LMFDB - Embedded newform 4275.2.a.ba.1.3

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:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.65109$ 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)

$f(q)$	$=$	$q+1.65109 q^{2} +0.726109 q^{4} -0.377203 q^{7} -2.10331 q^{8} +O(q^{10})$	$q+1.65109 q^{2} +0.726109 q^{4} -0.377203 q^{7} -2.10331 q^{8} +1.37720 q^{11} +2.82167 q^{13} -0.622797 q^{14} -4.92498 q^{16} -6.37720 q^{17} +1.00000 q^{19} +2.27389 q^{22} -6.19887 q^{23} +4.65884 q^{26} -0.273891 q^{28} +3.37720 q^{29} +2.48052 q^{31} -3.92498 q^{32} -10.5294 q^{34} +5.58383 q^{37} +1.65109 q^{38} -8.50106 q^{41} -12.1522 q^{43} +1.00000 q^{44} -10.2349 q^{46} +6.87826 q^{47} -6.85772 q^{49} +2.04884 q^{52} -11.5478 q^{53} +0.793375 q^{56} +5.57608 q^{58} -6.05659 q^{59} +5.02830 q^{61} +4.09556 q^{62} +3.36945 q^{64} +3.22717 q^{67} -4.63055 q^{68} +2.30219 q^{71} -3.19887 q^{73} +9.21942 q^{74} +0.726109 q^{76} -0.519485 q^{77} -6.71836 q^{79} -14.0360 q^{82} -18.2165 q^{83} -20.0643 q^{86} -2.89669 q^{88} -1.50106 q^{89} -1.06434 q^{91} -4.50106 q^{92} +11.3567 q^{94} -11.7827 q^{97} -11.3227 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 4 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 4 * q^4 + 4 * q^7 - 3 * q^8	$ 3 q - 2 q^{2} + 4 q^{4} + 4 q^{7} - 3 q^{8} - q^{11} + 3 q^{13} - 7 q^{14} - 6 q^{16} - 14 q^{17} + 3 q^{19} + 5 q^{22} - 8 q^{23} + 11 q^{26} + q^{28} + 5 q^{29} - q^{31} - 3 q^{32} + 5 q^{34} + 5 q^{37} - 2 q^{38} - q^{41} - 5 q^{43} + 3 q^{44} - 12 q^{46} - 9 q^{47} - 7 q^{49} - 22 q^{52} - 31 q^{53} + 9 q^{56} + q^{58} + 6 q^{59} + 3 q^{61} + 5 q^{62} + q^{64} - 13 q^{67} - 23 q^{68} - 7 q^{71} + q^{73} + q^{74} + 4 q^{76} - 10 q^{77} - 18 q^{79} - 34 q^{82} - 3 q^{83} - 40 q^{86} - 12 q^{88} + 20 q^{89} + 17 q^{91} + 11 q^{92} + 45 q^{94} - 13 q^{97} - 4 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 4 * q^4 + 4 * q^7 - 3 * q^8 - q^11 + 3 * q^13 - 7 * q^14 - 6 * q^16 - 14 * q^17 + 3 * q^19 + 5 * q^22 - 8 * q^23 + 11 * q^26 + q^28 + 5 * q^29 - q^31 - 3 * q^32 + 5 * q^34 + 5 * q^37 - 2 * q^38 - q^41 - 5 * q^43 + 3 * q^44 - 12 * q^46 - 9 * q^47 - 7 * q^49 - 22 * q^52 - 31 * q^53 + 9 * q^56 + q^58 + 6 * q^59 + 3 * q^61 + 5 * q^62 + q^64 - 13 * q^67 - 23 * q^68 - 7 * q^71 + q^73 + q^74 + 4 * q^76 - 10 * q^77 - 18 * q^79 - 34 * q^82 - 3 * q^83 - 40 * q^86 - 12 * q^88 + 20 * q^89 + 17 * q^91 + 11 * q^92 + 45 * q^94 - 13 * q^97 - 4 * 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 $

$2$	1.65109	1.16750	0.583750	−	0.811934i	$-0.301585\pi$
$2$	1.65109	1.16750	0.583750	+	0.811934i	$0.301585\pi$
$3$	0	0
$3$	0	0
$4$	0.726109	0.363055
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.377203	−0.142569	−0.0712846	−	0.997456i	$-0.522710\pi$
$7$	−0.377203	−0.142569	−0.0712846	+	0.997456i	$0.522710\pi$
$8$	−2.10331	−0.743633
$9$	0	0
$10$	0	0
$11$	1.37720	0.415242	0.207621	−	0.978209i	$-0.433428\pi$
$11$	1.37720	0.415242	0.207621	+	0.978209i	$0.433428\pi$
$12$	0	0
$13$	2.82167	0.782591	0.391295	−	0.920265i	$-0.372027\pi$
$13$	2.82167	0.782591	0.391295	+	0.920265i	$0.372027\pi$
$14$	−0.622797	−0.166450
$15$	0	0
$16$	−4.92498	−1.23125
$17$	−6.37720	−1.54670	−0.773349	−	0.633980i	$-0.781420\pi$
$17$	−6.37720	−1.54670	−0.773349	+	0.633980i	$0.781420\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	2.27389	0.484795
$23$	−6.19887	−1.29255	−0.646277	−	0.763103i	$-0.723675\pi$
$23$	−6.19887	−1.29255	−0.646277	+	0.763103i	$0.723675\pi$
$24$	0	0
$25$	0	0
$26$	4.65884	0.913674
$27$	0	0
$28$	−0.273891	−0.0517604
$29$	3.37720	0.627131	0.313565	−	0.949567i	$-0.398476\pi$
$29$	3.37720	0.627131	0.313565	+	0.949567i	$0.398476\pi$
$30$	0	0
$31$	2.48052	0.445514	0.222757	−	0.974874i	$-0.428494\pi$
$31$	2.48052	0.445514	0.222757	+	0.974874i	$0.428494\pi$
$32$	−3.92498	−0.693846
$33$	0	0
$34$	−10.5294	−1.80577
$35$	0	0
$36$	0	0
$37$	5.58383	0.917976	0.458988	−	0.888443i	$-0.348212\pi$
$37$	5.58383	0.917976	0.458988	+	0.888443i	$0.348212\pi$
$38$	1.65109	0.267843
$39$	0	0
$40$	0	0
$41$	−8.50106	−1.32764	−0.663821	−	0.747891i	$-0.731066\pi$
$41$	−8.50106	−1.32764	−0.663821	+	0.747891i	$0.731066\pi$
$42$	0	0
$43$	−12.1522	−1.85319	−0.926593	−	0.376065i	$-0.877277\pi$
$43$	−12.1522	−1.85319	−0.926593	+	0.376065i	$0.877277\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−10.2349	−1.50906
$47$	6.87826	1.00330	0.501649	−	0.865071i	$-0.332727\pi$
$47$	6.87826	1.00330	0.501649	+	0.865071i	$0.332727\pi$
$48$	0	0
$49$	−6.85772	−0.979674
$50$	0	0
$51$	0	0
$52$	2.04884	0.284123
$53$	−11.5478	−1.58621	−0.793105	−	0.609085i	$-0.791537\pi$
$53$	−11.5478	−1.58621	−0.793105	+	0.609085i	$0.791537\pi$
$54$	0	0
$55$	0	0
$56$	0.793375	0.106019
$57$	0	0
$58$	5.57608	0.732175
$59$	−6.05659	−0.788501	−0.394251	−	0.919003i	$-0.628996\pi$
$59$	−6.05659	−0.788501	−0.394251	+	0.919003i	$0.628996\pi$
$60$	0	0
