LMFDB - Embedded newform 4275.2.a.bx.1.12

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 19x^{10} + 115x^{8} - 288x^{6} + 295x^{4} - 111x^{2} + 9 $ x^12 - 19x^10 + 115x^8 - 288x^6 + 295x^4 - 111*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 855)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.331466$ 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)

$f(q)$	$=$	$q+2.67125 q^{2} +5.13555 q^{4} +2.04966 q^{7} +8.37582 q^{8} -3.94053 q^{11} +5.35075 q^{13} +5.47514 q^{14} +12.1028 q^{16} +6.45329 q^{17} -1.00000 q^{19} -10.5261 q^{22} +3.48297 q^{23} +14.2932 q^{26} +10.5261 q^{28} -8.34955 q^{29} -2.96722 q^{31} +15.5778 q^{32} +17.2383 q^{34} -6.42681 q^{37} -2.67125 q^{38} -11.4188 q^{41} +7.22503 q^{43} -20.2368 q^{44} +9.30388 q^{46} -1.11080 q^{47} -2.79890 q^{49} +27.4791 q^{52} -0.362087 q^{53} +17.1676 q^{56} -22.3037 q^{58} +2.87441 q^{59} -0.135550 q^{61} -7.92618 q^{62} +17.4066 q^{64} +5.17537 q^{67} +33.1412 q^{68} -2.87441 q^{71} -1.87428 q^{73} -17.1676 q^{74} -5.13555 q^{76} -8.07675 q^{77} +13.9344 q^{79} -30.5023 q^{82} +7.20201 q^{83} +19.2998 q^{86} -33.0052 q^{88} +6.41211 q^{89} +10.9672 q^{91} +17.8870 q^{92} -2.96722 q^{94} +5.35075 q^{97} -7.47654 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{4} + 40 q^{16} - 12 q^{19} + 24 q^{31} + 56 q^{34} + 80 q^{46} + 40 q^{49} + 44 q^{61} + 72 q^{64} - 16 q^{76} + 48 q^{79} + 72 q^{91} + 24 q^{94}+O(q^{100}) $ 12 * q + 16 * q^4 + 40 * q^16 - 12 * q^19 + 24 * q^31 + 56 * q^34 + 80 * q^46 + 40 * q^49 + 44 * q^61 + 72 * q^64 - 16 * q^76 + 48 * q^79 + 72 * q^91 + 24 * q^94

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 $

$2$	2.67125	1.88886	0.944428	−	0.328719i	$-0.106617\pi$
$2$	2.67125	1.88886	0.944428	+	0.328719i	$0.106617\pi$
$3$	0	0
$3$	0	0
$4$	5.13555	2.56777
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.04966	0.774698	0.387349	−	0.921933i	$-0.373391\pi$
$7$	2.04966	0.774698	0.387349	+	0.921933i	$0.373391\pi$
$8$	8.37582	2.96130
$9$	0	0
$10$	0	0
$11$	−3.94053	−1.18812	−0.594058	−	0.804422i	$-0.702475\pi$
$11$	−3.94053	−1.18812	−0.594058	+	0.804422i	$0.702475\pi$
$12$	0	0
$13$	5.35075	1.48403	0.742016	−	0.670382i	$-0.233870\pi$
$13$	5.35075	1.48403	0.742016	+	0.670382i	$0.233870\pi$
$14$	5.47514	1.46329
$15$	0	0
$16$	12.1028	3.02569
$17$	6.45329	1.56515	0.782577	−	0.622554i	$-0.213905\pi$
$17$	6.45329	1.56515	0.782577	+	0.622554i	$0.213905\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−10.5261	−2.24418
$23$	3.48297	0.726250	0.363125	−	0.931740i	$-0.381710\pi$
$23$	3.48297	0.726250	0.363125	+	0.931740i	$0.381710\pi$
$24$	0	0
$25$	0	0
$26$	14.2932	2.80312
$27$	0	0
$28$	10.5261	1.98925
$29$	−8.34955	−1.55047	−0.775236	−	0.631672i	$-0.782369\pi$
$29$	−8.34955	−1.55047	−0.775236	+	0.631672i	$0.782369\pi$
$30$	0	0
$31$	−2.96722	−0.532929	−0.266465	−	0.963845i	$-0.585856\pi$
$31$	−2.96722	−0.532929	−0.266465	+	0.963845i	$0.585856\pi$
$32$	15.5778	2.75380
$33$	0	0
$34$	17.2383	2.95635
$35$	0	0
$36$	0	0
$37$	−6.42681	−1.05656	−0.528280	−	0.849070i	$-0.677163\pi$
$37$	−6.42681	−1.05656	−0.528280	+	0.849070i	$0.677163\pi$
$38$	−2.67125	−0.433333
$39$	0	0
$40$	0	0
$41$	−11.4188	−1.78331	−0.891656	−	0.452714i	$-0.850456\pi$
$41$	−11.4188	−1.78331	−0.891656	+	0.452714i	$0.850456\pi$
$42$	0	0
$43$	7.22503	1.10181	0.550904	−	0.834569i	$-0.314283\pi$
$43$	7.22503	1.10181	0.550904	+	0.834569i	$0.314283\pi$
$44$	−20.2368	−3.05081
$45$	0	0
$46$	9.30388	1.37178
$47$	−1.11080	−0.162027	−0.0810135	−	0.996713i	$-0.525816\pi$
$47$	−1.11080	−0.162027	−0.0810135	+	0.996713i	$0.525816\pi$
$48$	0	0
$49$	−2.79890	−0.399842
$50$	0	0
$51$	0	0
$52$	27.4791	3.81066
$53$	−0.362087	−0.0497365	−0.0248682	−	0.999691i	$-0.507917\pi$
$53$	−0.362087	−0.0497365	−0.0248682	+	0.999691i	$0.507917\pi$
$54$	0	0
$55$	0	0
$56$	17.1676	2.29411
$57$	0	0
$58$	−22.3037	−2.92862
$59$	2.87441	0.374216	0.187108	−	0.982339i	$-0.440089\pi$
$59$	2.87441	0.374216	0.187108	+	0.982339i	$0.440089\pi$
$60$	0	0
$61$	−0.135550	−0.0173554	−0.00867769	−	0.999962i	$-0.502762\pi$
$61$	−0.135550	−0.0173554	−0.00867769	+	0.999962i	$0.502762\pi$
$62$	−7.92618	−1.00663
$63$	0	0
$64$	17.4066	2.17583
$65$	0	0
$66$	0	0
$67$	5.17537	0.632273	0.316136	−	0.948714i	$-0.397614\pi$
$67$	5.17537	0.632273	0.316136	+	0.948714i	$0.397614\pi$
$68$	33.1412	4.01896
$69$	0	0
$70$	0	0
$71$	−2.87441	−0.341129	−0.170565	−	0.985346i	$-0.554559\pi$
$71$	−2.87441	−0.341129	−0.170565	+	0.985346i	$0.554559\pi$
$72$	0	0
$73$	−1.87428	−0.219368	−0.109684	−	0.993967i	$-0.534984\pi$
$73$	−1.87428	−0.219368	−0.109684	+	0.993967i	$0.534984\pi$
$74$	−17.1676	−1.99569
