LMFDB - Embedded newform 4275.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.167449$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.167449 q^{2} -1.97196 q^{4} -4.13941 q^{7} -0.665102 q^{8} +4.80451 q^{11} -4.00000 q^{13} -0.693141 q^{14} +3.83255 q^{16} +5.94392 q^{17} -1.00000 q^{19} +0.804512 q^{22} -7.13941 q^{23} -0.669797 q^{26} +8.16275 q^{28} +5.00000 q^{29} +8.80451 q^{31} +1.97196 q^{32} +0.995305 q^{34} -3.66510 q^{37} -0.167449 q^{38} -0.195488 q^{41} -4.00000 q^{43} -9.47431 q^{44} -1.19549 q^{46} +11.6090 q^{47} +10.1347 q^{49} +7.88784 q^{52} -2.33020 q^{53} +2.75313 q^{56} +0.837246 q^{58} -7.46961 q^{59} -6.60902 q^{61} +1.47431 q^{62} -7.33490 q^{64} -1.19549 q^{67} -11.7212 q^{68} -9.46961 q^{71} -0.330203 q^{73} -0.613718 q^{74} +1.97196 q^{76} -19.8878 q^{77} +13.4182 q^{79} -0.0327344 q^{82} -2.80451 q^{83} -0.669797 q^{86} -3.19549 q^{88} -7.27882 q^{89} +16.5576 q^{91} +14.0786 q^{92} +1.94392 q^{94} +11.2741 q^{97} +1.69705 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{8} + 3 q^{11} - 12 q^{13} - 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} - 9 q^{22} - 9 q^{23} + 27 q^{28} + 15 q^{29} + 15 q^{31} - 6 q^{32} + 6 q^{34} - 12 q^{37} - 12 q^{41} - 12 q^{43}+ \cdots - 57 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^8 + 3 * q^11 - 12 * q^13 - 15 * q^14 + 12 * q^16 - 6 * q^17 - 3 * q^19 - 9 * q^22 - 9 * q^23 + 27 * q^28 + 15 * q^29 + 15 * q^31 - 6 * q^32 + 6 * q^34 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 15 * q^44 - 15 * q^46 + 12 * q^47 + 21 * q^49 - 24 * q^52 - 9 * q^53 - 30 * q^56 - 12 * q^59 + 3 * q^61 - 9 * q^62 - 21 * q^64 - 15 * q^67 - 60 * q^68 - 18 * q^71 - 3 * q^73 + 24 * q^74 - 6 * q^76 - 12 * q^77 + 3 * q^79 - 9 * q^82 + 3 * q^83 - 21 * q^88 + 3 * q^89 + 9 * q^92 - 18 * q^94 + 12 * q^97 - 57 * q^98

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 $

$2$	0.167449	0.118404	0.0592022	−	0.998246i	$-0.481144\pi$
$2$	0.167449	0.118404	0.0592022	+	0.998246i	$0.481144\pi$
$3$	0	0
$3$	0	0
$4$	−1.97196	−0.985980
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.13941	−1.56455	−0.782275	−	0.622933i	$-0.785941\pi$
$7$	−4.13941	−1.56455	−0.782275	+	0.622933i	$0.785941\pi$
$8$	−0.665102	−0.235149
$9$	0	0
$10$	0	0
$11$	4.80451	1.44861	0.724307	−	0.689477i	$-0.242160\pi$
$11$	4.80451	1.44861	0.724307	+	0.689477i	$0.242160\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−0.693141	−0.185250
$15$	0	0
$16$	3.83255	0.958138
$17$	5.94392	1.44161	0.720806	−	0.693136i	$-0.243772\pi$
$17$	5.94392	1.44161	0.720806	+	0.693136i	$0.243772\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0.804512	0.171522
$23$	−7.13941	−1.48867	−0.744335	−	0.667806i	$-0.767233\pi$
$23$	−7.13941	−1.48867	−0.744335	+	0.667806i	$0.767233\pi$
$24$	0	0
$25$	0	0
$26$	−0.669797	−0.131358
$27$	0	0
$28$	8.16275	1.54262
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	8.80451	1.58134	0.790668	−	0.612245i	$-0.209733\pi$
$31$	8.80451	1.58134	0.790668	+	0.612245i	$0.209733\pi$
$32$	1.97196	0.348597
$33$	0	0
$34$	0.995305	0.170693
$35$	0	0
$36$	0	0
$37$	−3.66510	−0.602539	−0.301269	−	0.953539i	$-0.597410\pi$
$37$	−3.66510	−0.602539	−0.301269	+	0.953539i	$0.597410\pi$
$38$	−0.167449	−0.0271638
$39$	0	0
$40$	0	0
$41$	−0.195488	−0.0305302	−0.0152651	−	0.999883i	$-0.504859\pi$
$41$	−0.195488	−0.0305302	−0.0152651	+	0.999883i	$0.504859\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−9.47431	−1.42831
$45$	0	0
$46$	−1.19549	−0.176265
$47$	11.6090	1.69335	0.846675	−	0.532110i	$-0.178601\pi$
$47$	11.6090	1.69335	0.846675	+	0.532110i	$0.178601\pi$
$48$	0	0
$49$	10.1347	1.44782
$50$	0	0
$51$	0	0
$52$	7.88784	1.09385
$53$	−2.33020	−0.320078	−0.160039	−	0.987111i	$-0.551162\pi$
$53$	−2.33020	−0.320078	−0.160039	+	0.987111i	$0.551162\pi$
$54$	0	0
$55$	0	0
$56$	2.75313	0.367902
$57$	0	0
$58$	0.837246	0.109936
$59$	−7.46961	−0.972461	−0.486230	−	0.873831i	$-0.661628\pi$
$59$	−7.46961	−0.972461	−0.486230	+	0.873831i	$0.661628\pi$
$60$	0	0
$61$	−6.60902	−0.846199	−0.423099	−	0.906083i	$-0.639058\pi$
$61$	−6.60902	−0.846199	−0.423099	+	0.906083i	$0.639058\pi$
$62$	1.47431	0.187237
$63$	0	0
$64$	−7.33490	−0.916862
$65$	0	0
$66$	0	0
$67$	−1.19549	−0.146052	−0.0730261	−	0.997330i	$-0.523266\pi$
$67$	−1.19549	−0.146052	−0.0730261	+	0.997330i	$0.523266\pi$
$68$	−11.7212	−1.42140
$69$	0	0
$70$	0	0
$71$	−9.46961	−1.12384	−0.561918	−	0.827193i	$-0.689936\pi$
$71$	−9.46961	−1.12384	−0.561918	+	0.827193i	$0.689936\pi$
$72$	0	0
$73$	−0.330203	−0.0386474	−0.0193237	−	0.999813i	$-0.506151\pi$
$73$	−0.330203	−0.0386474	−0.0193237	+	0.999813i	$0.506151\pi$
$74$	−0.613718	−0.0713433
$75$	0	0
$76$	1.97196	0.226199
