} +0.381966 q^{68} -8.18034 q^{71} +10.3820 q^{73} -16.5623 q^{74} -4.14590 q^{76} +2.85410 q^{77} +5.85410 q^{79} -4.85410 q^{82} +10.1459 q^{83} +9.70820 q^{86} -2.23607 q^{88} +6.90983 q^{89} -17.7984 q^{91} +2.52786 q^{92} +19.3262 q^{94} +1.61803 q^{97} -1.85410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.85410	−1.07875	−0.539375	−	0.842066i	$-0.681339\pi$
$7$	−2.85410	−1.07875	−0.539375	+	0.842066i	$0.681339\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.23607	1.72957	0.864787	−	0.502139i	$-0.167453\pi$
$13$	6.23607	1.72957	0.864787	+	0.502139i	$0.167453\pi$
$14$	4.61803	1.23422
$15$	0	0
$16$	−4.85410	−1.21353
$17$	0.618034	0.149895	0.0749476	−	0.997187i	$-0.476121\pi$
$17$	0.618034	0.149895	0.0749476	+	0.997187i	$0.476121\pi$
$18$	0	0
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	0	0
$22$	1.61803	0.344966
$23$	4.09017	0.852859	0.426430	−	0.904521i	$-0.359771\pi$
$23$	4.09017	0.852859	0.426430	+	0.904521i	$0.359771\pi$
$24$	0	0
$25$	0	0
$26$	−10.0902	−1.97885
$27$	0	0
$28$	−1.76393	−0.333352
$29$	1.38197	0.256625	0.128312	−	0.991734i	$-0.459044\pi$
$29$	1.38197	0.256625	0.128312	+	0.991734i	$0.459044\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	10.2361	1.68280	0.841400	−	0.540413i	$-0.181732\pi$
$37$	10.2361	1.68280	0.841400	+	0.540413i	$0.181732\pi$
$38$	10.8541	1.76077
$39$	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	$0.651253\pi$
$44$	−0.618034	−0.0931721
$45$	0	0
$46$	−6.61803	−0.975776
$47$	−11.9443	−1.74225	−0.871126	−	0.491060i	$-0.836609\pi$
$47$	−11.9443	−1.74225	−0.871126	+	0.491060i	$0.836609\pi$
$48$	0	0
$49$	1.14590	0.163700
$50$	0	0
$51$	0	0
$52$	3.85410	0.534468
$53$	−9.32624	−1.28106	−0.640529	−	0.767934i	$-0.721285\pi$
$53$	−9.32624	−1.28106	−0.640529	+	0.767934i	$0.721285\pi$
$54$	0	0
$55$	0	0
$56$	−6.38197	−0.852826
$57$	0	0
$58$	−2.23607	−0.293610
$59$	−0.527864	−0.0687220	−0.0343610	−	0.999409i	$-0.510940\pi$
$59$	−0.527864	−0.0687220	−0.0343610	+	0.999409i	$0.510940\pi$
$60$	0	0
$61$	0.0901699	0.0115451	0.00577254	−	0.999983i	$-0.498163\pi$
$61$	0.0901699	0.0115451	0.00577254	+	0.999983i	$0.498163\pi$
$62$	4.85410	0.616472
$63$	0	0
$64$	4.23607	0.529508
$65$	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	$0.337479\pi$
$68$	0.381966	0.0463202
$69$	0	0
$70$	0	0
$71$	−8.18034	−0.970828	−0.485414	−	0.874284i	$-0.661331\pi$
$71$	−8.18034	−0.970828	−0.485414	+	0.874284i	$0.661331\pi$
$72$	0	0
$73$	10.3820	1.21512	0.607559	−	0.794275i	$-0.292149\pi$
$73$	10.3820	1.21512	0.607559	+	0.794275i	$0.292149\pi$
$74$	−16.5623	−1.92533
$75$	0	0
$76$	−4.14590	−0.475567
$77$	2.85410	0.325255
$78$	0	0
$79$	5.85410	0.658638	0.329319	−	0.944219i	$-0.393181\pi$
$79$	5.85410	0.658638	0.329319	+	0.944219i	$0.393181\pi$
$80$	0	0
$81$	0	0
$82$	−4.85410	−0.536046
$83$	10.1459	1.11366	0.556828	−	0.830627i	$-0.312018\pi$
$83$	10.1459	1.11366	0.556828	+	0.830627i	$0.312018\pi$
$84$	0	0
$85$	0	0
$86$	9.70820	1.04686
$87$	0	0
$88$	−2.23607	−0.238366
$89$	6.90983	0.732441	0.366220	−	0.930528i	$-0.380652\pi$
$89$	6.90983	0.732441	0.366220	+	0.930528i	$0.380652\pi$
$90$	0	0
$91$	−17.7984	−1.86578
$92$	2.52786	0.263548
$93$	0	0
$94$	19.3262	1.99335
$95$	0	0
$96$	0	0
$97$	1.61803	0.164286	0.0821432	−	0.996621i	$-0.473824\pi$
$97$	1.61803	0.164286	0.0821432	+	0.996621i	$0.473824\pi$
$98$	−1.85410	−0.187293
$99$	0	0
$100$	0	0
$101$	6.09017	0.605995	0.302997	−	0.952991i	$-0.402013\pi$
$101$	6.09017	0.605995	0.302997	+	0.952991i	$0.402013\pi$
$102$	0	0
$103$	5.38197	0.530301	0.265150	−	0.964207i	$-0.414578\pi$
$103$	5.38197	0.530301	0.265150	+	0.964207i	$0.414578\pi$
$104$	13.9443	1.36735
$105$	0	0
$106$	15.0902	1.46569
$107$	4.23607	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$107$	4.23607	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$108$	0	0
$109$	−3.09017	−0.295985	−0.147992	−	0.988989i	$-0.547281\pi$
$109$	−3.09017	−0.295985	−0.147992	+	0.988989i	$0.547281\pi$
$110$	0	0
$111$	0	0
$112$	13.8541	1.30909
$113$	11.6525	1.09617	0.548086	−	0.836422i	$-0.315357\pi$
$113$	11.6525	1.09617	0.548086	+	0.836422i	$0.315357\pi$
$114$	0	0
$115$	0	0
$116$	0.854102	0.0793014
$117$	0	0
$118$	0.854102	0.0786265
$119$	−1.76393	−0.161699
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.145898	−0.0132090
$123$	0	0
$124$	−1.85410	−0.166503
