LMFDB - Embedded newform 1875.2.b.d.1249.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.b (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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^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			1249.5
Root			$1.33826i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.d.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.209057i q^{2} -1.00000i q^{3} +1.95630 q^{4} +0.209057 q^{6} -0.591023i q^{7} +0.827091i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.209057i q^{2} -1.00000i q^{3} +1.95630 q^{4} +0.209057 q^{6} -0.591023i q^{7} +0.827091i q^{8} -1.00000 q^{9} -0.870796 q^{11} -1.95630i q^{12} +1.15057i q^{13} +0.123557 q^{14} +3.73968 q^{16} +4.93395i q^{17} -0.209057i q^{18} +2.84943 q^{19} -0.591023 q^{21} -0.182046i q^{22} +6.91259i q^{23} +0.827091 q^{24} -0.240534 q^{26} +1.00000i q^{27} -1.15622i q^{28} +7.48883 q^{29} +3.45991 q^{31} +2.43599i q^{32} +0.870796i q^{33} -1.03148 q^{34} -1.95630 q^{36} -10.1681i q^{37} +0.595693i q^{38} +1.15057 q^{39} +9.11409 q^{41} -0.123557i q^{42} -2.81486i q^{43} -1.70353 q^{44} -1.44512 q^{46} -6.68842i q^{47} -3.73968i q^{48} +6.65069 q^{49} +4.93395 q^{51} +2.25085i q^{52} +3.87238i q^{53} -0.209057 q^{54} +0.488830 q^{56} -2.84943i q^{57} +1.56559i q^{58} +11.5277 q^{59} -12.7564 q^{61} +0.723318i q^{62} +0.591023i q^{63} +6.97010 q^{64} -0.182046 q^{66} +7.60292i q^{67} +9.65227i q^{68} +6.91259 q^{69} -15.0566 q^{71} -0.827091i q^{72} -2.98798i q^{73} +2.12571 q^{74} +5.57433 q^{76} +0.514660i q^{77} +0.240534i q^{78} -3.33728 q^{79} +1.00000 q^{81} +1.90536i q^{82} -9.73310i q^{83} -1.15622 q^{84} +0.588467 q^{86} -7.48883i q^{87} -0.720227i q^{88} -0.645045 q^{89} +0.680012 q^{91} +13.5231i q^{92} -3.45991i q^{93} +1.39826 q^{94} +2.43599 q^{96} -11.6970i q^{97} +1.39037i q^{98} +0.870796 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.209057i	0.147826i	0.997265	+	0.0739128i	$0.0235486\pi$
$2$	0.209057i	0.147826i	−0.997265	+	0.0739128i	$0.976451\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.95630	0.978148
$5$	0	0
$5$	0	0
$6$	0.209057	0.0853471
$7$	− 0.591023i	− 0.223386i	−0.993743	−	0.111693i	$-0.964373\pi$
$7$	− 0.591023i	− 0.223386i	0.993743	−	0.111693i	$-0.0356273\pi$
$8$	0.827091i	0.292421i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.870796	−0.262555	−0.131277	−	0.991346i	$-0.541908\pi$
$11$	−0.870796	−0.262555	−0.131277	+	0.991346i	$0.541908\pi$
$12$	− 1.95630i	− 0.564734i
$13$	1.15057i	0.319110i	0.987189	+	0.159555i	$0.0510060\pi$
$13$	1.15057i	0.319110i	−0.987189	+	0.159555i	$0.948994\pi$
$14$	0.123557	0.0330221
$15$	0	0
$16$	3.73968	0.934920
$17$	4.93395i	1.19666i	0.801250	+	0.598330i	$0.204169\pi$
$17$	4.93395i	1.19666i	−0.801250	+	0.598330i	$0.795831\pi$
$18$	− 0.209057i	− 0.0492752i
$19$	2.84943	0.653704	0.326852	−	0.945075i	$-0.394012\pi$
$19$	2.84943	0.653704	0.326852	+	0.945075i	$0.394012\pi$
$20$	0	0
$21$	−0.591023	−0.128972
$22$	− 0.182046i	− 0.0388123i
$23$	6.91259i	1.44137i	0.693260	+	0.720687i	$0.256174\pi$
$23$	6.91259i	1.44137i	−0.693260	+	0.720687i	$0.743826\pi$
$24$	0.827091	0.168829
$25$	0	0
$26$	−0.240534	−0.0471727
$27$	1.00000i	0.192450i
$28$	− 1.15622i	− 0.218504i