$77$	−19.8878	−2.26643
$78$	0	0
$79$	13.4182	1.50967	0.754834	−	0.655915i	$-0.227717\pi$
$79$	13.4182	1.50967	0.754834	+	0.655915i	$0.227717\pi$
$80$	0	0
$81$	0	0
$82$	−0.0327344	−0.00361491
$83$	−2.80451	−0.307835	−0.153918	−	0.988084i	$-0.549189\pi$
$83$	−2.80451	−0.307835	−0.153918	+	0.988084i	$0.549189\pi$
$84$	0	0
$85$	0	0
$86$	−0.669797	−0.0722260
$87$	0	0
$88$	−3.19549	−0.340640
$89$	−7.27882	−0.771553	−0.385777	−	0.922592i	$-0.626066\pi$
$89$	−7.27882	−0.771553	−0.385777	+	0.922592i	$0.626066\pi$
$90$	0	0
$91$	16.5576	1.73571
$92$	14.0786	1.46780
$93$	0	0
$94$	1.94392	0.200500
$95$	0	0
$96$	0	0
$97$	11.2741	1.14471	0.572357	−	0.820005i	$-0.306029\pi$
$97$	11.2741	1.14471	0.572357	+	0.820005i	$0.306029\pi$
$98$	1.69705	0.171428
$99$	0	0
$100$	0	0
$101$	−9.94392	−0.989457	−0.494729	−	0.869048i	$-0.664733\pi$
$101$	−9.94392	−0.989457	−0.494729	+	0.869048i	$0.664733\pi$
$102$	0	0
$103$	10.1441	0.999528	0.499764	−	0.866162i	$-0.333420\pi$
$103$	10.1441	0.999528	0.499764	+	0.866162i	$0.333420\pi$
$104$	2.66041	0.260874
$105$	0	0
$106$	−0.390191	−0.0378987
$107$	−6.80921	−0.658271	−0.329135	−	0.944283i	$-0.606757\pi$
$107$	−6.80921	−0.658271	−0.329135	+	0.944283i	$0.606757\pi$
$108$	0	0
$109$	−20.2227	−1.93699	−0.968494	−	0.249038i	$-0.919886\pi$
$109$	−20.2227	−1.93699	−0.968494	+	0.249038i	$0.919886\pi$
$110$	0	0
$111$	0	0
$112$	−15.8645	−1.49905
$113$	18.0226	1.69542	0.847710	−	0.530460i	$-0.177981\pi$
$113$	18.0226	1.69542	0.847710	+	0.530460i	$0.177981\pi$
$114$	0	0
$115$	0	0
$116$	−9.85980	−0.915460
$117$	0	0
$118$	−1.25078	−0.115144
$119$	−24.6043	−2.25548
$120$	0	0
$121$	12.0833	1.09848
$122$	−1.10668	−0.100194
$123$	0	0
$124$	−17.3622	−1.55917
$125$	0	0
$126$	0	0
$127$	−14.0786	−1.24928	−0.624638	−	0.780914i	$-0.714754\pi$
$127$	−14.0786	−1.24928	−0.624638	+	0.780914i	$0.714754\pi$
$128$	−5.17214	−0.457157
$129$	0	0
$130$	0	0
$131$	3.47431	0.303552	0.151776	−	0.988415i	$-0.451501\pi$
$131$	3.47431	0.303552	0.151776	+	0.988415i	$0.451501\pi$
$132$	0	0
$133$	4.13941	0.358932
$134$	−0.200184	−0.0172932
$135$	0	0
$136$	−3.95331	−0.338994
$137$	−18.6137	−1.59028	−0.795139	−	0.606428i	$-0.792602\pi$
$137$	−18.6137	−1.59028	−0.795139	+	0.606428i	$0.792602\pi$
$138$	0	0
$139$	−6.13941	−0.520738	−0.260369	−	0.965509i	$-0.583844\pi$
$139$	−6.13941	−0.520738	−0.260369	+	0.965509i	$0.583844\pi$
$140$	0	0
$141$	0	0
$142$	−1.58568	−0.133067
$143$	−19.2180	−1.60709
$144$	0	0
$145$	0	0
$146$	−0.0552923	−0.00457602
$147$	0	0
$148$	7.22744	0.594092
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	2.72588	0.221829	0.110914	−	0.993830i	$-0.464622\pi$
$151$	2.72588	0.221829	0.110914	+	0.993830i	$0.464622\pi$
$152$	0.665102	0.0539469
$153$	0	0
$154$	−3.33020	−0.268355
$155$	0	0
$156$	0	0
$157$	−15.6924	−1.25239	−0.626193	−	0.779668i	$-0.715388\pi$
$157$	−15.6924	−1.25239	−0.626193	+	0.779668i	$0.715388\pi$
$158$	2.24687	0.178752
$159$	0	0
$160$	0	0
$161$	29.5529	2.32910
$162$	0	0
$163$	−16.4088	−1.28524	−0.642620	−	0.766185i	$-0.722152\pi$
$163$	−16.4088	−1.28524	−0.642620	+	0.766185i	$0.722152\pi$
$164$	0.385496	0.0301021
$165$	0	0
$166$	−0.469613	−0.0364491
$167$	−18.1394	−1.40367	−0.701835	−	0.712340i	$-0.747636\pi$
$167$	−18.1394	−1.40367	−0.701835	+	0.712340i	$0.747636\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	7.88784	0.601442
$173$	−8.21805	−0.624806	−0.312403	−	0.949950i	$-0.601134\pi$
$173$	−8.21805	−0.624806	−0.312403	+	0.949950i	$0.601134\pi$
$174$	0	0
$175$	0	0
$176$	18.4135	1.38797
$177$	0	0
$178$	−1.21883	−0.0913554
$179$	−1.20018	−0.0897059	−0.0448530	−	0.998994i	$-0.514282\pi$
$179$	−1.20018	−0.0897059	−0.0448530	+	0.998994i	$0.514282\pi$
$180$	0	0
$181$	−26.5482	−1.97332	−0.986658	−	0.162807i	$-0.947945\pi$
$181$	−26.5482	−1.97332	−0.986658	+	0.162807i	$0.947945\pi$
$182$	2.77256	0.205516
$183$	0	0
$184$	4.74843	0.350059
$185$	0	0
$186$	0	0
$187$	28.5576	2.08834
$188$	−22.8925	−1.66961
$189$	0	0
$190$	0	0
$191$	22.9712	1.66214	0.831068	−	0.556171i	$-0.187730\pi$
$191$	22.9712	1.66214	0.831068	+	0.556171i	$0.187730\pi$
$192$	0	0
$193$	−4.72588	−0.340176	−0.170088	−	0.985429i	$-0.554405\pi$
$193$	−4.72588	−0.340176	−0.170088	+	0.985429i	$0.554405\pi$
$194$	1.88784	0.135539
$195$	0	0
$196$	−19.9853	−1.42752
$197$	−3.67449	−0.261797	−0.130898	−	0.991396i	$-0.541786\pi$
$197$	−3.67449	−0.261797	−0.130898	+	0.991396i	$0.541786\pi$
$198$	0	0
$199$	5.19079	0.367966	0.183983	−	0.982929i	$-0.441101\pi$
$199$	5.19079	0.367966	0.183983	+	0.982929i	$0.441101\pi$
$200$	0	0
$201$	0	0
$202$	−1.66510	−0.117156
$203$	−20.6970	−1.45265
$204$	0	0
$205$	0	0
$206$	1.69862	0.118349
$207$	0	0
$208$	−15.3302	−1.06296
$209$	−4.80451	−0.332335
$210$	0	0
