LMFDB - Embedded newform 7350.2.a.de.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -6.24264 q^{11} +1.00000 q^{12} -0.171573 q^{13} +1.00000 q^{16} +4.65685 q^{17} -1.00000 q^{18} +2.58579 q^{19} +6.24264 q^{22} -1.58579 q^{23} -1.00000 q^{24} +0.171573 q^{26} +1.00000 q^{27} -10.6569 q^{29} +7.24264 q^{31} -1.00000 q^{32} -6.24264 q^{33} -4.65685 q^{34} +1.00000 q^{36} -0.242641 q^{37} -2.58579 q^{38} -0.171573 q^{39} +7.00000 q^{41} +3.58579 q^{43} -6.24264 q^{44} +1.58579 q^{46} -8.00000 q^{47} +1.00000 q^{48} +4.65685 q^{51} -0.171573 q^{52} -2.65685 q^{53} -1.00000 q^{54} +2.58579 q^{57} +10.6569 q^{58} -1.58579 q^{59} +2.65685 q^{61} -7.24264 q^{62} +1.00000 q^{64} +6.24264 q^{66} -9.65685 q^{67} +4.65685 q^{68} -1.58579 q^{69} -2.00000 q^{71} -1.00000 q^{72} -8.24264 q^{73} +0.242641 q^{74} +2.58579 q^{76} +0.171573 q^{78} +6.24264 q^{79} +1.00000 q^{81} -7.00000 q^{82} +5.58579 q^{83} -3.58579 q^{86} -10.6569 q^{87} +6.24264 q^{88} +15.3137 q^{89} -1.58579 q^{92} +7.24264 q^{93} +8.00000 q^{94} -1.00000 q^{96} -14.0000 q^{97} -6.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 8 q^{19} + 4 q^{22} - 6 q^{23} - 2 q^{24} + 6 q^{26} + 2 q^{27} - 10 q^{29} + 6 q^{31} - 2 q^{32} - 4 q^{33} + 2 q^{34} + 2 q^{36} + 8 q^{37} - 8 q^{38} - 6 q^{39} + 14 q^{41} + 10 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} + 2 q^{48} - 2 q^{51} - 6 q^{52} + 6 q^{53} - 2 q^{54} + 8 q^{57} + 10 q^{58} - 6 q^{59} - 6 q^{61} - 6 q^{62} + 2 q^{64} + 4 q^{66} - 8 q^{67} - 2 q^{68} - 6 q^{69} - 4 q^{71} - 2 q^{72} - 8 q^{73} - 8 q^{74} + 8 q^{76} + 6 q^{78} + 4 q^{79} + 2 q^{81} - 14 q^{82} + 14 q^{83} - 10 q^{86} - 10 q^{87} + 4 q^{88} + 8 q^{89} - 6 q^{92} + 6 q^{93} + 16 q^{94} - 2 q^{96} - 28 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 - 6 * q^13 + 2 * q^16 - 2 * q^17 - 2 * q^18 + 8 * q^19 + 4 * q^22 - 6 * q^23 - 2 * q^24 + 6 * q^26 + 2 * q^27 - 10 * q^29 + 6 * q^31 - 2 * q^32 - 4 * q^33 + 2 * q^34 + 2 * q^36 + 8 * q^37 - 8 * q^38 - 6 * q^39 + 14 * q^41 + 10 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 + 2 * q^48 - 2 * q^51 - 6 * q^52 + 6 * q^53 - 2 * q^54 + 8 * q^57 + 10 * q^58 - 6 * q^59 - 6 * q^61 - 6 * q^62 + 2 * q^64 + 4 * q^66 - 8 * q^67 - 2 * q^68 - 6 * q^69 - 4 * q^71 - 2 * q^72 - 8 * q^73 - 8 * q^74 + 8 * q^76 + 6 * q^78 + 4 * q^79 + 2 * q^81 - 14 * q^82 + 14 * q^83 - 10 * q^86 - 10 * q^87 + 4 * q^88 + 8 * q^89 - 6 * q^92 + 6 * q^93 + 16 * q^94 - 2 * q^96 - 28 * q^97 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.24264	−1.88223	−0.941113	−	0.338091i	$-0.890219\pi$
$11$	−6.24264	−1.88223	−0.941113	+	0.338091i	$0.890219\pi$
$12$	1.00000	0.288675
$13$	−0.171573	−0.0475858	−0.0237929	−	0.999717i	$-0.507574\pi$
$13$	−0.171573	−0.0475858	−0.0237929	+	0.999717i	$0.507574\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.65685	1.12945	0.564727	−	0.825278i	$-0.308982\pi$
$17$	4.65685	1.12945	0.564727	+	0.825278i	$0.308982\pi$
$18$	−1.00000	−0.235702
$19$	2.58579	0.593220	0.296610	−	0.954999i	$-0.404144\pi$
$19$	2.58579	0.593220	0.296610	+	0.954999i	$0.404144\pi$
$20$	0	0
$21$	0	0
$22$	6.24264	1.33094
$23$	−1.58579	−0.330659	−0.165330	−	0.986238i	$-0.552869\pi$
$23$	−1.58579	−0.330659	−0.165330	+	0.986238i	$0.552869\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	0.171573	0.0336482
$27$	1.00000	0.192450
$28$	0	0
$29$	−10.6569	−1.97893	−0.989464	−	0.144779i	$-0.953753\pi$
$29$	−10.6569	−1.97893	−0.989464	+	0.144779i	$0.953753\pi$
$30$	0	0
$31$	7.24264	1.30082	0.650408	−	0.759585i	$-0.274598\pi$
$31$	7.24264	1.30082	0.650408	+	0.759585i	$0.274598\pi$
$32$	−1.00000	−0.176777
$33$	−6.24264	−1.08670
$34$	−4.65685	−0.798644
$35$	0	0
$36$	1.00000	0.166667
$37$	−0.242641	−0.0398899	−0.0199449	−	0.999801i	$-0.506349\pi$
$37$	−0.242641	−0.0398899	−0.0199449	+	0.999801i	$0.506349\pi$
$38$	−2.58579	−0.419470
$39$	−0.171573	−0.0274736
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	3.58579	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$43$	3.58579	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$44$	−6.24264	−0.941113
$45$	0	0
$46$	1.58579	0.233811
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	0	0
$51$	4.65685	0.652090
$52$	−0.171573	−0.0237929
$53$	−2.65685	−0.364947	−0.182473	−	0.983211i	$-0.558410\pi$
$53$	−2.65685	−0.364947	−0.182473	+	0.983211i	$0.558410\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	2.58579	0.342496
$58$	10.6569	1.39931
$59$	−1.58579	−0.206452	−0.103226	−	0.994658i	$-0.532916\pi$
$59$	−1.58579	−0.206452	−0.103226	+	0.994658i	$0.532916\pi$
$60$	0	0
$61$	2.65685	0.340175	0.170088	−	0.985429i	$-0.445595\pi$
$61$	2.65685	0.340175	0.170088	+	0.985429i	$0.445595\pi$
$62$	−7.24264	−0.919816
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.24264	0.768416
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	4.65685	0.564727
$69$	−1.58579	−0.190906
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−1.00000	−0.117851
$73$	−8.24264	−0.964728	−0.482364	−	0.875971i	$-0.660222\pi$
