LMFDB - Embedded newform 4012.2.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.18786$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.460784 q^{3} -0.730164 q^{5} +3.58746 q^{7} -2.78768 q^{9} +O(q^{10})$	$q-0.460784 q^{3} -0.730164 q^{5} +3.58746 q^{7} -2.78768 q^{9} +3.02705 q^{11} +0.246566 q^{13} +0.336448 q^{15} +1.00000 q^{17} -3.29765 q^{19} -1.65304 q^{21} -9.34754 q^{23} -4.46686 q^{25} +2.66687 q^{27} -0.632935 q^{29} +1.50261 q^{31} -1.39482 q^{33} -2.61943 q^{35} -6.91272 q^{37} -0.113614 q^{39} -5.28210 q^{41} +2.83103 q^{43} +2.03546 q^{45} +7.15459 q^{47} +5.86984 q^{49} -0.460784 q^{51} -11.4716 q^{53} -2.21024 q^{55} +1.51950 q^{57} -1.00000 q^{59} -2.09602 q^{61} -10.0007 q^{63} -0.180034 q^{65} +6.17171 q^{67} +4.30720 q^{69} +9.38615 q^{71} -2.44514 q^{73} +2.05826 q^{75} +10.8594 q^{77} +7.46383 q^{79} +7.13418 q^{81} -12.8110 q^{83} -0.730164 q^{85} +0.291646 q^{87} -16.0018 q^{89} +0.884545 q^{91} -0.692376 q^{93} +2.40782 q^{95} +6.15215 q^{97} -8.43844 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

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	0
$2$	0	0
$3$	−0.460784	−0.266034	−0.133017	−	0.991114i	$-0.542466\pi$
$3$	−0.460784	−0.266034	−0.133017	+	0.991114i	$0.542466\pi$
$4$	0	0
$5$	−0.730164	−0.326539	−0.163270	−	0.986582i	$-0.552204\pi$
$5$	−0.730164	−0.326539	−0.163270	+	0.986582i	$0.552204\pi$
$6$	0	0
$7$	3.58746	1.35593	0.677965	−	0.735094i	$-0.262862\pi$
$7$	3.58746	1.35593	0.677965	+	0.735094i	$0.262862\pi$
$8$	0	0
$9$	−2.78768	−0.929226
$10$	0	0
$11$	3.02705	0.912689	0.456345	−	0.889803i	$-0.349158\pi$
$11$	3.02705	0.912689	0.456345	+	0.889803i	$0.349158\pi$
$12$	0	0
$13$	0.246566	0.0683851	0.0341926	−	0.999415i	$-0.489114\pi$
$13$	0.246566	0.0683851	0.0341926	+	0.999415i	$0.489114\pi$
$14$	0	0
$15$	0.336448	0.0868704
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.29765	−0.756532	−0.378266	−	0.925697i	$-0.623480\pi$
$19$	−3.29765	−0.756532	−0.378266	+	0.925697i	$0.623480\pi$
$20$	0	0
$21$	−1.65304	−0.360723
$22$	0	0
$23$	−9.34754	−1.94910	−0.974549	−	0.224175i	$-0.928031\pi$
$23$	−9.34754	−1.94910	−0.974549	+	0.224175i	$0.928031\pi$
$24$	0	0
$25$	−4.46686	−0.893372
$26$	0	0
$27$	2.66687	0.513239
$28$	0	0
$29$	−0.632935	−0.117533	−0.0587666	−	0.998272i	$-0.518717\pi$
$29$	−0.632935	−0.117533	−0.0587666	+	0.998272i	$0.518717\pi$
$30$	0	0
$31$	1.50261	0.269876	0.134938	−	0.990854i	$-0.456916\pi$
$31$	1.50261	0.269876	0.134938	+	0.990854i	$0.456916\pi$
$32$	0	0
$33$	−1.39482	−0.242806
$34$	0	0
$35$	−2.61943	−0.442764
$36$	0	0
$37$	−6.91272	−1.13644	−0.568222	−	0.822875i	$-0.692369\pi$
$37$	−6.91272	−1.13644	−0.568222	+	0.822875i	$0.692369\pi$
$38$	0	0
$39$	−0.113614	−0.0181928
$40$	0	0
$41$	−5.28210	−0.824926	−0.412463	−	0.910974i	$-0.635331\pi$
$41$	−5.28210	−0.824926	−0.412463	+	0.910974i	$0.635331\pi$
$42$	0	0
$43$	2.83103	0.431728	0.215864	−	0.976423i	$-0.430743\pi$
$43$	2.83103	0.431728	0.215864	+	0.976423i	$0.430743\pi$
$44$	0	0
$45$	2.03546	0.303429
$46$	0	0
$47$	7.15459	1.04360	0.521802	−	0.853067i	$-0.325260\pi$
$47$	7.15459	1.04360	0.521802	+	0.853067i	$0.325260\pi$
$48$	0	0
$49$	5.86984	0.838549
$50$	0	0
$51$	−0.460784	−0.0645227
$52$	0	0
$53$	−11.4716	−1.57575	−0.787876	−	0.615834i	$-0.788819\pi$
$53$	−11.4716	−1.57575	−0.787876	+	0.615834i	$0.788819\pi$
$54$	0	0
$55$	−2.21024	−0.298029
$56$	0	0
$57$	1.51950	0.201263
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−2.09602	−0.268368	−0.134184	−	0.990956i	$-0.542841\pi$
$61$	−2.09602	−0.268368	−0.134184	+	0.990956i	$0.542841\pi$
$62$	0	0
$63$	−10.0007	−1.25997
$64$	0	0
$65$	−0.180034	−0.0223304
$66$	0	0
$67$	6.17171	0.753995	0.376997	−	0.926214i	$-0.376957\pi$
$67$	6.17171	0.753995	0.376997	+	0.926214i	$0.376957\pi$
$68$	0	0
$69$	4.30720	0.518526
$70$	0	0
$71$	9.38615	1.11393	0.556965	−	0.830536i	$-0.311966\pi$
$71$	9.38615	1.11393	0.556965	+	0.830536i	$0.311966\pi$
$72$	0	0
$73$	−2.44514	−0.286182	−0.143091	−	0.989710i	$-0.545704\pi$
$73$	−2.44514	−0.286182	−0.143091	+	0.989710i	$0.545704\pi$
$74$	0	0
$75$	2.05826	0.237667
$76$	0	0
$77$	10.8594	1.23754
$78$	0	0
$79$	7.46383	0.839746	0.419873	−	0.907583i	$-0.362075\pi$
$79$	7.46383	0.839746	0.419873	+	0.907583i	$0.362075\pi$
$80$	0	0
$81$	7.13418	0.792687
$82$	0	0
$83$	−12.8110	−1.40619	−0.703093	−	0.711097i	$-0.748198\pi$
$83$	−12.8110	−1.40619	−0.703093	+	0.711097i	$0.748198\pi$
$84$	0	0
$85$	−0.730164	−0.0791974
