LMFDB - Embedded newform 4275.2.a.bw.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4275,2,Mod(1,4275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [7,3,0,11,0,0,-8,9,0,0,4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 8x^{5} + 26x^{4} + 11x^{3} - 51x^{2} + 12x + 4 $ x^7 - 3x^6 - 8x^5 + 26x^4 + 11x^3 - 51x^2 + 12x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.57229$ 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)

$f(q)$	$=$	$q+1.57229 q^{2} +0.472094 q^{4} -1.87913 q^{7} -2.40231 q^{8} -1.92081 q^{11} -0.248324 q^{13} -2.95453 q^{14} -4.72132 q^{16} +7.06596 q^{17} +1.00000 q^{19} -3.02007 q^{22} -1.20286 q^{23} -0.390437 q^{26} -0.887125 q^{28} -2.58170 q^{29} +8.69676 q^{31} -2.61865 q^{32} +11.1097 q^{34} +7.86387 q^{37} +1.57229 q^{38} -1.52589 q^{41} -4.15375 q^{43} -0.906802 q^{44} -1.89124 q^{46} -5.96879 q^{47} -3.46888 q^{49} -0.117232 q^{52} +7.11918 q^{53} +4.51425 q^{56} -4.05918 q^{58} +11.8573 q^{59} +5.87781 q^{61} +13.6738 q^{62} +5.32535 q^{64} -9.53623 q^{67} +3.33580 q^{68} +9.53623 q^{71} +6.69123 q^{73} +12.3643 q^{74} +0.472094 q^{76} +3.60944 q^{77} -0.348185 q^{79} -2.39913 q^{82} +2.41705 q^{83} -6.53090 q^{86} +4.61438 q^{88} +17.3414 q^{89} +0.466632 q^{91} -0.567862 q^{92} -9.38467 q^{94} +7.70088 q^{97} -5.45408 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 11 q^{4} - 8 q^{7} + 9 q^{8} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 19 q^{16} + 4 q^{17} + 7 q^{19} - 12 q^{22} + 10 q^{23} + 20 q^{26} - 14 q^{28} + 6 q^{29} + 4 q^{31} + 31 q^{32} + 2 q^{34}+ \cdots + 5 q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 11 * q^4 - 8 * q^7 + 9 * q^8 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 19 * q^16 + 4 * q^17 + 7 * q^19 - 12 * q^22 + 10 * q^23 + 20 * q^26 - 14 * q^28 + 6 * q^29 + 4 * q^31 + 31 * q^32 + 2 * q^34 - 14 * q^37 + 3 * q^38 - 2 * q^41 + 2 * q^43 + 32 * q^44 - 4 * q^46 + 30 * q^47 + 17 * q^49 - 18 * q^52 + 22 * q^56 + 40 * q^58 + 18 * q^59 + 12 * q^61 + 18 * q^62 + 11 * q^64 - 18 * q^67 - 12 * q^68 + 18 * q^71 - 10 * q^73 - 6 * q^74 + 11 * q^76 - 18 * q^77 - 4 * q^79 + 16 * q^82 + 18 * q^83 + 46 * q^86 + 18 * q^88 + 8 * q^89 + 12 * q^91 + 34 * q^92 - 20 * q^94 - 20 * q^97 + 5 * 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 $

$2$	1.57229	1.11178	0.555888	−	0.831257i	$-0.312378\pi$
$2$	1.57229	1.11178	0.555888	+	0.831257i	$0.312378\pi$
$3$	0	0
$3$	0	0
$4$	0.472094	0.236047
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.87913	−0.710244	−0.355122	−	0.934820i	$-0.615561\pi$
$7$	−1.87913	−0.710244	−0.355122	+	0.934820i	$0.615561\pi$
$8$	−2.40231	−0.849345
$9$	0	0
$10$	0	0
$11$	−1.92081	−0.579146	−0.289573	−	0.957156i	$-0.593513\pi$
$11$	−1.92081	−0.579146	−0.289573	+	0.957156i	$0.593513\pi$
$12$	0	0
$13$	−0.248324	−0.0688726	−0.0344363	−	0.999407i	$-0.510964\pi$
$13$	−0.248324	−0.0688726	−0.0344363	+	0.999407i	$0.510964\pi$
$14$	−2.95453	−0.789632
$15$	0	0
$16$	−4.72132	−1.18033
$17$	7.06596	1.71375	0.856873	−	0.515527i	$-0.172404\pi$
$17$	7.06596	1.71375	0.856873	+	0.515527i	$0.172404\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−3.02007	−0.643880
$23$	−1.20286	−0.250813	−0.125406	−	0.992105i	$-0.540023\pi$
$23$	−1.20286	−0.250813	−0.125406	+	0.992105i	$0.540023\pi$
$24$	0	0
$25$	0	0
$26$	−0.390437	−0.0765709
$27$	0	0
$28$	−0.887125	−0.167651
$29$	−2.58170	−0.479409	−0.239705	−	0.970846i	$-0.577051\pi$
$29$	−2.58170	−0.479409	−0.239705	+	0.970846i	$0.577051\pi$
$30$	0	0
$31$	8.69676	1.56198	0.780992	−	0.624541i	$-0.214714\pi$
$31$	8.69676	1.56198	0.780992	+	0.624541i	$0.214714\pi$
$32$	−2.61865	−0.462917
$33$	0	0
$34$	11.1097	1.90530
$35$	0	0
$36$	0	0
$37$	7.86387	1.29281	0.646406	−	0.762993i	$-0.276271\pi$
$37$	7.86387	1.29281	0.646406	+	0.762993i	$0.276271\pi$
$38$	1.57229	0.255059
$39$	0	0
$40$	0	0
$41$	−1.52589	−0.238303	−0.119152	−	0.992876i	$-0.538017\pi$
$41$	−1.52589	−0.238303	−0.119152	+	0.992876i	$0.538017\pi$
$42$	0	0
$43$	−4.15375	−0.633441	−0.316720	−	0.948519i	$-0.602582\pi$
$43$	−4.15375	−0.633441	−0.316720	+	0.948519i	$0.602582\pi$
$44$	−0.906802	−0.136706
$45$	0	0
$46$	−1.89124	−0.278848
$47$	−5.96879	−0.870638	−0.435319	−	0.900276i	$-0.643364\pi$
$47$	−5.96879	−0.870638	−0.435319	+	0.900276i	$0.643364\pi$
$48$	0	0
$49$	−3.46888	−0.495554
$50$	0	0
$51$	0	0
$52$	−0.117232	−0.0162572
$53$	7.11918	0.977894	0.488947	−	0.872313i	$-0.337381\pi$
$53$	7.11918	0.977894	0.488947	+	0.872313i	$0.337381\pi$
$54$	0	0
$55$	0	0
$56$	4.51425	0.603242
$57$	0	0
$58$	−4.05918	−0.532996
$59$	11.8573	1.54369	0.771844	−	0.635812i	$-0.219335\pi$
$59$	11.8573	1.54369	0.771844	+	0.635812i	$0.219335\pi$
$60$	0	0
$61$	5.87781	0.752576	0.376288	−	0.926503i	$-0.377200\pi$
$61$	5.87781	0.752576	0.376288	+	0.926503i	$0.377200\pi$
$62$	13.6738	1.73658