$73$	−8.24264	−0.964728	−0.482364	+	0.875971i	$0.660222\pi$
$74$	0.242641	0.0282064
$75$	0	0
$76$	2.58579	0.296610
$77$	0	0
$78$	0.171573	0.0194268
$79$	6.24264	0.702352	0.351176	−	0.936309i	$-0.385782\pi$
$79$	6.24264	0.702352	0.351176	+	0.936309i	$0.385782\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	5.58579	0.613120	0.306560	−	0.951851i	$-0.400822\pi$
$83$	5.58579	0.613120	0.306560	+	0.951851i	$0.400822\pi$
$84$	0	0
$85$	0	0
$86$	−3.58579	−0.386665
$87$	−10.6569	−1.14253
$88$	6.24264	0.665468
$89$	15.3137	1.62325	0.811625	−	0.584179i	$-0.198583\pi$
$89$	15.3137	1.62325	0.811625	+	0.584179i	$0.198583\pi$
$90$	0	0
$91$	0	0
$92$	−1.58579	−0.165330
$93$	7.24264	0.751027
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−6.24264	−0.627409
$100$	0	0
$101$	−17.6569	−1.75692	−0.878461	−	0.477813i	$-0.841429\pi$
$101$	−17.6569	−1.75692	−0.878461	+	0.477813i	$0.841429\pi$
$102$	−4.65685	−0.461097
$103$	−1.58579	−0.156252	−0.0781261	−	0.996943i	$-0.524894\pi$
$103$	−1.58579	−0.156252	−0.0781261	+	0.996943i	$0.524894\pi$
$104$	0.171573	0.0168241
$105$	0	0
$106$	2.65685	0.258056
$107$	12.1421	1.17382	0.586912	−	0.809651i	$-0.300343\pi$
$107$	12.1421	1.17382	0.586912	+	0.809651i	$0.300343\pi$
$108$	1.00000	0.0962250
$109$	7.31371	0.700526	0.350263	−	0.936651i	$-0.386092\pi$
$109$	7.31371	0.700526	0.350263	+	0.936651i	$0.386092\pi$
$110$	0	0
$111$	−0.242641	−0.0230304
$112$	0	0
$113$	−12.2426	−1.15169	−0.575845	−	0.817559i	$-0.695327\pi$
$113$	−12.2426	−1.15169	−0.575845	+	0.817559i	$0.695327\pi$
$114$	−2.58579	−0.242181
$115$	0	0
$116$	−10.6569	−0.989464
$117$	−0.171573	−0.0158619
$118$	1.58579	0.145983
$119$	0	0
$120$	0	0
$121$	27.9706	2.54278
$122$	−2.65685	−0.240540
$123$	7.00000	0.631169
$124$	7.24264	0.650408
$125$	0	0
$126$	0	0
$127$	−15.1716	−1.34626	−0.673130	−	0.739524i	$-0.735050\pi$
$127$	−15.1716	−1.34626	−0.673130	+	0.739524i	$0.735050\pi$
$128$	−1.00000	−0.0883883
$129$	3.58579	0.315711
$130$	0	0
$131$	−2.82843	−0.247121	−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843	−0.247121	−0.123560	+	0.992337i	$0.539431\pi$
$132$	−6.24264	−0.543352
$133$	0	0
$134$	9.65685	0.834225
$135$	0	0
$136$	−4.65685	−0.399322
$137$	−5.07107	−0.433251	−0.216625	−	0.976255i	$-0.569505\pi$
$137$	−5.07107	−0.433251	−0.216625	+	0.976255i	$0.569505\pi$
$138$	1.58579	0.134991
$139$	6.58579	0.558599	0.279300	−	0.960204i	$-0.409898\pi$
$139$	6.58579	0.558599	0.279300	+	0.960204i	$0.409898\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	2.00000	0.167836
$143$	1.07107	0.0895672
$144$	1.00000	0.0833333
$145$	0	0
$146$	8.24264	0.682166
$147$	0	0
$148$	−0.242641	−0.0199449
$149$	−5.34315	−0.437728	−0.218864	−	0.975755i	$-0.570235\pi$
$149$	−5.34315	−0.437728	−0.218864	+	0.975755i	$0.570235\pi$
$150$	0	0
$151$	−21.5563	−1.75423	−0.877115	−	0.480280i	$-0.840535\pi$
$151$	−21.5563	−1.75423	−0.877115	+	0.480280i	$0.840535\pi$
$152$	−2.58579	−0.209735
$153$	4.65685	0.376484
$154$	0	0
$155$	0	0
$156$	−0.171573	−0.0137368
$157$	−13.1716	−1.05121	−0.525603	−	0.850730i	$-0.676160\pi$
$157$	−13.1716	−1.05121	−0.525603	+	0.850730i	$0.676160\pi$
$158$	−6.24264	−0.496638
$159$	−2.65685	−0.210702
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	8.89949	0.697062	0.348531	−	0.937297i	$-0.386681\pi$
$163$	8.89949	0.697062	0.348531	+	0.937297i	$0.386681\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	−5.58579	−0.433541
$167$	−18.4853	−1.43043	−0.715217	−	0.698902i	$-0.753672\pi$
$167$	−18.4853	−1.43043	−0.715217	+	0.698902i	$0.753672\pi$
$168$	0	0
$169$	−12.9706	−0.997736
$170$	0	0
$171$	2.58579	0.197740
$172$	3.58579	0.273414
$173$	−9.55635	−0.726556	−0.363278	−	0.931681i	$-0.618342\pi$
$173$	−9.55635	−0.726556	−0.363278	+	0.931681i	$0.618342\pi$
$174$	10.6569	0.807894
$175$	0	0
$176$	−6.24264	−0.470557
$177$	−1.58579	−0.119195
$178$	−15.3137	−1.14781
$179$	−9.75736	−0.729299	−0.364650	−	0.931145i	$-0.618811\pi$
$179$	−9.75736	−0.729299	−0.364650	+	0.931145i	$0.618811\pi$
$180$	0	0
$181$	−20.9706	−1.55873	−0.779365	−	0.626570i	$-0.784458\pi$
$181$	−20.9706	−1.55873	−0.779365	+	0.626570i	$0.784458\pi$
$182$	0	0
$183$	2.65685	0.196400
$184$	1.58579	0.116906
$185$	0	0
$186$	−7.24264	−0.531056
$187$	−29.0711	−2.12589
$188$	−8.00000	−0.583460
$189$	0	0
$190$	0	0
$191$	7.72792	0.559173	0.279586	−	0.960121i	$-0.409803\pi$
$191$	7.72792	0.559173	0.279586	+	0.960121i	$0.409803\pi$
$192$	1.00000	0.0721688
$193$	−15.7990	−1.13724	−0.568618	−	0.822602i	$-0.692522\pi$
$193$	−15.7990	−1.13724	−0.568618	+	0.822602i	$0.692522\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	0	0
$197$	11.1421	0.793844	0.396922	−	0.917852i	$-0.370078\pi$
$197$	11.1421	0.793844	0.396922	+	0.917852i	$0.370078\pi$
$198$	6.24264	0.443645
$199$	14.9706	1.06124	0.530618	−	0.847611i	$-0.321960\pi$
$199$	14.9706	1.06124	0.530618	+	0.847611i	$0.321960\pi$
$200$	0	0
$201$	−9.65685	−0.681142
$202$	17.6569	1.24233
$203$	0	0
$204$	4.65685	0.326045
$205$	0	0
$206$	1.58579	0.110487
$207$	−1.58579	−0.110220
