LMFDB - Embedded newform 1425.2.c.o.799.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.44836416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 1 $ x^6 + 12x^4 + 36x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.6
Root			$2.52892i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.o.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.52892i q^{2} +1.00000i q^{3} -4.39543 q^{4} -2.52892 q^{6} +4.92434i q^{7} -6.05784i q^{8} -1.00000 q^{9} -1.13349 q^{11} -4.39543i q^{12} +4.00000i q^{13} -12.4533 q^{14} +6.52892 q^{16} +6.79085i q^{17} -2.52892i q^{18} +1.00000 q^{19} -4.92434 q^{21} -2.86651i q^{22} +1.92434i q^{23} +6.05784 q^{24} -10.1157 q^{26} -1.00000i q^{27} -21.6446i q^{28} +5.00000 q^{29} +5.13349 q^{31} +4.39543i q^{32} -1.13349i q^{33} -17.1735 q^{34} +4.39543 q^{36} -9.05784i q^{37} +2.52892i q^{38} -4.00000 q^{39} +3.86651 q^{41} -12.4533i q^{42} +4.00000i q^{43} +4.98218 q^{44} -4.86651 q^{46} -4.26698i q^{47} +6.52892i q^{48} -17.2492 q^{49} -6.79085 q^{51} -17.5817i q^{52} -13.1157i q^{53} +2.52892 q^{54} +29.8309 q^{56} +1.00000i q^{57} +12.6446i q^{58} -9.19133 q^{59} +0.733016 q^{61} +12.9822i q^{62} -4.92434i q^{63} +1.94216 q^{64} +2.86651 q^{66} -4.86651i q^{67} -29.8487i q^{68} -1.92434 q^{69} +11.1913 q^{71} +6.05784i q^{72} +11.1157i q^{73} +22.9065 q^{74} -4.39543 q^{76} -5.58170i q^{77} -10.1157i q^{78} +13.7730 q^{79} +1.00000 q^{81} +9.77808i q^{82} +0.866508i q^{83} +21.6446 q^{84} -10.1157 q^{86} +5.00000i q^{87} +6.86651i q^{88} +10.8487 q^{89} -19.6974 q^{91} -8.45831i q^{92} +5.13349i q^{93} +10.7909 q^{94} -4.39543 q^{96} +9.32482i q^{97} -43.6217i q^{98} +1.13349 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 30 q^{14} + 24 q^{16} + 6 q^{19} + 6 q^{24} + 30 q^{29} + 30 q^{31} - 12 q^{34} + 12 q^{36} - 24 q^{39} + 24 q^{41} - 30 q^{44} - 30 q^{46} - 42 q^{49} - 12 q^{51}+ \cdots + 6 q^{99}+O(q^{100}) $ 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 30 * q^14 + 24 * q^16 + 6 * q^19 + 6 * q^24 + 30 * q^29 + 30 * q^31 - 12 * q^34 + 12 * q^36 - 24 * q^39 + 24 * q^41 - 30 * q^44 - 30 * q^46 - 42 * q^49 - 12 * q^51 + 60 * q^56 - 24 * q^59 + 6 * q^61 + 42 * q^64 + 18 * q^66 + 18 * q^69 + 36 * q^71 + 48 * q^74 - 12 * q^76 - 6 * q^79 + 6 * q^81 + 54 * q^84 + 6 * q^89 + 36 * q^94 - 12 * q^96 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	2.52892i	1.78822i	0.447852	+	0.894108i	$0.352189\pi$
$2$	2.52892i	1.78822i	−0.447852	+	0.894108i	$0.647811\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.39543	−2.19771
$5$	0	0
$5$	0	0
$6$	−2.52892	−1.03243
$7$	4.92434i	1.86123i	0.366003	+	0.930614i	$0.380726\pi$
$7$	4.92434i	1.86123i	−0.366003	+	0.930614i	$0.619274\pi$
$8$	− 6.05784i	− 2.14177i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.13349	−0.341761	−0.170880	−	0.985292i	$-0.554661\pi$
$11$	−1.13349	−0.341761	−0.170880	+	0.985292i	$0.554661\pi$
$12$	− 4.39543i	− 1.26885i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−12.4533	−3.32827
$15$	0	0
$16$	6.52892	1.63223
$17$	6.79085i	1.64702i	0.567299	+	0.823512i	$0.307988\pi$
$17$	6.79085i	1.64702i	−0.567299	+	0.823512i	$0.692012\pi$
$18$	− 2.52892i	− 0.596072i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.92434	−1.07458
$22$	− 2.86651i	− 0.611142i
$23$	1.92434i	0.401253i	0.979668	+	0.200627i	$0.0642978\pi$
$23$	1.92434i	0.401253i	−0.979668	+	0.200627i	$0.935702\pi$
$24$	6.05784	1.23655
$25$	0	0
$26$	−10.1157	−1.98385
$27$	− 1.00000i	− 0.192450i
$28$	− 21.6446i	− 4.09044i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	5.13349	0.922002	0.461001	−	0.887400i	$-0.347490\pi$
$31$	5.13349	0.922002	0.461001	+	0.887400i	$0.347490\pi$
$32$	4.39543i	0.777009i
$33$	− 1.13349i	− 0.197316i
$34$	−17.1735	−2.94523
$35$	0	0
$36$	4.39543	0.732571
$37$	− 9.05784i	− 1.48910i	−0.667567	−	0.744550i	$-0.732664\pi$
$37$	− 9.05784i	− 1.48910i	0.667567	−	0.744550i	$-0.267336\pi$
$38$	2.52892i	0.410245i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	3.86651	0.603847	0.301924	−	0.953332i	$-0.402371\pi$
$41$	3.86651	0.603847	0.301924	+	0.953332i	$0.402371\pi$
$42$	− 12.4533i	− 1.92158i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	4.98218	0.751092
$45$	0	0
$46$	−4.86651	−0.717527
$47$	− 4.26698i	− 0.622404i	−0.950344	−	0.311202i	$-0.899268\pi$
$47$	− 4.26698i	− 0.622404i	0.950344	−	0.311202i	$-0.100732\pi$
$48$	6.52892i	0.942368i
$49$	−17.2492	−2.46417
$50$	0	0
$51$	−6.79085	−0.950909
$52$	− 17.5817i	− 2.43814i
$53$	− 13.1157i	− 1.80158i	−0.434259	−	0.900788i	$-0.642990\pi$
$53$	− 13.1157i	− 1.80158i	0.434259	−	0.900788i	$-0.357010\pi$
$54$	2.52892	0.344142
$55$	0	0
$56$	29.8309	3.98632
$57$	1.00000i	0.132453i
$58$	12.6446i	1.66032i
$59$	−9.19133	−1.19661	−0.598304	−	0.801269i	$-0.704159\pi$
$59$	−9.19133	−1.19661	−0.598304	+	0.801269i	$0.704159\pi$
$60$	0	0
$61$	0.733016	0.0938531	0.0469266	−	0.998898i	$-0.485057\pi$
$61$	0.733016	0.0938531	0.0469266	+	0.998898i	$0.485057\pi$
$62$	12.9822i	1.64874i
$63$	− 4.92434i	− 0.620409i
$64$	1.94216	0.242771
$65$	0	0
$66$	2.86651	0.352843
$67$	− 4.86651i	− 0.594539i	−0.954794	−	0.297269i	$-0.903924\pi$
$67$	− 4.86651i	− 0.594539i	0.954794	−	0.297269i	$-0.0960759\pi$
$68$	− 29.8487i	− 3.61969i
$69$	−1.92434	−0.231664
$70$	0	0
$71$	11.1913	1.32817	0.664083	−	0.747659i	$-0.268822\pi$
$71$	11.1913	1.32817	0.664083	+	0.747659i	$0.268822\pi$
$72$	6.05784i	0.713923i
$73$	11.1157i	1.30099i	0.759510	+	0.650495i	$0.225439\pi$