$211$	15.4182	1.06143	0.530717	−	0.847549i	$-0.321923\pi$
$211$	15.4182	1.06143	0.530717	+	0.847549i	$0.321923\pi$
$212$	4.59507	0.315591
$213$	0	0
$214$	−1.14020	−0.0779422
$215$	0	0
$216$	0	0
$217$	−36.4455	−2.47408
$218$	−3.38628	−0.229348
$219$	0	0
$220$	0	0
$221$	−23.7757	−1.59933
$222$	0	0
$223$	−5.13941	−0.344160	−0.172080	−	0.985083i	$-0.555049\pi$
$223$	−5.13941	−0.344160	−0.172080	+	0.985083i	$0.555049\pi$
$224$	−8.16275	−0.545397
$225$	0	0
$226$	3.01786	0.200745
$227$	−25.3575	−1.68303	−0.841517	−	0.540231i	$-0.818337\pi$
$227$	−25.3575	−1.68303	−0.841517	+	0.540231i	$0.818337\pi$
$228$	0	0
$229$	−27.6316	−1.82595	−0.912973	−	0.408020i	$-0.866219\pi$
$229$	−27.6316	−1.82595	−0.912973	+	0.408020i	$0.866219\pi$
$230$	0	0
$231$	0	0
$232$	−3.32551	−0.218330
$233$	16.2788	1.06646	0.533230	−	0.845970i	$-0.320978\pi$
$233$	16.2788	1.06646	0.533230	+	0.845970i	$0.320978\pi$
$234$	0	0
$235$	0	0
$236$	14.7298	0.958827
$237$	0	0
$238$	−4.11997	−0.267058
$239$	25.1620	1.62759	0.813796	−	0.581150i	$-0.197397\pi$
$239$	25.1620	1.62759	0.813796	+	0.581150i	$0.197397\pi$
$240$	0	0
$241$	−13.6651	−0.880247	−0.440123	−	0.897937i	$-0.645065\pi$
$241$	−13.6651	−0.880247	−0.440123	+	0.897937i	$0.645065\pi$
$242$	2.02334	0.130065
$243$	0	0
$244$	13.0327	0.834335
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−5.85589	−0.371850
$249$	0	0
$250$	0	0
$251$	−14.0561	−0.887212	−0.443606	−	0.896222i	$-0.646301\pi$
$251$	−14.0561	−0.887212	−0.443606	+	0.896222i	$0.646301\pi$
$252$	0	0
$253$	−34.3014	−2.15651
$254$	−2.35746	−0.147920
$255$	0	0
$256$	13.8037	0.862733
$257$	−16.4969	−1.02905	−0.514523	−	0.857477i	$-0.672031\pi$
$257$	−16.4969	−1.02905	−0.514523	+	0.857477i	$0.672031\pi$
$258$	0	0
$259$	15.1714	0.942702
$260$	0	0
$261$	0	0
$262$	0.581770	0.0359419
$263$	−7.13941	−0.440235	−0.220117	−	0.975473i	$-0.570644\pi$
$263$	−7.13941	−0.440235	−0.220117	+	0.975473i	$0.570644\pi$
$264$	0	0
$265$	0	0
$266$	0.693141	0.0424992
$267$	0	0
$268$	2.35746	0.144005
$269$	5.88784	0.358988	0.179494	−	0.983759i	$-0.442554\pi$
$269$	5.88784	0.358988	0.179494	+	0.983759i	$0.442554\pi$
$270$	0	0
$271$	9.46961	0.575238	0.287619	−	0.957745i	$-0.407136\pi$
$271$	9.46961	0.575238	0.287619	+	0.957745i	$0.407136\pi$
$272$	22.7804	1.38126
$273$	0	0
$274$	−3.11685	−0.188296
$275$	0	0
$276$	0	0
$277$	16.4969	0.991201	0.495600	−	0.868551i	$-0.334948\pi$
$277$	16.4969	0.991201	0.495600	+	0.868551i	$0.334948\pi$
$278$	−1.02804	−0.0616577
$279$	0	0
$280$	0	0
$281$	9.86529	0.588514	0.294257	−	0.955726i	$-0.404928\pi$
$281$	9.86529	0.588514	0.294257	+	0.955726i	$0.404928\pi$
$282$	0	0
$283$	−1.33959	−0.0796306	−0.0398153	−	0.999207i	$-0.512677\pi$
$283$	−1.33959	−0.0796306	−0.0398153	+	0.999207i	$0.512677\pi$
$284$	18.6737	1.10808
$285$	0	0
$286$	−3.21805	−0.190287
$287$	0.809207	0.0477660
$288$	0	0
$289$	18.3302	1.07825
$290$	0	0
$291$	0	0
$292$	0.651148	0.0381055
$293$	8.52569	0.498076	0.249038	−	0.968494i	$-0.419886\pi$
$293$	8.52569	0.498076	0.249038	+	0.968494i	$0.419886\pi$
$294$	0	0
$295$	0	0
$296$	2.43767	0.141686
$297$	0	0
$298$	0	0
$299$	28.5576	1.65153
$300$	0	0
$301$	16.5576	0.954366
$302$	0.456446	0.0262655
$303$	0	0
$304$	−3.83255	−0.219812
$305$	0	0
$306$	0	0
$307$	2.41353	0.137748	0.0688739	−	0.997625i	$-0.478059\pi$
$307$	2.41353	0.137748	0.0688739	+	0.997625i	$0.478059\pi$
$308$	39.2180	2.23466
$309$	0	0
$310$	0	0
$311$	−9.33959	−0.529600	−0.264800	−	0.964303i	$-0.585306\pi$
$311$	−9.33959	−0.529600	−0.264800	+	0.964303i	$0.585306\pi$
$312$	0	0
$313$	7.74374	0.437702	0.218851	−	0.975758i	$-0.429769\pi$
$313$	7.74374	0.437702	0.218851	+	0.975758i	$0.429769\pi$
$314$	−2.62767	−0.148288
$315$	0	0
$316$	−26.4602	−1.48850
$317$	−1.11216	−0.0624650	−0.0312325	−	0.999512i	$-0.509943\pi$
$317$	−1.11216	−0.0624650	−0.0312325	+	0.999512i	$0.509943\pi$
$318$	0	0
$319$	24.0226	1.34501
$320$	0	0
$321$	0	0
$322$	4.94862	0.275776
$323$	−5.94392	−0.330729
$324$	0	0
$325$	0	0
$326$	−2.74765	−0.152178
$327$	0	0
$328$	0.130020	0.00717914
$329$	−48.0545	−2.64933
$330$	0	0
$331$	1.46492	0.0805192	0.0402596	−	0.999189i	$-0.487182\pi$
$331$	1.46492	0.0805192	0.0402596	+	0.999189i	$0.487182\pi$
$332$	5.53039	0.303519
$333$	0	0
$334$	−3.03743	−0.166201
$335$	0	0
$336$	0	0
$337$	−0.278820	−0.0151883	−0.00759414	−	0.999971i	$-0.502417\pi$
$337$	−0.278820	−0.0151883	−0.00759414	+	0.999971i	$0.502417\pi$
$338$	0.502348	0.0273241
$339$	0	0
$340$	0	0
$341$	42.3014	2.29075
$342$	0	0
$343$	−12.9759	−0.700631
$344$	2.66041	0.143440
$345$	0	0
$346$	−1.37611	−0.0739799
$347$	13.6651	0.733581	0.366791	−	0.930304i	$-0.380457\pi$
$347$	13.6651	0.733581	0.366791	+	0.930304i	$0.380457\pi$
$348$	0	0
