LMFDB - Embedded newform 7600.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1900)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18194 q^{3} +3.70370 q^{7} +1.76088 q^{9} +O(q^{10})$	$q-2.18194 q^{3} +3.70370 q^{7} +1.76088 q^{9} +3.18194 q^{11} +1.94282 q^{13} -4.66019 q^{17} -1.00000 q^{19} -8.08126 q^{21} +5.23912 q^{23} +2.70370 q^{27} -1.66019 q^{29} +3.46457 q^{31} -6.94282 q^{33} +7.18194 q^{37} -4.23912 q^{39} -0.717370 q^{41} -1.36389 q^{43} +5.37756 q^{47} +6.71737 q^{49} +10.1683 q^{51} -0.0435069 q^{53} +2.18194 q^{57} +7.88564 q^{59} -4.98633 q^{61} +6.52175 q^{63} +14.5023 q^{67} -11.4315 q^{69} -4.88564 q^{71} -10.6030 q^{73} +11.7850 q^{77} -14.9097 q^{79} -11.1819 q^{81} +9.76088 q^{83} +3.62244 q^{87} +10.0813 q^{89} +7.19562 q^{91} -7.55950 q^{93} +14.6706 q^{97} +5.60301 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9	$ 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9} + q^{11} - 3 q^{13} - 6 q^{17} - 3 q^{19} - 8 q^{21} + 16 q^{23} - q^{27} + 3 q^{29} + q^{31} - 12 q^{33} + 13 q^{37} - 13 q^{39} - 3 q^{41} + 13 q^{43} + 9 q^{47} + 21 q^{49} + 12 q^{51} + q^{53} - 2 q^{57} + 6 q^{59} - 5 q^{61} + 19 q^{63} + 19 q^{67} + 9 q^{69} + 3 q^{71} - 15 q^{73} + 10 q^{77} - 2 q^{79} - 25 q^{81} + 29 q^{83} + 18 q^{87} + 14 q^{89} + 23 q^{91} - 7 q^{93} + q^{97}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9 + q^11 - 3 * q^13 - 6 * q^17 - 3 * q^19 - 8 * q^21 + 16 * q^23 - q^27 + 3 * q^29 + q^31 - 12 * q^33 + 13 * q^37 - 13 * q^39 - 3 * q^41 + 13 * q^43 + 9 * q^47 + 21 * q^49 + 12 * q^51 + q^53 - 2 * q^57 + 6 * q^59 - 5 * q^61 + 19 * q^63 + 19 * q^67 + 9 * q^69 + 3 * q^71 - 15 * q^73 + 10 * q^77 - 2 * q^79 - 25 * q^81 + 29 * q^83 + 18 * q^87 + 14 * q^89 + 23 * q^91 - 7 * q^93 + q^97

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
$3$	−2.18194	−1.25975	−0.629873	−	0.776698i	$-0.716893\pi$
$3$	−2.18194	−1.25975	−0.629873	+	0.776698i	$0.716893\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.70370	1.39987	0.699933	−	0.714209i	$-0.253213\pi$
$7$	3.70370	1.39987	0.699933	+	0.714209i	$0.253213\pi$
$8$	0	0
$9$	1.76088	0.586959
$10$	0	0
$11$	3.18194	0.959392	0.479696	−	0.877435i	$-0.340747\pi$
$11$	3.18194	0.959392	0.479696	+	0.877435i	$0.340747\pi$
$12$	0	0
$13$	1.94282	0.538841	0.269421	−	0.963023i	$-0.413168\pi$
$13$	1.94282	0.538841	0.269421	+	0.963023i	$0.413168\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.66019	−1.13026	−0.565131	−	0.825001i	$-0.691174\pi$
$17$	−4.66019	−1.13026	−0.565131	+	0.825001i	$0.691174\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.08126	−1.76347
$22$	0	0
$23$	5.23912	1.09243	0.546216	−	0.837644i	$-0.316068\pi$
$23$	5.23912	1.09243	0.546216	+	0.837644i	$0.316068\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.70370	0.520327
$28$	0	0
$29$	−1.66019	−0.308290	−0.154145	−	0.988048i	$-0.549262\pi$
$29$	−1.66019	−0.308290	−0.154145	+	0.988048i	$0.549262\pi$
$30$	0	0
$31$	3.46457	0.622256	0.311128	−	0.950368i	$-0.399293\pi$
$31$	3.46457	0.622256	0.311128	+	0.950368i	$0.399293\pi$
$32$	0	0
$33$	−6.94282	−1.20859
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.18194	1.18070	0.590352	−	0.807146i	$-0.298989\pi$
$37$	7.18194	1.18070	0.590352	+	0.807146i	$0.298989\pi$
$38$	0	0
$39$	−4.23912	−0.678803
$40$	0	0
$41$	−0.717370	−0.112034	−0.0560172	−	0.998430i	$-0.517840\pi$
$41$	−0.717370	−0.112034	−0.0560172	+	0.998430i	$0.517840\pi$
$42$	0	0
$43$	−1.36389	−0.207991	−0.103995	−	0.994578i	$-0.533163\pi$
$43$	−1.36389	−0.207991	−0.103995	+	0.994578i	$0.533163\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.37756	0.784398	0.392199	−	0.919880i	$-0.371715\pi$
$47$	5.37756	0.784398	0.392199	+	0.919880i	$0.371715\pi$
$48$	0	0
$49$	6.71737	0.959624
$50$	0	0
$51$	10.1683	1.42384
$52$	0	0
$53$	−0.0435069	−0.00597613	−0.00298807	−	0.999996i	$-0.500951\pi$
$53$	−0.0435069	−0.00597613	−0.00298807	+	0.999996i	$0.500951\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.18194	0.289005
$58$	0	0
$59$	7.88564	1.02662	0.513311	−	0.858202i	$-0.328419\pi$
$59$	7.88564	1.02662	0.513311	+	0.858202i	$0.328419\pi$
$60$	0	0
$61$	−4.98633	−0.638434	−0.319217	−	0.947682i	$-0.603420\pi$
$61$	−4.98633	−0.638434	−0.319217	+	0.947682i	$0.603420\pi$
$62$	0	0
$63$	6.52175	0.821664
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.5023	1.77174	0.885870	−	0.463933i	$-0.153562\pi$
$67$	14.5023	1.77174	0.885870	+	0.463933i	$0.153562\pi$
$68$	0	0
$69$	−11.4315	−1.37619
$70$	0	0
$71$	−4.88564	−0.579819	−0.289909	−	0.957054i	$-0.593625\pi$
$71$	−4.88564	−0.579819	−0.289909	+	0.957054i	$0.593625\pi$