$29$	7.48883	1.39064	0.695320	−	0.718700i	$-0.255262\pi$
$29$	7.48883	1.39064	0.695320	+	0.718700i	$0.255262\pi$
$30$	0	0
$31$	3.45991	0.621418	0.310709	−	0.950505i	$-0.399434\pi$
$31$	3.45991	0.621418	0.310709	+	0.950505i	$0.399434\pi$
$32$	2.43599i	0.430626i
$33$	0.870796i	0.151586i
$34$	−1.03148	−0.176897
$35$	0	0
$36$	−1.95630	−0.326049
$37$	− 10.1681i	− 1.67163i	−0.549013	−	0.835814i	$-0.684996\pi$
$37$	− 10.1681i	− 1.67163i	0.549013	−	0.835814i	$-0.315004\pi$
$38$	0.595693i	0.0966342i
$39$	1.15057	0.184238
$40$	0	0
$41$	9.11409	1.42338	0.711691	−	0.702493i	$-0.247930\pi$
$41$	9.11409	1.42338	0.711691	+	0.702493i	$0.247930\pi$
$42$	− 0.123557i	− 0.0190653i
$43$	− 2.81486i	− 0.429263i	−0.976695	−	0.214631i	$-0.931145\pi$
$43$	− 2.81486i	− 0.429263i	0.976695	−	0.214631i	$-0.0688550\pi$
$44$	−1.70353	−0.256817
$45$	0	0
$46$	−1.44512	−0.213072
$47$	− 6.68842i	− 0.975606i	−0.872954	−	0.487803i	$-0.837798\pi$
$47$	− 6.68842i	− 0.975606i	0.872954	−	0.487803i	$-0.162202\pi$
$48$	− 3.73968i	− 0.539777i
$49$	6.65069	0.950099
$50$	0	0
$51$	4.93395	0.690892
$52$	2.25085i	0.312137i
$53$	3.87238i	0.531912i	0.963985	+	0.265956i	$0.0856875\pi$
$53$	3.87238i	0.531912i	−0.963985	+	0.265956i	$0.914312\pi$
$54$	−0.209057	−0.0284490
$55$	0	0
$56$	0.488830	0.0653226
$57$	− 2.84943i	− 0.377416i
$58$	1.56559i	0.205572i
$59$	11.5277	1.50078	0.750392	−	0.660993i	$-0.229865\pi$
$59$	11.5277	1.50078	0.750392	+	0.660993i	$0.229865\pi$
$60$	0	0
$61$	−12.7564	−1.63329	−0.816643	−	0.577143i	$-0.804168\pi$
$61$	−12.7564	−1.63329	−0.816643	+	0.577143i	$0.804168\pi$
$62$	0.723318i	0.0918615i
$63$	0.591023i	0.0744619i
$64$	6.97010	0.871263
$65$	0	0
$66$	−0.182046	−0.0224083
$67$	7.60292i	0.928845i	0.885614	+	0.464422i	$0.153738\pi$
$67$	7.60292i	0.928845i	−0.885614	+	0.464422i	$0.846262\pi$
$68$	9.65227i	1.17051i
$69$	6.91259	0.832178
$70$	0	0
$71$	−15.0566	−1.78689	−0.893444	−	0.449175i	$-0.851718\pi$
$71$	−15.0566	−1.78689	−0.893444	+	0.449175i	$0.851718\pi$
$72$	− 0.827091i	− 0.0974736i
$73$	− 2.98798i	− 0.349716i	−0.984594	−	0.174858i	$-0.944053\pi$
$73$	− 2.98798i	− 0.349716i	0.984594	−	0.174858i	$-0.0559467\pi$
$74$	2.12571	0.247109
$75$	0	0
$76$	5.57433	0.639419
$77$	0.514660i	0.0586510i
$78$	0.240534i	0.0272351i
$79$	−3.33728	−0.375474	−0.187737	−	0.982219i	$-0.560115\pi$
$79$	−3.33728	−0.375474	−0.187737	+	0.982219i	$0.560115\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.90536i	0.210412i
$83$	− 9.73310i	− 1.06835i	−0.845375	−	0.534173i	$-0.820623\pi$
$83$	− 9.73310i	− 1.06835i	0.845375	−	0.534173i	$-0.179377\pi$
$84$	−1.15622	−0.126153
$85$	0	0
$86$	0.588467	0.0634560
$87$	− 7.48883i	− 0.802887i
$88$	− 0.720227i	− 0.0767765i
$89$	−0.645045	−0.0683746	−0.0341873	−	0.999415i	$-0.510884\pi$
$89$	−0.645045	−0.0683746	−0.0341873	+	0.999415i	$0.510884\pi$
$90$	0	0
$91$	0.680012	0.0712847
$92$	13.5231i	1.40988i
$93$	− 3.45991i	− 0.358776i
$94$	1.39826	0.144220
$95$	0	0
$96$	2.43599	0.248622
$97$	− 11.6970i	− 1.18765i	−0.804596	−	0.593823i	$-0.797618\pi$
$97$	− 11.6970i	− 1.18765i	0.804596	−	0.593823i	$-0.202382\pi$
$98$	1.39037i	0.140449i
$99$	0.870796	0.0875183
$100$	0	0
$101$	−14.7636	−1.46903	−0.734517	−	0.678591i	$-0.762591\pi$
$101$	−14.7636	−1.46903	−0.734517	+	0.678591i	$0.762591\pi$