$73$	11.1157i	1.30099i	−0.759510	+	0.650495i	$0.774561\pi$
$74$	22.9065	2.66283
$75$	0	0
$76$	−4.39543	−0.504190
$77$	− 5.58170i	− 0.636094i
$78$	− 10.1157i	− 1.14537i
$79$	13.7730	1.54959	0.774794	−	0.632214i	$-0.217854\pi$
$79$	13.7730	1.54959	0.774794	+	0.632214i	$0.217854\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.77808i	1.07981i
$83$	0.866508i	0.0951116i	0.998869	+	0.0475558i	$0.0151432\pi$
$83$	0.866508i	0.0951116i	−0.998869	+	0.0475558i	$0.984857\pi$
$84$	21.6446	2.36162
$85$	0	0
$86$	−10.1157	−1.09080
$87$	5.00000i	0.536056i
$88$	6.86651i	0.731972i
$89$	10.8487	1.14996	0.574979	−	0.818168i	$-0.305010\pi$
$89$	10.8487	1.14996	0.574979	+	0.818168i	$0.305010\pi$
$90$	0	0
$91$	−19.6974	−2.06485
$92$	− 8.45831i	− 0.881840i
$93$	5.13349i	0.532318i
$94$	10.7909	1.11299
$95$	0	0
$96$	−4.39543	−0.448606
$97$	9.32482i	0.946792i	0.880850	+	0.473396i	$0.156972\pi$
$97$	9.32482i	0.946792i	−0.880850	+	0.473396i	$0.843028\pi$
$98$	− 43.6217i	− 4.40646i
$99$	1.13349	0.113920
$100$	0	0
$101$	−2.79085	−0.277700	−0.138850	−	0.990313i	$-0.544341\pi$
$101$	−2.79085	−0.277700	−0.138850	+	0.990313i	$0.544341\pi$
$102$	− 17.1735i	− 1.70043i
$103$	15.0979i	1.48764i	0.668382	+	0.743818i	$0.266987\pi$
$103$	15.0979i	1.48764i	−0.668382	+	0.743818i	$0.733013\pi$
$104$	24.2313	2.37608
$105$	0	0
$106$	33.1685	3.22161
$107$	− 13.0400i	− 1.26063i	−0.776341	−	0.630313i	$-0.782927\pi$
$107$	− 13.0400i	− 1.26063i	0.776341	−	0.630313i	$-0.217073\pi$
$108$	4.39543i	0.422950i
$109$	−10.6395	−1.01908	−0.509542	−	0.860446i	$-0.670185\pi$
$109$	−10.6395	−1.01908	−0.509542	+	0.860446i	$0.670185\pi$
$110$	0	0
$111$	9.05784	0.859732
$112$	32.1506i	3.03795i
$113$	− 0.332540i	− 0.0312828i	−0.999878	−	0.0156414i	$-0.995021\pi$
$113$	− 0.332540i	− 0.0312828i	0.999878	−	0.0156414i	$-0.00497901\pi$
$114$	−2.52892	−0.236855
$115$	0	0
$116$	−21.9771	−2.04053
$117$	− 4.00000i	− 0.369800i
$118$	− 23.2441i	− 2.13979i
$119$	−33.4405	−3.06548
$120$	0	0
$121$	−9.71520	−0.883200
$122$	1.85374i	0.167830i
$123$	3.86651i	0.348631i
$124$	−22.5639	−2.02630
$125$	0	0
$126$	12.4533	1.10942
$127$	− 8.45831i	− 0.750554i	−0.926913	−	0.375277i	$-0.877548\pi$
$127$	− 8.45831i	− 0.750554i	0.926913	−	0.375277i	$-0.122452\pi$
$128$	13.7024i	1.21113i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	10.9822	0.959518	0.479759	−	0.877400i	$-0.340724\pi$
$131$	10.9822	0.959518	0.479759	+	0.877400i	$0.340724\pi$
$132$	4.98218i	0.433643i
$133$	4.92434i	0.426995i
$134$	12.3070	1.06316
$135$	0	0
$136$	41.1379	3.52754
$137$	− 4.90652i	− 0.419193i	−0.977788	−	0.209596i	$-0.932785\pi$
$137$	− 4.90652i	− 0.419193i	0.977788	−	0.209596i	$-0.0672150\pi$
$138$	− 4.86651i	− 0.414265i
$139$	−2.92434	−0.248040	−0.124020	−	0.992280i	$-0.539579\pi$
$139$	−2.92434	−0.248040	−0.124020	+	0.992280i	$0.539579\pi$
$140$	0	0
$141$	4.26698	0.359345
$142$	28.3019i	2.37505i
$143$	− 4.53397i	− 0.379149i
$144$	−6.52892	−0.544076
$145$	0	0
$146$	−28.1106	−2.32645
$147$	− 17.2492i	− 1.42269i
$148$	39.8130i	3.27261i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	4.67518	0.380461	0.190230	−	0.981739i	$-0.439077\pi$
$151$	4.67518	0.380461	0.190230	+	0.981739i	$0.439077\pi$
$152$	− 6.05784i	− 0.491355i
$153$	− 6.79085i	− 0.549008i
$154$	14.1157	1.13747
$155$	0	0
$156$	17.5817	1.40766
$157$	13.4482i	1.07328i	0.843810	+	0.536642i	$0.180307\pi$
$157$	13.4482i	1.07328i	−0.843810	+	0.536642i	$0.819693\pi$
$158$	34.8309i	2.77100i
$159$	13.1157	1.04014
$160$	0	0
$161$	−9.47613	−0.746824
$162$	2.52892i	0.198691i
$163$	21.5740i	1.68980i	0.534920	+	0.844902i	$0.320342\pi$
$163$	21.5740i	1.68980i	−0.534920	+	0.844902i	$0.679658\pi$
$164$	−16.9950	−1.32708
$165$	0	0
$166$	−2.19133	−0.170080
$167$	9.07566i	0.702295i	0.936320	+	0.351148i	$0.114208\pi$
$167$	9.07566i	0.702295i	−0.936320	+	0.351148i	$0.885792\pi$
$168$	29.8309i	2.30150i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 17.5817i	− 1.34059i
$173$	6.46603i	0.491603i	0.969320	+	0.245802i	$0.0790512\pi$
$173$	6.46603i	0.491603i	−0.969320	+	0.245802i	$0.920949\pi$
$174$	−12.6446	−0.958584
$175$	0	0
$176$	−7.40048	−0.557832
$177$	− 9.19133i	− 0.690863i
$178$	27.4354i	2.05637i
$179$	11.3070	0.845125	0.422562	−	0.906334i	$-0.361131\pi$
$179$	11.3070	0.845125	0.422562	+	0.906334i	$0.361131\pi$
$180$	0	0
$181$	−22.6496	−1.68353	−0.841767	−	0.539841i	$-0.818484\pi$
$181$	−22.6496	−1.68353	−0.841767	+	0.539841i	$0.818484\pi$
$182$	− 49.8130i	− 3.69239i
$183$	0.733016i	0.0541861i
$184$	11.6574	0.859392
$185$	0	0
$186$	−12.9822	−0.951900
$187$	− 7.69738i	− 0.562888i
$188$	18.7552i	1.36786i
$189$	4.92434	0.358193
$190$	0	0
$191$	24.2969	1.75806	0.879031	−	0.476765i	$-0.158191\pi$
$191$	24.2969	1.75806	0.879031	+	0.476765i	$0.158191\pi$
$192$	1.94216i	0.140164i
$193$	6.67518i	0.480490i	0.970712	+	0.240245i	$0.0772278\pi$
$193$	6.67518i	0.480490i	−0.970712	+	0.240245i	$0.922772\pi$
$194$	−23.5817	−1.69307
$195$	0	0
$196$	75.8174	5.41553
$197$	− 23.2892i	− 1.65929i	−0.558295	−	0.829643i	$-0.688544\pi$
$197$	− 23.2892i	− 1.65929i	0.558295	−	0.829643i	$-0.311456\pi$
$198$	2.86651i	0.203714i
$199$	−25.0400	−1.77504	−0.887520	−	0.460770i	$-0.847573\pi$
$199$	−25.0400	−1.77504	−0.887520	+	0.460770i	$0.847573\pi$
$200$	0	0
$201$	4.86651	0.343257
$202$	− 7.05784i	− 0.496588i
$203$	24.6217i	1.72811i
$204$	29.8487	2.08983
$205$	0	0
$206$	−38.1812	−2.66021
$207$	− 1.92434i	− 0.133751i