$61$	5.02830	0.643807	0.321904	−	0.946772i	$-0.395677\pi$
$61$	5.02830	0.643807	0.321904	+	0.946772i	$0.395677\pi$
$62$	4.09556	0.520137
$63$	0	0
$64$	3.36945	0.421182
$65$	0	0
$66$	0	0
$67$	3.22717	0.394262	0.197131	−	0.980377i	$-0.436838\pi$
$67$	3.22717	0.394262	0.197131	+	0.980377i	$0.436838\pi$
$68$	−4.63055	−0.561536
$69$	0	0
$70$	0	0
$71$	2.30219	0.273219	0.136610	−	0.990625i	$-0.456379\pi$
$71$	2.30219	0.273219	0.136610	+	0.990625i	$0.456379\pi$
$72$	0	0
$73$	−3.19887	−0.374400	−0.187200	−	0.982322i	$-0.559941\pi$
$73$	−3.19887	−0.374400	−0.187200	+	0.982322i	$0.559941\pi$
$74$	9.21942	1.07174
$75$	0	0
$76$	0.726109	0.0832905
$77$	−0.519485	−0.0592008
$78$	0	0
$79$	−6.71836	−0.755874	−0.377937	−	0.925831i	$-0.623366\pi$
$79$	−6.71836	−0.755874	−0.377937	+	0.925831i	$0.623366\pi$
$80$	0	0
$81$	0	0
$82$	−14.0360	−1.55002
$83$	−18.2165	−1.99952	−0.999760	−	0.0218996i	$-0.993029\pi$
$83$	−18.2165	−1.99952	−0.999760	+	0.0218996i	$0.993029\pi$
$84$	0	0
$85$	0	0
$86$	−20.0643	−2.16359
$87$	0	0
$88$	−2.89669	−0.308788
$89$	−1.50106	−0.159112	−0.0795561	−	0.996830i	$-0.525350\pi$
$89$	−1.50106	−0.159112	−0.0795561	+	0.996830i	$0.525350\pi$
$90$	0	0
$91$	−1.06434	−0.111573
$92$	−4.50106	−0.469268
$93$	0	0
$94$	11.3567	1.17135
$95$	0	0
$96$	0	0
$97$	−11.7827	−1.19635	−0.598176	−	0.801365i	$-0.704108\pi$
$97$	−11.7827	−1.19635	−0.598176	+	0.801365i	$0.704108\pi$
$98$	−11.3227	−1.14377
$99$	0	0
$100$	0	0
$101$	9.18820	0.914260	0.457130	−	0.889400i	$-0.348877\pi$
$101$	9.18820	0.914260	0.457130	+	0.889400i	$0.348877\pi$
$102$	0	0
$103$	−9.47277	−0.933379	−0.466690	−	0.884421i	$-0.654553\pi$
$103$	−9.47277	−0.933379	−0.466690	+	0.884421i	$0.654553\pi$
$104$	−5.93486	−0.581961
$105$	0	0
$106$	−19.0665	−1.85190
$107$	15.3510	1.48404	0.742020	−	0.670378i	$-0.233868\pi$
$107$	15.3510	1.48404	0.742020	+	0.670378i	$0.233868\pi$
$108$	0	0
$109$	−16.7643	−1.60573	−0.802863	−	0.596163i	$-0.796691\pi$
$109$	−16.7643	−1.60573	−0.802863	+	0.596163i	$0.796691\pi$
$110$	0	0
$111$	0	0
$112$	1.85772	0.175538
$113$	13.3305	1.25403	0.627013	−	0.779009i	$-0.284277\pi$
$113$	13.3305	1.25403	0.627013	+	0.779009i	$0.284277\pi$
$114$	0	0
$115$	0	0
$116$	2.45222	0.227683
$117$	0	0
$118$	−10.0000	−0.920575
$119$	2.40550	0.220512
$120$	0	0
$121$	−9.10331	−0.827574
$122$	8.30219	0.751645
$123$	0	0
$124$	1.80113	0.161746
$125$	0	0
$126$	0	0
$127$	−3.04672	−0.270353	−0.135176	−	0.990822i	$-0.543160\pi$
$127$	−3.04672	−0.270353	−0.135176	+	0.990822i	$0.543160\pi$
$128$	13.4132	1.18557
$129$	0	0
$130$	0	0
$131$	8.70769	0.760794	0.380397	−	0.924823i	$-0.375787\pi$
$131$	8.70769	0.760794	0.380397	+	0.924823i	$0.375787\pi$
$132$	0	0
$133$	−0.377203	−0.0327076
$134$	5.32836	0.460300
$135$	0	0
$136$	13.4132	1.15018
$137$	−6.59450	−0.563406	−0.281703	−	0.959502i	$-0.590899\pi$
$137$	−6.59450	−0.563406	−0.281703	+	0.959502i	$0.590899\pi$
$138$	0	0
$139$	−10.1677	−0.862409	−0.431205	−	0.902254i	$-0.641911\pi$
$139$	−10.1677	−0.862409	−0.431205	+	0.902254i	$0.641911\pi$
$140$	0	0
$141$	0	0
$142$	3.80113	0.318983
$143$	3.88601	0.324965
$144$	0	0
$145$	0	0
$146$	−5.28164	−0.437112
$147$	0	0
$148$	4.05447	0.333275
$149$	5.54003	0.453857	0.226929	−	0.973911i	$-0.427132\pi$
$149$	5.54003	0.453857	0.226929	+	0.973911i	$0.427132\pi$
$150$	0	0
$151$	5.12386	0.416974	0.208487	−	0.978025i	$-0.433146\pi$
$151$	5.12386	0.416974	0.208487	+	0.978025i	$0.433146\pi$
$152$	−2.10331	−0.170601
$153$	0	0
$154$	−0.857718	−0.0691169
$155$	0	0
$156$	0	0
$157$	19.7643	1.57736	0.788681	−	0.614803i	$-0.210765\pi$
$157$	19.7643	1.57736	0.788681	+	0.614803i	$0.210765\pi$
$158$	−11.0926	−0.882483
$159$	0	0
$160$	0	0
$161$	2.33823	0.184279
$162$	0	0
$163$	13.5195	1.05893	0.529464	−	0.848332i	$-0.322393\pi$
$163$	13.5195	1.05893	0.529464	+	0.848332i	$0.322393\pi$
$164$	−6.17270	−0.482007
$165$	0	0
$166$	−30.0771	−2.33444
$167$	−13.1054	−1.01413	−0.507065	−	0.861908i	$-0.669269\pi$
$167$	−13.1054	−1.01413	−0.507065	+	0.861908i	$0.669269\pi$
$168$	0	0
$169$	−5.03817	−0.387551
$170$	0	0
$171$	0	0
$172$	−8.82379	−0.672808
$173$	−1.48827	−0.113151	−0.0565754	−	0.998398i	$-0.518018\pi$
$173$	−1.48827	−0.113151	−0.0565754	+	0.998398i	$0.518018\pi$
$174$	0	0
$175$	0	0
$176$	−6.78270	−0.511265
$177$	0	0
$178$	−2.47839	−0.185763
$179$	17.1132	1.27910	0.639550	−	0.768750i	$-0.279121\pi$
$179$	17.1132	1.27910	0.639550	+	0.768750i	$0.279121\pi$
$180$	0	0
$181$	−5.73891	−0.426569	−0.213285	−	0.976990i	$-0.568416\pi$
$181$	−5.73891	−0.426569	−0.213285	+	0.976990i	$0.568416\pi$
$182$	−1.75733	−0.130262
$183$	0	0
$184$	13.0382	0.961187
$185$	0	0
$186$	0	0
$187$	−8.78270	−0.642255
$188$	4.99437	0.364252
$189$	0	0
$190$	0	0
$191$	−22.3948	−1.62043	−0.810216	−	0.586131i	$-0.800650\pi$
$191$	−22.3948	−1.62043	−0.810216	+	0.586131i	$0.800650\pi$
$192$	0	0
$193$	−21.3687	−1.53815	−0.769075	−	0.639159i	$-0.779283\pi$
$193$	−21.3687	−1.53815	−0.769075	+	0.639159i	$0.779283\pi$