$63$	0	0
$64$	5.32535	0.665669
$65$	0	0
$66$	0	0
$67$	−9.53623	−1.16504	−0.582518	−	0.812818i	$-0.697932\pi$
$67$	−9.53623	−1.16504	−0.582518	+	0.812818i	$0.697932\pi$
$68$	3.33580	0.404525
$69$	0	0
$70$	0	0
$71$	9.53623	1.13174	0.565871	−	0.824494i	$-0.308540\pi$
$71$	9.53623	1.13174	0.565871	+	0.824494i	$0.308540\pi$
$72$	0	0
$73$	6.69123	0.783149	0.391575	−	0.920146i	$-0.371930\pi$
$73$	6.69123	0.783149	0.391575	+	0.920146i	$0.371930\pi$
$74$	12.3643	1.43732
$75$	0	0
$76$	0.472094	0.0541529
$77$	3.60944	0.411334
$78$	0	0
$79$	−0.348185	−0.0391739	−0.0195869	−	0.999808i	$-0.506235\pi$
$79$	−0.348185	−0.0391739	−0.0195869	+	0.999808i	$0.506235\pi$
$80$	0	0
$81$	0	0
$82$	−2.39913	−0.264940
$83$	2.41705	0.265306	0.132653	−	0.991163i	$-0.457650\pi$
$83$	2.41705	0.265306	0.132653	+	0.991163i	$0.457650\pi$
$84$	0	0
$85$	0	0
$86$	−6.53090	−0.704245
$87$	0	0
$88$	4.61438	0.491894
$89$	17.3414	1.83818	0.919090	−	0.394048i	$-0.128926\pi$
$89$	17.3414	1.83818	0.919090	+	0.394048i	$0.128926\pi$
$90$	0	0
$91$	0.466632	0.0489163
$92$	−0.567862	−0.0592037
$93$	0	0
$94$	−9.38467	−0.967955
$95$	0	0
$96$	0	0
$97$	7.70088	0.781905	0.390953	−	0.920411i	$-0.372146\pi$
$97$	7.70088	0.781905	0.390953	+	0.920411i	$0.372146\pi$
$98$	−5.45408	−0.550945
$99$	0	0
$100$	0	0
$101$	8.41297	0.837122	0.418561	−	0.908189i	$-0.362535\pi$
$101$	8.41297	0.837122	0.418561	+	0.908189i	$0.362535\pi$
$102$	0	0
$103$	9.64843	0.950688	0.475344	−	0.879800i	$-0.342324\pi$
$103$	9.64843	0.950688	0.475344	+	0.879800i	$0.342324\pi$
$104$	0.596550	0.0584966
$105$	0	0
$106$	11.1934	1.08720
$107$	0.241744	0.0233703	0.0116852	−	0.999932i	$-0.496280\pi$
$107$	0.241744	0.0233703	0.0116852	+	0.999932i	$0.496280\pi$
$108$	0	0
$109$	3.05581	0.292694	0.146347	−	0.989233i	$-0.453248\pi$
$109$	3.05581	0.292694	0.146347	+	0.989233i	$0.453248\pi$
$110$	0	0
$111$	0	0
$112$	8.87196	0.838321
$113$	16.5767	1.55940	0.779701	−	0.626152i	$-0.215371\pi$
$113$	16.5767	1.55940	0.779701	+	0.626152i	$0.215371\pi$
$114$	0	0
$115$	0	0
$116$	−1.21880	−0.113163
$117$	0	0
$118$	18.6431	1.71624
$119$	−13.2778	−1.21718
$120$	0	0
$121$	−7.31050	−0.664590
$122$	9.24161	0.836696
$123$	0	0
$124$	4.10569	0.368702
$125$	0	0
$126$	0	0
$127$	−21.5565	−1.91283	−0.956416	−	0.292007i	$-0.905677\pi$
$127$	−21.5565	−1.91283	−0.956416	+	0.292007i	$0.905677\pi$
$128$	13.6103	1.20299
$129$	0	0
$130$	0	0
$131$	18.3504	1.60328	0.801642	−	0.597804i	$-0.203960\pi$
$131$	18.3504	1.60328	0.801642	+	0.597804i	$0.203960\pi$
$132$	0	0
$133$	−1.87913	−0.162941
$134$	−14.9937	−1.29526
$135$	0	0
$136$	−16.9746	−1.45556
$137$	−9.63542	−0.823209	−0.411605	−	0.911362i	$-0.635032\pi$
$137$	−9.63542	−0.823209	−0.411605	+	0.911362i	$0.635032\pi$
$138$	0	0
$139$	−14.9629	−1.26914	−0.634568	−	0.772867i	$-0.718822\pi$
$139$	−14.9629	−1.26914	−0.634568	+	0.772867i	$0.718822\pi$
$140$	0	0
$141$	0	0
$142$	14.9937	1.25824
$143$	0.476982	0.0398872
$144$	0	0
$145$	0	0
$146$	10.5206	0.870687
$147$	0	0
$148$	3.71249	0.305165
$149$	−14.2063	−1.16383	−0.581915	−	0.813250i	$-0.697696\pi$
$149$	−14.2063	−1.16383	−0.581915	+	0.813250i	$0.697696\pi$
$150$	0	0
$151$	−2.26811	−0.184577	−0.0922883	−	0.995732i	$-0.529418\pi$
$151$	−2.26811	−0.184577	−0.0922883	+	0.995732i	$0.529418\pi$
$152$	−2.40231	−0.194853
$153$	0	0
$154$	5.67509	0.457312
$155$	0	0
$156$	0	0
$157$	1.23885	0.0988706	0.0494353	−	0.998777i	$-0.484258\pi$
$157$	1.23885	0.0988706	0.0494353	+	0.998777i	$0.484258\pi$
$158$	−0.547448	−0.0435526
$159$	0	0
$160$	0	0
$161$	2.26032	0.178138
$162$	0	0
$163$	4.19752	0.328775	0.164388	−	0.986396i	$-0.447435\pi$
$163$	4.19752	0.328775	0.164388	+	0.986396i	$0.447435\pi$
$164$	−0.720362	−0.0562508
$165$	0	0
$166$	3.80030	0.294961
$167$	16.5980	1.28439	0.642195	−	0.766541i	$-0.278024\pi$
$167$	16.5980	1.28439	0.642195	+	0.766541i	$0.278024\pi$
$168$	0	0
$169$	−12.9383	−0.995257
$170$	0	0
$171$	0	0
$172$	−1.96096	−0.149522
$173$	−0.784955	−0.0596790	−0.0298395	−	0.999555i	$-0.509500\pi$
$173$	−0.784955	−0.0596790	−0.0298395	+	0.999555i	$0.509500\pi$
$174$	0	0
$175$	0	0
$176$	9.06874	0.683582
$177$	0	0
$178$	27.2656	2.04365
$179$	23.1704	1.73183	0.865917	−	0.500188i	$-0.166736\pi$
$179$	23.1704	1.73183	0.865917	+	0.500188i	$0.166736\pi$
$180$	0	0
$181$	8.11162	0.602932	0.301466	−	0.953477i	$-0.402524\pi$
$181$	8.11162	0.602932	0.301466	+	0.953477i	$0.402524\pi$
$182$	0.733680	0.0543840
$183$	0	0
$184$	2.88963	0.213027
$185$	0	0
$186$	0	0
$187$	−13.5724	−0.992509
$188$	−2.81783	−0.205512
$189$	0	0
$190$	0	0
$191$	20.7192	1.49919	0.749595	−	0.661897i	$-0.230248\pi$
$191$	20.7192	1.49919	0.749595	+	0.661897i	$0.230248\pi$
$192$	0	0
$193$	−3.71249	−0.267231	−0.133615	−	0.991033i	$-0.542659\pi$
$193$	−3.71249	−0.267231	−0.133615	+	0.991033i	$0.542659\pi$
$194$	12.1080	0.869304
$195$	0	0