$208$	26.1157i	1.81080i
$209$	−1.13349	−0.0784053
$210$	0	0
$211$	−11.7730	−0.810489	−0.405244	−	0.914208i	$-0.632814\pi$
$211$	−11.7730	−0.810489	−0.405244	+	0.914208i	$0.632814\pi$
$212$	57.6490i	3.95935i
$213$	11.1913i	0.766817i
$214$	32.9771	2.25427
$215$	0	0
$216$	−6.05784	−0.412184
$217$	25.2791i	1.71606i
$218$	− 26.9065i	− 1.82234i
$219$	−11.1157	−0.751127
$220$	0	0
$221$	−27.1634	−1.82721
$222$	22.9065i	1.53739i
$223$	− 3.92434i	− 0.262794i	−0.991330	−	0.131397i	$-0.958054\pi$
$223$	− 3.92434i	− 0.262794i	0.991330	−	0.131397i	$-0.0419462\pi$
$224$	−21.6446	−1.44619
$225$	0	0
$226$	0.840967	0.0559403
$227$	1.60962i	0.106834i	0.998572	+	0.0534172i	$0.0170113\pi$
$227$	1.60962i	0.106834i	−0.998572	+	0.0534172i	$0.982989\pi$
$228$	− 4.39543i	− 0.291094i
$229$	1.93444	0.127832	0.0639158	−	0.997955i	$-0.479641\pi$
$229$	1.93444	0.127832	0.0639158	+	0.997955i	$0.479641\pi$
$230$	0	0
$231$	5.58170	0.367249
$232$	− 30.2892i	− 1.98858i
$233$	− 1.84869i	− 0.121112i	−0.998165	−	0.0605558i	$-0.980713\pi$
$233$	− 1.84869i	− 0.121112i	0.998165	−	0.0605558i	$-0.0192873\pi$
$234$	10.1157	0.661282
$235$	0	0
$236$	40.3998	2.62980
$237$	13.7730i	0.894655i
$238$	− 84.5683i	− 5.48175i
$239$	−2.25688	−0.145986	−0.0729929	−	0.997332i	$-0.523255\pi$
$239$	−2.25688	−0.145986	−0.0729929	+	0.997332i	$0.523255\pi$
$240$	0	0
$241$	−19.0578	−1.22762	−0.613812	−	0.789453i	$-0.710365\pi$
$241$	−19.0578	−1.22762	−0.613812	+	0.789453i	$0.710365\pi$
$242$	− 24.5689i	− 1.57935i
$243$	1.00000i	0.0641500i
$244$	−3.22192	−0.206262
$245$	0	0
$246$	−9.77808	−0.623428
$247$	4.00000i	0.254514i
$248$	− 31.0979i	− 1.97472i
$249$	−0.866508	−0.0549127
$250$	0	0
$251$	26.7909	1.69102	0.845512	−	0.533957i	$-0.179295\pi$
$251$	26.7909	1.69102	0.845512	+	0.533957i	$0.179295\pi$
$252$	21.6446i	1.36348i
$253$	− 2.18123i	− 0.137133i
$254$	21.3904	1.34215
$255$	0	0
$256$	−30.7680	−1.92300
$257$	− 16.3147i	− 1.01768i	−0.860860	−	0.508842i	$-0.830074\pi$
$257$	− 16.3147i	− 1.01768i	0.860860	−	0.508842i	$-0.169926\pi$
$258$	− 10.1157i	− 0.629774i
$259$	44.6039	2.77155
$260$	0	0
$261$	−5.00000	−0.309492
$262$	27.7730i	1.71582i
$263$	1.92434i	0.118660i	0.998238	+	0.0593301i	$0.0188964\pi$
$263$	1.92434i	0.118660i	−0.998238	+	0.0593301i	$0.981104\pi$
$264$	−6.86651	−0.422604
$265$	0	0
$266$	−12.4533	−0.763558
$267$	10.8487i	0.663929i
$268$	21.3904i	1.30663i
$269$	−19.5817	−1.19392	−0.596959	−	0.802272i	$-0.703624\pi$
$269$	−19.5817	−1.19392	−0.596959	+	0.802272i	$0.703624\pi$
$270$	0	0
$271$	11.1913	0.679825	0.339912	−	0.940457i	$-0.389603\pi$
$271$	11.1913	0.679825	0.339912	+	0.940457i	$0.389603\pi$
$272$	44.3369i	2.68832i
$273$	− 19.6974i	− 1.19214i
$274$	12.4082	0.749607
$275$	0	0
$276$	8.45831	0.509131
$277$	− 16.3147i	− 0.980257i	−0.871650	−	0.490128i	$-0.836950\pi$
$277$	− 16.3147i	− 0.980257i	0.871650	−	0.490128i	$-0.163050\pi$
$278$	− 7.39543i	− 0.443548i
$279$	−5.13349	−0.307334
$280$	0	0
$281$	−2.75084	−0.164101	−0.0820506	−	0.996628i	$-0.526147\pi$
$281$	−2.75084	−0.164101	−0.0820506	+	0.996628i	$0.526147\pi$
$282$	10.7909i	0.642586i
$283$	− 20.2313i	− 1.20263i	−0.799013	−	0.601314i	$-0.794644\pi$
$283$	− 20.2313i	− 1.20263i	0.799013	−	0.601314i	$-0.205356\pi$
$284$	−49.1907	−2.91893
$285$	0	0
$286$	11.4660	0.678001
$287$	19.0400i	1.12390i
$288$	− 4.39543i	− 0.259003i
$289$	−29.1157	−1.71269
$290$	0	0
$291$	−9.32482	−0.546631
$292$	− 48.8581i	− 2.85920i
$293$	22.9822i	1.34263i	0.741171	+	0.671317i	$0.234271\pi$
$293$	22.9822i	1.34263i	−0.741171	+	0.671317i	$0.765729\pi$
$294$	43.6217	2.54407
$295$	0	0
$296$	−54.8709	−3.18931
$297$	1.13349i	0.0657719i
$298$	0	0
$299$	−7.69738	−0.445151
$300$	0	0
$301$	−19.6974	−1.13534
$302$	11.8231i	0.680346i
$303$	− 2.79085i	− 0.160330i
$304$	6.52892	0.374459
$305$	0	0
$306$	17.1735	0.981744
$307$	− 8.59952i	− 0.490801i	−0.969422	−	0.245400i	$-0.921081\pi$
$307$	− 8.59952i	− 0.490801i	0.969422	−	0.245400i	$-0.0789194\pi$
$308$	24.5340i	1.39795i
$309$	−15.0979	−0.858887
$310$	0	0
$311$	−12.2313	−0.693576	−0.346788	−	0.937944i	$-0.612728\pi$
$311$	−12.2313	−0.693576	−0.346788	+	0.937944i	$0.612728\pi$
$312$	24.2313i	1.37183i
$313$	− 7.51615i	− 0.424838i	−0.977179	−	0.212419i	$-0.931866\pi$
$313$	− 7.51615i	− 0.424838i	0.977179	−	0.212419i	$-0.0681341\pi$
$314$	−34.0094	−1.91926
$315$	0	0
$316$	−60.5383	−3.40555
$317$	26.5817i	1.49298i	0.665398	+	0.746489i	$0.268262\pi$
$317$	26.5817i	1.49298i	−0.665398	+	0.746489i	$0.731738\pi$
$318$	33.1685i	1.85999i
$319$	−5.66746	−0.317317
$320$	0	0
$321$	13.0400	0.727823
$322$	− 23.9644i	− 1.33548i
$323$	6.79085i	0.377853i
$324$	−4.39543	−0.244190
$325$	0	0
$326$	−54.5588	−3.02173
$327$	− 10.6395i	− 0.588368i
$328$	− 23.4227i	− 1.29330i
$329$	21.0121	1.15843
$330$	0	0
$331$	19.3648	1.06439	0.532194	−	0.846623i	$-0.321368\pi$
$331$	19.3648	1.06439	0.532194	+	0.846623i	$0.321368\pi$
$332$	− 3.80867i	− 0.209028i
$333$	9.05784i	0.496366i
$334$	−22.9516	−1.25586
$335$	0	0
$336$	−32.1506	−1.75396
$337$	17.8487i	0.972280i	0.873881	+	0.486140i	$0.161595\pi$
$337$	17.8487i	0.972280i	−0.873881	+	0.486140i	$0.838405\pi$
$338$	− 7.58675i	− 0.412665i
$339$	0.332540	0.0180611
$340$	0	0
$341$	−5.81877	−0.315104
$342$	− 2.52892i	− 0.136748i
$343$	− 50.4704i	− 2.72515i
$344$	24.2313	1.30647
$345$	0	0
$346$	−16.3521	−0.879092
$347$	− 19.0578i	− 1.02308i	−0.859260	−	0.511539i	$-0.829076\pi$
$347$	− 19.0578i	− 1.02308i	0.859260	−	0.511539i	$-0.170924\pi$