$75$	0	0
$76$	−5.13555	−0.589088
$77$	−8.07675	−0.920431
$78$	0	0
$79$	13.9344	1.56775	0.783874	−	0.620920i	$-0.213241\pi$
$79$	13.9344	1.56775	0.783874	+	0.620920i	$0.213241\pi$
$80$	0	0
$81$	0	0
$82$	−30.5023	−3.36842
$83$	7.20201	0.790523	0.395261	−	0.918569i	$-0.370654\pi$
$83$	7.20201	0.790523	0.395261	+	0.918569i	$0.370654\pi$
$84$	0	0
$85$	0	0
$86$	19.2998	2.08115
$87$	0	0
$88$	−33.0052	−3.51837
$89$	6.41211	0.679682	0.339841	−	0.940483i	$-0.389627\pi$
$89$	6.41211	0.679682	0.339841	+	0.940483i	$0.389627\pi$
$90$	0	0
$91$	10.9672	1.14968
$92$	17.8870	1.86485
$93$	0	0
$94$	−2.96722	−0.306046
$95$	0	0
$96$	0	0
$97$	5.35075	0.543287	0.271643	−	0.962398i	$-0.412433\pi$
$97$	5.35075	0.543287	0.271643	+	0.962398i	$0.412433\pi$
$98$	−7.47654	−0.755244
$99$	0	0
$100$	0	0
$101$	2.87441	0.286014	0.143007	−	0.989722i	$-0.454323\pi$
$101$	2.87441	0.286014	0.143007	+	0.989722i	$0.454323\pi$
$102$	0	0
$103$	16.9529	1.67042	0.835211	−	0.549929i	$-0.185345\pi$
$103$	16.9529	1.67042	0.835211	+	0.549929i	$0.185345\pi$
$104$	44.8170	4.39466
$105$	0	0
$106$	−0.967223	−0.0939450
$107$	−16.6256	−1.60726	−0.803629	−	0.595130i	$-0.797101\pi$
$107$	−16.6256	−1.60726	−0.803629	+	0.595130i	$0.797101\pi$
$108$	0	0
$109$	14.2711	1.36692	0.683462	−	0.729986i	$-0.260474\pi$
$109$	14.2711	1.36692	0.683462	+	0.729986i	$0.260474\pi$
$110$	0	0
$111$	0	0
$112$	24.8066	2.34400
$113$	−0.362087	−0.0340623	−0.0170311	−	0.999855i	$-0.505421\pi$
$113$	−0.362087	−0.0340623	−0.0170311	+	0.999855i	$0.505421\pi$
$114$	0	0
$115$	0	0
$116$	−42.8795	−3.98126
$117$	0	0
$118$	7.67824	0.706840
$119$	13.2270	1.21252
$120$	0	0
$121$	4.52780	0.411618
$122$	−0.362087	−0.0327818
$123$	0	0
$124$	−15.2383	−1.36844
$125$	0	0
$126$	0	0
$127$	−11.7776	−1.04509	−0.522545	−	0.852612i	$-0.675017\pi$
$127$	−11.7776	−1.04509	−0.522545	+	0.852612i	$0.675017\pi$
$128$	15.3418	1.35603
$129$	0	0
$130$	0	0
$131$	1.06613	0.0931478	0.0465739	−	0.998915i	$-0.485170\pi$
$131$	1.06613	0.0931478	0.0465739	+	0.998915i	$0.485170\pi$
$132$	0	0
$133$	−2.04966	−0.177728
$134$	13.8247	1.19427
$135$	0	0
$136$	54.0516	4.63489
$137$	−19.1238	−1.63386	−0.816929	−	0.576738i	$-0.804325\pi$
$137$	−19.1238	−1.63386	−0.816929	+	0.576738i	$0.804325\pi$
$138$	0	0
$139$	−13.3739	−1.13436	−0.567179	−	0.823595i	$-0.691965\pi$
$139$	−13.3739	−1.13436	−0.567179	+	0.823595i	$0.691965\pi$
$140$	0	0
$141$	0	0
$142$	−7.67824	−0.644344
$143$	−21.0848	−1.76320
$144$	0	0
$145$	0	0
$146$	−5.00666	−0.414354
$147$	0	0
$148$	−33.0052	−2.71301
$149$	−2.47158	−0.202480	−0.101240	−	0.994862i	$-0.532281\pi$
$149$	−2.47158	−0.202480	−0.101240	+	0.994862i	$0.532281\pi$
$150$	0	0
$151$	−7.23832	−0.589046	−0.294523	−	0.955644i	$-0.595161\pi$
$151$	−7.23832	−0.589046	−0.294523	+	0.955644i	$0.595161\pi$
$152$	−8.37582	−0.679369
$153$	0	0
$154$	−21.5750	−1.73856
$155$	0	0
$156$	0	0
$157$	7.67824	0.612791	0.306395	−	0.951904i	$-0.400877\pi$
$157$	7.67824	0.612791	0.306395	+	0.951904i	$0.400877\pi$
$158$	37.2223	2.96125
$159$	0	0
$160$	0	0
$161$	7.13891	0.562625
$162$	0	0
$163$	−4.99999	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−4.99999	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	−58.6417	−4.57914
$165$	0	0
$166$	19.2383	1.49318
$167$	−7.97526	−0.617144	−0.308572	−	0.951201i	$-0.599851\pi$
$167$	−7.97526	−0.617144	−0.308572	+	0.951201i	$0.599851\pi$
$168$	0	0
$169$	15.6306	1.20235
$170$	0	0
$171$	0	0
$172$	37.1045	2.82919
$173$	14.7661	1.12265	0.561323	−	0.827597i	$-0.310293\pi$
$173$	14.7661	1.12265	0.561323	+	0.827597i	$0.310293\pi$
$174$	0	0
$175$	0	0
$176$	−47.6914	−3.59487
$177$	0	0
$178$	17.1283	1.28382
$179$	17.8309	1.33274	0.666371	−	0.745620i	$-0.267847\pi$
$179$	17.8309	1.33274	0.666371	+	0.745620i	$0.267847\pi$
$180$	0	0
$181$	−6.54220	−0.486278	−0.243139	−	0.969991i	$-0.578177\pi$
$181$	−6.54220	−0.486278	−0.243139	+	0.969991i	$0.578177\pi$
$182$	29.2961	2.17157
$183$	0	0
$184$	29.1728	2.15065
$185$	0	0
$186$	0	0
$187$	−25.4294	−1.85958
$188$	−5.70458	−0.416049
$189$	0	0
$190$	0	0
$191$	−14.8908	−1.07746	−0.538731	−	0.842478i	$-0.681096\pi$
$191$	−14.8908	−1.07746	−0.538731	+	0.842478i	$0.681096\pi$
$192$	0	0
$193$	11.6022	0.835143	0.417572	−	0.908644i	$-0.362881\pi$
$193$	11.6022	0.835143	0.417572	+	0.908644i	$0.362881\pi$
$194$	14.2932	1.02619
$195$	0	0
$196$	−14.3739	−1.02671
$197$	8.05221	0.573696	0.286848	−	0.957976i	$-0.407393\pi$
$197$	8.05221	0.573696	0.286848	+	0.957976i	$0.407393\pi$
$198$	0	0
$199$	−25.3739	−1.79871	−0.899353	−	0.437223i	$-0.855962\pi$
$199$	−25.3739	−1.79871	−0.899353	+	0.437223i	$0.855962\pi$
$200$	0	0
$201$	0	0
$202$	7.67824	0.540239