$208$	−0.171573	−0.0118964
$209$	−16.1421	−1.11657
$210$	0	0
$211$	13.9289	0.958907	0.479454	−	0.877567i	$-0.340835\pi$
$211$	13.9289	0.958907	0.479454	+	0.877567i	$0.340835\pi$
$212$	−2.65685	−0.182473
$213$	−2.00000	−0.137038
$214$	−12.1421	−0.830019
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−7.31371	−0.495347
$219$	−8.24264	−0.556986
$220$	0	0
$221$	−0.798990	−0.0537459
$222$	0.242641	0.0162850
$223$	18.0711	1.21013	0.605064	−	0.796177i	$-0.293147\pi$
$223$	18.0711	1.21013	0.605064	+	0.796177i	$0.293147\pi$
$224$	0	0
$225$	0	0
$226$	12.2426	0.814368
$227$	−29.2426	−1.94090	−0.970451	−	0.241298i	$-0.922427\pi$
$227$	−29.2426	−1.94090	−0.970451	+	0.241298i	$0.922427\pi$
$228$	2.58579	0.171248
$229$	−9.31371	−0.615467	−0.307734	−	0.951473i	$-0.599571\pi$
$229$	−9.31371	−0.615467	−0.307734	+	0.951473i	$0.599571\pi$
$230$	0	0
$231$	0	0
$232$	10.6569	0.699657
$233$	14.7279	0.964858	0.482429	−	0.875935i	$-0.339755\pi$
$233$	14.7279	0.964858	0.482429	+	0.875935i	$0.339755\pi$
$234$	0.171573	0.0112161
$235$	0	0
$236$	−1.58579	−0.103226
$237$	6.24264	0.405503
$238$	0	0
$239$	−19.3137	−1.24930	−0.624650	−	0.780905i	$-0.714758\pi$
$239$	−19.3137	−1.24930	−0.624650	+	0.780905i	$0.714758\pi$
$240$	0	0
$241$	−19.6569	−1.26621	−0.633105	−	0.774066i	$-0.718220\pi$
$241$	−19.6569	−1.26621	−0.633105	+	0.774066i	$0.718220\pi$
$242$	−27.9706	−1.79802
$243$	1.00000	0.0641500
$244$	2.65685	0.170088
$245$	0	0
$246$	−7.00000	−0.446304
$247$	−0.443651	−0.0282288
$248$	−7.24264	−0.459908
$249$	5.58579	0.353985
$250$	0	0
$251$	14.2132	0.897129	0.448565	−	0.893750i	$-0.351935\pi$
$251$	14.2132	0.897129	0.448565	+	0.893750i	$0.351935\pi$
$252$	0	0
$253$	9.89949	0.622376
$254$	15.1716	0.951949
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.1716	−1.38302	−0.691512	−	0.722365i	$-0.743055\pi$
$257$	−22.1716	−1.38302	−0.691512	+	0.722365i	$0.743055\pi$
$258$	−3.58579	−0.223241
$259$	0	0
$260$	0	0
$261$	−10.6569	−0.659643
$262$	2.82843	0.174741
$263$	−32.4142	−1.99875	−0.999373	−	0.0354058i	$-0.988728\pi$
$263$	−32.4142	−1.99875	−0.999373	+	0.0354058i	$0.988728\pi$
$264$	6.24264	0.384208
$265$	0	0
$266$	0	0
$267$	15.3137	0.937184
$268$	−9.65685	−0.589886
$269$	1.41421	0.0862261	0.0431131	−	0.999070i	$-0.486272\pi$
$269$	1.41421	0.0862261	0.0431131	+	0.999070i	$0.486272\pi$
$270$	0	0
$271$	29.4558	1.78932	0.894658	−	0.446753i	$-0.147420\pi$
$271$	29.4558	1.78932	0.894658	+	0.446753i	$0.147420\pi$
$272$	4.65685	0.282363
$273$	0	0
$274$	5.07107	0.306354
$275$	0	0
$276$	−1.58579	−0.0954531
$277$	−24.0416	−1.44452	−0.722261	−	0.691621i	$-0.756897\pi$
$277$	−24.0416	−1.44452	−0.722261	+	0.691621i	$0.756897\pi$
$278$	−6.58579	−0.394989
$279$	7.24264	0.433606
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	8.00000	0.476393
$283$	19.7574	1.17445	0.587227	−	0.809423i	$-0.300220\pi$
$283$	19.7574	1.17445	0.587227	+	0.809423i	$0.300220\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−1.07107	−0.0633336
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	4.68629	0.275664
$290$	0	0
$291$	−14.0000	−0.820695
$292$	−8.24264	−0.482364
$293$	−3.31371	−0.193589	−0.0967945	−	0.995304i	$-0.530859\pi$
$293$	−3.31371	−0.193589	−0.0967945	+	0.995304i	$0.530859\pi$
$294$	0	0
$295$	0	0
$296$	0.242641	0.0141032
$297$	−6.24264	−0.362235
$298$	5.34315	0.309520
$299$	0.272078	0.0157347
$300$	0	0
$301$	0	0
$302$	21.5563	1.24043
$303$	−17.6569	−1.01436
$304$	2.58579	0.148305
$305$	0	0
$306$	−4.65685	−0.266215
$307$	−11.4142	−0.651444	−0.325722	−	0.945466i	$-0.605607\pi$
$307$	−11.4142	−0.651444	−0.325722	+	0.945466i	$0.605607\pi$
$308$	0	0
$309$	−1.58579	−0.0902122
$310$	0	0
$311$	17.5563	0.995529	0.497765	−	0.867312i	$-0.334154\pi$
$311$	17.5563	0.995529	0.497765	+	0.867312i	$0.334154\pi$
$312$	0.171573	0.00971340
$313$	17.2132	0.972948	0.486474	−	0.873695i	$-0.338283\pi$
$313$	17.2132	0.972948	0.486474	+	0.873695i	$0.338283\pi$
$314$	13.1716	0.743315
$315$	0	0
$316$	6.24264	0.351176
$317$	−4.31371	−0.242282	−0.121141	−	0.992635i	$-0.538655\pi$
$317$	−4.31371	−0.242282	−0.121141	+	0.992635i	$0.538655\pi$
$318$	2.65685	0.148989
$319$	66.5269	3.72479
$320$	0	0
$321$	12.1421	0.677708
$322$	0	0
$323$	12.0416	0.670014
$324$	1.00000	0.0555556
$325$	0	0
$326$	−8.89949	−0.492897
$327$	7.31371	0.404449
$328$	−7.00000	−0.386510
$329$	0	0
$330$	0	0
$331$	−7.58579	−0.416953	−0.208476	−	0.978027i	$-0.566850\pi$
$331$	−7.58579	−0.416953	−0.208476	+	0.978027i	$0.566850\pi$
$332$	5.58579	0.306560
$333$	−0.242641	−0.0132966
$334$	18.4853	1.01147
$335$	0	0
$336$	0	0
$337$	20.1716	1.09882	0.549408	−	0.835554i	$-0.314853\pi$
$337$	20.1716	1.09882	0.549408	+	0.835554i	$0.314853\pi$
$338$	12.9706	0.705506
$339$	−12.2426	−0.664929
$340$	0	0
$341$	−45.2132	−2.44843
$342$	−2.58579	−0.139823
$343$	0	0
$344$	−3.58579	−0.193333
$345$	0	0
$346$	9.55635	0.513753
$347$	−29.7990	−1.59969	−0.799847	−	0.600204i	$-0.795086\pi$
$347$	−29.7990	−1.59969	−0.799847	+	0.600204i	$0.795086\pi$
$348$	−10.6569	−0.571267
