LMFDB - Embedded newform 1575.4.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.0765073$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0765073 q^{2} -7.99415 q^{4} +7.00000 q^{7} -1.22367 q^{8} +O(q^{10})$	$q+0.0765073 q^{2} -7.99415 q^{4} +7.00000 q^{7} -1.22367 q^{8} -10.8947 q^{11} -26.5468 q^{13} +0.535551 q^{14} +63.8596 q^{16} +95.1237 q^{17} -35.4649 q^{19} -0.833522 q^{22} -62.8303 q^{23} -2.03102 q^{26} -55.9590 q^{28} -117.823 q^{29} -171.090 q^{31} +14.6751 q^{32} +7.27766 q^{34} +203.813 q^{37} -2.71332 q^{38} +428.705 q^{41} -96.5851 q^{43} +87.0936 q^{44} -4.80698 q^{46} +407.806 q^{47} +49.0000 q^{49} +212.219 q^{52} +380.874 q^{53} -8.56568 q^{56} -9.01429 q^{58} +287.149 q^{59} -823.988 q^{61} -13.0897 q^{62} -509.754 q^{64} +585.549 q^{67} -760.433 q^{68} +653.856 q^{71} -1051.77 q^{73} +15.5932 q^{74} +283.512 q^{76} -76.2627 q^{77} -751.268 q^{79} +32.7991 q^{82} -844.677 q^{83} -7.38946 q^{86} +13.3315 q^{88} +262.334 q^{89} -185.828 q^{91} +502.275 q^{92} +31.2001 q^{94} -814.908 q^{97} +3.74886 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8	$ 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8} + 66 q^{11} - 102 q^{13} + 7 q^{14} - 69 q^{16} + 152 q^{17} - 138 q^{19} - 186 q^{22} - 180 q^{23} + 98 q^{26} + 21 q^{28} - 170 q^{29} - 366 q^{31} - 151 q^{32} - 36 q^{34} - 252 q^{37} - 234 q^{38} + 206 q^{41} + 108 q^{43} + 306 q^{44} - 672 q^{46} - 24 q^{47} + 147 q^{49} - 78 q^{52} + 354 q^{53} + 147 q^{56} + 858 q^{58} + 880 q^{59} - 870 q^{61} - 1366 q^{62} - 813 q^{64} + 96 q^{67} - 512 q^{68} + 1018 q^{71} - 1554 q^{73} - 980 q^{74} - 450 q^{76} + 462 q^{77} - 1620 q^{79} + 1638 q^{82} - 872 q^{83} + 2932 q^{86} + 1326 q^{88} + 1938 q^{89} - 714 q^{91} - 708 q^{92} + 2112 q^{94} - 1878 q^{97} + 49 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8 + 66 * q^11 - 102 * q^13 + 7 * q^14 - 69 * q^16 + 152 * q^17 - 138 * q^19 - 186 * q^22 - 180 * q^23 + 98 * q^26 + 21 * q^28 - 170 * q^29 - 366 * q^31 - 151 * q^32 - 36 * q^34 - 252 * q^37 - 234 * q^38 + 206 * q^41 + 108 * q^43 + 306 * q^44 - 672 * q^46 - 24 * q^47 + 147 * q^49 - 78 * q^52 + 354 * q^53 + 147 * q^56 + 858 * q^58 + 880 * q^59 - 870 * q^61 - 1366 * q^62 - 813 * q^64 + 96 * q^67 - 512 * q^68 + 1018 * q^71 - 1554 * q^73 - 980 * q^74 - 450 * q^76 + 462 * q^77 - 1620 * q^79 + 1638 * q^82 - 872 * q^83 + 2932 * q^86 + 1326 * q^88 + 1938 * q^89 - 714 * q^91 - 708 * q^92 + 2112 * q^94 - 1878 * q^97 + 49 * 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 $

$2$	0.0765073	0.0270494	0.0135247	−	0.999909i	$-0.495695\pi$
$2$	0.0765073	0.0270494	0.0135247	+	0.999909i	$0.495695\pi$
$3$	0	0
$3$	0	0
$4$	−7.99415	−0.999268
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−1.22367	−0.0540790
$9$	0	0
$10$	0	0
$11$	−10.8947	−0.298624	−0.149312	−	0.988790i	$-0.547706\pi$
$11$	−10.8947	−0.298624	−0.149312	+	0.988790i	$0.547706\pi$
$12$	0	0
$13$	−26.5468	−0.566366	−0.283183	−	0.959066i	$-0.591390\pi$
$13$	−26.5468	−0.566366	−0.283183	+	0.959066i	$0.591390\pi$
$14$	0.535551	0.0102237
$15$	0	0
$16$	63.8596	0.997806
$17$	95.1237	1.35711	0.678556	−	0.734549i	$-0.262606\pi$
$17$	95.1237	1.35711	0.678556	+	0.734549i	$0.262606\pi$
$18$	0	0
$19$	−35.4649	−0.428221	−0.214111	−	0.976809i	$-0.568685\pi$
$19$	−35.4649	−0.428221	−0.214111	+	0.976809i	$0.568685\pi$
$20$	0	0
$21$	0	0
$22$	−0.833522	−0.00807761
$23$	−62.8303	−0.569610	−0.284805	−	0.958585i	$-0.591929\pi$
$23$	−62.8303	−0.569610	−0.284805	+	0.958585i	$0.591929\pi$
$24$	0	0
$25$	0	0
$26$	−2.03102	−0.0153199
$27$	0	0
$28$	−55.9590	−0.377688
$29$	−117.823	−0.754452	−0.377226	−	0.926121i	$-0.623122\pi$
$29$	−117.823	−0.754452	−0.377226	+	0.926121i	$0.623122\pi$
$30$	0	0
$31$	−171.090	−0.991250	−0.495625	−	0.868537i	$-0.665061\pi$
$31$	−171.090	−0.991250	−0.495625	+	0.868537i	$0.665061\pi$
$32$	14.6751	0.0810691
$33$	0	0
$34$	7.27766	0.0367091
$35$	0	0
$36$	0	0
$37$	203.813	0.905584	0.452792	−	0.891616i	$-0.350428\pi$
$37$	203.813	0.905584	0.452792	+	0.891616i	$0.350428\pi$
$38$	−2.71332	−0.0115831
$39$	0	0
$40$	0	0
$41$	428.705	1.63299	0.816494	−	0.577354i	$-0.195915\pi$
$41$	428.705	1.63299	0.816494	+	0.577354i	$0.195915\pi$
$42$	0	0
$43$	−96.5851	−0.342537	−0.171268	−	0.985224i	$-0.554787\pi$
$43$	−96.5851	−0.342537	−0.171268	+	0.985224i	$0.554787\pi$
$44$	87.0936	0.298406
$45$	0	0
$46$	−4.80698	−0.0154076
$47$	407.806	1.26563	0.632816	−	0.774303i	$-0.281899\pi$
$47$	407.806	1.26563	0.632816	+	0.774303i	$0.281899\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	212.219	0.565952
$53$	380.874	0.987115	0.493558	−	0.869713i	$-0.335696\pi$
$53$	380.874	0.987115	0.493558	+	0.869713i	$0.335696\pi$
$54$	0	0
$55$	0	0
$56$	−8.56568	−0.0204399
$57$	0	0
$58$	−9.01429	−0.0204075
$59$	287.149	0.633620	0.316810	−	0.948489i	$-0.397388\pi$
$59$	287.149	0.633620	0.316810	+	0.948489i	$0.397388\pi$
$60$	0	0