$125$	0	0
$126$	0	0
$127$	−0.618034	−0.0548416	−0.0274208	−	0.999624i	$-0.508729\pi$
$127$	−0.618034	−0.0548416	−0.0274208	+	0.999624i	$0.508729\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	0	0
$131$	−10.0902	−0.881582	−0.440791	−	0.897610i	$-0.645302\pi$
$131$	−10.0902	−0.881582	−0.440791	+	0.897610i	$0.645302\pi$
$132$	0	0
$133$	19.1459	1.66016
$134$	−12.9443	−1.11821
$135$	0	0
$136$	1.38197	0.118503
$137$	−5.56231	−0.475220	−0.237610	−	0.971361i	$-0.576364\pi$
$137$	−5.56231	−0.475220	−0.237610	+	0.971361i	$0.576364\pi$
$138$	0	0
$139$	−3.29180	−0.279206	−0.139603	−	0.990208i	$-0.544583\pi$
$139$	−3.29180	−0.279206	−0.139603	+	0.990208i	$0.544583\pi$
$140$	0	0
$141$	0	0
$142$	13.2361	1.11075
$143$	−6.23607	−0.521486
$144$	0	0
$145$	0	0
$146$	−16.7984	−1.39024
$147$	0	0
$148$	6.32624	0.520014
$149$	8.94427	0.732743	0.366372	−	0.930469i	$-0.380600\pi$
$149$	8.94427	0.732743	0.366372	+	0.930469i	$0.380600\pi$
$150$	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
$154$	−4.61803	−0.372132
$155$	0	0
$156$	0	0
$157$	−5.41641	−0.432276	−0.216138	−	0.976363i	$-0.569346\pi$
$157$	−5.41641	−0.432276	−0.216138	+	0.976363i	$0.569346\pi$
$158$	−9.47214	−0.753563
$159$	0	0
$160$	0	0
$161$	−11.6738	−0.920021
$162$	0	0
$163$	−6.85410	−0.536855	−0.268427	−	0.963300i	$-0.586504\pi$
$163$	−6.85410	−0.536855	−0.268427	+	0.963300i	$0.586504\pi$
$164$	1.85410	0.144781
$165$	0	0
$166$	−16.4164	−1.27416
$167$	5.29180	0.409491	0.204746	−	0.978815i	$-0.434363\pi$
$167$	5.29180	0.409491	0.204746	+	0.978815i	$0.434363\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	0	0
$172$	−3.70820	−0.282748
$173$	5.47214	0.416039	0.208019	−	0.978125i	$-0.433298\pi$
$173$	5.47214	0.416039	0.208019	+	0.978125i	$0.433298\pi$
$174$	0	0
$175$	0	0
$176$	4.85410	0.365892
$177$	0	0
$178$	−11.1803	−0.838002
$179$	13.6180	1.01786	0.508930	−	0.860808i	$-0.330041\pi$
$179$	13.6180	1.01786	0.508930	+	0.860808i	$0.330041\pi$
$180$	0	0
$181$	−6.09017	−0.452679	−0.226339	−	0.974049i	$-0.572676\pi$
$181$	−6.09017	−0.452679	−0.226339	+	0.974049i	$0.572676\pi$
$182$	28.7984	2.13468
$183$	0	0
$184$	9.14590	0.674245
$185$	0	0
$186$	0	0
$187$	−0.618034	−0.0451951
$188$	−7.38197	−0.538385
$189$	0	0
$190$	0	0
$191$	21.0902	1.52603	0.763016	−	0.646380i	$-0.223718\pi$
$191$	21.0902	1.52603	0.763016	+	0.646380i	$0.223718\pi$
$192$	0	0
$193$	12.9443	0.931749	0.465875	−	0.884851i	$-0.345740\pi$
$193$	12.9443	0.931749	0.465875	+	0.884851i	$0.345740\pi$
$194$	−2.61803	−0.187964
$195$	0	0
$196$	0.708204	0.0505860
$197$	20.0902	1.43137	0.715683	−	0.698426i	$-0.246116\pi$
$197$	20.0902	1.43137	0.715683	+	0.698426i	$0.246116\pi$
$198$	0	0
$199$	−3.09017	−0.219056	−0.109528	−	0.993984i	$-0.534934\pi$
$199$	−3.09017	−0.219056	−0.109528	+	0.993984i	$0.534934\pi$
$200$	0	0
$201$	0	0
$202$	−9.85410	−0.693332
$203$	−3.94427	−0.276834
$204$	0	0
$205$	0	0
$206$	−8.70820	−0.606729
$207$	0	0
$208$	−30.2705	−2.09888
$209$	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	$0.301030\pi$
$212$	−5.76393	−0.395868
$213$	0	0
$214$	−6.85410	−0.468537
$215$	0	0
$216$	0	0
$217$	8.56231	0.581247
$218$	5.00000	0.338643
$219$	0	0
$220$	0	0
$221$	3.85410	0.259255
$222$	0	0
$223$	16.8885	1.13094	0.565470	−	0.824769i	$-0.308695\pi$
$223$	16.8885	1.13094	0.565470	+	0.824769i	$0.308695\pi$
$224$	−9.65248	−0.644933
$225$	0	0
$226$	−18.8541	−1.25416
$227$	24.0344	1.59522	0.797611	−	0.603172i	$-0.206097\pi$
$227$	24.0344	1.59522	0.797611	+	0.603172i	$0.206097\pi$
$228$	0	0
$229$	12.0344	0.795258	0.397629	−	0.917546i	$-0.369833\pi$
$229$	12.0344	0.795258	0.397629	+	0.917546i	$0.369833\pi$
$230$	0	0
$231$	0	0
$232$	3.09017	0.202880
$233$	−15.5066	−1.01587	−0.507935	−	0.861395i	$-0.669591\pi$
$233$	−15.5066	−1.01587	−0.507935	+	0.861395i	$0.669591\pi$
$234$	0	0
$235$	0	0
$236$	−0.326238	−0.0212363
$237$	0	0
$238$	2.85410	0.185004
$239$	−6.38197	−0.412815	−0.206408	−	0.978466i	$-0.566177\pi$
$239$	−6.38197	−0.412815	−0.206408	+	0.978466i	$0.566177\pi$
$240$	0	0
$241$	21.2705	1.37015	0.685077	−	0.728471i	$-0.259769\pi$
$241$	21.2705	1.37015	0.685077	+	0.728471i	$0.259769\pi$
$242$	−1.61803	−0.104011
$243$	0	0
$244$	0.0557281	0.00356763
$245$	0	0
$246$	0	0
$247$	−41.8328	−2.66176
$248$	−6.70820	−0.425971
$249$	0	0
$250$	0	0
$251$	27.2705	1.72130	0.860650	−	0.509198i	$-0.170058\pi$