$86$	0	0
$87$	0.291646	0.0312678
$88$	0	0
$89$	−16.0018	−1.69619	−0.848094	−	0.529846i	$-0.822250\pi$
$89$	−16.0018	−1.69619	−0.848094	+	0.529846i	$0.822250\pi$
$90$	0	0
$91$	0.884545	0.0927255
$92$	0	0
$93$	−0.692376	−0.0717961
$94$	0	0
$95$	2.40782	0.247037
$96$	0	0
$97$	6.15215	0.624656	0.312328	−	0.949974i	$-0.398891\pi$
$97$	6.15215	0.624656	0.312328	+	0.949974i	$0.398891\pi$
$98$	0	0
$99$	−8.43844	−0.848095
$100$	0	0
$101$	9.57276	0.952525	0.476262	−	0.879303i	$-0.341991\pi$
$101$	9.57276	0.952525	0.476262	+	0.879303i	$0.341991\pi$
$102$	0	0
$103$	−10.8901	−1.07303	−0.536517	−	0.843889i	$-0.680260\pi$
$103$	−10.8901	−1.07303	−0.536517	+	0.843889i	$0.680260\pi$
$104$	0	0
$105$	1.20699	0.117790
$106$	0	0
$107$	2.25952	0.218436	0.109218	−	0.994018i	$-0.465165\pi$
$107$	2.25952	0.218436	0.109218	+	0.994018i	$0.465165\pi$
$108$	0	0
$109$	−2.83175	−0.271233	−0.135616	−	0.990761i	$-0.543301\pi$
$109$	−2.83175	−0.271233	−0.135616	+	0.990761i	$0.543301\pi$
$110$	0	0
$111$	3.18527	0.302332
$112$	0	0
$113$	1.90955	0.179636	0.0898179	−	0.995958i	$-0.471371\pi$
$113$	1.90955	0.179636	0.0898179	+	0.995958i	$0.471371\pi$
$114$	0	0
$115$	6.82524	0.636457
$116$	0	0
$117$	−0.687347	−0.0635452
$118$	0	0
$119$	3.58746	0.328862
$120$	0	0
$121$	−1.83698	−0.166998
$122$	0	0
$123$	2.43391	0.219458
$124$	0	0
$125$	6.91236	0.618260
$126$	0	0
$127$	−2.41498	−0.214295	−0.107147	−	0.994243i	$-0.534172\pi$
$127$	−2.41498	−0.214295	−0.107147	+	0.994243i	$0.534172\pi$
$128$	0	0
$129$	−1.30449	−0.114854
$130$	0	0
$131$	−9.69389	−0.846959	−0.423480	−	0.905906i	$-0.639191\pi$
$131$	−9.69389	−0.846959	−0.423480	+	0.905906i	$0.639191\pi$
$132$	0	0
$133$	−11.8302	−1.02581
$134$	0	0
$135$	−1.94725	−0.167593
$136$	0	0
$137$	−18.0213	−1.53967	−0.769833	−	0.638245i	$-0.779661\pi$
$137$	−18.0213	−1.53967	−0.769833	+	0.638245i	$0.779661\pi$
$138$	0	0
$139$	9.41812	0.798834	0.399417	−	0.916769i	$-0.369213\pi$
$139$	9.41812	0.798834	0.399417	+	0.916769i	$0.369213\pi$
$140$	0	0
$141$	−3.29672	−0.277634
$142$	0	0
$143$	0.746367	0.0624144
$144$	0	0
$145$	0.462146	0.0383792
$146$	0	0
$147$	−2.70473	−0.223082
$148$	0	0
$149$	3.83474	0.314154	0.157077	−	0.987586i	$-0.449793\pi$
$149$	3.83474	0.314154	0.157077	+	0.987586i	$0.449793\pi$
$150$	0	0
$151$	−10.8793	−0.885343	−0.442671	−	0.896684i	$-0.645969\pi$
$151$	−10.8793	−0.885343	−0.442671	+	0.896684i	$0.645969\pi$
$152$	0	0
$153$	−2.78768	−0.225370
$154$	0	0
$155$	−1.09715	−0.0881250
$156$	0	0
$157$	−3.51455	−0.280492	−0.140246	−	0.990117i	$-0.544789\pi$
$157$	−3.51455	−0.280492	−0.140246	+	0.990117i	$0.544789\pi$
$158$	0	0
$159$	5.28595	0.419203
$160$	0	0
$161$	−33.5339	−2.64284
$162$	0	0
$163$	−9.02497	−0.706890	−0.353445	−	0.935455i	$-0.614990\pi$
$163$	−9.02497	−0.706890	−0.353445	+	0.935455i	$0.614990\pi$
$164$	0	0
$165$	1.01844	0.0792857
$166$	0	0
$167$	11.5178	0.891274	0.445637	−	0.895214i	$-0.352977\pi$
$167$	11.5178	0.891274	0.445637	+	0.895214i	$0.352977\pi$
$168$	0	0
$169$	−12.9392	−0.995323
$170$	0	0
$171$	9.19278	0.702989
$172$	0	0
$173$	−21.1297	−1.60646	−0.803231	−	0.595668i	$-0.796888\pi$
$173$	−21.1297	−1.60646	−0.803231	+	0.595668i	$0.796888\pi$
$174$	0	0
$175$	−16.0247	−1.21135
$176$	0	0
$177$	0.460784	0.0346346
$178$	0	0
$179$	−0.801336	−0.0598947	−0.0299473	−	0.999551i	$-0.509534\pi$
$179$	−0.801336	−0.0598947	−0.0299473	+	0.999551i	$0.509534\pi$
$180$	0	0
$181$	−2.06063	−0.153165	−0.0765826	−	0.997063i	$-0.524401\pi$
$181$	−2.06063	−0.153165	−0.0765826	+	0.997063i	$0.524401\pi$
$182$	0	0
$183$	0.965813	0.0713950
$184$	0	0
$185$	5.04742	0.371093
$186$	0	0
$187$	3.02705	0.221360
$188$	0	0
$189$	9.56728	0.695917
$190$	0	0
$191$	−7.09740	−0.513550	−0.256775	−	0.966471i	$-0.582660\pi$
$191$	−7.09740	−0.513550	−0.256775	+	0.966471i	$0.582660\pi$
$192$	0	0
$193$	25.4816	1.83421	0.917103	−	0.398651i	$-0.130521\pi$
$193$	25.4816	1.83421	0.917103	+	0.398651i	$0.130521\pi$
$194$	0	0
$195$	0.0829566	0.00594064
$196$	0	0
$197$	−11.9188	−0.849181	−0.424590	−	0.905386i	$-0.639582\pi$
$197$	−11.9188	−0.849181	−0.424590	+	0.905386i	$0.639582\pi$
$198$	0	0
$199$	−12.0007	−0.850710	−0.425355	−	0.905027i	$-0.639851\pi$
$199$	−12.0007	−0.850710	−0.425355	+	0.905027i	$0.639851\pi$
$200$	0	0
$201$	−2.84383	−0.200588
$202$	0	0
$203$	−2.27063	−0.159367
$204$	0	0
$205$	3.85680	0.269371
$206$	0	0
$207$	26.0579	1.81115
$208$	0	0
$209$	−9.98214	−0.690479
$210$	0	0
$211$	−17.4785	−1.20327	−0.601636	−	0.798770i	$-0.705484\pi$