$348$	− 21.9771i	− 1.17810i
$349$	−22.9644	−1.22925	−0.614627	−	0.788818i	$-0.710693\pi$
$349$	−22.9644	−1.22925	−0.614627	+	0.788818i	$0.710693\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	− 4.98218i	− 0.265551i
$353$	− 25.5817i	− 1.36158i	−0.732480	−	0.680788i	$-0.761637\pi$
$353$	− 25.5817i	− 1.36158i	0.732480	−	0.680788i	$-0.238363\pi$
$354$	23.2441	1.23541
$355$	0	0
$356$	−47.6846	−2.52728
$357$	− 33.4405i	− 1.76986i
$358$	28.5945i	1.51126i
$359$	27.3248	1.44215	0.721074	−	0.692858i	$-0.243649\pi$
$359$	27.3248	1.44215	0.721074	+	0.692858i	$0.243649\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 57.2791i	− 3.01052i
$363$	− 9.71520i	− 0.509916i
$364$	86.5784	4.53794
$365$	0	0
$366$	−1.85374	−0.0968964
$367$	20.3827i	1.06397i	0.846755	+	0.531983i	$0.178553\pi$
$367$	20.3827i	1.06397i	−0.846755	+	0.531983i	$0.821447\pi$
$368$	12.5639i	0.654938i
$369$	−3.86651	−0.201282
$370$	0	0
$371$	64.5861	3.35314
$372$	− 22.5639i	− 1.16988i
$373$	2.52387i	0.130681i	0.997863	+	0.0653405i	$0.0208133\pi$
$373$	2.52387i	0.130681i	−0.997863	+	0.0653405i	$0.979187\pi$
$374$	19.4660	1.00656
$375$	0	0
$376$	−25.8487	−1.33304
$377$	20.0000i	1.03005i
$378$	12.4533i	0.640527i
$379$	9.44049	0.484925	0.242463	−	0.970161i	$-0.422045\pi$
$379$	9.44049	0.484925	0.242463	+	0.970161i	$0.422045\pi$
$380$	0	0
$381$	8.45831	0.433332
$382$	61.4449i	3.14379i
$383$	11.1557i	0.570029i	0.958523	+	0.285015i	$0.0919984\pi$
$383$	11.1557i	0.570029i	−0.958523	+	0.285015i	$0.908002\pi$
$384$	−13.7024	−0.699249
$385$	0	0
$386$	−16.8810	−0.859219
$387$	− 4.00000i	− 0.203331i
$388$	− 40.9866i	− 2.08078i
$389$	−28.3827	−1.43906	−0.719529	−	0.694463i	$-0.755642\pi$
$389$	−28.3827	−1.43906	−0.719529	+	0.694463i	$0.755642\pi$
$390$	0	0
$391$	−13.0679	−0.660874
$392$	104.493i	5.27767i
$393$	10.9822i	0.553978i
$394$	58.8964	2.96716
$395$	0	0
$396$	−4.98218	−0.250364
$397$	1.40048i	0.0702879i	0.999382	+	0.0351439i	$0.0111890\pi$
$397$	1.40048i	0.0702879i	−0.999382	+	0.0351439i	$0.988811\pi$
$398$	− 63.3241i	− 3.17415i
$399$	−4.92434	−0.246526
$400$	0	0
$401$	3.51615	0.175588	0.0877940	−	0.996139i	$-0.472018\pi$
$401$	3.51615	0.175588	0.0877940	+	0.996139i	$0.472018\pi$
$402$	12.3070i	0.613817i
$403$	20.5340i	1.02287i
$404$	12.2670	0.610305
$405$	0	0
$406$	−62.2663	−3.09023
$407$	10.2670i	0.508915i
$408$	41.1379i	2.03663i
$409$	31.1379	1.53967	0.769834	−	0.638244i	$-0.220339\pi$
$409$	31.1379	1.53967	0.769834	+	0.638244i	$0.220339\pi$
$410$	0	0
$411$	4.90652	0.242021
$412$	− 66.3615i	− 3.26940i
$413$	− 45.2613i	− 2.22716i
$414$	4.86651	0.239176
$415$	0	0
$416$	−17.5817	−0.862014
$417$	− 2.92434i	− 0.143206i
$418$	− 2.86651i	− 0.140205i
$419$	−20.4983	−1.00141	−0.500704	−	0.865618i	$-0.666926\pi$
$419$	−20.4983	−1.00141	−0.500704	+	0.865618i	$0.666926\pi$
$420$	0	0
$421$	−3.06794	−0.149522	−0.0747610	−	0.997201i	$-0.523819\pi$
$421$	−3.06794	−0.149522	−0.0747610	+	0.997201i	$0.523819\pi$
$422$	− 29.7730i	− 1.44933i
$423$	4.26698i	0.207468i
$424$	−79.4526	−3.85856
$425$	0	0
$426$	−28.3019	−1.37123
$427$	3.60962i	0.174682i
$428$	57.3164i	2.77049i
$429$	4.53397	0.218902
$430$	0	0
$431$	5.34264	0.257346	0.128673	−	0.991687i	$-0.458928\pi$
$431$	5.34264	0.257346	0.128673	+	0.991687i	$0.458928\pi$
$432$	− 6.52892i	− 0.314123i
$433$	− 6.00000i	− 0.288342i	−0.989553	−	0.144171i	$-0.953949\pi$
$433$	− 6.00000i	− 0.288342i	0.989553	−	0.144171i	$-0.0460515\pi$
$434$	−63.9287	−3.06868
$435$	0	0
$436$	46.7653	2.23965
$437$	1.92434i	0.0920539i
$438$	− 28.1106i	− 1.34318i
$439$	−3.27470	−0.156293	−0.0781466	−	0.996942i	$-0.524900\pi$
$439$	−3.27470	−0.156293	−0.0781466	+	0.996942i	$0.524900\pi$
$440$	0	0
$441$	17.2492	0.821389
$442$	− 68.6940i	− 3.26744i
$443$	− 35.8988i	− 1.70560i	−0.522235	−	0.852802i	$-0.674902\pi$
$443$	− 35.8988i	− 1.70560i	0.522235	−	0.852802i	$-0.325098\pi$
$444$	−39.8130	−1.88944
$445$	0	0
$446$	9.92434	0.469931
$447$	0	0
$448$	9.56388i	0.451851i
$449$	−1.26698	−0.0597927	−0.0298963	−	0.999553i	$-0.509518\pi$
$449$	−1.26698	−0.0597927	−0.0298963	+	0.999553i	$0.509518\pi$
$450$	0	0
$451$	−4.38266	−0.206371
$452$	1.46166i	0.0687505i
$453$	4.67518i	0.219659i
$454$	−4.07061	−0.191043
$455$	0	0
$456$	6.05784	0.283684
$457$	− 1.48385i	− 0.0694117i	−0.999398	−	0.0347058i	$-0.988951\pi$
$457$	− 1.48385i	− 0.0694117i	0.999398	−	0.0347058i	$-0.0110494\pi$
$458$	4.89205i	0.228590i
$459$	6.79085	0.316970
$460$	0	0
$461$	−4.90652	−0.228520	−0.114260	−	0.993451i	$-0.536450\pi$
$461$	−4.90652	−0.228520	−0.114260	+	0.993451i	$0.536450\pi$
$462$	14.1157i	0.656720i
$463$	− 13.8843i	− 0.645259i	−0.946525	−	0.322630i	$-0.895433\pi$
$463$	− 13.8843i	− 0.645259i	0.946525	−	0.322630i	$-0.104567\pi$
$464$	32.6446	1.51549
$465$	0	0
$466$	4.67518	0.216574
$467$	29.5060i	1.36538i	0.730709	+	0.682689i	$0.239189\pi$
$467$	29.5060i	1.36538i	−0.730709	+	0.682689i	$0.760811\pi$
$468$	17.5817i	0.812715i
$469$	23.9644	1.10657
$470$	0	0
$471$	−13.4482	−0.619661
$472$	55.6796i	2.56286i
$473$	− 4.53397i	− 0.208472i
$474$	−34.8309	−1.59983
$475$	0	0
$476$	146.985	6.73706
$477$	13.1157i	0.600525i
$478$	− 5.70748i	− 0.261054i
$479$	−10.8766	−0.496965	−0.248482	−	0.968636i	$-0.579932\pi$
$479$	−10.8766	−0.496965	−0.248482	+	0.968636i	$0.579932\pi$
$480$	0	0
$481$	36.2313	1.65201
$482$	− 48.1957i	− 2.19525i
$483$	− 9.47613i	− 0.431179i
$484$	42.7024	1.94102
$485$	0	0
$486$	−2.52892	−0.114714
$487$	33.2791i	1.50802i	0.656863	+	0.754010i	$0.271883\pi$