$61$	−823.988	−1.72952	−0.864761	−	0.502183i	$-0.832530\pi$
$61$	−823.988	−1.72952	−0.864761	+	0.502183i	$0.832530\pi$
$62$	−13.0897	−0.0268127
$63$	0	0
$64$	−509.754	−0.995613
$65$	0	0
$66$	0	0
$67$	585.549	1.06770	0.533852	−	0.845578i	$-0.320744\pi$
$67$	585.549	1.06770	0.533852	+	0.845578i	$0.320744\pi$
$68$	−760.433	−1.35612
$69$	0	0
$70$	0	0
$71$	653.856	1.09294	0.546468	−	0.837480i	$-0.315972\pi$
$71$	653.856	1.09294	0.546468	+	0.837480i	$0.315972\pi$
$72$	0	0
$73$	−1051.77	−1.68630	−0.843152	−	0.537676i	$-0.819302\pi$
$73$	−1051.77	−1.68630	−0.843152	+	0.537676i	$0.819302\pi$
$74$	15.5932	0.0244955
$75$	0	0
$76$	283.512	0.427908
$77$	−76.2627	−0.112869
$78$	0	0
$79$	−751.268	−1.06993	−0.534964	−	0.844875i	$-0.679675\pi$
$79$	−751.268	−1.06993	−0.534964	+	0.844875i	$0.679675\pi$
$80$	0	0
$81$	0	0
$82$	32.7991	0.0441713
$83$	−844.677	−1.11705	−0.558526	−	0.829487i	$-0.688633\pi$
$83$	−844.677	−1.11705	−0.558526	+	0.829487i	$0.688633\pi$
$84$	0	0
$85$	0	0
$86$	−7.38946	−0.00926542
$87$	0	0
$88$	13.3315	0.0161493
$89$	262.334	0.312442	0.156221	−	0.987722i	$-0.450069\pi$
$89$	262.334	0.312442	0.156221	+	0.987722i	$0.450069\pi$
$90$	0	0
$91$	−185.828	−0.214066
$92$	502.275	0.569193
$93$	0	0
$94$	31.2001	0.0342346
$95$	0	0
$96$	0	0
$97$	−814.908	−0.853004	−0.426502	−	0.904487i	$-0.640254\pi$
$97$	−814.908	−0.853004	−0.426502	+	0.904487i	$0.640254\pi$
$98$	3.74886	0.00386420
$99$	0	0
$100$	0	0
$101$	−315.660	−0.310984	−0.155492	−	0.987837i	$-0.549696\pi$
$101$	−315.660	−0.310984	−0.155492	+	0.987837i	$0.549696\pi$
$102$	0	0
$103$	1858.26	1.77767	0.888835	−	0.458228i	$-0.151516\pi$
$103$	1858.26	1.77767	0.888835	+	0.458228i	$0.151516\pi$
$104$	32.4845	0.0306285
$105$	0	0
$106$	29.1397	0.0267009
$107$	−1202.75	−1.08668	−0.543338	−	0.839514i	$-0.682840\pi$
$107$	−1202.75	−1.08668	−0.543338	+	0.839514i	$0.682840\pi$
$108$	0	0
$109$	262.450	0.230625	0.115313	−	0.993329i	$-0.463213\pi$
$109$	262.450	0.230625	0.115313	+	0.993329i	$0.463213\pi$
$110$	0	0
$111$	0	0
$112$	447.017	0.377135
$113$	138.803	0.115553	0.0577765	−	0.998330i	$-0.481599\pi$
$113$	138.803	0.115553	0.0577765	+	0.998330i	$0.481599\pi$
$114$	0	0
$115$	0	0
$116$	941.892	0.753900
$117$	0	0
$118$	21.9690	0.0171390
$119$	665.866	0.512940
$120$	0	0
$121$	−1212.31	−0.910824
$122$	−63.0411	−0.0467826
$123$	0	0
$124$	1367.72	0.990525
$125$	0	0
$126$	0	0
$127$	−2632.46	−1.83931	−0.919657	−	0.392722i	$-0.871534\pi$
$127$	−2632.46	−1.83931	−0.919657	+	0.392722i	$0.871534\pi$
$128$	−156.400	−0.108000
$129$	0	0
$130$	0	0
$131$	301.676	0.201203	0.100601	−	0.994927i	$-0.467923\pi$
$131$	301.676	0.201203	0.100601	+	0.994927i	$0.467923\pi$
$132$	0	0
$133$	−248.254	−0.161852
$134$	44.7987	0.0288808
$135$	0	0
$136$	−116.400	−0.0733913
$137$	−2131.17	−1.32904	−0.664520	−	0.747271i	$-0.731364\pi$
$137$	−2131.17	−1.32904	−0.664520	+	0.747271i	$0.731364\pi$
$138$	0	0
$139$	1691.63	1.03225	0.516123	−	0.856515i	$-0.327375\pi$
$139$	1691.63	1.03225	0.516123	+	0.856515i	$0.327375\pi$
$140$	0	0
$141$	0	0
$142$	50.0247	0.0295633
$143$	289.219	0.169131
$144$	0	0
$145$	0	0
$146$	−80.4679	−0.0456135
$147$	0	0
$148$	−1629.31	−0.904921
$149$	80.4827	0.0442510	0.0221255	−	0.999755i	$-0.492957\pi$
$149$	80.4827	0.0442510	0.0221255	+	0.999755i	$0.492957\pi$
$150$	0	0
$151$	833.698	0.449307	0.224654	−	0.974439i	$-0.427875\pi$
$151$	833.698	0.449307	0.224654	+	0.974439i	$0.427875\pi$
$152$	43.3973	0.0231578
$153$	0	0
$154$	−5.83465	−0.00305305
$155$	0	0
$156$	0	0
$157$	−2108.68	−1.07192	−0.535960	−	0.844244i	$-0.680050\pi$
$157$	−2108.68	−1.07192	−0.535960	+	0.844244i	$0.680050\pi$
$158$	−57.4775	−0.0289409
$159$	0	0
$160$	0	0
$161$	−439.812	−0.215292
$162$	0	0
$163$	−174.223	−0.0837189	−0.0418594	−	0.999124i	$-0.513328\pi$
$163$	−174.223	−0.0837189	−0.0418594	+	0.999124i	$0.513328\pi$
$164$	−3427.13	−1.63179
$165$	0	0
$166$	−64.6239	−0.0302156
$167$	−1009.94	−0.467972	−0.233986	−	0.972240i	$-0.575177\pi$
$167$	−1009.94	−0.467972	−0.233986	+	0.972240i	$0.575177\pi$
$168$	0	0
$169$	−1492.27	−0.679229
$170$	0	0
$171$	0	0
$172$	772.115	0.342286
$173$	−3404.03	−1.49598	−0.747988	−	0.663712i	$-0.768980\pi$
$173$	−3404.03	−1.49598	−0.747988	+	0.663712i	$0.768980\pi$
$174$	0	0
$175$	0	0
$176$	−695.729	−0.297969
$177$	0	0
$178$	20.0704	0.00845136
$179$	1493.22	0.623509	0.311755	−	0.950163i	$-0.399083\pi$
$179$	1493.22	0.623509	0.311755	+	0.950163i	$0.399083\pi$
$180$	0	0
$181$	−1293.52	−0.531196	−0.265598	−	0.964084i	$-0.585569\pi$
$181$	−1293.52	−0.531196	−0.265598	+	0.964084i	$0.585569\pi$
$182$	−14.2172	−0.00579037
$183$	0	0
$184$	76.8835	0.0308039
$185$	0	0
$186$	0	0
$187$	−1036.34	−0.405267
$188$	−3260.06	−1.26471
$189$	0	0
$190$	0	0
$191$	−2004.86	−0.759510	−0.379755	−	0.925087i	$-0.623992\pi$
$191$	−2004.86	−0.759510	−0.379755	+	0.925087i	$0.623992\pi$
$192$	0	0
$193$	−446.924	−0.166686	−0.0833428	−	0.996521i	$-0.526560\pi$
$193$	−446.924	−0.166686	−0.0833428	+	0.996521i	$0.526560\pi$