$72$	0	0
$73$	−10.6030	−1.24099	−0.620494	−	0.784211i	$-0.713068\pi$
$73$	−10.6030	−1.24099	−0.620494	+	0.784211i	$0.713068\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.7850	1.34302
$78$	0	0
$79$	−14.9097	−1.67747	−0.838737	−	0.544537i	$-0.816706\pi$
$79$	−14.9097	−1.67747	−0.838737	+	0.544537i	$0.816706\pi$
$80$	0	0
$81$	−11.1819	−1.24244
$82$	0	0
$83$	9.76088	1.07140	0.535698	−	0.844410i	$-0.320049\pi$
$83$	9.76088	1.07140	0.535698	+	0.844410i	$0.320049\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.62244	0.388366
$88$	0	0
$89$	10.0813	1.06861	0.534306	−	0.845291i	$-0.320573\pi$
$89$	10.0813	1.06861	0.534306	+	0.845291i	$0.320573\pi$
$90$	0	0
$91$	7.19562	0.754306
$92$	0	0
$93$	−7.55950	−0.783884
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.6706	1.48957	0.744787	−	0.667303i	$-0.232551\pi$
$97$	14.6706	1.48957	0.744787	+	0.667303i	$0.232551\pi$
$98$	0	0
$99$	5.60301	0.563124
$100$	0	0
$101$	15.9532	1.58741	0.793703	−	0.608306i	$-0.208151\pi$
$101$	15.9532	1.58741	0.793703	+	0.608306i	$0.208151\pi$
$102$	0	0
$103$	−10.6328	−1.04769	−0.523843	−	0.851815i	$-0.675502\pi$
$103$	−10.6328	−1.04769	−0.523843	+	0.851815i	$0.675502\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.6465	1.41593	0.707966	−	0.706246i	$-0.249613\pi$
$107$	14.6465	1.41593	0.707966	+	0.706246i	$0.249613\pi$
$108$	0	0
$109$	−3.55950	−0.340939	−0.170469	−	0.985363i	$-0.554528\pi$
$109$	−3.55950	−0.340939	−0.170469	+	0.985363i	$0.554528\pi$
$110$	0	0
$111$	−15.6706	−1.48739
$112$	0	0
$113$	−14.8285	−1.39494	−0.697472	−	0.716612i	$-0.745692\pi$
$113$	−14.8285	−1.39494	−0.697472	+	0.716612i	$0.745692\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.42107	0.316278
$118$	0	0
$119$	−17.2599	−1.58222
$120$	0	0
$121$	−0.875237	−0.0795670
$122$	0	0
$123$	1.56526	0.141135
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.69002	0.682379	0.341190	−	0.939994i	$-0.389170\pi$
$127$	7.69002	0.682379	0.341190	+	0.939994i	$0.389170\pi$
$128$	0	0
$129$	2.97592	0.262015
$130$	0	0
$131$	−19.4451	−1.69893	−0.849465	−	0.527645i	$-0.823075\pi$
$131$	−19.4451	−1.69893	−0.849465	+	0.527645i	$0.823075\pi$
$132$	0	0
$133$	−3.70370	−0.321151
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.39699	−0.290224	−0.145112	−	0.989415i	$-0.546354\pi$
$137$	−3.39699	−0.290224	−0.145112	+	0.989415i	$0.546354\pi$
$138$	0	0
$139$	−10.5699	−0.896528	−0.448264	−	0.893901i	$-0.647958\pi$
$139$	−10.5699	−0.896528	−0.448264	+	0.893901i	$0.647958\pi$
$140$	0	0
$141$	−11.7335	−0.988142
$142$	0	0
$143$	6.18194	0.516960
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−14.6569	−1.20888
$148$	0	0
$149$	19.5322	1.60014	0.800068	−	0.599909i	$-0.204797\pi$
$149$	19.5322	1.60014	0.800068	+	0.599909i	$0.204797\pi$
$150$	0	0
$151$	1.17154	0.0953386	0.0476693	−	0.998863i	$-0.484821\pi$
$151$	1.17154	0.0953386	0.0476693	+	0.998863i	$0.484821\pi$
$152$	0	0
$153$	−8.20602	−0.663417
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.01040	−0.399874	−0.199937	−	0.979809i	$-0.564074\pi$
$157$	−5.01040	−0.399874	−0.199937	+	0.979809i	$0.564074\pi$
$158$	0	0
$159$	0.0949296	0.00752840
$160$	0	0
$161$	19.4041	1.52926
$162$	0	0
$163$	−6.78495	−0.531439	−0.265719	−	0.964050i	$-0.585609\pi$
$163$	−6.78495	−0.531439	−0.265719	+	0.964050i	$0.585609\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.64652	−0.436941	−0.218470	−	0.975844i	$-0.570107\pi$
$167$	−5.64652	−0.436941	−0.218470	+	0.975844i	$0.570107\pi$
$168$	0	0
$169$	−9.22545	−0.709650
$170$	0	0
$171$	−1.76088	−0.134658
$172$	0	0
$173$	−5.38796	−0.409639	−0.204820	−	0.978800i	$-0.565661\pi$
$173$	−5.38796	−0.409639	−0.204820	+	0.978800i	$0.565661\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−17.2060	−1.29328
$178$	0	0
$179$	−3.27687	−0.244925	−0.122462	−	0.992473i	$-0.539079\pi$
$179$	−3.27687	−0.244925	−0.122462	+	0.992473i	$0.539079\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	10.8799	0.804264
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−14.8285	−1.08436
$188$	0	0
$189$	10.0137	0.728388
$190$	0	0
$191$	−0.225450	−0.0163130	−0.00815650	−	0.999967i	$-0.502596\pi$
$191$	−0.225450	−0.0163130	−0.00815650	+	0.999967i	$0.502596\pi$
$192$	0	0
$193$	8.12476	0.584833	0.292417	−	0.956291i	$-0.405541\pi$
$193$	8.12476	0.584833	0.292417	+	0.956291i	$0.405541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.25280	−0.302999	−0.151500	−	0.988457i	$-0.548410\pi$
$197$	−4.25280	−0.302999	−0.151500	+	0.988457i	$0.548410\pi$
$198$	0	0
$199$	−26.1592	−1.85438	−0.927190	−	0.374592i	$-0.877783\pi$
$199$	−26.1592	−1.85438	−0.927190	+	0.374592i	$0.877783\pi$
