LMFDB - Embedded newform 825.2.c.f.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$0.432320 + 0.432320i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.f.199.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.76156i q^{2} +1.00000i q^{3} -5.62620 q^{4} +2.76156 q^{6} +1.86464i q^{7} +10.0140i q^{8} -1.00000 q^{9} +1.00000 q^{11} -5.62620i q^{12} -4.62620i q^{13} +5.14931 q^{14} +16.4017 q^{16} +2.49084i q^{17} +2.76156i q^{18} +5.38776 q^{19} -1.86464 q^{21} -2.76156i q^{22} -7.14931i q^{23} -10.0140 q^{24} -12.7755 q^{26} -1.00000i q^{27} -10.4908i q^{28} +3.52311 q^{29} +8.62620 q^{31} -25.2663i q^{32} +1.00000i q^{33} +6.87859 q^{34} +5.62620 q^{36} -8.87859i q^{37} -14.8786i q^{38} +4.62620 q^{39} -0.761557 q^{41} +5.14931i q^{42} +7.40171i q^{43} -5.62620 q^{44} -19.7432 q^{46} -0.373802i q^{47} +16.4017i q^{48} +3.52311 q^{49} -2.49084 q^{51} +26.0279i q^{52} +5.45856i q^{53} -2.76156 q^{54} -18.6724 q^{56} +5.38776i q^{57} -9.72928i q^{58} -5.14931 q^{59} +4.42003 q^{61} -23.8217i q^{62} -1.86464i q^{63} -36.9711 q^{64} +2.76156 q^{66} +11.9431i q^{67} -14.0140i q^{68} +7.14931 q^{69} +11.6262 q^{71} -10.0140i q^{72} -6.77551i q^{73} -24.5187 q^{74} -30.3126 q^{76} +1.86464i q^{77} -12.7755i q^{78} +6.01395 q^{79} +1.00000 q^{81} +2.10308i q^{82} -14.5693i q^{83} +10.4908 q^{84} +20.4402 q^{86} +3.52311i q^{87} +10.0140i q^{88} -9.04623 q^{89} +8.62620 q^{91} +40.2234i q^{92} +8.62620i q^{93} -1.03228 q^{94} +25.2663 q^{96} +16.3169i q^{97} -9.72928i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46}+ \cdots - 6 q^{99}+O(q^{100}) $ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 + 6 * q^11 - 12 * q^14 + 20 * q^16 + 2 * q^19 - 6 * q^21 - 12 * q^24 - 16 * q^26 - 4 * q^29 + 34 * q^31 - 12 * q^34 + 16 * q^36 + 10 * q^39 + 8 * q^41 - 16 * q^44 - 60 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 - 44 * q^56 + 12 * q^59 - 6 * q^61 - 68 * q^64 + 4 * q^66 + 52 * q^71 - 28 * q^74 - 48 * q^76 - 12 * q^79 + 6 * q^81 + 40 * q^84 + 56 * q^86 - 4 * q^89 + 34 * q^91 - 4 * q^94 + 68 * q^96 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.76156i	− 1.95272i	−0.216160	−	0.976358i	$-0.569353\pi$
$2$	− 2.76156i	− 1.95272i	0.216160	−	0.976358i	$-0.430647\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−5.62620	−2.81310
$5$	0	0
$5$	0	0
$6$	2.76156	1.12740
$7$	1.86464i	0.704768i	0.935855	+	0.352384i	$0.114629\pi$
$7$	1.86464i	0.704768i	−0.935855	+	0.352384i	$0.885371\pi$
$8$	10.0140i	3.54047i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	− 5.62620i	− 1.62414i
$13$	− 4.62620i	− 1.28308i	−0.767091	−	0.641538i	$-0.778297\pi$
$13$	− 4.62620i	− 1.28308i	0.767091	−	0.641538i	$-0.221703\pi$
$14$	5.14931	1.37621
$15$	0	0
$16$	16.4017	4.10043
$17$	2.49084i	0.604117i	0.953289	+	0.302059i	$0.0976738\pi$
$17$	2.49084i	0.604117i	−0.953289	+	0.302059i	$0.902326\pi$
$18$	2.76156i	0.650905i
$19$	5.38776	1.23604	0.618018	−	0.786164i	$-0.287936\pi$
$19$	5.38776	1.23604	0.618018	+	0.786164i	$0.287936\pi$
$20$	0	0
$21$	−1.86464	−0.406898
$22$	− 2.76156i	− 0.588766i
$23$	− 7.14931i	− 1.49073i	−0.666654	−	0.745367i	$-0.732274\pi$
$23$	− 7.14931i	− 1.49073i	0.666654	−	0.745367i	$-0.267726\pi$
$24$	−10.0140	−2.04409
$25$	0	0
$26$	−12.7755	−2.50548
$27$	− 1.00000i	− 0.192450i
$28$	− 10.4908i	− 1.98258i
$29$	3.52311	0.654226	0.327113	−	0.944985i	$-0.393924\pi$
$29$	3.52311	0.654226	0.327113	+	0.944985i	$0.393924\pi$
$30$	0	0
$31$	8.62620	1.54931	0.774655	−	0.632384i	$-0.217923\pi$
$31$	8.62620	1.54931	0.774655	+	0.632384i	$0.217923\pi$
$32$	− 25.2663i	− 4.46650i
$33$	1.00000i	0.174078i
$34$	6.87859	1.17967
$35$	0	0
$36$	5.62620	0.937700
$37$	− 8.87859i	− 1.45963i	−0.683644	−	0.729816i	$-0.739606\pi$
$37$	− 8.87859i	− 1.45963i	0.683644	−	0.729816i	$-0.260394\pi$
$38$	− 14.8786i	− 2.41363i
$39$	4.62620	0.740785
$40$	0	0
$41$	−0.761557	−0.118935	−0.0594676	−	0.998230i	$-0.518940\pi$
$41$	−0.761557	−0.118935	−0.0594676	+	0.998230i	$0.518940\pi$
$42$	5.14931i	0.794556i
$43$	7.40171i	1.12875i	0.825519	+	0.564375i	$0.190883\pi$
$43$	7.40171i	1.12875i	−0.825519	+	0.564375i	$0.809117\pi$
$44$	−5.62620	−0.848181
$45$	0	0
$46$	−19.7432	−2.91098
$47$	− 0.373802i	− 0.0545246i	−0.999628	−	0.0272623i	$-0.991321\pi$
$47$	− 0.373802i	− 0.0545246i	0.999628	−	0.0272623i	$-0.00867894\pi$
$48$	16.4017i	2.36738i
$49$	3.52311	0.503302
$50$	0	0
$51$	−2.49084	−0.348787
$52$	26.0279i	3.60942i
$53$	5.45856i	0.749791i	0.927067	+	0.374896i	$0.122321\pi$
$53$	5.45856i	0.749791i	−0.927067	+	0.374896i	$0.877679\pi$
$54$	−2.76156	−0.375800
$55$	0	0
$56$	−18.6724	−2.49521
$57$	5.38776i	0.713626i
$58$	− 9.72928i	− 1.27752i
$59$	−5.14931	−0.670383	−0.335192	−	0.942150i	$-0.608801\pi$
$59$	−5.14931	−0.670383	−0.335192	+	0.942150i	$0.608801\pi$
$60$	0	0
$61$	4.42003	0.565927	0.282963	−	0.959131i	$-0.408682\pi$
$61$	4.42003	0.565927	0.282963	+	0.959131i	$0.408682\pi$
$62$	− 23.8217i	− 3.02536i
$63$	− 1.86464i	− 0.234923i
$64$	−36.9711	−4.62138
$65$	0	0
$66$	2.76156	0.339924
$67$	11.9431i	1.45909i	0.683934	+	0.729544i	$0.260268\pi$
$67$	11.9431i	1.45909i	−0.683934	+	0.729544i	$0.739732\pi$
$68$	− 14.0140i	− 1.69944i
$69$	7.14931	0.860676
$70$	0	0
$71$	11.6262	1.37978	0.689888	−	0.723916i	$-0.257660\pi$
$71$	11.6262	1.37978	0.689888	+	0.723916i	$0.257660\pi$
$72$	− 10.0140i	− 1.18016i
$73$	− 6.77551i	− 0.793014i	−0.918032	−	0.396507i	$-0.870222\pi$
$73$	− 6.77551i	− 0.793014i	0.918032	−	0.396507i	$-0.129778\pi$