$194$	−19.4543	−1.39674
$195$	0	0
$196$	−4.97945	−0.355675
$197$	−1.11399	−0.0793682	−0.0396841	−	0.999212i	$-0.512635\pi$
$197$	−1.11399	−0.0793682	−0.0396841	+	0.999212i	$0.512635\pi$
$198$	0	0
$199$	−2.22505	−0.157729	−0.0788647	−	0.996885i	$-0.525130\pi$
$199$	−2.22505	−0.157729	−0.0788647	+	0.996885i	$0.525130\pi$
$200$	0	0
$201$	0	0
$202$	15.1706	1.06740
$203$	−1.27389	−0.0894096
$204$	0	0
$205$	0	0
$206$	−15.6404	−1.08972
$207$	0	0
$208$	−13.8967	−0.963562
$209$	1.37720	0.0952631
$210$	0	0
$211$	−9.75441	−0.671521	−0.335760	−	0.941947i	$-0.608993\pi$
$211$	−9.75441	−0.671521	−0.335760	+	0.941947i	$0.608993\pi$
$212$	−8.38495	−0.575881
$213$	0	0
$214$	25.3460	1.73262
$215$	0	0
$216$	0	0
$217$	−0.935657	−0.0635166
$218$	−27.6794	−1.87468
$219$	0	0
$220$	0	0
$221$	−17.9944	−1.21043
$222$	0	0
$223$	−18.6433	−1.24845	−0.624225	−	0.781244i	$-0.714585\pi$
$223$	−18.6433	−1.24845	−0.624225	+	0.781244i	$0.714585\pi$
$224$	1.48052	0.0989211
$225$	0	0
$226$	22.0099	1.46407
$227$	8.04672	0.534080	0.267040	−	0.963686i	$-0.413954\pi$
$227$	8.04672	0.534080	0.267040	+	0.963686i	$0.413954\pi$
$228$	0	0
$229$	20.1415	1.33099	0.665493	−	0.746404i	$-0.268221\pi$
$229$	20.1415	1.33099	0.665493	+	0.746404i	$0.268221\pi$
$230$	0	0
$231$	0	0
$232$	−7.10331	−0.466355
$233$	1.73386	0.113589	0.0567945	−	0.998386i	$-0.481912\pi$
$233$	1.73386	0.113589	0.0567945	+	0.998386i	$0.481912\pi$
$234$	0	0
$235$	0	0
$236$	−4.39775	−0.286269
$237$	0	0
$238$	3.97170	0.257447
$239$	18.3150	1.18470	0.592349	−	0.805682i	$-0.298201\pi$
$239$	18.3150	1.18470	0.592349	+	0.805682i	$0.298201\pi$
$240$	0	0
$241$	−2.19675	−0.141505	−0.0707526	−	0.997494i	$-0.522540\pi$
$241$	−2.19675	−0.141505	−0.0707526	+	0.997494i	$0.522540\pi$
$242$	−15.0304	−0.966192
$243$	0	0
$244$	3.65109	0.233737
$245$	0	0
$246$	0	0
$247$	2.82167	0.179539
$248$	−5.21730	−0.331299
$249$	0	0
$250$	0	0
$251$	13.6716	0.862946	0.431473	−	0.902126i	$-0.357994\pi$
$251$	13.6716	0.862946	0.431473	+	0.902126i	$0.357994\pi$
$252$	0	0
$253$	−8.53711	−0.536723
$254$	−5.03042	−0.315637
$255$	0	0
$256$	15.4076	0.962976
$257$	−28.4904	−1.77718	−0.888591	−	0.458701i	$-0.848315\pi$
$257$	−28.4904	−1.77718	−0.888591	+	0.458701i	$0.848315\pi$
$258$	0	0
$259$	−2.10624	−0.130875
$260$	0	0
$261$	0	0
$262$	14.3772	0.888227
$263$	−5.36945	−0.331095	−0.165547	−	0.986202i	$-0.552939\pi$
$263$	−5.36945	−0.331095	−0.165547	+	0.986202i	$0.552939\pi$
$264$	0	0
$265$	0	0
$266$	−0.622797	−0.0381861
$267$	0	0
$268$	2.34328	0.143139
$269$	−7.06727	−0.430899	−0.215449	−	0.976515i	$-0.569122\pi$
$269$	−7.06727	−0.430899	−0.215449	+	0.976515i	$0.569122\pi$
$270$	0	0
$271$	21.6970	1.31800	0.659000	−	0.752143i	$-0.270980\pi$
$271$	21.6970	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$272$	31.4076	1.90437
$273$	0	0
$274$	−10.8881	−0.657776
$275$	0	0
$276$	0	0
$277$	25.1132	1.50891	0.754453	−	0.656355i	$-0.227902\pi$
$277$	25.1132	1.50891	0.754453	+	0.656355i	$0.227902\pi$
$278$	−16.7877	−1.00686
$279$	0	0
$280$	0	0
$281$	10.0411	0.599001	0.299501	−	0.954096i	$-0.403180\pi$
$281$	10.0411	0.599001	0.299501	+	0.954096i	$0.403180\pi$
$282$	0	0
$283$	−20.9143	−1.24323	−0.621613	−	0.783324i	$-0.713522\pi$
$283$	−20.9143	−1.24323	−0.621613	+	0.783324i	$0.713522\pi$
$284$	1.67164	0.0991936
$285$	0	0
$286$	6.41617	0.379396
$287$	3.20662	0.189281
$288$	0	0
$289$	23.6687	1.39228
$290$	0	0
$291$	0	0
$292$	−2.32273	−0.135928
$293$	0.818748	0.0478318	0.0239159	−	0.999714i	$-0.492387\pi$
$293$	0.818748	0.0478318	0.0239159	+	0.999714i	$0.492387\pi$
$294$	0	0
$295$	0	0
$296$	−11.7445	−0.682637
$297$	0	0
$298$	9.14711	0.529878
$299$	−17.4912	−1.01154
$300$	0	0
$301$	4.58383	0.264207
$302$	8.45997	0.486817
$303$	0	0
$304$	−4.92498	−0.282467
$305$	0	0
$306$	0	0
$307$	−7.44447	−0.424878	−0.212439	−	0.977174i	$-0.568141\pi$
$307$	−7.44447	−0.424878	−0.212439	+	0.977174i	$0.568141\pi$
$308$	−0.377203	−0.0214931
$309$	0	0
$310$	0	0
$311$	0.956204	0.0542213	0.0271107	−	0.999632i	$-0.491369\pi$
$311$	0.956204	0.0542213	0.0271107	+	0.999632i	$0.491369\pi$
$312$	0	0
$313$	5.98158	0.338099	0.169049	−	0.985608i	$-0.445930\pi$
$313$	5.98158	0.338099	0.169049	+	0.985608i	$0.445930\pi$
$314$	32.6327	1.84157
$315$	0	0
$316$	−4.87826	−0.274424
$317$	−22.6511	−1.27221	−0.636106	−	0.771602i	$-0.719456\pi$
$317$	−22.6511	−1.27221	−0.636106	+	0.771602i	$0.719456\pi$
$318$	0	0
$319$	4.65109	0.260411
$320$	0	0
$321$	0	0
$322$	3.86064	0.215145
$323$	−6.37720	−0.354837
$324$	0	0
$325$	0	0
$326$	22.3219	1.23630
$327$	0	0
$328$	17.8804	0.987279
$329$	−2.59450	−0.143039
$330$	0	0
$331$	4.16283	0.228810	0.114405	−	0.993434i	$-0.463504\pi$
$331$	4.16283	0.228810	0.114405	+	0.993434i	$0.463504\pi$
$332$	−13.2272	−0.725935
$333$	0	0
$334$	−21.6383	−1.18399
$335$	0	0
$336$	0	0
$337$	−18.0304	−0.982180	−0.491090	−	0.871109i	$-0.663401\pi$
$337$	−18.0304	−0.982180	−0.491090	+	0.871109i	$0.663401\pi$