$194$	−62.3464	−0.0230733
$195$	0	0
$196$	−391.713	−0.142753
$197$	−713.335	−0.257985	−0.128992	−	0.991646i	$-0.541174\pi$
$197$	−713.335	−0.257985	−0.128992	+	0.991646i	$0.541174\pi$
$198$	0	0
$199$	−4073.28	−1.45099	−0.725496	−	0.688227i	$-0.758389\pi$
$199$	−4073.28	−1.45099	−0.725496	+	0.688227i	$0.758389\pi$
$200$	0	0
$201$	0	0
$202$	−24.1503	−0.00841192
$203$	−824.759	−0.285156
$204$	0	0
$205$	0	0
$206$	142.170	0.0480849
$207$	0	0
$208$	−1695.27	−0.565123
$209$	386.378	0.127877
$210$	0	0
$211$	4996.44	1.63018	0.815092	−	0.579331i	$-0.196686\pi$
$211$	4996.44	1.63018	0.815092	+	0.579331i	$0.196686\pi$
$212$	−3044.77	−0.986393
$213$	0	0
$214$	−92.0192	−0.0293939
$215$	0	0
$216$	0	0
$217$	−1197.63	−0.374657
$218$	20.0793	0.00623827
$219$	0	0
$220$	0	0
$221$	−2525.23	−0.768622
$222$	0	0
$223$	−2326.57	−0.698650	−0.349325	−	0.937002i	$-0.613589\pi$
$223$	−2326.57	−0.698650	−0.349325	+	0.937002i	$0.613589\pi$
$224$	102.725	0.0306412
$225$	0	0
$226$	10.6194	0.00312564
$227$	−731.574	−0.213904	−0.106952	−	0.994264i	$-0.534109\pi$
$227$	−731.574	−0.213904	−0.106952	+	0.994264i	$0.534109\pi$
$228$	0	0
$229$	−913.230	−0.263528	−0.131764	−	0.991281i	$-0.542064\pi$
$229$	−913.230	−0.263528	−0.131764	+	0.991281i	$0.542064\pi$
$230$	0	0
$231$	0	0
$232$	144.176	0.0408000
$233$	−5301.18	−1.49052	−0.745262	−	0.666772i	$-0.767675\pi$
$233$	−5301.18	−1.49052	−0.745262	+	0.666772i	$0.767675\pi$
$234$	0	0
$235$	0	0
$236$	−2295.51	−0.633156
$237$	0	0
$238$	50.9436	0.0138747
$239$	2696.95	0.729920	0.364960	−	0.931023i	$-0.381083\pi$
$239$	2696.95	0.729920	0.364960	+	0.931023i	$0.381083\pi$
$240$	0	0
$241$	4299.38	1.14916	0.574579	−	0.818449i	$-0.305166\pi$
$241$	4299.38	1.14916	0.574579	+	0.818449i	$0.305166\pi$
$242$	−92.7502	−0.0246372
$243$	0	0
$244$	6587.08	1.72826
$245$	0	0
$246$	0	0
$247$	941.480	0.242530
$248$	209.358	0.0536058
$249$	0	0
$250$	0	0
$251$	−4712.81	−1.18514	−0.592570	−	0.805519i	$-0.701887\pi$
$251$	−4712.81	−1.18514	−0.592570	+	0.805519i	$0.701887\pi$
$252$	0	0
$253$	684.516	0.170099
$254$	−201.402	−0.0497524
$255$	0	0
$256$	4066.06	0.992691
$257$	−6833.53	−1.65861	−0.829307	−	0.558793i	$-0.811265\pi$
$257$	−6833.53	−1.65861	−0.829307	+	0.558793i	$0.811265\pi$
$258$	0	0
$259$	1426.69	0.342278
$260$	0	0
$261$	0	0
$262$	23.0804	0.00544242
$263$	−8221.38	−1.92757	−0.963787	−	0.266674i	$-0.914075\pi$
$263$	−8221.38	−1.92757	−0.963787	+	0.266674i	$0.914075\pi$
$264$	0	0
$265$	0	0
$266$	−18.9933	−0.00437801
$267$	0	0
$268$	−4680.96	−1.06692
$269$	−6319.20	−1.43230	−0.716149	−	0.697947i	$-0.754097\pi$
$269$	−6319.20	−1.43230	−0.716149	+	0.697947i	$0.754097\pi$
$270$	0	0
$271$	−1127.67	−0.252771	−0.126386	−	0.991981i	$-0.540338\pi$
$271$	−1127.67	−0.252771	−0.126386	+	0.991981i	$0.540338\pi$
$272$	6074.56	1.35413
$273$	0	0
$274$	−163.050	−0.0359497
$275$	0	0
$276$	0	0
$277$	2522.90	0.547243	0.273622	−	0.961837i	$-0.411778\pi$
$277$	2522.90	0.547243	0.273622	+	0.961837i	$0.411778\pi$
$278$	129.422	0.0279216
$279$	0	0
$280$	0	0
$281$	9070.33	1.92559	0.962794	−	0.270235i	$-0.0871015\pi$
$281$	9070.33	1.92559	0.962794	+	0.270235i	$0.0871015\pi$
$282$	0	0
$283$	145.080	0.0304739	0.0152369	−	0.999884i	$-0.495150\pi$
$283$	145.080	0.0304739	0.0152369	+	0.999884i	$0.495150\pi$
$284$	−5227.02	−1.09214
$285$	0	0
$286$	22.1273	0.00457489
$287$	3000.94	0.617211
$288$	0	0
$289$	4135.53	0.841752
$290$	0	0
$291$	0	0
$292$	8407.99	1.68507
$293$	5876.23	1.17165	0.585824	−	0.810438i	$-0.300771\pi$
$293$	5876.23	1.17165	0.585824	+	0.810438i	$0.300771\pi$
$294$	0	0
$295$	0	0
$296$	−249.399	−0.0489731
$297$	0	0
$298$	6.15751	0.00119696
$299$	1667.94	0.322608
$300$	0	0
$301$	−676.095	−0.129467
$302$	63.7839	0.0121535
$303$	0	0
$304$	−2264.77	−0.427282
$305$	0	0
$306$	0	0
$307$	2894.45	0.538094	0.269047	−	0.963127i	$-0.413291\pi$
$307$	2894.45	0.538094	0.269047	+	0.963127i	$0.413291\pi$
$308$	609.655	0.112787
$309$	0	0
$310$	0	0
$311$	−3009.61	−0.548744	−0.274372	−	0.961624i	$-0.588470\pi$
$311$	−3009.61	−0.548744	−0.274372	+	0.961624i	$0.588470\pi$
$312$	0	0
$313$	−10064.0	−1.81741	−0.908706	−	0.417436i	$-0.862929\pi$
$313$	−10064.0	−1.81741	−0.908706	+	0.417436i	$0.862929\pi$
$314$	−161.330	−0.0289948
$315$	0	0
$316$	6005.75	1.06914
$317$	8304.18	1.47132	0.735661	−	0.677349i	$-0.236871\pi$
$317$	8304.18	1.47132	0.735661	+	0.677349i	$0.236871\pi$
$318$	0	0
$319$	1283.64	0.225298
$320$	0	0
$321$	0	0
$322$	−33.6488	−0.00582353
$323$	−3373.55	−0.581144
$324$	0	0
$325$	0	0
$326$	−13.3293	−0.00226455
$327$	0	0
$328$	−524.593	−0.0883103
$329$	2854.64	0.478364
$330$	0	0
$331$	−7066.25	−1.17340	−0.586702	−	0.809803i	$-0.699574\pi$
$331$	−7066.25	−1.17340	−0.586702	+	0.809803i	$0.699574\pi$
$332$	6752.47	1.11623
$333$	0	0
$334$	−77.2677	−0.0126584
$335$	0	0
$336$	0	0
$337$	−1510.59	−0.244175	−0.122087	−	0.992519i	$-0.538959\pi$
$337$	−1510.59	−0.244175	−0.122087	+	0.992519i	$0.538959\pi$