$487$	33.2791i	1.50802i	−0.656863	+	0.754010i	$0.728117\pi$
$488$	− 4.44049i	− 0.201012i
$489$	−21.5740	−0.975609
$490$	0	0
$491$	−4.90652	−0.221428	−0.110714	−	0.993852i	$-0.535314\pi$
$491$	−4.90652	−0.221428	−0.110714	+	0.993852i	$0.535314\pi$
$492$	− 16.9950i	− 0.766192i
$493$	33.9543i	1.52922i
$494$	−10.1157	−0.455126
$495$	0	0
$496$	33.5161	1.50492
$497$	55.1099i	2.47202i
$498$	− 2.19133i	− 0.0981957i
$499$	−2.00772	−0.0898779	−0.0449390	−	0.998990i	$-0.514309\pi$
$499$	−2.00772	−0.0898779	−0.0449390	+	0.998990i	$0.514309\pi$
$500$	0	0
$501$	−9.07566	−0.405470
$502$	67.7519i	3.02391i
$503$	− 7.69738i	− 0.343209i	−0.985166	−	0.171605i	$-0.945105\pi$
$503$	− 7.69738i	− 0.343209i	0.985166	−	0.171605i	$-0.0548951\pi$
$504$	−29.8309	−1.32877
$505$	0	0
$506$	5.51615	0.245223
$507$	− 3.00000i	− 0.133235i
$508$	37.1779i	1.64950i
$509$	−27.1157	−1.20188	−0.600941	−	0.799294i	$-0.705207\pi$
$509$	−27.1157	−1.20188	−0.600941	+	0.799294i	$0.705207\pi$
$510$	0	0
$511$	−54.7374	−2.42144
$512$	− 50.4049i	− 2.22760i
$513$	− 1.00000i	− 0.0441511i
$514$	41.2586	1.81984
$515$	0	0
$516$	17.5817	0.773991
$517$	4.83659i	0.212713i
$518$	112.800i	4.95613i
$519$	−6.46603	−0.283827
$520$	0	0
$521$	17.0800	0.748290	0.374145	−	0.927370i	$-0.377936\pi$
$521$	17.0800	0.748290	0.374145	+	0.927370i	$0.377936\pi$
$522$	− 12.6446i	− 0.553439i
$523$	9.70748i	0.424478i	0.977218	+	0.212239i	$0.0680756\pi$
$523$	9.70748i	0.424478i	−0.977218	+	0.212239i	$0.931924\pi$
$524$	−48.2714	−2.10874
$525$	0	0
$526$	−4.86651	−0.212190
$527$	34.8608i	1.51856i
$528$	− 7.40048i	− 0.322064i
$529$	19.2969	0.838996
$530$	0	0
$531$	9.19133	0.398870
$532$	− 21.6446i	− 0.938412i
$533$	15.4660i	0.669908i
$534$	−27.4354	−1.18725
$535$	0	0
$536$	−29.4805	−1.27336
$537$	11.3070i	0.487933i
$538$	− 49.5205i	− 2.13498i
$539$	19.5518	0.842155
$540$	0	0
$541$	27.7475	1.19296	0.596479	−	0.802629i	$-0.296566\pi$
$541$	27.7475	1.19296	0.596479	+	0.802629i	$0.296566\pi$
$542$	28.3019i	1.21567i
$543$	− 22.6496i	− 0.971989i
$544$	−29.8487	−1.27975
$545$	0	0
$546$	49.8130	2.13180
$547$	2.04002i	0.0872248i	0.999049	+	0.0436124i	$0.0138867\pi$
$547$	2.04002i	0.0872248i	−0.999049	+	0.0436124i	$0.986113\pi$
$548$	21.5663i	0.921265i
$549$	−0.733016	−0.0312844
$550$	0	0
$551$	5.00000	0.213007
$552$	11.6574i	0.496170i
$553$	67.8231i	2.88413i
$554$	41.2586	1.75291
$555$	0	0
$556$	12.8537	0.545120
$557$	24.5340i	1.03954i	0.854307	+	0.519769i	$0.173982\pi$
$557$	24.5340i	1.03954i	−0.854307	+	0.519769i	$0.826018\pi$
$558$	− 12.9822i	− 0.549579i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	7.69738	0.324983
$562$	− 6.95664i	− 0.293448i
$563$	− 38.0878i	− 1.60521i	−0.596513	−	0.802604i	$-0.703447\pi$
$563$	− 38.0878i	− 1.60521i	0.596513	−	0.802604i	$-0.296553\pi$
$564$	−18.7552	−0.789737
$565$	0	0
$566$	51.1634	2.15056
$567$	4.92434i	0.206803i
$568$	− 67.7952i	− 2.84462i
$569$	24.4005	1.02292	0.511461	−	0.859307i	$-0.329105\pi$
$569$	24.4005	1.02292	0.511461	+	0.859307i	$0.329105\pi$
$570$	0	0
$571$	−13.0400	−0.545708	−0.272854	−	0.962055i	$-0.587968\pi$
$571$	−13.0400	−0.545708	−0.272854	+	0.962055i	$0.587968\pi$
$572$	19.9287i	0.833262i
$573$	24.2969i	1.01502i
$574$	−48.1506	−2.00977
$575$	0	0
$576$	−1.94216	−0.0809235
$577$	13.0979i	0.545271i	0.962117	+	0.272635i	$0.0878953\pi$
$577$	13.0979i	0.545271i	−0.962117	+	0.272635i	$0.912105\pi$
$578$	− 73.6311i	− 3.06265i
$579$	−6.67518	−0.277411
$580$	0	0
$581$	−4.26698	−0.177024
$582$	− 23.5817i	− 0.977493i
$583$	14.8665i	0.615708i
$584$	67.3369	2.78642
$585$	0	0
$586$	−58.1200	−2.40092
$587$	30.4227i	1.25568i	0.778343	+	0.627839i	$0.216060\pi$
$587$	30.4227i	1.25568i	−0.778343	+	0.627839i	$0.783940\pi$
$588$	75.8174i	3.12666i
$589$	5.13349	0.211522
$590$	0	0
$591$	23.2892	0.957989
$592$	− 59.1379i	− 2.43055i
$593$	30.3470i	1.24620i	0.782141	+	0.623101i	$0.214128\pi$
$593$	30.3470i	1.24620i	−0.782141	+	0.623101i	$0.785872\pi$
$594$	−2.86651	−0.117614
$595$	0	0
$596$	0	0
$597$	− 25.0400i	− 1.02482i
$598$	− 19.4660i	− 0.796025i
$599$	43.3948	1.77306	0.886531	−	0.462670i	$-0.153108\pi$
$599$	43.3948	1.77306	0.886531	+	0.462670i	$0.153108\pi$
$600$	0	0
$601$	7.84869	0.320155	0.160077	−	0.987104i	$-0.448826\pi$
$601$	7.84869	0.320155	0.160077	+	0.987104i	$0.448826\pi$
$602$	− 49.8130i	− 2.03023i
$603$	4.86651i	0.198180i
$604$	−20.5494	−0.836144
$605$	0	0
$606$	7.05784	0.286705
$607$	− 13.8087i	− 0.560477i	−0.959930	−	0.280238i	$-0.909586\pi$
$607$	− 13.8087i	− 0.560477i	0.959930	−	0.280238i	$-0.0904135\pi$
$608$	4.39543i	0.178258i
$609$	−24.6217	−0.997722
$610$	0	0
$611$	17.0679	0.690495
$612$	29.8487i	1.20656i
$613$	− 9.98218i	− 0.403176i	−0.979470	−	0.201588i	$-0.935390\pi$
$613$	− 9.98218i	− 0.403176i	0.979470	−	0.201588i	$-0.0646103\pi$
$614$	21.7475	0.877657
$615$	0	0
$616$	−33.8130	−1.36237
$617$	− 0.115672i	− 0.00465677i	−0.999997	−	0.00232839i	$-0.999259\pi$
$617$	− 0.115672i	− 0.00465677i	0.999997	−	0.00232839i	$-0.000741149\pi$
$618$	− 38.1812i	− 1.53587i
$619$	5.45831	0.219388	0.109694	−	0.993965i	$-0.465013\pi$
$619$	5.45831	0.219388	0.109694	+	0.993965i	$0.465013\pi$
$620$	0	0
$621$	1.92434	0.0772213
$622$	− 30.9321i	− 1.24026i
$623$	53.4227i	2.14033i
$624$	−26.1157	−1.04546
$625$	0	0
$626$	19.0077	0.759701
$627$	− 1.13349i	− 0.0452673i
$628$	− 59.1106i	− 2.35877i
$629$	61.5104	2.45258
$630$	0	0
$631$	45.8130	1.82379	0.911894	−	0.410425i	$-0.134620\pi$