$196$	−1.63764	−0.116974
$197$	−7.81160	−0.556554	−0.278277	−	0.960501i	$-0.589763\pi$
$197$	−7.81160	−0.556554	−0.278277	+	0.960501i	$0.589763\pi$
$198$	0	0
$199$	5.87477	0.416451	0.208226	−	0.978081i	$-0.433231\pi$
$199$	5.87477	0.416451	0.208226	+	0.978081i	$0.433231\pi$
$200$	0	0
$201$	0	0
$202$	13.2276	0.930692
$203$	4.85134	0.340497
$204$	0	0
$205$	0	0
$206$	15.1701	1.05695
$207$	0	0
$208$	1.17241	0.0812923
$209$	−1.92081	−0.132865
$210$	0	0
$211$	−9.79251	−0.674144	−0.337072	−	0.941479i	$-0.609437\pi$
$211$	−9.79251	−0.674144	−0.337072	+	0.941479i	$0.609437\pi$
$212$	3.36092	0.230829
$213$	0	0
$214$	0.380092	0.0259826
$215$	0	0
$216$	0	0
$217$	−16.3423	−1.10939
$218$	4.80462	0.325410
$219$	0	0
$220$	0	0
$221$	−1.75464	−0.118030
$222$	0	0
$223$	−20.4327	−1.36827	−0.684137	−	0.729354i	$-0.739821\pi$
$223$	−20.4327	−1.36827	−0.684137	+	0.729354i	$0.739821\pi$
$224$	4.92079	0.328784
$225$	0	0
$226$	26.0633	1.73371
$227$	−10.0931	−0.669903	−0.334952	−	0.942235i	$-0.608720\pi$
$227$	−10.0931	−0.669903	−0.334952	+	0.942235i	$0.608720\pi$
$228$	0	0
$229$	−14.5637	−0.962397	−0.481198	−	0.876612i	$-0.659798\pi$
$229$	−14.5637	−0.962397	−0.481198	+	0.876612i	$0.659798\pi$
$230$	0	0
$231$	0	0
$232$	6.20204	0.407184
$233$	−25.8921	−1.69625	−0.848125	−	0.529797i	$-0.822268\pi$
$233$	−25.8921	−1.69625	−0.848125	+	0.529797i	$0.822268\pi$
$234$	0	0
$235$	0	0
$236$	5.59776	0.364383
$237$	0	0
$238$	−20.8766	−1.35323
$239$	−18.7389	−1.21212	−0.606061	−	0.795418i	$-0.707251\pi$
$239$	−18.7389	−1.21212	−0.606061	+	0.795418i	$0.707251\pi$
$240$	0	0
$241$	28.4586	1.83318	0.916589	−	0.399831i	$-0.130931\pi$
$241$	28.4586	1.83318	0.916589	+	0.399831i	$0.130931\pi$
$242$	−11.4942	−0.738876
$243$	0	0
$244$	2.77488	0.177643
$245$	0	0
$246$	0	0
$247$	−0.248324	−0.0158005
$248$	−20.8923	−1.32666
$249$	0	0
$250$	0	0
$251$	−1.22886	−0.0775650	−0.0387825	−	0.999248i	$-0.512348\pi$
$251$	−1.22886	−0.0775650	−0.0387825	+	0.999248i	$0.512348\pi$
$252$	0	0
$253$	2.31046	0.145257
$254$	−33.8931	−2.12664
$255$	0	0
$256$	10.7486	0.671789
$257$	−5.56841	−0.347348	−0.173674	−	0.984803i	$-0.555564\pi$
$257$	−5.56841	−0.347348	−0.173674	+	0.984803i	$0.555564\pi$
$258$	0	0
$259$	−14.7772	−0.918212
$260$	0	0
$261$	0	0
$262$	28.8522	1.78249
$263$	−10.7855	−0.665063	−0.332531	−	0.943092i	$-0.607903\pi$
$263$	−10.7855	−0.665063	−0.332531	+	0.943092i	$0.607903\pi$
$264$	0	0
$265$	0	0
$266$	−2.95453	−0.181154
$267$	0	0
$268$	−4.50200	−0.275003
$269$	22.8529	1.39337	0.696684	−	0.717378i	$-0.254658\pi$
$269$	22.8529	1.39337	0.696684	+	0.717378i	$0.254658\pi$
$270$	0	0
$271$	−9.95413	−0.604670	−0.302335	−	0.953202i	$-0.597766\pi$
$271$	−9.95413	−0.604670	−0.302335	+	0.953202i	$0.597766\pi$
$272$	−33.3606	−2.02278
$273$	0	0
$274$	−15.1497	−0.915225
$275$	0	0
$276$	0	0
$277$	14.4026	0.865371	0.432686	−	0.901545i	$-0.357566\pi$
$277$	14.4026	0.865371	0.432686	+	0.901545i	$0.357566\pi$
$278$	−23.5260	−1.41100
$279$	0	0
$280$	0	0
$281$	−5.34285	−0.318728	−0.159364	−	0.987220i	$-0.550944\pi$
$281$	−5.34285	−0.318728	−0.159364	+	0.987220i	$0.550944\pi$
$282$	0	0
$283$	−5.73067	−0.340653	−0.170326	−	0.985388i	$-0.554482\pi$
$283$	−5.73067	−0.340653	−0.170326	+	0.985388i	$0.554482\pi$
$284$	4.50200	0.267144
$285$	0	0
$286$	0.749954	0.0443457
$287$	2.86733	0.169253
$288$	0	0
$289$	32.9278	1.93693
$290$	0	0
$291$	0	0
$292$	3.15889	0.184860
$293$	27.4793	1.60536	0.802678	−	0.596412i	$-0.203408\pi$
$293$	27.4793	1.60536	0.802678	+	0.596412i	$0.203408\pi$
$294$	0	0
$295$	0	0
$296$	−18.8915	−1.09804
$297$	0	0
$298$	−22.3365	−1.29392
$299$	0.298698	0.0172741
$300$	0	0
$301$	7.80543	0.449897
$302$	−3.56613	−0.205208
$303$	0	0
$304$	−4.72132	−0.270786
$305$	0	0
$306$	0	0
$307$	−10.7031	−0.610858	−0.305429	−	0.952215i	$-0.598800\pi$
$307$	−10.7031	−0.610858	−0.305429	+	0.952215i	$0.598800\pi$
$308$	1.70400	0.0970943
$309$	0	0
$310$	0	0
$311$	11.5209	0.653288	0.326644	−	0.945148i	$-0.394082\pi$
$311$	11.5209	0.653288	0.326644	+	0.945148i	$0.394082\pi$
$312$	0	0
$313$	−18.1596	−1.02644	−0.513221	−	0.858257i	$-0.671548\pi$
$313$	−18.1596	−1.02644	−0.513221	+	0.858257i	$0.671548\pi$
$314$	1.94782	0.109922
$315$	0	0
$316$	−0.164376	−0.00924688
$317$	−19.7109	−1.10707	−0.553537	−	0.832824i	$-0.686722\pi$
$317$	−19.7109	−1.10707	−0.553537	+	0.832824i	$0.686722\pi$
$318$	0	0
$319$	4.95895	0.277648
$320$	0	0
$321$	0	0
$322$	3.55388	0.198050
$323$	7.06596	0.393160
$324$	0	0
$325$	0	0
$326$	6.59972	0.365525
$327$	0	0
$328$	3.66565	0.202402
$329$	11.2161	0.618365
$330$	0	0
$331$	−1.88788	−0.103767	−0.0518836	−	0.998653i	$-0.516522\pi$
$331$	−1.88788	−0.103767	−0.0518836	+	0.998653i	$0.516522\pi$
$332$	1.14108	0.0626247
$333$	0	0
$334$	26.0968	1.42796
$335$	0	0
$336$	0	0
$337$	−14.0147	−0.763429	−0.381714	−	0.924280i	$-0.624666\pi$