$338$	−114.169	−0.0183727
$339$	0	0
$340$	0	0
$341$	1863.97	0.296011
$342$	0	0
$343$	343.000	0.0539949
$344$	118.188	0.0185241
$345$	0	0
$346$	−260.433	−0.0404653
$347$	−11921.4	−1.84431	−0.922154	−	0.386822i	$-0.873573\pi$
$347$	−11921.4	−1.84431	−0.922154	+	0.386822i	$0.873573\pi$
$348$	0	0
$349$	−3329.00	−0.510594	−0.255297	−	0.966863i	$-0.582173\pi$
$349$	−3329.00	−0.510594	−0.255297	+	0.966863i	$0.582173\pi$
$350$	0	0
$351$	0	0
$352$	−159.880	−0.0242092
$353$	−5883.87	−0.887158	−0.443579	−	0.896235i	$-0.646291\pi$
$353$	−5883.87	−0.887158	−0.443579	+	0.896235i	$0.646291\pi$
$354$	0	0
$355$	0	0
$356$	−2097.13	−0.312213
$357$	0	0
$358$	114.242	0.0168656
$359$	−822.114	−0.120862	−0.0604311	−	0.998172i	$-0.519248\pi$
$359$	−822.114	−0.120862	−0.0604311	+	0.998172i	$0.519248\pi$
$360$	0	0
$361$	−5601.24	−0.816626
$362$	−98.9636	−0.0143685
$363$	0	0
$364$	1485.53	0.213910
$365$	0	0
$366$	0	0
$367$	−9873.06	−1.40428	−0.702139	−	0.712040i	$-0.747771\pi$
$367$	−9873.06	−1.40428	−0.702139	+	0.712040i	$0.747771\pi$
$368$	−4012.32	−0.568360
$369$	0	0
$370$	0	0
$371$	2666.12	0.373095
$372$	0	0
$373$	7152.24	0.992840	0.496420	−	0.868083i	$-0.334648\pi$
$373$	7152.24	0.992840	0.496420	+	0.868083i	$0.334648\pi$
$374$	−79.2877	−0.0109622
$375$	0	0
$376$	−499.020	−0.0684441
$377$	3127.82	0.427296
$378$	0	0
$379$	−14117.4	−1.91336	−0.956680	−	0.291140i	$-0.905965\pi$
$379$	−14117.4	−1.91336	−0.956680	+	0.291140i	$0.905965\pi$
$380$	0	0
$381$	0	0
$382$	−153.386	−0.0205443
$383$	5293.18	0.706185	0.353093	−	0.935588i	$-0.385130\pi$
$383$	5293.18	0.706185	0.353093	+	0.935588i	$0.385130\pi$
$384$	0	0
$385$	0	0
$386$	−34.1930	−0.00450875
$387$	0	0
$388$	6514.50	0.852380
$389$	8910.01	1.16132	0.580662	−	0.814144i	$-0.302794\pi$
$389$	8910.01	1.16132	0.580662	+	0.814144i	$0.302794\pi$
$390$	0	0
$391$	−5976.66	−0.773024
$392$	−59.9598	−0.00772557
$393$	0	0
$394$	−54.5753	−0.00697834
$395$	0	0
$396$	0	0
$397$	3986.11	0.503922	0.251961	−	0.967737i	$-0.418925\pi$
$397$	3986.11	0.503922	0.251961	+	0.967737i	$0.418925\pi$
$398$	−311.636	−0.0392485
$399$	0	0
$400$	0	0
$401$	−3119.10	−0.388429	−0.194215	−	0.980959i	$-0.562216\pi$
$401$	−3119.10	−0.388429	−0.194215	+	0.980959i	$0.562216\pi$
$402$	0	0
$403$	4541.90	0.561410
$404$	2523.43	0.310756
$405$	0	0
$406$	−63.1000	−0.00771331
$407$	−2220.47	−0.270429
$408$	0	0
$409$	−4125.99	−0.498819	−0.249410	−	0.968398i	$-0.580237\pi$
$409$	−4125.99	−0.498819	−0.249410	+	0.968398i	$0.580237\pi$
$410$	0	0
$411$	0	0
$412$	−14855.2	−1.77637
$413$	2010.04	0.239486
$414$	0	0
$415$	0	0
$416$	−389.576	−0.0459148
$417$	0	0
$418$	29.5608	0.00345901
$419$	13550.2	1.57988	0.789939	−	0.613185i	$-0.210112\pi$
$419$	13550.2	1.57988	0.789939	+	0.613185i	$0.210112\pi$
$420$	0	0
$421$	6464.49	0.748361	0.374180	−	0.927356i	$-0.377924\pi$
$421$	6464.49	0.748361	0.374180	+	0.927356i	$0.377924\pi$
$422$	382.264	0.0440955
$423$	0	0
$424$	−466.064	−0.0533822
$425$	0	0
$426$	0	0
$427$	−5767.92	−0.653698
$428$	9614.97	1.08588
$429$	0	0
$430$	0	0
$431$	8652.70	0.967020	0.483510	−	0.875339i	$-0.339362\pi$
$431$	8652.70	0.967020	0.483510	+	0.875339i	$0.339362\pi$
$432$	0	0
$433$	−10201.3	−1.13221	−0.566103	−	0.824334i	$-0.691550\pi$
$433$	−10201.3	−1.13221	−0.566103	+	0.824334i	$0.691550\pi$
$434$	−91.6276	−0.0101343
$435$	0	0
$436$	−2098.06	−0.230456
$437$	2228.27	0.243919
$438$	0	0
$439$	13063.9	1.42029	0.710143	−	0.704057i	$-0.248630\pi$
$439$	13063.9	1.42029	0.710143	+	0.704057i	$0.248630\pi$
$440$	0	0
$441$	0	0
$442$	−193.199	−0.0207908
$443$	−2032.27	−0.217959	−0.108980	−	0.994044i	$-0.534758\pi$
$443$	−2032.27	−0.217959	−0.108980	+	0.994044i	$0.534758\pi$
$444$	0	0
$445$	0	0
$446$	−178.000	−0.0188981
$447$	0	0
$448$	−3568.28	−0.376306
$449$	−1640.75	−0.172454	−0.0862271	−	0.996276i	$-0.527481\pi$
$449$	−1640.75	−0.172454	−0.0862271	+	0.996276i	$0.527481\pi$
$450$	0	0
$451$	−4670.60	−0.487650
$452$	−1109.61	−0.115468
$453$	0	0
$454$	−55.9707	−0.00578598
$455$	0	0
$456$	0	0
$457$	5578.74	0.571033	0.285517	−	0.958374i	$-0.407835\pi$
$457$	5578.74	0.571033	0.285517	+	0.958374i	$0.407835\pi$
$458$	−69.8687	−0.00712828
$459$	0	0
$460$	0	0
$461$	15124.5	1.52803	0.764013	−	0.645201i	$-0.223227\pi$
$461$	15124.5	1.52803	0.764013	+	0.645201i	$0.223227\pi$
$462$	0	0
$463$	14882.8	1.49387	0.746934	−	0.664898i	$-0.231525\pi$
$463$	14882.8	1.49387	0.746934	+	0.664898i	$0.231525\pi$
$464$	−7524.10	−0.752797
$465$	0	0
$466$	−405.579	−0.0403178
$467$	−5926.96	−0.587296	−0.293648	−	0.955914i	$-0.594869\pi$
$467$	−5926.96	−0.587296	−0.293648	+	0.955914i	$0.594869\pi$
$468$	0	0
$469$	4098.84	0.403554
$470$	0	0
$471$	0	0
$472$	−351.375	−0.0342655
$473$	1052.26	0.102290
$474$	0	0
$475$	0	0
$476$	−5323.03	−0.512565
$477$	0	0
$478$	206.336	0.0197439
$479$	−13214.0	−1.26046	−0.630231	−	0.776407i	$-0.717040\pi$
$479$	−13214.0	−1.26046	−0.630231	+	0.776407i	$0.717040\pi$