$203$	−17.1137	−1.20115
$204$	0	0
$205$	0	0
$206$	45.2854	3.15519
$207$	0	0
$208$	64.7590	4.49023
$209$	3.94053	0.272572
$210$	0	0
$211$	−1.93445	−0.133173	−0.0665864	−	0.997781i	$-0.521211\pi$
$211$	−1.93445	−0.133173	−0.0665864	+	0.997781i	$0.521211\pi$
$212$	−1.85952	−0.127712
$213$	0	0
$214$	−44.4111	−3.03588
$215$	0	0
$216$	0	0
$217$	−6.08180	−0.412859
$218$	38.1216	2.58192
$219$	0	0
$220$	0	0
$221$	34.5300	2.32274
$222$	0	0
$223$	−13.2044	−0.884231	−0.442115	−	0.896958i	$-0.645772\pi$
$223$	−13.2044	−0.884231	−0.442115	+	0.896958i	$0.645772\pi$
$224$	31.9292	2.13336
$225$	0	0
$226$	−0.967223	−0.0643387
$227$	1.26137	0.0837201	0.0418601	−	0.999123i	$-0.486672\pi$
$227$	1.26137	0.0837201	0.0418601	+	0.999123i	$0.486672\pi$
$228$	0	0
$229$	−12.6777	−0.837769	−0.418885	−	0.908039i	$-0.637579\pi$
$229$	−12.6777	−0.837769	−0.418885	+	0.908039i	$0.637579\pi$
$230$	0	0
$231$	0	0
$232$	−69.9343	−4.59141
$233$	−19.8480	−1.30029	−0.650143	−	0.759812i	$-0.725291\pi$
$233$	−19.8480	−1.30029	−0.650143	+	0.759812i	$0.725291\pi$
$234$	0	0
$235$	0	0
$236$	14.7617	0.960902
$237$	0	0
$238$	35.3327	2.29028
$239$	−8.01022	−0.518138	−0.259069	−	0.965859i	$-0.583416\pi$
$239$	−8.01022	−0.518138	−0.259069	+	0.965859i	$0.583416\pi$
$240$	0	0
$241$	−12.8789	−0.829600	−0.414800	−	0.909913i	$-0.636148\pi$
$241$	−12.8789	−0.829600	−0.414800	+	0.909913i	$0.636148\pi$
$242$	12.0949	0.777487
$243$	0	0
$244$	−0.696123	−0.0445647
$245$	0	0
$246$	0	0
$247$	−5.35075	−0.340460
$248$	−24.8529	−1.57816
$249$	0	0
$250$	0	0
$251$	−24.4510	−1.54333	−0.771667	−	0.636027i	$-0.780577\pi$
$251$	−24.4510	−1.54333	−0.771667	+	0.636027i	$0.780577\pi$
$252$	0	0
$253$	−13.7248	−0.862869
$254$	−31.4608	−1.97402
$255$	0	0
$256$	6.16833	0.385520
$257$	−4.13020	−0.257635	−0.128817	−	0.991668i	$-0.541118\pi$
$257$	−4.13020	−0.257635	−0.128817	+	0.991668i	$0.541118\pi$
$258$	0	0
$259$	−13.1728	−0.818516
$260$	0	0
$261$	0	0
$262$	2.84788	0.175943
$263$	5.08189	0.313363	0.156681	−	0.987649i	$-0.449920\pi$
$263$	5.08189	0.313363	0.156681	+	0.987649i	$0.449920\pi$
$264$	0	0
$265$	0	0
$266$	−5.47514	−0.335703
$267$	0	0
$268$	26.5784	1.62353
$269$	−17.3624	−1.05860	−0.529302	−	0.848434i	$-0.677546\pi$
$269$	−17.3624	−1.05860	−0.529302	+	0.848434i	$0.677546\pi$
$270$	0	0
$271$	−24.5422	−1.49083	−0.745416	−	0.666599i	$-0.767749\pi$
$271$	−24.5422	−1.49083	−0.745416	+	0.666599i	$0.767749\pi$
$272$	78.1027	4.73567
$273$	0	0
$274$	−51.0844	−3.08612
$275$	0	0
$276$	0	0
$277$	26.8562	1.61363	0.806817	−	0.590801i	$-0.201188\pi$
$277$	26.8562	1.61363	0.806817	+	0.590801i	$0.201188\pi$
$278$	−35.7249	−2.14264
$279$	0	0
$280$	0	0
$281$	−6.41211	−0.382514	−0.191257	−	0.981540i	$-0.561256\pi$
$281$	−6.41211	−0.382514	−0.191257	+	0.981540i	$0.561256\pi$
$282$	0	0
$283$	−17.0554	−1.01384	−0.506919	−	0.861994i	$-0.669216\pi$
$283$	−17.0554	−1.01384	−0.506919	+	0.861994i	$0.669216\pi$
$284$	−14.7617	−0.875943
$285$	0	0
$286$	−56.3227	−3.33043
$287$	−23.4046	−1.38153
$288$	0	0
$289$	24.6450	1.44970
$290$	0	0
$291$	0	0
$292$	−9.62545	−0.563287
$293$	17.9639	1.04946	0.524732	−	0.851267i	$-0.324165\pi$
$293$	17.9639	1.04946	0.524732	+	0.851267i	$0.324165\pi$
$294$	0	0
$295$	0	0
$296$	−53.8298	−3.12879
$297$	0	0
$298$	−6.60219	−0.382455
$299$	18.6365	1.07778
$300$	0	0
$301$	14.8089	0.853568
$302$	−19.3353	−1.11262
$303$	0	0
$304$	−12.1028	−0.694142
$305$	0	0
$306$	0	0
$307$	23.2044	1.32434	0.662172	−	0.749352i	$-0.269635\pi$
$307$	23.2044	1.32434	0.662172	+	0.749352i	$0.269635\pi$
$308$	−41.4785	−2.36346
$309$	0	0
$310$	0	0
$311$	−13.8903	−0.787649	−0.393825	−	0.919186i	$-0.628848\pi$
$311$	−13.8903	−0.787649	−0.393825	+	0.919186i	$0.628848\pi$
$312$	0	0
$313$	−26.2276	−1.48247	−0.741237	−	0.671244i	$-0.765760\pi$
$313$	−26.2276	−1.48247	−0.741237	+	0.671244i	$0.765760\pi$
$314$	20.5105	1.15747
$315$	0	0
$316$	71.5610	4.02562
$317$	−6.55478	−0.368153	−0.184077	−	0.982912i	$-0.558929\pi$
$317$	−6.55478	−0.368153	−0.184077	+	0.982912i	$0.558929\pi$
$318$	0	0
$319$	32.9017	1.84214
$320$	0	0
$321$	0	0
$322$	19.0698	1.06272
$323$	−6.45329	−0.359071
$324$	0	0
$325$	0	0
$326$	−13.3562	−0.739732
$327$	0	0
$328$	−95.6416	−5.28092
$329$	−2.27676	−0.125522
$330$	0	0
$331$	−7.91163	−0.434863	−0.217431	−	0.976076i	$-0.569768\pi$
$331$	−7.91163	−0.434863	−0.217431	+	0.976076i	$0.569768\pi$
$332$	36.9863	2.02988
$333$	0	0
$334$	−21.3039	−1.16570
$335$	0	0
$336$	0	0
$337$	2.67825	0.145894	0.0729468	−	0.997336i	$-0.476760\pi$
$337$	2.67825	0.145894	0.0729468	+	0.997336i	$0.476760\pi$
$338$	41.7531	2.27107
$339$	0	0
$340$	0	0
$341$	11.6924	0.633181
$342$	0	0
$343$	−20.0844	−1.08446