$200$	0	0
$201$	−31.6432	−2.23194
$202$	0	0
$203$	−6.14884	−0.431564
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.22545	0.641213
$208$	0	0
$209$	−3.18194	−0.220100
$210$	0	0
$211$	−4.27223	−0.294112	−0.147056	−	0.989128i	$-0.546980\pi$
$211$	−4.27223	−0.294112	−0.147056	+	0.989128i	$0.546980\pi$
$212$	0	0
$213$	10.6602	0.730424
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.8317	0.871075
$218$	0	0
$219$	23.1352	1.56333
$220$	0	0
$221$	−9.05391	−0.609032
$222$	0	0
$223$	23.0208	1.54159	0.770794	−	0.637085i	$-0.219860\pi$
$223$	23.0208	1.54159	0.770794	+	0.637085i	$0.219860\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.96690	0.595154	0.297577	−	0.954698i	$-0.403822\pi$
$227$	8.96690	0.595154	0.297577	+	0.954698i	$0.403822\pi$
$228$	0	0
$229$	−8.62709	−0.570094	−0.285047	−	0.958514i	$-0.592009\pi$
$229$	−8.62709	−0.570094	−0.285047	+	0.958514i	$0.592009\pi$
$230$	0	0
$231$	−25.7141	−1.69186
$232$	0	0
$233$	29.0014	1.89994	0.949972	−	0.312336i	$-0.101111\pi$
$233$	29.0014	1.89994	0.949972	+	0.312336i	$0.101111\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.5322	2.11319
$238$	0	0
$239$	23.7414	1.53571	0.767853	−	0.640626i	$-0.221325\pi$
$239$	23.7414	1.53571	0.767853	+	0.640626i	$0.221325\pi$
$240$	0	0
$241$	−6.32614	−0.407502	−0.203751	−	0.979023i	$-0.565313\pi$
$241$	−6.32614	−0.407502	−0.203751	+	0.979023i	$0.565313\pi$
$242$	0	0
$243$	16.2873	1.04483
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.94282	−0.123619
$248$	0	0
$249$	−21.2977	−1.34969
$250$	0	0
$251$	21.9201	1.38359	0.691793	−	0.722096i	$-0.256821\pi$
$251$	21.9201	1.38359	0.691793	+	0.722096i	$0.256821\pi$
$252$	0	0
$253$	16.6706	1.04807
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.31246	−0.206626	−0.103313	−	0.994649i	$-0.532944\pi$
$257$	−3.31246	−0.206626	−0.103313	+	0.994649i	$0.532944\pi$
$258$	0	0
$259$	26.5997	1.65283
$260$	0	0
$261$	−2.92339	−0.180953
$262$	0	0
$263$	−2.33405	−0.143924	−0.0719619	−	0.997407i	$-0.522926\pi$
$263$	−2.33405	−0.143924	−0.0719619	+	0.997407i	$0.522926\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−21.9967	−1.34618
$268$	0	0
$269$	−24.5789	−1.49860	−0.749302	−	0.662228i	$-0.769611\pi$
$269$	−24.5789	−1.49860	−0.749302	+	0.662228i	$0.769611\pi$
$270$	0	0
$271$	−16.3880	−0.995498	−0.497749	−	0.867321i	$-0.665840\pi$
$271$	−16.3880	−0.995498	−0.497749	+	0.867321i	$0.665840\pi$
$272$	0	0
$273$	−15.7004	−0.950233
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.68427	−0.521787	−0.260894	−	0.965368i	$-0.584017\pi$
$277$	−8.68427	−0.521787	−0.260894	+	0.965368i	$0.584017\pi$
$278$	0	0
$279$	6.10069	0.365239
$280$	0	0
$281$	−0.450900	−0.0268985	−0.0134492	−	0.999910i	$-0.504281\pi$
$281$	−0.450900	−0.0268985	−0.0134492	+	0.999910i	$0.504281\pi$
$282$	0	0
$283$	−28.6512	−1.70313	−0.851567	−	0.524245i	$-0.824348\pi$
$283$	−28.6512	−1.70313	−0.851567	+	0.524245i	$0.824348\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.65692	−0.156833
$288$	0	0
$289$	4.71737	0.277492
$290$	0	0
$291$	−32.0104	−1.87648
$292$	0	0
$293$	20.7335	1.21127	0.605633	−	0.795744i	$-0.292920\pi$
$293$	20.7335	1.21127	0.605633	+	0.795744i	$0.292920\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.60301	0.499197
$298$	0	0
$299$	10.1787	0.588648
$300$	0	0
$301$	−5.05142	−0.291159
$302$	0	0
$303$	−34.8090	−1.99973
$304$	0	0
$305$	0	0
$306$	0	0
$307$	34.4933	1.96864	0.984318	−	0.176402i	$-0.0564459\pi$
$307$	34.4933	1.96864	0.984318	+	0.176402i	$0.0564459\pi$
$308$	0	0
$309$	23.2003	1.31982
$310$	0	0
$311$	−15.5789	−0.883400	−0.441700	−	0.897163i	$-0.645624\pi$
$311$	−15.5789	−0.883400	−0.441700	+	0.897163i	$0.645624\pi$
$312$	0	0
$313$	2.98057	0.168472	0.0842359	−	0.996446i	$-0.473155\pi$
$313$	2.98057	0.168472	0.0842359	+	0.996446i	$0.473155\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.5803	−0.987409	−0.493704	−	0.869630i	$-0.664357\pi$
$317$	−17.5803	−0.987409	−0.493704	+	0.869630i	$0.664357\pi$
$318$	0	0
$319$	−5.28263	−0.295771
$320$	0	0
$321$	−31.9579	−1.78371
$322$	0	0
$323$	4.66019	0.259300
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.76663	0.429496
$328$	0	0
$329$	19.9169	1.09805
$330$	0	0
$331$	35.2405	1.93699	0.968497	−	0.249027i	$-0.0801108\pi$
$331$	35.2405	1.93699	0.968497	+	0.249027i	$0.0801108\pi$
$332$	0	0
$333$	12.6465	0.693025
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.74145	0.530650	0.265325	−	0.964159i	$-0.414521\pi$
$337$	9.74145	0.530650	0.265325	+	0.964159i	$0.414521\pi$
$338$	0	0
$339$	32.3549	1.75727
$340$	0	0
$341$	11.0241	0.596987
$342$	0	0
$343$	−1.04678	−0.0565206