$211$	−17.4785	−1.20327	−0.601636	+	0.798770i	$0.705484\pi$
$212$	0	0
$213$	−4.32499	−0.296343
$214$	0	0
$215$	−2.06712	−0.140976
$216$	0	0
$217$	5.39053	0.365933
$218$	0	0
$219$	1.12668	0.0761340
$220$	0	0
$221$	0.246566	0.0165858
$222$	0	0
$223$	−1.15686	−0.0774692	−0.0387346	−	0.999250i	$-0.512333\pi$
$223$	−1.15686	−0.0774692	−0.0387346	+	0.999250i	$0.512333\pi$
$224$	0	0
$225$	12.4522	0.830145
$226$	0	0
$227$	−8.30390	−0.551149	−0.275575	−	0.961280i	$-0.588868\pi$
$227$	−8.30390	−0.551149	−0.275575	+	0.961280i	$0.588868\pi$
$228$	0	0
$229$	−1.89289	−0.125086	−0.0625429	−	0.998042i	$-0.519921\pi$
$229$	−1.89289	−0.125086	−0.0625429	+	0.998042i	$0.519921\pi$
$230$	0	0
$231$	−5.00384	−0.329228
$232$	0	0
$233$	19.9586	1.30753	0.653765	−	0.756698i	$-0.273189\pi$
$233$	19.9586	1.30753	0.653765	+	0.756698i	$0.273189\pi$
$234$	0	0
$235$	−5.22402	−0.340777
$236$	0	0
$237$	−3.43921	−0.223401
$238$	0	0
$239$	−14.8152	−0.958317	−0.479159	−	0.877728i	$-0.659058\pi$
$239$	−14.8152	−0.958317	−0.479159	+	0.877728i	$0.659058\pi$
$240$	0	0
$241$	12.4606	0.802659	0.401329	−	0.915934i	$-0.368548\pi$
$241$	12.4606	0.802659	0.401329	+	0.915934i	$0.368548\pi$
$242$	0	0
$243$	−11.2879	−0.724121
$244$	0	0
$245$	−4.28594	−0.273819
$246$	0	0
$247$	−0.813088	−0.0517355
$248$	0	0
$249$	5.90309	0.374093
$250$	0	0
$251$	9.01178	0.568819	0.284409	−	0.958703i	$-0.408203\pi$
$251$	9.01178	0.568819	0.284409	+	0.958703i	$0.408203\pi$
$252$	0	0
$253$	−28.2955	−1.77892
$254$	0	0
$255$	0.336448	0.0210692
$256$	0	0
$257$	−12.6924	−0.791733	−0.395866	−	0.918308i	$-0.629556\pi$
$257$	−12.6924	−0.791733	−0.395866	+	0.918308i	$0.629556\pi$
$258$	0	0
$259$	−24.7991	−1.54094
$260$	0	0
$261$	1.76442	0.109215
$262$	0	0
$263$	−4.11094	−0.253492	−0.126746	−	0.991935i	$-0.540453\pi$
$263$	−4.11094	−0.253492	−0.126746	+	0.991935i	$0.540453\pi$
$264$	0	0
$265$	8.37618	0.514544
$266$	0	0
$267$	7.37338	0.451243
$268$	0	0
$269$	0.418438	0.0255126	0.0127563	−	0.999919i	$-0.495939\pi$
$269$	0.418438	0.0255126	0.0127563	+	0.999919i	$0.495939\pi$
$270$	0	0
$271$	0.0474225	0.00288071	0.00144036	−	0.999999i	$-0.499542\pi$
$271$	0.0474225	0.00288071	0.00144036	+	0.999999i	$0.499542\pi$
$272$	0	0
$273$	−0.407584	−0.0246681
$274$	0	0
$275$	−13.5214	−0.815371
$276$	0	0
$277$	8.65391	0.519963	0.259982	−	0.965614i	$-0.416284\pi$
$277$	8.65391	0.519963	0.259982	+	0.965614i	$0.416284\pi$
$278$	0	0
$279$	−4.18878	−0.250776
$280$	0	0
$281$	−25.9356	−1.54719	−0.773593	−	0.633683i	$-0.781542\pi$
$281$	−25.9356	−1.54719	−0.773593	+	0.633683i	$0.781542\pi$
$282$	0	0
$283$	19.4240	1.15463	0.577317	−	0.816520i	$-0.304100\pi$
$283$	19.4240	1.15463	0.577317	+	0.816520i	$0.304100\pi$
$284$	0	0
$285$	−1.10949	−0.0657203
$286$	0	0
$287$	−18.9493	−1.11854
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.83481	−0.166180
$292$	0	0
$293$	14.9316	0.872316	0.436158	−	0.899870i	$-0.356339\pi$
$293$	14.9316	0.872316	0.436158	+	0.899870i	$0.356339\pi$
$294$	0	0
$295$	0.730164	0.0425118
$296$	0	0
$297$	8.07274	0.468428
$298$	0	0
$299$	−2.30479	−0.133289
$300$	0	0
$301$	10.1562	0.585394
$302$	0	0
$303$	−4.41097	−0.253404
$304$	0	0
$305$	1.53044	0.0876327
$306$	0	0
$307$	23.0048	1.31295	0.656476	−	0.754347i	$-0.272046\pi$
$307$	23.0048	1.31295	0.656476	+	0.754347i	$0.272046\pi$
$308$	0	0
$309$	5.01799	0.285463
$310$	0	0
$311$	−7.73030	−0.438345	−0.219173	−	0.975686i	$-0.570336\pi$
$311$	−7.73030	−0.438345	−0.219173	+	0.975686i	$0.570336\pi$
$312$	0	0
$313$	0.755472	0.0427018	0.0213509	−	0.999772i	$-0.493203\pi$
$313$	0.755472	0.0427018	0.0213509	+	0.999772i	$0.493203\pi$
$314$	0	0
$315$	7.30213	0.411428
$316$	0	0
$317$	−3.47493	−0.195172	−0.0975858	−	0.995227i	$-0.531112\pi$
$317$	−3.47493	−0.195172	−0.0975858	+	0.995227i	$0.531112\pi$
$318$	0	0
$319$	−1.91593	−0.107271
$320$	0	0
$321$	−1.04115	−0.0581114
$322$	0	0
$323$	−3.29765	−0.183486
$324$	0	0
$325$	−1.10138	−0.0610934
$326$	0	0
$327$	1.30483	0.0721571
$328$	0	0
$329$	25.6668	1.41505
$330$	0	0
$331$	−9.88263	−0.543199	−0.271599	−	0.962410i	$-0.587553\pi$
$331$	−9.88263	−0.543199	−0.271599	+	0.962410i	$0.587553\pi$
$332$	0	0
$333$	19.2704	1.05601
$334$	0	0
$335$	−4.50636	−0.246209
$336$	0	0
$337$	18.9833	1.03409	0.517043	−	0.855959i	$-0.327033\pi$
$337$	18.9833	1.03409	0.517043	+	0.855959i	$0.327033\pi$
$338$	0	0
$339$	−0.879892	−0.0477892
$340$	0	0
$341$	4.54846	0.246313
$342$	0	0
$343$	−4.05440	−0.218917