$337$	−14.0147	−0.763429	−0.381714	+	0.924280i	$0.624666\pi$
$338$	−20.3428	−1.10650
$339$	0	0
$340$	0	0
$341$	−16.7048	−0.904616
$342$	0	0
$343$	19.6724	1.06221
$344$	9.97859	0.538010
$345$	0	0
$346$	−1.23418	−0.0663498
$347$	−1.05107	−0.0564245	−0.0282122	−	0.999602i	$-0.508981\pi$
$347$	−1.05107	−0.0564245	−0.0282122	+	0.999602i	$0.508981\pi$
$348$	0	0
$349$	35.4633	1.89831	0.949153	−	0.314815i	$-0.101943\pi$
$349$	35.4633	1.89831	0.949153	+	0.314815i	$0.101943\pi$
$350$	0	0
$351$	0	0
$352$	5.02993	0.268096
$353$	5.28721	0.281410	0.140705	−	0.990052i	$-0.455063\pi$
$353$	5.28721	0.281410	0.140705	+	0.990052i	$0.455063\pi$
$354$	0	0
$355$	0	0
$356$	8.18675	0.433897
$357$	0	0
$358$	36.4305	1.92541
$359$	−16.0220	−0.845607	−0.422803	−	0.906221i	$-0.638954\pi$
$359$	−16.0220	−0.845607	−0.422803	+	0.906221i	$0.638954\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	12.7538	0.670326
$363$	0	0
$364$	0.220294	0.0115466
$365$	0	0
$366$	0	0
$367$	−27.7038	−1.44613	−0.723063	−	0.690782i	$-0.757267\pi$
$367$	−27.7038	−1.44613	−0.723063	+	0.690782i	$0.757267\pi$
$368$	5.67907	0.296042
$369$	0	0
$370$	0	0
$371$	−13.3778	−0.694543
$372$	0	0
$373$	−11.7532	−0.608559	−0.304279	−	0.952583i	$-0.598416\pi$
$373$	−11.7532	−0.608559	−0.304279	+	0.952583i	$0.598416\pi$
$374$	−21.3397	−1.10345
$375$	0	0
$376$	14.3389	0.739472
$377$	0.641096	0.0330181
$378$	0	0
$379$	−7.04988	−0.362128	−0.181064	−	0.983471i	$-0.557954\pi$
$379$	−7.04988	−0.362128	−0.181064	+	0.983471i	$0.557954\pi$
$380$	0	0
$381$	0	0
$382$	32.5766	1.66676
$383$	7.01682	0.358543	0.179271	−	0.983800i	$-0.442626\pi$
$383$	7.01682	0.358543	0.179271	+	0.983800i	$0.442626\pi$
$384$	0	0
$385$	0	0
$386$	−5.83711	−0.297101
$387$	0	0
$388$	3.63554	0.184567
$389$	38.1712	1.93536	0.967679	−	0.252185i	$-0.0811492\pi$
$389$	38.1712	1.93536	0.967679	+	0.252185i	$0.0811492\pi$
$390$	0	0
$391$	−8.49933	−0.429830
$392$	8.33332	0.420896
$393$	0	0
$394$	−12.2821	−0.618763
$395$	0	0
$396$	0	0
$397$	−7.84847	−0.393904	−0.196952	−	0.980413i	$-0.563104\pi$
$397$	−7.84847	−0.393904	−0.196952	+	0.980413i	$0.563104\pi$
$398$	9.23683	0.463001
$399$	0	0
$400$	0	0
$401$	−0.879277	−0.0439090	−0.0219545	−	0.999759i	$-0.506989\pi$
$401$	−0.879277	−0.0439090	−0.0219545	+	0.999759i	$0.506989\pi$
$402$	0	0
$403$	−2.15961	−0.107578
$404$	3.97171	0.197600
$405$	0	0
$406$	7.62771	0.378557
$407$	−15.1050	−0.748727
$408$	0	0
$409$	38.8153	1.91929	0.959647	−	0.281207i	$-0.0907349\pi$
$409$	38.8153	1.91929	0.959647	+	0.281207i	$0.0907349\pi$
$410$	0	0
$411$	0	0
$412$	4.55497	0.224407
$413$	−22.2814	−1.09639
$414$	0	0
$415$	0	0
$416$	0.650274	0.0318823
$417$	0	0
$418$	−3.02007	−0.147716
$419$	31.2248	1.52543	0.762716	−	0.646734i	$-0.223866\pi$
$419$	31.2248	1.52543	0.762716	+	0.646734i	$0.223866\pi$
$420$	0	0
$421$	−4.46904	−0.217808	−0.108904	−	0.994052i	$-0.534734\pi$
$421$	−4.46904	−0.217808	−0.108904	+	0.994052i	$0.534734\pi$
$422$	−15.3967	−0.749498
$423$	0	0
$424$	−17.1025	−0.830570
$425$	0	0
$426$	0	0
$427$	−11.0452	−0.534512
$428$	0.114126	0.00551649
$429$	0	0
$430$	0	0
$431$	−26.0794	−1.25620	−0.628101	−	0.778132i	$-0.716167\pi$
$431$	−26.0794	−1.25620	−0.628101	+	0.778132i	$0.716167\pi$
$432$	0	0
$433$	15.4199	0.741035	0.370518	−	0.928825i	$-0.379180\pi$
$433$	15.4199	0.741035	0.370518	+	0.928825i	$0.379180\pi$
$434$	−25.6949	−1.23339
$435$	0	0
$436$	1.44263	0.0690895
$437$	−1.20286	−0.0575404
$438$	0	0
$439$	−26.1326	−1.24724	−0.623620	−	0.781728i	$-0.714339\pi$
$439$	−26.1326	−1.24724	−0.623620	+	0.781728i	$0.714339\pi$
$440$	0	0
$441$	0	0
$442$	−2.75881	−0.131223
$443$	−18.7686	−0.891721	−0.445861	−	0.895102i	$-0.647102\pi$
$443$	−18.7686	−0.891721	−0.445861	+	0.895102i	$0.647102\pi$
$444$	0	0
$445$	0	0
$446$	−32.1261	−1.52121
$447$	0	0
$448$	−10.0070	−0.472787
$449$	−18.5078	−0.873438	−0.436719	−	0.899598i	$-0.643860\pi$
$449$	−18.5078	−0.873438	−0.436719	+	0.899598i	$0.643860\pi$
$450$	0	0
$451$	2.93093	0.138012
$452$	7.82575	0.368092
$453$	0	0
$454$	−15.8693	−0.744783
$455$	0	0
$456$	0	0
$457$	35.7154	1.67070	0.835348	−	0.549722i	$-0.185266\pi$
$457$	35.7154	1.67070	0.835348	+	0.549722i	$0.185266\pi$
$458$	−22.8984	−1.06997
$459$	0	0
$460$	0	0
$461$	30.7787	1.43351	0.716753	−	0.697327i	$-0.245628\pi$
$461$	30.7787	1.43351	0.716753	+	0.697327i	$0.245628\pi$
$462$	0	0
$463$	−31.1807	−1.44909	−0.724546	−	0.689226i	$-0.757951\pi$
$463$	−31.1807	−1.44909	−0.724546	+	0.689226i	$0.757951\pi$
$464$	12.1890	0.565860
$465$	0	0
$466$	−40.7099	−1.88585
$467$	−8.01621	−0.370946	−0.185473	−	0.982649i	$-0.559382\pi$
$467$	−8.01621	−0.370946	−0.185473	+	0.982649i	$0.559382\pi$
$468$	0	0
$469$	17.9198	0.827459
$470$	0	0
$471$	0	0
$472$	−28.4849	−1.31112
$473$	7.97856	0.366854
$474$	0	0
$475$	0	0
$476$	−6.26839	−0.287311
$477$	0	0