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.2028	1.78242	0.891209	−	0.453594i	$-0.149858\pi$
$347$	33.2028	1.78242	0.891209	+	0.453594i	$0.149858\pi$
$348$	0	0
$349$	−14.7141	−0.787628	−0.393814	−	0.919190i	$-0.628845\pi$
$349$	−14.7141	−0.787628	−0.393814	+	0.919190i	$0.628845\pi$
$350$	0	0
$351$	5.25280	0.280374
$352$	0	0
$353$	−3.44841	−0.183540	−0.0917702	−	0.995780i	$-0.529253\pi$
$353$	−3.44841	−0.183540	−0.0917702	+	0.995780i	$0.529253\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	37.6602	1.99319
$358$	0	0
$359$	−6.55623	−0.346025	−0.173012	−	0.984920i	$-0.555350\pi$
$359$	−6.55623	−0.346025	−0.173012	+	0.984920i	$0.555350\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.90972	0.100234
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.5641	1.69983	0.849917	−	0.526916i	$-0.176652\pi$
$367$	32.5641	1.69983	0.849917	+	0.526916i	$0.176652\pi$
$368$	0	0
$369$	−1.26320	−0.0657596
$370$	0	0
$371$	−0.161136	−0.00836578
$372$	0	0
$373$	15.0137	0.777379	0.388689	−	0.921369i	$-0.372928\pi$
$373$	15.0137	0.777379	0.388689	+	0.921369i	$0.372928\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.22545	−0.166119
$378$	0	0
$379$	−12.4807	−0.641092	−0.320546	−	0.947233i	$-0.603866\pi$
$379$	−12.4807	−0.641092	−0.320546	+	0.947233i	$0.603866\pi$
$380$	0	0
$381$	−16.7792	−0.859624
$382$	0	0
$383$	12.5354	0.640530	0.320265	−	0.947328i	$-0.396228\pi$
$383$	12.5354	0.640530	0.320265	+	0.947328i	$0.396228\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.40164	−0.122082
$388$	0	0
$389$	37.4328	1.89792	0.948960	−	0.315395i	$-0.102137\pi$
$389$	37.4328	1.89792	0.948960	+	0.315395i	$0.102137\pi$
$390$	0	0
$391$	−24.4153	−1.23474
$392$	0	0
$393$	42.4282	2.14022
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.05718	0.153435	0.0767177	−	0.997053i	$-0.475556\pi$
$397$	3.05718	0.153435	0.0767177	+	0.997053i	$0.475556\pi$
$398$	0	0
$399$	8.08126	0.404569
$400$	0	0
$401$	18.5458	0.926135	0.463067	−	0.886323i	$-0.346749\pi$
$401$	18.5458	0.926135	0.463067	+	0.886323i	$0.346749\pi$
$402$	0	0
$403$	6.73104	0.335297
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.8525	1.13276
$408$	0	0
$409$	13.9669	0.690619	0.345309	−	0.938489i	$-0.387774\pi$
$409$	13.9669	0.690619	0.345309	+	0.938489i	$0.387774\pi$
$410$	0	0
$411$	7.41204	0.365609
$412$	0	0
$413$	29.2060	1.43713
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.0629	1.12940
$418$	0	0
$419$	32.7623	1.60054	0.800270	−	0.599639i	$-0.204689\pi$
$419$	32.7623	1.60054	0.800270	+	0.599639i	$0.204689\pi$
$420$	0	0
$421$	29.5127	1.43836	0.719181	−	0.694823i	$-0.244517\pi$
$421$	29.5127	1.43836	0.719181	+	0.694823i	$0.244517\pi$
$422$	0	0
$423$	9.46922	0.460409
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−18.4678	−0.893722
$428$	0	0
$429$	−13.4887	−0.651238
$430$	0	0
$431$	2.90507	0.139932	0.0699662	−	0.997549i	$-0.477711\pi$
$431$	2.90507	0.139932	0.0699662	+	0.997549i	$0.477711\pi$
$432$	0	0
$433$	29.4887	1.41713	0.708567	−	0.705643i	$-0.249342\pi$
$433$	29.4887	1.41713	0.708567	+	0.705643i	$0.249342\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.23912	−0.250621
$438$	0	0
$439$	30.4088	1.45133	0.725666	−	0.688047i	$-0.241532\pi$
$439$	30.4088	1.45133	0.725666	+	0.688047i	$0.241532\pi$
$440$	0	0
$441$	11.8285	0.563260
$442$	0	0
$443$	−1.80903	−0.0859496	−0.0429748	−	0.999076i	$-0.513684\pi$
$443$	−1.80903	−0.0859496	−0.0429748	+	0.999076i	$0.513684\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−42.6181	−2.01577
$448$	0	0
$449$	−21.4257	−1.01114	−0.505571	−	0.862785i	$-0.668718\pi$
$449$	−21.4257	−1.01114	−0.505571	+	0.862785i	$0.668718\pi$
$450$	0	0
$451$	−2.28263	−0.107485
$452$	0	0
$453$	−2.55623	−0.120102
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.5789	0.915864	0.457932	−	0.888987i	$-0.348590\pi$
$457$	19.5789	0.915864	0.457932	+	0.888987i	$0.348590\pi$
$458$	0	0
$459$	−12.5997	−0.588106
$460$	0	0
$461$	7.92666	0.369181	0.184591	−	0.982815i	$-0.440904\pi$
$461$	7.92666	0.369181	0.184591	+	0.982815i	$0.440904\pi$
$462$	0	0
$463$	−11.0298	−0.512600	−0.256300	−	0.966597i	$-0.582503\pi$
$463$	−11.0298	−0.512600	−0.256300	+	0.966597i	$0.582503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.48400	0.161220	0.0806102	−	0.996746i	$-0.474313\pi$
$467$	3.48400	0.161220	0.0806102	+	0.996746i	$0.474313\pi$
$468$	0	0
$469$	53.7122	2.48020
$470$	0	0
$471$	10.9324	0.503739
$472$	0	0
$473$	−4.33981	−0.199545
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.0766103	−0.00350774
$478$	0	0
$479$	29.6569	1.35506	0.677530	−	0.735495i	$-0.263051\pi$
$479$	29.6569	1.35506	0.677530	+	0.735495i	$0.263051\pi$