$344$	0	0
$345$	−3.14496	−0.169319
$346$	0	0
$347$	−2.95522	−0.158645	−0.0793223	−	0.996849i	$-0.525276\pi$
$347$	−2.95522	−0.158645	−0.0793223	+	0.996849i	$0.525276\pi$
$348$	0	0
$349$	23.6175	1.26421	0.632107	−	0.774881i	$-0.282190\pi$
$349$	23.6175	1.26421	0.632107	+	0.774881i	$0.282190\pi$
$350$	0	0
$351$	0.657559	0.0350979
$352$	0	0
$353$	12.1935	0.648997	0.324499	−	0.945886i	$-0.394804\pi$
$353$	12.1935	0.648997	0.324499	+	0.945886i	$0.394804\pi$
$354$	0	0
$355$	−6.85342	−0.363742
$356$	0	0
$357$	−1.65304	−0.0874883
$358$	0	0
$359$	−18.6189	−0.982668	−0.491334	−	0.870971i	$-0.663490\pi$
$359$	−18.6189	−0.982668	−0.491334	+	0.870971i	$0.663490\pi$
$360$	0	0
$361$	−8.12552	−0.427659
$362$	0	0
$363$	0.846451	0.0444272
$364$	0	0
$365$	1.78535	0.0934496
$366$	0	0
$367$	−12.9392	−0.675423	−0.337712	−	0.941250i	$-0.609653\pi$
$367$	−12.9392	−0.675423	−0.337712	+	0.941250i	$0.609653\pi$
$368$	0	0
$369$	14.7248	0.766543
$370$	0	0
$371$	−41.1540	−2.13661
$372$	0	0
$373$	−14.3949	−0.745339	−0.372669	−	0.927964i	$-0.621557\pi$
$373$	−14.3949	−0.745339	−0.372669	+	0.927964i	$0.621557\pi$
$374$	0	0
$375$	−3.18510	−0.164478
$376$	0	0
$377$	−0.156060	−0.00803752
$378$	0	0
$379$	−5.49864	−0.282446	−0.141223	−	0.989978i	$-0.545103\pi$
$379$	−5.49864	−0.282446	−0.141223	+	0.989978i	$0.545103\pi$
$380$	0	0
$381$	1.11278	0.0570097
$382$	0	0
$383$	−34.4463	−1.76013	−0.880063	−	0.474858i	$-0.842500\pi$
$383$	−34.4463	−1.76013	−0.880063	+	0.474858i	$0.842500\pi$
$384$	0	0
$385$	−7.92914	−0.404106
$386$	0	0
$387$	−7.89200	−0.401173
$388$	0	0
$389$	−23.4972	−1.19136	−0.595678	−	0.803223i	$-0.703117\pi$
$389$	−23.4972	−1.19136	−0.595678	+	0.803223i	$0.703117\pi$
$390$	0	0
$391$	−9.34754	−0.472726
$392$	0	0
$393$	4.46679	0.225320
$394$	0	0
$395$	−5.44981	−0.274210
$396$	0	0
$397$	−26.7637	−1.34323	−0.671615	−	0.740901i	$-0.734399\pi$
$397$	−26.7637	−1.34323	−0.671615	+	0.740901i	$0.734399\pi$
$398$	0	0
$399$	5.45115	0.272899
$400$	0	0
$401$	15.5833	0.778193	0.389096	−	0.921197i	$-0.372787\pi$
$401$	15.5833	0.778193	0.389096	+	0.921197i	$0.372787\pi$
$402$	0	0
$403$	0.370491	0.0184555
$404$	0	0
$405$	−5.20912	−0.258843
$406$	0	0
$407$	−20.9251	−1.03722
$408$	0	0
$409$	8.01222	0.396179	0.198089	−	0.980184i	$-0.436526\pi$
$409$	8.01222	0.396179	0.198089	+	0.980184i	$0.436526\pi$
$410$	0	0
$411$	8.30394	0.409603
$412$	0	0
$413$	−3.58746	−0.176527
$414$	0	0
$415$	9.35411	0.459175
$416$	0	0
$417$	−4.33972	−0.212517
$418$	0	0
$419$	8.81791	0.430783	0.215392	−	0.976528i	$-0.430897\pi$
$419$	8.81791	0.430783	0.215392	+	0.976528i	$0.430897\pi$
$420$	0	0
$421$	36.9729	1.80195	0.900976	−	0.433870i	$-0.142852\pi$
$421$	36.9729	1.80195	0.900976	+	0.433870i	$0.142852\pi$
$422$	0	0
$423$	−19.9447	−0.969744
$424$	0	0
$425$	−4.46686	−0.216675
$426$	0	0
$427$	−7.51939	−0.363889
$428$	0	0
$429$	−0.343914	−0.0166043
$430$	0	0
$431$	−15.9266	−0.767159	−0.383580	−	0.923508i	$-0.625309\pi$
$431$	−15.9266	−0.767159	−0.383580	+	0.923508i	$0.625309\pi$
$432$	0	0
$433$	3.41602	0.164164	0.0820818	−	0.996626i	$-0.473843\pi$
$433$	3.41602	0.164164	0.0820818	+	0.996626i	$0.473843\pi$
$434$	0	0
$435$	−0.212950	−0.0102101
$436$	0	0
$437$	30.8249	1.47455
$438$	0	0
$439$	−15.5643	−0.742844	−0.371422	−	0.928464i	$-0.621130\pi$
$439$	−15.5643	−0.742844	−0.371422	+	0.928464i	$0.621130\pi$
$440$	0	0
$441$	−16.3632	−0.779201
$442$	0	0
$443$	−23.4245	−1.11293	−0.556465	−	0.830871i	$-0.687843\pi$
$443$	−23.4245	−1.11293	−0.556465	+	0.830871i	$0.687843\pi$
$444$	0	0
$445$	11.6839	0.553872
$446$	0	0
$447$	−1.76699	−0.0835755
$448$	0	0
$449$	−12.8925	−0.608437	−0.304218	−	0.952602i	$-0.598395\pi$
$449$	−12.8925	−0.608437	−0.304218	+	0.952602i	$0.598395\pi$
$450$	0	0
$451$	−15.9892	−0.752901
$452$	0	0
$453$	5.01299	0.235531
$454$	0	0
$455$	−0.645863	−0.0302785
$456$	0	0
$457$	−11.3779	−0.532237	−0.266118	−	0.963940i	$-0.585741\pi$
$457$	−11.3779	−0.532237	−0.266118	+	0.963940i	$0.585741\pi$
$458$	0	0
$459$	2.66687	0.124479
$460$	0	0
$461$	17.5554	0.817635	0.408817	−	0.912616i	$-0.365941\pi$
$461$	17.5554	0.817635	0.408817	+	0.912616i	$0.365941\pi$
$462$	0	0
$463$	−7.72624	−0.359069	−0.179534	−	0.983752i	$-0.557459\pi$
$463$	−7.72624	−0.359069	−0.179534	+	0.983752i	$0.557459\pi$
$464$	0	0
$465$	0.505548	0.0234442
$466$	0	0
$467$	38.0061	1.75871	0.879356	−	0.476165i	$-0.157974\pi$
$467$	38.0061	1.75871	0.879356	+	0.476165i	$0.157974\pi$
$468$	0	0
$469$	22.1407	1.02236