$349$	−5.94862	−0.318422	−0.159211	−	0.987245i	$-0.550895\pi$
$349$	−5.94862	−0.318422	−0.159211	+	0.987245i	$0.550895\pi$
$350$	0	0
$351$	0	0
$352$	9.47431	0.504982
$353$	−0.112157	−0.00596951	−0.00298476	−	0.999996i	$-0.500950\pi$
$353$	−0.112157	−0.00596951	−0.00298476	+	0.999996i	$0.500950\pi$
$354$	0	0
$355$	0	0
$356$	14.3535	0.760736
$357$	0	0
$358$	−0.200970	−0.0106216
$359$	29.2741	1.54503	0.772515	−	0.634997i	$-0.218999\pi$
$359$	29.2741	1.54503	0.772515	+	0.634997i	$0.218999\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.44548	−0.233649
$363$	0	0
$364$	−32.6510	−1.71138
$365$	0	0
$366$	0	0
$367$	16.9392	0.884220	0.442110	−	0.896961i	$-0.354230\pi$
$367$	16.9392	0.884220	0.442110	+	0.896961i	$0.354230\pi$
$368$	−27.3622	−1.42635
$369$	0	0
$370$	0	0
$371$	9.64567	0.500778
$372$	0	0
$373$	17.5529	0.908857	0.454429	−	0.890783i	$-0.349844\pi$
$373$	17.5529	0.908857	0.454429	+	0.890783i	$0.349844\pi$
$374$	4.78195	0.247269
$375$	0	0
$376$	−7.72118	−0.398189
$377$	−20.0000	−1.03005
$378$	0	0
$379$	−0.604328	−0.0310422	−0.0155211	−	0.999880i	$-0.504941\pi$
$379$	−0.604328	−0.0310422	−0.0155211	+	0.999880i	$0.504941\pi$
$380$	0	0
$381$	0	0
$382$	3.84650	0.196804
$383$	−19.4790	−0.995331	−0.497665	−	0.867369i	$-0.665809\pi$
$383$	−19.4790	−0.995331	−0.497665	+	0.867369i	$0.665809\pi$
$384$	0	0
$385$	0	0
$386$	−0.791344	−0.0402783
$387$	0	0
$388$	−22.2321	−1.12867
$389$	−24.9392	−1.26447	−0.632234	−	0.774777i	$-0.717862\pi$
$389$	−24.9392	−1.26447	−0.632234	+	0.774777i	$0.717862\pi$
$390$	0	0
$391$	−42.4361	−2.14609
$392$	−6.74062	−0.340452
$393$	0	0
$394$	−0.615291	−0.0309979
$395$	0	0
$396$	0	0
$397$	12.4135	0.623017	0.311509	−	0.950243i	$-0.399166\pi$
$397$	12.4135	0.623017	0.311509	+	0.950243i	$0.399166\pi$
$398$	0.869194	0.0435688
$399$	0	0
$400$	0	0
$401$	−3.74374	−0.186953	−0.0934767	−	0.995621i	$-0.529798\pi$
$401$	−3.74374	−0.186953	−0.0934767	+	0.995621i	$0.529798\pi$
$402$	0	0
$403$	−35.2180	−1.75434
$404$	19.6090	0.975585
$405$	0	0
$406$	−3.46570	−0.172000
$407$	−17.6090	−0.872847
$408$	0	0
$409$	13.9533	0.689947	0.344973	−	0.938612i	$-0.387888\pi$
$409$	13.9533	0.689947	0.344973	+	0.938612i	$0.387888\pi$
$410$	0	0
$411$	0	0
$412$	−20.0038	−0.985515
$413$	30.9198	1.52146
$414$	0	0
$415$	0	0
$416$	−7.88784	−0.386733
$417$	0	0
$418$	−0.804512	−0.0393499
$419$	−6.26943	−0.306282	−0.153141	−	0.988204i	$-0.548939\pi$
$419$	−6.26943	−0.306282	−0.153141	+	0.988204i	$0.548939\pi$
$420$	0	0
$421$	−32.4361	−1.58084	−0.790419	−	0.612566i	$-0.790137\pi$
$421$	−32.4361	−1.58084	−0.790419	+	0.612566i	$0.790137\pi$
$422$	2.58177	0.125679
$423$	0	0
$424$	1.54982	0.0752660
$425$	0	0
$426$	0	0
$427$	27.3575	1.32392
$428$	13.4275	0.649042
$429$	0	0
$430$	0	0
$431$	−21.7484	−1.04759	−0.523793	−	0.851846i	$-0.675483\pi$
$431$	−21.7484	−1.04759	−0.523793	+	0.851846i	$0.675483\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−6.10277	−0.292942
$435$	0	0
$436$	39.8785	1.90983
$437$	7.13941	0.341524
$438$	0	0
$439$	−9.68766	−0.462367	−0.231183	−	0.972910i	$-0.574260\pi$
$439$	−9.68766	−0.462367	−0.231183	+	0.972910i	$0.574260\pi$
$440$	0	0
$441$	0	0
$442$	−3.98122	−0.189367
$443$	−32.6830	−1.55281	−0.776407	−	0.630232i	$-0.782960\pi$
$443$	−32.6830	−1.55281	−0.776407	+	0.630232i	$0.782960\pi$
$444$	0	0
$445$	0	0
$446$	−0.860590	−0.0407501
$447$	0	0
$448$	30.3622	1.43448
$449$	−8.60902	−0.406285	−0.203142	−	0.979149i	$-0.565115\pi$
$449$	−8.60902	−0.406285	−0.203142	+	0.979149i	$0.565115\pi$
$450$	0	0
$451$	−0.939226	−0.0442264
$452$	−35.5398	−1.67165
$453$	0	0
$454$	−4.24609	−0.199279
$455$	0	0
$456$	0	0
$457$	−1.25626	−0.0587655	−0.0293827	−	0.999568i	$-0.509354\pi$
$457$	−1.25626	−0.0587655	−0.0293827	+	0.999568i	$0.509354\pi$
$458$	−4.62689	−0.216200
$459$	0	0
$460$	0	0
$461$	−18.6137	−0.866927	−0.433464	−	0.901171i	$-0.642709\pi$
$461$	−18.6137	−0.866927	−0.433464	+	0.901171i	$0.642709\pi$
$462$	0	0
$463$	24.6698	1.14650	0.573251	−	0.819380i	$-0.305682\pi$
$463$	24.6698	1.14650	0.573251	+	0.819380i	$0.305682\pi$
$464$	19.1628	0.889609
$465$	0	0
$466$	2.72588	0.126274
$467$	5.02725	0.232634	0.116317	−	0.993212i	$-0.462891\pi$
$467$	5.02725	0.232634	0.116317	+	0.993212i	$0.462891\pi$
$468$	0	0
$469$	4.94862	0.228506
$470$	0	0
$471$	0	0
$472$	4.96805	0.228673
$473$	−19.2180	−0.883647
$474$	0	0
$475$	0	0
$476$	48.5188	2.22385
$477$	0	0
$478$	4.21335	0.192714
$479$	−41.9665	−1.91750	−0.958749	−	0.284255i	$-0.908254\pi$
$479$	−41.9665	−1.91750	−0.958749	+	0.284255i	$0.908254\pi$
$480$	0	0
$481$	14.6604	0.668457
$482$	−2.28821	−0.104225
$483$	0	0
$484$	−23.8279	−1.08308
$485$	0	0
$486$	0	0
$487$	−28.4455	−1.28899	−0.644494	−	0.764609i	$-0.722932\pi$