$102$	1.03148i	0.102131i
$103$	− 1.41969i	− 0.139887i	−0.997551	−	0.0699433i	$-0.977718\pi$
$103$	− 1.41969i	− 0.139887i	0.997551	−	0.0699433i	$-0.0222818\pi$
$104$	−0.951625	−0.0933145
$105$	0	0
$106$	−0.809547	−0.0786301
$107$	14.9579i	1.44603i	0.690831	+	0.723016i	$0.257245\pi$
$107$	14.9579i	1.44603i	−0.690831	+	0.723016i	$0.742755\pi$
$108$	1.95630i	0.188245i
$109$	11.0977	1.06297	0.531485	−	0.847068i	$-0.321634\pi$
$109$	11.0977	1.06297	0.531485	+	0.847068i	$0.321634\pi$
$110$	0	0
$111$	−10.1681	−0.965115
$112$	− 2.21024i	− 0.208848i
$113$	− 12.1069i	− 1.13892i	−0.822020	−	0.569459i	$-0.807153\pi$
$113$	− 12.1069i	− 1.13892i	0.822020	−	0.569459i	$-0.192847\pi$
$114$	0.595693	0.0557918
$115$	0	0
$116$	14.6504	1.36025
$117$	− 1.15057i	− 0.106370i
$118$	2.40995i	0.221854i
$119$	2.91608	0.267317
$120$	0	0
$121$	−10.2417	−0.931065
$122$	− 2.66681i	− 0.241442i
$123$	− 9.11409i	− 0.821790i
$124$	6.76860	0.607838
$125$	0	0
$126$	−0.123557	−0.0110074
$127$	− 0.541671i	− 0.0480656i	−0.999711	−	0.0240328i	$-0.992349\pi$
$127$	− 0.541671i	− 0.0480656i	0.999711	−	0.0240328i	$-0.00765061\pi$
$128$	6.32912i	0.559421i
$129$	−2.81486	−0.247835
$130$	0	0
$131$	−2.46747	−0.215583	−0.107792	−	0.994173i	$-0.534378\pi$
$131$	−2.46747	−0.215583	−0.107792	+	0.994173i	$0.534378\pi$
$132$	1.70353i	0.148274i
$133$	− 1.68408i	− 0.146028i
$134$	−1.58944	−0.137307
$135$	0	0
$136$	−4.08083	−0.349928
$137$	5.64471i	0.482261i	0.970493	+	0.241130i	$0.0775181\pi$
$137$	5.64471i	0.482261i	−0.970493	+	0.241130i	$0.922482\pi$
$138$	1.44512i	0.123017i
$139$	0.326034	0.0276538	0.0138269	−	0.999904i	$-0.495599\pi$
$139$	0.326034	0.0276538	0.0138269	+	0.999904i	$0.495599\pi$
$140$	0	0
$141$	−6.68842	−0.563267
$142$	− 3.14768i	− 0.264148i
$143$	− 1.00191i	− 0.0837839i
$144$	−3.73968	−0.311640
$145$	0	0
$146$	0.624657	0.0516970
$147$	− 6.65069i	− 0.548540i
$148$	− 19.8918i	− 1.63510i
$149$	0.637614	0.0522354	0.0261177	−	0.999659i	$-0.491686\pi$
$149$	0.637614	0.0522354	0.0261177	+	0.999659i	$0.491686\pi$
$150$	0	0
$151$	11.3948	0.927299	0.463649	−	0.886019i	$-0.346540\pi$
$151$	11.3948	0.927299	0.463649	+	0.886019i	$0.346540\pi$
$152$	2.35674i	0.191157i
$153$	− 4.93395i	− 0.398887i
$154$	−0.107593	−0.00867011
$155$	0	0
$156$	2.25085	0.180212
$157$	− 2.86068i	− 0.228307i	−0.993463	−	0.114154i	$-0.963584\pi$
$157$	− 2.86068i	− 0.228307i	0.993463	−	0.114154i	$-0.0364156\pi$
$158$	− 0.697683i	− 0.0555046i
$159$	3.87238	0.307099
$160$	0	0
$161$	4.08550	0.321982
$162$	0.209057i	0.0164251i
$163$	14.1106i	1.10523i	0.833437	+	0.552614i	$0.186370\pi$
$163$	14.1106i	1.10523i	−0.833437	+	0.552614i	$0.813630\pi$
$164$	17.8299	1.39228
$165$	0	0
$166$	2.03477	0.157929
$167$	7.49041i	0.579625i	0.957083	+	0.289813i	$0.0935930\pi$
$167$	7.49041i	0.579625i	−0.957083	+	0.289813i	$0.906407\pi$
$168$	− 0.488830i	− 0.0377140i
$169$	11.6762	0.898169
$170$	0	0
$171$	−2.84943	−0.217901
$172$	− 5.50670i	− 0.419882i
$173$	20.3231i	1.54514i	0.634929	+	0.772570i	$0.281029\pi$
$173$	20.3231i	1.54514i	−0.634929	+	0.772570i	$0.718971\pi$
$174$	1.56559	0.118687
$175$	0	0
$176$	−3.25650	−0.245468
$177$	− 11.5277i	− 0.866478i
$178$	− 0.134851i	− 0.0101075i
$179$	−1.13209	−0.0846164	−0.0423082	−	0.999105i	$-0.513471\pi$