$478$	−29.4630	−1.34761
$479$	4.00338	0.182919	0.0914596	−	0.995809i	$-0.470847\pi$
$479$	4.00338	0.182919	0.0914596	+	0.995809i	$0.470847\pi$
$480$	0	0
$481$	−1.95279	−0.0890394
$482$	44.7451	2.03808
$483$	0	0
$484$	−3.45124	−0.156875
$485$	0	0
$486$	0	0
$487$	17.5299	0.794358	0.397179	−	0.917741i	$-0.369989\pi$
$487$	17.5299	0.794358	0.397179	+	0.917741i	$0.369989\pi$
$488$	−14.1203	−0.639197
$489$	0	0
$490$	0	0
$491$	−15.7747	−0.711901	−0.355951	−	0.934505i	$-0.615843\pi$
$491$	−15.7747	−0.711901	−0.355951	+	0.934505i	$0.615843\pi$
$492$	0	0
$493$	−18.2422	−0.821586
$494$	−0.390437	−0.0175666
$495$	0	0
$496$	−41.0602	−1.84366
$497$	−17.9198	−0.803813
$498$	0	0
$499$	31.4231	1.40669	0.703347	−	0.710847i	$-0.251688\pi$
$499$	31.4231	1.40669	0.703347	+	0.710847i	$0.251688\pi$
$500$	0	0
$501$	0	0
$502$	−1.93212	−0.0862349
$503$	15.7396	0.701796	0.350898	−	0.936414i	$-0.385876\pi$
$503$	15.7396	0.701796	0.350898	+	0.936414i	$0.385876\pi$
$504$	0	0
$505$	0	0
$506$	3.63271	0.161494
$507$	0	0
$508$	−10.1767	−0.451518
$509$	−17.2396	−0.764131	−0.382066	−	0.924135i	$-0.624787\pi$
$509$	−17.2396	−0.764131	−0.382066	+	0.924135i	$0.624787\pi$
$510$	0	0
$511$	−12.5737	−0.556227
$512$	−10.3206	−0.456112
$513$	0	0
$514$	−8.75516	−0.386173
$515$	0	0
$516$	0	0
$517$	11.4649	0.504226
$518$	−23.2341	−1.02085
$519$	0	0
$520$	0	0
$521$	28.3700	1.24291	0.621457	−	0.783448i	$-0.286541\pi$
$521$	28.3700	1.24291	0.621457	+	0.783448i	$0.286541\pi$
$522$	0	0
$523$	−31.3645	−1.37148	−0.685738	−	0.727849i	$-0.740520\pi$
$523$	−31.3645	−1.37148	−0.685738	+	0.727849i	$0.740520\pi$
$524$	8.66313	0.378451
$525$	0	0
$526$	−16.9579	−0.739401
$527$	61.4510	2.67685
$528$	0	0
$529$	−21.5531	−0.937093
$530$	0	0
$531$	0	0
$532$	−0.887125	−0.0384618
$533$	0.378913	0.0164126
$534$	0	0
$535$	0	0
$536$	22.9090	0.989517
$537$	0	0
$538$	35.9314	1.54911
$539$	6.66305	0.286998
$540$	0	0
$541$	−12.6888	−0.545536	−0.272768	−	0.962080i	$-0.587939\pi$
$541$	−12.6888	−0.545536	−0.272768	+	0.962080i	$0.587939\pi$
$542$	−15.6508	−0.672258
$543$	0	0
$544$	−18.5033	−0.793323
$545$	0	0
$546$	0	0
$547$	19.2634	0.823644	0.411822	−	0.911264i	$-0.364893\pi$
$547$	19.2634	0.823644	0.411822	+	0.911264i	$0.364893\pi$
$548$	−4.54883	−0.194316
$549$	0	0
$550$	0	0
$551$	−2.58170	−0.109984
$552$	0	0
$553$	0.654284	0.0278230
$554$	22.6451	0.962099
$555$	0	0
$556$	−7.06389	−0.299576
$557$	−21.6253	−0.916294	−0.458147	−	0.888877i	$-0.651487\pi$
$557$	−21.6253	−0.916294	−0.458147	+	0.888877i	$0.651487\pi$
$558$	0	0
$559$	1.03147	0.0436267
$560$	0	0
$561$	0	0
$562$	−8.40051	−0.354354
$563$	39.1301	1.64914	0.824569	−	0.565761i	$-0.191418\pi$
$563$	39.1301	1.64914	0.824569	+	0.565761i	$0.191418\pi$
$564$	0	0
$565$	0	0
$566$	−9.01027	−0.378730
$567$	0	0
$568$	−22.9090	−0.961240
$569$	−10.5271	−0.441320	−0.220660	−	0.975351i	$-0.570821\pi$
$569$	−10.5271	−0.441320	−0.220660	+	0.975351i	$0.570821\pi$
$570$	0	0
$571$	43.1731	1.80674	0.903368	−	0.428866i	$-0.141087\pi$
$571$	43.1731	1.80674	0.903368	+	0.428866i	$0.141087\pi$
$572$	0.225180	0.00941527
$573$	0	0
$574$	4.50828	0.188172
$575$	0	0
$576$	0	0
$577$	13.8196	0.575316	0.287658	−	0.957733i	$-0.407123\pi$
$577$	13.8196	0.575316	0.287658	+	0.957733i	$0.407123\pi$
$578$	51.7720	2.15343
$579$	0	0
$580$	0	0
$581$	−4.54195	−0.188432
$582$	0	0
$583$	−13.6746	−0.566343
$584$	−16.0744	−0.665164
$585$	0	0
$586$	43.2054	1.78480
$587$	−16.3981	−0.676822	−0.338411	−	0.940998i	$-0.609889\pi$
$587$	−16.3981	−0.676822	−0.338411	+	0.940998i	$0.609889\pi$
$588$	0	0
$589$	8.69676	0.358344
$590$	0	0
$591$	0	0
$592$	−37.1278	−1.52594
$593$	−26.5182	−1.08897	−0.544485	−	0.838770i	$-0.683275\pi$
$593$	−26.5182	−1.08897	−0.544485	+	0.838770i	$0.683275\pi$
$594$	0	0
$595$	0	0
$596$	−6.70674	−0.274719
$597$	0	0
$598$	0.469639	0.0192050
$599$	7.85156	0.320806	0.160403	−	0.987052i	$-0.448721\pi$
$599$	7.85156	0.320806	0.160403	+	0.987052i	$0.448721\pi$
$600$	0	0
$601$	−8.31507	−0.339179	−0.169589	−	0.985515i	$-0.554244\pi$
$601$	−8.31507	−0.339179	−0.169589	+	0.985515i	$0.554244\pi$
$602$	12.2724	0.500185
$603$	0	0
$604$	−1.07076	−0.0435688
$605$	0	0
$606$	0	0
$607$	−26.6353	−1.08109	−0.540546	−	0.841314i	$-0.681782\pi$
$607$	−26.6353	−1.08109	−0.540546	+	0.841314i	$0.681782\pi$
$608$	−2.61865	−0.106200
$609$	0	0
$610$	0	0
$611$	1.48219	0.0599631
$612$	0	0
$613$	−38.7710	−1.56595	−0.782973	−	0.622056i	$-0.786298\pi$
$613$	−38.7710	−1.56595	−0.782973	+	0.622056i	$0.786298\pi$
$614$	−16.8284	−0.679138
$615$	0	0
$616$	−8.67101	−0.349365
$617$	23.1535	0.932125	0.466063	−	0.884752i	$-0.345672\pi$
$617$	23.1535	0.932125	0.466063	+	0.884752i	$0.345672\pi$
$618$	0	0
$619$	47.6444	1.91499	0.957495	−	0.288449i	$-0.0931396\pi$
$619$	47.6444	1.91499	0.957495	+	0.288449i	$0.0931396\pi$
$620$	0	0
$621$	0	0
$622$	18.1141	0.726310