$480$	0	0
$481$	13.9532	0.636212
$482$	0	0
$483$	−42.3387	−1.92648
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.65116	−0.210764	−0.105382	−	0.994432i	$-0.533607\pi$
$487$	−4.65116	−0.210764	−0.105382	+	0.994432i	$0.533607\pi$
$488$	0	0
$489$	14.8044	0.669477
$490$	0	0
$491$	21.6947	0.979067	0.489533	−	0.871985i	$-0.337167\pi$
$491$	21.6947	0.979067	0.489533	+	0.871985i	$0.337167\pi$
$492$	0	0
$493$	7.73680	0.348448
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.0949	−0.811669
$498$	0	0
$499$	−7.83886	−0.350916	−0.175458	−	0.984487i	$-0.556141\pi$
$499$	−7.83886	−0.350916	−0.175458	+	0.984487i	$0.556141\pi$
$500$	0	0
$501$	12.3204	0.550434
$502$	0	0
$503$	−32.8824	−1.46615	−0.733076	−	0.680146i	$-0.761916\pi$
$503$	−32.8824	−1.46615	−0.733076	+	0.680146i	$0.761916\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.1294	0.893978
$508$	0	0
$509$	−43.5530	−1.93045	−0.965226	−	0.261418i	$-0.915810\pi$
$509$	−43.5530	−1.93045	−0.965226	+	0.261418i	$0.915810\pi$
$510$	0	0
$511$	−39.2703	−1.73722
$512$	0	0
$513$	−2.70370	−0.119371
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17.1111	0.752545
$518$	0	0
$519$	11.7562	0.516041
$520$	0	0
$521$	−6.34308	−0.277895	−0.138948	−	0.990300i	$-0.544372\pi$
$521$	−6.34308	−0.277895	−0.138948	+	0.990300i	$0.544372\pi$
$522$	0	0
$523$	31.0449	1.35750	0.678749	−	0.734370i	$-0.262522\pi$
$523$	31.0449	1.35750	0.678749	+	0.734370i	$0.262522\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.1456	−0.703312
$528$	0	0
$529$	4.44841	0.193409
$530$	0	0
$531$	13.8856	0.602585
$532$	0	0
$533$	−1.39372	−0.0603687
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.14995	0.308543
$538$	0	0
$539$	21.3743	0.920656
$540$	0	0
$541$	34.3218	1.47561	0.737804	−	0.675015i	$-0.235863\pi$
$541$	34.3218	1.47561	0.737804	+	0.675015i	$0.235863\pi$
$542$	0	0
$543$	15.2736	0.655453
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23.5836	−1.00836	−0.504181	−	0.863598i	$-0.668205\pi$
$547$	−23.5836	−1.00836	−0.504181	+	0.863598i	$0.668205\pi$
$548$	0	0
$549$	−8.78031	−0.374734
$550$	0	0
$551$	1.66019	0.0707265
$552$	0	0
$553$	−55.2211	−2.34824
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−44.8824	−1.90173	−0.950864	−	0.309610i	$-0.899801\pi$
$557$	−44.8824	−1.90173	−0.950864	+	0.309610i	$0.899801\pi$
$558$	0	0
$559$	−2.64979	−0.112074
$560$	0	0
$561$	32.3549	1.36602
$562$	0	0
$563$	6.13844	0.258704	0.129352	−	0.991599i	$-0.458710\pi$
$563$	6.13844	0.258704	0.129352	+	0.991599i	$0.458710\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−41.4145	−1.73925
$568$	0	0
$569$	42.9384	1.80007	0.900037	−	0.435814i	$-0.143540\pi$
$569$	42.9384	1.80007	0.900037	+	0.435814i	$0.143540\pi$
$570$	0	0
$571$	3.00576	0.125787	0.0628935	−	0.998020i	$-0.479967\pi$
$571$	3.00576	0.125787	0.0628935	+	0.998020i	$0.479967\pi$
$572$	0	0
$573$	0.491920	0.0205502
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.2028	0.674529	0.337265	−	0.941410i	$-0.390498\pi$
$577$	16.2028	0.674529	0.337265	+	0.941410i	$0.390498\pi$
$578$	0	0
$579$	−17.7278	−0.736741
$580$	0	0
$581$	36.1513	1.49981
$582$	0	0
$583$	−0.138436	−0.00573345
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.4542	1.09188	0.545940	−	0.837824i	$-0.316173\pi$
$587$	26.4542	1.09188	0.545940	+	0.837824i	$0.316173\pi$
$588$	0	0
$589$	−3.46457	−0.142755
$590$	0	0
$591$	9.27936	0.381702
$592$	0	0
$593$	−25.3710	−1.04186	−0.520931	−	0.853599i	$-0.674415\pi$
$593$	−25.3710	−1.04186	−0.520931	+	0.853599i	$0.674415\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	57.0780	2.33605
$598$	0	0
$599$	3.26896	0.133566	0.0667830	−	0.997768i	$-0.478726\pi$
$599$	3.26896	0.133566	0.0667830	+	0.997768i	$0.478726\pi$
$600$	0	0
$601$	−27.6465	−1.12772	−0.563862	−	0.825869i	$-0.690685\pi$
$601$	−27.6465	−1.12772	−0.563862	+	0.825869i	$0.690685\pi$
$602$	0	0
$603$	25.5368	1.03994
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.6739	−1.28560	−0.642801	−	0.766033i	$-0.722228\pi$
$607$	−31.6739	−1.28560	−0.642801	+	0.766033i	$0.722228\pi$
$608$	0	0
$609$	13.4164	0.543661
$610$	0	0
$611$	10.4476	0.422666
$612$	0	0
$613$	20.1970	0.815749	0.407874	−	0.913038i	$-0.366270\pi$
$613$	20.1970	0.815749	0.407874	+	0.913038i	$0.366270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.4717	−1.46830	−0.734148	−	0.678990i	$-0.762418\pi$
$617$	−36.4717	−1.46830	−0.734148	+	0.678990i	$0.762418\pi$
$618$	0	0
$619$	−2.99424	−0.120349	−0.0601744	−	0.998188i	$-0.519166\pi$
$619$	−2.99424	−0.120349	−0.0601744	+	0.998188i	$0.519166\pi$
$620$	0	0
$621$	14.1650	0.568422
$622$	0	0