$480$	0	0
$481$	−5410.58	−0.512892
$482$	328.933	0.0310840
$483$	0	0
$484$	9691.35	0.910157
$485$	0	0
$486$	0	0
$487$	17646.0	1.64192	0.820961	−	0.570984i	$-0.193438\pi$
$487$	17646.0	1.64192	0.820961	+	0.570984i	$0.193438\pi$
$488$	1008.29	0.0935309
$489$	0	0
$490$	0	0
$491$	11540.7	1.06074	0.530371	−	0.847766i	$-0.322053\pi$
$491$	11540.7	1.06074	0.530371	+	0.847766i	$0.322053\pi$
$492$	0	0
$493$	−11207.7	−1.02388
$494$	72.0301	0.00656029
$495$	0	0
$496$	−10925.8	−0.989075
$497$	4576.99	0.413091
$498$	0	0
$499$	−6820.43	−0.611873	−0.305936	−	0.952052i	$-0.598969\pi$
$499$	−6820.43	−0.611873	−0.305936	+	0.952052i	$0.598969\pi$
$500$	0	0
$501$	0	0
$502$	−360.564	−0.0320573
$503$	7174.71	0.635993	0.317996	−	0.948092i	$-0.396990\pi$
$503$	7174.71	0.635993	0.317996	+	0.948092i	$0.396990\pi$
$504$	0	0
$505$	0	0
$506$	52.3704	0.00460109
$507$	0	0
$508$	21044.3	1.83797
$509$	3889.82	0.338729	0.169365	−	0.985553i	$-0.445828\pi$
$509$	3889.82	0.338729	0.169365	+	0.985553i	$0.445828\pi$
$510$	0	0
$511$	−7362.38	−0.637363
$512$	1562.29	0.134851
$513$	0	0
$514$	−522.815	−0.0448645
$515$	0	0
$516$	0	0
$517$	−4442.92	−0.377948
$518$	109.152	0.00925843
$519$	0	0
$520$	0	0
$521$	−3687.49	−0.310080	−0.155040	−	0.987908i	$-0.549551\pi$
$521$	−3687.49	−0.310080	−0.155040	+	0.987908i	$0.549551\pi$
$522$	0	0
$523$	8280.56	0.692320	0.346160	−	0.938175i	$-0.387485\pi$
$523$	8280.56	0.692320	0.346160	+	0.938175i	$0.387485\pi$
$524$	−2411.65	−0.201056
$525$	0	0
$526$	−628.995	−0.0521397
$527$	−16274.8	−1.34524
$528$	0	0
$529$	−8219.35	−0.675545
$530$	0	0
$531$	0	0
$532$	1984.58	0.161734
$533$	−11380.8	−0.924869
$534$	0	0
$535$	0	0
$536$	−716.518	−0.0577404
$537$	0	0
$538$	−483.465	−0.0387428
$539$	−533.839	−0.0426606
$540$	0	0
$541$	9002.84	0.715457	0.357728	−	0.933826i	$-0.383551\pi$
$541$	9002.84	0.715457	0.357728	+	0.933826i	$0.383551\pi$
$542$	−86.2748	−0.00683731
$543$	0	0
$544$	1395.95	0.110020
$545$	0	0
$546$	0	0
$547$	−17066.3	−1.33401	−0.667005	−	0.745053i	$-0.732424\pi$
$547$	−17066.3	−1.33401	−0.667005	+	0.745053i	$0.732424\pi$
$548$	17036.9	1.32807
$549$	0	0
$550$	0	0
$551$	4178.57	0.323073
$552$	0	0
$553$	−5258.88	−0.404395
$554$	193.020	0.0148026
$555$	0	0
$556$	−13523.1	−1.03149
$557$	−23487.4	−1.78670	−0.893351	−	0.449360i	$-0.851652\pi$
$557$	−23487.4	−1.78670	−0.893351	+	0.449360i	$0.851652\pi$
$558$	0	0
$559$	2564.03	0.194001
$560$	0	0
$561$	0	0
$562$	693.946	0.0520860
$563$	15357.7	1.14964	0.574821	−	0.818279i	$-0.305072\pi$
$563$	15357.7	1.14964	0.574821	+	0.818279i	$0.305072\pi$
$564$	0	0
$565$	0	0
$566$	11.0997	0.000824300 0
$567$	0	0
$568$	−800.103	−0.0591049
$569$	−8871.22	−0.653604	−0.326802	−	0.945093i	$-0.605971\pi$
$569$	−8871.22	−0.653604	−0.326802	+	0.945093i	$0.605971\pi$
$570$	0	0
$571$	−6428.16	−0.471121	−0.235561	−	0.971860i	$-0.575693\pi$
$571$	−6428.16	−0.471121	−0.235561	+	0.971860i	$0.575693\pi$
$572$	−2312.06	−0.169007
$573$	0	0
$574$	229.593	0.0166952
$575$	0	0
$576$	0	0
$577$	−8519.56	−0.614686	−0.307343	−	0.951599i	$-0.599440\pi$
$577$	−8519.56	−0.614686	−0.307343	+	0.951599i	$0.599440\pi$
$578$	316.398	0.0227689
$579$	0	0
$580$	0	0
$581$	−5912.74	−0.422206
$582$	0	0
$583$	−4149.50	−0.294777
$584$	1287.02	0.0911936
$585$	0	0
$586$	449.574	0.0316924
$587$	13458.2	0.946300	0.473150	−	0.880982i	$-0.343117\pi$
$587$	13458.2	0.946300	0.473150	+	0.880982i	$0.343117\pi$
$588$	0	0
$589$	6067.70	0.424474
$590$	0	0
$591$	0	0
$592$	13015.4	0.903597
$593$	−15744.8	−1.09032	−0.545162	−	0.838331i	$-0.683532\pi$
$593$	−15744.8	−1.09032	−0.545162	+	0.838331i	$0.683532\pi$
$594$	0	0
$595$	0	0
$596$	−643.390	−0.0442186
$597$	0	0
$598$	127.610	0.00872635
$599$	9108.67	0.621319	0.310660	−	0.950521i	$-0.399450\pi$
$599$	9108.67	0.621319	0.310660	+	0.950521i	$0.399450\pi$
$600$	0	0
$601$	25596.1	1.73725	0.868625	−	0.495470i	$-0.165004\pi$
$601$	25596.1	1.73725	0.868625	+	0.495470i	$0.165004\pi$
$602$	−51.7262	−0.00350200
$603$	0	0
$604$	−6664.70	−0.448978
$605$	0	0
$606$	0	0
$607$	4719.62	0.315590	0.157795	−	0.987472i	$-0.449561\pi$
$607$	4719.62	0.315590	0.157795	+	0.987472i	$0.449561\pi$
$608$	−520.450	−0.0347155
$609$	0	0
$610$	0	0
$611$	−10826.0	−0.716811
$612$	0	0
$613$	−8663.56	−0.570829	−0.285414	−	0.958404i	$-0.592131\pi$
$613$	−8663.56	−0.570829	−0.285414	+	0.958404i	$0.592131\pi$
$614$	221.446	0.0145551
$615$	0	0
$616$	93.3203	0.00610387
$617$	22393.6	1.46115	0.730577	−	0.682830i	$-0.239251\pi$
$617$	22393.6	1.46115	0.730577	+	0.682830i	$0.239251\pi$
$618$	0	0
$619$	−22053.3	−1.43198	−0.715991	−	0.698109i	$-0.754025\pi$
$619$	−22053.3	−1.43198	−0.715991	+	0.698109i	$0.754025\pi$
$620$	0	0
$621$	0	0
$622$	−230.257	−0.0148432
$623$	1836.34	0.118092
$624$	0	0
$625$	0	0
$626$	−769.968	−0.0491599
$627$	0	0
$628$	16857.1	1.07114
$629$	19387.4	1.22898
$630$	0	0
$631$	−1637.02	−0.103278	−0.0516392	−	0.998666i	$-0.516445\pi$
$631$	−1637.02	−0.103278	−0.0516392	+	0.998666i	$0.516445\pi$