$623$	−32.5866	−1.30556
$624$	0	0
$625$	0	0
$626$	−28.5522	−1.14117
$627$	0	0
$628$	0.584852	0.0233381
$629$	55.5658	2.21555
$630$	0	0
$631$	29.4039	1.17055	0.585275	−	0.810835i	$-0.300987\pi$
$631$	29.4039	1.17055	0.585275	+	0.810835i	$0.300987\pi$
$632$	0.836449	0.0332721
$633$	0	0
$634$	−30.9912	−1.23082
$635$	0	0
$636$	0	0
$637$	0.861404	0.0341301
$638$	7.79690	0.308682
$639$	0	0
$640$	0	0
$641$	11.0401	0.436059	0.218029	−	0.975942i	$-0.430037\pi$
$641$	11.0401	0.436059	0.218029	+	0.975942i	$0.430037\pi$
$642$	0	0
$643$	19.6424	0.774622	0.387311	−	0.921949i	$-0.373404\pi$
$643$	19.6424	0.774622	0.387311	+	0.921949i	$0.373404\pi$
$644$	1.06708	0.0420490
$645$	0	0
$646$	11.1097	0.437107
$647$	32.7753	1.28853	0.644265	−	0.764803i	$-0.277163\pi$
$647$	32.7753	1.28853	0.644265	+	0.764803i	$0.277163\pi$
$648$	0	0
$649$	−22.7756	−0.894020
$650$	0	0
$651$	0	0
$652$	1.98163	0.0776065
$653$	10.5395	0.412442	0.206221	−	0.978505i	$-0.433883\pi$
$653$	10.5395	0.412442	0.206221	+	0.978505i	$0.433883\pi$
$654$	0	0
$655$	0	0
$656$	7.20419	0.281276
$657$	0	0
$658$	17.6350	0.687484
$659$	−29.1597	−1.13590	−0.567950	−	0.823063i	$-0.692263\pi$
$659$	−29.1597	−1.13590	−0.567950	+	0.823063i	$0.692263\pi$
$660$	0	0
$661$	−48.4368	−1.88397	−0.941987	−	0.335649i	$-0.891044\pi$
$661$	−48.4368	−1.88397	−0.941987	+	0.335649i	$0.891044\pi$
$662$	−2.96829	−0.115366
$663$	0	0
$664$	−5.80651	−0.225336
$665$	0	0
$666$	0	0
$667$	3.10541	0.120242
$668$	7.83581	0.303177
$669$	0	0
$670$	0	0
$671$	−11.2901	−0.435851
$672$	0	0
$673$	−0.819338	−0.0315831	−0.0157916	−	0.999875i	$-0.505027\pi$
$673$	−0.819338	−0.0315831	−0.0157916	+	0.999875i	$0.505027\pi$
$674$	−22.0351	−0.848762
$675$	0	0
$676$	−6.10811	−0.234927
$677$	−3.76524	−0.144710	−0.0723549	−	0.997379i	$-0.523051\pi$
$677$	−3.76524	−0.144710	−0.0723549	+	0.997379i	$0.523051\pi$
$678$	0	0
$679$	−14.4709	−0.555343
$680$	0	0
$681$	0	0
$682$	−26.2648	−1.00573
$683$	−36.9829	−1.41511	−0.707556	−	0.706657i	$-0.750202\pi$
$683$	−36.9829	−1.41511	−0.707556	+	0.706657i	$0.750202\pi$
$684$	0	0
$685$	0	0
$686$	30.9306	1.18094
$687$	0	0
$688$	19.6112	0.747669
$689$	−1.76786	−0.0673501
$690$	0	0
$691$	10.0198	0.381171	0.190585	−	0.981671i	$-0.438961\pi$
$691$	10.0198	0.381171	0.190585	+	0.981671i	$0.438961\pi$
$692$	−0.370573	−0.0140871
$693$	0	0
$694$	−1.65259	−0.0627314
$695$	0	0
$696$	0	0
$697$	−10.7818	−0.408391
$698$	55.7585	2.11049
$699$	0	0
$700$	0	0
$701$	15.6128	0.589688	0.294844	−	0.955545i	$-0.404732\pi$
$701$	15.6128	0.589688	0.294844	+	0.955545i	$0.404732\pi$
$702$	0	0
$703$	7.86387	0.296592
$704$	−10.2290	−0.385519
$705$	0	0
$706$	8.31302	0.312865
$707$	−15.8090	−0.594560
$708$	0	0
$709$	11.5966	0.435519	0.217759	−	0.976002i	$-0.430125\pi$
$709$	11.5966	0.435519	0.217759	+	0.976002i	$0.430125\pi$
$710$	0	0
$711$	0	0
$712$	−41.6593	−1.56125
$713$	−10.4610	−0.391766
$714$	0	0
$715$	0	0
$716$	10.9386	0.408794
$717$	0	0
$718$	−25.1912	−0.940126
$719$	13.1532	0.490532	0.245266	−	0.969456i	$-0.421125\pi$
$719$	13.1532	0.490532	0.245266	+	0.969456i	$0.421125\pi$
$720$	0	0
$721$	−18.1306	−0.675220
$722$	1.57229	0.0585146
$723$	0	0
$724$	3.82945	0.142320
$725$	0	0
$726$	0	0
$727$	−26.7258	−0.991203	−0.495602	−	0.868550i	$-0.665052\pi$
$727$	−26.7258	−0.991203	−0.495602	+	0.868550i	$0.665052\pi$
$728$	−1.12099	−0.0415468
$729$	0	0
$730$	0	0
$731$	−29.3502	−1.08556
$732$	0	0
$733$	1.01889	0.0376337	0.0188169	−	0.999823i	$-0.494010\pi$
$733$	1.01889	0.0376337	0.0188169	+	0.999823i	$0.494010\pi$
$734$	−43.5584	−1.60777
$735$	0	0
$736$	3.14987	0.116106
$737$	18.3173	0.674725
$738$	0	0
$739$	−30.1705	−1.10984	−0.554920	−	0.831904i	$-0.687251\pi$
$739$	−30.1705	−1.10984	−0.554920	+	0.831904i	$0.687251\pi$
$740$	0	0
$741$	0	0
$742$	−21.0338	−0.772177
$743$	8.28423	0.303919	0.151959	−	0.988387i	$-0.451442\pi$
$743$	8.28423	0.303919	0.151959	+	0.988387i	$0.451442\pi$
$744$	0	0
$745$	0	0
$746$	−18.4795	−0.676581
$747$	0	0
$748$	−6.40743	−0.234279
$749$	−0.454268	−0.0165986
$750$	0	0
$751$	−3.74254	−0.136567	−0.0682836	−	0.997666i	$-0.521752\pi$
$751$	−3.74254	−0.136567	−0.0682836	+	0.997666i	$0.521752\pi$
$752$	28.1806	1.02764
$753$	0	0
$754$	1.00799	0.0367088
$755$	0	0
$756$	0	0
$757$	29.6308	1.07695	0.538475	−	0.842642i	$-0.319001\pi$
$757$	29.6308	1.07695	0.538475	+	0.842642i	$0.319001\pi$
$758$	−11.0845	−0.402605
$759$	0	0
$760$	0	0
$761$	43.2167	1.56660	0.783302	−	0.621642i	$-0.213534\pi$
$761$	43.2167	1.56660	0.783302	+	0.621642i	$0.213534\pi$
$762$	0	0
$763$	−5.74226	−0.207884
$764$	9.78142	0.353880
$765$	0	0
$766$	11.0325	0.398619
$767$	−2.94445	−0.106318
$768$	0	0
$769$	−7.30244	−0.263333	−0.131666	−	0.991294i	$-0.542033\pi$
$769$	−7.30244	−0.263333	−0.131666	+	0.991294i	$0.542033\pi$
$770$	0	0
$771$	0	0
$772$	−1.75264	−0.0630791