$338$	−8.31849	−0.452466
$339$	0	0
$340$	0	0
$341$	3.41617	0.184996
$342$	0	0
$343$	5.22717	0.282241
$344$	25.5598	1.37809
$345$	0	0
$346$	−2.45726	−0.132103
$347$	−7.43380	−0.399067	−0.199534	−	0.979891i	$-0.563943\pi$
$347$	−7.43380	−0.399067	−0.199534	+	0.979891i	$0.563943\pi$
$348$	0	0
$349$	21.9194	1.17332	0.586658	−	0.809835i	$-0.300443\pi$
$349$	21.9194	1.17332	0.586658	+	0.809835i	$0.300443\pi$
$350$	0	0
$351$	0	0
$352$	−5.40550	−0.288114
$353$	−10.3695	−0.551910	−0.275955	−	0.961171i	$-0.588994\pi$
$353$	−10.3695	−0.551910	−0.275955	+	0.961171i	$0.588994\pi$
$354$	0	0
$355$	0	0
$356$	−1.08993	−0.0577664
$357$	0	0
$358$	28.2555	1.49335
$359$	23.9893	1.26611	0.633054	−	0.774108i	$-0.281801\pi$
$359$	23.9893	1.26611	0.633054	+	0.774108i	$0.281801\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−9.47547	−0.498020
$363$	0	0
$364$	−0.772829	−0.0405073
$365$	0	0
$366$	0	0
$367$	35.0275	1.82842	0.914210	−	0.405240i	$-0.132812\pi$
$367$	35.0275	1.82842	0.914210	+	0.405240i	$0.132812\pi$
$368$	30.5294	1.59145
$369$	0	0
$370$	0	0
$371$	4.35586	0.226145
$372$	0	0
$373$	30.8003	1.59478	0.797390	−	0.603464i	$-0.206213\pi$
$373$	30.8003	1.59478	0.797390	+	0.603464i	$0.206213\pi$
$374$	−14.5011	−0.749832
$375$	0	0
$376$	−14.4671	−0.746086
$377$	9.52936	0.490787
$378$	0	0
$379$	0.671640	0.0344998	0.0172499	−	0.999851i	$-0.494509\pi$
$379$	0.671640	0.0344998	0.0172499	+	0.999851i	$0.494509\pi$
$380$	0	0
$381$	0	0
$382$	−36.9759	−1.89185
$383$	30.1805	1.54215	0.771075	−	0.636745i	$-0.219720\pi$
$383$	30.1805	1.54215	0.771075	+	0.636745i	$0.219720\pi$
$384$	0	0
$385$	0	0
$386$	−35.2816	−1.79579
$387$	0	0
$388$	−8.55553	−0.434341
$389$	31.5109	1.59767	0.798834	−	0.601552i	$-0.205451\pi$
$389$	31.5109	1.59767	0.798834	+	0.601552i	$0.205451\pi$
$390$	0	0
$391$	39.5315	1.99919
$392$	14.4239	0.728518
$393$	0	0
$394$	−1.83929	−0.0926623
$395$	0	0
$396$	0	0
$397$	−12.5761	−0.631175	−0.315588	−	0.948896i	$-0.602202\pi$
$397$	−12.5761	−0.631175	−0.315588	+	0.948896i	$0.602202\pi$
$398$	−3.67376	−0.184149
$399$	0	0
$400$	0	0
$401$	−24.6036	−1.22864	−0.614322	−	0.789056i	$-0.710570\pi$
$401$	−24.6036	−1.22864	−0.614322	+	0.789056i	$0.710570\pi$
$402$	0	0
$403$	6.99920	0.348655
$404$	6.67164	0.331926
$405$	0	0
$406$	−2.10331	−0.104386
$407$	7.69006	0.381182
$408$	0	0
$409$	13.6666	0.675770	0.337885	−	0.941187i	$-0.390289\pi$
$409$	13.6666	0.675770	0.337885	+	0.941187i	$0.390289\pi$
$410$	0	0
$411$	0	0
$412$	−6.87826	−0.338868
$413$	2.28456	0.112416
$414$	0	0
$415$	0	0
$416$	−11.0750	−0.542997
$417$	0	0
$418$	2.27389	0.111220
$419$	−12.6893	−0.619911	−0.309956	−	0.950751i	$-0.600314\pi$
$419$	−12.6893	−0.619911	−0.309956	+	0.950751i	$0.600314\pi$
$420$	0	0
$421$	29.1826	1.42227	0.711136	−	0.703055i	$-0.248181\pi$
$421$	29.1826	1.42227	0.711136	+	0.703055i	$0.248181\pi$
$422$	−16.1054	−0.784000
$423$	0	0
$424$	24.2886	1.17956
$425$	0	0
$426$	0	0
$427$	−1.89669	−0.0917872
$428$	11.1465	0.538788
$429$	0	0
$430$	0	0
$431$	29.9816	1.44416	0.722081	−	0.691809i	$-0.243186\pi$
$431$	29.9816	1.44416	0.722081	+	0.691809i	$0.243186\pi$
$432$	0	0
$433$	−4.76216	−0.228855	−0.114427	−	0.993432i	$-0.536503\pi$
$433$	−4.76216	−0.228855	−0.114427	+	0.993432i	$0.536503\pi$
$434$	−1.54486	−0.0741555
$435$	0	0
$436$	−12.1727	−0.582967
$437$	−6.19887	−0.296532
$438$	0	0
$439$	6.88601	0.328652	0.164326	−	0.986406i	$-0.447455\pi$
$439$	6.88601	0.328652	0.164326	+	0.986406i	$0.447455\pi$
$440$	0	0
$441$	0	0
$442$	−29.7104	−1.41318
$443$	−8.65109	−0.411026	−0.205513	−	0.978654i	$-0.565886\pi$
$443$	−8.65109	−0.411026	−0.205513	+	0.978654i	$0.565886\pi$
$444$	0	0
$445$	0	0
$446$	−30.7819	−1.45757
$447$	0	0
$448$	−1.27097	−0.0600476
$449$	−8.63055	−0.407301	−0.203650	−	0.979044i	$-0.565281\pi$
$449$	−8.63055	−0.407301	−0.203650	+	0.979044i	$0.565281\pi$
$450$	0	0
$451$	−11.7077	−0.551293
$452$	9.67939	0.455280
$453$	0	0
$454$	13.2859	0.623538
$455$	0	0
$456$	0	0
$457$	−28.6092	−1.33828	−0.669141	−	0.743135i	$-0.733338\pi$
$457$	−28.6092	−1.33828	−0.669141	+	0.743135i	$0.733338\pi$
$458$	33.2555	1.55393
$459$	0	0
$460$	0	0
$461$	39.1025	1.82119	0.910593	−	0.413305i	$-0.135626\pi$
$461$	39.1025	1.82119	0.910593	+	0.413305i	$0.135626\pi$
$462$	0	0
$463$	−8.04380	−0.373827	−0.186913	−	0.982376i	$-0.559848\pi$
$463$	−8.04380	−0.373827	−0.186913	+	0.982376i	$0.559848\pi$
$464$	−16.6327	−0.772152
$465$	0	0
$466$	2.86276	0.132615
$467$	−17.0488	−0.788926	−0.394463	−	0.918912i	$-0.629069\pi$
$467$	−17.0488	−0.788926	−0.394463	+	0.918912i	$0.629069\pi$
$468$	0	0
$469$	−1.21730	−0.0562096
$470$	0	0
$471$	0	0
$472$	12.7389	0.586356
$473$	−16.7360	−0.769521
$474$	0	0
$475$	0	0
$476$	1.74666	0.0800578
$477$	0	0
$478$	30.2397	1.38313
$479$	−22.5478	−1.03023	−0.515117	−	0.857120i	$-0.672252\pi$
$479$	−22.5478	−1.03023	−0.515117	+	0.857120i	$0.672252\pi$
$480$	0	0
$481$	15.7557	0.718399