$632$	919.303	0.0578606
$633$	0	0
$634$	635.330	0.0397984
$635$	0	0
$636$	0	0
$637$	−1300.79	−0.0809095
$638$	98.2078	0.00609417
$639$	0	0
$640$	0	0
$641$	−25331.3	−1.56089	−0.780443	−	0.625227i	$-0.785006\pi$
$641$	−25331.3	−1.56089	−0.780443	+	0.625227i	$0.785006\pi$
$642$	0	0
$643$	2162.69	0.132641	0.0663205	−	0.997798i	$-0.478874\pi$
$643$	2162.69	0.132641	0.0663205	+	0.997798i	$0.478874\pi$
$644$	3515.92	0.215135
$645$	0	0
$646$	−258.101	−0.0157196
$647$	−247.980	−0.0150682	−0.00753408	−	0.999972i	$-0.502398\pi$
$647$	−247.980	−0.0150682	−0.00753408	+	0.999972i	$0.502398\pi$
$648$	0	0
$649$	−3128.39	−0.189214
$650$	0	0
$651$	0	0
$652$	1392.76	0.0836576
$653$	−1225.37	−0.0734339	−0.0367169	−	0.999326i	$-0.511690\pi$
$653$	−1225.37	−0.0734339	−0.0367169	+	0.999326i	$0.511690\pi$
$654$	0	0
$655$	0	0
$656$	27376.9	1.62940
$657$	0	0
$658$	218.401	0.0129394
$659$	2284.95	0.135067	0.0675335	−	0.997717i	$-0.478487\pi$
$659$	2284.95	0.135067	0.0675335	+	0.997717i	$0.478487\pi$
$660$	0	0
$661$	24079.0	1.41689	0.708445	−	0.705766i	$-0.249397\pi$
$661$	24079.0	1.41689	0.708445	+	0.705766i	$0.249397\pi$
$662$	−540.620	−0.0317399
$663$	0	0
$664$	1033.60	0.0604091
$665$	0	0
$666$	0	0
$667$	7402.84	0.429744
$668$	8073.60	0.467630
$669$	0	0
$670$	0	0
$671$	8977.08	0.516478
$672$	0	0
$673$	−31756.3	−1.81889	−0.909447	−	0.415819i	$-0.863495\pi$
$673$	−31756.3	−1.81889	−0.909447	+	0.415819i	$0.863495\pi$
$674$	−115.571	−0.00660479
$675$	0	0
$676$	11929.4	0.678732
$677$	19214.7	1.09081	0.545406	−	0.838172i	$-0.316375\pi$
$677$	19214.7	1.09081	0.545406	+	0.838172i	$0.316375\pi$
$678$	0	0
$679$	−5704.36	−0.322405
$680$	0	0
$681$	0	0
$682$	142.608	0.00800693
$683$	16969.5	0.950688	0.475344	−	0.879800i	$-0.342324\pi$
$683$	16969.5	0.950688	0.475344	+	0.879800i	$0.342324\pi$
$684$	0	0
$685$	0	0
$686$	26.2420	0.00146053
$687$	0	0
$688$	−6167.88	−0.341785
$689$	−10111.0	−0.559069
$690$	0	0
$691$	9412.73	0.518202	0.259101	−	0.965850i	$-0.416574\pi$
$691$	9412.73	0.518202	0.259101	+	0.965850i	$0.416574\pi$
$692$	27212.3	1.49488
$693$	0	0
$694$	−912.075	−0.0498875
$695$	0	0
$696$	0	0
$697$	40780.0	2.21615
$698$	−254.693	−0.0138113
$699$	0	0
$700$	0	0
$701$	−14947.9	−0.805384	−0.402692	−	0.915335i	$-0.631925\pi$
$701$	−14947.9	−0.805384	−0.402692	+	0.915335i	$0.631925\pi$
$702$	0	0
$703$	−7228.20	−0.387790
$704$	5553.60	0.297314
$705$	0	0
$706$	−450.159	−0.0239971
$707$	−2209.62	−0.117541
$708$	0	0
$709$	14067.4	0.745152	0.372576	−	0.928002i	$-0.378475\pi$
$709$	14067.4	0.745152	0.372576	+	0.928002i	$0.378475\pi$
$710$	0	0
$711$	0	0
$712$	−321.009	−0.0168965
$713$	10749.7	0.564626
$714$	0	0
$715$	0	0
$716$	−11937.0	−0.623053
$717$	0	0
$718$	−62.8977	−0.00326925
$719$	−20536.2	−1.06519	−0.532595	−	0.846370i	$-0.678783\pi$
$719$	−20536.2	−1.06519	−0.532595	+	0.846370i	$0.678783\pi$
$720$	0	0
$721$	13007.8	0.671896
$722$	−428.536	−0.0220893
$723$	0	0
$724$	10340.6	0.530807
$725$	0	0
$726$	0	0
$727$	−15008.2	−0.765643	−0.382821	−	0.923822i	$-0.625047\pi$
$727$	−15008.2	−0.765643	−0.382821	+	0.923822i	$0.625047\pi$
$728$	227.391	0.0115765
$729$	0	0
$730$	0	0
$731$	−9187.53	−0.464861
$732$	0	0
$733$	26461.7	1.33340	0.666701	−	0.745325i	$-0.267706\pi$
$733$	26461.7	1.33340	0.666701	+	0.745325i	$0.267706\pi$
$734$	−755.361	−0.0379849
$735$	0	0
$736$	−922.039	−0.0461777
$737$	−6379.36	−0.318842
$738$	0	0
$739$	21868.2	1.08854	0.544272	−	0.838909i	$-0.316806\pi$
$739$	21868.2	1.08854	0.544272	+	0.838909i	$0.316806\pi$
$740$	0	0
$741$	0	0
$742$	203.978	0.0100920
$743$	23832.7	1.17677	0.588383	−	0.808582i	$-0.299765\pi$
$743$	23832.7	1.17677	0.588383	+	0.808582i	$0.299765\pi$
$744$	0	0
$745$	0	0
$746$	547.199	0.0268557
$747$	0	0
$748$	8284.67	0.404970
$749$	−8419.26	−0.410725
$750$	0	0
$751$	−17221.2	−0.836763	−0.418381	−	0.908271i	$-0.637402\pi$
$751$	−17221.2	−0.836763	−0.418381	+	0.908271i	$0.637402\pi$
$752$	26042.3	1.26285
$753$	0	0
$754$	239.301	0.0115581
$755$	0	0
$756$	0	0
$757$	−1461.19	−0.0701557	−0.0350779	−	0.999385i	$-0.511168\pi$
$757$	−1461.19	−0.0701557	−0.0350779	+	0.999385i	$0.511168\pi$
$758$	−1080.09	−0.0517553
$759$	0	0
$760$	0	0
$761$	6949.62	0.331043	0.165521	−	0.986206i	$-0.447069\pi$
$761$	6949.62	0.331043	0.165521	+	0.986206i	$0.447069\pi$
$762$	0	0
$763$	1837.15	0.0871681
$764$	16027.1	0.758954
$765$	0	0
$766$	404.967	0.0191019
$767$	−7622.88	−0.358861
$768$	0	0
$769$	−12778.2	−0.599209	−0.299605	−	0.954063i	$-0.596855\pi$
$769$	−12778.2	−0.599209	−0.299605	+	0.954063i	$0.596855\pi$
$770$	0	0
$771$	0	0
$772$	3572.78	0.166564
$773$	20178.8	0.938915	0.469458	−	0.882955i	$-0.344449\pi$
$773$	20178.8	0.938915	0.469458	+	0.882955i	$0.344449\pi$
$774$	0	0
$775$	0	0
$776$	997.178	0.0461296
$777$	0	0
$778$	681.680	0.0314131
$779$	−15204.0	−0.699280
$780$	0	0
$781$	−7123.55	−0.326377
$782$	−457.258	−0.0209098
$783$	0	0
$784$	3129.12	0.142544
$785$	0	0