$74$	−24.5187	−2.85025
$75$	0	0
$76$	−30.3126	−3.47709
$77$	1.86464i	0.212496i
$78$	− 12.7755i	− 1.44654i
$79$	6.01395	0.676623	0.338311	−	0.941034i	$-0.390144\pi$
$79$	6.01395	0.676623	0.338311	+	0.941034i	$0.390144\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.10308i	0.232247i
$83$	− 14.5693i	− 1.59919i	−0.600538	−	0.799597i	$-0.705047\pi$
$83$	− 14.5693i	− 1.59919i	0.600538	−	0.799597i	$-0.294953\pi$
$84$	10.4908	1.14464
$85$	0	0
$86$	20.4402	2.20413
$87$	3.52311i	0.377718i
$88$	10.0140i	1.06749i
$89$	−9.04623	−0.958898	−0.479449	−	0.877570i	$-0.659164\pi$
$89$	−9.04623	−0.958898	−0.479449	+	0.877570i	$0.659164\pi$
$90$	0	0
$91$	8.62620	0.904271
$92$	40.2234i	4.19358i
$93$	8.62620i	0.894495i
$94$	−1.03228	−0.106471
$95$	0	0
$96$	25.2663	2.57874
$97$	16.3169i	1.65673i	0.560185	+	0.828367i	$0.310730\pi$
$97$	16.3169i	1.65673i	−0.560185	+	0.828367i	$0.689270\pi$
$98$	− 9.72928i	− 0.982806i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−1.03228	−0.102715	−0.0513576	−	0.998680i	$-0.516355\pi$
$101$	−1.03228	−0.102715	−0.0513576	+	0.998680i	$0.516355\pi$
$102$	6.87859i	0.681082i
$103$	− 3.04623i	− 0.300154i	−0.988674	−	0.150077i	$-0.952048\pi$
$103$	− 3.04623i	− 0.300154i	0.988674	−	0.150077i	$-0.0479521\pi$
$104$	46.3265	4.54269
$105$	0	0
$106$	15.0741	1.46413
$107$	− 8.50479i	− 0.822189i	−0.911593	−	0.411095i	$-0.865147\pi$
$107$	− 8.50479i	− 0.822189i	0.911593	−	0.411095i	$-0.134853\pi$
$108$	5.62620i	0.541381i
$109$	6.14931	0.588997	0.294499	−	0.955652i	$-0.404847\pi$
$109$	6.14931	0.588997	0.294499	+	0.955652i	$0.404847\pi$
$110$	0	0
$111$	8.87859	0.842719
$112$	30.5833i	2.88985i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	14.8786	1.39351
$115$	0	0
$116$	−19.8217	−1.84040
$117$	4.62620i	0.427692i
$118$	14.2201i	1.30907i
$119$	−4.64452	−0.425762
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 12.2062i	− 1.10509i
$123$	− 0.761557i	− 0.0686673i
$124$	−48.5327	−4.35837
$125$	0	0
$126$	−5.14931	−0.458737
$127$	− 9.26635i	− 0.822256i	−0.911578	−	0.411128i	$-0.865135\pi$
$127$	− 9.26635i	− 0.822256i	0.911578	−	0.411128i	$-0.134865\pi$
$128$	51.5650i	4.55774i
$129$	−7.40171	−0.651684
$130$	0	0
$131$	4.06455	0.355121	0.177561	−	0.984110i	$-0.443179\pi$
$131$	4.06455	0.355121	0.177561	+	0.984110i	$0.443179\pi$
$132$	− 5.62620i	− 0.489698i
$133$	10.0462i	0.871119i
$134$	32.9817	2.84918
$135$	0	0
$136$	−24.9431	−2.13886
$137$	5.93545i	0.507100i	0.967322	+	0.253550i	$0.0815982\pi$
$137$	5.93545i	0.507100i	−0.967322	+	0.253550i	$0.918402\pi$
$138$	− 19.7432i	− 1.68066i
$139$	−6.71096	−0.569216	−0.284608	−	0.958644i	$-0.591863\pi$
$139$	−6.71096	−0.569216	−0.284608	+	0.958644i	$0.591863\pi$
$140$	0	0
$141$	0.373802	0.0314798
$142$	− 32.1064i	− 2.69431i
$143$	− 4.62620i	− 0.386862i
$144$	−16.4017	−1.36681
$145$	0	0
$146$	−18.7110	−1.54853
$147$	3.52311i	0.290582i
$148$	49.9527i	4.10609i
$149$	−5.74324	−0.470504	−0.235252	−	0.971934i	$-0.575592\pi$
$149$	−5.74324	−0.470504	−0.235252	+	0.971934i	$0.575592\pi$
$150$	0	0
$151$	10.5616	0.859495	0.429747	−	0.902949i	$-0.358603\pi$
$151$	10.5616	0.859495	0.429747	+	0.902949i	$0.358603\pi$
$152$	53.9527i	4.37614i
$153$	− 2.49084i	− 0.201372i
$154$	5.14931	0.414943
$155$	0	0
$156$	−26.0279	−2.08390
$157$	3.10308i	0.247653i	0.992304	+	0.123827i	$0.0395166\pi$
$157$	3.10308i	0.247653i	−0.992304	+	0.123827i	$0.960483\pi$
$158$	− 16.6079i	− 1.32125i
$159$	−5.45856	−0.432892
$160$	0	0
$161$	13.3309	1.05062
$162$	− 2.76156i	− 0.216968i
$163$	− 15.6724i	− 1.22756i	−0.789477	−	0.613780i	$-0.789648\pi$
$163$	− 15.6724i	− 1.22756i	0.789477	−	0.613780i	$-0.210352\pi$
$164$	4.28467	0.334577
$165$	0	0
$166$	−40.2341	−3.12277
$167$	− 8.98168i	− 0.695023i	−0.937676	−	0.347512i	$-0.887027\pi$
$167$	− 8.98168i	− 0.695023i	0.937676	−	0.347512i	$-0.112973\pi$
$168$	− 18.6724i	− 1.44061i
$169$	−8.40171	−0.646285
$170$	0	0
$171$	−5.38776	−0.412012
$172$	− 41.6435i	− 3.17529i
$173$	11.5092i	0.875025i	0.899212	+	0.437513i	$0.144140\pi$
$173$	11.5092i	0.875025i	−0.899212	+	0.437513i	$0.855860\pi$
$174$	9.72928	0.737575
$175$	0	0
$176$	16.4017	1.23633
$177$	− 5.14931i	− 0.387046i
$178$	24.9817i	1.87246i
$179$	−10.3738	−0.775374	−0.387687	−	0.921791i	$-0.626726\pi$
$179$	−10.3738	−0.775374	−0.387687	+	0.921791i	$0.626726\pi$
$180$	0	0
$181$	−2.66473	−0.198068	−0.0990339	−	0.995084i	$-0.531575\pi$
$181$	−2.66473	−0.198068	−0.0990339	+	0.995084i	$0.531575\pi$
$182$	− 23.8217i	− 1.76578i
$183$	4.42003i	0.326738i
$184$	71.5929	5.27790
$185$	0	0
$186$	23.8217	1.74669
$187$	2.49084i	0.182148i
$188$	2.10308i	0.153383i
$189$	1.86464	0.135633
$190$	0	0
$191$	14.6724	1.06166	0.530830	−	0.847478i	$-0.321880\pi$
$191$	14.6724	1.06166	0.530830	+	0.847478i	$0.321880\pi$
$192$	− 36.9711i	− 2.66816i
$193$	− 5.10308i	− 0.367328i	−0.982989	−	0.183664i	$-0.941204\pi$
$193$	− 5.10308i	− 0.367328i	0.982989	−	0.183664i	$-0.0587958\pi$
$194$	45.0602	3.23513
$195$	0	0
$196$	−19.8217	−1.41584
$197$	3.74324i	0.266694i	0.991069	+	0.133347i	$0.0425725\pi$
$197$	3.74324i	0.266694i	−0.991069	+	0.133347i	$0.957427\pi$
$198$	2.76156i	0.196255i
$199$	8.08476	0.573114	0.286557	−	0.958063i	$-0.407489\pi$
$199$	8.08476	0.573114	0.286557	+	0.958063i	$0.407489\pi$
$200$	0	0
$201$	−11.9431	−0.842404
$202$	2.85069i	0.200574i
$203$	6.56934i	0.461077i
$204$	14.0140	0.981173
$205$	0	0
$206$	−8.41233	−0.586115
$207$	7.14931i	0.496912i
$208$	− 75.8776i	− 5.26116i