$482$	−3.62704	−0.165207
$483$	0	0
$484$	−6.61000	−0.300455
$485$	0	0
$486$	0	0
$487$	−24.5577	−1.11281	−0.556407	−	0.830910i	$-0.687820\pi$
$487$	−24.5577	−1.11281	−0.556407	+	0.830910i	$0.687820\pi$
$488$	−10.5761	−0.478757
$489$	0	0
$490$	0	0
$491$	16.9298	0.764032	0.382016	−	0.924156i	$-0.375230\pi$
$491$	16.9298	0.764032	0.382016	+	0.924156i	$0.375230\pi$
$492$	0	0
$493$	−21.5371	−0.969983
$494$	4.65884	0.209611
$495$	0	0
$496$	−12.2165	−0.548537
$497$	−0.868391	−0.0389527
$498$	0	0
$499$	0.418295	0.0187255	0.00936273	−	0.999956i	$-0.497020\pi$
$499$	0.418295	0.0187255	0.00936273	+	0.999956i	$0.497020\pi$
$500$	0	0
$501$	0	0
$502$	22.5732	1.00749
$503$	12.5993	0.561776	0.280888	−	0.959741i	$-0.409371\pi$
$503$	12.5993	0.561776	0.280888	+	0.959741i	$0.409371\pi$
$504$	0	0
$505$	0	0
$506$	−14.0956	−0.626624
$507$	0	0
$508$	−2.21225	−0.0981528
$509$	−12.8209	−0.568275	−0.284138	−	0.958784i	$-0.591707\pi$
$509$	−12.8209	−0.568275	−0.284138	+	0.958784i	$0.591707\pi$
$510$	0	0
$511$	1.20662	0.0533779
$512$	−1.38708	−0.0613007
$513$	0	0
$514$	−47.0403	−2.07486
$515$	0	0
$516$	0	0
$517$	9.47277	0.416612
$518$	−3.47759	−0.152797
$519$	0	0
$520$	0	0
$521$	−11.3743	−0.498316	−0.249158	−	0.968463i	$-0.580154\pi$
$521$	−11.3743	−0.498316	−0.249158	+	0.968463i	$0.580154\pi$
$522$	0	0
$523$	33.0948	1.44713	0.723566	−	0.690255i	$-0.242502\pi$
$523$	33.0948	1.44713	0.723566	+	0.690255i	$0.242502\pi$
$524$	6.32273	0.276210
$525$	0	0
$526$	−8.86547	−0.386553
$527$	−15.8187	−0.689075
$528$	0	0
$529$	15.4260	0.670698
$530$	0	0
$531$	0	0
$532$	−0.273891	−0.0118747
$533$	−23.9872	−1.03900
$534$	0	0
$535$	0	0
$536$	−6.78775	−0.293186
$537$	0	0
$538$	−11.6687	−0.503074
$539$	−9.44447	−0.406802
$540$	0	0
$541$	31.4338	1.35144	0.675722	−	0.737156i	$-0.263832\pi$
$541$	31.4338	1.35144	0.675722	+	0.737156i	$0.263832\pi$
$542$	35.8238	1.53876
$543$	0	0
$544$	25.0304	1.07317
$545$	0	0
$546$	0	0
$547$	−23.3460	−0.998202	−0.499101	−	0.866544i	$-0.666336\pi$
$547$	−23.3460	−0.998202	−0.499101	+	0.866544i	$0.666336\pi$
$548$	−4.78833	−0.204547
$549$	0	0
$550$	0	0
$551$	3.37720	0.143874
$552$	0	0
$553$	2.53418	0.107764
$554$	41.4642	1.76165
$555$	0	0
$556$	−7.38283	−0.313102
$557$	−15.9229	−0.674673	−0.337337	−	0.941384i	$-0.609526\pi$
$557$	−15.9229	−0.674673	−0.337337	+	0.941384i	$0.609526\pi$
$558$	0	0
$559$	−34.2894	−1.45029
$560$	0	0
$561$	0	0
$562$	16.5788	0.699334
$563$	29.6404	1.24919	0.624597	−	0.780947i	$-0.285263\pi$
$563$	29.6404	1.24919	0.624597	+	0.780947i	$0.285263\pi$
$564$	0	0
$565$	0	0
$566$	−34.5315	−1.45147
$567$	0	0
$568$	−4.84222	−0.203175
$569$	7.64334	0.320426	0.160213	−	0.987082i	$-0.448782\pi$
$569$	7.64334	0.320426	0.160213	+	0.987082i	$0.448782\pi$
$570$	0	0
$571$	−10.5577	−0.441824	−0.220912	−	0.975294i	$-0.570903\pi$
$571$	−10.5577	−0.441824	−0.220912	+	0.975294i	$0.570903\pi$
$572$	2.82167	0.117980
$573$	0	0
$574$	5.29444	0.220986
$575$	0	0
$576$	0	0
$577$	38.9447	1.62129	0.810645	−	0.585538i	$-0.199117\pi$
$577$	38.9447	1.62129	0.810645	+	0.585538i	$0.199117\pi$
$578$	39.0793	1.62548
$579$	0	0
$580$	0	0
$581$	6.87131	0.285070
$582$	0	0
$583$	−15.9036	−0.658661
$584$	6.72823	0.278416
$585$	0	0
$586$	1.35183	0.0558436
$587$	10.9992	0.453986	0.226993	−	0.973896i	$-0.427111\pi$
$587$	10.9992	0.453986	0.226993	+	0.973896i	$0.427111\pi$
$588$	0	0
$589$	2.48052	0.102208
$590$	0	0
$591$	0	0
$592$	−27.5003	−1.13025
$593$	−26.7848	−1.09992	−0.549960	−	0.835191i	$-0.685357\pi$
$593$	−26.7848	−1.09992	−0.549960	+	0.835191i	$0.685357\pi$
$594$	0	0
$595$	0	0
$596$	4.02267	0.164775
$597$	0	0
$598$	−28.8796	−1.18097
$599$	3.44930	0.140934	0.0704672	−	0.997514i	$-0.477551\pi$
$599$	3.44930	0.140934	0.0704672	+	0.997514i	$0.477551\pi$
$600$	0	0
$601$	−37.0686	−1.51206	−0.756030	−	0.654537i	$-0.772863\pi$
$601$	−37.0686	−1.51206	−0.756030	+	0.654537i	$0.772863\pi$
$602$	7.56833	0.308462
$603$	0	0
$604$	3.72048	0.151384
$605$	0	0
$606$	0	0
$607$	−14.8732	−0.603685	−0.301843	−	0.953358i	$-0.597602\pi$
$607$	−14.8732	−0.603685	−0.301843	+	0.953358i	$0.597602\pi$
$608$	−3.92498	−0.159179
$609$	0	0
$610$	0	0
$611$	19.4082	0.785172
$612$	0	0
$613$	−40.4671	−1.63445	−0.817226	−	0.576317i	$-0.804489\pi$
$613$	−40.4671	−1.63445	−0.817226	+	0.576317i	$0.804489\pi$
$614$	−12.2915	−0.496045
$615$	0	0
$616$	1.09264	0.0440237
$617$	−8.76991	−0.353063	−0.176532	−	0.984295i	$-0.556488\pi$
$617$	−8.76991	−0.353063	−0.176532	+	0.984295i	$0.556488\pi$
$618$	0	0
$619$	32.0510	1.28824	0.644119	−	0.764926i	$-0.277224\pi$
$619$	32.0510	1.28824	0.644119	+	0.764926i	$0.277224\pi$
$620$	0	0
$621$	0	0
$622$	1.57878	0.0633034
$623$	0.566205	0.0226845
$624$	0	0
$625$	0	0
$626$	9.87614	0.394730
$627$	0	0
$628$	14.3510	0.572668
$629$	−35.6092	−1.41983
$630$	0	0
$631$	−18.9709	−0.755220	−0.377610	−	0.925965i	$-0.623254\pi$
$631$	−18.9709	−0.755220	−0.377610	+	0.925965i	$0.623254\pi$
$632$	14.1308	0.562093