$179$	−1.13209	−0.0846164	−0.0423082	+	0.999105i	$0.513471\pi$
$180$	0	0
$181$	4.96450	0.369008	0.184504	−	0.982832i	$-0.440932\pi$
$181$	4.96450	0.369008	0.184504	+	0.982832i	$0.440932\pi$
$182$	0.142161i	0.0105377i
$183$	12.7564i	0.942978i
$184$	−5.71734	−0.421488
$185$	0	0
$186$	0.723318	0.0530362
$187$	− 4.29647i	− 0.314189i
$188$	− 13.0845i	− 0.954287i
$189$	0.591023	0.0429906
$190$	0	0
$191$	6.44610	0.466424	0.233212	−	0.972426i	$-0.425076\pi$
$191$	6.44610	0.466424	0.233212	+	0.972426i	$0.425076\pi$
$192$	− 6.97010i	− 0.503024i
$193$	− 18.4312i	− 1.32671i	−0.748307	−	0.663353i	$-0.769133\pi$
$193$	− 18.4312i	− 1.32671i	0.748307	−	0.663353i	$-0.230867\pi$
$194$	2.44533	0.175564
$195$	0	0
$196$	13.0107	0.929337
$197$	16.8343i	1.19940i	0.800227	+	0.599698i	$0.204713\pi$
$197$	16.8343i	1.19940i	−0.800227	+	0.599698i	$0.795287\pi$
$198$	0.182046i	0.0129374i
$199$	−16.9968	−1.20487	−0.602435	−	0.798168i	$-0.705803\pi$
$199$	−16.9968	−1.20487	−0.602435	+	0.798168i	$0.705803\pi$
$200$	0	0
$201$	7.60292	0.536269
$202$	− 3.08643i	− 0.217161i
$203$	− 4.42607i	− 0.310649i
$204$	9.65227	0.675794
$205$	0	0
$206$	0.296797	0.0206788
$207$	− 6.91259i	− 0.480458i
$208$	4.30276i	0.298343i
$209$	−2.48127	−0.171633
$210$	0	0
$211$	5.00874	0.344816	0.172408	−	0.985026i	$-0.444845\pi$
$211$	5.00874	0.344816	0.172408	+	0.985026i	$0.444845\pi$
$212$	7.57551i	0.520288i
$213$	15.0566i	1.03166i
$214$	−3.12705	−0.213761
$215$	0	0
$216$	−0.827091	−0.0562764
$217$	− 2.04489i	− 0.138816i
$218$	2.32006i	0.157134i
$219$	−2.98798	−0.201909
$220$	0	0
$221$	−5.67685	−0.381866
$222$	− 2.12571i	− 0.142669i
$223$	15.9452i	1.06777i	0.845556	+	0.533886i	$0.179269\pi$
$223$	15.9452i	1.06777i	−0.845556	+	0.533886i	$0.820731\pi$
$224$	1.43972	0.0961957
$225$	0	0
$226$	2.53102	0.168361
$227$	26.8967i	1.78520i	0.450850	+	0.892600i	$0.351121\pi$
$227$	26.8967i	1.78520i	−0.450850	+	0.892600i	$0.648879\pi$
$228$	− 5.57433i	− 0.369169i
$229$	2.94598	0.194676	0.0973378	−	0.995251i	$-0.468967\pi$
$229$	2.94598	0.194676	0.0973378	+	0.995251i	$0.468967\pi$
$230$	0	0
$231$	0.514660	0.0338622
$232$	6.19394i	0.406652i
$233$	− 23.4054i	− 1.53334i	−0.642041	−	0.766670i	$-0.721912\pi$
$233$	− 23.4054i	− 1.53334i	0.642041	−	0.766670i	$-0.278088\pi$
$234$	0.240534	0.0157242
$235$	0	0
$236$	22.5517	1.46799
$237$	3.33728i	0.216780i
$238$	0.609627i	0.0395162i
$239$	−14.7790	−0.955973	−0.477987	−	0.878367i	$-0.658633\pi$
$239$	−14.7790	−0.955973	−0.477987	+	0.878367i	$0.658633\pi$
$240$	0	0
$241$	−14.3894	−0.926903	−0.463451	−	0.886122i	$-0.653389\pi$
$241$	−14.3894	−0.926903	−0.463451	+	0.886122i	$0.653389\pi$
$242$	− 2.14110i	− 0.137635i
$243$	− 1.00000i	− 0.0641500i
$244$	−24.9552	−1.59760
$245$	0	0
$246$	1.90536	0.121482
$247$	3.27847i	0.208604i
$248$	2.86166i	0.181716i
$249$	−9.73310	−0.616810
$250$	0	0
$251$	−18.7342	−1.18249	−0.591247	−	0.806490i	$-0.701364\pi$
$251$	−18.7342	−1.18249	−0.591247	+	0.806490i	$0.701364\pi$
$252$	1.15622i	0.0728347i
$253$	− 6.01945i	− 0.378440i
$254$	0.113240	0.00710532
$255$	0	0
$256$	12.6171	0.788566
$257$	6.42218i	0.400605i	0.979734	+	0.200302i	$0.0641924\pi$
$257$	6.42218i	0.400605i	−0.979734	+	0.200302i	$0.935808\pi$
$258$	− 0.588467i	− 0.0366363i