$487$	−28.4455	−1.28899	−0.644494	+	0.764609i	$0.722932\pi$
$488$	4.39567	0.198983
$489$	0	0
$490$	0	0
$491$	−18.6137	−0.840025	−0.420013	−	0.907518i	$-0.637974\pi$
$491$	−18.6137	−0.840025	−0.420013	+	0.907518i	$0.637974\pi$
$492$	0	0
$493$	29.7196	1.33850
$494$	0.669797	0.0301356
$495$	0	0
$496$	33.7437	1.51514
$497$	39.1986	1.75830
$498$	0	0
$499$	−18.2967	−0.819072	−0.409536	−	0.912294i	$-0.634309\pi$
$499$	−18.2967	−0.819072	−0.409536	+	0.912294i	$0.634309\pi$
$500$	0	0
$501$	0	0
$502$	−2.35368	−0.105050
$503$	28.5576	1.27332	0.636661	−	0.771144i	$-0.280315\pi$
$503$	28.5576	1.27332	0.636661	+	0.771144i	$0.280315\pi$
$504$	0	0
$505$	0	0
$506$	−5.74374	−0.255340
$507$	0	0
$508$	27.7625	1.23176
$509$	−16.3302	−0.723824	−0.361912	−	0.932212i	$-0.617876\pi$
$509$	−16.3302	−0.723824	−0.361912	+	0.932212i	$0.617876\pi$
$510$	0	0
$511$	1.36685	0.0604657
$512$	12.6557	0.559309
$513$	0	0
$514$	−2.76239	−0.121844
$515$	0	0
$516$	0	0
$517$	55.7757	2.45301
$518$	2.54043	0.111620
$519$	0	0
$520$	0	0
$521$	22.6184	0.990931	0.495465	−	0.868628i	$-0.334998\pi$
$521$	22.6184	0.990931	0.495465	+	0.868628i	$0.334998\pi$
$522$	0	0
$523$	−8.21335	−0.359145	−0.179572	−	0.983745i	$-0.557471\pi$
$523$	−8.21335	−0.359145	−0.179572	+	0.983745i	$0.557471\pi$
$524$	−6.85120	−0.299296
$525$	0	0
$526$	−1.19549	−0.0521258
$527$	52.3333	2.27968
$528$	0	0
$529$	27.9712	1.21614
$530$	0	0
$531$	0	0
$532$	−8.16275	−0.353900
$533$	0.781954	0.0338702
$534$	0	0
$535$	0	0
$536$	0.795121	0.0343440
$537$	0	0
$538$	0.985915	0.0425058
$539$	48.6924	2.09733
$540$	0	0
$541$	6.40414	0.275336	0.137668	−	0.990478i	$-0.456039\pi$
$541$	6.40414	0.275336	0.137668	+	0.990478i	$0.456039\pi$
$542$	1.58568	0.0681107
$543$	0	0
$544$	11.7212	0.502541
$545$	0	0
$546$	0	0
$547$	−17.8092	−0.761467	−0.380733	−	0.924685i	$-0.624328\pi$
$547$	−17.8092	−0.761467	−0.380733	+	0.924685i	$0.624328\pi$
$548$	36.7055	1.56798
$549$	0	0
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	−55.5436	−2.36195
$554$	2.76239	0.117363
$555$	0	0
$556$	12.1067	0.513437
$557$	−39.2180	−1.66172	−0.830861	−	0.556480i	$-0.812152\pi$
$557$	−39.2180	−1.66172	−0.830861	+	0.556480i	$0.812152\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	1.65193	0.0696826
$563$	21.9151	0.923611	0.461806	−	0.886981i	$-0.347202\pi$
$563$	21.9151	0.923611	0.461806	+	0.886981i	$0.347202\pi$
$564$	0	0
$565$	0	0
$566$	−0.224314	−0.00942861
$567$	0	0
$568$	6.29826	0.264269
$569$	35.4135	1.48461	0.742306	−	0.670061i	$-0.233732\pi$
$569$	35.4135	1.48461	0.742306	+	0.670061i	$0.233732\pi$
$570$	0	0
$571$	6.80921	0.284956	0.142478	−	0.989798i	$-0.454493\pi$
$571$	6.80921	0.284956	0.142478	+	0.989798i	$0.454493\pi$
$572$	37.8972	1.58456
$573$	0	0
$574$	0.135501	0.00565570
$575$	0	0
$576$	0	0
$577$	−12.1441	−0.505566	−0.252783	−	0.967523i	$-0.581346\pi$
$577$	−12.1441	−0.505566	−0.252783	+	0.967523i	$0.581346\pi$
$578$	3.06938	0.127669
$579$	0	0
$580$	0	0
$581$	11.6090	0.481623
$582$	0	0
$583$	−11.1955	−0.463670
$584$	0.219619	0.00908789
$585$	0	0
$586$	1.42762	0.0589744
$587$	−7.13002	−0.294287	−0.147144	−	0.989115i	$-0.547008\pi$
$587$	−7.13002	−0.294287	−0.147144	+	0.989115i	$0.547008\pi$
$588$	0	0
$589$	−8.80451	−0.362784
$590$	0	0
$591$	0	0
$592$	−14.0467	−0.577315
$593$	−2.00939	−0.0825158	−0.0412579	−	0.999149i	$-0.513137\pi$
$593$	−2.00939	−0.0825158	−0.0412579	+	0.999149i	$0.513137\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	4.78195	0.195549
$599$	−29.1153	−1.18962	−0.594809	−	0.803867i	$-0.702772\pi$
$599$	−29.1153	−1.18962	−0.594809	+	0.803867i	$0.702772\pi$
$600$	0	0
$601$	−10.2788	−0.419282	−0.209641	−	0.977778i	$-0.567229\pi$
$601$	−10.2788	−0.419282	−0.209641	+	0.977778i	$0.567229\pi$
$602$	2.77256	0.113001
$603$	0	0
$604$	−5.37532	−0.218719
$605$	0	0
$606$	0	0
$607$	−15.5304	−0.630359	−0.315179	−	0.949032i	$-0.602065\pi$
$607$	−15.5304	−0.630359	−0.315179	+	0.949032i	$0.602065\pi$
$608$	−1.97196	−0.0799736
$609$	0	0
$610$	0	0
$611$	−46.4361	−1.87860
$612$	0	0
$613$	−4.47431	−0.180716	−0.0903578	−	0.995909i	$-0.528801\pi$
$613$	−4.47431	−0.180716	−0.0903578	+	0.995909i	$0.528801\pi$
$614$	0.404144	0.0163099
$615$	0	0
$616$	13.2274	0.532949
$617$	−10.6698	−0.429550	−0.214775	−	0.976664i	$-0.568902\pi$
$617$	−10.6698	−0.429550	−0.214775	+	0.976664i	$0.568902\pi$
$618$	0	0
$619$	−11.0786	−0.445288	−0.222644	−	0.974900i	$-0.571469\pi$
$619$	−11.0786	−0.445288	−0.222644	+	0.974900i	$0.571469\pi$
$620$	0	0
$621$	0	0
$622$	−1.56391	−0.0627070
$623$	30.1300	1.20713
$624$	0	0
$625$	0	0
$626$	1.29668	0.0518259
$627$	0	0
$628$	30.9447	1.23483
$629$	−21.7851	−0.868628
$630$	0	0
$631$	−1.22744	−0.0488635	−0.0244317	−	0.999702i	$-0.507778\pi$