$251$	27.2705	1.72130	0.860650	+	0.509198i	$0.170058\pi$
$252$	0	0
$253$	−4.09017	−0.257147
$254$	1.00000	0.0627456
$255$	0	0
$256$	13.5623	0.847644
$257$	10.9443	0.682685	0.341342	−	0.939939i	$-0.389118\pi$
$257$	10.9443	0.682685	0.341342	+	0.939939i	$0.389118\pi$
$258$	0	0
$259$	−29.2148	−1.81532
$260$	0	0
$261$	0	0
$262$	16.3262	1.00864
$263$	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
$266$	−30.9787	−1.89943
$267$	0	0
$268$	4.94427	0.302019
$269$	−30.3262	−1.84902	−0.924512	−	0.381154i	$-0.875527\pi$
$269$	−30.3262	−1.84902	−0.924512	+	0.381154i	$0.875527\pi$
$270$	0	0
$271$	13.1803	0.800649	0.400324	−	0.916374i	$-0.368897\pi$
$271$	13.1803	0.800649	0.400324	+	0.916374i	$0.368897\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	9.00000	0.543710
$275$	0	0
$276$	0	0
$277$	−11.4721	−0.689294	−0.344647	−	0.938732i	$-0.612001\pi$
$277$	−11.4721	−0.689294	−0.344647	+	0.938732i	$0.612001\pi$
$278$	5.32624	0.319447
$279$	0	0
$280$	0	0
$281$	25.3607	1.51289	0.756446	−	0.654057i	$-0.226934\pi$
$281$	25.3607	1.51289	0.756446	+	0.654057i	$0.226934\pi$
$282$	0	0
$283$	−7.38197	−0.438812	−0.219406	−	0.975634i	$-0.570412\pi$
$283$	−7.38197	−0.438812	−0.219406	+	0.975634i	$0.570412\pi$
$284$	−5.05573	−0.300002
$285$	0	0
$286$	10.0902	0.596644
$287$	−8.56231	−0.505417
$288$	0	0
$289$	−16.6180	−0.977531
$290$	0	0
$291$	0	0
$292$	6.41641	0.375492
$293$	23.8885	1.39558	0.697792	−	0.716301i	$-0.254166\pi$
$293$	23.8885	1.39558	0.697792	+	0.716301i	$0.254166\pi$
$294$	0	0
$295$	0	0
$296$	22.8885	1.33037
$297$	0	0
$298$	−14.4721	−0.838348
$299$	25.5066	1.47508
$300$	0	0
$301$	17.1246	0.987046
$302$	4.85410	0.279322
$303$	0	0
$304$	32.5623	1.86758
$305$	0	0
$306$	0	0
$307$	33.4508	1.90914	0.954570	−	0.297985i	$-0.0963147\pi$
$307$	33.4508	1.90914	0.954570	+	0.297985i	$0.0963147\pi$
$308$	1.76393	0.100509
$309$	0	0
$310$	0	0
$311$	19.1803	1.08762	0.543809	−	0.839209i	$-0.316982\pi$
$311$	19.1803	1.08762	0.543809	+	0.839209i	$0.316982\pi$
$312$	0	0
$313$	−3.23607	−0.182913	−0.0914567	−	0.995809i	$-0.529152\pi$
$313$	−3.23607	−0.182913	−0.0914567	+	0.995809i	$0.529152\pi$
$314$	8.76393	0.494577
$315$	0	0
$316$	3.61803	0.203530
$317$	−16.6180	−0.933362	−0.466681	−	0.884426i	$-0.654550\pi$
$317$	−16.6180	−0.933362	−0.466681	+	0.884426i	$0.654550\pi$
$318$	0	0
$319$	−1.38197	−0.0773752
$320$	0	0
$321$	0	0
$322$	18.8885	1.05262
$323$	−4.14590	−0.230684
$324$	0	0
$325$	0	0
$326$	11.0902	0.614228
$327$	0	0
$328$	6.70820	0.370399
$329$	34.0902	1.87945
$330$	0	0
$331$	−19.1803	−1.05425	−0.527123	−	0.849789i	$-0.676729\pi$
$331$	−19.1803	−1.05425	−0.527123	+	0.849789i	$0.676729\pi$
$332$	6.27051	0.344139
$333$	0	0
$334$	−8.56231	−0.468509
$335$	0	0
$336$	0	0
$337$	1.41641	0.0771567	0.0385783	−	0.999256i	$-0.487717\pi$
$337$	1.41641	0.0771567	0.0385783	+	0.999256i	$0.487717\pi$
$338$	−41.8885	−2.27844
$339$	0	0
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	16.7082	0.902158
$344$	−13.4164	−0.723364
$345$	0	0
$346$	−8.85410	−0.475999
$347$	−20.5623	−1.10384	−0.551921	−	0.833896i	$-0.686105\pi$
$347$	−20.5623	−1.10384	−0.551921	+	0.833896i	$0.686105\pi$
$348$	0	0
$349$	21.8328	1.16868	0.584342	−	0.811508i	$-0.301353\pi$
$349$	21.8328	1.16868	0.584342	+	0.811508i	$0.301353\pi$
$350$	0	0
$351$	0	0
$352$	−3.38197	−0.180259
$353$	−21.3607	−1.13691	−0.568457	−	0.822713i	$-0.692459\pi$
$353$	−21.3607	−1.13691	−0.568457	+	0.822713i	$0.692459\pi$
$354$	0	0
$355$	0	0
$356$	4.27051	0.226337
$357$	0	0
$358$	−22.0344	−1.16456
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	9.85410	0.517920
$363$	0	0
$364$	−11.0000	−0.576557
$365$	0	0
$366$	0	0
$367$	7.14590	0.373013	0.186506	−	0.982454i	$-0.440283\pi$
$367$	7.14590	0.373013	0.186506	+	0.982454i	$0.440283\pi$
$368$	−19.8541	−1.03497
$369$	0	0
$370$	0	0
$371$	26.6180	1.38194
$372$	0	0
$373$	2.81966	0.145996	0.0729982	−	0.997332i	$-0.476743\pi$
$373$	2.81966	0.145996	0.0729982	+	0.997332i	$0.476743\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	−26.7082	−1.37737
$377$	8.61803	0.443851
$378$	0	0
$379$	−17.7639	−0.912472	−0.456236	−	0.889859i	$-0.650803\pi$
$379$	−17.7639	−0.912472	−0.456236	+	0.889859i	$0.650803\pi$
$380$	0	0
$381$	0	0
$382$	−34.1246	−1.74597
$383$	−5.05573	−0.258336	−0.129168	−	0.991623i	$-0.541231\pi$