$209$	5.38776	0.372679
$210$	0	0
$211$	−15.9431	−1.09757	−0.548786	−	0.835963i	$-0.684910\pi$
$211$	−15.9431	−1.09757	−0.548786	+	0.835963i	$0.684910\pi$
$212$	− 30.7110i	− 2.10924i
$213$	11.6262i	0.796614i
$214$	−23.4865	−1.60550
$215$	0	0
$216$	10.0140	0.681363
$217$	16.0848i	1.09190i
$218$	− 16.9817i	− 1.15014i
$219$	6.77551	0.457847
$220$	0	0
$221$	11.5231	0.775129
$222$	− 24.5187i	− 1.64559i
$223$	22.1772i	1.48510i	0.669793	+	0.742548i	$0.266383\pi$
$223$	22.1772i	1.48510i	−0.669793	+	0.742548i	$0.733617\pi$
$224$	47.1127	3.14785
$225$	0	0
$226$	16.5693	1.10218
$227$	5.93545i	0.393950i	0.980409	+	0.196975i	$0.0631117\pi$
$227$	5.93545i	0.393950i	−0.980409	+	0.196975i	$0.936888\pi$
$228$	− 30.3126i	− 2.00750i
$229$	8.31695	0.549599	0.274800	−	0.961502i	$-0.411388\pi$
$229$	8.31695	0.549599	0.274800	+	0.961502i	$0.411388\pi$
$230$	0	0
$231$	−1.86464	−0.122684
$232$	35.2803i	2.31627i
$233$	− 28.5187i	− 1.86833i	−0.356848	−	0.934163i	$-0.616148\pi$
$233$	− 28.5187i	− 1.86833i	0.356848	−	0.934163i	$-0.383852\pi$
$234$	12.7755	0.835161
$235$	0	0
$236$	28.9711	1.88585
$237$	6.01395i	0.390648i
$238$	12.8261i	0.831393i
$239$	−24.0925	−1.55841	−0.779206	−	0.626768i	$-0.784377\pi$
$239$	−24.0925	−1.55841	−0.779206	+	0.626768i	$0.784377\pi$
$240$	0	0
$241$	−5.16763	−0.332877	−0.166438	−	0.986052i	$-0.553227\pi$
$241$	−5.16763	−0.332877	−0.166438	+	0.986052i	$0.553227\pi$
$242$	− 2.76156i	− 0.177520i
$243$	1.00000i	0.0641500i
$244$	−24.8680	−1.59201
$245$	0	0
$246$	−2.10308	−0.134088
$247$	− 24.9248i	− 1.58593i
$248$	86.3823i	5.48528i
$249$	14.5693	0.923295
$250$	0	0
$251$	−9.25240	−0.584006	−0.292003	−	0.956417i	$-0.594322\pi$
$251$	−9.25240	−0.584006	−0.292003	+	0.956417i	$0.594322\pi$
$252$	10.4908i	0.660861i
$253$	− 7.14931i	− 0.449473i
$254$	−25.5896	−1.60563
$255$	0	0
$256$	68.4575	4.27860
$257$	− 25.2158i	− 1.57292i	−0.617644	−	0.786458i	$-0.711913\pi$
$257$	− 25.2158i	− 1.57292i	0.617644	−	0.786458i	$-0.288087\pi$
$258$	20.4402i	1.27255i
$259$	16.5554	1.02870
$260$	0	0
$261$	−3.52311	−0.218075
$262$	− 11.2245i	− 0.693451i
$263$	− 5.45856i	− 0.336589i	−0.985737	−	0.168295i	$-0.946174\pi$
$263$	− 5.45856i	− 0.336589i	0.985737	−	0.168295i	$-0.0538260\pi$
$264$	−10.0140	−0.616316
$265$	0	0
$266$	27.7432	1.70105
$267$	− 9.04623i	− 0.553620i
$268$	− 67.1945i	− 4.10456i
$269$	−17.0741	−1.04103	−0.520514	−	0.853853i	$-0.674260\pi$
$269$	−17.0741	−1.04103	−0.520514	+	0.853853i	$0.674260\pi$
$270$	0	0
$271$	9.77988	0.594085	0.297043	−	0.954864i	$-0.404000\pi$
$271$	9.77988	0.594085	0.297043	+	0.954864i	$0.404000\pi$
$272$	40.8540i	2.47714i
$273$	8.62620i	0.522081i
$274$	16.3911	0.990221
$275$	0	0
$276$	−40.2234	−2.42117
$277$	5.91524i	0.355412i	0.984084	+	0.177706i	$0.0568676\pi$
$277$	5.91524i	0.355412i	−0.984084	+	0.177706i	$0.943132\pi$
$278$	18.5327i	1.11152i
$279$	−8.62620	−0.516437
$280$	0	0
$281$	−6.76156	−0.403361	−0.201680	−	0.979451i	$-0.564640\pi$
$281$	−6.76156	−0.403361	−0.201680	+	0.979451i	$0.564640\pi$
$282$	− 1.03228i	− 0.0614711i
$283$	− 8.81841i	− 0.524200i	−0.965041	−	0.262100i	$-0.915585\pi$
$283$	− 8.81841i	− 0.524200i	0.965041	−	0.262100i	$-0.0844151\pi$
$284$	−65.4113	−3.88145
$285$	0	0
$286$	−12.7755	−0.755432
$287$	− 1.42003i	− 0.0838218i
$288$	25.2663i	1.48883i
$289$	10.7957	0.635042
$290$	0	0
$291$	−16.3169	−0.956516
$292$	38.1204i	2.23083i
$293$	30.4908i	1.78129i	0.454696	+	0.890647i	$0.349748\pi$
$293$	30.4908i	1.78129i	−0.454696	+	0.890647i	$0.650252\pi$
$294$	9.72928	0.567423
$295$	0	0
$296$	88.9098	5.16778
$297$	− 1.00000i	− 0.0580259i
$298$	15.8603i	0.918761i
$299$	−33.0741	−1.91273
$300$	0	0
$301$	−13.8015	−0.795507
$302$	− 29.1666i	− 1.67835i
$303$	− 1.03228i	− 0.0593027i
$304$	88.3684	5.06827
$305$	0	0
$306$	−6.87859	−0.393223
$307$	0.767815i	0.0438215i	0.999760	+	0.0219107i	$0.00697497\pi$
$307$	0.767815i	0.0438215i	−0.999760	+	0.0219107i	$0.993025\pi$
$308$	− 10.4908i	− 0.597771i
$309$	3.04623	0.173294
$310$	0	0
$311$	−33.8496	−1.91944	−0.959719	−	0.280963i	$-0.909346\pi$
$311$	−33.8496	−1.91944	−0.959719	+	0.280963i	$0.909346\pi$
$312$	46.3265i	2.62272i
$313$	8.11078i	0.458448i	0.973374	+	0.229224i	$0.0736189\pi$
$313$	8.11078i	0.458448i	−0.973374	+	0.229224i	$0.926381\pi$
$314$	8.56934	0.483596
$315$	0	0
$316$	−33.8357	−1.90341
$317$	8.06455i	0.452950i	0.974017	+	0.226475i	$0.0727202\pi$
$317$	8.06455i	0.452950i	−0.974017	+	0.226475i	$0.927280\pi$
$318$	15.0741i	0.845316i
$319$	3.52311	0.197257
$320$	0	0
$321$	8.50479	0.474691
$322$	− 36.8140i	− 2.05157i
$323$	13.4200i	0.746710i
$324$	−5.62620	−0.312567
$325$	0	0
$326$	−43.2803	−2.39707
$327$	6.14931i	0.340058i
$328$	− 7.62620i	− 0.421086i
$329$	0.697006	0.0384272
$330$	0	0
$331$	22.4769	1.23544	0.617721	−	0.786398i	$-0.288056\pi$
$331$	22.4769	1.23544	0.617721	+	0.786398i	$0.288056\pi$
$332$	81.9700i	4.49869i
$333$	8.87859i	0.486544i
$334$	−24.8034	−1.35718
$335$	0	0
$336$	−30.5833	−1.66846
$337$	6.32757i	0.344685i	0.985037	+	0.172342i	$0.0551336\pi$
$337$	6.32757i	0.344685i	−0.985037	+	0.172342i	$0.944866\pi$
$338$	23.2018i	1.26201i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	8.62620	0.467135
$342$	14.8786i	0.804542i
$343$	19.6218i	1.05948i
$344$	−74.1204	−3.99630
$345$	0	0
$346$	31.7832	1.70868
$347$	− 0.178261i	− 0.00956954i	−0.999989	−	0.00478477i	$-0.998477\pi$
$347$	− 0.178261i	− 0.00956954i	0.999989	−	0.00478477i	$-0.00152305\pi$
$348$	− 19.8217i	− 1.06256i