$631$	−1.22744	−0.0488635	−0.0244317	+	0.999702i	$0.507778\pi$
$632$	−8.92449	−0.354997
$633$	0	0
$634$	−0.186230	−0.00739613
$635$	0	0
$636$	0	0
$637$	−40.5389	−1.60621
$638$	4.02256	0.159255
$639$	0	0
$640$	0	0
$641$	−13.7757	−0.544107	−0.272053	−	0.962282i	$-0.587703\pi$
$641$	−13.7757	−0.544107	−0.272053	+	0.962282i	$0.587703\pi$
$642$	0	0
$643$	17.0786	0.673516	0.336758	−	0.941591i	$-0.390670\pi$
$643$	17.0786	0.673516	0.336758	+	0.941591i	$0.390670\pi$
$644$	−58.2772	−2.29645
$645$	0	0
$646$	−0.995305	−0.0391597
$647$	26.7017	1.04975	0.524877	−	0.851178i	$-0.324111\pi$
$647$	26.7017	1.04975	0.524877	+	0.851178i	$0.324111\pi$
$648$	0	0
$649$	−35.8878	−1.40872
$650$	0	0
$651$	0	0
$652$	32.3576	1.26722
$653$	−36.4455	−1.42622	−0.713111	−	0.701051i	$-0.752714\pi$
$653$	−36.4455	−1.42622	−0.713111	+	0.701051i	$0.752714\pi$
$654$	0	0
$655$	0	0
$656$	−0.749219	−0.0292521
$657$	0	0
$658$	−8.04669	−0.313693
$659$	46.4549	1.80962	0.904812	−	0.425810i	$-0.140011\pi$
$659$	46.4549	1.80962	0.904812	+	0.425810i	$0.140011\pi$
$660$	0	0
$661$	34.5016	1.34196	0.670978	−	0.741478i	$-0.265875\pi$
$661$	34.5016	1.34196	0.670978	+	0.741478i	$0.265875\pi$
$662$	0.245299	0.00953383
$663$	0	0
$664$	1.86529	0.0723871
$665$	0	0
$666$	0	0
$667$	−35.6970	−1.38220
$668$	35.7702	1.38399
$669$	0	0
$670$	0	0
$671$	−31.7531	−1.22582
$672$	0	0
$673$	20.5576	0.792439	0.396219	−	0.918156i	$-0.370322\pi$
$673$	20.5576	0.792439	0.396219	+	0.918156i	$0.370322\pi$
$674$	−0.0466882	−0.00179836
$675$	0	0
$676$	−5.91588	−0.227534
$677$	−20.2274	−0.777404	−0.388702	−	0.921364i	$-0.627076\pi$
$677$	−20.2274	−0.777404	−0.388702	+	0.921364i	$0.627076\pi$
$678$	0	0
$679$	−46.6682	−1.79096
$680$	0	0
$681$	0	0
$682$	7.08333	0.271235
$683$	26.9759	1.03220	0.516101	−	0.856527i	$-0.327383\pi$
$683$	26.9759	1.03220	0.516101	+	0.856527i	$0.327383\pi$
$684$	0	0
$685$	0	0
$686$	−2.17280	−0.0829578
$687$	0	0
$688$	−15.3302	−0.584459
$689$	9.32081	0.355095
$690$	0	0
$691$	−8.82707	−0.335798	−0.167899	−	0.985804i	$-0.553698\pi$
$691$	−8.82707	−0.335798	−0.167899	+	0.985804i	$0.553698\pi$
$692$	16.2057	0.616047
$693$	0	0
$694$	2.28821	0.0868593
$695$	0	0
$696$	0	0
$697$	−1.16197	−0.0440127
$698$	−0.996091	−0.0377026
$699$	0	0
$700$	0	0
$701$	−36.2134	−1.36776	−0.683880	−	0.729595i	$-0.739709\pi$
$701$	−36.2134	−1.36776	−0.683880	+	0.729595i	$0.739709\pi$
$702$	0	0
$703$	3.66510	0.138232
$704$	−35.2406	−1.32818
$705$	0	0
$706$	−0.0187806	−0.000706817 0
$707$	41.1620	1.54806
$708$	0	0
$709$	27.1573	1.01991	0.509956	−	0.860200i	$-0.329662\pi$
$709$	27.1573	1.01991	0.509956	+	0.860200i	$0.329662\pi$
$710$	0	0
$711$	0	0
$712$	4.84115	0.181430
$713$	−62.8590	−2.35409
$714$	0	0
$715$	0	0
$716$	2.36671	0.0884483
$717$	0	0
$718$	4.90193	0.182938
$719$	−41.3622	−1.54255	−0.771274	−	0.636503i	$-0.780380\pi$
$719$	−41.3622	−1.54255	−0.771274	+	0.636503i	$0.780380\pi$
$720$	0	0
$721$	−41.9906	−1.56381
$722$	0.167449	0.00623181
$723$	0	0
$724$	52.3521	1.94565
$725$	0	0
$726$	0	0
$727$	47.2547	1.75258	0.876290	−	0.481785i	$-0.160011\pi$
$727$	47.2547	1.75258	0.876290	+	0.481785i	$0.160011\pi$
$728$	−11.0125	−0.408151
$729$	0	0
$730$	0	0
$731$	−23.7757	−0.879376
$732$	0	0
$733$	17.9580	0.663294	0.331647	−	0.943404i	$-0.392396\pi$
$733$	17.9580	0.663294	0.331647	+	0.943404i	$0.392396\pi$
$734$	2.83646	0.104696
$735$	0	0
$736$	−14.0786	−0.518945
$737$	−5.74374	−0.211573
$738$	0	0
$739$	31.0786	1.14325	0.571623	−	0.820516i	$-0.306314\pi$
$739$	31.0786	1.14325	0.571623	+	0.820516i	$0.306314\pi$
$740$	0	0
$741$	0	0
$742$	1.61516	0.0592944
$743$	19.6363	0.720385	0.360193	−	0.932878i	$-0.382711\pi$
$743$	19.6363	0.720385	0.360193	+	0.932878i	$0.382711\pi$
$744$	0	0
$745$	0	0
$746$	2.93923	0.107613
$747$	0	0
$748$	−56.3145	−2.05906
$749$	28.1861	1.02990
$750$	0	0
$751$	25.4875	0.930051	0.465026	−	0.885297i	$-0.346045\pi$
$751$	25.4875	0.930051	0.465026	+	0.885297i	$0.346045\pi$
$752$	44.4922	1.62246
$753$	0	0
$754$	−3.34898	−0.121963
$755$	0	0
$756$	0	0
$757$	−29.3910	−1.06823	−0.534117	−	0.845411i	$-0.679356\pi$
$757$	−29.3910	−1.06823	−0.534117	+	0.845411i	$0.679356\pi$
$758$	−0.101194	−0.00367554
$759$	0	0
$760$	0	0
$761$	17.1526	0.621780	0.310890	−	0.950446i	$-0.399373\pi$
$761$	17.1526	0.621780	0.310890	+	0.950446i	$0.399373\pi$
$762$	0	0
$763$	83.7102	3.03051
$764$	−45.2983	−1.63883
$765$	0	0
$766$	−3.26174	−0.117852
$767$	29.8785	1.07885
$768$	0	0
$769$	−9.11216	−0.328593	−0.164296	−	0.986411i	$-0.552535\pi$
$769$	−9.11216	−0.328593	−0.164296	+	0.986411i	$0.552535\pi$
$770$	0	0
$771$	0	0
$772$	9.31924	0.335407
$773$	−26.0927	−0.938490	−0.469245	−	0.883068i	$-0.655474\pi$