$773$	42.4163	1.52561	0.762804	−	0.646630i	$-0.223822\pi$
$773$	42.4163	1.52561	0.762804	+	0.646630i	$0.223822\pi$
$774$	0	0
$775$	0	0
$776$	−18.4999	−0.664107
$777$	0	0
$778$	60.0162	2.15169
$779$	−1.52589	−0.0546705
$780$	0	0
$781$	−18.3173	−0.655443
$782$	−13.3634	−0.477875
$783$	0	0
$784$	16.3777	0.584917
$785$	0	0
$786$	0	0
$787$	−34.5998	−1.23335	−0.616675	−	0.787218i	$-0.711521\pi$
$787$	−34.5998	−1.23335	−0.616675	+	0.787218i	$0.711521\pi$
$788$	−3.68781	−0.131373
$789$	0	0
$790$	0	0
$791$	−31.1497	−1.10755
$792$	0	0
$793$	−1.45960	−0.0518319
$794$	−12.3401	−0.437933
$795$	0	0
$796$	2.77344	0.0983021
$797$	31.9016	1.13001	0.565006	−	0.825087i	$-0.308874\pi$
$797$	31.9016	1.13001	0.565006	+	0.825087i	$0.308874\pi$
$798$	0	0
$799$	−42.1752	−1.49205
$800$	0	0
$801$	0	0
$802$	−1.38248	−0.0488170
$803$	−12.8526	−0.453557
$804$	0	0
$805$	0	0
$806$	−3.39553	−0.119603
$807$	0	0
$808$	−20.2106	−0.711005
$809$	−7.96516	−0.280040	−0.140020	−	0.990149i	$-0.544717\pi$
$809$	−7.96516	−0.280040	−0.140020	+	0.990149i	$0.544717\pi$
$810$	0	0
$811$	−8.45805	−0.297002	−0.148501	−	0.988912i	$-0.547445\pi$
$811$	−8.45805	−0.297002	−0.148501	+	0.988912i	$0.547445\pi$
$812$	2.29029	0.0803734
$813$	0	0
$814$	−23.7494	−0.832417
$815$	0	0
$816$	0	0
$817$	−4.15375	−0.145321
$818$	61.0289	2.13383
$819$	0	0
$820$	0	0
$821$	−29.8649	−1.04229	−0.521147	−	0.853467i	$-0.674496\pi$
$821$	−29.8649	−1.04229	−0.521147	+	0.853467i	$0.674496\pi$
$822$	0	0
$823$	29.4632	1.02702	0.513512	−	0.858083i	$-0.328344\pi$
$823$	29.4632	1.02702	0.513512	+	0.858083i	$0.328344\pi$
$824$	−23.1785	−0.807462
$825$	0	0
$826$	−35.0328	−1.21895
$827$	35.7491	1.24312	0.621558	−	0.783368i	$-0.286500\pi$
$827$	35.7491	1.24312	0.621558	+	0.783368i	$0.286500\pi$
$828$	0	0
$829$	30.6073	1.06303	0.531517	−	0.847047i	$-0.321622\pi$
$829$	30.6073	1.06303	0.531517	+	0.847047i	$0.321622\pi$
$830$	0	0
$831$	0	0
$832$	−1.32241	−0.0458463
$833$	−24.5110	−0.849254
$834$	0	0
$835$	0	0
$836$	−0.906802	−0.0313624
$837$	0	0
$838$	49.0944	1.69594
$839$	−20.2431	−0.698868	−0.349434	−	0.936961i	$-0.613626\pi$
$839$	−20.2431	−0.698868	−0.349434	+	0.936961i	$0.613626\pi$
$840$	0	0
$841$	−22.3348	−0.770167
$842$	−7.02662	−0.242153
$843$	0	0
$844$	−4.62299	−0.159130
$845$	0	0
$846$	0	0
$847$	13.7374	0.472021
$848$	−33.6119	−1.15424
$849$	0	0
$850$	0	0
$851$	−9.45911	−0.324254
$852$	0	0
$853$	36.5320	1.25083	0.625415	−	0.780292i	$-0.284930\pi$
$853$	36.5320	1.25083	0.625415	+	0.780292i	$0.284930\pi$
$854$	−17.3662	−0.594258
$855$	0	0
$856$	−0.580745	−0.0198495
$857$	−22.5292	−0.769584	−0.384792	−	0.923003i	$-0.625727\pi$
$857$	−22.5292	−0.769584	−0.384792	+	0.923003i	$0.625727\pi$
$858$	0	0
$859$	4.59482	0.156773	0.0783866	−	0.996923i	$-0.475023\pi$
$859$	4.59482	0.156773	0.0783866	+	0.996923i	$0.475023\pi$
$860$	0	0
$861$	0	0
$862$	−41.0044	−1.39662
$863$	11.5034	0.391579	0.195789	−	0.980646i	$-0.437273\pi$
$863$	11.5034	0.391579	0.195789	+	0.980646i	$0.437273\pi$
$864$	0	0
$865$	0	0
$866$	24.2446	0.823866
$867$	0	0
$868$	−7.71512	−0.261868
$869$	0.668797	0.0226874
$870$	0	0
$871$	2.36807	0.0802390
$872$	−7.34101	−0.248598
$873$	0	0
$874$	−1.89124	−0.0639721
$875$	0	0
$876$	0	0
$877$	9.18451	0.310139	0.155069	−	0.987904i	$-0.450440\pi$
$877$	9.18451	0.310139	0.155069	+	0.987904i	$0.450440\pi$
$878$	−41.0880	−1.38665
$879$	0	0
$880$	0	0
$881$	−46.8437	−1.57821	−0.789103	−	0.614261i	$-0.789454\pi$
$881$	−46.8437	−1.57821	−0.789103	+	0.614261i	$0.789454\pi$
$882$	0	0
$883$	−29.8269	−1.00375	−0.501877	−	0.864939i	$-0.667357\pi$
$883$	−29.8269	−1.00375	−0.501877	+	0.864939i	$0.667357\pi$
$884$	−0.828357	−0.0278607
$885$	0	0
$886$	−29.5096	−0.991395
$887$	20.4681	0.687252	0.343626	−	0.939107i	$-0.388345\pi$
$887$	20.4681	0.687252	0.343626	+	0.939107i	$0.388345\pi$
$888$	0	0
$889$	40.5075	1.35858
$890$	0	0
$891$	0	0
$892$	−9.64615	−0.322977
$893$	−5.96879	−0.199738
$894$	0	0
$895$	0	0
$896$	−25.5755	−0.854417
$897$	0	0
$898$	−29.0996	−0.971068
$899$	−22.4524	−0.748830
$900$	0	0
$901$	50.3038	1.67586
$902$	4.60828	0.153439
$903$	0	0
$904$	−39.8223	−1.32447
$905$	0	0
$906$	0	0
$907$	16.6266	0.552076	0.276038	−	0.961147i	$-0.410978\pi$
$907$	16.6266	0.552076	0.276038	+	0.961147i	$0.410978\pi$
$908$	−4.76490	−0.158129
$909$	0	0
$910$	0	0
$911$	40.6554	1.34697	0.673487	−	0.739199i	$-0.264796\pi$
$911$	40.6554	1.34697	0.673487	+	0.739199i	$0.264796\pi$
$912$	0	0
$913$	−4.64269	−0.153651
$914$	56.1549	1.85744
$915$	0	0
$916$	−6.87544	−0.227171
$917$	−34.4828	−1.13872
$918$	0	0
$919$	23.9633	0.790477	0.395238	−	0.918579i	$-0.370662\pi$
$919$	23.9633	0.790477	0.395238	+	0.918579i	$0.370662\pi$
$920$	0	0
$921$	0	0
$922$	48.3930	1.59374
$923$	−2.36807	−0.0779460
$924$	0	0
$925$	0	0
$926$	−49.0252	−1.61107
$927$	0	0
$928$	6.76057	0.221927