$349$	−21.4853	−1.15008	−0.575040	−	0.818125i	$-0.695014\pi$
$349$	−21.4853	−1.15008	−0.575040	+	0.818125i	$0.695014\pi$
$350$	0	0
$351$	−0.171573	−0.00915788
$352$	6.24264	0.332734
$353$	34.1421	1.81720	0.908601	−	0.417665i	$-0.137151\pi$
$353$	34.1421	1.81720	0.908601	+	0.417665i	$0.137151\pi$
$354$	1.58579	0.0842836
$355$	0	0
$356$	15.3137	0.811625
$357$	0	0
$358$	9.75736	0.515692
$359$	−23.7279	−1.25231	−0.626156	−	0.779698i	$-0.715373\pi$
$359$	−23.7279	−1.25231	−0.626156	+	0.779698i	$0.715373\pi$
$360$	0	0
$361$	−12.3137	−0.648090
$362$	20.9706	1.10219
$363$	27.9706	1.46807
$364$	0	0
$365$	0	0
$366$	−2.65685	−0.138876
$367$	14.5563	0.759835	0.379918	−	0.925020i	$-0.375952\pi$
$367$	14.5563	0.759835	0.379918	+	0.925020i	$0.375952\pi$
$368$	−1.58579	−0.0826648
$369$	7.00000	0.364405
$370$	0	0
$371$	0	0
$372$	7.24264	0.375513
$373$	−9.65685	−0.500013	−0.250006	−	0.968244i	$-0.580433\pi$
$373$	−9.65685	−0.500013	−0.250006	+	0.968244i	$0.580433\pi$
$374$	29.0711	1.50323
$375$	0	0
$376$	8.00000	0.412568
$377$	1.82843	0.0941688
$378$	0	0
$379$	6.27208	0.322175	0.161088	−	0.986940i	$-0.448500\pi$
$379$	6.27208	0.322175	0.161088	+	0.986940i	$0.448500\pi$
$380$	0	0
$381$	−15.1716	−0.777263
$382$	−7.72792	−0.395395
$383$	−4.92893	−0.251857	−0.125928	−	0.992039i	$-0.540191\pi$
$383$	−4.92893	−0.251857	−0.125928	+	0.992039i	$0.540191\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	15.7990	0.804147
$387$	3.58579	0.182276
$388$	−14.0000	−0.710742
$389$	20.2843	1.02845	0.514227	−	0.857654i	$-0.328079\pi$
$389$	20.2843	1.02845	0.514227	+	0.857654i	$0.328079\pi$
$390$	0	0
$391$	−7.38478	−0.373464
$392$	0	0
$393$	−2.82843	−0.142675
$394$	−11.1421	−0.561333
$395$	0	0
$396$	−6.24264	−0.313704
$397$	−34.1127	−1.71207	−0.856034	−	0.516920i	$-0.827078\pi$
$397$	−34.1127	−1.71207	−0.856034	+	0.516920i	$0.827078\pi$
$398$	−14.9706	−0.750407
$399$	0	0
$400$	0	0
$401$	35.6985	1.78270	0.891349	−	0.453318i	$-0.149760\pi$
$401$	35.6985	1.78270	0.891349	+	0.453318i	$0.149760\pi$
$402$	9.65685	0.481640
$403$	−1.24264	−0.0619003
$404$	−17.6569	−0.878461
$405$	0	0
$406$	0	0
$407$	1.51472	0.0750818
$408$	−4.65685	−0.230549
$409$	−33.2132	−1.64229	−0.821144	−	0.570722i	$-0.806664\pi$
$409$	−33.2132	−1.64229	−0.821144	+	0.570722i	$0.806664\pi$
$410$	0	0
$411$	−5.07107	−0.250137
$412$	−1.58579	−0.0781261
$413$	0	0
$414$	1.58579	0.0779372
$415$	0	0
$416$	0.171573	0.00841205
$417$	6.58579	0.322507
$418$	16.1421	0.789538
$419$	11.7279	0.572946	0.286473	−	0.958088i	$-0.407517\pi$
$419$	11.7279	0.572946	0.286473	+	0.958088i	$0.407517\pi$
$420$	0	0
$421$	8.34315	0.406620	0.203310	−	0.979114i	$-0.434830\pi$
$421$	8.34315	0.406620	0.203310	+	0.979114i	$0.434830\pi$
$422$	−13.9289	−0.678050
$423$	−8.00000	−0.388973
$424$	2.65685	0.129028
$425$	0	0
$426$	2.00000	0.0969003
$427$	0	0
$428$	12.1421	0.586912
$429$	1.07107	0.0517116
$430$	0	0
$431$	32.6985	1.57503	0.787515	−	0.616295i	$-0.211367\pi$
$431$	32.6985	1.57503	0.787515	+	0.616295i	$0.211367\pi$
$432$	1.00000	0.0481125
$433$	4.34315	0.208718	0.104359	−	0.994540i	$-0.466721\pi$
$433$	4.34315	0.208718	0.104359	+	0.994540i	$0.466721\pi$
$434$	0	0
$435$	0	0
$436$	7.31371	0.350263
$437$	−4.10051	−0.196154
$438$	8.24264	0.393849
$439$	36.2132	1.72836	0.864181	−	0.503181i	$-0.167837\pi$
$439$	36.2132	1.72836	0.864181	+	0.503181i	$0.167837\pi$
$440$	0	0
$441$	0	0
$442$	0.798990	0.0380041
$443$	−14.6274	−0.694970	−0.347485	−	0.937686i	$-0.612964\pi$
$443$	−14.6274	−0.694970	−0.347485	+	0.937686i	$0.612964\pi$
$444$	−0.242641	−0.0115152
$445$	0	0
$446$	−18.0711	−0.855690
$447$	−5.34315	−0.252722
$448$	0	0
$449$	−26.7279	−1.26137	−0.630684	−	0.776039i	$-0.717226\pi$
$449$	−26.7279	−1.26137	−0.630684	+	0.776039i	$0.717226\pi$
$450$	0	0
$451$	−43.6985	−2.05768
$452$	−12.2426	−0.575845
$453$	−21.5563	−1.01281
$454$	29.2426	1.37243
$455$	0	0
$456$	−2.58579	−0.121091
$457$	10.4558	0.489104	0.244552	−	0.969636i	$-0.421359\pi$
$457$	10.4558	0.489104	0.244552	+	0.969636i	$0.421359\pi$
$458$	9.31371	0.435201
$459$	4.65685	0.217363
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	0	0
$463$	31.5563	1.46655	0.733274	−	0.679933i	$-0.237991\pi$
$463$	31.5563	1.46655	0.733274	+	0.679933i	$0.237991\pi$
$464$	−10.6569	−0.494732
$465$	0	0
$466$	−14.7279	−0.682258
$467$	−0.272078	−0.0125903	−0.00629513	−	0.999980i	$-0.502004\pi$
$467$	−0.272078	−0.0125903	−0.00629513	+	0.999980i	$0.502004\pi$
$468$	−0.171573	−0.00793096
$469$	0	0
$470$	0	0
$471$	−13.1716	−0.606914
$472$	1.58579	0.0729917
$473$	−22.3848	−1.02925
$474$	−6.24264	−0.286734
$475$	0	0
$476$	0	0
$477$	−2.65685	−0.121649
$478$	19.3137	0.883388
$479$	−0.727922	−0.0332596	−0.0166298	−	0.999862i	$-0.505294\pi$
$479$	−0.727922	−0.0332596	−0.0166298	+	0.999862i	$0.505294\pi$
$480$	0	0
$481$	0.0416306	0.00189819
$482$	19.6569	0.895345
$483$	0	0
$484$	27.9706	1.27139
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−0.142136	−0.00644078	−0.00322039	−	0.999995i	$-0.501025\pi$
$487$	−0.142136	−0.00644078	−0.00322039	+	0.999995i	$0.501025\pi$