$259$	−6.00959	−0.373418
$260$	0	0
$261$	−7.48883	−0.463547
$262$	− 0.515841i	− 0.0318687i
$263$	− 14.2071i	− 0.876050i	−0.898963	−	0.438025i	$-0.855678\pi$
$263$	− 14.2071i	− 0.876050i	0.898963	−	0.438025i	$-0.144322\pi$
$264$	−0.720227	−0.0443269
$265$	0	0
$266$	0.352068	0.0215867
$267$	0.645045i	0.0394761i
$268$	14.8736i	0.908547i
$269$	8.67115	0.528689	0.264345	−	0.964428i	$-0.414844\pi$
$269$	8.67115	0.528689	0.264345	+	0.964428i	$0.414844\pi$
$270$	0	0
$271$	11.5697	0.702807	0.351403	−	0.936224i	$-0.385705\pi$
$271$	11.5697	0.702807	0.351403	+	0.936224i	$0.385705\pi$
$272$	18.4514i	1.11878i
$273$	− 0.680012i	− 0.0411562i
$274$	−1.18007	−0.0712904
$275$	0	0
$276$	13.5231	0.813993
$277$	− 19.2540i	− 1.15686i	−0.815732	−	0.578431i	$-0.803665\pi$
$277$	− 19.2540i	− 1.15686i	0.815732	−	0.578431i	$-0.196335\pi$
$278$	0.0681596i	0.00408794i
$279$	−3.45991	−0.207139
$280$	0	0
$281$	−3.21917	−0.192040	−0.0960198	−	0.995379i	$-0.530611\pi$
$281$	−3.21917	−0.192040	−0.0960198	+	0.995379i	$0.530611\pi$
$282$	− 1.39826i	− 0.0832652i
$283$	30.6086i	1.81949i	0.415167	+	0.909745i	$0.363723\pi$
$283$	30.6086i	1.81949i	−0.415167	+	0.909745i	$0.636277\pi$
$284$	−29.4551	−1.74784
$285$	0	0
$286$	0.209456	0.0123854
$287$	− 5.38664i	− 0.317963i
$288$	− 2.43599i	− 0.143542i
$289$	−7.34391	−0.431995
$290$	0	0
$291$	−11.6970	−0.685688
$292$	− 5.84536i	− 0.342074i
$293$	− 12.6725i	− 0.740333i	−0.928965	−	0.370166i	$-0.879301\pi$
$293$	− 12.6725i	− 0.740333i	0.928965	−	0.370166i	$-0.120699\pi$
$294$	1.39037	0.0810882
$295$	0	0
$296$	8.40995	0.488819
$297$	− 0.870796i	− 0.0505287i
$298$	0.133298i	0.00772173i
$299$	−7.95341	−0.459958
$300$	0	0
$301$	−1.66365	−0.0958911
$302$	2.38217i	0.137078i
$303$	14.7636i	0.848147i
$304$	10.6560	0.611162
$305$	0	0
$306$	1.03148	0.0589656
$307$	− 28.0055i	− 1.59836i	−0.601092	−	0.799180i	$-0.705268\pi$
$307$	− 28.0055i	− 1.59836i	0.601092	−	0.799180i	$-0.294732\pi$
$308$	1.00683i	0.0573693i
$309$	−1.41969	−0.0807636
$310$	0	0
$311$	29.1437	1.65259	0.826293	−	0.563241i	$-0.190446\pi$
$311$	29.1437	1.65259	0.826293	+	0.563241i	$0.190446\pi$
$312$	0.951625i	0.0538751i
$313$	19.1782i	1.08402i	0.840373	+	0.542009i	$0.182336\pi$
$313$	19.1782i	1.08402i	−0.840373	+	0.542009i	$0.817664\pi$
$314$	0.598045	0.0337497
$315$	0	0
$316$	−6.52871	−0.367269
$317$	− 19.8985i	− 1.11761i	−0.829300	−	0.558804i	$-0.811260\pi$
$317$	− 19.8985i	− 1.11761i	0.829300	−	0.558804i	$-0.188740\pi$
$318$	0.809547i	0.0453971i
$319$	−6.52124	−0.365119
$320$	0	0
$321$	14.9579	0.834867
$322$	0.854102i	0.0475972i
$323$	14.0590i	0.782262i
$324$	1.95630	0.108683
$325$	0	0
$326$	−2.94992	−0.163381
$327$	− 11.0977i	− 0.613706i
$328$	7.53818i	0.416226i
$329$	−3.95301	−0.217936
$330$	0	0
$331$	−27.8128	−1.52873	−0.764366	−	0.644783i	$-0.776948\pi$
$331$	−27.8128	−1.52873	−0.764366	+	0.644783i	$0.776948\pi$
$332$	− 19.0408i	− 1.04500i
$333$	10.1681i	0.557209i
$334$	−1.56592	−0.0856834
$335$	0	0
$336$	−2.21024	−0.120578
$337$	31.7776i	1.73104i	0.500877	+	0.865518i	$0.333011\pi$
$337$	31.7776i	1.73104i	−0.500877	+	0.865518i	$0.666989\pi$
$338$	2.44099i	0.132772i
$339$	−12.1069	−0.657555
$340$	0	0
$341$	−3.01287	−0.163156
$342$	− 0.595693i	− 0.0322114i
$343$	− 8.06787i	− 0.435624i