$383$	−5.05573	−0.258336	−0.129168	+	0.991623i	$0.541231\pi$
$384$	0	0
$385$	0	0
$386$	−20.9443	−1.06604
$387$	0	0
$388$	1.00000	0.0507673
$389$	−5.52786	−0.280274	−0.140137	−	0.990132i	$-0.544754\pi$
$389$	−5.52786	−0.280274	−0.140137	+	0.990132i	$0.544754\pi$
$390$	0	0
$391$	2.52786	0.127840
$392$	2.56231	0.129416
$393$	0	0
$394$	−32.5066	−1.63766
$395$	0	0
$396$	0	0
$397$	14.9098	0.748303	0.374151	−	0.927368i	$-0.377934\pi$
$397$	14.9098	0.748303	0.374151	+	0.927368i	$0.377934\pi$
$398$	5.00000	0.250627
$399$	0	0
$400$	0	0
$401$	−34.3607	−1.71589	−0.857945	−	0.513741i	$-0.828259\pi$
$401$	−34.3607	−1.71589	−0.857945	+	0.513741i	$0.828259\pi$
$402$	0	0
$403$	−18.7082	−0.931922
$404$	3.76393	0.187263
$405$	0	0
$406$	6.38197	0.316732
$407$	−10.2361	−0.507383
$408$	0	0
$409$	20.1246	0.995098	0.497549	−	0.867436i	$-0.334233\pi$
$409$	20.1246	0.995098	0.497549	+	0.867436i	$0.334233\pi$
$410$	0	0
$411$	0	0
$412$	3.32624	0.163872
$413$	1.50658	0.0741338
$414$	0	0
$415$	0	0
$416$	21.0902	1.03403
$417$	0	0
$418$	−10.8541	−0.530891
$419$	−21.1803	−1.03473	−0.517364	−	0.855766i	$-0.673087\pi$
$419$	−21.1803	−1.03473	−0.517364	+	0.855766i	$0.673087\pi$
$420$	0	0
$421$	−12.2705	−0.598028	−0.299014	−	0.954249i	$-0.596658\pi$
$421$	−12.2705	−0.598028	−0.299014	+	0.954249i	$0.596658\pi$
$422$	−27.5066	−1.33900
$423$	0	0
$424$	−20.8541	−1.01276
$425$	0	0
$426$	0	0
$427$	−0.257354	−0.0124542
$428$	2.61803	0.126547
$429$	0	0
$430$	0	0
$431$	−0.819660	−0.0394816	−0.0197408	−	0.999805i	$-0.506284\pi$
$431$	−0.819660	−0.0394816	−0.0197408	+	0.999805i	$0.506284\pi$
$432$	0	0
$433$	−18.8885	−0.907725	−0.453863	−	0.891072i	$-0.649954\pi$
$433$	−18.8885	−0.907725	−0.453863	+	0.891072i	$0.649954\pi$
$434$	−13.8541	−0.665018
$435$	0	0
$436$	−1.90983	−0.0914643
$437$	−27.4377	−1.31252
$438$	0	0
$439$	−0.729490	−0.0348167	−0.0174083	−	0.999848i	$-0.505542\pi$
$439$	−0.729490	−0.0348167	−0.0174083	+	0.999848i	$0.505542\pi$
$440$	0	0
$441$	0	0
$442$	−6.23607	−0.296620
$443$	36.6525	1.74141	0.870706	−	0.491804i	$-0.163662\pi$
$443$	36.6525	1.74141	0.870706	+	0.491804i	$0.163662\pi$
$444$	0	0
$445$	0	0
$446$	−27.3262	−1.29393
$447$	0	0
$448$	−12.0902	−0.571207
$449$	15.3262	0.723290	0.361645	−	0.932316i	$-0.382215\pi$
$449$	15.3262	0.723290	0.361645	+	0.932316i	$0.382215\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	7.20163	0.338736
$453$	0	0
$454$	−38.8885	−1.82513
$455$	0	0
$456$	0	0
$457$	−7.97871	−0.373228	−0.186614	−	0.982433i	$-0.559751\pi$
$457$	−7.97871	−0.373228	−0.186614	+	0.982433i	$0.559751\pi$
$458$	−19.4721	−0.909873
$459$	0	0
$460$	0	0
$461$	9.18034	0.427571	0.213786	−	0.976881i	$-0.431421\pi$
$461$	9.18034	0.427571	0.213786	+	0.976881i	$0.431421\pi$
$462$	0	0
$463$	11.3607	0.527976	0.263988	−	0.964526i	$-0.414962\pi$
$463$	11.3607	0.527976	0.263988	+	0.964526i	$0.414962\pi$
$464$	−6.70820	−0.311421
$465$	0	0
$466$	25.0902	1.16228
$467$	−37.4721	−1.73400	−0.867002	−	0.498305i	$-0.833956\pi$
$467$	−37.4721	−1.73400	−0.867002	+	0.498305i	$0.833956\pi$
$468$	0	0
$469$	−22.8328	−1.05432
$470$	0	0
$471$	0	0
$472$	−1.18034	−0.0543295
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	−1.09017	−0.0499679
$477$	0	0
$478$	10.3262	0.472311
$479$	−28.4164	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$479$	−28.4164	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$480$	0	0
$481$	63.8328	2.91053
$482$	−34.4164	−1.56762
$483$	0	0
$484$	0.618034	0.0280925
$485$	0	0
$486$	0	0
$487$	−30.4164	−1.37830	−0.689150	−	0.724619i	$-0.742016\pi$
$487$	−30.4164	−1.37830	−0.689150	+	0.724619i	$0.742016\pi$
$488$	0.201626	0.00912719
$489$	0	0
$490$	0	0
$491$	6.81966	0.307767	0.153883	−	0.988089i	$-0.450822\pi$
$491$	6.81966	0.307767	0.153883	+	0.988089i	$0.450822\pi$
$492$	0	0
$493$	0.854102	0.0384668
$494$	67.6869	3.04538
$495$	0	0
$496$	14.5623	0.653867
$497$	23.3475	1.04728
$498$	0	0
$499$	29.7984	1.33396	0.666979	−	0.745076i	$-0.267587\pi$
$499$	29.7984	1.33396	0.666979	+	0.745076i	$0.267587\pi$
$500$	0	0
$501$	0	0
$502$	−44.1246	−1.96938
$503$	6.65248	0.296619	0.148310	−	0.988941i	$-0.452617\pi$
$503$	6.65248	0.296619	0.148310	+	0.988941i	$0.452617\pi$
$504$	0	0
$505$	0	0
$506$	6.61803	0.294207
$507$	0	0
$508$	−0.381966	−0.0169470
$509$	−16.3820	−0.726118	−0.363059	−	0.931766i	$-0.618268\pi$