$488$	−2.65685	−0.120270
$489$	8.89949	0.402449
$490$	0	0
$491$	−0.828427	−0.0373864	−0.0186932	−	0.999825i	$-0.505951\pi$
$491$	−0.828427	−0.0373864	−0.0186932	+	0.999825i	$0.505951\pi$
$492$	7.00000	0.315584
$493$	−49.6274	−2.23511
$494$	0.443651	0.0199608
$495$	0	0
$496$	7.24264	0.325204
$497$	0	0
$498$	−5.58579	−0.250305
$499$	20.0711	0.898504	0.449252	−	0.893405i	$-0.351690\pi$
$499$	20.0711	0.898504	0.449252	+	0.893405i	$0.351690\pi$
$500$	0	0
$501$	−18.4853	−0.825861
$502$	−14.2132	−0.634366
$503$	−36.3848	−1.62232	−0.811158	−	0.584826i	$-0.801163\pi$
$503$	−36.3848	−1.62232	−0.811158	+	0.584826i	$0.801163\pi$
$504$	0	0
$505$	0	0
$506$	−9.89949	−0.440086
$507$	−12.9706	−0.576043
$508$	−15.1716	−0.673130
$509$	−18.7279	−0.830101	−0.415050	−	0.909798i	$-0.636236\pi$
$509$	−18.7279	−0.830101	−0.415050	+	0.909798i	$0.636236\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	2.58579	0.114165
$514$	22.1716	0.977946
$515$	0	0
$516$	3.58579	0.157855
$517$	49.9411	2.19641
$518$	0	0
$519$	−9.55635	−0.419477
$520$	0	0
$521$	−1.82843	−0.0801048	−0.0400524	−	0.999198i	$-0.512752\pi$
$521$	−1.82843	−0.0801048	−0.0400524	+	0.999198i	$0.512752\pi$
$522$	10.6569	0.466438
$523$	−3.31371	−0.144898	−0.0724492	−	0.997372i	$-0.523082\pi$
$523$	−3.31371	−0.144898	−0.0724492	+	0.997372i	$0.523082\pi$
$524$	−2.82843	−0.123560
$525$	0	0
$526$	32.4142	1.41333
$527$	33.7279	1.46921
$528$	−6.24264	−0.271676
$529$	−20.4853	−0.890664
$530$	0	0
$531$	−1.58579	−0.0688173
$532$	0	0
$533$	−1.20101	−0.0520215
$534$	−15.3137	−0.662689
$535$	0	0
$536$	9.65685	0.417113
$537$	−9.75736	−0.421061
$538$	−1.41421	−0.0609711
$539$	0	0
$540$	0	0
$541$	18.9289	0.813818	0.406909	−	0.913469i	$-0.366607\pi$
$541$	18.9289	0.813818	0.406909	+	0.913469i	$0.366607\pi$
$542$	−29.4558	−1.26524
$543$	−20.9706	−0.899933
$544$	−4.65685	−0.199661
$545$	0	0
$546$	0	0
$547$	−32.4142	−1.38593	−0.692966	−	0.720970i	$-0.743696\pi$
$547$	−32.4142	−1.38593	−0.692966	+	0.720970i	$0.743696\pi$
$548$	−5.07107	−0.216625
$549$	2.65685	0.113392
$550$	0	0
$551$	−27.5563	−1.17394
$552$	1.58579	0.0674956
$553$	0	0
$554$	24.0416	1.02143
$555$	0	0
$556$	6.58579	0.279300
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	−7.24264	−0.306605
$559$	−0.615224	−0.0260212
$560$	0	0
$561$	−29.0711	−1.22738
$562$	4.00000	0.168730
$563$	38.6985	1.63095	0.815473	−	0.578795i	$-0.196477\pi$
$563$	38.6985	1.63095	0.815473	+	0.578795i	$0.196477\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−19.7574	−0.830464
$567$	0	0
$568$	2.00000	0.0839181
$569$	−3.27208	−0.137173	−0.0685863	−	0.997645i	$-0.521849\pi$
$569$	−3.27208	−0.137173	−0.0685863	+	0.997645i	$0.521849\pi$
$570$	0	0
$571$	28.8995	1.20941	0.604703	−	0.796451i	$-0.293292\pi$
$571$	28.8995	1.20941	0.604703	+	0.796451i	$0.293292\pi$
$572$	1.07107	0.0447836
$573$	7.72792	0.322839
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−28.6274	−1.19177	−0.595887	−	0.803068i	$-0.703200\pi$
$577$	−28.6274	−1.19177	−0.595887	+	0.803068i	$0.703200\pi$
$578$	−4.68629	−0.194924
$579$	−15.7990	−0.656584
$580$	0	0
$581$	0	0
$582$	14.0000	0.580319
$583$	16.5858	0.686913
$584$	8.24264	0.341083
$585$	0	0
$586$	3.31371	0.136888
$587$	−33.5269	−1.38380	−0.691902	−	0.721992i	$-0.743227\pi$
$587$	−33.5269	−1.38380	−0.691902	+	0.721992i	$0.743227\pi$
$588$	0	0
$589$	18.7279	0.771671
$590$	0	0
$591$	11.1421	0.458326
$592$	−0.242641	−0.00997247
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	6.24264	0.256139
$595$	0	0
$596$	−5.34315	−0.218864
$597$	14.9706	0.612704
$598$	−0.272078	−0.0111261
$599$	22.2132	0.907607	0.453803	−	0.891102i	$-0.350067\pi$
$599$	22.2132	0.907607	0.453803	+	0.891102i	$0.350067\pi$
$600$	0	0
$601$	−34.2426	−1.39679	−0.698393	−	0.715714i	$-0.746101\pi$
$601$	−34.2426	−1.39679	−0.698393	+	0.715714i	$0.746101\pi$
$602$	0	0
$603$	−9.65685	−0.393258
$604$	−21.5563	−0.877115
$605$	0	0
$606$	17.6569	0.717261
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−2.58579	−0.104867
$609$	0	0
$610$	0	0
$611$	1.37258	0.0555288
$612$	4.65685	0.188242
$613$	2.68629	0.108498	0.0542491	−	0.998527i	$-0.482723\pi$
$613$	2.68629	0.108498	0.0542491	+	0.998527i	$0.482723\pi$
$614$	11.4142	0.460640
$615$	0	0
$616$	0	0
$617$	−12.7279	−0.512407	−0.256203	−	0.966623i	$-0.582472\pi$
$617$	−12.7279	−0.512407	−0.256203	+	0.966623i	$0.582472\pi$
$618$	1.58579	0.0637897
$619$	3.85786	0.155061	0.0775303	−	0.996990i	$-0.475297\pi$
$619$	3.85786	0.155061	0.0775303	+	0.996990i	$0.475297\pi$
$620$	0	0
$621$	−1.58579	−0.0636354
$622$	−17.5563	−0.703945
$623$	0	0
$624$	−0.171573	−0.00686841
$625$	0	0
$626$	−17.2132	−0.687978
$627$	−16.1421	−0.644655
$628$	−13.1716	−0.525603
$629$	−1.12994	−0.0450538
$630$	0	0
$631$	−3.17157	−0.126258	−0.0631292	−	0.998005i	$-0.520108\pi$
$631$	−3.17157	−0.126258	−0.0631292	+	0.998005i	$0.520108\pi$
$632$	−6.24264	−0.248319
$633$	13.9289	0.553625
$634$	4.31371	0.171319
$635$	0	0
$636$	−2.65685	−0.105351
$637$	0	0
$638$	−66.5269	−2.63383