$929$	17.6807	0.580085	0.290043	−	0.957014i	$-0.406331\pi$
$929$	17.6807	0.580085	0.290043	+	0.957014i	$0.406331\pi$
$930$	0	0
$931$	−3.46888	−0.113688
$932$	−12.2235	−0.400395
$933$	0	0
$934$	−12.6038	−0.412409
$935$	0	0
$936$	0	0
$937$	−45.8986	−1.49944	−0.749721	−	0.661754i	$-0.769812\pi$
$937$	−45.8986	−1.49944	−0.749721	+	0.661754i	$0.769812\pi$
$938$	28.1751	0.919950
$939$	0	0
$940$	0	0
$941$	−7.89440	−0.257350	−0.128675	−	0.991687i	$-0.541072\pi$
$941$	−7.89440	−0.257350	−0.128675	+	0.991687i	$0.541072\pi$
$942$	0	0
$943$	1.83542	0.0597695
$944$	−55.9820	−1.82206
$945$	0	0
$946$	12.5446	0.407860
$947$	−51.7953	−1.68312	−0.841561	−	0.540162i	$-0.818363\pi$
$947$	−51.7953	−1.68312	−0.841561	+	0.540162i	$0.818363\pi$
$948$	0	0
$949$	−1.66159	−0.0539375
$950$	0	0
$951$	0	0
$952$	31.8975	1.03380
$953$	−11.5807	−0.375137	−0.187568	−	0.982252i	$-0.560061\pi$
$953$	−11.5807	−0.375137	−0.187568	+	0.982252i	$0.560061\pi$
$954$	0	0
$955$	0	0
$956$	−8.84655	−0.286118
$957$	0	0
$958$	6.29447	0.203365
$959$	18.1062	0.584679
$960$	0	0
$961$	44.6337	1.43980
$962$	−3.07034	−0.0989919
$963$	0	0
$964$	13.4351	0.432716
$965$	0	0
$966$	0	0
$967$	−60.5547	−1.94731	−0.973654	−	0.228032i	$-0.926771\pi$
$967$	−60.5547	−1.94731	−0.973654	+	0.228032i	$0.926771\pi$
$968$	17.5621	0.564467
$969$	0	0
$970$	0	0
$971$	4.19412	0.134596	0.0672978	−	0.997733i	$-0.478562\pi$
$971$	4.19412	0.134596	0.0672978	+	0.997733i	$0.478562\pi$
$972$	0	0
$973$	28.1172	0.901395
$974$	27.5621	0.883148
$975$	0	0
$976$	−27.7510	−0.888287
$977$	−12.7754	−0.408720	−0.204360	−	0.978896i	$-0.565511\pi$
$977$	−12.7754	−0.408720	−0.204360	+	0.978896i	$0.565511\pi$
$978$	0	0
$979$	−33.3094	−1.06457
$980$	0	0
$981$	0	0
$982$	−24.8024	−0.791475
$983$	35.8198	1.14247	0.571237	−	0.820785i	$-0.306464\pi$
$983$	35.8198	1.14247	0.571237	+	0.820785i	$0.306464\pi$
$984$	0	0
$985$	0	0
$986$	−28.6820	−0.913420
$987$	0	0
$988$	−0.117232	−0.00372965
$989$	4.99636	0.158875
$990$	0	0
$991$	−19.1653	−0.608807	−0.304404	−	0.952543i	$-0.598457\pi$
$991$	−19.1653	−0.608807	−0.304404	+	0.952543i	$0.598457\pi$
$992$	−22.7738	−0.723069
$993$	0	0
$994$	−28.1751	−0.893660
$995$	0	0
$996$	0	0
$997$	−9.88660	−0.313112	−0.156556	−	0.987669i	$-0.550039\pi$
$997$	−9.88660	−0.313112	−0.156556	+	0.987669i	$0.550039\pi$
$998$	49.4063	1.56393
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bw.1.5		7
3.2	odd	2		1425.2.a.y.1.3		7
5.2	odd	4		855.2.c.g.514.10		14
5.3	odd	4		855.2.c.g.514.5		14
5.4	even	2		4275.2.a.bv.1.3		7
15.2	even	4		285.2.c.b.229.5	✓	14
15.8	even	4		285.2.c.b.229.10	yes	14
15.14	odd	2		1425.2.a.z.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.c.b.229.5	✓	14	15.2	even	4
285.2.c.b.229.10	yes	14	15.8	even	4
855.2.c.g.514.5		14	5.3	odd	4
855.2.c.g.514.10		14	5.2	odd	4
1425.2.a.y.1.3		7	3.2	odd	2
1425.2.a.z.1.5		7	15.14	odd	2
4275.2.a.bv.1.3		7	5.4	even	2
4275.2.a.bw.1.5		7	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.57229 q^{2} +0.472094 q^{4} -1.87913 q^{7} -2.40231 q^{8} -1.92081 q^{11} -0.248324 q^{13} -2.95453 q^{14} -4.72132 q^{16} +7.06596 q^{17} +1.00000 q^{19} -3.02007 q^{22} -1.20286 q^{23} -0.390437 q^{26} -0.887125 q^{28} -2.58170 q^{29} +8.69676 q^{31} -2.61865 q^{32} +11.1097 q^{34} +7.86387 q^{37} +1.57229 q^{38} -1.52589 q^{41} -4.15375 q^{43} -0.906802 q^{44} -1.89124 q^{46} -5.96879 q^{47} -3.46888 q^{49} -0.117232 q^{52} +7.11918 q^{53} +4.51425 q^{56} -4.05918 q^{58} +11.8573 q^{59} +5.87781 q^{61} +13.6738 q^{62} +5.32535 q^{64} -9.53623 q^{67} +3.33580 q^{68} +9.53623 q^{71} +6.69123 q^{73} +12.3643 q^{74} +0.472094 q^{76} +3.60944 q^{77} -0.348185 q^{79} -2.39913 q^{82} +2.41705 q^{83} -6.53090 q^{86} +4.61438 q^{88} +17.3414 q^{89} +0.466632 q^{91} -0.567862 q^{92} -9.38467 q^{94} +7.70088 q^{97} -5.45408 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 11 q^{4} - 8 q^{7} + 9 q^{8} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 19 q^{16} + 4 q^{17} + 7 q^{19} - 12 q^{22} + 10 q^{23} + 20 q^{26} - 14 q^{28} + 6 q^{29} + 4 q^{31} + 31 q^{32} + 2 q^{34}+ \cdots + 5 q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 11 * q^4 - 8 * q^7 + 9 * q^8 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 19 * q^16 + 4 * q^17 + 7 * q^19 - 12 * q^22 + 10 * q^23 + 20 * q^26 - 14 * q^28 + 6 * q^29 + 4 * q^31 + 31 * q^32 + 2 * q^34 - 14 * q^37 + 3 * q^38 - 2 * q^41 + 2 * q^43 + 32 * q^44 - 4 * q^46 + 30 * q^47 + 17 * q^49 - 18 * q^52 + 22 * q^56 + 40 * q^58 + 18 * q^59 + 12 * q^61 + 18 * q^62 + 11 * q^64 - 18 * q^67 - 12 * q^68 + 18 * q^71 - 10 * q^73 - 6 * q^74 + 11 * q^76 - 18 * q^77 - 4 * q^79 + 16 * q^82 + 18 * q^83 + 46 * q^86 + 18 * q^88 + 8 * q^89 + 12 * q^91 + 34 * q^92 - 20 * q^94 - 20 * q^97 + 5 * q^98

Introduction

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