$470$	0	0
$471$	1.61945	0.0746202
$472$	0	0
$473$	8.56967	0.394034
$474$	0	0
$475$	14.7301	0.675865
$476$	0	0
$477$	31.9793	1.46423
$478$	0	0
$479$	−19.1244	−0.873817	−0.436908	−	0.899506i	$-0.643927\pi$
$479$	−19.1244	−0.873817	−0.436908	+	0.899506i	$0.643927\pi$
$480$	0	0
$481$	−1.70444	−0.0777159
$482$	0	0
$483$	15.4519	0.703085
$484$	0	0
$485$	−4.49208	−0.203975
$486$	0	0
$487$	−0.573377	−0.0259822	−0.0129911	−	0.999916i	$-0.504135\pi$
$487$	−0.573377	−0.0259822	−0.0129911	+	0.999916i	$0.504135\pi$
$488$	0	0
$489$	4.15856	0.188057
$490$	0	0
$491$	−20.7805	−0.937809	−0.468905	−	0.883249i	$-0.655351\pi$
$491$	−20.7805	−0.937809	−0.468905	+	0.883249i	$0.655351\pi$
$492$	0	0
$493$	−0.632935	−0.0285060
$494$	0	0
$495$	6.16144	0.276936
$496$	0	0
$497$	33.6724	1.51041
$498$	0	0
$499$	21.8442	0.977881	0.488941	−	0.872317i	$-0.337384\pi$
$499$	21.8442	0.977881	0.488941	+	0.872317i	$0.337384\pi$
$500$	0	0
$501$	−5.30722	−0.237109
$502$	0	0
$503$	−1.74549	−0.0778274	−0.0389137	−	0.999243i	$-0.512390\pi$
$503$	−1.74549	−0.0778274	−0.0389137	+	0.999243i	$0.512390\pi$
$504$	0	0
$505$	−6.98968	−0.311037
$506$	0	0
$507$	5.96218	0.264790
$508$	0	0
$509$	32.6471	1.44706	0.723528	−	0.690295i	$-0.242519\pi$
$509$	32.6471	1.44706	0.723528	+	0.690295i	$0.242519\pi$
$510$	0	0
$511$	−8.77183	−0.388043
$512$	0	0
$513$	−8.79439	−0.388282
$514$	0	0
$515$	7.95156	0.350388
$516$	0	0
$517$	21.6573	0.952486
$518$	0	0
$519$	9.73623	0.427373
$520$	0	0
$521$	−17.8552	−0.782249	−0.391125	−	0.920338i	$-0.627914\pi$
$521$	−17.8552	−0.782249	−0.391125	+	0.920338i	$0.627914\pi$
$522$	0	0
$523$	31.4320	1.37443	0.687213	−	0.726456i	$-0.258834\pi$
$523$	31.4320	1.37443	0.687213	+	0.726456i	$0.258834\pi$
$524$	0	0
$525$	7.38391	0.322260
$526$	0	0
$527$	1.50261	0.0654545
$528$	0	0
$529$	64.3766	2.79898
$530$	0	0
$531$	2.78768	0.120975
$532$	0	0
$533$	−1.30239	−0.0564127
$534$	0	0
$535$	−1.64982	−0.0713280
$536$	0	0
$537$	0.369243	0.0159340
$538$	0	0
$539$	17.7683	0.765334
$540$	0	0
$541$	25.4853	1.09570	0.547850	−	0.836577i	$-0.315447\pi$
$541$	25.4853	1.09570	0.547850	+	0.836577i	$0.315447\pi$
$542$	0	0
$543$	0.949504	0.0407471
$544$	0	0
$545$	2.06764	0.0885681
$546$	0	0
$547$	−10.8260	−0.462885	−0.231443	−	0.972849i	$-0.574345\pi$
$547$	−10.8260	−0.462885	−0.231443	+	0.972849i	$0.574345\pi$
$548$	0	0
$549$	5.84303	0.249375
$550$	0	0
$551$	2.08720	0.0889176
$552$	0	0
$553$	26.7761	1.13864
$554$	0	0
$555$	−2.32577	−0.0987234
$556$	0	0
$557$	35.2776	1.49476	0.747380	−	0.664397i	$-0.231312\pi$
$557$	35.2776	1.49476	0.747380	+	0.664397i	$0.231312\pi$
$558$	0	0
$559$	0.698036	0.0295238
$560$	0	0
$561$	−1.39482	−0.0588891
$562$	0	0
$563$	−21.8279	−0.919935	−0.459967	−	0.887936i	$-0.652139\pi$
$563$	−21.8279	−0.919935	−0.459967	+	0.887936i	$0.652139\pi$
$564$	0	0
$565$	−1.39429	−0.0586581
$566$	0	0
$567$	25.5936	1.07483
$568$	0	0
$569$	−42.7780	−1.79335	−0.896673	−	0.442694i	$-0.854023\pi$
$569$	−42.7780	−1.79335	−0.896673	+	0.442694i	$0.854023\pi$
$570$	0	0
$571$	−2.13885	−0.0895081	−0.0447540	−	0.998998i	$-0.514250\pi$
$571$	−2.13885	−0.0895081	−0.0447540	+	0.998998i	$0.514250\pi$
$572$	0	0
$573$	3.27037	0.136622
$574$	0	0
$575$	41.7542	1.74127
$576$	0	0
$577$	−20.6451	−0.859466	−0.429733	−	0.902956i	$-0.641392\pi$
$577$	−20.6451	−0.859466	−0.429733	+	0.902956i	$0.641392\pi$
$578$	0	0
$579$	−11.7415	−0.487961
$580$	0	0
$581$	−45.9588	−1.90669
$582$	0	0
$583$	−34.7252	−1.43817
$584$	0	0
$585$	0.501876	0.0207500
$586$	0	0
$587$	2.40288	0.0991775	0.0495887	−	0.998770i	$-0.484209\pi$
$587$	2.40288	0.0991775	0.0495887	+	0.998770i	$0.484209\pi$
$588$	0	0
$589$	−4.95506	−0.204170
$590$	0	0
$591$	5.49200	0.225911
$592$	0	0
$593$	−10.2998	−0.422961	−0.211480	−	0.977382i	$-0.567828\pi$
$593$	−10.2998	−0.422961	−0.211480	+	0.977382i	$0.567828\pi$
$594$	0	0
$595$	−2.61943	−0.107386
$596$	0	0
$597$	5.52975	0.226318
$598$	0	0
$599$	38.2784	1.56401	0.782006	−	0.623271i	$-0.214197\pi$
$599$	38.2784	1.56401	0.782006	+	0.623271i	$0.214197\pi$
$600$	0	0
$601$	24.5870	1.00292	0.501462	−	0.865180i	$-0.332796\pi$
$601$	24.5870	1.00292	0.501462	+	0.865180i	$0.332796\pi$
$602$	0	0
$603$	−17.2047	−0.700632
$604$	0	0
$605$	1.34130	0.0545314
$606$	0	0
$607$	7.35623	0.298580	0.149290	−	0.988793i	$-0.452301\pi$
$607$	7.35623	0.298580	0.149290	+	0.988793i	$0.452301\pi$
$608$	0	0
$609$	1.04627	0.0423969
$610$	0	0
$611$	1.76408	0.0713670
$612$	0	0