$633$	0	0
$634$	−37.3991	−1.48531
$635$	0	0
$636$	0	0
$637$	−19.3502	−0.766684
$638$	7.67939	0.304030
$639$	0	0
$640$	0	0
$641$	44.3433	1.75145	0.875727	−	0.482806i	$-0.160383\pi$
$641$	44.3433	1.75145	0.875727	+	0.482806i	$0.160383\pi$
$642$	0	0
$643$	42.9397	1.69338	0.846688	−	0.532090i	$-0.178593\pi$
$643$	42.9397	1.69338	0.846688	+	0.532090i	$0.178593\pi$
$644$	1.69781	0.0669032
$645$	0	0
$646$	−10.5294	−0.414272
$647$	−17.9864	−0.707118	−0.353559	−	0.935412i	$-0.615029\pi$
$647$	−17.9864	−0.707118	−0.353559	+	0.935412i	$0.615029\pi$
$648$	0	0
$649$	−8.34116	−0.327419
$650$	0	0
$651$	0	0
$652$	9.81663	0.384449
$653$	21.7274	0.850260	0.425130	−	0.905132i	$-0.360228\pi$
$653$	21.7274	0.850260	0.425130	+	0.905132i	$0.360228\pi$
$654$	0	0
$655$	0	0
$656$	41.8676	1.63465
$657$	0	0
$658$	−4.28376	−0.166998
$659$	40.8590	1.59164	0.795821	−	0.605532i	$-0.207040\pi$
$659$	40.8590	1.59164	0.795821	+	0.605532i	$0.207040\pi$
$660$	0	0
$661$	33.7282	1.31188	0.655938	−	0.754815i	$-0.272273\pi$
$661$	33.7282	1.31188	0.655938	+	0.754815i	$0.272273\pi$
$662$	6.87322	0.267135
$663$	0	0
$664$	38.3150	1.48691
$665$	0	0
$666$	0	0
$667$	−20.9349	−0.810601
$668$	−9.51598	−0.368184
$669$	0	0
$670$	0	0
$671$	6.92498	0.267336
$672$	0	0
$673$	−11.0622	−0.426417	−0.213209	−	0.977007i	$-0.568391\pi$
$673$	−11.0622	−0.426417	−0.213209	+	0.977007i	$0.568391\pi$
$674$	−29.7699	−1.14669
$675$	0	0
$676$	−3.65826	−0.140702
$677$	6.14711	0.236253	0.118126	−	0.992999i	$-0.462311\pi$
$677$	6.14711	0.236253	0.118126	+	0.992999i	$0.462311\pi$
$678$	0	0
$679$	4.44447	0.170563
$680$	0	0
$681$	0	0
$682$	5.64042	0.215983
$683$	13.6073	0.520669	0.260334	−	0.965519i	$-0.416167\pi$
$683$	13.6073	0.520669	0.260334	+	0.965519i	$0.416167\pi$
$684$	0	0
$685$	0	0
$686$	8.63055	0.329516
$687$	0	0
$688$	59.8492	2.28173
$689$	−32.5840	−1.24135
$690$	0	0
$691$	−40.3043	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$691$	−40.3043	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$692$	−1.08064	−0.0410799
$693$	0	0
$694$	−12.2739	−0.465911
$695$	0	0
$696$	0	0
$697$	54.2130	2.05346
$698$	36.1909	1.36985
$699$	0	0
$700$	0	0
$701$	3.23704	0.122261	0.0611307	−	0.998130i	$-0.480529\pi$
$701$	3.23704	0.122261	0.0611307	+	0.998130i	$0.480529\pi$
$702$	0	0
$703$	5.58383	0.210598
$704$	4.64042	0.174892
$705$	0	0
$706$	−17.1209	−0.644355
$707$	−3.46582	−0.130345
$708$	0	0
$709$	20.2370	0.760018	0.380009	−	0.924983i	$-0.375921\pi$
$709$	20.2370	0.760018	0.380009	+	0.924983i	$0.375921\pi$
$710$	0	0
$711$	0	0
$712$	3.15720	0.118321
$713$	−15.3764	−0.575851
$714$	0	0
$715$	0	0
$716$	12.4260	0.464383
$717$	0	0
$718$	39.6086	1.47818
$719$	−47.8804	−1.78564	−0.892819	−	0.450416i	$-0.851276\pi$
$719$	−47.8804	−1.78564	−0.892819	+	0.450416i	$0.851276\pi$
$720$	0	0
$721$	3.57315	0.133071
$722$	1.65109	0.0614473
$723$	0	0
$724$	−4.16707	−0.154868
$725$	0	0
$726$	0	0
$727$	17.7926	0.659890	0.329945	−	0.944000i	$-0.392970\pi$
$727$	17.7926	0.659890	0.329945	+	0.944000i	$0.392970\pi$
$728$	2.23864	0.0829697
$729$	0	0
$730$	0	0
$731$	77.4968	2.86632
$732$	0	0
$733$	−5.66097	−0.209093	−0.104546	−	0.994520i	$-0.533339\pi$
$733$	−5.66097	−0.209093	−0.104546	+	0.994520i	$0.533339\pi$
$734$	57.8337	2.13468
$735$	0	0
$736$	24.3305	0.896834
$737$	4.44447	0.163714
$738$	0	0
$739$	−5.59955	−0.205983	−0.102991	−	0.994682i	$-0.532841\pi$
$739$	−5.59955	−0.205983	−0.102991	+	0.994682i	$0.532841\pi$
$740$	0	0
$741$	0	0
$742$	7.19193	0.264024
$743$	−5.81663	−0.213391	−0.106696	−	0.994292i	$-0.534027\pi$
$743$	−5.81663	−0.213391	−0.106696	+	0.994292i	$0.534027\pi$
$744$	0	0
$745$	0	0
$746$	50.8542	1.86191
$747$	0	0
$748$	−6.37720	−0.233174
$749$	−5.79045	−0.211579
$750$	0	0
$751$	−41.1797	−1.50267	−0.751333	−	0.659923i	$-0.770589\pi$
$751$	−41.1797	−1.50267	−0.751333	+	0.659923i	$0.770589\pi$
$752$	−33.8753	−1.23531
$753$	0	0
$754$	15.7339	0.572993
$755$	0	0
$756$	0	0
$757$	−33.5208	−1.21833	−0.609167	−	0.793042i	$-0.708496\pi$
$757$	−33.5208	−1.21833	−0.609167	+	0.793042i	$0.708496\pi$
$758$	1.10894	0.0402785
$759$	0	0
$760$	0	0
$761$	−31.5032	−1.14199	−0.570995	−	0.820954i	$-0.693442\pi$
$761$	−31.5032	−1.14199	−0.570995	+	0.820954i	$0.693442\pi$
$762$	0	0
$763$	6.32353	0.228927
$764$	−16.2611	−0.588306
$765$	0	0
$766$	49.8307	1.80046
$767$	−17.0897	−0.617074
$768$	0	0
$769$	−1.22425	−0.0441475	−0.0220737	−	0.999756i	$-0.507027\pi$
$769$	−1.22425	−0.0441475	−0.0220737	+	0.999756i	$0.507027\pi$
$770$	0	0
$771$	0	0
$772$	−15.5160	−0.558432
$773$	−40.6687	−1.46275	−0.731376	−	0.681974i	$-0.761122\pi$
$773$	−40.6687	−1.46275	−0.731376	+	0.681974i	$0.761122\pi$
$774$	0	0
$775$	0	0
$776$	24.7827	0.889647
$777$	0	0
$778$	52.0275	1.86528
$779$	−8.50106	−0.304582
$780$	0	0
$781$	3.17058	0.113452
$782$	65.2702	2.33406
$783$	0	0
$784$	33.7742	1.20622
$785$	0	0
$786$	0	0
$787$	−39.9525	−1.42415	−0.712076	−	0.702102i	$-0.752245\pi$