$344$	60.5156	3.26278
$345$	0	0
$346$	39.4439	2.12052
$347$	−9.57418	−0.513969	−0.256984	−	0.966416i	$-0.582729\pi$
$347$	−9.57418	−0.513969	−0.256984	+	0.966416i	$0.582729\pi$
$348$	0	0
$349$	−17.9816	−0.962534	−0.481267	−	0.876574i	$-0.659823\pi$
$349$	−17.9816	−0.962534	−0.481267	+	0.876574i	$0.659823\pi$
$350$	0	0
$351$	0	0
$352$	−61.3849	−3.27183
$353$	4.56924	0.243196	0.121598	−	0.992579i	$-0.461198\pi$
$353$	4.56924	0.243196	0.121598	+	0.992579i	$0.461198\pi$
$354$	0	0
$355$	0	0
$356$	32.9297	1.74527
$357$	0	0
$358$	47.6306	2.51736
$359$	−4.87750	−0.257425	−0.128712	−	0.991682i	$-0.541084\pi$
$359$	−4.87750	−0.257425	−0.128712	+	0.991682i	$0.541084\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.4758	−0.918509
$363$	0	0
$364$	56.3227	2.95211
$365$	0	0
$366$	0	0
$367$	25.8826	1.35106	0.675531	−	0.737332i	$-0.263915\pi$
$367$	25.8826	1.35106	0.675531	+	0.737332i	$0.263915\pi$
$368$	42.1536	2.19741
$369$	0	0
$370$	0	0
$371$	−0.742155	−0.0385308
$372$	0	0
$373$	0.900675	0.0466352	0.0233176	−	0.999728i	$-0.492577\pi$
$373$	0.900675	0.0466352	0.0233176	+	0.999728i	$0.492577\pi$
$374$	−67.9282	−3.51248
$375$	0	0
$376$	−9.30388	−0.479811
$377$	−44.6764	−2.30095
$378$	0	0
$379$	14.9672	0.768815	0.384407	−	0.923164i	$-0.374406\pi$
$379$	14.9672	0.768815	0.384407	+	0.923164i	$0.374406\pi$
$380$	0	0
$381$	0	0
$382$	−39.7770	−2.03517
$383$	−26.8385	−1.37138	−0.685691	−	0.727893i	$-0.740500\pi$
$383$	−26.8385	−1.37138	−0.685691	+	0.727893i	$0.740500\pi$
$384$	0	0
$385$	0	0
$386$	30.9923	1.57747
$387$	0	0
$388$	27.4791	1.39504
$389$	11.3531	0.575626	0.287813	−	0.957687i	$-0.407072\pi$
$389$	11.3531	0.575626	0.287813	+	0.957687i	$0.407072\pi$
$390$	0	0
$391$	22.4766	1.13669
$392$	−23.4431	−1.18405
$393$	0	0
$394$	21.5094	1.08363
$395$	0	0
$396$	0	0
$397$	−4.89754	−0.245801	−0.122900	−	0.992419i	$-0.539220\pi$
$397$	−4.89754	−0.245801	−0.122900	+	0.992419i	$0.539220\pi$
$398$	−67.7798	−3.39750
$399$	0	0
$400$	0	0
$401$	29.0548	1.45093	0.725465	−	0.688259i	$-0.241625\pi$
$401$	29.0548	1.45093	0.725465	+	0.688259i	$0.241625\pi$
$402$	0	0
$403$	−15.8769	−0.790884
$404$	14.7617	0.734420
$405$	0	0
$406$	−45.7150	−2.26880
$407$	25.3251	1.25532
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	87.0627	4.28927
$413$	5.89155	0.289904
$414$	0	0
$415$	0	0
$416$	83.3531	4.08672
$417$	0	0
$418$	10.5261	0.514850
$419$	−24.0482	−1.17483	−0.587415	−	0.809286i	$-0.699854\pi$
$419$	−24.0482	−1.17483	−0.587415	+	0.809286i	$0.699854\pi$
$420$	0	0
$421$	33.1500	1.61563	0.807815	−	0.589436i	$-0.200650\pi$
$421$	33.1500	1.61563	0.807815	+	0.589436i	$0.200650\pi$
$422$	−5.16738	−0.251544
$423$	0	0
$424$	−3.03278	−0.147285
$425$	0	0
$426$	0	0
$427$	−0.277831	−0.0134452
$428$	−85.3817	−4.12708
$429$	0	0
$430$	0	0
$431$	−35.5304	−1.71144	−0.855721	−	0.517438i	$-0.826886\pi$
$431$	−35.5304	−1.71144	−0.855721	+	0.517438i	$0.826886\pi$
$432$	0	0
$433$	34.6016	1.66285	0.831425	−	0.555638i	$-0.187526\pi$
$433$	34.6016	1.66285	0.831425	+	0.555638i	$0.187526\pi$
$434$	−16.2460	−0.779831
$435$	0	0
$436$	73.2899	3.50995
$437$	−3.48297	−0.166613
$438$	0	0
$439$	−1.66335	−0.0793872	−0.0396936	−	0.999212i	$-0.512638\pi$
$439$	−1.66335	−0.0793872	−0.0396936	+	0.999212i	$0.512638\pi$
$440$	0	0
$441$	0	0
$442$	92.2380	4.38732
$443$	36.3636	1.72769	0.863843	−	0.503761i	$-0.168051\pi$
$443$	36.3636	1.72769	0.863843	+	0.503761i	$0.168051\pi$
$444$	0	0
$445$	0	0
$446$	−35.2721	−1.67018
$447$	0	0
$448$	35.6777	1.68561
$449$	34.2563	1.61666	0.808328	−	0.588733i	$-0.200373\pi$
$449$	34.2563	1.61666	0.808328	+	0.588733i	$0.200373\pi$
$450$	0	0
$451$	44.9960	2.11878
$452$	−1.85952	−0.0874643
$453$	0	0
$454$	3.36943	0.158135
$455$	0	0
$456$	0	0
$457$	21.3301	0.997779	0.498890	−	0.866665i	$-0.333741\pi$
$457$	21.3301	0.997779	0.498890	+	0.866665i	$0.333741\pi$
$458$	−33.8654	−1.58243
$459$	0	0
$460$	0	0
$461$	30.9266	1.44040	0.720198	−	0.693768i	$-0.244051\pi$
$461$	30.9266	1.44040	0.720198	+	0.693768i	$0.244051\pi$
$462$	0	0
$463$	37.3823	1.73731	0.868653	−	0.495422i	$-0.164986\pi$
$463$	37.3823	1.73731	0.868653	+	0.495422i	$0.164986\pi$
$464$	−101.053	−4.69125
$465$	0	0
$466$	−53.0188	−2.45605
$467$	−17.7855	−0.823015	−0.411507	−	0.911406i	$-0.634998\pi$
$467$	−17.7855	−0.823015	−0.411507	+	0.911406i	$0.634998\pi$
$468$	0	0
$469$	10.6078	0.489821
$470$	0	0
$471$	0	0
$472$	24.0755	1.10817
$473$	−28.4705	−1.30907
$474$	0	0
$475$	0	0
$476$	67.9282	3.11348
$477$	0	0
$478$	−21.3973	−0.978688
$479$	−2.21111	−0.101028	−0.0505141	−	0.998723i	$-0.516086\pi$
$479$	−2.21111	−0.101028	−0.0505141	+	0.998723i	$0.516086\pi$
$480$	0	0