$613$	17.0629	0.689164	0.344582	−	0.938756i	$-0.388021\pi$
$613$	17.0629	0.689164	0.344582	+	0.938756i	$0.388021\pi$
$614$	0	0
$615$	−1.77715	−0.0716617
$616$	0	0
$617$	20.7525	0.835465	0.417732	−	0.908570i	$-0.362825\pi$
$617$	20.7525	0.835465	0.417732	+	0.908570i	$0.362825\pi$
$618$	0	0
$619$	40.1578	1.61408	0.807039	−	0.590498i	$-0.201069\pi$
$619$	40.1578	1.61408	0.807039	+	0.590498i	$0.201069\pi$
$620$	0	0
$621$	−24.9287	−1.00035
$622$	0	0
$623$	−57.4058	−2.29991
$624$	0	0
$625$	17.2872	0.691486
$626$	0	0
$627$	4.59961	0.183691
$628$	0	0
$629$	−6.91272	−0.275628
$630$	0	0
$631$	−17.1145	−0.681317	−0.340658	−	0.940187i	$-0.610650\pi$
$631$	−17.1145	−0.681317	−0.340658	+	0.940187i	$0.610650\pi$
$632$	0	0
$633$	8.05383	0.320111
$634$	0	0
$635$	1.76333	0.0699756
$636$	0	0
$637$	1.44730	0.0573443
$638$	0	0
$639$	−26.1656	−1.03509
$640$	0	0
$641$	−1.23231	−0.0486734	−0.0243367	−	0.999704i	$-0.507747\pi$
$641$	−1.23231	−0.0486734	−0.0243367	+	0.999704i	$0.507747\pi$
$642$	0	0
$643$	10.3579	0.408477	0.204238	−	0.978921i	$-0.434528\pi$
$643$	10.3579	0.408477	0.204238	+	0.978921i	$0.434528\pi$
$644$	0	0
$645$	0.952494	0.0375044
$646$	0	0
$647$	28.2491	1.11059	0.555293	−	0.831655i	$-0.312606\pi$
$647$	28.2491	1.11059	0.555293	+	0.831655i	$0.312606\pi$
$648$	0	0
$649$	−3.02705	−0.118822
$650$	0	0
$651$	−2.48387	−0.0973505
$652$	0	0
$653$	−6.27302	−0.245482	−0.122741	−	0.992439i	$-0.539168\pi$
$653$	−6.27302	−0.245482	−0.122741	+	0.992439i	$0.539168\pi$
$654$	0	0
$655$	7.07813	0.276565
$656$	0	0
$657$	6.81626	0.265928
$658$	0	0
$659$	26.2529	1.02267	0.511333	−	0.859382i	$-0.329152\pi$
$659$	26.2529	1.02267	0.511333	+	0.859382i	$0.329152\pi$
$660$	0	0
$661$	16.2578	0.632354	0.316177	−	0.948700i	$-0.397601\pi$
$661$	16.2578	0.632354	0.316177	+	0.948700i	$0.397601\pi$
$662$	0	0
$663$	−0.113614	−0.00441239
$664$	0	0
$665$	8.63795	0.334965
$666$	0	0
$667$	5.91639	0.229084
$668$	0	0
$669$	0.533064	0.0206094
$670$	0	0
$671$	−6.34476	−0.244937
$672$	0	0
$673$	5.56894	0.214667	0.107333	−	0.994223i	$-0.465769\pi$
$673$	5.56894	0.214667	0.107333	+	0.994223i	$0.465769\pi$
$674$	0	0
$675$	−11.9125	−0.458514
$676$	0	0
$677$	−16.9781	−0.652520	−0.326260	−	0.945280i	$-0.605788\pi$
$677$	−16.9781	−0.652520	−0.326260	+	0.945280i	$0.605788\pi$
$678$	0	0
$679$	22.0706	0.846991
$680$	0	0
$681$	3.82630	0.146624
$682$	0	0
$683$	−31.0449	−1.18790	−0.593950	−	0.804502i	$-0.702432\pi$
$683$	−31.0449	−1.18790	−0.593950	+	0.804502i	$0.702432\pi$
$684$	0	0
$685$	13.1585	0.502761
$686$	0	0
$687$	0.872214	0.0332770
$688$	0	0
$689$	−2.82852	−0.107758
$690$	0	0
$691$	−11.9628	−0.455088	−0.227544	−	0.973768i	$-0.573069\pi$
$691$	−11.9628	−0.455088	−0.227544	+	0.973768i	$0.573069\pi$
$692$	0	0
$693$	−30.2725	−1.14996
$694$	0	0
$695$	−6.87676	−0.260851
$696$	0	0
$697$	−5.28210	−0.200074
$698$	0	0
$699$	−9.19659	−0.347847
$700$	0	0
$701$	13.0871	0.494293	0.247147	−	0.968978i	$-0.420507\pi$
$701$	13.0871	0.494293	0.247147	+	0.968978i	$0.420507\pi$
$702$	0	0
$703$	22.7957	0.859756
$704$	0	0
$705$	2.40714	0.0906583
$706$	0	0
$707$	34.3418	1.29156
$708$	0	0
$709$	6.13830	0.230529	0.115264	−	0.993335i	$-0.463228\pi$
$709$	6.13830	0.230529	0.115264	+	0.993335i	$0.463228\pi$
$710$	0	0
$711$	−20.8067	−0.780314
$712$	0	0
$713$	−14.0457	−0.526014
$714$	0	0
$715$	−0.544970	−0.0203807
$716$	0	0
$717$	6.82662	0.254945
$718$	0	0
$719$	21.6983	0.809211	0.404606	−	0.914491i	$-0.367409\pi$
$719$	21.6983	0.809211	0.404606	+	0.914491i	$0.367409\pi$
$720$	0	0
$721$	−39.0678	−1.45496
$722$	0	0
$723$	−5.74165	−0.213534
$724$	0	0
$725$	2.82723	0.105001
$726$	0	0
$727$	19.3740	0.718542	0.359271	−	0.933233i	$-0.383025\pi$
$727$	19.3740	0.718542	0.359271	+	0.933233i	$0.383025\pi$
$728$	0	0
$729$	−16.2013	−0.600047
$730$	0	0
$731$	2.83103	0.104709
$732$	0	0
$733$	15.0353	0.555342	0.277671	−	0.960676i	$-0.410437\pi$
$733$	15.0353	0.555342	0.277671	+	0.960676i	$0.410437\pi$
$734$	0	0
$735$	1.97489	0.0728451
$736$	0	0
$737$	18.6821	0.688163
$738$	0	0
$739$	−18.9931	−0.698671	−0.349336	−	0.936998i	$-0.613593\pi$
$739$	−18.9931	−0.698671	−0.349336	+	0.936998i	$0.613593\pi$
$740$	0	0
$741$	0.374658	0.0137634
$742$	0	0
$743$	−34.0861	−1.25050	−0.625249	−	0.780425i	$-0.715003\pi$
$743$	−34.0861	−1.25050	−0.625249	+	0.780425i	$0.715003\pi$
$744$	0	0
$745$	−2.79999	−0.102584
$746$	0	0
$747$	35.7129	1.30667
$748$	0	0
$749$	8.10594	0.296185
$750$	0	0
$751$	25.5082	0.930807	0.465404	−	0.885099i	$-0.345909\pi$