$631$	45.8130	1.82379	0.911894	+	0.410425i	$0.134620\pi$
$632$	− 83.4348i	− 3.31886i
$633$	− 11.7730i	− 0.467936i
$634$	−67.2229	−2.66976
$635$	0	0
$636$	−57.6490	−2.28593
$637$	− 68.9967i	− 2.73375i
$638$	− 14.3325i	− 0.567431i
$639$	−11.1913	−0.442722
$640$	0	0
$641$	−37.1634	−1.46787	−0.733933	−	0.679222i	$-0.762317\pi$
$641$	−37.1634	−1.46787	−0.733933	+	0.679222i	$0.762317\pi$
$642$	32.9771i	1.30150i
$643$	− 11.4583i	− 0.451872i	−0.974142	−	0.225936i	$-0.927456\pi$
$643$	− 11.4583i	− 0.451872i	0.974142	−	0.225936i	$-0.0725440\pi$
$644$	41.6516	1.64130
$645$	0	0
$646$	−17.1735	−0.675683
$647$	34.7952i	1.36794i	0.729509	+	0.683971i	$0.239748\pi$
$647$	34.7952i	1.36794i	−0.729509	+	0.683971i	$0.760252\pi$
$648$	− 6.05784i	− 0.237974i
$649$	10.4183	0.408954
$650$	0	0
$651$	−25.2791	−0.990765
$652$	− 94.8268i	− 3.71371i
$653$	25.2791i	0.989247i	0.869107	+	0.494623i	$0.164694\pi$
$653$	25.2791i	0.989247i	−0.869107	+	0.494623i	$0.835306\pi$
$654$	26.9065	1.05213
$655$	0	0
$656$	25.2441	0.985617
$657$	− 11.1157i	− 0.433664i
$658$	53.1379i	2.07153i
$659$	−47.6261	−1.85525	−0.927625	−	0.373514i	$-0.878153\pi$
$659$	−47.6261	−1.85525	−0.927625	+	0.373514i	$0.878153\pi$
$660$	0	0
$661$	−14.4882	−0.563527	−0.281763	−	0.959484i	$-0.590919\pi$
$661$	−14.4882	−0.563527	−0.281763	+	0.959484i	$0.590919\pi$
$662$	48.9721i	1.90335i
$663$	− 27.1634i	− 1.05494i
$664$	5.24916	0.203707
$665$	0	0
$666$	−22.9065	−0.887610
$667$	9.62172i	0.372554i
$668$	− 39.8914i	− 1.54344i
$669$	3.92434	0.151724
$670$	0	0
$671$	−0.830868	−0.0320753
$672$	− 21.6446i	− 0.834958i
$673$	15.6974i	0.605089i	0.953135	+	0.302545i	$0.0978362\pi$
$673$	15.6974i	0.605089i	−0.953135	+	0.302545i	$0.902164\pi$
$674$	−45.1379	−1.73865
$675$	0	0
$676$	13.1863	0.507165
$677$	− 26.8130i	− 1.03051i	−0.857037	−	0.515255i	$-0.827697\pi$
$677$	− 26.8130i	− 1.03051i	0.857037	−	0.515255i	$-0.172303\pi$
$678$	0.840967i	0.0322972i
$679$	−45.9186	−1.76219
$680$	0	0
$681$	−1.60962	−0.0616809
$682$	− 14.7152i	− 0.563474i
$683$	− 36.4704i	− 1.39550i	−0.716341	−	0.697751i	$-0.754184\pi$
$683$	− 36.4704i	− 1.39550i	0.716341	−	0.697751i	$-0.245816\pi$
$684$	4.39543	0.168063
$685$	0	0
$686$	127.636	4.87315
$687$	1.93444i	0.0738036i
$688$	26.1157i	0.995651i
$689$	52.4627	1.99867
$690$	0	0
$691$	13.1990	0.502115	0.251058	−	0.967972i	$-0.419222\pi$
$691$	13.1990	0.502115	0.251058	+	0.967972i	$0.419222\pi$
$692$	− 28.4210i	− 1.08040i
$693$	5.58170i	0.212031i
$694$	48.1957	1.82948
$695$	0	0
$696$	30.2892	1.14811
$697$	26.2569i	0.994550i
$698$	− 58.0750i	− 2.19817i
$699$	1.84869	0.0699238
$700$	0	0
$701$	37.7075	1.42419	0.712096	−	0.702082i	$-0.247746\pi$
$701$	37.7075	1.42419	0.712096	+	0.702082i	$0.247746\pi$
$702$	10.1157i	0.381791i
$703$	− 9.05784i	− 0.341623i
$704$	−2.20143	−0.0829694
$705$	0	0
$706$	64.6940	2.43479
$707$	− 13.7431i	− 0.516863i
$708$	40.3998i	1.51832i
$709$	−15.9166	−0.597761	−0.298881	−	0.954290i	$-0.596613\pi$
$709$	−15.9166	−0.597761	−0.298881	+	0.954290i	$0.596613\pi$
$710$	0	0
$711$	−13.7730	−0.516529
$712$	− 65.7196i	− 2.46295i
$713$	9.87860i	0.369957i
$714$	84.5683	3.16489
$715$	0	0
$716$	−49.6991	−1.85734
$717$	− 2.25688i	− 0.0842849i
$718$	69.1022i	2.57887i
$719$	−1.43612	−0.0535581	−0.0267790	−	0.999641i	$-0.508525\pi$
$719$	−1.43612	−0.0535581	−0.0267790	+	0.999641i	$0.508525\pi$
$720$	0	0
$721$	−74.3470	−2.76883
$722$	2.52892i	0.0941166i
$723$	− 19.0578i	− 0.708769i
$724$	99.5548	3.69993
$725$	0	0
$726$	24.5689	0.911839
$727$	− 34.3191i	− 1.27282i	−0.771349	−	0.636412i	$-0.780418\pi$
$727$	− 34.3191i	− 1.27282i	0.771349	−	0.636412i	$-0.219582\pi$
$728$	119.323i	4.42242i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−27.1634	−1.00467
$732$	− 3.22192i	− 0.119086i
$733$	43.3114i	1.59974i	0.600172	+	0.799871i	$0.295099\pi$
$733$	43.3114i	1.59974i	−0.600172	+	0.799871i	$0.704901\pi$
$734$	−51.5461	−1.90260
$735$	0	0
$736$	−8.45831	−0.311778
$737$	5.51615i	0.203190i
$738$	− 9.77808i	− 0.359936i
$739$	−25.4583	−0.936499	−0.468250	−	0.883596i	$-0.655115\pi$
$739$	−25.4583	−0.936499	−0.468250	+	0.883596i	$0.655115\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	163.333i	5.99614i
$743$	− 22.2391i	− 0.815872i	−0.913010	−	0.407936i	$-0.866249\pi$
$743$	− 22.2391i	− 0.815872i	0.913010	−	0.407936i	$-0.133751\pi$
$744$	31.0979	1.14010
$745$	0	0
$746$	−6.38266	−0.233686
$747$	− 0.866508i	− 0.0317039i
$748$	33.8332i	1.23707i
$749$	64.2135	2.34631
$750$	0	0
$751$	25.0323	0.913441	0.456721	−	0.889610i	$-0.349024\pi$
$751$	25.0323	0.913441	0.456721	+	0.889610i	$0.349024\pi$
$752$	− 27.8588i	− 1.01591i
$753$	26.7909i	0.976313i
$754$	−50.5784	−1.84196
$755$	0	0
$756$	−21.6446	−0.787206
$757$	− 36.7330i	− 1.33508i	−0.744572	−	0.667542i	$-0.767346\pi$
$757$	− 36.7330i	− 1.33508i	0.744572	−	0.667542i	$-0.232654\pi$
$758$	23.8742i	0.867151i
$759$	2.18123	0.0791736
$760$	0	0
$761$	−22.0901	−0.800767	−0.400383	−	0.916348i	$-0.631123\pi$
$761$	−22.0901	−0.800767	−0.400383	+	0.916348i	$0.631123\pi$
$762$	21.3904i	0.774892i
$763$	− 52.3928i	− 1.89675i
$764$	−106.795	−3.86372
$765$	0	0
$766$	−28.2118	−1.01933
$767$	− 36.7653i	− 1.32752i
$768$	− 30.7680i	− 1.11024i
$769$	34.5817	1.24705	0.623524	−	0.781804i	$-0.285700\pi$
$769$	34.5817	1.24705	0.623524	+	0.781804i	$0.285700\pi$
$770$	0	0
$771$	16.3147	0.587560
$772$	− 29.3403i	− 1.05598i
$773$	28.0622i	1.00933i	0.863316	+	0.504664i	$0.168384\pi$