$349$	19.7293	1.05608	0.528042	−	0.849218i	$-0.322926\pi$
$349$	19.7293	1.05608	0.528042	+	0.849218i	$0.322926\pi$
$350$	0	0
$351$	−4.62620	−0.246928
$352$	− 25.2663i	− 1.34670i
$353$	19.5231i	1.03911i	0.854437	+	0.519555i	$0.173902\pi$
$353$	19.5231i	1.03911i	−0.854437	+	0.519555i	$0.826098\pi$
$354$	−14.2201	−0.755791
$355$	0	0
$356$	50.8959	2.69748
$357$	− 4.64452i	− 0.245814i
$358$	28.6478i	1.51409i
$359$	−35.6926	−1.88379	−0.941893	−	0.335914i	$-0.890955\pi$
$359$	−35.6926	−1.88379	−0.941893	+	0.335914i	$0.890955\pi$
$360$	0	0
$361$	10.0279	0.527785
$362$	7.35881i	0.386770i
$363$	1.00000i	0.0524864i
$364$	−48.5327	−2.54380
$365$	0	0
$366$	12.2062	0.638027
$367$	20.1127i	1.04987i	0.851141	+	0.524936i	$0.175911\pi$
$367$	20.1127i	1.04987i	−0.851141	+	0.524936i	$0.824089\pi$
$368$	− 117.261i	− 6.11265i
$369$	0.761557	0.0396451
$370$	0	0
$371$	−10.1783	−0.528429
$372$	− 48.5327i	− 2.51630i
$373$	2.56165i	0.132637i	0.997799	+	0.0663185i	$0.0211253\pi$
$373$	2.56165i	0.132637i	−0.997799	+	0.0663185i	$0.978875\pi$
$374$	6.87859	0.355684
$375$	0	0
$376$	3.74324	0.193043
$377$	− 16.2986i	− 0.839422i
$378$	− 5.14931i	− 0.264852i
$379$	−23.2601	−1.19479	−0.597395	−	0.801947i	$-0.703798\pi$
$379$	−23.2601	−1.19479	−0.597395	+	0.801947i	$0.703798\pi$
$380$	0	0
$381$	9.26635	0.474729
$382$	− 40.5187i	− 2.07312i
$383$	− 25.7938i	− 1.31800i	−0.752142	−	0.659002i	$-0.770979\pi$
$383$	− 25.7938i	− 1.31800i	0.752142	−	0.659002i	$-0.229021\pi$
$384$	−51.5650	−2.63141
$385$	0	0
$386$	−14.0925	−0.717287
$387$	− 7.40171i	− 0.376250i
$388$	− 91.8024i	− 4.66056i
$389$	−2.33527	−0.118403	−0.0592014	−	0.998246i	$-0.518855\pi$
$389$	−2.33527	−0.118403	−0.0592014	+	0.998246i	$0.518855\pi$
$390$	0	0
$391$	17.8078	0.900578
$392$	35.2803i	1.78192i
$393$	4.06455i	0.205029i
$394$	10.3372	0.520778
$395$	0	0
$396$	5.62620	0.282727
$397$	17.1955i	0.863019i	0.902108	+	0.431510i	$0.142019\pi$
$397$	17.1955i	0.863019i	−0.902108	+	0.431510i	$0.857981\pi$
$398$	− 22.3265i	− 1.11913i
$399$	−10.0462	−0.502941
$400$	0	0
$401$	−26.5693	−1.32681	−0.663405	−	0.748261i	$-0.730889\pi$
$401$	−26.5693	−1.32681	−0.663405	+	0.748261i	$0.730889\pi$
$402$	32.9817i	1.64498i
$403$	− 39.9065i	− 1.98788i
$404$	5.80779	0.288948
$405$	0	0
$406$	18.1416	0.900353
$407$	− 8.87859i	− 0.440096i
$408$	− 24.9431i	− 1.23487i
$409$	20.4200	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$409$	20.4200	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$410$	0	0
$411$	−5.93545	−0.292774
$412$	17.1387i	0.844362i
$413$	− 9.60162i	− 0.472465i
$414$	19.7432	0.970327
$415$	0	0
$416$	−116.887	−5.73086
$417$	− 6.71096i	− 0.328637i
$418$	− 14.8786i	− 0.727736i
$419$	13.5896	0.663893	0.331947	−	0.943298i	$-0.392295\pi$
$419$	13.5896	0.663893	0.331947	+	0.943298i	$0.392295\pi$
$420$	0	0
$421$	−39.6175	−1.93084	−0.965418	−	0.260705i	$-0.916045\pi$
$421$	−39.6175	−1.93084	−0.965418	+	0.260705i	$0.916045\pi$
$422$	44.0279i	2.14324i
$423$	0.373802i	0.0181749i
$424$	−54.6618	−2.65461
$425$	0	0
$426$	32.1064	1.55556
$427$	8.24177i	0.398847i
$428$	47.8496i	2.31290i
$429$	4.62620	0.223355
$430$	0	0
$431$	−6.56934	−0.316434	−0.158217	−	0.987404i	$-0.550575\pi$
$431$	−6.56934	−0.316434	−0.158217	+	0.987404i	$0.550575\pi$
$432$	− 16.4017i	− 0.789128i
$433$	36.4113i	1.74982i	0.484290	+	0.874908i	$0.339078\pi$
$433$	36.4113i	1.74982i	−0.484290	+	0.874908i	$0.660922\pi$
$434$	44.4190	2.13218
$435$	0	0
$436$	−34.5972	−1.65691
$437$	− 38.5187i	− 1.84260i
$438$	− 18.7110i	− 0.894044i
$439$	−25.3878	−1.21169	−0.605846	−	0.795582i	$-0.707165\pi$
$439$	−25.3878	−1.21169	−0.605846	+	0.795582i	$0.707165\pi$
$440$	0	0
$441$	−3.52311	−0.167767
$442$	− 31.8217i	− 1.51361i
$443$	20.9065i	0.993298i	0.867952	+	0.496649i	$0.165436\pi$
$443$	20.9065i	0.993298i	−0.867952	+	0.496649i	$0.834564\pi$
$444$	−49.9527	−2.37065
$445$	0	0
$446$	61.2437	2.89997
$447$	− 5.74324i	− 0.271646i
$448$	− 68.9377i	− 3.25700i
$449$	34.9450	1.64916	0.824579	−	0.565747i	$-0.191412\pi$
$449$	34.9450	1.64916	0.824579	+	0.565747i	$0.191412\pi$
$450$	0	0
$451$	−0.761557	−0.0358603
$452$	− 33.7572i	− 1.58780i
$453$	10.5616i	0.496229i
$454$	16.3911	0.769272
$455$	0	0
$456$	−53.9527	−2.52657
$457$	7.31695i	0.342272i	0.985247	+	0.171136i	$0.0547438\pi$
$457$	7.31695i	0.342272i	−0.985247	+	0.171136i	$0.945256\pi$
$458$	− 22.9677i	− 1.07321i
$459$	2.49084	0.116262
$460$	0	0
$461$	−14.0279	−0.653345	−0.326672	−	0.945138i	$-0.605927\pi$
$461$	−14.0279	−0.653345	−0.326672	+	0.945138i	$0.605927\pi$
$462$	5.14931i	0.239568i
$463$	4.33527i	0.201477i	0.994913	+	0.100739i	$0.0321206\pi$
$463$	4.33527i	0.201477i	−0.994913	+	0.100739i	$0.967879\pi$
$464$	57.7851	2.68261
$465$	0	0
$466$	−78.7561	−3.64831
$467$	− 3.70138i	− 0.171279i	−0.996326	−	0.0856396i	$-0.972707\pi$
$467$	− 3.70138i	− 0.171279i	0.996326	−	0.0856396i	$-0.0272934\pi$
$468$	− 26.0279i	− 1.20314i
$469$	−22.2697	−1.02832
$470$	0	0
$471$	−3.10308	−0.142983
$472$	− 51.5650i	− 2.37347i
$473$	7.40171i	0.340331i
$474$	16.6079	0.762825
$475$	0	0
$476$	26.1310	1.19771
$477$	− 5.45856i	− 0.249930i
$478$	66.5327i	3.04313i
$479$	22.8680	1.04486	0.522432	−	0.852681i	$-0.325025\pi$
$479$	22.8680	1.04486	0.522432	+	0.852681i	$0.325025\pi$
$480$	0	0
$481$	−41.0741	−1.87282
$482$	14.2707i	0.650013i
$483$	13.3309i	0.606577i
$484$	−5.62620	−0.255736
$485$	0	0
$486$	2.76156	0.125267
$487$	5.42962i	0.246039i	0.992404	+	0.123020i	$0.0392578\pi$