$751$	25.5082	0.930807	0.465404	+	0.885099i	$0.345909\pi$
$752$	0	0
$753$	−4.15248	−0.151325
$754$	0	0
$755$	7.94365	0.289099
$756$	0	0
$757$	0.0372384	0.00135345	0.000676727	−	1.00000i	$-0.499785\pi$
$757$	0.0372384	0.00135345	0.000676727		1.00000i	$0.499785\pi$
$758$	0	0
$759$	13.0381	0.473253
$760$	0	0
$761$	−26.2584	−0.951867	−0.475933	−	0.879481i	$-0.657890\pi$
$761$	−26.2584	−0.951867	−0.475933	+	0.879481i	$0.657890\pi$
$762$	0	0
$763$	−10.1588	−0.367773
$764$	0	0
$765$	2.03546	0.0735922
$766$	0	0
$767$	−0.246566	−0.00890299
$768$	0	0
$769$	−40.8728	−1.47391	−0.736956	−	0.675940i	$-0.763738\pi$
$769$	−40.8728	−1.47391	−0.736956	+	0.675940i	$0.763738\pi$
$770$	0	0
$771$	5.84847	0.210628
$772$	0	0
$773$	40.5570	1.45874	0.729368	−	0.684122i	$-0.239814\pi$
$773$	40.5570	1.45874	0.729368	+	0.684122i	$0.239814\pi$
$774$	0	0
$775$	−6.71193	−0.241100
$776$	0	0
$777$	11.4270	0.409942
$778$	0	0
$779$	17.4185	0.624083
$780$	0	0
$781$	28.4123	1.01667
$782$	0	0
$783$	−1.68796	−0.0603226
$784$	0	0
$785$	2.56620	0.0915915
$786$	0	0
$787$	−2.18689	−0.0779541	−0.0389771	−	0.999240i	$-0.512410\pi$
$787$	−2.18689	−0.0779541	−0.0389771	+	0.999240i	$0.512410\pi$
$788$	0	0
$789$	1.89426	0.0674373
$790$	0	0
$791$	6.85044	0.243574
$792$	0	0
$793$	−0.516808	−0.0183524
$794$	0	0
$795$	−3.85961	−0.136886
$796$	0	0
$797$	−6.26665	−0.221976	−0.110988	−	0.993822i	$-0.535402\pi$
$797$	−6.26665	−0.221976	−0.110988	+	0.993822i	$0.535402\pi$
$798$	0	0
$799$	7.15459	0.253111
$800$	0	0
$801$	44.6079	1.57614
$802$	0	0
$803$	−7.40155	−0.261195
$804$	0	0
$805$	24.4852	0.862991
$806$	0	0
$807$	−0.192809	−0.00678721
$808$	0	0
$809$	−14.1334	−0.496903	−0.248451	−	0.968644i	$-0.579922\pi$
$809$	−14.1334	−0.496903	−0.248451	+	0.968644i	$0.579922\pi$
$810$	0	0
$811$	−10.3432	−0.363200	−0.181600	−	0.983372i	$-0.558128\pi$
$811$	−10.3432	−0.363200	−0.181600	+	0.983372i	$0.558128\pi$
$812$	0	0
$813$	−0.0218515	−0.000766366 0
$814$	0	0
$815$	6.58970	0.230827
$816$	0	0
$817$	−9.33574	−0.326616
$818$	0	0
$819$	−2.46583	−0.0861630
$820$	0	0
$821$	6.63288	0.231489	0.115744	−	0.993279i	$-0.463075\pi$
$821$	6.63288	0.231489	0.115744	+	0.993279i	$0.463075\pi$
$822$	0	0
$823$	21.3078	0.742743	0.371372	−	0.928484i	$-0.378888\pi$
$823$	21.3078	0.742743	0.371372	+	0.928484i	$0.378888\pi$
$824$	0	0
$825$	6.23045	0.216916
$826$	0	0
$827$	24.1096	0.838374	0.419187	−	0.907900i	$-0.362315\pi$
$827$	24.1096	0.838374	0.419187	+	0.907900i	$0.362315\pi$
$828$	0	0
$829$	40.7139	1.41405	0.707027	−	0.707187i	$-0.250036\pi$
$829$	40.7139	1.41405	0.707027	+	0.707187i	$0.250036\pi$
$830$	0	0
$831$	−3.98758	−0.138328
$832$	0	0
$833$	5.86984	0.203378
$834$	0	0
$835$	−8.40988	−0.291036
$836$	0	0
$837$	4.00725	0.138511
$838$	0	0
$839$	−25.9413	−0.895595	−0.447797	−	0.894135i	$-0.647791\pi$
$839$	−25.9413	−0.895595	−0.447797	+	0.894135i	$0.647791\pi$
$840$	0	0
$841$	−28.5994	−0.986186
$842$	0	0
$843$	11.9507	0.411604
$844$	0	0
$845$	9.44774	0.325012
$846$	0	0
$847$	−6.59009	−0.226438
$848$	0	0
$849$	−8.95025	−0.307172
$850$	0	0
$851$	64.6170	2.21504
$852$	0	0
$853$	51.2128	1.75349	0.876746	−	0.480953i	$-0.159709\pi$
$853$	51.2128	1.75349	0.876746	+	0.480953i	$0.159709\pi$
$854$	0	0
$855$	−6.71223	−0.229553
$856$	0	0
$857$	−28.7595	−0.982405	−0.491203	−	0.871045i	$-0.663443\pi$
$857$	−28.7595	−0.982405	−0.491203	+	0.871045i	$0.663443\pi$
$858$	0	0
$859$	19.8391	0.676903	0.338451	−	0.940984i	$-0.390097\pi$
$859$	19.8391	0.676903	0.338451	+	0.940984i	$0.390097\pi$
$860$	0	0
$861$	8.73154	0.297570
$862$	0	0
$863$	−42.6777	−1.45277	−0.726384	−	0.687289i	$-0.758800\pi$
$863$	−42.6777	−1.45277	−0.726384	+	0.687289i	$0.758800\pi$
$864$	0	0
$865$	15.4281	0.524573
$866$	0	0
$867$	−0.460784	−0.0156490
$868$	0	0
$869$	22.5934	0.766427
$870$	0	0
$871$	1.52174	0.0515620
$872$	0	0
$873$	−17.1502	−0.580447
$874$	0	0
$875$	24.7978	0.838318
$876$	0	0
$877$	−44.9566	−1.51808	−0.759038	−	0.651046i	$-0.774331\pi$
$877$	−44.9566	−1.51808	−0.759038	+	0.651046i	$0.774331\pi$
$878$	0	0
$879$	−6.88026	−0.232066
$880$	0	0
$881$	22.1003	0.744577	0.372288	−	0.928117i	$-0.378573\pi$
$881$	22.1003	0.744577	0.372288	+	0.928117i	$0.378573\pi$
$882$	0	0
$883$	19.3716	0.651908	0.325954	−	0.945386i	$-0.394315\pi$
$883$	19.3716	0.651908	0.325954	+	0.945386i	$0.394315\pi$
$884$	0	0
$885$	−0.336448	−0.0113096
$886$	0	0
$887$	−50.9927	−1.71217	−0.856084	−	0.516836i	$-0.827110\pi$