$481$	−34.3883	−1.56797
$482$	−34.4026	−1.56699
$483$	0	0
$484$	23.2527	1.05694
$485$	0	0
$486$	0	0
$487$	−21.9234	−0.993445	−0.496722	−	0.867910i	$-0.665463\pi$
$487$	−21.9234	−0.993445	−0.496722	+	0.867910i	$0.665463\pi$
$488$	−1.13534	−0.0513945
$489$	0	0
$490$	0	0
$491$	−15.4885	−0.698984	−0.349492	−	0.936939i	$-0.613646\pi$
$491$	−15.4885	−0.698984	−0.349492	+	0.936939i	$0.613646\pi$
$492$	0	0
$493$	−53.8821	−2.42673
$494$	−14.2932	−0.643080
$495$	0	0
$496$	−35.9116	−1.61248
$497$	−5.89155	−0.264272
$498$	0	0
$499$	21.3739	0.956826	0.478413	−	0.878135i	$-0.341212\pi$
$499$	21.3739	0.956826	0.478413	+	0.878135i	$0.341212\pi$
$500$	0	0
$501$	0	0
$502$	−65.3146	−2.91513
$503$	−23.3064	−1.03918	−0.519591	−	0.854415i	$-0.673916\pi$
$503$	−23.3064	−1.03918	−0.519591	+	0.854415i	$0.673916\pi$
$504$	0	0
$505$	0	0
$506$	−36.6622	−1.62983
$507$	0	0
$508$	−60.4843	−2.68356
$509$	−3.53770	−0.156806	−0.0784030	−	0.996922i	$-0.524982\pi$
$509$	−3.53770	−0.156806	−0.0784030	+	0.996922i	$0.524982\pi$
$510$	0	0
$511$	−3.84163	−0.169944
$512$	−14.2064	−0.627841
$513$	0	0
$514$	−11.0328	−0.486635
$515$	0	0
$516$	0	0
$517$	4.37715	0.192507
$518$	−35.1877	−1.54606
$519$	0	0
$520$	0	0
$521$	16.1671	0.708294	0.354147	−	0.935190i	$-0.384771\pi$
$521$	16.1671	0.708294	0.354147	+	0.935190i	$0.384771\pi$
$522$	0	0
$523$	−9.27469	−0.405554	−0.202777	−	0.979225i	$-0.564997\pi$
$523$	−9.27469	−0.405554	−0.202777	+	0.979225i	$0.564997\pi$
$524$	5.47514	0.239183
$525$	0	0
$526$	13.5750	0.591897
$527$	−19.1484	−0.834116
$528$	0	0
$529$	−10.8689	−0.472561
$530$	0	0
$531$	0	0
$532$	−10.5261	−0.456366
$533$	−61.0990	−2.64649
$534$	0	0
$535$	0	0
$536$	43.3480	1.87235
$537$	0	0
$538$	−46.3792	−1.99955
$539$	11.0291	0.475059
$540$	0	0
$541$	29.9816	1.28901	0.644505	−	0.764600i	$-0.277063\pi$
$541$	29.9816	1.28901	0.644505	+	0.764600i	$0.277063\pi$
$542$	−65.5582	−2.81597
$543$	0	0
$544$	100.528	4.31011
$545$	0	0
$546$	0	0
$547$	10.1458	0.433805	0.216903	−	0.976193i	$-0.430405\pi$
$547$	10.1458	0.433805	0.216903	+	0.976193i	$0.430405\pi$
$548$	−98.2113	−4.19538
$549$	0	0
$550$	0	0
$551$	8.34955	0.355703
$552$	0	0
$553$	28.5609	1.21453
$554$	71.7395	3.04792
$555$	0	0
$556$	−68.6822	−2.91277
$557$	28.8326	1.22167	0.610837	−	0.791756i	$-0.290833\pi$
$557$	28.8326	1.22167	0.610837	+	0.791756i	$0.290833\pi$
$558$	0	0
$559$	38.6594	1.63512
$560$	0	0
$561$	0	0
$562$	−17.1283	−0.722514
$563$	−2.03463	−0.0857492	−0.0428746	−	0.999080i	$-0.513652\pi$
$563$	−2.03463	−0.0857492	−0.0428746	+	0.999080i	$0.513652\pi$
$564$	0	0
$565$	0	0
$566$	−45.5591	−1.91499
$567$	0	0
$568$	−24.0755	−1.01019
$569$	−20.2368	−0.848371	−0.424185	−	0.905575i	$-0.639440\pi$
$569$	−20.2368	−0.848371	−0.424185	+	0.905575i	$0.639440\pi$
$570$	0	0
$571$	−47.1500	−1.97316	−0.986582	−	0.163266i	$-0.947797\pi$
$571$	−47.1500	−1.97316	−0.986582	+	0.163266i	$0.947797\pi$
$572$	−108.282	−4.52750
$573$	0	0
$574$	−62.5194	−2.60951
$575$	0	0
$576$	0	0
$577$	7.23078	0.301021	0.150511	−	0.988608i	$-0.451908\pi$
$577$	7.23078	0.301021	0.150511	+	0.988608i	$0.451908\pi$
$578$	65.8328	2.73828
$579$	0	0
$580$	0	0
$581$	14.7617	0.612417
$582$	0	0
$583$	1.42682	0.0590927
$584$	−15.6986	−0.649614
$585$	0	0
$586$	47.9861	1.98229
$587$	−31.6192	−1.30507	−0.652533	−	0.757761i	$-0.726293\pi$
$587$	−31.6192	−1.30507	−0.652533	+	0.757761i	$0.726293\pi$
$588$	0	0
$589$	2.96722	0.122262
$590$	0	0
$591$	0	0
$592$	−77.7822	−3.19683
$593$	15.7263	0.645803	0.322902	−	0.946433i	$-0.395342\pi$
$593$	15.7263	0.645803	0.322902	+	0.946433i	$0.395342\pi$
$594$	0	0
$595$	0	0
$596$	−12.6929	−0.519922
$597$	0	0
$598$	49.7828	2.03577
$599$	−0.0635030	−0.00259466	−0.00129733	−	0.999999i	$-0.500413\pi$
$599$	−0.0635030	−0.00259466	−0.00129733	+	0.999999i	$0.500413\pi$
$600$	0	0
$601$	28.8789	1.17799	0.588996	−	0.808136i	$-0.299523\pi$
$601$	28.8789	1.17799	0.588996	+	0.808136i	$0.299523\pi$
$602$	39.5581	1.61227
$603$	0	0
$604$	−37.1728	−1.51254
$605$	0	0
$606$	0	0
$607$	−3.22817	−0.131027	−0.0655136	−	0.997852i	$-0.520869\pi$
$607$	−3.22817	−0.131027	−0.0655136	+	0.997852i	$0.520869\pi$
$608$	−15.5778	−0.631764
$609$	0	0
$610$	0	0
$611$	−5.94363	−0.240453
$612$	0	0
$613$	−17.0259	−0.687668	−0.343834	−	0.939030i	$-0.611726\pi$
$613$	−17.0259	−0.687668	−0.343834	+	0.939030i	$0.611726\pi$
$614$	61.9845	2.50149
$615$	0	0
$616$	−67.6494	−2.72567
$617$	−0.984773	−0.0396455	−0.0198227	−	0.999804i	$-0.506310\pi$
$617$	−0.984773	−0.0396455	−0.0198227	+	0.999804i	$0.506310\pi$
$618$	0	0
$619$	34.4766	1.38573	0.692867	−	0.721066i	$-0.256347\pi$
$619$	34.4766	1.38573	0.692867	+	0.721066i	$0.256347\pi$
$620$	0	0