$487$	5.42962i	0.246039i	−0.992404	+	0.123020i	$0.960742\pi$
$488$	44.2620i	2.00365i
$489$	15.6724	0.708732
$490$	0	0
$491$	21.2803	0.960367	0.480183	−	0.877168i	$-0.340570\pi$
$491$	21.2803	0.960367	0.480183	+	0.877168i	$0.340570\pi$
$492$	4.28467i	0.193168i
$493$	8.77551i	0.395229i
$494$	−68.8313	−3.09687
$495$	0	0
$496$	141.484	6.35284
$497$	21.6787i	0.972422i
$498$	− 40.2341i	− 1.80293i
$499$	16.0077	0.716603	0.358301	−	0.933606i	$-0.383356\pi$
$499$	16.0077	0.716603	0.358301	+	0.933606i	$0.383356\pi$
$500$	0	0
$501$	8.98168	0.401272
$502$	25.5510i	1.14040i
$503$	12.0925i	0.539176i	0.962976	+	0.269588i	$0.0868875\pi$
$503$	12.0925i	0.539176i	−0.962976	+	0.269588i	$0.913112\pi$
$504$	18.6724	0.831736
$505$	0	0
$506$	−19.7432	−0.877694
$507$	− 8.40171i	− 0.373133i
$508$	52.1343i	2.31309i
$509$	−14.7110	−0.652052	−0.326026	−	0.945361i	$-0.605710\pi$
$509$	−14.7110	−0.652052	−0.326026	+	0.945361i	$0.605710\pi$
$510$	0	0
$511$	12.6339	0.558891
$512$	− 85.9194i	− 3.79714i
$513$	− 5.38776i	− 0.237875i
$514$	−69.6347	−3.07146
$515$	0	0
$516$	41.6435	1.83325
$517$	− 0.373802i	− 0.0164398i
$518$	− 45.7187i	− 2.00876i
$519$	−11.5092	−0.505196
$520$	0	0
$521$	−17.2803	−0.757064	−0.378532	−	0.925588i	$-0.623571\pi$
$521$	−17.2803	−0.757064	−0.378532	+	0.925588i	$0.623571\pi$
$522$	9.72928i	0.425839i
$523$	− 42.7326i	− 1.86857i	−0.356532	−	0.934283i	$-0.616041\pi$
$523$	− 42.7326i	− 1.86857i	0.356532	−	0.934283i	$-0.383959\pi$
$524$	−22.8680	−0.998992
$525$	0	0
$526$	−15.0741	−0.657264
$527$	21.4865i	0.935965i
$528$	16.4017i	0.713793i
$529$	−28.1127	−1.22229
$530$	0	0
$531$	5.14931	0.223461
$532$	− 56.5221i	− 2.45054i
$533$	3.52311i	0.152603i
$534$	−24.9817	−1.08106
$535$	0	0
$536$	−119.598	−5.16585
$537$	− 10.3738i	− 0.447663i
$538$	47.1512i	2.03283i
$539$	3.52311	0.151751
$540$	0	0
$541$	8.69075	0.373644	0.186822	−	0.982394i	$-0.440181\pi$
$541$	8.69075	0.373644	0.186822	+	0.982394i	$0.440181\pi$
$542$	− 27.0077i	− 1.16008i
$543$	− 2.66473i	− 0.114355i
$544$	62.9344	2.69829
$545$	0	0
$546$	23.8217	1.01948
$547$	− 44.1064i	− 1.88585i	−0.333000	−	0.942927i	$-0.608061\pi$
$547$	− 44.1064i	− 1.88585i	0.333000	−	0.942927i	$-0.391939\pi$
$548$	− 33.3940i	− 1.42652i
$549$	−4.42003	−0.188642
$550$	0	0
$551$	18.9817	0.808647
$552$	71.5929i	3.04720i
$553$	11.2139i	0.476862i
$554$	16.3353	0.694019
$555$	0	0
$556$	37.7572	1.60126
$557$	− 7.69264i	− 0.325948i	−0.986630	−	0.162974i	$-0.947891\pi$
$557$	− 7.69264i	− 0.325948i	0.986630	−	0.162974i	$-0.0521086\pi$
$558$	23.8217i	1.00845i
$559$	34.2418	1.44827
$560$	0	0
$561$	−2.49084	−0.105163
$562$	18.6724i	0.787649i
$563$	− 9.25240i	− 0.389942i	−0.980809	−	0.194971i	$-0.937539\pi$
$563$	− 9.25240i	− 0.389942i	0.980809	−	0.194971i	$-0.0624613\pi$
$564$	−2.10308	−0.0885558
$565$	0	0
$566$	−24.3525	−1.02361
$567$	1.86464i	0.0783076i
$568$	116.424i	4.88505i
$569$	−4.96772	−0.208258	−0.104129	−	0.994564i	$-0.533205\pi$
$569$	−4.96772	−0.208258	−0.104129	+	0.994564i	$0.533205\pi$
$570$	0	0
$571$	6.02021	0.251938	0.125969	−	0.992034i	$-0.459796\pi$
$571$	6.02021	0.251938	0.125969	+	0.992034i	$0.459796\pi$
$572$	26.0279i	1.08828i
$573$	14.6724i	0.612949i
$574$	−3.92150	−0.163680
$575$	0	0
$576$	36.9711	1.54046
$577$	8.25240i	0.343552i	0.985136	+	0.171776i	$0.0549505\pi$
$577$	8.25240i	0.343552i	−0.985136	+	0.171776i	$0.945050\pi$
$578$	− 29.8130i	− 1.24006i
$579$	5.10308	0.212077
$580$	0	0
$581$	27.1666	1.12706
$582$	45.0602i	1.86780i
$583$	5.45856i	0.226071i
$584$	67.8496	2.80764
$585$	0	0
$586$	84.2022	3.47836
$587$	− 11.4846i	− 0.474019i	−0.971507	−	0.237010i	$-0.923833\pi$
$587$	− 11.4846i	− 0.474019i	0.971507	−	0.237010i	$-0.0761673\pi$
$588$	− 19.8217i	− 0.817435i
$589$	46.4758	1.91500
$590$	0	0
$591$	−3.74324	−0.153976
$592$	− 145.624i	− 5.98511i
$593$	39.0375i	1.60308i	0.597943	+	0.801539i	$0.295985\pi$
$593$	39.0375i	1.60308i	−0.597943	+	0.801539i	$0.704015\pi$
$594$	−2.76156	−0.113308
$595$	0	0
$596$	32.3126	1.32357
$597$	8.08476i	0.330887i
$598$	91.3361i	3.73501i
$599$	30.1589	1.23226	0.616130	−	0.787645i	$-0.288700\pi$
$599$	30.1589	1.23226	0.616130	+	0.787645i	$0.288700\pi$
$600$	0	0
$601$	19.1310	0.780369	0.390185	−	0.920737i	$-0.372411\pi$
$601$	19.1310	0.780369	0.390185	+	0.920737i	$0.372411\pi$
$602$	38.1137i	1.55340i
$603$	− 11.9431i	− 0.486362i
$604$	−59.4219	−2.41784
$605$	0	0
$606$	−2.85069	−0.115801
$607$	4.20617i	0.170723i	0.996350	+	0.0853615i	$0.0272045\pi$
$607$	4.20617i	0.170723i	−0.996350	+	0.0853615i	$0.972795\pi$
$608$	− 136.129i	− 5.52076i
$609$	−6.56934	−0.266203
$610$	0	0
$611$	−1.72928	−0.0699593
$612$	14.0140i	0.566480i
$613$	− 5.45856i	− 0.220469i	−0.993906	−	0.110235i	$-0.964840\pi$
$613$	− 5.45856i	− 0.220469i	0.993906	−	0.110235i	$-0.0351602\pi$
$614$	2.12036	0.0855709
$615$	0	0
$616$	−18.6724	−0.752334
$617$	30.1974i	1.21570i	0.794051	+	0.607851i	$0.207968\pi$
$617$	30.1974i	1.21570i	−0.794051	+	0.607851i	$0.792032\pi$
$618$	− 8.41233i	− 0.338394i
$619$	−23.1955	−0.932308	−0.466154	−	0.884704i	$-0.654361\pi$
$619$	−23.1955	−0.932308	−0.466154	+	0.884704i	$0.654361\pi$
$620$	0	0
$621$	−7.14931	−0.286892
$622$	93.4777i	3.74812i
$623$	− 16.8680i	− 0.675801i
$624$	75.8776	3.03753
$625$	0	0
$626$	22.3984	0.895219
$627$	5.38776i	0.215166i
$628$	− 17.4586i	− 0.696673i
$629$	22.1151	0.881789
$630$	0	0
$631$	−3.19554	−0.127212	−0.0636062	−	0.997975i	$-0.520260\pi$
$631$	−3.19554	−0.127212	−0.0636062	+	0.997975i	$0.520260\pi$