$887$	−50.9927	−1.71217	−0.856084	+	0.516836i	$0.827110\pi$
$888$	0	0
$889$	−8.66363	−0.290569
$890$	0	0
$891$	21.5955	0.723477
$892$	0	0
$893$	−23.5933	−0.789520
$894$	0	0
$895$	0.585107	0.0195580
$896$	0	0
$897$	1.06201	0.0354594
$898$	0	0
$899$	−0.951052	−0.0317194
$900$	0	0
$901$	−11.4716	−0.382176
$902$	0	0
$903$	−4.67981	−0.155734
$904$	0	0
$905$	1.50460	0.0500145
$906$	0	0
$907$	−35.3821	−1.17484	−0.587421	−	0.809282i	$-0.699857\pi$
$907$	−35.3821	−1.17484	−0.587421	+	0.809282i	$0.699857\pi$
$908$	0	0
$909$	−26.6858	−0.885111
$910$	0	0
$911$	5.45651	0.180782	0.0903911	−	0.995906i	$-0.471188\pi$
$911$	5.45651	0.180782	0.0903911	+	0.995906i	$0.471188\pi$
$912$	0	0
$913$	−38.7794	−1.28341
$914$	0	0
$915$	−0.705202	−0.0233132
$916$	0	0
$917$	−34.7764	−1.14842
$918$	0	0
$919$	−26.4690	−0.873130	−0.436565	−	0.899673i	$-0.643805\pi$
$919$	−26.4690	−0.873130	−0.436565	+	0.899673i	$0.643805\pi$
$920$	0	0
$921$	−10.6002	−0.349289
$922$	0	0
$923$	2.31431	0.0761763
$924$	0	0
$925$	30.8782	1.01527
$926$	0	0
$927$	30.3581	0.997092
$928$	0	0
$929$	−26.5891	−0.872359	−0.436180	−	0.899860i	$-0.643669\pi$
$929$	−26.5891	−0.872359	−0.436180	+	0.899860i	$0.643669\pi$
$930$	0	0
$931$	−19.3567	−0.634389
$932$	0	0
$933$	3.56200	0.116615
$934$	0	0
$935$	−2.21024	−0.0722826
$936$	0	0
$937$	−2.79458	−0.0912949	−0.0456475	−	0.998958i	$-0.514535\pi$
$937$	−2.79458	−0.0912949	−0.0456475	+	0.998958i	$0.514535\pi$
$938$	0	0
$939$	−0.348109	−0.0113601
$940$	0	0
$941$	37.7467	1.23051	0.615254	−	0.788329i	$-0.289054\pi$
$941$	37.7467	1.23051	0.615254	+	0.788329i	$0.289054\pi$
$942$	0	0
$943$	49.3747	1.60786
$944$	0	0
$945$	−6.98568	−0.227244
$946$	0	0
$947$	17.8014	0.578469	0.289234	−	0.957258i	$-0.406599\pi$
$947$	17.8014	0.578469	0.289234	+	0.957258i	$0.406599\pi$
$948$	0	0
$949$	−0.602888	−0.0195706
$950$	0	0
$951$	1.60119	0.0519222
$952$	0	0
$953$	18.1287	0.587247	0.293624	−	0.955921i	$-0.405139\pi$
$953$	18.1287	0.587247	0.293624	+	0.955921i	$0.405139\pi$
$954$	0	0
$955$	5.18227	0.167694
$956$	0	0
$957$	0.882828	0.0285378
$958$	0	0
$959$	−64.6508	−2.08768
$960$	0	0
$961$	−28.7422	−0.927167
$962$	0	0
$963$	−6.29882	−0.202977
$964$	0	0
$965$	−18.6057	−0.598940
$966$	0	0
$967$	53.1217	1.70828	0.854139	−	0.520044i	$-0.174085\pi$
$967$	53.1217	1.70828	0.854139	+	0.520044i	$0.174085\pi$
$968$	0	0
$969$	1.51950	0.0488135
$970$	0	0
$971$	−15.1364	−0.485751	−0.242875	−	0.970057i	$-0.578091\pi$
$971$	−15.1364	−0.485751	−0.242875	+	0.970057i	$0.578091\pi$
$972$	0	0
$973$	33.7871	1.08316
$974$	0	0
$975$	0.507497	0.0162529
$976$	0	0
$977$	12.4909	0.399618	0.199809	−	0.979835i	$-0.435968\pi$
$977$	12.4909	0.399618	0.199809	+	0.979835i	$0.435968\pi$
$978$	0	0
$979$	−48.4382	−1.54809
$980$	0	0
$981$	7.89402	0.252037
$982$	0	0
$983$	−37.8821	−1.20825	−0.604125	−	0.796889i	$-0.706477\pi$
$983$	−37.8821	−1.20825	−0.604125	+	0.796889i	$0.706477\pi$
$984$	0	0
$985$	8.70269	0.277291
$986$	0	0
$987$	−11.8268	−0.376452
$988$	0	0
$989$	−26.4632	−0.841480
$990$	0	0
$991$	55.0553	1.74889	0.874445	−	0.485124i	$-0.161226\pi$
$991$	55.0553	1.74889	0.874445	+	0.485124i	$0.161226\pi$
$992$	0	0
$993$	4.55376	0.144509
$994$	0	0
$995$	8.76250	0.277790
$996$	0	0
$997$	−40.0927	−1.26975	−0.634874	−	0.772616i	$-0.718948\pi$
$997$	−40.0927	−1.26975	−0.634874	+	0.772616i	$0.718948\pi$
$998$	0	0
$999$	−18.4353	−0.583268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.7	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-0.460784 q^{3} -0.730164 q^{5} +3.58746 q^{7} -2.78768 q^{9} +O(q^{10})\)	\(q-0.460784 q^{3} -0.730164 q^{5} +3.58746 q^{7} -2.78768 q^{9} +3.02705 q^{11} +0.246566 q^{13} +0.336448 q^{15} +1.00000 q^{17} -3.29765 q^{19} -1.65304 q^{21} -9.34754 q^{23} -4.46686 q^{25} +2.66687 q^{27} -0.632935 q^{29} +1.50261 q^{31} -1.39482 q^{33} -2.61943 q^{35} -6.91272 q^{37} -0.113614 q^{39} -5.28210 q^{41} +2.83103 q^{43} +2.03546 q^{45} +7.15459 q^{47} +5.86984 q^{49} -0.460784 q^{51} -11.4716 q^{53} -2.21024 q^{55} +1.51950 q^{57} -1.00000 q^{59} -2.09602 q^{61} -10.0007 q^{63} -0.180034 q^{65} +6.17171 q^{67} +4.30720 q^{69} +9.38615 q^{71} -2.44514 q^{73} +2.05826 q^{75} +10.8594 q^{77} +7.46383 q^{79} +7.13418 q^{81} -12.8110 q^{83} -0.730164 q^{85} +0.291646 q^{87} -16.0018 q^{89} +0.884545 q^{91} -0.692376 q^{93} +2.40782 q^{95} +6.15215 q^{97} -8.43844 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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