$621$	0	0
$622$	−37.1045	−1.48776
$623$	13.1426	0.526549
$624$	0	0
$625$	0	0
$626$	−70.0604	−2.80018
$627$	0	0
$628$	39.4320	1.57351
$629$	−41.4741	−1.65368
$630$	0	0
$631$	−18.4950	−0.736275	−0.368138	−	0.929771i	$-0.620004\pi$
$631$	−18.4950	−0.736275	−0.368138	+	0.929771i	$0.620004\pi$
$632$	116.712	4.64257
$633$	0	0
$634$	−17.5094	−0.695388
$635$	0	0
$636$	0	0
$637$	−14.9762	−0.593379
$638$	87.8884	3.47954
$639$	0	0
$640$	0	0
$641$	−38.8733	−1.53540	−0.767702	−	0.640807i	$-0.778600\pi$
$641$	−38.8733	−1.53540	−0.767702	+	0.640807i	$0.778600\pi$
$642$	0	0
$643$	45.2302	1.78371	0.891853	−	0.452325i	$-0.149405\pi$
$643$	45.2302	1.78371	0.891853	+	0.452325i	$0.149405\pi$
$644$	36.6622	1.44469
$645$	0	0
$646$	−17.2383	−0.678233
$647$	−10.5995	−0.416709	−0.208354	−	0.978053i	$-0.566811\pi$
$647$	−10.5995	−0.416709	−0.208354	+	0.978053i	$0.566811\pi$
$648$	0	0
$649$	−11.3267	−0.444612
$650$	0	0
$651$	0	0
$652$	−25.6777	−1.00562
$653$	9.70021	0.379598	0.189799	−	0.981823i	$-0.439216\pi$
$653$	9.70021	0.379598	0.189799	+	0.981823i	$0.439216\pi$
$654$	0	0
$655$	0	0
$656$	−138.199	−5.39576
$657$	0	0
$658$	−6.08180	−0.237093
$659$	12.6294	0.491972	0.245986	−	0.969273i	$-0.420888\pi$
$659$	12.6294	0.491972	0.245986	+	0.969273i	$0.420888\pi$
$660$	0	0
$661$	18.9444	0.736852	0.368426	−	0.929657i	$-0.379897\pi$
$661$	18.9444	0.736852	0.368426	+	0.929657i	$0.379897\pi$
$662$	−21.1339	−0.821392
$663$	0	0
$664$	60.3227	2.34098
$665$	0	0
$666$	0	0
$667$	−29.0813	−1.12603
$668$	−40.9574	−1.58469
$669$	0	0
$670$	0	0
$671$	0.534139	0.0206202
$672$	0	0
$673$	9.28044	0.357735	0.178867	−	0.983873i	$-0.442757\pi$
$673$	9.28044	0.357735	0.178867	+	0.983873i	$0.442757\pi$
$674$	7.15427	0.275572
$675$	0	0
$676$	80.2716	3.08737
$677$	22.9283	0.881208	0.440604	−	0.897702i	$-0.354764\pi$
$677$	22.9283	0.881208	0.440604	+	0.897702i	$0.354764\pi$
$678$	0	0
$679$	10.9672	0.420883
$680$	0	0
$681$	0	0
$682$	31.2334	1.19599
$683$	32.0059	1.22467	0.612335	−	0.790598i	$-0.290230\pi$
$683$	32.0059	1.22467	0.612335	+	0.790598i	$0.290230\pi$
$684$	0	0
$685$	0	0
$686$	−53.6504	−2.04838
$687$	0	0
$688$	87.4429	3.33373
$689$	−1.93744	−0.0738105
$690$	0	0
$691$	−47.5794	−1.81001	−0.905003	−	0.425405i	$-0.860132\pi$
$691$	−47.5794	−1.81001	−0.905003	+	0.425405i	$0.860132\pi$
$692$	75.8320	2.88270
$693$	0	0
$694$	−25.5750	−0.970813
$695$	0	0
$696$	0	0
$697$	−73.6886	−2.79116
$698$	−48.0333	−1.81809
$699$	0	0
$700$	0	0
$701$	−0.389626	−0.0147160	−0.00735798	−	0.999973i	$-0.502342\pi$
$701$	−0.389626	−0.0147160	−0.00735798	+	0.999973i	$0.502342\pi$
$702$	0	0
$703$	6.42681	0.242392
$704$	−68.5915	−2.58514
$705$	0	0
$706$	12.2055	0.459362
$707$	5.89155	0.221575
$708$	0	0
$709$	26.6306	1.00013	0.500066	−	0.865987i	$-0.333309\pi$
$709$	26.6306	1.00013	0.500066	+	0.865987i	$0.333309\pi$
$710$	0	0
$711$	0	0
$712$	53.7067	2.01274
$713$	−10.3348	−0.387040
$714$	0	0
$715$	0	0
$716$	91.5714	3.42218
$717$	0	0
$718$	−13.0290	−0.486238
$719$	−4.13535	−0.154222	−0.0771112	−	0.997023i	$-0.524570\pi$
$719$	−4.13535	−0.154222	−0.0771112	+	0.997023i	$0.524570\pi$
$720$	0	0
$721$	34.7477	1.29407
$722$	2.67125	0.0994134
$723$	0	0
$724$	−33.5978	−1.24865
$725$	0	0
$726$	0	0
$727$	−33.8034	−1.25370	−0.626850	−	0.779140i	$-0.715656\pi$
$727$	−33.8034	−1.25370	−0.626850	+	0.779140i	$0.715656\pi$
$728$	91.8595	3.40454
$729$	0	0
$730$	0	0
$731$	46.6252	1.72450
$732$	0	0
$733$	−26.5784	−0.981695	−0.490848	−	0.871245i	$-0.663313\pi$
$733$	−26.5784	−0.981695	−0.490848	+	0.871245i	$0.663313\pi$
$734$	69.1388	2.55196
$735$	0	0
$736$	54.2572	1.99995
$737$	−20.3937	−0.751213
$738$	0	0
$739$	1.50498	0.0553616	0.0276808	−	0.999617i	$-0.491188\pi$
$739$	1.50498	0.0553616	0.0276808	+	0.999617i	$0.491188\pi$
$740$	0	0
$741$	0	0
$742$	−1.98248	−0.0727791
$743$	−14.6561	−0.537679	−0.268840	−	0.963185i	$-0.586640\pi$
$743$	−14.6561	−0.537679	−0.268840	+	0.963185i	$0.586640\pi$
$744$	0	0
$745$	0	0
$746$	2.40592	0.0880871
$747$	0	0
$748$	−130.594	−4.77499
$749$	−34.0768	−1.24514
$750$	0	0
$751$	2.02282	0.0738136	0.0369068	−	0.999319i	$-0.488250\pi$
$751$	2.02282	0.0738136	0.0369068	+	0.999319i	$0.488250\pi$
$752$	−13.4438	−0.490244
$753$	0	0
$754$	−119.342	−4.34616
$755$	0	0
$756$	0	0
$757$	21.4997	0.781421	0.390710	−	0.920514i	$-0.372229\pi$
$757$	21.4997	0.781421	0.390710	+	0.920514i	$0.372229\pi$
$758$	39.9811	1.45218
$759$	0	0
$760$	0	0
$761$	10.1578	0.368221	0.184111	−	0.982906i	$-0.441060\pi$
$761$	10.1578	0.368221	0.184111	+	0.982906i	$0.441060\pi$
$762$	0	0
$763$	29.2509	1.05895
$764$	−76.4725	−2.76668
$765$	0	0
$766$	−71.6922	−2.59034
$767$	15.3802	0.555348