$786$	0	0
$787$	782.952	0.0354628	0.0177314	−	0.999843i	$-0.494356\pi$
$787$	782.952	0.0354628	0.0177314	+	0.999843i	$0.494356\pi$
$788$	5702.51	0.257796
$789$	0	0
$790$	0	0
$791$	971.620	0.0436749
$792$	0	0
$793$	21874.3	0.979543
$794$	304.966	0.0136308
$795$	0	0
$796$	32562.4	1.44993
$797$	−18184.7	−0.808200	−0.404100	−	0.914715i	$-0.632415\pi$
$797$	−18184.7	−0.808200	−0.404100	+	0.914715i	$0.632415\pi$
$798$	0	0
$799$	38792.1	1.71760
$800$	0	0
$801$	0	0
$802$	−238.634	−0.0105068
$803$	11458.7	0.503571
$804$	0	0
$805$	0	0
$806$	347.489	0.0151858
$807$	0	0
$808$	386.263	0.0168177
$809$	−33601.8	−1.46029	−0.730146	−	0.683291i	$-0.760548\pi$
$809$	−33601.8	−1.46029	−0.730146	+	0.683291i	$0.760548\pi$
$810$	0	0
$811$	−4640.05	−0.200905	−0.100453	−	0.994942i	$-0.532029\pi$
$811$	−4640.05	−0.200905	−0.100453	+	0.994942i	$0.532029\pi$
$812$	6593.24	0.284948
$813$	0	0
$814$	−169.882	−0.00731495
$815$	0	0
$816$	0	0
$817$	3425.38	0.146682
$818$	−315.668	−0.0134928
$819$	0	0
$820$	0	0
$821$	−34248.4	−1.45588	−0.727939	−	0.685641i	$-0.759522\pi$
$821$	−34248.4	−1.45588	−0.727939	+	0.685641i	$0.759522\pi$
$822$	0	0
$823$	5076.24	0.215002	0.107501	−	0.994205i	$-0.465715\pi$
$823$	5076.24	0.215002	0.107501	+	0.994205i	$0.465715\pi$
$824$	−2273.90	−0.0961346
$825$	0	0
$826$	153.783	0.00647795
$827$	10769.6	0.452835	0.226418	−	0.974030i	$-0.427299\pi$
$827$	10769.6	0.452835	0.226418	+	0.974030i	$0.427299\pi$
$828$	0	0
$829$	25104.1	1.05175	0.525876	−	0.850561i	$-0.323738\pi$
$829$	25104.1	1.05175	0.525876	+	0.850561i	$0.323738\pi$
$830$	0	0
$831$	0	0
$832$	13532.3	0.563881
$833$	4661.06	0.193873
$834$	0	0
$835$	0	0
$836$	−3088.77	−0.127784
$837$	0	0
$838$	1036.69	0.0427348
$839$	−37028.0	−1.52366	−0.761829	−	0.647778i	$-0.775699\pi$
$839$	−37028.0	−1.52366	−0.761829	+	0.647778i	$0.775699\pi$
$840$	0	0
$841$	−10506.8	−0.430801
$842$	494.580	0.0202427
$843$	0	0
$844$	−39942.3	−1.62899
$845$	0	0
$846$	0	0
$847$	−8486.14	−0.344259
$848$	24322.5	0.984949
$849$	0	0
$850$	0	0
$851$	−12805.6	−0.515829
$852$	0	0
$853$	22334.8	0.896518	0.448259	−	0.893904i	$-0.352044\pi$
$853$	22334.8	0.896518	0.448259	+	0.893904i	$0.352044\pi$
$854$	−441.288	−0.0176821
$855$	0	0
$856$	1471.77	0.0587664
$857$	18241.5	0.727090	0.363545	−	0.931577i	$-0.381566\pi$
$857$	18241.5	0.727090	0.363545	+	0.931577i	$0.381566\pi$
$858$	0	0
$859$	−12413.0	−0.493045	−0.246523	−	0.969137i	$-0.579288\pi$
$859$	−12413.0	−0.493045	−0.246523	+	0.969137i	$0.579288\pi$
$860$	0	0
$861$	0	0
$862$	661.994	0.0261573
$863$	−4676.16	−0.184448	−0.0922238	−	0.995738i	$-0.529398\pi$
$863$	−4676.16	−0.184448	−0.0922238	+	0.995738i	$0.529398\pi$
$864$	0	0
$865$	0	0
$866$	−780.477	−0.0306255
$867$	0	0
$868$	9574.05	0.374383
$869$	8184.82	0.319506
$870$	0	0
$871$	−15544.5	−0.604711
$872$	−321.152	−0.0124720
$873$	0	0
$874$	170.479	0.00659787
$875$	0	0
$876$	0	0
$877$	7649.96	0.294551	0.147275	−	0.989096i	$-0.452950\pi$
$877$	7649.96	0.294551	0.147275	+	0.989096i	$0.452950\pi$
$878$	999.483	0.0384179
$879$	0	0
$880$	0	0
$881$	−6816.13	−0.260660	−0.130330	−	0.991471i	$-0.541604\pi$
$881$	−6816.13	−0.260660	−0.130330	+	0.991471i	$0.541604\pi$
$882$	0	0
$883$	22141.7	0.843860	0.421930	−	0.906628i	$-0.361353\pi$
$883$	22141.7	0.843860	0.421930	+	0.906628i	$0.361353\pi$
$884$	20187.1	0.768060
$885$	0	0
$886$	−155.483	−0.00589566
$887$	−28729.2	−1.08752	−0.543762	−	0.839240i	$-0.683001\pi$
$887$	−28729.2	−1.08752	−0.543762	+	0.839240i	$0.683001\pi$
$888$	0	0
$889$	−18427.2	−0.695196
$890$	0	0
$891$	0	0
$892$	18599.0	0.698139
$893$	−14462.8	−0.541970
$894$	0	0
$895$	0	0
$896$	−1094.80	−0.0408201
$897$	0	0
$898$	−125.530	−0.00466478
$899$	20158.3	0.747851
$900$	0	0
$901$	36230.2	1.33963
$902$	−357.335	−0.0131906
$903$	0	0
$904$	−169.849	−0.00624899
$905$	0	0
$906$	0	0
$907$	−47268.0	−1.73044	−0.865219	−	0.501395i	$-0.832820\pi$
$907$	−47268.0	−1.73044	−0.865219	+	0.501395i	$0.832820\pi$
$908$	5848.31	0.213748
$909$	0	0
$910$	0	0
$911$	32904.1	1.19666	0.598332	−	0.801248i	$-0.295830\pi$
$911$	32904.1	1.19666	0.598332	+	0.801248i	$0.295830\pi$
$912$	0	0
$913$	9202.48	0.333579
$914$	426.814	0.0154461
$915$	0	0
$916$	7300.49	0.263335
$917$	2111.73	0.0760476
$918$	0	0
$919$	−2561.48	−0.0919426	−0.0459713	−	0.998943i	$-0.514638\pi$
$919$	−2561.48	−0.0919426	−0.0459713	+	0.998943i	$0.514638\pi$
$920$	0	0
$921$	0	0
$922$	1157.14	0.0413322
$923$	−17357.8	−0.619002
$924$	0	0
$925$	0	0
$926$	1138.64	0.0404082
$927$	0	0
$928$	−1729.06	−0.0611628
$929$	44532.6	1.57273	0.786365	−	0.617762i	$-0.211960\pi$
$929$	44532.6	1.57273	0.786365	+	0.617762i	$0.211960\pi$
$930$	0	0
$931$	−1737.78	−0.0611745
$932$	42378.4	1.48943
$933$	0	0
$934$	−453.456	−0.0158860
$935$	0	0
$936$	0	0
$937$	−6891.28	−0.240265	−0.120133	−	0.992758i	$-0.538332\pi$
$937$	−6891.28	−0.240265	−0.120133	+	0.992758i	$0.538332\pi$
$938$	313.591	0.0109159
$939$	0	0
$940$	0	0