$773$	−26.0927	−0.938490	−0.469245	+	0.883068i	$0.655474\pi$
$774$	0	0
$775$	0	0
$776$	−7.49844	−0.269178
$777$	0	0
$778$	−4.17605	−0.149719
$779$	0.195488	0.00700410
$780$	0	0
$781$	−45.4969	−1.62801
$782$	−7.10589	−0.254106
$783$	0	0
$784$	38.8418	1.38721
$785$	0	0
$786$	0	0
$787$	−1.92136	−0.0684892	−0.0342446	−	0.999413i	$-0.510903\pi$
$787$	−1.92136	−0.0684892	−0.0342446	+	0.999413i	$0.510903\pi$
$788$	7.24595	0.258126
$789$	0	0
$790$	0	0
$791$	−74.6028	−2.65257
$792$	0	0
$793$	26.4361	0.938773
$794$	2.07864	0.0737680
$795$	0	0
$796$	−10.2360	−0.362807
$797$	−40.6410	−1.43958	−0.719789	−	0.694193i	$-0.755761\pi$
$797$	−40.6410	−1.43958	−0.719789	+	0.694193i	$0.755761\pi$
$798$	0	0
$799$	69.0031	2.44115
$800$	0	0
$801$	0	0
$802$	−0.626886	−0.0221361
$803$	−1.58647	−0.0559851
$804$	0	0
$805$	0	0
$806$	−5.89723	−0.207721
$807$	0	0
$808$	6.61372	0.232670
$809$	−12.2788	−0.431700	−0.215850	−	0.976426i	$-0.569252\pi$
$809$	−12.2788	−0.431700	−0.215850	+	0.976426i	$0.569252\pi$
$810$	0	0
$811$	50.5802	1.77611	0.888055	−	0.459736i	$-0.152056\pi$
$811$	50.5802	1.77611	0.888055	+	0.459736i	$0.152056\pi$
$812$	40.8138	1.43228
$813$	0	0
$814$	−2.94862	−0.103349
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	2.33647	0.0816928
$819$	0	0
$820$	0	0
$821$	−9.82237	−0.342803	−0.171402	−	0.985201i	$-0.554830\pi$
$821$	−9.82237	−0.342803	−0.171402	+	0.985201i	$0.554830\pi$
$822$	0	0
$823$	−10.1394	−0.353438	−0.176719	−	0.984261i	$-0.556548\pi$
$823$	−10.1394	−0.353438	−0.176719	+	0.984261i	$0.556548\pi$
$824$	−6.74686	−0.235038
$825$	0	0
$826$	5.17749	0.180148
$827$	−9.33959	−0.324769	−0.162385	−	0.986728i	$-0.551919\pi$
$827$	−9.33959	−0.324769	−0.162385	+	0.986728i	$0.551919\pi$
$828$	0	0
$829$	20.5670	0.714322	0.357161	−	0.934043i	$-0.383745\pi$
$829$	20.5670	0.714322	0.357161	+	0.934043i	$0.383745\pi$
$830$	0	0
$831$	0	0
$832$	29.3396	1.01717
$833$	60.2399	2.08719
$834$	0	0
$835$	0	0
$836$	9.47431	0.327676
$837$	0	0
$838$	−1.04981	−0.0362651
$839$	43.8029	1.51225	0.756123	−	0.654430i	$-0.227091\pi$
$839$	43.8029	1.51225	0.756123	+	0.654430i	$0.227091\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−5.43140	−0.187178
$843$	0	0
$844$	−30.4041	−1.04655
$845$	0	0
$846$	0	0
$847$	−50.0179	−1.71863
$848$	−8.93062	−0.306679
$849$	0	0
$850$	0	0
$851$	26.1667	0.896982
$852$	0	0
$853$	−32.6410	−1.11761	−0.558803	−	0.829301i	$-0.688739\pi$
$853$	−32.6410	−1.11761	−0.558803	+	0.829301i	$0.688739\pi$
$854$	4.58098	0.156758
$855$	0	0
$856$	4.52881	0.154792
$857$	17.1441	0.585631	0.292816	−	0.956169i	$-0.405408\pi$
$857$	17.1441	0.585631	0.292816	+	0.956169i	$0.405408\pi$
$858$	0	0
$859$	21.5241	0.734393	0.367197	−	0.930143i	$-0.380318\pi$
$859$	21.5241	0.734393	0.367197	+	0.930143i	$0.380318\pi$
$860$	0	0
$861$	0	0
$862$	−3.64176	−0.124039
$863$	−38.8543	−1.32262	−0.661308	−	0.750114i	$-0.729998\pi$
$863$	−38.8543	−1.32262	−0.661308	+	0.750114i	$0.729998\pi$
$864$	0	0
$865$	0	0
$866$	1.00470	0.0341409
$867$	0	0
$868$	71.8691	2.43939
$869$	64.4680	2.18693
$870$	0	0
$871$	4.78195	0.162030
$872$	13.4502	0.455481
$873$	0	0
$874$	1.19549	0.0404380
$875$	0	0
$876$	0	0
$877$	24.1573	0.815733	0.407867	−	0.913042i	$-0.366273\pi$
$877$	24.1573	0.815733	0.407867	+	0.913042i	$0.366273\pi$
$878$	−1.62219	−0.0547463
$879$	0	0
$880$	0	0
$881$	−3.33020	−0.112197	−0.0560987	−	0.998425i	$-0.517866\pi$
$881$	−3.33020	−0.112197	−0.0560987	+	0.998425i	$0.517866\pi$
$882$	0	0
$883$	48.8543	1.64408	0.822039	−	0.569431i	$-0.192836\pi$
$883$	48.8543	1.64408	0.822039	+	0.569431i	$0.192836\pi$
$884$	46.8847	1.57690
$885$	0	0
$886$	−5.47274	−0.183860
$887$	−30.9937	−1.04067	−0.520334	−	0.853963i	$-0.674192\pi$
$887$	−30.9937	−1.04067	−0.520334	+	0.853963i	$0.674192\pi$
$888$	0	0
$889$	58.2772	1.95456
$890$	0	0
$891$	0	0
$892$	10.1347	0.339335
$893$	−11.6090	−0.388481
$894$	0	0
$895$	0	0
$896$	21.4096	0.715245
$897$	0	0
$898$	−1.44157	−0.0481059
$899$	44.0226	1.46823
$900$	0	0
$901$	−13.8505	−0.461429
$902$	−0.157273	−0.00523661
$903$	0	0
$904$	−11.9868	−0.398676
$905$	0	0
$906$	0	0
$907$	−46.3894	−1.54033	−0.770167	−	0.637842i	$-0.779827\pi$
$907$	−46.3894	−1.54033	−0.770167	+	0.637842i	$0.779827\pi$
$908$	50.0039	1.65944
$909$	0	0
$910$	0	0
$911$	−46.3061	−1.53419	−0.767094	−	0.641534i	$-0.778298\pi$
$911$	−46.3061	−1.53419	−0.767094	+	0.641534i	$0.778298\pi$
$912$	0	0
$913$	−13.4743	−0.445935
$914$	−0.210360	−0.00695809
$915$	0	0
$916$	54.4884	1.80035
$917$	−14.3816	−0.474922
$918$	0	0
$919$	−27.0880	−0.893552	−0.446776	−	0.894646i	$-0.647428\pi$
$919$	−27.0880	−0.893552	−0.446776	+	0.894646i	$0.647428\pi$
$920$	0	0
$921$	0	0
$922$	−3.11685	−0.102648
$923$	37.8785	1.24678
$924$	0	0
$925$	0	0
$926$	4.13094	0.135751