$632$	60.2234i	2.39556i
$633$	− 15.9431i	− 0.633683i
$634$	22.2707	0.884483
$635$	0	0
$636$	30.7110	1.21777
$637$	− 16.2986i	− 0.645775i
$638$	− 9.72928i	− 0.385186i
$639$	−11.6262	−0.459925
$640$	0	0
$641$	28.1974	1.11373	0.556866	−	0.830603i	$-0.312004\pi$
$641$	28.1974	1.11373	0.556866	+	0.830603i	$0.312004\pi$
$642$	− 23.4865i	− 0.926937i
$643$	− 17.9634i	− 0.708406i	−0.935169	−	0.354203i	$-0.884752\pi$
$643$	− 17.9634i	− 0.708406i	0.935169	−	0.354203i	$-0.115248\pi$
$644$	−75.0023	−2.95550
$645$	0	0
$646$	37.0602	1.45811
$647$	− 40.9990i	− 1.61184i	−0.592028	−	0.805918i	$-0.701672\pi$
$647$	− 40.9990i	− 1.61184i	0.592028	−	0.805918i	$-0.298328\pi$
$648$	10.0140i	0.393385i
$649$	−5.14931	−0.202128
$650$	0	0
$651$	−16.0848	−0.630412
$652$	88.1762i	3.45325i
$653$	− 11.6926i	− 0.457568i	−0.973477	−	0.228784i	$-0.926525\pi$
$653$	− 11.6926i	− 0.457568i	0.973477	−	0.228784i	$-0.0734750\pi$
$654$	16.9817	0.664036
$655$	0	0
$656$	−12.4908	−0.487685
$657$	6.77551i	0.264338i
$658$	− 1.92482i	− 0.0750374i
$659$	1.72928	0.0673633	0.0336816	−	0.999433i	$-0.489277\pi$
$659$	1.72928	0.0673633	0.0336816	+	0.999433i	$0.489277\pi$
$660$	0	0
$661$	0.232185	0.00903096	0.00451548	−	0.999990i	$-0.498563\pi$
$661$	0.232185	0.00903096	0.00451548	+	0.999990i	$0.498563\pi$
$662$	− 62.0712i	− 2.41247i
$663$	11.5231i	0.447521i
$664$	145.897	5.66189
$665$	0	0
$666$	24.5187	0.950082
$667$	− 25.1878i	− 0.975277i
$668$	50.5327i	1.95517i
$669$	−22.1772	−0.857421
$670$	0	0
$671$	4.42003	0.170633
$672$	47.1127i	1.81741i
$673$	− 33.5144i	− 1.29188i	−0.763386	−	0.645942i	$-0.776465\pi$
$673$	− 33.5144i	− 1.29188i	0.763386	−	0.645942i	$-0.223535\pi$
$674$	17.4740	0.673072
$675$	0	0
$676$	47.2697	1.81806
$677$	− 13.0183i	− 0.500335i	−0.968203	−	0.250167i	$-0.919514\pi$
$677$	− 13.0183i	− 0.500335i	0.968203	−	0.250167i	$-0.0804857\pi$
$678$	16.5693i	0.636342i
$679$	−30.4252	−1.16761
$680$	0	0
$681$	−5.93545	−0.227447
$682$	− 23.8217i	− 0.912182i
$683$	4.54333i	0.173846i	0.996215	+	0.0869228i	$0.0277033\pi$
$683$	4.54333i	0.173846i	−0.996215	+	0.0869228i	$0.972297\pi$
$684$	30.3126	1.15903
$685$	0	0
$686$	54.1868	2.06886
$687$	8.31695i	0.317311i
$688$	121.401i	4.62836i
$689$	25.2524	0.962040
$690$	0	0
$691$	−19.8988	−0.756986	−0.378493	−	0.925604i	$-0.623558\pi$
$691$	−19.8988	−0.756986	−0.378493	+	0.925604i	$0.623558\pi$
$692$	− 64.7528i	− 2.46153i
$693$	− 1.86464i	− 0.0708319i
$694$	−0.492277	−0.0186866
$695$	0	0
$696$	−35.2803	−1.33730
$697$	− 1.89692i	− 0.0718508i
$698$	− 54.4835i	− 2.06223i
$699$	28.5187	1.07868
$700$	0	0
$701$	−15.8444	−0.598436	−0.299218	−	0.954185i	$-0.596726\pi$
$701$	−15.8444	−0.598436	−0.299218	+	0.954185i	$0.596726\pi$
$702$	12.7755i	0.482181i
$703$	− 47.8357i	− 1.80416i
$704$	−36.9711	−1.39340
$705$	0	0
$706$	53.9142	2.02909
$707$	− 1.92482i	− 0.0723904i
$708$	28.9711i	1.08880i
$709$	−8.82174	−0.331307	−0.165654	−	0.986184i	$-0.552973\pi$
$709$	−8.82174	−0.331307	−0.165654	+	0.986184i	$0.552973\pi$
$710$	0	0
$711$	−6.01395	−0.225541
$712$	− 90.5885i	− 3.39495i
$713$	− 61.6714i	− 2.30961i
$714$	−12.8261	−0.480005
$715$	0	0
$716$	58.3651	2.18120
$717$	− 24.0925i	− 0.899749i
$718$	98.5673i	3.67850i
$719$	−36.7668	−1.37117	−0.685585	−	0.727993i	$-0.740453\pi$
$719$	−36.7668	−1.37117	−0.685585	+	0.727993i	$0.740453\pi$
$720$	0	0
$721$	5.68012	0.211539
$722$	− 27.6926i	− 1.03061i
$723$	− 5.16763i	− 0.192186i
$724$	14.9923	0.557185
$725$	0	0
$726$	2.76156	0.102491
$727$	− 5.27261i	− 0.195550i	−0.995209	−	0.0977751i	$-0.968827\pi$
$727$	− 5.27261i	− 0.195550i	0.995209	−	0.0977751i	$-0.0311726\pi$
$728$	86.3823i	3.20154i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−18.4365	−0.681897
$732$	− 24.8680i	− 0.919147i
$733$	− 37.9508i	− 1.40175i	−0.713286	−	0.700873i	$-0.752794\pi$
$733$	− 37.9508i	− 1.40175i	0.713286	−	0.700873i	$-0.247206\pi$
$734$	55.5423	2.05010
$735$	0	0
$736$	−180.637	−6.65837
$737$	11.9431i	0.439931i
$738$	− 2.10308i	− 0.0774156i
$739$	−37.2943	−1.37189	−0.685946	−	0.727653i	$-0.740611\pi$
$739$	−37.2943	−1.37189	−0.685946	+	0.727653i	$0.740611\pi$
$740$	0	0
$741$	24.9248	0.915636
$742$	28.1078i	1.03187i
$743$	− 22.5693i	− 0.827989i	−0.910279	−	0.413994i	$-0.864133\pi$
$743$	− 22.5693i	− 0.827989i	0.910279	−	0.413994i	$-0.135867\pi$
$744$	−86.3823	−3.16693
$745$	0	0
$746$	7.07414	0.259002
$747$	14.5693i	0.533064i
$748$	− 14.0140i	− 0.512401i
$749$	15.8584	0.579453
$750$	0	0
$751$	−9.55102	−0.348522	−0.174261	−	0.984700i	$-0.555754\pi$
$751$	−9.55102	−0.348522	−0.174261	+	0.984700i	$0.555754\pi$
$752$	− 6.13099i	− 0.223574i
$753$	− 9.25240i	− 0.337176i
$754$	−45.0096	−1.63915
$755$	0	0
$756$	−10.4908	−0.381548
$757$	30.3188i	1.10196i	0.834519	+	0.550978i	$0.185745\pi$
$757$	30.3188i	1.10196i	−0.834519	+	0.550978i	$0.814255\pi$
$758$	64.2341i	2.33309i
$759$	7.14931	0.259504
$760$	0	0
$761$	−1.43066	−0.0518613	−0.0259306	−	0.999664i	$-0.508255\pi$
$761$	−1.43066	−0.0518613	−0.0259306	+	0.999664i	$0.508255\pi$
$762$	− 25.5896i	− 0.927012i
$763$	11.4663i	0.415106i
$764$	−82.5500	−2.98655
$765$	0	0
$766$	−71.2311	−2.57369
$767$	23.8217i	0.860153i
$768$	68.4575i	2.47025i
$769$	35.4017	1.27662	0.638309	−	0.769780i	$-0.279634\pi$
$769$	35.4017	1.27662	0.638309	+	0.769780i	$0.279634\pi$
$770$	0	0
$771$	25.2158	0.908123
$772$	28.7110i	1.03333i
$773$	13.4094i	0.482303i	0.970488	+	0.241151i	$0.0775250\pi$