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−13.2132	−0.521890	−0.260945	−	0.965354i	$-0.584034\pi$
$641$	−13.2132	−0.521890	−0.260945	+	0.965354i	$0.584034\pi$
$642$	−12.1421	−0.479212
$643$	−50.0416	−1.97345	−0.986725	−	0.162402i	$-0.948076\pi$
$643$	−50.0416	−1.97345	−0.986725	+	0.162402i	$0.948076\pi$
$644$	0	0
$645$	0	0
$646$	−12.0416	−0.473772
$647$	−10.9289	−0.429661	−0.214830	−	0.976651i	$-0.568920\pi$
$647$	−10.9289	−0.429661	−0.214830	+	0.976651i	$0.568920\pi$
$648$	−1.00000	−0.0392837
$649$	9.89949	0.388589
$650$	0	0
$651$	0	0
$652$	8.89949	0.348531
$653$	−18.3431	−0.717823	−0.358911	−	0.933372i	$-0.616852\pi$
$653$	−18.3431	−0.717823	−0.358911	+	0.933372i	$0.616852\pi$
$654$	−7.31371	−0.285989
$655$	0	0
$656$	7.00000	0.273304
$657$	−8.24264	−0.321576
$658$	0	0
$659$	−5.79899	−0.225897	−0.112948	−	0.993601i	$-0.536029\pi$
$659$	−5.79899	−0.225897	−0.112948	+	0.993601i	$0.536029\pi$
$660$	0	0
$661$	−24.9706	−0.971242	−0.485621	−	0.874169i	$-0.661407\pi$
$661$	−24.9706	−0.971242	−0.485621	+	0.874169i	$0.661407\pi$
$662$	7.58579	0.294830
$663$	−0.798990	−0.0310302
$664$	−5.58579	−0.216771
$665$	0	0
$666$	0.242641	0.00940214
$667$	16.8995	0.654351
$668$	−18.4853	−0.715217
$669$	18.0711	0.698668
$670$	0	0
$671$	−16.5858	−0.640287
$672$	0	0
$673$	43.1421	1.66301	0.831504	−	0.555519i	$-0.187481\pi$
$673$	43.1421	1.66301	0.831504	+	0.555519i	$0.187481\pi$
$674$	−20.1716	−0.776980
$675$	0	0
$676$	−12.9706	−0.498868
$677$	21.7574	0.836203	0.418102	−	0.908400i	$-0.362696\pi$
$677$	21.7574	0.836203	0.418102	+	0.908400i	$0.362696\pi$
$678$	12.2426	0.470176
$679$	0	0
$680$	0	0
$681$	−29.2426	−1.12058
$682$	45.2132	1.73130
$683$	22.9289	0.877351	0.438676	−	0.898645i	$-0.355448\pi$
$683$	22.9289	0.877351	0.438676	+	0.898645i	$0.355448\pi$
$684$	2.58579	0.0988700
$685$	0	0
$686$	0	0
$687$	−9.31371	−0.355340
$688$	3.58579	0.136707
$689$	0.455844	0.0173663
$690$	0	0
$691$	23.4558	0.892302	0.446151	−	0.894958i	$-0.352794\pi$
$691$	23.4558	0.892302	0.446151	+	0.894958i	$0.352794\pi$
$692$	−9.55635	−0.363278
$693$	0	0
$694$	29.7990	1.13115
$695$	0	0
$696$	10.6569	0.403947
$697$	32.5980	1.23474
$698$	21.4853	0.813230
$699$	14.7279	0.557061
$700$	0	0
$701$	19.0000	0.717620	0.358810	−	0.933411i	$-0.383183\pi$
$701$	19.0000	0.717620	0.358810	+	0.933411i	$0.383183\pi$
$702$	0.171573	0.00647560
$703$	−0.627417	−0.0236635
$704$	−6.24264	−0.235278
$705$	0	0
$706$	−34.1421	−1.28496
$707$	0	0
$708$	−1.58579	−0.0595975
$709$	4.10051	0.153998	0.0769989	−	0.997031i	$-0.475466\pi$
$709$	4.10051	0.153998	0.0769989	+	0.997031i	$0.475466\pi$
$710$	0	0
$711$	6.24264	0.234117
$712$	−15.3137	−0.573905
$713$	−11.4853	−0.430127
$714$	0	0
$715$	0	0
$716$	−9.75736	−0.364650
$717$	−19.3137	−0.721284
$718$	23.7279	0.885518
$719$	48.1838	1.79695	0.898476	−	0.439023i	$-0.144675\pi$
$719$	48.1838	1.79695	0.898476	+	0.439023i	$0.144675\pi$
$720$	0	0
$721$	0	0
$722$	12.3137	0.458269
$723$	−19.6569	−0.731046
$724$	−20.9706	−0.779365
$725$	0	0
$726$	−27.9706	−1.03808
$727$	25.8701	0.959467	0.479734	−	0.877414i	$-0.340733\pi$
$727$	25.8701	0.959467	0.479734	+	0.877414i	$0.340733\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.6985	0.617616
$732$	2.65685	0.0982002
$733$	34.9411	1.29058	0.645290	−	0.763938i	$-0.276737\pi$
$733$	34.9411	1.29058	0.645290	+	0.763938i	$0.276737\pi$
$734$	−14.5563	−0.537285
$735$	0	0
$736$	1.58579	0.0584529
$737$	60.2843	2.22060
$738$	−7.00000	−0.257674
$739$	3.44365	0.126677	0.0633384	−	0.997992i	$-0.479825\pi$
$739$	3.44365	0.126677	0.0633384	+	0.997992i	$0.479825\pi$
$740$	0	0
$741$	−0.443651	−0.0162979
$742$	0	0
$743$	−27.3848	−1.00465	−0.502325	−	0.864679i	$-0.667522\pi$
$743$	−27.3848	−1.00465	−0.502325	+	0.864679i	$0.667522\pi$
$744$	−7.24264	−0.265528
$745$	0	0
$746$	9.65685	0.353563
$747$	5.58579	0.204373
$748$	−29.0711	−1.06294
$749$	0	0
$750$	0	0
$751$	9.37258	0.342010	0.171005	−	0.985270i	$-0.445299\pi$
$751$	9.37258	0.342010	0.171005	+	0.985270i	$0.445299\pi$
$752$	−8.00000	−0.291730
$753$	14.2132	0.517958
$754$	−1.82843	−0.0665874
$755$	0	0
$756$	0	0
$757$	−47.5563	−1.72846	−0.864232	−	0.503093i	$-0.832195\pi$
$757$	−47.5563	−1.72846	−0.864232	+	0.503093i	$0.832195\pi$
$758$	−6.27208	−0.227812
$759$	9.89949	0.359329
$760$	0	0
$761$	−17.9411	−0.650365	−0.325183	−	0.945651i	$-0.605426\pi$
$761$	−17.9411	−0.650365	−0.325183	+	0.945651i	$0.605426\pi$
$762$	15.1716	0.549608
$763$	0	0
$764$	7.72792	0.279586
$765$	0	0
$766$	4.92893	0.178090
$767$	0.272078	0.00982416
$768$	1.00000	0.0360844
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	0	0
$771$	−22.1716	−0.798490
$772$	−15.7990	−0.568618
$773$	21.9411	0.789167	0.394584	−	0.918860i	$-0.370889\pi$
$773$	21.9411	0.789167	0.394584	+	0.918860i	$0.370889\pi$
$774$	−3.58579	−0.128888
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	−20.2843	−0.727226
$779$	18.1005	0.648518
$780$	0	0
$781$	12.4853	0.446758
$782$	7.38478	0.264079
$783$	−10.6569	−0.380845
$784$	0	0
$785$	0	0
$786$	2.82843	0.100887
$787$	−23.8579	−0.850441	−0.425221	−	0.905090i	$-0.639803\pi$