$773$	28.0622i	1.00933i	−0.863316	+	0.504664i	$0.831616\pi$
$774$	10.1157	0.363600
$775$	0	0
$776$	56.4882	2.02781
$777$	44.6039i	1.60016i
$778$	− 71.7774i	− 2.57334i
$779$	3.86651	0.138532
$780$	0	0
$781$	−12.6853	−0.453915
$782$	− 33.0477i	− 1.18178i
$783$	− 5.00000i	− 0.178685i
$784$	−112.618	−4.02208
$785$	0	0
$786$	−27.7730	−0.990631
$787$	− 7.54169i	− 0.268832i	−0.990925	−	0.134416i	$-0.957084\pi$
$787$	− 7.54169i	− 0.268832i	0.990925	−	0.134416i	$-0.0429159\pi$
$788$	102.366i	3.64663i
$789$	−1.92434	−0.0685085
$790$	0	0
$791$	1.63754	0.0582243
$792$	− 6.86651i	− 0.243991i
$793$	2.93206i	0.104121i
$794$	−3.54169	−0.125690
$795$	0	0
$796$	110.062	3.90103
$797$	− 17.4126i	− 0.616785i	−0.951259	−	0.308392i	$-0.900209\pi$
$797$	− 17.4126i	− 0.616785i	0.951259	−	0.308392i	$-0.0997910\pi$
$798$	− 12.4533i	− 0.440841i
$799$	28.9765	1.02511
$800$	0	0
$801$	−10.8487	−0.383320
$802$	8.89205i	0.313989i
$803$	− 12.5995i	− 0.444628i
$804$	−21.3904	−0.754380
$805$	0	0
$806$	−51.9287	−1.82911
$807$	− 19.5817i	− 0.689309i
$808$	16.9065i	0.594769i
$809$	5.84869	0.205629	0.102814	−	0.994701i	$-0.467215\pi$
$809$	5.84869	0.205629	0.102814	+	0.994701i	$0.467215\pi$
$810$	0	0
$811$	−4.02992	−0.141510	−0.0707548	−	0.997494i	$-0.522541\pi$
$811$	−4.02992	−0.141510	−0.0707548	+	0.997494i	$0.522541\pi$
$812$	− 108.223i	− 3.79788i
$813$	11.1913i	0.392497i
$814$	−25.9644	−0.910050
$815$	0	0
$816$	−44.3369	−1.55210
$817$	4.00000i	0.139942i
$818$	78.7451i	2.75326i
$819$	19.6974	0.688282
$820$	0	0
$821$	3.97446	0.138710	0.0693548	−	0.997592i	$-0.477906\pi$
$821$	3.97446	0.138710	0.0693548	+	0.997592i	$0.477906\pi$
$822$	12.4082i	0.432786i
$823$	1.07566i	0.0374950i	0.999824	+	0.0187475i	$0.00596787\pi$
$823$	1.07566i	0.0374950i	−0.999824	+	0.0187475i	$0.994032\pi$
$824$	91.4603	3.18617
$825$	0	0
$826$	114.462	3.98264
$827$	− 12.2313i	− 0.425325i	−0.977126	−	0.212663i	$-0.931786\pi$
$827$	− 12.2313i	− 0.425325i	0.977126	−	0.212663i	$-0.0682136\pi$
$828$	8.45831i	0.293947i
$829$	48.0444	1.66865	0.834325	−	0.551272i	$-0.185857\pi$
$829$	48.0444	1.66865	0.834325	+	0.551272i	$0.185857\pi$
$830$	0	0
$831$	16.3147	0.565951
$832$	7.76866i	0.269330i
$833$	− 117.137i	− 4.05854i
$834$	7.39543	0.256083
$835$	0	0
$836$	4.98218	0.172312
$837$	− 5.13349i	− 0.177439i
$838$	− 51.8386i	− 1.79073i
$839$	−41.6695	−1.43859	−0.719295	−	0.694705i	$-0.755535\pi$
$839$	−41.6695	−1.43859	−0.719295	+	0.694705i	$0.755535\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	− 7.75856i	− 0.267378i
$843$	− 2.75084i	− 0.0947438i
$844$	51.7475	1.78122
$845$	0	0
$846$	−10.7909	−0.370997
$847$	− 47.8410i	− 1.64384i
$848$	− 85.6311i	− 2.94059i
$849$	20.2313	0.694338
$850$	0	0
$851$	17.4304	0.597506
$852$	− 49.1907i	− 1.68524i
$853$	− 25.4126i	− 0.870110i	−0.900404	−	0.435055i	$-0.856729\pi$
$853$	− 25.4126i	− 0.870110i	0.900404	−	0.435055i	$-0.143271\pi$
$854$	−9.12844	−0.312369
$855$	0	0
$856$	−78.9943	−2.69997
$857$	8.09785i	0.276617i	0.990389	+	0.138309i	$0.0441666\pi$
$857$	8.09785i	0.276617i	−0.990389	+	0.138309i	$0.955833\pi$
$858$	11.4660i	0.391444i
$859$	45.8208	1.56338	0.781692	−	0.623664i	$-0.214357\pi$
$859$	45.8208	1.56338	0.781692	+	0.623664i	$0.214357\pi$
$860$	0	0
$861$	−19.0400	−0.648882
$862$	13.5111i	0.460190i
$863$	17.7051i	0.602688i	0.953515	+	0.301344i	$0.0974353\pi$
$863$	17.7051i	0.602688i	−0.953515	+	0.301344i	$0.902565\pi$
$864$	4.39543	0.149535
$865$	0	0
$866$	15.1735	0.515617
$867$	− 29.1157i	− 0.988820i
$868$	− 111.112i	− 3.77140i
$869$	−15.6116	−0.529588
$870$	0	0
$871$	19.4660	0.659581
$872$	64.4526i	2.18264i
$873$	− 9.32482i	− 0.315597i
$874$	−4.86651	−0.164612
$875$	0	0
$876$	48.8581	1.65076
$877$	12.9166i	0.436163i	0.975931	+	0.218082i	$0.0699799\pi$
$877$	12.9166i	0.436163i	−0.975931	+	0.218082i	$0.930020\pi$
$878$	− 8.28146i	− 0.279486i
$879$	−22.9822	−0.775170
$880$	0	0
$881$	14.1157	0.475569	0.237785	−	0.971318i	$-0.423579\pi$
$881$	14.1157	0.475569	0.237785	+	0.971318i	$0.423579\pi$
$882$	43.6217i	1.46882i
$883$	7.70510i	0.259297i	0.991560	+	0.129649i	$0.0413849\pi$
$883$	7.70510i	0.259297i	−0.991560	+	0.129649i	$0.958615\pi$
$884$	119.395	4.01568
$885$	0	0
$886$	90.7851	3.04999
$887$	− 34.6294i	− 1.16274i	−0.813638	−	0.581371i	$-0.802516\pi$
$887$	− 34.6294i	− 1.16274i	0.813638	−	0.581371i	$-0.197484\pi$
$888$	− 54.8709i	− 1.84135i
$889$	41.6516	1.39695
$890$	0	0
$891$	−1.13349	−0.0379734
$892$	17.2492i	0.577545i
$893$	− 4.26698i	− 0.142789i
$894$	0	0
$895$	0	0
$896$	−67.4755	−2.25420
$897$	− 7.69738i	− 0.257008i
$898$	− 3.20410i	− 0.106922i
$899$	25.6675	0.856058
$900$	0	0
$901$	89.0666	2.96724
$902$	− 11.0834i	− 0.369036i
$903$	− 19.6974i	− 0.655488i
$904$	−2.01448	−0.0670004
$905$	0	0
$906$	−11.8231	−0.392798
$907$	28.0699i	0.932047i	0.884772	+	0.466023i	$0.154314\pi$
$907$	28.0699i	0.932047i	−0.884772	+	0.466023i	$0.845686\pi$
$908$	− 7.07498i	− 0.234792i
$909$	2.79085	0.0925667
$910$	0	0
$911$	−6.35474	−0.210542	−0.105271	−	0.994444i	$-0.533571\pi$
$911$	−6.35474	−0.210542	−0.105271	+	0.994444i	$0.533571\pi$
$912$	6.52892i	0.216194i
$913$	− 0.982180i	− 0.0325054i
$914$	3.75254	0.124123
$915$	0	0
$916$	−8.50270	−0.280937
$917$	54.0800i	1.78588i
$918$	17.1735i	0.566810i
$919$	−10.8887	−0.359185	−0.179593	−	0.983741i	$-0.557478\pi$
$919$	−10.8887	−0.359185	−0.179593	+	0.983741i	$0.557478\pi$
$920$	0	0
$921$	8.59952	0.283364
$922$	− 12.4082i	− 0.408642i
$923$	44.7653i	1.47347i
$924$	−24.5340	−0.807108