$623$	37.3379	1.49591
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.94282	0.277270
$628$	0	0
$629$	−33.4692	−1.33451
$630$	0	0
$631$	−33.1683	−1.32041	−0.660204	−	0.751086i	$-0.729530\pi$
$631$	−33.1683	−1.32041	−0.660204	+	0.751086i	$0.729530\pi$
$632$	0	0
$633$	9.32176	0.370507
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.0506	0.517085
$638$	0	0
$639$	−8.60301	−0.340330
$640$	0	0
$641$	12.0435	0.475690	0.237845	−	0.971303i	$-0.423559\pi$
$641$	12.0435	0.475690	0.237845	+	0.971303i	$0.423559\pi$
$642$	0	0
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.0150	−1.10139	−0.550693	−	0.834708i	$-0.685636\pi$
$647$	−28.0150	−1.10139	−0.550693	+	0.834708i	$0.685636\pi$
$648$	0	0
$649$	25.0917	0.984934
$650$	0	0
$651$	−27.9981	−1.09733
$652$	0	0
$653$	−7.85254	−0.307294	−0.153647	−	0.988126i	$-0.549102\pi$
$653$	−7.85254	−0.307294	−0.153647	+	0.988126i	$0.549102\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−18.6706	−0.728409
$658$	0	0
$659$	29.9590	1.16704	0.583518	−	0.812100i	$-0.301676\pi$
$659$	29.9590	1.16704	0.583518	+	0.812100i	$0.301676\pi$
$660$	0	0
$661$	−3.42107	−0.133064	−0.0665320	−	0.997784i	$-0.521193\pi$
$661$	−3.42107	−0.133064	−0.0665320	+	0.997784i	$0.521193\pi$
$662$	0	0
$663$	19.7551	0.767225
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.69794	−0.336786
$668$	0	0
$669$	−50.2301	−1.94201
$670$	0	0
$671$	−15.8662	−0.612508
$672$	0	0
$673$	−45.5095	−1.75426	−0.877130	−	0.480252i	$-0.840545\pi$
$673$	−45.5095	−1.75426	−0.877130	+	0.480252i	$0.840545\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.6375	0.639431	0.319715	−	0.947514i	$-0.396413\pi$
$677$	16.6375	0.639431	0.319715	+	0.947514i	$0.396413\pi$
$678$	0	0
$679$	54.3354	2.08520
$680$	0	0
$681$	−19.5653	−0.749742
$682$	0	0
$683$	−15.2463	−0.583382	−0.291691	−	0.956513i	$-0.594218\pi$
$683$	−15.2463	−0.583382	−0.291691	+	0.956513i	$0.594218\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18.8238	0.718173
$688$	0	0
$689$	−0.0845261	−0.00322019
$690$	0	0
$691$	45.8676	1.74489	0.872443	−	0.488717i	$-0.162535\pi$
$691$	45.8676	1.74489	0.872443	+	0.488717i	$0.162535\pi$
$692$	0	0
$693$	20.7518	0.788298
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.34308	0.126628
$698$	0	0
$699$	−63.2794	−2.39345
$700$	0	0
$701$	−6.59974	−0.249269	−0.124634	−	0.992203i	$-0.539776\pi$
$701$	−6.59974	−0.249269	−0.124634	+	0.992203i	$0.539776\pi$
$702$	0	0
$703$	−7.18194	−0.270872
$704$	0	0
$705$	0	0
$706$	0	0
$707$	59.0859	2.22215
$708$	0	0
$709$	22.0118	0.826670	0.413335	−	0.910579i	$-0.364364\pi$
$709$	22.0118	0.826670	0.413335	+	0.910579i	$0.364364\pi$
$710$	0	0
$711$	−26.2542	−0.984608
$712$	0	0
$713$	18.1513	0.679773
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−51.8025	−1.93460
$718$	0	0
$719$	22.2405	0.829431	0.414715	−	0.909951i	$-0.363881\pi$
$719$	22.2405	0.829431	0.414715	+	0.909951i	$0.363881\pi$
$720$	0	0
$721$	−39.3808	−1.46662
$722$	0	0
$723$	13.8033	0.513349
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.8662	−1.40438	−0.702190	−	0.711990i	$-0.747794\pi$
$727$	−37.8662	−1.40438	−0.702190	+	0.711990i	$0.747794\pi$
$728$	0	0
$729$	−1.99208	−0.0737809
$730$	0	0
$731$	6.35597	0.235084
$732$	0	0
$733$	5.02735	0.185689	0.0928446	−	0.995681i	$-0.470404\pi$
$733$	5.02735	0.185689	0.0928446	+	0.995681i	$0.470404\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.1456	1.69979
$738$	0	0
$739$	−31.6159	−1.16301	−0.581505	−	0.813543i	$-0.697536\pi$
$739$	−31.6159	−1.16301	−0.581505	+	0.813543i	$0.697536\pi$
$740$	0	0
$741$	4.23912	0.155728
$742$	0	0
$743$	33.8252	1.24093	0.620463	−	0.784236i	$-0.286945\pi$
$743$	33.8252	1.24093	0.620463	+	0.784236i	$0.286945\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	17.1877	0.628865
$748$	0	0
$749$	54.2463	1.98212
$750$	0	0
$751$	45.5368	1.66166	0.830831	−	0.556525i	$-0.187866\pi$
$751$	45.5368	1.66166	0.830831	+	0.556525i	$0.187866\pi$
$752$	0	0
$753$	−47.8285	−1.74297
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.55950	−0.165718	−0.0828590	−	0.996561i	$-0.526405\pi$
$757$	−4.55950	−0.165718	−0.0828590	+	0.996561i	$0.526405\pi$
$758$	0	0
$759$	−36.3743	−1.32030
$760$	0	0
$761$	−51.4386	−1.86465	−0.932324	−	0.361624i	$-0.882222\pi$
$761$	−51.4386	−1.86465	−0.932324	+	0.361624i	$0.882222\pi$
$762$	0	0
$763$	−13.1833	−0.477268
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.3204	0.553187
$768$	0	0
$769$	9.34773	0.337088	0.168544	−	0.985694i	$-0.446094\pi$
$769$	9.34773	0.337088	0.168544	+	0.985694i	$0.446094\pi$
$770$	0	0
$771$	7.22761	0.260296
$772$	0	0
$773$	19.4887	0.700958	0.350479	−	0.936571i	$-0.386019\pi$