$773$	13.4094i	0.482303i	−0.970488	+	0.241151i	$0.922475\pi$
$774$	−20.4402	−0.734709
$775$	0	0
$776$	−163.397	−5.86562
$777$	16.5554i	0.593921i
$778$	6.44898i	0.231207i
$779$	−4.10308	−0.147008
$780$	0	0
$781$	11.6262	0.416018
$782$	− 49.1772i	− 1.75857i
$783$	− 3.52311i	− 0.125906i
$784$	57.7851	2.06375
$785$	0	0
$786$	11.2245	0.400364
$787$	− 37.3232i	− 1.33043i	−0.746653	−	0.665214i	$-0.768340\pi$
$787$	− 37.3232i	− 1.33043i	0.746653	−	0.665214i	$-0.231660\pi$
$788$	− 21.0602i	− 0.750238i
$789$	5.45856	0.194330
$790$	0	0
$791$	−11.1878	−0.397794
$792$	− 10.0140i	− 0.355830i
$793$	− 20.4479i	− 0.726128i
$794$	47.4865	1.68523
$795$	0	0
$796$	−45.4865	−1.61223
$797$	− 8.05581i	− 0.285352i	−0.989769	−	0.142676i	$-0.954429\pi$
$797$	− 8.05581i	− 0.285352i	0.989769	−	0.142676i	$-0.0455706\pi$
$798$	27.7432i	0.982100i
$799$	0.931080	0.0329393
$800$	0	0
$801$	9.04623	0.319633
$802$	73.3728i	2.59088i
$803$	− 6.77551i	− 0.239103i
$804$	67.1945	2.36977
$805$	0	0
$806$	−110.204	−3.88177
$807$	− 17.0741i	− 0.601038i
$808$	− 10.3372i	− 0.363660i
$809$	42.3771	1.48990	0.744950	−	0.667120i	$-0.232473\pi$
$809$	42.3771	1.48990	0.744950	+	0.667120i	$0.232473\pi$
$810$	0	0
$811$	−21.3878	−0.751026	−0.375513	−	0.926817i	$-0.622533\pi$
$811$	−21.3878	−0.751026	−0.375513	+	0.926817i	$0.622533\pi$
$812$	− 36.9604i	− 1.29706i
$813$	9.77988i	0.342995i
$814$	−24.5187	−0.859382
$815$	0	0
$816$	−40.8540	−1.43018
$817$	39.8786i	1.39518i
$818$	− 56.3911i	− 1.97167i
$819$	−8.62620	−0.301424
$820$	0	0
$821$	−47.8776	−1.67094	−0.835469	−	0.549537i	$-0.814804\pi$
$821$	−47.8776	−1.67094	−0.835469	+	0.549537i	$0.814804\pi$
$822$	16.3911i	0.571705i
$823$	− 16.3555i	− 0.570116i	−0.958510	−	0.285058i	$-0.907987\pi$
$823$	− 16.3555i	− 0.570116i	0.958510	−	0.285058i	$-0.0920129\pi$
$824$	30.5048	1.06268
$825$	0	0
$826$	−26.5154	−0.922589
$827$	− 44.7110i	− 1.55475i	−0.629036	−	0.777376i	$-0.716550\pi$
$827$	− 44.7110i	− 1.55475i	0.629036	−	0.777376i	$-0.283450\pi$
$828$	− 40.2234i	− 1.39786i
$829$	38.5972	1.34054	0.670269	−	0.742118i	$-0.266179\pi$
$829$	38.5972	1.34054	0.670269	+	0.742118i	$0.266179\pi$
$830$	0	0
$831$	−5.91524	−0.205197
$832$	171.035i	5.92959i
$833$	8.77551i	0.304053i
$834$	−18.5327	−0.641735
$835$	0	0
$836$	−30.3126	−1.04838
$837$	− 8.62620i	− 0.298165i
$838$	− 37.5283i	− 1.29639i
$839$	0.710960	0.0245451	0.0122725	−	0.999925i	$-0.496093\pi$
$839$	0.710960	0.0245451	0.0122725	+	0.999925i	$0.496093\pi$
$840$	0	0
$841$	−16.5877	−0.571988
$842$	109.406i	3.77038i
$843$	− 6.76156i	− 0.232880i
$844$	89.6993	3.08758
$845$	0	0
$846$	1.03228	0.0354904
$847$	1.86464i	0.0640698i
$848$	89.5298i	3.07446i
$849$	8.81841	0.302647
$850$	0	0
$851$	−63.4758	−2.17592
$852$	− 65.4113i	− 2.24095i
$853$	− 52.7187i	− 1.80505i	−0.430635	−	0.902526i	$-0.641710\pi$
$853$	− 52.7187i	− 1.80505i	0.430635	−	0.902526i	$-0.358290\pi$
$854$	22.7601	0.778835
$855$	0	0
$856$	85.1666	2.91093
$857$	16.7124i	0.570885i	0.958396	+	0.285442i	$0.0921405\pi$
$857$	16.7124i	0.570885i	−0.958396	+	0.285442i	$0.907859\pi$
$858$	− 12.7755i	− 0.436149i
$859$	41.9142	1.43009	0.715047	−	0.699076i	$-0.246405\pi$
$859$	41.9142	1.43009	0.715047	+	0.699076i	$0.246405\pi$
$860$	0	0
$861$	1.42003	0.0483945
$862$	18.1416i	0.617906i
$863$	41.4219i	1.41002i	0.709198	+	0.705009i	$0.249057\pi$
$863$	41.4219i	1.41002i	−0.709198	+	0.705009i	$0.750943\pi$
$864$	−25.2663	−0.859579
$865$	0	0
$866$	100.552	3.41689
$867$	10.7957i	0.366642i
$868$	− 90.4961i	− 3.07164i
$869$	6.01395	0.204009
$870$	0	0
$871$	55.2514	1.87212
$872$	61.5789i	2.08533i
$873$	− 16.3169i	− 0.552245i
$874$	−106.372	−3.59808
$875$	0	0
$876$	−38.1204	−1.28797
$877$	− 12.0848i	− 0.408073i	−0.978963	−	0.204037i	$-0.934594\pi$
$877$	− 12.0848i	− 0.408073i	0.978963	−	0.204037i	$-0.0654062\pi$
$878$	70.1097i	2.36609i
$879$	−30.4908	−1.02843
$880$	0	0
$881$	−31.8130	−1.07181	−0.535904	−	0.844279i	$-0.680029\pi$
$881$	−31.8130	−1.07181	−0.535904	+	0.844279i	$0.680029\pi$
$882$	9.72928i	0.327602i
$883$	20.9894i	0.706349i	0.935558	+	0.353174i	$0.114898\pi$
$883$	20.9894i	0.706349i	−0.935558	+	0.353174i	$0.885102\pi$
$884$	−64.8313	−2.18051
$885$	0	0
$886$	57.7345	1.93963
$887$	49.2032i	1.65208i	0.563609	+	0.826042i	$0.309412\pi$
$887$	49.2032i	1.65208i	−0.563609	+	0.826042i	$0.690588\pi$
$888$	88.9098i	2.98362i
$889$	17.2784	0.579499
$890$	0	0
$891$	1.00000	0.0335013
$892$	− 124.773i	− 4.17772i
$893$	− 2.01395i	− 0.0673944i
$894$	−15.8603	−0.530447
$895$	0	0
$896$	−96.1502	−3.21215
$897$	− 33.0741i	− 1.10431i
$898$	− 96.5027i	− 3.22034i
$899$	30.3911	1.01360
$900$	0	0
$901$	−13.5964	−0.452962
$902$	2.10308i	0.0700250i
$903$	− 13.8015i	− 0.459286i
$904$	−60.0837	−1.99835
$905$	0	0
$906$	29.1666	0.968995
$907$	− 24.8526i	− 0.825216i	−0.910909	−	0.412608i	$-0.864618\pi$
$907$	− 24.8526i	− 0.825216i	0.910909	−	0.412608i	$-0.135382\pi$
$908$	− 33.3940i	− 1.10822i
$909$	1.03228	0.0342384
$910$	0	0
$911$	45.4758	1.50668	0.753341	−	0.657630i	$-0.228441\pi$
$911$	45.4758	1.50668	0.753341	+	0.657630i	$0.228441\pi$
$912$	88.3684i	2.92617i
$913$	− 14.5693i	− 0.482175i
$914$	20.2062	0.668361
$915$	0	0
$916$	−46.7928	−1.54608
$917$	7.57893i	0.250278i
$918$	− 6.87859i	− 0.227027i
$919$	15.7355	0.519068	0.259534	−	0.965734i	$-0.416431\pi$
$919$	15.7355	0.519068	0.259534	+	0.965734i	$0.416431\pi$
$920$	0	0
$921$	−0.767815	−0.0253004
$922$	38.7389i	1.27580i
$923$	− 53.7851i	− 1.77036i
$924$	10.4908	0.345123
$925$	0	0