$509$	−16.3820	−0.726118	−0.363059	+	0.931766i	$0.618268\pi$
$510$	0	0
$511$	−29.6312	−1.31081
$512$	5.29180	0.233867
$513$	0	0
$514$	−17.7082	−0.781075
$515$	0	0
$516$	0	0
$517$	11.9443	0.525308
$518$	47.2705	2.07695
$519$	0	0
$520$	0	0
$521$	1.81966	0.0797208	0.0398604	−	0.999205i	$-0.487309\pi$
$521$	1.81966	0.0797208	0.0398604	+	0.999205i	$0.487309\pi$
$522$	0	0
$523$	22.9443	1.00328	0.501641	−	0.865076i	$-0.332730\pi$
$523$	22.9443	1.00328	0.501641	+	0.865076i	$0.332730\pi$
$524$	−6.23607	−0.272424
$525$	0	0
$526$	−33.9787	−1.48154
$527$	−1.85410	−0.0807660
$528$	0	0
$529$	−6.27051	−0.272631
$530$	0	0
$531$	0	0
$532$	11.8328	0.513018
$533$	18.7082	0.810342
$534$	0	0
$535$	0	0
$536$	17.8885	0.772667
$537$	0	0
$538$	49.0689	2.11551
$539$	−1.14590	−0.0493573
$540$	0	0
$541$	−42.2705	−1.81735	−0.908676	−	0.417503i	$-0.862905\pi$
$541$	−42.2705	−1.81735	−0.908676	+	0.417503i	$0.862905\pi$
$542$	−21.3262	−0.916040
$543$	0	0
$544$	2.09017	0.0896153
$545$	0	0
$546$	0	0
$547$	−21.1459	−0.904133	−0.452067	−	0.891984i	$-0.649313\pi$
$547$	−21.1459	−0.904133	−0.452067	+	0.891984i	$0.649313\pi$
$548$	−3.43769	−0.146851
$549$	0	0
$550$	0	0
$551$	−9.27051	−0.394937
$552$	0	0
$553$	−16.7082	−0.710505
$554$	18.5623	0.788637
$555$	0	0
$556$	−2.03444	−0.0862796
$557$	9.76393	0.413711	0.206856	−	0.978371i	$-0.433677\pi$
$557$	9.76393	0.413711	0.206856	+	0.978371i	$0.433677\pi$
$558$	0	0
$559$	−37.4164	−1.58255
$560$	0	0
$561$	0	0
$562$	−41.0344	−1.73093
$563$	13.0344	0.549336	0.274668	−	0.961539i	$-0.411432\pi$
$563$	13.0344	0.549336	0.274668	+	0.961539i	$0.411432\pi$
$564$	0	0
$565$	0	0
$566$	11.9443	0.502055
$567$	0	0
$568$	−18.2918	−0.767507
$569$	−26.3820	−1.10599	−0.552995	−	0.833185i	$-0.686515\pi$
$569$	−26.3820	−1.10599	−0.552995	+	0.833185i	$0.686515\pi$
$570$	0	0
$571$	36.2705	1.51787	0.758937	−	0.651164i	$-0.225719\pi$
$571$	36.2705	1.51787	0.758937	+	0.651164i	$0.225719\pi$
$572$	−3.85410	−0.161148
$573$	0	0
$574$	13.8541	0.578259
$575$	0	0
$576$	0	0
$577$	15.5623	0.647867	0.323934	−	0.946080i	$-0.394995\pi$
$577$	15.5623	0.647867	0.323934	+	0.946080i	$0.394995\pi$
$578$	26.8885	1.11842
$579$	0	0
$580$	0	0
$581$	−28.9574	−1.20136
$582$	0	0
$583$	9.32624	0.386253
$584$	23.2148	0.960635
$585$	0	0
$586$	−38.6525	−1.59672
$587$	4.03444	0.166519	0.0832596	−	0.996528i	$-0.473467\pi$
$587$	4.03444	0.166519	0.0832596	+	0.996528i	$0.473467\pi$
$588$	0	0
$589$	20.1246	0.829220
$590$	0	0
$591$	0	0
$592$	−49.6869	−2.04212
$593$	−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
$596$	5.52786	0.226430
$597$	0	0
$598$	−41.2705	−1.68768
$599$	−0.326238	−0.0133297	−0.00666486	−	0.999978i	$-0.502122\pi$
$599$	−0.326238	−0.0133297	−0.00666486	+	0.999978i	$0.502122\pi$
$600$	0	0
$601$	−22.2705	−0.908433	−0.454217	−	0.890891i	$-0.650081\pi$
$601$	−22.2705	−0.908433	−0.454217	+	0.890891i	$0.650081\pi$
$602$	−27.7082	−1.12930
$603$	0	0
$604$	−1.85410	−0.0754423
$605$	0	0
$606$	0	0
$607$	3.52786	0.143192	0.0715958	−	0.997434i	$-0.477191\pi$
$607$	3.52786	0.143192	0.0715958	+	0.997434i	$0.477191\pi$
$608$	−22.6869	−0.920076
$609$	0	0
$610$	0	0
$611$	−74.4853	−3.01335
$612$	0	0
$613$	6.43769	0.260016	0.130008	−	0.991513i	$-0.458500\pi$
$613$	6.43769	0.260016	0.130008	+	0.991513i	$0.458500\pi$
$614$	−54.1246	−2.18429
$615$	0	0
$616$	6.38197	0.257137
$617$	38.1803	1.53708	0.768541	−	0.639800i	$-0.220983\pi$
$617$	38.1803	1.53708	0.768541	+	0.639800i	$0.220983\pi$
$618$	0	0
$619$	−22.8885	−0.919968	−0.459984	−	0.887927i	$-0.652145\pi$
$619$	−22.8885	−0.919968	−0.459984	+	0.887927i	$0.652145\pi$
$620$	0	0
$621$	0	0
$622$	−31.0344	−1.24437
$623$	−19.7214	−0.790120
$624$	0	0
$625$	0	0
$626$	5.23607	0.209275
$627$	0	0
$628$	−3.34752	−0.133581
$629$	6.32624	0.252244
$630$	0	0
$631$	36.2705	1.44391	0.721953	−	0.691942i	$-0.243245\pi$
$631$	36.2705	1.44391	0.721953	+	0.691942i	$0.243245\pi$
$632$	13.0902	0.520699
$633$	0	0
$634$	26.8885	1.06788
$635$	0	0
$636$	0	0
$637$	7.14590	0.283131
$638$	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
$643$	10.5836	0.417376	0.208688	−	0.977982i	$-0.433081\pi$
$643$	10.5836	0.417376	0.208688	+	0.977982i	$0.433081\pi$
$644$	−7.21478	−0.284302
$645$	0	0
$646$	6.70820	0.263931
$647$	−12.3475	−0.485431	−0.242716	−	0.970097i	$-0.578038\pi$