$768$	0	0
$769$	4.76612	0.171871	0.0859353	−	0.996301i	$-0.472612\pi$
$769$	4.76612	0.171871	0.0859353	+	0.996301i	$0.472612\pi$
$770$	0	0
$771$	0	0
$772$	59.5836	2.14446
$773$	−23.9537	−0.861553	−0.430776	−	0.902459i	$-0.641760\pi$
$773$	−23.9537	−0.861553	−0.430776	+	0.902459i	$0.641760\pi$
$774$	0	0
$775$	0	0
$776$	44.8170	1.60884
$777$	0	0
$778$	30.3269	1.08727
$779$	11.4188	0.409120
$780$	0	0
$781$	11.3267	0.405301
$782$	60.0406	2.14705
$783$	0	0
$784$	−33.8744	−1.20980
$785$	0	0
$786$	0	0
$787$	13.7248	0.489235	0.244618	−	0.969620i	$-0.421338\pi$
$787$	13.7248	0.489235	0.244618	+	0.969620i	$0.421338\pi$
$788$	41.3525	1.47312
$789$	0	0
$790$	0	0
$791$	−0.742155	−0.0263880
$792$	0	0
$793$	−0.725294	−0.0257560
$794$	−13.0825	−0.464282
$795$	0	0
$796$	−130.309	−4.61867
$797$	−2.83575	−0.100447	−0.0502237	−	0.998738i	$-0.515993\pi$
$797$	−2.83575	−0.100447	−0.0502237	+	0.998738i	$0.515993\pi$
$798$	0	0
$799$	−7.16833	−0.253597
$800$	0	0
$801$	0	0
$802$	77.6126	2.74060
$803$	7.38566	0.260634
$804$	0	0
$805$	0	0
$806$	−42.4110	−1.49387
$807$	0	0
$808$	24.0755	0.846974
$809$	−12.4849	−0.438946	−0.219473	−	0.975619i	$-0.570434\pi$
$809$	−12.4849	−0.438946	−0.219473	+	0.975619i	$0.570434\pi$
$810$	0	0
$811$	−21.6633	−0.760703	−0.380351	−	0.924842i	$-0.624197\pi$
$811$	−21.6633	−0.760703	−0.380351	+	0.924842i	$0.624197\pi$
$812$	−87.8884	−3.08428
$813$	0	0
$814$	67.6494	2.37111
$815$	0	0
$816$	0	0
$817$	−7.22503	−0.252772
$818$	58.7674	2.05475
$819$	0	0
$820$	0	0
$821$	38.7442	1.35218	0.676091	−	0.736818i	$-0.263673\pi$
$821$	38.7442	1.35218	0.676091	+	0.736818i	$0.263673\pi$
$822$	0	0
$823$	33.0781	1.15303	0.576516	−	0.817086i	$-0.304412\pi$
$823$	33.0781	1.15303	0.576516	+	0.817086i	$0.304412\pi$
$824$	141.995	4.94662
$825$	0	0
$826$	15.7378	0.547588
$827$	−6.94995	−0.241673	−0.120837	−	0.992672i	$-0.538558\pi$
$827$	−6.94995	−0.241673	−0.120837	+	0.992672i	$0.538558\pi$
$828$	0	0
$829$	38.2711	1.32921	0.664605	−	0.747195i	$-0.268600\pi$
$829$	38.2711	1.32921	0.664605	+	0.747195i	$0.268600\pi$
$830$	0	0
$831$	0	0
$832$	93.1387	3.22900
$833$	−18.0621	−0.625815
$834$	0	0
$835$	0	0
$836$	20.2368	0.699904
$837$	0	0
$838$	−64.2386	−2.21908
$839$	−17.4413	−0.602139	−0.301069	−	0.953602i	$-0.597344\pi$
$839$	−17.4413	−0.602139	−0.301069	+	0.953602i	$0.597344\pi$
$840$	0	0
$841$	40.7150	1.40396
$842$	88.5516	3.05169
$843$	0	0
$844$	−9.93445	−0.341958
$845$	0	0
$846$	0	0
$847$	9.28044	0.318880
$848$	−4.38226	−0.150487
$849$	0	0
$850$	0	0
$851$	−22.3844	−0.767328
$852$	0	0
$853$	33.5551	1.14891	0.574453	−	0.818538i	$-0.305215\pi$
$853$	33.5551	1.14891	0.574453	+	0.818538i	$0.305215\pi$
$854$	−0.742155	−0.0253960
$855$	0	0
$856$	−139.253	−4.75958
$857$	26.3782	0.901063	0.450532	−	0.892760i	$-0.351234\pi$
$857$	26.3782	0.901063	0.450532	+	0.892760i	$0.351234\pi$
$858$	0	0
$859$	−11.8873	−0.405588	−0.202794	−	0.979221i	$-0.565002\pi$
$859$	−11.8873	−0.405588	−0.202794	+	0.979221i	$0.565002\pi$
$860$	0	0
$861$	0	0
$862$	−94.9105	−3.23267
$863$	21.6220	0.736022	0.368011	−	0.929821i	$-0.380039\pi$
$863$	21.6220	0.736022	0.368011	+	0.929821i	$0.380039\pi$
$864$	0	0
$865$	0	0
$866$	92.4295	3.14088
$867$	0	0
$868$	−31.2334	−1.06013
$869$	−54.9091	−1.86266
$870$	0	0
$871$	27.6922	0.938313
$872$	119.532	4.04787
$873$	0	0
$874$	−9.30388	−0.314708
$875$	0	0
$876$	0	0
$877$	−34.0813	−1.15084	−0.575421	−	0.817857i	$-0.695162\pi$
$877$	−34.0813	−1.15084	−0.575421	+	0.817857i	$0.695162\pi$
$878$	−4.44321	−0.149951
$879$	0	0
$880$	0	0
$881$	−24.9195	−0.839559	−0.419779	−	0.907626i	$-0.637893\pi$
$881$	−24.9195	−0.839559	−0.419779	+	0.907626i	$0.637893\pi$
$882$	0	0
$883$	37.5520	1.26372	0.631862	−	0.775081i	$-0.282291\pi$
$883$	37.5520	1.26372	0.631862	+	0.775081i	$0.282291\pi$
$884$	177.330	5.96427
$885$	0	0
$886$	97.1360	3.26335
$887$	3.24692	0.109021	0.0545104	−	0.998513i	$-0.482640\pi$
$887$	3.24692	0.109021	0.0545104	+	0.998513i	$0.482640\pi$
$888$	0	0
$889$	−24.1400	−0.809629
$890$	0	0
$891$	0	0
$892$	−67.8117	−2.27051
$893$	1.11080	0.0371716
$894$	0	0
$895$	0	0
$896$	31.4454	1.05052
$897$	0	0
$898$	91.5070	3.05363
$899$	24.7750	0.826292
$900$	0	0
$901$	−2.33665	−0.0778452
$902$	120.195	4.00207
$903$	0	0
$904$	−3.03278	−0.100869
$905$	0	0
$906$	0	0
$907$	24.6312	0.817865	0.408932	−	0.912565i	$-0.365901\pi$
$907$	24.6312	0.817865	0.408932	+	0.912565i	$0.365901\pi$
$908$	6.47783	0.214974
$909$	0	0
$910$	0	0
$911$	−4.55353	−0.150865	−0.0754326	−	0.997151i	$-0.524034\pi$
$911$	−4.55353	−0.150865	−0.0754326	+	0.997151i	$0.524034\pi$
$912$	0	0
$913$	−28.3797	−0.939232
$914$	56.9779	1.88466
$915$	0	0
$916$	−65.1072	−2.15120