$941$	−36509.6	−1.26480	−0.632401	−	0.774641i	$-0.717931\pi$
$941$	−36509.6	−1.26480	−0.632401	+	0.774641i	$0.717931\pi$
$942$	0	0
$943$	−26935.7	−0.930166
$944$	18337.2	0.632229
$945$	0	0
$946$	80.5057	0.00276688
$947$	−23454.7	−0.804830	−0.402415	−	0.915457i	$-0.631829\pi$
$947$	−23454.7	−0.804830	−0.402415	+	0.915457i	$0.631829\pi$
$948$	0	0
$949$	27921.1	0.955065
$950$	0	0
$951$	0	0
$952$	−814.799	−0.0277393
$953$	8009.19	0.272238	0.136119	−	0.990692i	$-0.456537\pi$
$953$	8009.19	0.272238	0.136119	+	0.990692i	$0.456537\pi$
$954$	0	0
$955$	0	0
$956$	−21559.8	−0.729386
$957$	0	0
$958$	−1010.96	−0.0340948
$959$	−14918.2	−0.502330
$960$	0	0
$961$	−519.069	−0.0174237
$962$	−413.948	−0.0138734
$963$	0	0
$964$	−34369.8	−1.14832
$965$	0	0
$966$	0	0
$967$	1962.54	0.0652648	0.0326324	−	0.999467i	$-0.489611\pi$
$967$	1962.54	0.0652648	0.0326324	+	0.999467i	$0.489611\pi$
$968$	1483.46	0.0492564
$969$	0	0
$970$	0	0
$971$	52109.6	1.72222	0.861110	−	0.508419i	$-0.169770\pi$
$971$	52109.6	1.72222	0.861110	+	0.508419i	$0.169770\pi$
$972$	0	0
$973$	11841.4	0.390152
$974$	1350.05	0.0444130
$975$	0	0
$976$	−52619.5	−1.72573
$977$	10104.1	0.330868	0.165434	−	0.986221i	$-0.447097\pi$
$977$	10104.1	0.330868	0.165434	+	0.986221i	$0.447097\pi$
$978$	0	0
$979$	−2858.04	−0.0933027
$980$	0	0
$981$	0	0
$982$	882.947	0.0286924
$983$	18109.7	0.587599	0.293799	−	0.955867i	$-0.405080\pi$
$983$	18109.7	0.587599	0.293799	+	0.955867i	$0.405080\pi$
$984$	0	0
$985$	0	0
$986$	−857.473	−0.0276952
$987$	0	0
$988$	−7526.33	−0.242353
$989$	6068.47	0.195112
$990$	0	0
$991$	−48982.5	−1.57011	−0.785055	−	0.619426i	$-0.787366\pi$
$991$	−48982.5	−1.57011	−0.785055	+	0.619426i	$0.787366\pi$
$992$	−2510.76	−0.0803597
$993$	0	0
$994$	350.173	0.0111739
$995$	0	0
$996$	0	0
$997$	−41760.0	−1.32653	−0.663265	−	0.748384i	$-0.730830\pi$
$997$	−41760.0	−1.32653	−0.663265	+	0.748384i	$0.730830\pi$
$998$	−521.813	−0.0165508
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bd.1.2		3
3.2	odd	2		525.4.a.q.1.2		3
5.2	odd	4		315.4.d.a.64.4		6
5.3	odd	4		315.4.d.a.64.3		6
5.4	even	2		1575.4.a.bc.1.2		3
15.2	even	4		105.4.d.a.64.3	✓	6
15.8	even	4		105.4.d.a.64.4	yes	6
15.14	odd	2		525.4.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.d.a.64.3	✓	6	15.2	even	4
105.4.d.a.64.4	yes	6	15.8	even	4
315.4.d.a.64.3		6	5.3	odd	4
315.4.d.a.64.4		6	5.2	odd	4
525.4.a.q.1.2		3	3.2	odd	2
525.4.a.r.1.2		3	15.14	odd	2
1575.4.a.bc.1.2		3	5.4	even	2
1575.4.a.bd.1.2		3	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0765073 q^{2} -7.99415 q^{4} +7.00000 q^{7} -1.22367 q^{8} +O(q^{10})\)	\(q+0.0765073 q^{2} -7.99415 q^{4} +7.00000 q^{7} -1.22367 q^{8} -10.8947 q^{11} -26.5468 q^{13} +0.535551 q^{14} +63.8596 q^{16} +95.1237 q^{17} -35.4649 q^{19} -0.833522 q^{22} -62.8303 q^{23} -2.03102 q^{26} -55.9590 q^{28} -117.823 q^{29} -171.090 q^{31} +14.6751 q^{32} +7.27766 q^{34} +203.813 q^{37} -2.71332 q^{38} +428.705 q^{41} -96.5851 q^{43} +87.0936 q^{44} -4.80698 q^{46} +407.806 q^{47} +49.0000 q^{49} +212.219 q^{52} +380.874 q^{53} -8.56568 q^{56} -9.01429 q^{58} +287.149 q^{59} -823.988 q^{61} -13.0897 q^{62} -509.754 q^{64} +585.549 q^{67} -760.433 q^{68} +653.856 q^{71} -1051.77 q^{73} +15.5932 q^{74} +283.512 q^{76} -76.2627 q^{77} -751.268 q^{79} +32.7991 q^{82} -844.677 q^{83} -7.38946 q^{86} +13.3315 q^{88} +262.334 q^{89} -185.828 q^{91} +502.275 q^{92} +31.2001 q^{94} -814.908 q^{97} +3.74886 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8	\( 3 q + q^{2} + 3 q^{4} + 21 q^{7} + 21 q^{8} + 66 q^{11} - 102 q^{13} + 7 q^{14} - 69 q^{16} + 152 q^{17} - 138 q^{19} - 186 q^{22} - 180 q^{23} + 98 q^{26} + 21 q^{28} - 170 q^{29} - 366 q^{31} - 151 q^{32} - 36 q^{34} - 252 q^{37} - 234 q^{38} + 206 q^{41} + 108 q^{43} + 306 q^{44} - 672 q^{46} - 24 q^{47} + 147 q^{49} - 78 q^{52} + 354 q^{53} + 147 q^{56} + 858 q^{58} + 880 q^{59} - 870 q^{61} - 1366 q^{62} - 813 q^{64} + 96 q^{67} - 512 q^{68} + 1018 q^{71} - 1554 q^{73} - 980 q^{74} - 450 q^{76} + 462 q^{77} - 1620 q^{79} + 1638 q^{82} - 872 q^{83} + 2932 q^{86} + 1326 q^{88} + 1938 q^{89} - 714 q^{91} - 708 q^{92} + 2112 q^{94} - 1878 q^{97} + 49 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 + 21 * q^7 + 21 * q^8 + 66 * q^11 - 102 * q^13 + 7 * q^14 - 69 * q^16 + 152 * q^17 - 138 * q^19 - 186 * q^22 - 180 * q^23 + 98 * q^26 + 21 * q^28 - 170 * q^29 - 366 * q^31 - 151 * q^32 - 36 * q^34 - 252 * q^37 - 234 * q^38 + 206 * q^41 + 108 * q^43 + 306 * q^44 - 672 * q^46 - 24 * q^47 + 147 * q^49 - 78 * q^52 + 354 * q^53 + 147 * q^56 + 858 * q^58 + 880 * q^59 - 870 * q^61 - 1366 * q^62 - 813 * q^64 + 96 * q^67 - 512 * q^68 + 1018 * q^71 - 1554 * q^73 - 980 * q^74 - 450 * q^76 + 462 * q^77 - 1620 * q^79 + 1638 * q^82 - 872 * q^83 + 2932 * q^86 + 1326 * q^88 + 1938 * q^89 - 714 * q^91 - 708 * q^92 + 2112 * q^94 - 1878 * q^97 + 49 * q^98

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