$787$	−23.8579	−0.850441	−0.425221	+	0.905090i	$0.639803\pi$
$788$	11.1421	0.396922
$789$	−32.4142	−1.15398
$790$	0	0
$791$	0	0
$792$	6.24264	0.221823
$793$	−0.455844	−0.0161875
$794$	34.1127	1.21061
$795$	0	0
$796$	14.9706	0.530618
$797$	−25.8995	−0.917407	−0.458704	−	0.888589i	$-0.651686\pi$
$797$	−25.8995	−0.917407	−0.458704	+	0.888589i	$0.651686\pi$
$798$	0	0
$799$	−37.2548	−1.31798
$800$	0	0
$801$	15.3137	0.541083
$802$	−35.6985	−1.26056
$803$	51.4558	1.81584
$804$	−9.65685	−0.340571
$805$	0	0
$806$	1.24264	0.0437702
$807$	1.41421	0.0497827
$808$	17.6569	0.621166
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	−22.0416	−0.773986	−0.386993	−	0.922083i	$-0.626486\pi$
$811$	−22.0416	−0.773986	−0.386993	+	0.922083i	$0.626486\pi$
$812$	0	0
$813$	29.4558	1.03306
$814$	−1.51472	−0.0530909
$815$	0	0
$816$	4.65685	0.163023
$817$	9.27208	0.324389
$818$	33.2132	1.16127
$819$	0	0
$820$	0	0
$821$	−40.4853	−1.41295	−0.706473	−	0.707740i	$-0.749715\pi$
$821$	−40.4853	−1.41295	−0.706473	+	0.707740i	$0.749715\pi$
$822$	5.07107	0.176874
$823$	15.2132	0.530299	0.265149	−	0.964207i	$-0.414579\pi$
$823$	15.2132	0.530299	0.265149	+	0.964207i	$0.414579\pi$
$824$	1.58579	0.0552435
$825$	0	0
$826$	0	0
$827$	13.2721	0.461515	0.230758	−	0.973011i	$-0.425880\pi$
$827$	13.2721	0.461515	0.230758	+	0.973011i	$0.425880\pi$
$828$	−1.58579	−0.0551099
$829$	−31.3431	−1.08859	−0.544296	−	0.838893i	$-0.683203\pi$
$829$	−31.3431	−1.08859	−0.544296	+	0.838893i	$0.683203\pi$
$830$	0	0
$831$	−24.0416	−0.833995
$832$	−0.171573	−0.00594822
$833$	0	0
$834$	−6.58579	−0.228047
$835$	0	0
$836$	−16.1421	−0.558287
$837$	7.24264	0.250342
$838$	−11.7279	−0.405134
$839$	−21.2132	−0.732361	−0.366181	−	0.930544i	$-0.619335\pi$
$839$	−21.2132	−0.732361	−0.366181	+	0.930544i	$0.619335\pi$
$840$	0	0
$841$	84.5685	2.91616
$842$	−8.34315	−0.287524
$843$	−4.00000	−0.137767
$844$	13.9289	0.479454
$845$	0	0
$846$	8.00000	0.275046
$847$	0	0
$848$	−2.65685	−0.0912367
$849$	19.7574	0.678071
$850$	0	0
$851$	0.384776	0.0131900
$852$	−2.00000	−0.0685189
$853$	35.1421	1.20324	0.601622	−	0.798781i	$-0.294521\pi$
$853$	35.1421	1.20324	0.601622	+	0.798781i	$0.294521\pi$
$854$	0	0
$855$	0	0
$856$	−12.1421	−0.415010
$857$	44.4853	1.51959	0.759794	−	0.650164i	$-0.225300\pi$
$857$	44.4853	1.51959	0.759794	+	0.650164i	$0.225300\pi$
$858$	−1.07107	−0.0365657
$859$	−39.3137	−1.34137	−0.670683	−	0.741744i	$-0.733999\pi$
$859$	−39.3137	−1.34137	−0.670683	+	0.741744i	$0.733999\pi$
$860$	0	0
$861$	0	0
$862$	−32.6985	−1.11371
$863$	−19.3137	−0.657446	−0.328723	−	0.944426i	$-0.606618\pi$
$863$	−19.3137	−0.657446	−0.328723	+	0.944426i	$0.606618\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−4.34315	−0.147586
$867$	4.68629	0.159155
$868$	0	0
$869$	−38.9706	−1.32199
$870$	0	0
$871$	1.65685	0.0561404
$872$	−7.31371	−0.247673
$873$	−14.0000	−0.473828
$874$	4.10051	0.138702
$875$	0	0
$876$	−8.24264	−0.278493
$877$	−40.9706	−1.38348	−0.691739	−	0.722148i	$-0.743155\pi$
$877$	−40.9706	−1.38348	−0.691739	+	0.722148i	$0.743155\pi$
$878$	−36.2132	−1.22214
$879$	−3.31371	−0.111769
$880$	0	0
$881$	43.4264	1.46307	0.731536	−	0.681802i	$-0.238804\pi$
$881$	43.4264	1.46307	0.731536	+	0.681802i	$0.238804\pi$
$882$	0	0
$883$	10.7574	0.362014	0.181007	−	0.983482i	$-0.442064\pi$
$883$	10.7574	0.362014	0.181007	+	0.983482i	$0.442064\pi$
$884$	−0.798990	−0.0268729
$885$	0	0
$886$	14.6274	0.491418
$887$	19.4142	0.651865	0.325933	−	0.945393i	$-0.394322\pi$
$887$	19.4142	0.651865	0.325933	+	0.945393i	$0.394322\pi$
$888$	0.242641	0.00814249
$889$	0	0
$890$	0	0
$891$	−6.24264	−0.209136
$892$	18.0711	0.605064
$893$	−20.6863	−0.692240
$894$	5.34315	0.178702
$895$	0	0
$896$	0	0
$897$	0.272078	0.00908442
$898$	26.7279	0.891922
$899$	−77.1838	−2.57422
$900$	0	0
$901$	−12.3726	−0.412191
$902$	43.6985	1.45500
$903$	0	0
$904$	12.2426	0.407184
$905$	0	0
$906$	21.5563	0.716162
$907$	−44.8995	−1.49086	−0.745432	−	0.666582i	$-0.767757\pi$
$907$	−44.8995	−1.49086	−0.745432	+	0.666582i	$0.767757\pi$
$908$	−29.2426	−0.970451
$909$	−17.6569	−0.585641
$910$	0	0
$911$	−39.2426	−1.30017	−0.650083	−	0.759863i	$-0.725266\pi$
$911$	−39.2426	−1.30017	−0.650083	+	0.759863i	$0.725266\pi$
$912$	2.58579	0.0856239
$913$	−34.8701	−1.15403
$914$	−10.4558	−0.345849
$915$	0	0
$916$	−9.31371	−0.307734
$917$	0	0
$918$	−4.65685	−0.153699
$919$	14.8701	0.490518	0.245259	−	0.969458i	$-0.421127\pi$
$919$	14.8701	0.490518	0.245259	+	0.969458i	$0.421127\pi$
$920$	0	0
$921$	−11.4142	−0.376111
$922$	8.00000	0.263466
$923$	0.343146	0.0112948
$924$	0	0
$925$	0	0
$926$	−31.5563	−1.03701
$927$	−1.58579	−0.0520841
$928$	10.6569	0.349828
$929$	−31.1421	−1.02174	−0.510870	−	0.859658i	$-0.670677\pi$
$929$	−31.1421	−1.02174	−0.510870	+	0.859658i	$0.670677\pi$
$930$	0	0
$931$	0	0
$932$	14.7279	0.482429
$933$	17.5563	0.574769
$934$	0.272078	0.00890266
$935$	0	0
$936$	0.171573	0.00560803
$937$	38.8701	1.26983	0.634915	−	0.772582i	$-0.281035\pi$
$937$	38.8701	1.26983	0.634915	+	0.772582i	$0.281035\pi$