$787$	−39.9525	−1.42415	−0.712076	+	0.702102i	$0.752245\pi$
$788$	−0.808876	−0.0288150
$789$	0	0
$790$	0	0
$791$	−5.02830	−0.178786
$792$	0	0
$793$	14.1882	0.503838
$794$	−20.7643	−0.736897
$795$	0	0
$796$	−1.61563	−0.0572644
$797$	4.53791	0.160741	0.0803705	−	0.996765i	$-0.474390\pi$
$797$	4.53791	0.160741	0.0803705	+	0.996765i	$0.474390\pi$
$798$	0	0
$799$	−43.8641	−1.55180
$800$	0	0
$801$	0	0
$802$	−40.6228	−1.43444
$803$	−4.40550	−0.155467
$804$	0	0
$805$	0	0
$806$	11.5563	0.407054
$807$	0	0
$808$	−19.3257	−0.679874
$809$	−55.5958	−1.95465	−0.977323	−	0.211756i	$-0.932082\pi$
$809$	−55.5958	−1.95465	−0.977323	+	0.211756i	$0.932082\pi$
$810$	0	0
$811$	42.3326	1.48650	0.743249	−	0.669014i	$-0.233284\pi$
$811$	42.3326	1.48650	0.743249	+	0.669014i	$0.233284\pi$
$812$	−0.924984	−0.0324606
$813$	0	0
$814$	12.6970	0.445030
$815$	0	0
$816$	0	0
$817$	−12.1522	−0.425150
$818$	22.5648	0.788961
$819$	0	0
$820$	0	0
$821$	3.10543	0.108380	0.0541902	−	0.998531i	$-0.482742\pi$
$821$	3.10543	0.108380	0.0541902	+	0.998531i	$0.482742\pi$
$822$	0	0
$823$	−45.2002	−1.57558	−0.787790	−	0.615944i	$-0.788775\pi$
$823$	−45.2002	−1.57558	−0.787790	+	0.615944i	$0.788775\pi$
$824$	19.9242	0.694092
$825$	0	0
$826$	3.77203	0.131246
$827$	35.6503	1.23968	0.619841	−	0.784727i	$-0.287197\pi$
$827$	35.6503	1.23968	0.619841	+	0.784727i	$0.287197\pi$
$828$	0	0
$829$	−18.4330	−0.640204	−0.320102	−	0.947383i	$-0.603717\pi$
$829$	−18.4330	−0.640204	−0.320102	+	0.947383i	$0.603717\pi$
$830$	0	0
$831$	0	0
$832$	9.50749	0.329613
$833$	43.7331	1.51526
$834$	0	0
$835$	0	0
$836$	1.00000	0.0345857
$837$	0	0
$838$	−20.9512	−0.723746
$839$	14.1805	0.489564	0.244782	−	0.969578i	$-0.421284\pi$
$839$	14.1805	0.489564	0.244782	+	0.969578i	$0.421284\pi$
$840$	0	0
$841$	−17.5945	−0.606707
$842$	48.1832	1.66050
$843$	0	0
$844$	−7.08277	−0.243799
$845$	0	0
$846$	0	0
$847$	3.43380	0.117987
$848$	56.8726	1.95301
$849$	0	0
$850$	0	0
$851$	−34.6134	−1.18653
$852$	0	0
$853$	42.6639	1.46078	0.730392	−	0.683028i	$-0.239337\pi$
$853$	42.6639	1.46078	0.730392	+	0.683028i	$0.239337\pi$
$854$	−3.13161	−0.107161
$855$	0	0
$856$	−32.2880	−1.10358
$857$	1.42392	0.0486403	0.0243201	−	0.999704i	$-0.492258\pi$
$857$	1.42392	0.0486403	0.0243201	+	0.999704i	$0.492258\pi$
$858$	0	0
$859$	−8.27894	−0.282474	−0.141237	−	0.989976i	$-0.545108\pi$
$859$	−8.27894	−0.282474	−0.141237	+	0.989976i	$0.545108\pi$
$860$	0	0
$861$	0	0
$862$	49.5024	1.68606
$863$	−30.4154	−1.03535	−0.517676	−	0.855577i	$-0.673203\pi$
$863$	−30.4154	−1.03535	−0.517676	+	0.855577i	$0.673203\pi$
$864$	0	0
$865$	0	0
$866$	−7.86276	−0.267188
$867$	0	0
$868$	−0.679390	−0.0230600
$869$	−9.25254	−0.313871
$870$	0	0
$871$	9.10602	0.308546
$872$	35.2605	1.19407
$873$	0	0
$874$	−10.2349	−0.346201
$875$	0	0
$876$	0	0
$877$	57.8484	1.95340	0.976700	−	0.214608i	$-0.0688474\pi$
$877$	57.8484	1.95340	0.976700	+	0.214608i	$0.0688474\pi$
$878$	11.3695	0.383700
$879$	0	0
$880$	0	0
$881$	−18.4055	−0.620097	−0.310049	−	0.950721i	$-0.600345\pi$
$881$	−18.4055	−0.620097	−0.310049	+	0.950721i	$0.600345\pi$
$882$	0	0
$883$	−18.7947	−0.632492	−0.316246	−	0.948677i	$-0.602422\pi$
$883$	−18.7947	−0.632492	−0.316246	+	0.948677i	$0.602422\pi$
$884$	−13.0659	−0.439453
$885$	0	0
$886$	−14.2838	−0.479872
$887$	11.1471	0.374283	0.187142	−	0.982333i	$-0.440078\pi$
$887$	11.1471	0.374283	0.187142	+	0.982333i	$0.440078\pi$
$888$	0	0
$889$	1.14923	0.0385440
$890$	0	0
$891$	0	0
$892$	−13.5371	−0.453256
$893$	6.87826	0.230172
$894$	0	0
$895$	0	0
$896$	−5.05952	−0.169027
$897$	0	0
$898$	−14.2498	−0.475523
$899$	8.37720	0.279395
$900$	0	0
$901$	73.6425	2.45339
$902$	−19.3305	−0.643635
$903$	0	0
$904$	−28.0382	−0.932536
$905$	0	0
$906$	0	0
$907$	38.5598	1.28036	0.640178	−	0.768226i	$-0.278861\pi$
$907$	38.5598	1.28036	0.640178	+	0.768226i	$0.278861\pi$
$908$	5.84280	0.193900
$909$	0	0
$910$	0	0
$911$	1.79820	0.0595771	0.0297885	−	0.999556i	$-0.490517\pi$
$911$	1.79820	0.0595771	0.0297885	+	0.999556i	$0.490517\pi$
$912$	0	0
$913$	−25.0878	−0.830285
$914$	−47.2365	−1.56244
$915$	0	0
$916$	14.6249	0.483221
$917$	−3.28456	−0.108466
$918$	0	0
$919$	−32.4458	−1.07029	−0.535144	−	0.844761i	$-0.679743\pi$
$919$	−32.4458	−1.07029	−0.535144	+	0.844761i	$0.679743\pi$
$920$	0	0
$921$	0	0
$922$	64.5619	2.12623
$923$	6.49602	0.213819
$924$	0	0
$925$	0	0
$926$	−13.2811	−0.436443
$927$	0	0
$928$	−13.2555	−0.435132
$929$	18.4231	0.604443	0.302222	−	0.953238i	$-0.402272\pi$
$929$	18.4231	0.604443	0.302222	+	0.953238i	$0.402272\pi$
$930$	0	0
$931$	−6.85772	−0.224753
$932$	1.25897	0.0412390
$933$	0	0
$934$	−28.1492	−0.921071
$935$	0	0
$936$	0	0
$937$	8.29926	0.271125	0.135563	−	0.990769i	$-0.456716\pi$
$937$	8.29926	0.271125	0.135563	+	0.990769i	$0.456716\pi$
$938$	−2.00987	−0.0656247
$939$	0	0
$940$	0	0
$941$	−33.6687	−1.09757	−0.548784	−	0.835964i	$-0.684909\pi$
$941$	−33.6687	−1.09757	−0.548784	+	0.835964i	$0.684909\pi$