$917$	2.18519	0.0721615
$918$	0	0
$919$	−25.0844	−0.827458	−0.413729	−	0.910400i	$-0.635774\pi$
$919$	−25.0844	−0.827458	−0.413729	+	0.910400i	$0.635774\pi$
$920$	0	0
$921$	0	0
$922$	82.6126	2.72070
$923$	−15.3802	−0.506247
$924$	0	0
$925$	0	0
$926$	99.8574	3.28152
$927$	0	0
$928$	−130.068	−4.26969
$929$	−48.5495	−1.59286	−0.796428	−	0.604733i	$-0.793280\pi$
$929$	−48.5495	−1.59286	−0.796428	+	0.604733i	$0.793280\pi$
$930$	0	0
$931$	2.79890	0.0917301
$932$	−101.930	−3.33884
$933$	0	0
$934$	−47.5094	−1.55456
$935$	0	0
$936$	0	0
$937$	−4.54678	−0.148537	−0.0742684	−	0.997238i	$-0.523662\pi$
$937$	−4.54678	−0.148537	−0.0742684	+	0.997238i	$0.523662\pi$
$938$	28.3359	0.925200
$939$	0	0
$940$	0	0
$941$	−3.60121	−0.117396	−0.0586980	−	0.998276i	$-0.518695\pi$
$941$	−3.60121	−0.117396	−0.0586980	+	0.998276i	$0.518695\pi$
$942$	0	0
$943$	−39.7713	−1.29513
$944$	34.7883	1.13226
$945$	0	0
$946$	−76.0516	−2.47265
$947$	41.5276	1.34947	0.674733	−	0.738062i	$-0.264259\pi$
$947$	41.5276	1.34947	0.674733	+	0.738062i	$0.264259\pi$
$948$	0	0
$949$	−10.0288	−0.325549
$950$	0	0
$951$	0	0
$952$	110.787	3.59064
$953$	28.7471	0.931209	0.465605	−	0.884993i	$-0.345837\pi$
$953$	28.7471	0.931209	0.465605	+	0.884993i	$0.345837\pi$
$954$	0	0
$955$	0	0
$956$	−41.1369	−1.33046
$957$	0	0
$958$	−5.90642	−0.190828
$959$	−39.1973	−1.26575
$960$	0	0
$961$	−22.1956	−0.715987
$962$	−91.8595	−2.96167
$963$	0	0
$964$	−66.1400	−2.13023
$965$	0	0
$966$	0	0
$967$	−18.5551	−0.596693	−0.298346	−	0.954458i	$-0.596435\pi$
$967$	−18.5551	−0.596693	−0.298346	+	0.954458i	$0.596435\pi$
$968$	37.9240	1.21892
$969$	0	0
$970$	0	0
$971$	−4.20100	−0.134817	−0.0674083	−	0.997725i	$-0.521473\pi$
$971$	−4.20100	−0.134817	−0.0674083	+	0.997725i	$0.521473\pi$
$972$	0	0
$973$	−27.4119	−0.878785
$974$	−58.5628	−1.87647
$975$	0	0
$976$	−1.64053	−0.0525121
$977$	−43.7280	−1.39898	−0.699492	−	0.714641i	$-0.746590\pi$
$977$	−43.7280	−1.39898	−0.699492	+	0.714641i	$0.746590\pi$
$978$	0	0
$979$	−25.2671	−0.807541
$980$	0	0
$981$	0	0
$982$	−41.3735	−1.32028
$983$	18.6112	0.593604	0.296802	−	0.954939i	$-0.404080\pi$
$983$	18.6112	0.593604	0.296802	+	0.954939i	$0.404080\pi$
$984$	0	0
$985$	0	0
$986$	−143.932	−4.58374
$987$	0	0
$988$	−27.4791	−0.874226
$989$	25.1646	0.800188
$990$	0	0
$991$	33.6266	1.06818	0.534092	−	0.845426i	$-0.320654\pi$
$991$	33.6266	1.06818	0.534092	+	0.845426i	$0.320654\pi$
$992$	−46.2229	−1.46758
$993$	0	0
$994$	−15.7378	−0.499172
$995$	0	0
$996$	0	0
$997$	−31.6808	−1.00334	−0.501671	−	0.865058i	$-0.667281\pi$
$997$	−31.6808	−1.00334	−0.501671	+	0.865058i	$0.667281\pi$
$998$	57.0949	1.80731
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bx.1.12		12
3.2	odd	2	inner	4275.2.a.bx.1.2		12
5.2	odd	4		855.2.c.f.514.11	yes	12
5.3	odd	4		855.2.c.f.514.1	✓	12
5.4	even	2	inner	4275.2.a.bx.1.1		12
15.2	even	4		855.2.c.f.514.2	yes	12
15.8	even	4		855.2.c.f.514.12	yes	12
15.14	odd	2	inner	4275.2.a.bx.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.c.f.514.1	✓	12	5.3	odd	4
855.2.c.f.514.2	yes	12	15.2	even	4
855.2.c.f.514.11	yes	12	5.2	odd	4
855.2.c.f.514.12	yes	12	15.8	even	4
4275.2.a.bx.1.1		12	5.4	even	2	inner
4275.2.a.bx.1.2		12	3.2	odd	2	inner
4275.2.a.bx.1.11		12	15.14	odd	2	inner
4275.2.a.bx.1.12		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67125 q^{2} +5.13555 q^{4} +2.04966 q^{7} +8.37582 q^{8} -3.94053 q^{11} +5.35075 q^{13} +5.47514 q^{14} +12.1028 q^{16} +6.45329 q^{17} -1.00000 q^{19} -10.5261 q^{22} +3.48297 q^{23} +14.2932 q^{26} +10.5261 q^{28} -8.34955 q^{29} -2.96722 q^{31} +15.5778 q^{32} +17.2383 q^{34} -6.42681 q^{37} -2.67125 q^{38} -11.4188 q^{41} +7.22503 q^{43} -20.2368 q^{44} +9.30388 q^{46} -1.11080 q^{47} -2.79890 q^{49} +27.4791 q^{52} -0.362087 q^{53} +17.1676 q^{56} -22.3037 q^{58} +2.87441 q^{59} -0.135550 q^{61} -7.92618 q^{62} +17.4066 q^{64} +5.17537 q^{67} +33.1412 q^{68} -2.87441 q^{71} -1.87428 q^{73} -17.1676 q^{74} -5.13555 q^{76} -8.07675 q^{77} +13.9344 q^{79} -30.5023 q^{82} +7.20201 q^{83} +19.2998 q^{86} -33.0052 q^{88} +6.41211 q^{89} +10.9672 q^{91} +17.8870 q^{92} -2.96722 q^{94} +5.35075 q^{97} -7.47654 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{4} + 40 q^{16} - 12 q^{19} + 24 q^{31} + 56 q^{34} + 80 q^{46} + 40 q^{49} + 44 q^{61} + 72 q^{64} - 16 q^{76} + 48 q^{79} + 72 q^{91} + 24 q^{94}+O(q^{100}) \) 12 * q + 16 * q^4 + 40 * q^16 - 12 * q^19 + 24 * q^31 + 56 * q^34 + 80 * q^46 + 40 * q^49 + 44 * q^61 + 72 * q^64 - 16 * q^76 + 48 * q^79 + 72 * q^91 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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