$938$	0	0
$939$	17.2132	0.561732
$940$	0	0
$941$	−12.6863	−0.413561	−0.206781	−	0.978387i	$-0.566299\pi$
$941$	−12.6863	−0.413561	−0.206781	+	0.978387i	$0.566299\pi$
$942$	13.1716	0.429153
$943$	−11.1005	−0.361482
$944$	−1.58579	−0.0516130
$945$	0	0
$946$	22.3848	0.727792
$947$	27.5563	0.895461	0.447731	−	0.894169i	$-0.352232\pi$
$947$	27.5563	0.895461	0.447731	+	0.894169i	$0.352232\pi$
$948$	6.24264	0.202752
$949$	1.41421	0.0459073
$950$	0	0
$951$	−4.31371	−0.139882
$952$	0	0
$953$	55.5980	1.80100	0.900498	−	0.434861i	$-0.143202\pi$
$953$	55.5980	1.80100	0.900498	+	0.434861i	$0.143202\pi$
$954$	2.65685	0.0860188
$955$	0	0
$956$	−19.3137	−0.624650
$957$	66.5269	2.15051
$958$	0.727922	0.0235181
$959$	0	0
$960$	0	0
$961$	21.4558	0.692124
$962$	−0.0416306	−0.00134222
$963$	12.1421	0.391275
$964$	−19.6569	−0.633105
$965$	0	0
$966$	0	0
$967$	29.8995	0.961503	0.480751	−	0.876857i	$-0.340364\pi$
$967$	29.8995	0.961503	0.480751	+	0.876857i	$0.340364\pi$
$968$	−27.9706	−0.899008
$969$	12.0416	0.386833
$970$	0	0
$971$	−21.9411	−0.704124	−0.352062	−	0.935977i	$-0.614519\pi$
$971$	−21.9411	−0.704124	−0.352062	+	0.935977i	$0.614519\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	0.142136	0.00455432
$975$	0	0
$976$	2.65685	0.0850438
$977$	47.3553	1.51503	0.757516	−	0.652817i	$-0.226413\pi$
$977$	47.3553	1.51503	0.757516	+	0.652817i	$0.226413\pi$
$978$	−8.89949	−0.284574
$979$	−95.5980	−3.05532
$980$	0	0
$981$	7.31371	0.233509
$982$	0.828427	0.0264362
$983$	−30.6274	−0.976863	−0.488431	−	0.872602i	$-0.662431\pi$
$983$	−30.6274	−0.976863	−0.488431	+	0.872602i	$0.662431\pi$
$984$	−7.00000	−0.223152
$985$	0	0
$986$	49.6274	1.58046
$987$	0	0
$988$	−0.443651	−0.0141144
$989$	−5.68629	−0.180814
$990$	0	0
$991$	33.2548	1.05637	0.528187	−	0.849128i	$-0.322872\pi$
$991$	33.2548	1.05637	0.528187	+	0.849128i	$0.322872\pi$
$992$	−7.24264	−0.229954
$993$	−7.58579	−0.240728
$994$	0	0
$995$	0	0
$996$	5.58579	0.176992
$997$	−13.3137	−0.421649	−0.210825	−	0.977524i	$-0.567615\pi$
$997$	−13.3137	−0.421649	−0.210825	+	0.977524i	$0.567615\pi$
$998$	−20.0711	−0.635339
$999$	−0.242641	−0.00767681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7350.2.a.de.1.1	yes	2
5.4	even	2		7350.2.a.di.1.1	yes	2
7.6	odd	2		7350.2.a.db.1.1	✓	2
35.34	odd	2		7350.2.a.dk.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7350.2.a.db.1.1	✓	2	7.6	odd	2
7350.2.a.de.1.1	yes	2	1.1	even	1	trivial
7350.2.a.di.1.1	yes	2	5.4	even	2
7350.2.a.dk.1.1	yes	2	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -6.24264 q^{11} +1.00000 q^{12} -0.171573 q^{13} +1.00000 q^{16} +4.65685 q^{17} -1.00000 q^{18} +2.58579 q^{19} +6.24264 q^{22} -1.58579 q^{23} -1.00000 q^{24} +0.171573 q^{26} +1.00000 q^{27} -10.6569 q^{29} +7.24264 q^{31} -1.00000 q^{32} -6.24264 q^{33} -4.65685 q^{34} +1.00000 q^{36} -0.242641 q^{37} -2.58579 q^{38} -0.171573 q^{39} +7.00000 q^{41} +3.58579 q^{43} -6.24264 q^{44} +1.58579 q^{46} -8.00000 q^{47} +1.00000 q^{48} +4.65685 q^{51} -0.171573 q^{52} -2.65685 q^{53} -1.00000 q^{54} +2.58579 q^{57} +10.6569 q^{58} -1.58579 q^{59} +2.65685 q^{61} -7.24264 q^{62} +1.00000 q^{64} +6.24264 q^{66} -9.65685 q^{67} +4.65685 q^{68} -1.58579 q^{69} -2.00000 q^{71} -1.00000 q^{72} -8.24264 q^{73} +0.242641 q^{74} +2.58579 q^{76} +0.171573 q^{78} +6.24264 q^{79} +1.00000 q^{81} -7.00000 q^{82} +5.58579 q^{83} -3.58579 q^{86} -10.6569 q^{87} +6.24264 q^{88} +15.3137 q^{89} -1.58579 q^{92} +7.24264 q^{93} +8.00000 q^{94} -1.00000 q^{96} -14.0000 q^{97} -6.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 8 q^{19} + 4 q^{22} - 6 q^{23} - 2 q^{24} + 6 q^{26} + 2 q^{27} - 10 q^{29} + 6 q^{31} - 2 q^{32} - 4 q^{33} + 2 q^{34} + 2 q^{36} + 8 q^{37} - 8 q^{38} - 6 q^{39} + 14 q^{41} + 10 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} + 2 q^{48} - 2 q^{51} - 6 q^{52} + 6 q^{53} - 2 q^{54} + 8 q^{57} + 10 q^{58} - 6 q^{59} - 6 q^{61} - 6 q^{62} + 2 q^{64} + 4 q^{66} - 8 q^{67} - 2 q^{68} - 6 q^{69} - 4 q^{71} - 2 q^{72} - 8 q^{73} - 8 q^{74} + 8 q^{76} + 6 q^{78} + 4 q^{79} + 2 q^{81} - 14 q^{82} + 14 q^{83} - 10 q^{86} - 10 q^{87} + 4 q^{88} + 8 q^{89} - 6 q^{92} + 6 q^{93} + 16 q^{94} - 2 q^{96} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 - 4 * q^11 + 2 * q^12 - 6 * q^13 + 2 * q^16 - 2 * q^17 - 2 * q^18 + 8 * q^19 + 4 * q^22 - 6 * q^23 - 2 * q^24 + 6 * q^26 + 2 * q^27 - 10 * q^29 + 6 * q^31 - 2 * q^32 - 4 * q^33 + 2 * q^34 + 2 * q^36 + 8 * q^37 - 8 * q^38 - 6 * q^39 + 14 * q^41 + 10 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 + 2 * q^48 - 2 * q^51 - 6 * q^52 + 6 * q^53 - 2 * q^54 + 8 * q^57 + 10 * q^58 - 6 * q^59 - 6 * q^61 - 6 * q^62 + 2 * q^64 + 4 * q^66 - 8 * q^67 - 2 * q^68 - 6 * q^69 - 4 * q^71 - 2 * q^72 - 8 * q^73 - 8 * q^74 + 8 * q^76 + 6 * q^78 + 4 * q^79 + 2 * q^81 - 14 * q^82 + 14 * q^83 - 10 * q^86 - 10 * q^87 + 4 * q^88 + 8 * q^89 - 6 * q^92 + 6 * q^93 + 16 * q^94 - 2 * q^96 - 28 * q^97 - 4 * q^99

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