$942$	0	0
$943$	52.6970	1.71605
$944$	29.8286	0.970839
$945$	0	0
$946$	−27.6327	−0.898416
$947$	−1.35103	−0.0439026	−0.0219513	−	0.999759i	$-0.506988\pi$
$947$	−1.35103	−0.0439026	−0.0219513	+	0.999759i	$0.506988\pi$
$948$	0	0
$949$	−9.02617	−0.293002
$950$	0	0
$951$	0	0
$952$	−5.05952	−0.163980
$953$	6.70769	0.217283	0.108642	−	0.994081i	$-0.465350\pi$
$953$	6.70769	0.217283	0.108642	+	0.994081i	$0.465350\pi$
$954$	0	0
$955$	0	0
$956$	13.2987	0.430110
$957$	0	0
$958$	−37.2285	−1.20280
$959$	2.48746	0.0803244
$960$	0	0
$961$	−24.8470	−0.801518
$962$	26.0142	0.838731
$963$	0	0
$964$	−1.59508	−0.0513741
$965$	0	0
$966$	0	0
$967$	19.5174	0.627636	0.313818	−	0.949483i	$-0.398392\pi$
$967$	19.5174	0.627636	0.313818	+	0.949483i	$0.398392\pi$
$968$	19.1471	0.615411
$969$	0	0
$970$	0	0
$971$	8.50669	0.272993	0.136496	−	0.990641i	$-0.456416\pi$
$971$	8.50669	0.272993	0.136496	+	0.990641i	$0.456416\pi$
$972$	0	0
$973$	3.83527	0.122953
$974$	−40.5470	−1.29921
$975$	0	0
$976$	−24.7643	−0.792685
$977$	−19.7048	−0.630411	−0.315206	−	0.949023i	$-0.602073\pi$
$977$	−19.7048	−0.630411	−0.315206	+	0.949023i	$0.602073\pi$
$978$	0	0
$979$	−2.06727	−0.0660701
$980$	0	0
$981$	0	0
$982$	27.9527	0.892006
$983$	12.4677	0.397658	0.198829	−	0.980034i	$-0.436286\pi$
$983$	12.4677	0.397658	0.198829	+	0.980034i	$0.436286\pi$
$984$	0	0
$985$	0	0
$986$	−35.5598	−1.13245
$987$	0	0
$988$	2.04884	0.0651824
$989$	75.3297	2.39534
$990$	0	0
$991$	25.0841	0.796822	0.398411	−	0.917207i	$-0.369562\pi$
$991$	25.0841	0.796822	0.398411	+	0.917207i	$0.369562\pi$
$992$	−9.73598	−0.309118
$993$	0	0
$994$	−1.43380	−0.0454772
$995$	0	0
$996$	0	0
$997$	11.8775	0.376163	0.188082	−	0.982153i	$-0.439773\pi$
$997$	11.8775	0.376163	0.188082	+	0.982153i	$0.439773\pi$
$998$	0.690644	0.0218620
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.ba.1.3		3
3.2	odd	2		475.2.a.g.1.1	yes	3
5.4	even	2		4275.2.a.bm.1.1		3
12.11	even	2		7600.2.a.bh.1.1		3
15.2	even	4		475.2.b.b.324.2		6
15.8	even	4		475.2.b.b.324.5		6
15.14	odd	2		475.2.a.e.1.3	✓	3
57.56	even	2		9025.2.a.y.1.3		3
60.59	even	2		7600.2.a.cc.1.3		3
285.284	even	2		9025.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.3	✓	3	15.14	odd	2
475.2.a.g.1.1	yes	3	3.2	odd	2
475.2.b.b.324.2		6	15.2	even	4
475.2.b.b.324.5		6	15.8	even	4
4275.2.a.ba.1.3		3	1.1	even	1	trivial
4275.2.a.bm.1.1		3	5.4	even	2
7600.2.a.bh.1.1		3	12.11	even	2
7600.2.a.cc.1.3		3	60.59	even	2
9025.2.a.y.1.3		3	57.56	even	2
9025.2.a.bc.1.1		3	285.284	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)

\(f(q)\)	\(=\)	\(q+1.65109 q^{2} +0.726109 q^{4} -0.377203 q^{7} -2.10331 q^{8} +O(q^{10})\)	\(q+1.65109 q^{2} +0.726109 q^{4} -0.377203 q^{7} -2.10331 q^{8} +1.37720 q^{11} +2.82167 q^{13} -0.622797 q^{14} -4.92498 q^{16} -6.37720 q^{17} +1.00000 q^{19} +2.27389 q^{22} -6.19887 q^{23} +4.65884 q^{26} -0.273891 q^{28} +3.37720 q^{29} +2.48052 q^{31} -3.92498 q^{32} -10.5294 q^{34} +5.58383 q^{37} +1.65109 q^{38} -8.50106 q^{41} -12.1522 q^{43} +1.00000 q^{44} -10.2349 q^{46} +6.87826 q^{47} -6.85772 q^{49} +2.04884 q^{52} -11.5478 q^{53} +0.793375 q^{56} +5.57608 q^{58} -6.05659 q^{59} +5.02830 q^{61} +4.09556 q^{62} +3.36945 q^{64} +3.22717 q^{67} -4.63055 q^{68} +2.30219 q^{71} -3.19887 q^{73} +9.21942 q^{74} +0.726109 q^{76} -0.519485 q^{77} -6.71836 q^{79} -14.0360 q^{82} -18.2165 q^{83} -20.0643 q^{86} -2.89669 q^{88} -1.50106 q^{89} -1.06434 q^{91} -4.50106 q^{92} +11.3567 q^{94} -11.7827 q^{97} -11.3227 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 4 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 4 * q^4 + 4 * q^7 - 3 * q^8	\( 3 q - 2 q^{2} + 4 q^{4} + 4 q^{7} - 3 q^{8} - q^{11} + 3 q^{13} - 7 q^{14} - 6 q^{16} - 14 q^{17} + 3 q^{19} + 5 q^{22} - 8 q^{23} + 11 q^{26} + q^{28} + 5 q^{29} - q^{31} - 3 q^{32} + 5 q^{34} + 5 q^{37} - 2 q^{38} - q^{41} - 5 q^{43} + 3 q^{44} - 12 q^{46} - 9 q^{47} - 7 q^{49} - 22 q^{52} - 31 q^{53} + 9 q^{56} + q^{58} + 6 q^{59} + 3 q^{61} + 5 q^{62} + q^{64} - 13 q^{67} - 23 q^{68} - 7 q^{71} + q^{73} + q^{74} + 4 q^{76} - 10 q^{77} - 18 q^{79} - 34 q^{82} - 3 q^{83} - 40 q^{86} - 12 q^{88} + 20 q^{89} + 17 q^{91} + 11 q^{92} + 45 q^{94} - 13 q^{97} - 4 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 4 * q^4 + 4 * q^7 - 3 * q^8 - q^11 + 3 * q^13 - 7 * q^14 - 6 * q^16 - 14 * q^17 + 3 * q^19 + 5 * q^22 - 8 * q^23 + 11 * q^26 + q^28 + 5 * q^29 - q^31 - 3 * q^32 + 5 * q^34 + 5 * q^37 - 2 * q^38 - q^41 - 5 * q^43 + 3 * q^44 - 12 * q^46 - 9 * q^47 - 7 * q^49 - 22 * q^52 - 31 * q^53 + 9 * q^56 + q^58 + 6 * q^59 + 3 * q^61 + 5 * q^62 + q^64 - 13 * q^67 - 23 * q^68 - 7 * q^71 + q^73 + q^74 + 4 * q^76 - 10 * q^77 - 18 * q^79 - 34 * q^82 - 3 * q^83 - 40 * q^86 - 12 * q^88 + 20 * q^89 + 17 * q^91 + 11 * q^92 + 45 * q^94 - 13 * q^97 - 4 * q^98

Introduction

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