$773$	19.4887	0.700958	0.350479	+	0.936571i	$0.386019\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−58.0391	−2.08214
$778$	0	0
$779$	0.717370	0.0257024
$780$	0	0
$781$	−15.5458	−0.556274
$782$	0	0
$783$	−4.48865	−0.160411
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5.32941	−0.189973	−0.0949864	−	0.995479i	$-0.530281\pi$
$787$	−5.32941	−0.189973	−0.0949864	+	0.995479i	$0.530281\pi$
$788$	0	0
$789$	5.09277	0.181307
$790$	0	0
$791$	−54.9201	−1.95273
$792$	0	0
$793$	−9.68754	−0.344014
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.7220	1.33618	0.668091	−	0.744079i	$-0.267112\pi$
$797$	37.7220	1.33618	0.668091	+	0.744079i	$0.267112\pi$
$798$	0	0
$799$	−25.0604	−0.886575
$800$	0	0
$801$	17.7518	0.627231
$802$	0	0
$803$	−33.7382	−1.19059
$804$	0	0
$805$	0	0
$806$	0	0
$807$	53.6298	1.88786
$808$	0	0
$809$	31.0071	1.09015	0.545076	−	0.838386i	$-0.316501\pi$
$809$	31.0071	1.09015	0.545076	+	0.838386i	$0.316501\pi$
$810$	0	0
$811$	−26.7738	−0.940154	−0.470077	−	0.882625i	$-0.655774\pi$
$811$	−26.7738	−0.940154	−0.470077	+	0.882625i	$0.655774\pi$
$812$	0	0
$813$	35.7576	1.25407
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.36389	0.0477164
$818$	0	0
$819$	12.6706	0.442746
$820$	0	0
$821$	24.2873	0.847632	0.423816	−	0.905748i	$-0.360690\pi$
$821$	24.2873	0.847632	0.423816	+	0.905748i	$0.360690\pi$
$822$	0	0
$823$	−0.828460	−0.0288783	−0.0144392	−	0.999896i	$-0.504596\pi$
$823$	−0.828460	−0.0288783	−0.0144392	+	0.999896i	$0.504596\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.3387	1.47226	0.736130	−	0.676840i	$-0.236651\pi$
$827$	42.3387	1.47226	0.736130	+	0.676840i	$0.236651\pi$
$828$	0	0
$829$	−3.20602	−0.111350	−0.0556748	−	0.998449i	$-0.517731\pi$
$829$	−3.20602	−0.111350	−0.0556748	+	0.998449i	$0.517731\pi$
$830$	0	0
$831$	18.9486	0.657319
$832$	0	0
$833$	−31.3042	−1.08463
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.36716	0.323776
$838$	0	0
$839$	20.2520	0.699177	0.349589	−	0.936903i	$-0.386321\pi$
$839$	20.2520	0.699177	0.349589	+	0.936903i	$0.386321\pi$
$840$	0	0
$841$	−26.2438	−0.904958
$842$	0	0
$843$	0.983839	0.0338852
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.24161	−0.111383
$848$	0	0
$849$	62.5152	2.14552
$850$	0	0
$851$	37.6271	1.28984
$852$	0	0
$853$	−3.10644	−0.106363	−0.0531813	−	0.998585i	$-0.516936\pi$
$853$	−3.10644	−0.106363	−0.0531813	+	0.998585i	$0.516936\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.89467	−0.303836	−0.151918	−	0.988393i	$-0.548545\pi$
$857$	−8.89467	−0.303836	−0.151918	+	0.988393i	$0.548545\pi$
$858$	0	0
$859$	17.9989	0.614114	0.307057	−	0.951691i	$-0.400656\pi$
$859$	17.9989	0.614114	0.307057	+	0.951691i	$0.400656\pi$
$860$	0	0
$861$	5.79725	0.197570
$862$	0	0
$863$	43.9291	1.49537	0.747683	−	0.664056i	$-0.231166\pi$
$863$	43.9291	1.49537	0.747683	+	0.664056i	$0.231166\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.2930	−0.349570
$868$	0	0
$869$	−47.4419	−1.60936
$870$	0	0
$871$	28.1754	0.954687
$872$	0	0
$873$	25.8331	0.874318
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.55375	0.153769	0.0768845	−	0.997040i	$-0.475503\pi$
$877$	4.55375	0.153769	0.0768845	+	0.997040i	$0.475503\pi$
$878$	0	0
$879$	−45.2394	−1.52589
$880$	0	0
$881$	2.71272	0.0913940	0.0456970	−	0.998955i	$-0.485449\pi$
$881$	2.71272	0.0913940	0.0456970	+	0.998955i	$0.485449\pi$
$882$	0	0
$883$	−9.92588	−0.334032	−0.167016	−	0.985954i	$-0.553413\pi$
$883$	−9.92588	−0.334032	−0.167016	+	0.985954i	$0.553413\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.7018	−1.76955	−0.884777	−	0.466015i	$-0.845689\pi$
$887$	−52.7018	−1.76955	−0.884777	+	0.466015i	$0.845689\pi$
$888$	0	0
$889$	28.4815	0.955239
$890$	0	0
$891$	−35.5803	−1.19199
$892$	0	0
$893$	−5.37756	−0.179953
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−22.2093	−0.741547
$898$	0	0
$899$	−5.75185	−0.191835
$900$	0	0
$901$	0.202750	0.00675459
$902$	0	0
$903$	11.0219	0.366786
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−37.3743	−1.24099	−0.620496	−	0.784209i	$-0.713069\pi$
$907$	−37.3743	−1.24099	−0.620496	+	0.784209i	$0.713069\pi$
$908$	0	0
$909$	28.0917	0.931742
$910$	0	0
$911$	47.6752	1.57955	0.789776	−	0.613396i	$-0.210197\pi$
$911$	47.6752	1.57955	0.789776	+	0.613396i	$0.210197\pi$
$912$	0	0
$913$	31.0586	1.02789
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−72.0189	−2.37827
$918$	0	0
$919$	35.8605	1.18293	0.591464	−	0.806332i	$-0.298550\pi$
$919$	35.8605	1.18293	0.591464	+	0.806332i	$0.298550\pi$
$920$	0	0
$921$	−75.2624	−2.47998
$922$	0	0
$923$	−9.49192	−0.312430