$344$	2.32815	0.125525
$345$	0	0
$346$	−4.24869	−0.228411
$347$	27.6962i	1.48681i	0.668842	+	0.743404i	$0.266790\pi$
$347$	27.6962i	1.48681i	−0.668842	+	0.743404i	$0.733210\pi$
$348$	− 14.6504i	− 0.785342i
$349$	11.3435	0.607206	0.303603	−	0.952799i	$-0.401810\pi$
$349$	11.3435	0.607206	0.303603	+	0.952799i	$0.401810\pi$
$350$	0	0
$351$	−1.15057	−0.0614128
$352$	− 2.12125i	− 0.113063i
$353$	− 13.2761i	− 0.706614i	−0.935507	−	0.353307i	$-0.885057\pi$
$353$	− 13.2761i	− 0.706614i	0.935507	−	0.353307i	$-0.114943\pi$
$354$	2.40995	0.128088
$355$	0	0
$356$	−1.26190	−0.0668805
$357$	− 2.91608i	− 0.154335i
$358$	− 0.236671i	− 0.0125085i
$359$	−0.547922	−0.0289182	−0.0144591	−	0.999895i	$-0.504603\pi$
$359$	−0.547922	−0.0289182	−0.0144591	+	0.999895i	$0.504603\pi$
$360$	0	0
$361$	−10.8807	−0.572671
$362$	1.03786i	0.0545489i
$363$	10.2417i	0.537551i
$364$	1.33030	0.0697269
$365$	0	0
$366$	−2.66681	−0.139396
$367$	− 32.3147i	− 1.68681i	−0.537277	−	0.843406i	$-0.680547\pi$
$367$	− 32.3147i	− 1.68681i	0.537277	−	0.843406i	$-0.319453\pi$
$368$	25.8509i	1.34757i
$369$	−9.11409	−0.474461
$370$	0	0
$371$	2.28866	0.118821
$372$	− 6.76860i	− 0.350936i
$373$	− 15.2740i	− 0.790860i	−0.918496	−	0.395430i	$-0.870596\pi$
$373$	− 15.2740i	− 0.790860i	0.918496	−	0.395430i	$-0.129404\pi$
$374$	0.898206	0.0464451
$375$	0	0
$376$	5.53193	0.285288
$377$	8.61641i	0.443768i
$378$	0.123557i	0.00635511i
$379$	−11.5928	−0.595482	−0.297741	−	0.954647i	$-0.596233\pi$
$379$	−11.5928	−0.595482	−0.297741	+	0.954647i	$0.596233\pi$
$380$	0	0
$381$	−0.541671	−0.0277507
$382$	1.34760i	0.0689493i
$383$	− 27.1864i	− 1.38916i	−0.719416	−	0.694580i	$-0.755590\pi$
$383$	− 27.1864i	− 1.38916i	0.719416	−	0.694580i	$-0.244410\pi$
$384$	6.32912	0.322982
$385$	0	0
$386$	3.85317	0.196121
$387$	2.81486i	0.143088i
$388$	− 22.8827i	− 1.16169i
$389$	−36.2825	−1.83959	−0.919797	−	0.392394i	$-0.871647\pi$
$389$	−36.2825	−1.83959	−0.919797	+	0.392394i	$0.871647\pi$
$390$	0	0
$391$	−34.1064	−1.72484
$392$	5.50073i	0.277829i
$393$	2.46747i	0.124467i
$394$	−3.51933	−0.177301
$395$	0	0
$396$	1.70353	0.0856058
$397$	24.6879i	1.23905i	0.784977	+	0.619525i	$0.212675\pi$
$397$	24.6879i	1.23905i	−0.784977	+	0.619525i	$0.787325\pi$
$398$	− 3.55330i	− 0.178111i
$399$	−1.68408	−0.0843094
$400$	0	0
$401$	−34.4435	−1.72003	−0.860014	−	0.510271i	$-0.829545\pi$
$401$	−34.4435	−1.72003	−0.860014	+	0.510271i	$0.829545\pi$
$402$	1.58944i	0.0792742i
$403$	3.98086i	0.198301i
$404$	−28.8820	−1.43693
$405$	0	0
$406$	0.925301	0.0459219
$407$	8.85435i	0.438894i
$408$	4.08083i	0.202031i
$409$	12.9896	0.642294	0.321147	−	0.947029i	$-0.395932\pi$
$409$	12.9896	0.642294	0.321147	+	0.947029i	$0.395932\pi$
$410$	0	0
$411$	5.64471	0.278433
$412$	− 2.77734i	− 0.136830i
$413$	− 6.81316i	− 0.335254i
$414$	1.44512	0.0710240
$415$	0	0
$416$	−2.80277	−0.137417
$417$	− 0.326034i	− 0.0159659i
$418$	− 0.518727i	− 0.0253718i
$419$	−18.5116	−0.904353	−0.452177	−	0.891928i	$-0.649352\pi$
$419$	−18.5116	−0.904353	−0.452177	+	0.891928i	$0.649352\pi$
$420$	0	0
$421$	23.8357	1.16168	0.580841	−	0.814017i	$-0.302724\pi$
$421$	23.8357	1.16168	0.580841	+	0.814017i	$0.302724\pi$
$422$	1.04711i	0.0509726i
$423$	6.68842i	0.325202i
$424$	−3.20281	−0.155542