$927$	0	0
$928$	9.85980	0.323664
$929$	18.8925	0.619844	0.309922	−	0.950762i	$-0.399697\pi$
$929$	18.8925	0.619844	0.309922	+	0.950762i	$0.399697\pi$
$930$	0	0
$931$	−10.1347	−0.332152
$932$	−32.1012	−1.05151
$933$	0	0
$934$	0.841809	0.0275448
$935$	0	0
$936$	0	0
$937$	31.5257	1.02990	0.514950	−	0.857220i	$-0.327811\pi$
$937$	31.5257	1.02990	0.514950	+	0.857220i	$0.327811\pi$
$938$	0.828642	0.0270561
$939$	0	0
$940$	0	0
$941$	−23.3622	−0.761584	−0.380792	−	0.924661i	$-0.624349\pi$
$941$	−23.3622	−0.761584	−0.380792	+	0.924661i	$0.624349\pi$
$942$	0	0
$943$	1.39567	0.0454493
$944$	−28.6277	−0.931751
$945$	0	0
$946$	−3.21805	−0.104628
$947$	−19.2180	−0.624503	−0.312251	−	0.950000i	$-0.601083\pi$
$947$	−19.2180	−0.624503	−0.312251	+	0.950000i	$0.601083\pi$
$948$	0	0
$949$	1.32081	0.0428754
$950$	0	0
$951$	0	0
$952$	16.3644	0.530373
$953$	7.16666	0.232151	0.116075	−	0.993240i	$-0.462969\pi$
$953$	7.16666	0.232151	0.116075	+	0.993240i	$0.462969\pi$
$954$	0	0
$955$	0	0
$956$	−49.6184	−1.60477
$957$	0	0
$958$	−7.02725	−0.227040
$959$	77.0498	2.48807
$960$	0	0
$961$	46.5194	1.50063
$962$	2.45487	0.0791483
$963$	0	0
$964$	26.9470	0.867906
$965$	0	0
$966$	0	0
$967$	7.75782	0.249475	0.124737	−	0.992190i	$-0.460191\pi$
$967$	7.75782	0.249475	0.124737	+	0.992190i	$0.460191\pi$
$968$	−8.03664	−0.258307
$969$	0	0
$970$	0	0
$971$	42.1845	1.35377	0.676883	−	0.736091i	$-0.263330\pi$
$971$	42.1845	1.35377	0.676883	+	0.736091i	$0.263330\pi$
$972$	0	0
$973$	25.4135	0.814721
$974$	−4.76317	−0.152622
$975$	0	0
$976$	−25.3294	−0.810775
$977$	20.6604	0.660985	0.330492	−	0.943809i	$-0.392785\pi$
$977$	20.6604	0.660985	0.330492	+	0.943809i	$0.392785\pi$
$978$	0	0
$979$	−34.9712	−1.11768
$980$	0	0
$981$	0	0
$982$	−3.11685	−0.0994627
$983$	3.61841	0.115409	0.0577047	−	0.998334i	$-0.481622\pi$
$983$	3.61841	0.115409	0.0577047	+	0.998334i	$0.481622\pi$
$984$	0	0
$985$	0	0
$986$	4.97652	0.158485
$987$	0	0
$988$	−7.88784	−0.250946
$989$	28.5576	0.908080
$990$	0	0
$991$	1.46492	0.0465347	0.0232673	−	0.999729i	$-0.492593\pi$
$991$	1.46492	0.0465347	0.0232673	+	0.999729i	$0.492593\pi$
$992$	17.3622	0.551249
$993$	0	0
$994$	6.56378	0.208190
$995$	0	0
$996$	0	0
$997$	−19.3622	−0.613205	−0.306603	−	0.951838i	$-0.599192\pi$
$997$	−19.3622	−0.613205	−0.306603	+	0.951838i	$0.599192\pi$
$998$	−3.06376	−0.0969818
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bf.1.2		3
3.2	odd	2		1425.2.a.t.1.2	✓	3
5.4	even	2		4275.2.a.bg.1.2		3
15.2	even	4		1425.2.c.o.799.3		6
15.8	even	4		1425.2.c.o.799.4		6
15.14	odd	2		1425.2.a.w.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.t.1.2	✓	3	3.2	odd	2
1425.2.a.w.1.2	yes	3	15.14	odd	2
1425.2.c.o.799.3		6	15.2	even	4
1425.2.c.o.799.4		6	15.8	even	4
4275.2.a.bf.1.2		3	1.1	even	1	trivial
4275.2.a.bg.1.2		3	5.4	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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+0.167449 q^{2} -1.97196 q^{4} -4.13941 q^{7} -0.665102 q^{8} +4.80451 q^{11} -4.00000 q^{13} -0.693141 q^{14} +3.83255 q^{16} +5.94392 q^{17} -1.00000 q^{19} +0.804512 q^{22} -7.13941 q^{23} -0.669797 q^{26} +8.16275 q^{28} +5.00000 q^{29} +8.80451 q^{31} +1.97196 q^{32} +0.995305 q^{34} -3.66510 q^{37} -0.167449 q^{38} -0.195488 q^{41} -4.00000 q^{43} -9.47431 q^{44} -1.19549 q^{46} +11.6090 q^{47} +10.1347 q^{49} +7.88784 q^{52} -2.33020 q^{53} +2.75313 q^{56} +0.837246 q^{58} -7.46961 q^{59} -6.60902 q^{61} +1.47431 q^{62} -7.33490 q^{64} -1.19549 q^{67} -11.7212 q^{68} -9.46961 q^{71} -0.330203 q^{73} -0.613718 q^{74} +1.97196 q^{76} -19.8878 q^{77} +13.4182 q^{79} -0.0327344 q^{82} -2.80451 q^{83} -0.669797 q^{86} -3.19549 q^{88} -7.27882 q^{89} +16.5576 q^{91} +14.0786 q^{92} +1.94392 q^{94} +11.2741 q^{97} +1.69705 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{8} + 3 q^{11} - 12 q^{13} - 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} - 9 q^{22} - 9 q^{23} + 27 q^{28} + 15 q^{29} + 15 q^{31} - 6 q^{32} + 6 q^{34} - 12 q^{37} - 12 q^{41} - 12 q^{43}+ \cdots - 57 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^8 + 3 * q^11 - 12 * q^13 - 15 * q^14 + 12 * q^16 - 6 * q^17 - 3 * q^19 - 9 * q^22 - 9 * q^23 + 27 * q^28 + 15 * q^29 + 15 * q^31 - 6 * q^32 + 6 * q^34 - 12 * q^37 - 12 * q^41 - 12 * q^43 - 15 * q^44 - 15 * q^46 + 12 * q^47 + 21 * q^49 - 24 * q^52 - 9 * q^53 - 30 * q^56 - 12 * q^59 + 3 * q^61 - 9 * q^62 - 21 * q^64 - 15 * q^67 - 60 * q^68 - 18 * q^71 - 3 * q^73 + 24 * q^74 - 6 * q^76 - 12 * q^77 + 3 * q^79 - 9 * q^82 + 3 * q^83 - 21 * q^88 + 3 * q^89 + 9 * q^92 - 18 * q^94 + 12 * q^97 - 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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