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−18.7231	−0.614948
$928$	0	0
$929$	−55.1229	−1.80852	−0.904261	−	0.426979i	$-0.859578\pi$
$929$	−55.1229	−1.80852	−0.904261	+	0.426979i	$0.859578\pi$
$930$	0	0
$931$	−6.71737	−0.220153
$932$	0	0
$933$	33.9923	1.11286
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−53.3333	−1.74232	−0.871161	−	0.490997i	$-0.836632\pi$
$937$	−53.3333	−1.74232	−0.871161	+	0.490997i	$0.836632\pi$
$938$	0	0
$939$	−6.50343	−0.212232
$940$	0	0
$941$	29.7335	0.969285	0.484643	−	0.874712i	$-0.338950\pi$
$941$	29.7335	0.969285	0.484643	+	0.874712i	$0.338950\pi$
$942$	0	0
$943$	−3.75839	−0.122390
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.4821	1.28300	0.641498	−	0.767125i	$-0.278313\pi$
$947$	39.4821	1.28300	0.641498	+	0.767125i	$0.278313\pi$
$948$	0	0
$949$	−20.5997	−0.668696
$950$	0	0
$951$	38.3592	1.24388
$952$	0	0
$953$	−15.8479	−0.513364	−0.256682	−	0.966496i	$-0.582629\pi$
$953$	−15.8479	−0.513364	−0.256682	+	0.966496i	$0.582629\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	11.5264	0.372596
$958$	0	0
$959$	−12.5814	−0.406275
$960$	0	0
$961$	−18.9967	−0.612798
$962$	0	0
$963$	25.7907	0.831094
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.2150	0.650072	0.325036	−	0.945702i	$-0.394624\pi$
$967$	20.2150	0.650072	0.325036	+	0.945702i	$0.394624\pi$
$968$	0	0
$969$	−10.1683	−0.326652
$970$	0	0
$971$	26.8903	0.862950	0.431475	−	0.902125i	$-0.357993\pi$
$971$	26.8903	0.862950	0.431475	+	0.902125i	$0.357993\pi$
$972$	0	0
$973$	−39.1477	−1.25502
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.6432	0.596450	0.298225	−	0.954496i	$-0.403605\pi$
$977$	18.6432	0.596450	0.298225	+	0.954496i	$0.403605\pi$
$978$	0	0
$979$	32.0780	1.02522
$980$	0	0
$981$	−6.26785	−0.200117
$982$	0	0
$983$	36.7713	1.17282	0.586411	−	0.810014i	$-0.300540\pi$
$983$	36.7713	1.17282	0.586411	+	0.810014i	$0.300540\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−43.4574	−1.38327
$988$	0	0
$989$	−7.14557	−0.227216
$990$	0	0
$991$	60.0197	1.90659	0.953294	−	0.302043i	$-0.0976687\pi$
$991$	60.0197	1.90659	0.953294	+	0.302043i	$0.0976687\pi$
$992$	0	0
$993$	−76.8928	−2.44012
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−28.8364	−0.913257	−0.456629	−	0.889657i	$-0.650943\pi$
$997$	−28.8364	−0.913257	−0.456629	+	0.889657i	$0.650943\pi$
$998$	0	0
$999$	19.4178	0.614352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ca.1.1		3
4.3	odd	2		1900.2.a.g.1.3	✓	3
5.4	even	2		7600.2.a.bl.1.3		3
20.3	even	4		1900.2.c.f.1749.5		6
20.7	even	4		1900.2.c.f.1749.2		6
20.19	odd	2		1900.2.a.i.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.g.1.3	✓	3	4.3	odd	2
1900.2.a.i.1.1	yes	3	20.19	odd	2
1900.2.c.f.1749.2		6	20.7	even	4
1900.2.c.f.1749.5		6	20.3	even	4
7600.2.a.bl.1.3		3	5.4	even	2
7600.2.a.ca.1.1		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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18194 q^{3} +3.70370 q^{7} +1.76088 q^{9} +O(q^{10})\)	\(q-2.18194 q^{3} +3.70370 q^{7} +1.76088 q^{9} +3.18194 q^{11} +1.94282 q^{13} -4.66019 q^{17} -1.00000 q^{19} -8.08126 q^{21} +5.23912 q^{23} +2.70370 q^{27} -1.66019 q^{29} +3.46457 q^{31} -6.94282 q^{33} +7.18194 q^{37} -4.23912 q^{39} -0.717370 q^{41} -1.36389 q^{43} +5.37756 q^{47} +6.71737 q^{49} +10.1683 q^{51} -0.0435069 q^{53} +2.18194 q^{57} +7.88564 q^{59} -4.98633 q^{61} +6.52175 q^{63} +14.5023 q^{67} -11.4315 q^{69} -4.88564 q^{71} -10.6030 q^{73} +11.7850 q^{77} -14.9097 q^{79} -11.1819 q^{81} +9.76088 q^{83} +3.62244 q^{87} +10.0813 q^{89} +7.19562 q^{91} -7.55950 q^{93} +14.6706 q^{97} +5.60301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9	\( 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9} + q^{11} - 3 q^{13} - 6 q^{17} - 3 q^{19} - 8 q^{21} + 16 q^{23} - q^{27} + 3 q^{29} + q^{31} - 12 q^{33} + 13 q^{37} - 13 q^{39} - 3 q^{41} + 13 q^{43} + 9 q^{47} + 21 q^{49} + 12 q^{51} + q^{53} - 2 q^{57} + 6 q^{59} - 5 q^{61} + 19 q^{63} + 19 q^{67} + 9 q^{69} + 3 q^{71} - 15 q^{73} + 10 q^{77} - 2 q^{79} - 25 q^{81} + 29 q^{83} + 18 q^{87} + 14 q^{89} + 23 q^{91} - 7 q^{93} + q^{97}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9 + q^11 - 3 * q^13 - 6 * q^17 - 3 * q^19 - 8 * q^21 + 16 * q^23 - q^27 + 3 * q^29 + q^31 - 12 * q^33 + 13 * q^37 - 13 * q^39 - 3 * q^41 + 13 * q^43 + 9 * q^47 + 21 * q^49 + 12 * q^51 + q^53 - 2 * q^57 + 6 * q^59 - 5 * q^61 + 19 * q^63 + 19 * q^67 + 9 * q^69 + 3 * q^71 - 15 * q^73 + 10 * q^77 - 2 * q^79 - 25 * q^81 + 29 * q^83 + 18 * q^87 + 14 * q^89 + 23 * q^91 - 7 * q^93 + q^97

Introduction

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