$647$	−12.3475	−0.485431	−0.242716	+	0.970097i	$0.578038\pi$
$648$	0	0
$649$	0.527864	0.0207205
$650$	0	0
$651$	0	0
$652$	−4.23607	−0.165897
$653$	−7.74265	−0.302993	−0.151497	−	0.988458i	$-0.548409\pi$
$653$	−7.74265	−0.302993	−0.151497	+	0.988458i	$0.548409\pi$
$654$	0	0
$655$	0	0
$656$	−14.5623	−0.568563
$657$	0	0
$658$	−55.1591	−2.15032
$659$	−9.27051	−0.361128	−0.180564	−	0.983563i	$-0.557792\pi$
$659$	−9.27051	−0.361128	−0.180564	+	0.983563i	$0.557792\pi$
$660$	0	0
$661$	−49.1803	−1.91289	−0.956447	−	0.291907i	$-0.905710\pi$
$661$	−49.1803	−1.91289	−0.956447	+	0.291907i	$0.905710\pi$
$662$	31.0344	1.20619
$663$	0	0
$664$	22.6869	0.880423
$665$	0	0
$666$	0	0
$667$	5.65248	0.218865
$668$	3.27051	0.126540
$669$	0	0
$670$	0	0
$671$	−0.0901699	−0.00348097
$672$	0	0
$673$	2.41641	0.0931457	0.0465728	−	0.998915i	$-0.485170\pi$
$673$	2.41641	0.0931457	0.0465728	+	0.998915i	$0.485170\pi$
$674$	−2.29180	−0.0882767
$675$	0	0
$676$	16.0000	0.615385
$677$	−28.6525	−1.10120	−0.550602	−	0.834768i	$-0.685602\pi$
$677$	−28.6525	−1.10120	−0.550602	+	0.834768i	$0.685602\pi$
$678$	0	0
$679$	−4.61803	−0.177224
$680$	0	0
$681$	0	0
$682$	−4.85410	−0.185873
$683$	43.3607	1.65915	0.829575	−	0.558395i	$-0.188583\pi$
$683$	43.3607	1.65915	0.829575	+	0.558395i	$0.188583\pi$
$684$	0	0
$685$	0	0
$686$	−27.0344	−1.03218
$687$	0	0
$688$	29.1246	1.11037
$689$	−58.1591	−2.21568
$690$	0	0
$691$	30.0902	1.14468	0.572342	−	0.820015i	$-0.306035\pi$
$691$	30.0902	1.14468	0.572342	+	0.820015i	$0.306035\pi$
$692$	3.38197	0.128563
$693$	0	0
$694$	33.2705	1.26293
$695$	0	0
$696$	0	0
$697$	1.85410	0.0702291
$698$	−35.3262	−1.33712
$699$	0	0
$700$	0	0
$701$	0.360680	0.0136227	0.00681134	−	0.999977i	$-0.497832\pi$
$701$	0.360680	0.0136227	0.00681134	+	0.999977i	$0.497832\pi$
$702$	0	0
$703$	−68.6656	−2.58977
$704$	−4.23607	−0.159653
$705$	0	0
$706$	34.5623	1.30077
$707$	−17.3820	−0.653716
$708$	0	0
$709$	−41.3050	−1.55124	−0.775620	−	0.631200i	$-0.782563\pi$
$709$	−41.3050	−1.55124	−0.775620	+	0.631200i	$0.782563\pi$
$710$	0	0
$711$	0	0
$712$	15.4508	0.579045
$713$	−12.2705	−0.459534
$714$	0	0
$715$	0	0
$716$	8.41641	0.314536
$717$	0	0
$718$	−7.23607	−0.270048
$719$	15.6525	0.583739	0.291869	−	0.956458i	$-0.405723\pi$
$719$	15.6525	0.583739	0.291869	+	0.956458i	$0.405723\pi$
$720$	0	0
$721$	−15.3607	−0.572062
$722$	−42.0689	−1.56564
$723$	0	0
$724$	−3.76393	−0.139885
$725$	0	0
$726$	0	0
$727$	−41.2705	−1.53064	−0.765319	−	0.643651i	$-0.777419\pi$
$727$	−41.2705	−1.53064	−0.765319	+	0.643651i	$0.777419\pi$
$728$	−39.7984	−1.47503
$729$	0	0
$730$	0	0
$731$	−3.70820	−0.137153
$732$	0	0
$733$	45.8328	1.69287	0.846437	−	0.532489i	$-0.178743\pi$
$733$	45.8328	1.69287	0.846437	+	0.532489i	$0.178743\pi$
$734$	−11.5623	−0.426772
$735$	0	0
$736$	13.8328	0.509884
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−29.1459	−1.07215	−0.536075	−	0.844171i	$-0.680093\pi$
$739$	−29.1459	−1.07215	−0.536075	+	0.844171i	$0.680093\pi$
$740$	0	0
$741$	0	0
$742$	−43.0689	−1.58111
$743$	−18.6738	−0.685074	−0.342537	−	0.939504i	$-0.611286\pi$
$743$	−18.6738	−0.685074	−0.342537	+	0.939504i	$0.611286\pi$
$744$	0	0
$745$	0	0
$746$	−4.56231	−0.167038
$747$	0	0
$748$	−0.381966	−0.0139661
$749$	−12.0902	−0.441765
$750$	0	0
$751$	11.2705	0.411267	0.205633	−	0.978629i	$-0.434075\pi$
$751$	11.2705	0.411267	0.205633	+	0.978629i	$0.434075\pi$
$752$	57.9787	2.11427
$753$	0	0
$754$	−13.9443	−0.507820
$755$	0	0
$756$	0	0
$757$	−21.4721	−0.780418	−0.390209	−	0.920726i	$-0.627597\pi$
$757$	−21.4721	−0.780418	−0.390209	+	0.920726i	$0.627597\pi$
$758$	28.7426	1.04398
$759$	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
$763$	8.81966	0.319293
$764$	13.0344	0.471570
$765$	0	0
$766$	8.18034	0.295568
$767$	−3.29180	−0.118860
$768$	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
$773$	−33.7984	−1.21564	−0.607822	−	0.794074i	$-0.707956\pi$
$773$	−33.7984	−1.21564	−0.607822	+	0.794074i	$0.707956\pi$
$774$	0	0
$775$	0	0
$776$	3.61803	0.129880
$777$	0	0
$778$	8.94427	0.320668
$779$	−20.1246	−0.721039
$780$	0	0
$781$	8.18034	0.292716
$782$	−4.09017	−0.146264
$783$	0	0
$784$	−5.56231	−0.198654
$785$	0	0
$786$	0	0
$787$	21.2918	0.758971	0.379485	−	0.925198i	$-0.376101\pi$
$787$	21.2918	0.758971	0.379485	+	0.925198i	$0.376101\pi$
$788$	12.4164	0.442316
$789$	0	0
$790$	0	0