$425$	0	0
$426$	−3.14768	−0.152506
$427$	7.53931i	0.364853i
$428$	29.2620i	1.41443i
$429$	−1.00191	−0.0483727
$430$	0	0
$431$	−29.6800	−1.42963	−0.714817	−	0.699312i	$-0.753490\pi$
$431$	−29.6800	−1.42963	−0.714817	+	0.699312i	$0.753490\pi$
$432$	3.73968i	0.179926i
$433$	− 18.0164i	− 0.865811i	−0.901439	−	0.432906i	$-0.857488\pi$
$433$	− 18.0164i	− 0.865811i	0.901439	−	0.432906i	$-0.142512\pi$
$434$	0.427497	0.0205205
$435$	0	0
$436$	21.7104	1.03974
$437$	19.6970i	0.942233i
$438$	− 0.624657i	− 0.0298473i
$439$	−21.0442	−1.00438	−0.502191	−	0.864757i	$-0.667473\pi$
$439$	−21.0442	−1.00438	−0.502191	+	0.864757i	$0.667473\pi$
$440$	0	0
$441$	−6.65069	−0.316700
$442$	− 1.18679i	− 0.0564496i
$443$	− 4.15877i	− 0.197589i	−0.995108	−	0.0987946i	$-0.968501\pi$
$443$	− 4.15877i	− 0.197589i	0.995108	−	0.0987946i	$-0.0314987\pi$
$444$	−19.8918	−0.944024
$445$	0	0
$446$	−3.33346	−0.157844
$447$	− 0.637614i	− 0.0301581i
$448$	− 4.11949i	− 0.194628i
$449$	7.39256	0.348876	0.174438	−	0.984668i	$-0.444189\pi$
$449$	7.39256	0.348876	0.174438	+	0.984668i	$0.444189\pi$
$450$	0	0
$451$	−7.93651	−0.373716
$452$	− 23.6846i	− 1.11403i
$453$	− 11.3948i	− 0.535376i
$454$	−5.62295	−0.263898
$455$	0	0
$456$	2.35674	0.110364
$457$	0.573353i	0.0268203i	0.999910	+	0.0134102i	$0.00426871\pi$
$457$	0.573353i	0.0268203i	−0.999910	+	0.0134102i	$0.995731\pi$
$458$	0.615877i	0.0287780i
$459$	−4.93395	−0.230297
$460$	0	0
$461$	3.74666	0.174499	0.0872497	−	0.996186i	$-0.472192\pi$
$461$	3.74666	0.174499	0.0872497	+	0.996186i	$0.472192\pi$
$462$	0.107593i	0.00500569i
$463$	17.3803i	0.807732i	0.914818	+	0.403866i	$0.132334\pi$
$463$	17.3803i	0.807732i	−0.914818	+	0.403866i	$0.867666\pi$
$464$	28.0058	1.30014
$465$	0	0
$466$	4.89307	0.226667
$467$	− 10.0211i	− 0.463720i	−0.972749	−	0.231860i	$-0.925519\pi$
$467$	− 10.0211i	− 0.463720i	0.972749	−	0.231860i	$-0.0744811\pi$
$468$	− 2.25085i	− 0.104046i
$469$	4.49350	0.207491
$470$	0	0
$471$	−2.86068	−0.131813
$472$	9.53449i	0.438860i
$473$	2.45117i	0.112705i
$474$	−0.697683	−0.0320456
$475$	0	0
$476$	5.70471	0.261475
$477$	− 3.87238i	− 0.177304i
$478$	− 3.08965i	− 0.141317i
$479$	−18.6136	−0.850476	−0.425238	−	0.905082i	$-0.639810\pi$
$479$	−18.6136	−0.850476	−0.425238	+	0.905082i	$0.639810\pi$
$480$	0	0
$481$	11.6991	0.533433
$482$	− 3.00820i	− 0.137020i
$483$	− 4.08550i	− 0.185897i
$484$	−20.0358	−0.910719
$485$	0	0
$486$	0.209057	0.00948301
$487$	− 2.07209i	− 0.0938954i	−0.998897	−	0.0469477i	$-0.985051\pi$
$487$	− 2.07209i	− 0.0938954i	0.998897	−	0.0469477i	$-0.0149494\pi$
$488$	− 10.5507i	− 0.477607i
$489$	14.1106	0.638103
$490$	0	0
$491$	−22.2355	−1.00348	−0.501738	−	0.865020i	$-0.667306\pi$
$491$	−22.2355	−1.00348	−0.501738	+	0.865020i	$0.667306\pi$
$492$	− 17.8299i	− 0.803832i
$493$	36.9495i	1.66412i
$494$	−0.685386	−0.0308370
$495$	0	0
$496$	12.9390	0.580976
$497$	8.89878i	0.399165i
$498$	− 2.03477i	− 0.0911803i
$499$	−28.2713	−1.26560	−0.632798	−	0.774317i	$-0.718094\pi$
$499$	−28.2713	−1.26560	−0.632798	+	0.774317i	$0.718094\pi$
$500$	0	0
$501$	7.49041	0.334647
$502$	− 3.91652i	− 0.174803i
$503$	− 3.39793i	− 0.151506i	−0.997127	−	0.0757531i	$-0.975864\pi$
$503$	− 3.39793i	− 0.151506i	0.997127	−	0.0757531i	$-0.0241361\pi$