$925$	0	0
$926$	35.1123	1.15386
$927$	− 15.0979i	− 0.495879i
$928$	21.9771i	0.721435i
$929$	−22.7552	−0.746574	−0.373287	−	0.927716i	$-0.621769\pi$
$929$	−22.7552	−0.746574	−0.373287	+	0.927716i	$0.621769\pi$
$930$	0	0
$931$	−17.2492	−0.565319
$932$	8.12577i	0.266168i
$933$	− 12.2313i	− 0.400436i
$934$	−74.6184	−2.44159
$935$	0	0
$936$	−24.2313	−0.792026
$937$	45.9822i	1.50217i	0.660204	+	0.751086i	$0.270470\pi$
$937$	45.9822i	1.50217i	−0.660204	+	0.751086i	$0.729530\pi$
$938$	60.6039i	1.97879i
$939$	7.51615	0.245280
$940$	0	0
$941$	−16.5639	−0.539967	−0.269984	−	0.962865i	$-0.587018\pi$
$941$	−16.5639	−0.539967	−0.269984	+	0.962865i	$0.587018\pi$
$942$	− 34.0094i	− 1.10809i
$943$	7.44049i	0.242296i
$944$	−60.0094	−1.95314
$945$	0	0
$946$	11.4660	0.372793
$947$	4.53397i	0.147334i	0.997283	+	0.0736671i	$0.0234702\pi$
$947$	4.53397i	0.147334i	−0.997283	+	0.0736671i	$0.976530\pi$
$948$	− 60.5383i	− 1.96619i
$949$	−44.4627	−1.44332
$950$	0	0
$951$	−26.5817	−0.861971
$952$	202.577i	6.56556i
$953$	− 36.4304i	− 1.18010i	−0.807368	−	0.590048i	$-0.799109\pi$
$953$	− 36.4304i	− 1.18010i	0.807368	−	0.590048i	$-0.200891\pi$
$954$	−33.1685	−1.07387
$955$	0	0
$956$	9.91997	0.320835
$957$	− 5.66746i	− 0.183203i
$958$	− 27.5060i	− 0.888680i
$959$	24.1614	0.780213
$960$	0	0
$961$	−4.64726	−0.149912
$962$	91.6261i	2.95414i
$963$	13.0400i	0.420209i
$964$	83.7673	2.69796
$965$	0	0
$966$	23.9644	0.771041
$967$	− 41.0044i	− 1.31861i	−0.751875	−	0.659306i	$-0.770850\pi$
$967$	− 41.0044i	− 1.31861i	0.751875	−	0.659306i	$-0.229150\pi$
$968$	58.8531i	1.89161i
$969$	−6.79085	−0.218154
$970$	0	0
$971$	3.58942	0.115190	0.0575951	−	0.998340i	$-0.481657\pi$
$971$	3.58942	0.115190	0.0575951	+	0.998340i	$0.481657\pi$
$972$	− 4.39543i	− 0.140983i
$973$	− 14.4005i	− 0.461658i
$974$	−84.1601	−2.69666
$975$	0	0
$976$	4.78580	0.153190
$977$	− 42.2313i	− 1.35110i	−0.737314	−	0.675550i	$-0.763906\pi$
$977$	− 42.2313i	− 1.35110i	0.737314	−	0.675550i	$-0.236094\pi$
$978$	− 54.5588i	− 1.74460i
$979$	−12.2969	−0.393011
$980$	0	0
$981$	10.6395	0.339694
$982$	− 12.4082i	− 0.395961i
$983$	− 36.0800i	− 1.15077i	−0.817881	−	0.575387i	$-0.804851\pi$
$983$	− 36.0800i	− 1.15077i	0.817881	−	0.575387i	$-0.195149\pi$
$984$	23.4227	0.746687
$985$	0	0
$986$	−85.8675	−2.73458
$987$	21.0121i	0.668822i
$988$	− 17.5817i	− 0.559349i
$989$	−7.69738	−0.244762
$990$	0	0
$991$	19.3648	0.615144	0.307572	−	0.951525i	$-0.400483\pi$
$991$	19.3648	0.615144	0.307572	+	0.951525i	$0.400483\pi$
$992$	22.5639i	0.716404i
$993$	19.3648i	0.614524i
$994$	−139.369	−4.42050
$995$	0	0
$996$	3.80867	0.120682
$997$	20.5639i	0.651265i	0.945496	+	0.325632i	$0.105577\pi$
$997$	20.5639i	0.651265i	−0.945496	+	0.325632i	$0.894423\pi$
$998$	− 5.07736i	− 0.160721i
$999$	−9.05784	−0.286577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.o.799.6		6
5.2	odd	4		1425.2.a.w.1.1	yes	3
5.3	odd	4		1425.2.a.t.1.3	✓	3
5.4	even	2	inner	1425.2.c.o.799.1		6
15.2	even	4		4275.2.a.bg.1.3		3
15.8	even	4		4275.2.a.bf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.t.1.3	✓	3	5.3	odd	4
1425.2.a.w.1.1	yes	3	5.2	odd	4
1425.2.c.o.799.1		6	5.4	even	2	inner
1425.2.c.o.799.6		6	1.1	even	1	trivial
4275.2.a.bf.1.1		3	15.8	even	4
4275.2.a.bg.1.3		3	15.2	even	4

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:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.52892i q^{2} +1.00000i q^{3} -4.39543 q^{4} -2.52892 q^{6} +4.92434i q^{7} -6.05784i q^{8} -1.00000 q^{9} -1.13349 q^{11} -4.39543i q^{12} +4.00000i q^{13} -12.4533 q^{14} +6.52892 q^{16} +6.79085i q^{17} -2.52892i q^{18} +1.00000 q^{19} -4.92434 q^{21} -2.86651i q^{22} +1.92434i q^{23} +6.05784 q^{24} -10.1157 q^{26} -1.00000i q^{27} -21.6446i q^{28} +5.00000 q^{29} +5.13349 q^{31} +4.39543i q^{32} -1.13349i q^{33} -17.1735 q^{34} +4.39543 q^{36} -9.05784i q^{37} +2.52892i q^{38} -4.00000 q^{39} +3.86651 q^{41} -12.4533i q^{42} +4.00000i q^{43} +4.98218 q^{44} -4.86651 q^{46} -4.26698i q^{47} +6.52892i q^{48} -17.2492 q^{49} -6.79085 q^{51} -17.5817i q^{52} -13.1157i q^{53} +2.52892 q^{54} +29.8309 q^{56} +1.00000i q^{57} +12.6446i q^{58} -9.19133 q^{59} +0.733016 q^{61} +12.9822i q^{62} -4.92434i q^{63} +1.94216 q^{64} +2.86651 q^{66} -4.86651i q^{67} -29.8487i q^{68} -1.92434 q^{69} +11.1913 q^{71} +6.05784i q^{72} +11.1157i q^{73} +22.9065 q^{74} -4.39543 q^{76} -5.58170i q^{77} -10.1157i q^{78} +13.7730 q^{79} +1.00000 q^{81} +9.77808i q^{82} +0.866508i q^{83} +21.6446 q^{84} -10.1157 q^{86} +5.00000i q^{87} +6.86651i q^{88} +10.8487 q^{89} -19.6974 q^{91} -8.45831i q^{92} +5.13349i q^{93} +10.7909 q^{94} -4.39543 q^{96} +9.32482i q^{97} -43.6217i q^{98} +1.13349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 30 q^{14} + 24 q^{16} + 6 q^{19} + 6 q^{24} + 30 q^{29} + 30 q^{31} - 12 q^{34} + 12 q^{36} - 24 q^{39} + 24 q^{41} - 30 q^{44} - 30 q^{46} - 42 q^{49} - 12 q^{51}+ \cdots + 6 q^{99}+O(q^{100}) \) 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 30 * q^14 + 24 * q^16 + 6 * q^19 + 6 * q^24 + 30 * q^29 + 30 * q^31 - 12 * q^34 + 12 * q^36 - 24 * q^39 + 24 * q^41 - 30 * q^44 - 30 * q^46 - 42 * q^49 - 12 * q^51 + 60 * q^56 - 24 * q^59 + 6 * q^61 + 42 * q^64 + 18 * q^66 + 18 * q^69 + 36 * q^71 + 48 * q^74 - 12 * q^76 - 6 * q^79 + 6 * q^81 + 54 * q^84 + 6 * q^89 + 36 * q^94 - 12 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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