$791$	−33.2574	−1.18250
$792$	0	0
$793$	0.562306	0.0199681
$794$	−24.1246	−0.856150
$795$	0	0
$796$	−1.90983	−0.0676921
$797$	−3.20163	−0.113407	−0.0567037	−	0.998391i	$-0.518059\pi$
$797$	−3.20163	−0.113407	−0.0567037	+	0.998391i	$0.518059\pi$
$798$	0	0
$799$	−7.38197	−0.261155
$800$	0	0
$801$	0	0
$802$	55.5967	1.96319
$803$	−10.3820	−0.366372
$804$	0	0
$805$	0	0
$806$	30.2705	1.06623
$807$	0	0
$808$	13.6180	0.479081
$809$	16.5836	0.583048	0.291524	−	0.956564i	$-0.405838\pi$
$809$	16.5836	0.583048	0.291524	+	0.956564i	$0.405838\pi$
$810$	0	0
$811$	17.0000	0.596951	0.298475	−	0.954417i	$-0.403522\pi$
$811$	17.0000	0.596951	0.298475	+	0.954417i	$0.403522\pi$
$812$	−2.43769	−0.0855463
$813$	0	0
$814$	16.5623	0.580509
$815$	0	0
$816$	0	0
$817$	40.2492	1.40814
$818$	−32.5623	−1.13851
$819$	0	0
$820$	0	0
$821$	−4.36068	−0.152189	−0.0760944	−	0.997101i	$-0.524245\pi$
$821$	−4.36068	−0.152189	−0.0760944	+	0.997101i	$0.524245\pi$
$822$	0	0
$823$	27.4164	0.955676	0.477838	−	0.878448i	$-0.341421\pi$
$823$	27.4164	0.955676	0.477838	+	0.878448i	$0.341421\pi$
$824$	12.0344	0.419240
$825$	0	0
$826$	−2.43769	−0.0848182
$827$	−32.0689	−1.11514	−0.557572	−	0.830128i	$-0.688267\pi$
$827$	−32.0689	−1.11514	−0.557572	+	0.830128i	$0.688267\pi$
$828$	0	0
$829$	−36.1033	−1.25392	−0.626960	−	0.779051i	$-0.715701\pi$
$829$	−36.1033	−1.25392	−0.626960	+	0.779051i	$0.715701\pi$
$830$	0	0
$831$	0	0
$832$	26.4164	0.915824
$833$	0.708204	0.0245378
$834$	0	0
$835$	0	0
$836$	4.14590	0.143389
$837$	0	0
$838$	34.2705	1.18386
$839$	48.3394	1.66886	0.834431	−	0.551113i	$-0.185797\pi$
$839$	48.3394	1.66886	0.834431	+	0.551113i	$0.185797\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	19.8541	0.684218
$843$	0	0
$844$	10.5066	0.361651
$845$	0	0
$846$	0	0
$847$	−2.85410	−0.0980681
$848$	45.2705	1.55460
$849$	0	0
$850$	0	0
$851$	41.8673	1.43519
$852$	0	0
$853$	49.8541	1.70697	0.853486	−	0.521116i	$-0.174484\pi$
$853$	49.8541	1.70697	0.853486	+	0.521116i	$0.174484\pi$
$854$	0.416408	0.0142492
$855$	0	0
$856$	9.47214	0.323751
$857$	53.1803	1.81661	0.908303	−	0.418313i	$-0.137379\pi$
$857$	53.1803	1.81661	0.908303	+	0.418313i	$0.137379\pi$
$858$	0	0
$859$	16.1803	0.552066	0.276033	−	0.961148i	$-0.410980\pi$
$859$	16.1803	0.552066	0.276033	+	0.961148i	$0.410980\pi$
$860$	0	0
$861$	0	0
$862$	1.32624	0.0451718
$863$	0.596748	0.0203135	0.0101568	−	0.999948i	$-0.496767\pi$
$863$	0.596748	0.0203135	0.0101568	+	0.999948i	$0.496767\pi$
$864$	0	0
$865$	0	0
$866$	30.5623	1.03855
$867$	0	0
$868$	5.29180	0.179615
$869$	−5.85410	−0.198587
$870$	0	0
$871$	49.8885	1.69041
$872$	−6.90983	−0.233996
$873$	0	0
$874$	44.3951	1.50169
$875$	0	0
$876$	0	0
$877$	4.58359	0.154777	0.0773885	−	0.997001i	$-0.475342\pi$
$877$	4.58359	0.154777	0.0773885	+	0.997001i	$0.475342\pi$
$878$	1.18034	0.0398345
$879$	0	0
$880$	0	0
$881$	−10.0902	−0.339946	−0.169973	−	0.985449i	$-0.554368\pi$
$881$	−10.0902	−0.339946	−0.169973	+	0.985449i	$0.554368\pi$
$882$	0	0
$883$	−31.6525	−1.06519	−0.532595	−	0.846370i	$-0.678783\pi$
$883$	−31.6525	−1.06519	−0.532595	+	0.846370i	$0.678783\pi$
$884$	2.38197	0.0801142
$885$	0	0
$886$	−59.3050	−1.99239
$887$	−28.7771	−0.966240	−0.483120	−	0.875554i	$-0.660497\pi$
$887$	−28.7771	−0.966240	−0.483120	+	0.875554i	$0.660497\pi$
$888$	0	0
$889$	1.76393	0.0591604
$890$	0	0
$891$	0	0
$892$	10.4377	0.349480
$893$	80.1246	2.68127
$894$	0	0
$895$	0	0
$896$	38.8673	1.29846
$897$	0	0
$898$	−24.7984	−0.827532
$899$	−4.14590	−0.138273
$900$	0	0
$901$	−5.76393	−0.192024
$902$	4.85410	0.161624
$903$	0	0
$904$	26.0557	0.866601
$905$	0	0
$906$	0	0
$907$	14.8328	0.492516	0.246258	−	0.969204i	$-0.420799\pi$
$907$	14.8328	0.492516	0.246258	+	0.969204i	$0.420799\pi$
$908$	14.8541	0.492951
$909$	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
$913$	−10.1459	−0.335780
$914$	12.9098	0.427019
$915$	0	0
$916$	7.43769	0.245748
$917$	28.7984	0.951006
$918$	0	0
$919$	23.4164	0.772436	0.386218	−	0.922408i	$-0.373781\pi$
$919$	23.4164	0.772436	0.386218	+	0.922408i	$0.373781\pi$
$920$	0	0
$921$	0	0
$922$	−14.8541	−0.489194
$923$	−51.0132	−1.67912
$924$	0	0
$925$	0	0
$926$	−18.3820	−0.604069
$927$	0	0
$928$	4.67376	0.153424
$929$	−54.5967	−1.79126	−0.895631	−	0.444799i	$-0.853275\pi$
$929$	−54.5967	−1.79126	−0.895631	+	0.444799i	$0.853275\pi$
$930$	0	0