$926$	11.9721	0.393427
$927$	3.04623i	0.100051i
$928$	− 89.0162i	− 2.92210i
$929$	2.06455	0.0677357	0.0338679	−	0.999426i	$-0.489217\pi$
$929$	2.06455	0.0677357	0.0338679	+	0.999426i	$0.489217\pi$
$930$	0	0
$931$	18.9817	0.622099
$932$	160.452i	5.25578i
$933$	− 33.8496i	− 1.10819i
$934$	−10.2216	−0.334460
$935$	0	0
$936$	−46.3265	−1.51423
$937$	40.1493i	1.31162i	0.754926	+	0.655810i	$0.227673\pi$
$937$	40.1493i	1.31162i	−0.754926	+	0.655810i	$0.772327\pi$
$938$	61.4990i	2.00801i
$939$	−8.11078	−0.264685
$940$	0	0
$941$	26.4050	0.860780	0.430390	−	0.902643i	$-0.358376\pi$
$941$	26.4050	0.860780	0.430390	+	0.902643i	$0.358376\pi$
$942$	8.56934i	0.279204i
$943$	5.44461i	0.177301i
$944$	−84.4575	−2.74886
$945$	0	0
$946$	20.4402	0.664570
$947$	8.67243i	0.281816i	0.990023	+	0.140908i	$0.0450022\pi$
$947$	8.67243i	0.281816i	−0.990023	+	0.140908i	$0.954998\pi$
$948$	− 33.8357i	− 1.09893i
$949$	−31.3449	−1.01750
$950$	0	0
$951$	−8.06455	−0.261511
$952$	− 46.5100i	− 1.50740i
$953$	26.8540i	0.869887i	0.900458	+	0.434943i	$0.143232\pi$
$953$	26.8540i	0.869887i	−0.900458	+	0.434943i	$0.856768\pi$
$954$	−15.0741	−0.488043
$955$	0	0
$956$	135.549	4.38397
$957$	3.52311i	0.113886i
$958$	− 63.1512i	− 2.04032i
$959$	−11.0675	−0.357388
$960$	0	0
$961$	43.4113	1.40036
$962$	113.429i	3.65708i
$963$	8.50479i	0.274063i
$964$	29.0741	0.936415
$965$	0	0
$966$	36.8140	1.18447
$967$	− 55.8496i	− 1.79600i	−0.439992	−	0.898002i	$-0.645019\pi$
$967$	− 55.8496i	− 1.79600i	0.439992	−	0.898002i	$-0.354981\pi$
$968$	10.0140i	0.321861i
$969$	−13.4200	−0.431113
$970$	0	0
$971$	30.0664	0.964878	0.482439	−	0.875930i	$-0.339751\pi$
$971$	30.0664	0.964878	0.482439	+	0.875930i	$0.339751\pi$
$972$	− 5.62620i	− 0.180460i
$973$	− 12.5135i	− 0.401165i
$974$	14.9942	0.480445
$975$	0	0
$976$	72.4961	2.32054
$977$	15.7014i	0.502331i	0.967944	+	0.251166i	$0.0808139\pi$
$977$	15.7014i	0.502331i	−0.967944	+	0.251166i	$0.919186\pi$
$978$	− 43.2803i	− 1.38395i
$979$	−9.04623	−0.289119
$980$	0	0
$981$	−6.14931	−0.196332
$982$	− 58.7668i	− 1.87532i
$983$	51.9894i	1.65820i	0.559098	+	0.829102i	$0.311148\pi$
$983$	51.9894i	1.65820i	−0.559098	+	0.829102i	$0.688852\pi$
$984$	7.62620	0.243114
$985$	0	0
$986$	24.2341	0.771770
$987$	0.697006i	0.0221860i
$988$	140.232i	4.46137i
$989$	52.9171	1.68267
$990$	0	0
$991$	27.6445	0.878157	0.439079	−	0.898449i	$-0.355305\pi$
$991$	27.6445	0.878157	0.439079	+	0.898449i	$0.355305\pi$
$992$	− 217.953i	− 6.92000i
$993$	22.4769i	0.713282i
$994$	59.8669	1.89886
$995$	0	0
$996$	−81.9700	−2.59732
$997$	40.5048i	1.28280i	0.767207	+	0.641400i	$0.221646\pi$
$997$	40.5048i	1.28280i	−0.767207	+	0.641400i	$0.778354\pi$
$998$	− 44.2062i	− 1.39932i
$999$	−8.87859	−0.280906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.f.199.1		6
3.2	odd	2		2475.2.c.q.199.6		6
5.2	odd	4		825.2.a.m.1.3	yes	3
5.3	odd	4		825.2.a.i.1.1	✓	3
5.4	even	2	inner	825.2.c.f.199.6		6
15.2	even	4		2475.2.a.z.1.1		3
15.8	even	4		2475.2.a.bd.1.3		3
15.14	odd	2		2475.2.c.q.199.1		6
55.32	even	4		9075.2.a.cd.1.1		3
55.43	even	4		9075.2.a.cj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.1	✓	3	5.3	odd	4
825.2.a.m.1.3	yes	3	5.2	odd	4
825.2.c.f.199.1		6	1.1	even	1	trivial
825.2.c.f.199.6		6	5.4	even	2	inner
2475.2.a.z.1.1		3	15.2	even	4
2475.2.a.bd.1.3		3	15.8	even	4
2475.2.c.q.199.1		6	15.14	odd	2
2475.2.c.q.199.6		6	3.2	odd	2
9075.2.a.cd.1.1		3	55.32	even	4
9075.2.a.cj.1.3		3	55.43	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.76156i q^{2} +1.00000i q^{3} -5.62620 q^{4} +2.76156 q^{6} +1.86464i q^{7} +10.0140i q^{8} -1.00000 q^{9} +1.00000 q^{11} -5.62620i q^{12} -4.62620i q^{13} +5.14931 q^{14} +16.4017 q^{16} +2.49084i q^{17} +2.76156i q^{18} +5.38776 q^{19} -1.86464 q^{21} -2.76156i q^{22} -7.14931i q^{23} -10.0140 q^{24} -12.7755 q^{26} -1.00000i q^{27} -10.4908i q^{28} +3.52311 q^{29} +8.62620 q^{31} -25.2663i q^{32} +1.00000i q^{33} +6.87859 q^{34} +5.62620 q^{36} -8.87859i q^{37} -14.8786i q^{38} +4.62620 q^{39} -0.761557 q^{41} +5.14931i q^{42} +7.40171i q^{43} -5.62620 q^{44} -19.7432 q^{46} -0.373802i q^{47} +16.4017i q^{48} +3.52311 q^{49} -2.49084 q^{51} +26.0279i q^{52} +5.45856i q^{53} -2.76156 q^{54} -18.6724 q^{56} +5.38776i q^{57} -9.72928i q^{58} -5.14931 q^{59} +4.42003 q^{61} -23.8217i q^{62} -1.86464i q^{63} -36.9711 q^{64} +2.76156 q^{66} +11.9431i q^{67} -14.0140i q^{68} +7.14931 q^{69} +11.6262 q^{71} -10.0140i q^{72} -6.77551i q^{73} -24.5187 q^{74} -30.3126 q^{76} +1.86464i q^{77} -12.7755i q^{78} +6.01395 q^{79} +1.00000 q^{81} +2.10308i q^{82} -14.5693i q^{83} +10.4908 q^{84} +20.4402 q^{86} +3.52311i q^{87} +10.0140i q^{88} -9.04623 q^{89} +8.62620 q^{91} +40.2234i q^{92} +8.62620i q^{93} -1.03228 q^{94} +25.2663 q^{96} +16.3169i q^{97} -9.72928i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46}+ \cdots - 6 q^{99}+O(q^{100}) \) 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 + 6 * q^11 - 12 * q^14 + 20 * q^16 + 2 * q^19 - 6 * q^21 - 12 * q^24 - 16 * q^26 - 4 * q^29 + 34 * q^31 - 12 * q^34 + 16 * q^36 + 10 * q^39 + 8 * q^41 - 16 * q^44 - 60 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 - 44 * q^56 + 12 * q^59 - 6 * q^61 - 68 * q^64 + 4 * q^66 + 52 * q^71 - 28 * q^74 - 48 * q^76 - 12 * q^79 + 6 * q^81 + 40 * q^84 + 56 * q^86 - 4 * q^89 + 34 * q^91 - 4 * q^94 + 68 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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