$931$	−7.68692	−0.251929
$932$	−9.58359	−0.313921
$933$	0	0
$934$	60.6312	1.98391
$935$	0	0
$936$	0	0
$937$	−38.8328	−1.26861	−0.634306	−	0.773082i	$-0.718714\pi$
$937$	−38.8328	−1.26861	−0.634306	+	0.773082i	$0.718714\pi$
$938$	36.9443	1.20627
$939$	0	0
$940$	0	0
$941$	−41.7214	−1.36008	−0.680039	−	0.733176i	$-0.738037\pi$
$941$	−41.7214	−1.36008	−0.680039	+	0.733176i	$0.738037\pi$
$942$	0	0
$943$	12.2705	0.399583
$944$	2.56231	0.0833960
$945$	0	0
$946$	−9.70820	−0.315641
$947$	39.3607	1.27905	0.639525	−	0.768770i	$-0.279131\pi$
$947$	39.3607	1.27905	0.639525	+	0.768770i	$0.279131\pi$
$948$	0	0
$949$	64.7426	2.10164
$950$	0	0
$951$	0	0
$952$	−3.94427	−0.127835
$953$	−8.47214	−0.274439	−0.137220	−	0.990541i	$-0.543817\pi$
$953$	−8.47214	−0.274439	−0.137220	+	0.990541i	$0.543817\pi$
$954$	0	0
$955$	0	0
$956$	−3.94427	−0.127567
$957$	0	0
$958$	45.9787	1.48550
$959$	15.8754	0.512643
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−103.284	−3.33000
$963$	0	0
$964$	13.1459	0.423401
$965$	0	0
$966$	0	0
$967$	37.2705	1.19854	0.599269	−	0.800547i	$-0.295458\pi$
$967$	37.2705	1.19854	0.599269	+	0.800547i	$0.295458\pi$
$968$	2.23607	0.0718699
$969$	0	0
$970$	0	0
$971$	−23.9098	−0.767303	−0.383651	−	0.923478i	$-0.625334\pi$
$971$	−23.9098	−0.767303	−0.383651	+	0.923478i	$0.625334\pi$
$972$	0	0
$973$	9.39512	0.301194
$974$	49.2148	1.57694
$975$	0	0
$976$	−0.437694	−0.0140102
$977$	−57.0689	−1.82580	−0.912898	−	0.408188i	$-0.866161\pi$
$977$	−57.0689	−1.82580	−0.912898	+	0.408188i	$0.866161\pi$
$978$	0	0
$979$	−6.90983	−0.220839
$980$	0	0
$981$	0	0
$982$	−11.0344	−0.352123
$983$	1.52786	0.0487313	0.0243656	−	0.999703i	$-0.492243\pi$
$983$	1.52786	0.0487313	0.0243656	+	0.999703i	$0.492243\pi$
$984$	0	0
$985$	0	0
$986$	−1.38197	−0.0440108
$987$	0	0
$988$	−25.8541	−0.822529
$989$	−24.5410	−0.780359
$990$	0	0
$991$	21.2705	0.675680	0.337840	−	0.941204i	$-0.390304\pi$
$991$	21.2705	0.675680	0.337840	+	0.941204i	$0.390304\pi$
$992$	−10.1459	−0.322133
$993$	0	0
$994$	−37.7771	−1.19822
$995$	0	0
$996$	0	0
$997$	32.1459	1.01807	0.509035	−	0.860746i	$-0.330002\pi$
$997$	32.1459	1.01807	0.509035	+	0.860746i	$0.330002\pi$
$998$	−48.2148	−1.52621
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.n.1.1		2
3.2	odd	2		275.2.a.g.1.2	yes	2
5.2	odd	4		2475.2.c.p.199.1		4
5.3	odd	4		2475.2.c.p.199.4		4
5.4	even	2		2475.2.a.s.1.2		2
12.11	even	2		4400.2.a.bg.1.1		2
15.2	even	4		275.2.b.e.199.4		4
15.8	even	4		275.2.b.e.199.1		4
15.14	odd	2		275.2.a.d.1.1	✓	2
33.32	even	2		3025.2.a.i.1.1		2
60.23	odd	4		4400.2.b.x.4049.1		4
60.47	odd	4		4400.2.b.x.4049.4		4
60.59	even	2		4400.2.a.bv.1.2		2
165.164	even	2		3025.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.d.1.1	✓	2	15.14	odd	2
275.2.a.g.1.2	yes	2	3.2	odd	2
275.2.b.e.199.1		4	15.8	even	4
275.2.b.e.199.4		4	15.2	even	4
2475.2.a.n.1.1		2	1.1	even	1	trivial
2475.2.a.s.1.2		2	5.4	even	2
2475.2.c.p.199.1		4	5.2	odd	4
2475.2.c.p.199.4		4	5.3	odd	4
3025.2.a.i.1.1		2	33.32	even	2
3025.2.a.m.1.2		2	165.164	even	2
4400.2.a.bg.1.1		2	12.11	even	2
4400.2.a.bv.1.2		2	60.59	even	2
4400.2.b.x.4049.1		4	60.23	odd	4
4400.2.b.x.4049.4		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)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} -2.85410 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} -2.85410 q^{7} +2.23607 q^{8} -1.00000 q^{11} +6.23607 q^{13} +4.61803 q^{14} -4.85410 q^{16} +0.618034 q^{17} -6.70820 q^{19} +1.61803 q^{22} +4.09017 q^{23} -10.0902 q^{26} -1.76393 q^{28} +1.38197 q^{29} -3.00000 q^{31} +3.38197 q^{32} -1.00000 q^{34} +10.2361 q^{37} +10.8541 q^{38} +3.00000 q^{41} -6.00000 q^{43} -0.618034 q^{44} -6.61803 q^{46} -11.9443 q^{47} +1.14590 q^{49} +3.85410 q^{52} -9.32624 q^{53} -6.38197 q^{56} -2.23607 q^{58} -0.527864 q^{59} +0.0901699 q^{61} +4.85410 q^{62} +4.23607 q^{64} +8.00000 q^{67} +0.381966 q^{68} -8.18034 q^{71} +10.3820 q^{73} -16.5623 q^{74} -4.14590 q^{76} +2.85410 q^{77} +5.85410 q^{79} -4.85410 q^{82} +10.1459 q^{83} +9.70820 q^{86} -2.23607 q^{88} +6.90983 q^{89} -17.7984 q^{91} +2.52786 q^{92} +19.3262 q^{94} +1.61803 q^{97} -1.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + q^{7}+O(q^{10}) \) 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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