$504$	−0.488830	−0.0217742
$505$	0	0
$506$	1.25841	0.0559431
$507$	− 11.6762i	− 0.518558i
$508$	− 1.05967i	− 0.0470152i
$509$	10.8405	0.480497	0.240248	−	0.970711i	$-0.422771\pi$
$509$	10.8405	0.480497	0.240248	+	0.970711i	$0.422771\pi$
$510$	0	0
$511$	−1.76596	−0.0781216
$512$	15.2959i	0.675991i
$513$	2.84943i	0.125805i
$514$	−1.34260	−0.0592196
$515$	0	0
$516$	−5.50670	−0.242419
$517$	5.82425i	0.256150i
$518$	− 1.25635i	− 0.0552007i
$519$	20.3231	0.892087
$520$	0	0
$521$	15.6473	0.685519	0.342760	−	0.939423i	$-0.388638\pi$
$521$	15.6473	0.685519	0.342760	+	0.939423i	$0.388638\pi$
$522$	− 1.56559i	− 0.0685241i
$523$	− 18.3578i	− 0.802729i	−0.915918	−	0.401364i	$-0.868536\pi$
$523$	− 18.3578i	− 0.802729i	0.915918	−	0.401364i	$-0.131464\pi$
$524$	−4.82709	−0.210872
$525$	0	0
$526$	2.97010	0.129503
$527$	17.0710i	0.743626i
$528$	3.25650i	0.141721i
$529$	−24.7839	−1.07756
$530$	0	0
$531$	−11.5277	−0.500261
$532$	− 3.29456i	− 0.142837i
$533$	10.4864i	0.454216i
$534$	−0.134851	−0.00583558
$535$	0	0
$536$	−6.28831	−0.271613
$537$	1.13209i	0.0488533i
$538$	1.81276i	0.0781538i
$539$	−5.79139	−0.249453
$540$	0	0
$541$	−13.8856	−0.596988	−0.298494	−	0.954412i	$-0.596484\pi$
$541$	−13.8856	−0.596988	−0.298494	+	0.954412i	$0.596484\pi$
$542$	2.41872i	0.103893i
$543$	− 4.96450i	− 0.213047i
$544$	−12.0191	−0.515313
$545$	0	0
$546$	0.142161	0.00608394
$547$	− 41.9134i	− 1.79209i	−0.443966	−	0.896043i	$-0.646429\pi$
$547$	− 41.9134i	− 1.79209i	0.443966	−	0.896043i	$-0.353571\pi$
$548$	11.0427i	0.471722i
$549$	12.7564	0.544429
$550$	0	0
$551$	21.3389	0.909068
$552$	5.71734i	0.243346i
$553$	1.97241i	0.0838755i
$554$	4.02518	0.171014
$555$	0	0
$556$	0.637818	0.0270495
$557$	8.04257i	0.340774i	0.985377	+	0.170387i	$0.0545019\pi$
$557$	8.04257i	0.340774i	−0.985377	+	0.170387i	$0.945498\pi$
$558$	− 0.723318i	− 0.0306205i
$559$	3.23869	0.136982
$560$	0	0
$561$	−4.29647	−0.181397
$562$	− 0.672990i	− 0.0283884i
$563$	5.67464i	0.239158i	0.992825	+	0.119579i	$0.0381544\pi$
$563$	5.67464i	0.239158i	−0.992825	+	0.119579i	$0.961846\pi$
$564$	−13.0845	−0.550958
$565$	0	0
$566$	−6.39893	−0.268967
$567$	− 0.591023i	− 0.0248206i
$568$	− 12.4532i	− 0.522523i
$569$	−8.25303	−0.345985	−0.172993	−	0.984923i	$-0.555344\pi$
$569$	−8.25303	−0.345985	−0.172993	+	0.984923i	$0.555344\pi$
$570$	0	0
$571$	−17.5703	−0.735293	−0.367646	−	0.929966i	$-0.619836\pi$
$571$	−17.5703	−0.735293	−0.367646	+	0.929966i	$0.619836\pi$
$572$	− 1.96003i	− 0.0819531i
$573$	− 6.44610i	− 0.269290i
$574$	1.12611	0.0470031
$575$	0	0
$576$	−6.97010	−0.290421
$577$	− 29.6207i	− 1.23313i	−0.787305	−	0.616564i	$-0.788524\pi$
$577$	− 29.6207i	− 1.23313i	0.787305	−	0.616564i	$-0.211476\pi$
$578$	− 1.53529i	− 0.0638598i
$579$	−18.4312	−0.765974
$580$	0	0
$581$	−5.75249	−0.238653
$582$	− 2.44533i	− 0.101362i
$583$	− 3.37205i	− 0.139656i
$584$	2.47133	0.102264
$585$	0	0
$586$	2.64926	0.109440
$587$	18.1475i	0.749027i	0.927222	+	0.374513i	$0.122190\pi$
$587$	18.1475i	0.749027i	−0.927222	+	0.374513i	$0.877810\pi$
$588$	− 13.0107i	− 0.536553i
$589$	9.85877	0.406224
$590$	0	0
$591$	16.8343	0.692471
$592$	− 38.0255i	− 1.56284i
$593$	− 26.0022i	− 1.06778i	−0.845553	−	0.533891i	$-0.820729\pi$
$593$	− 26.0022i	− 1.06778i	0.845553	−	0.533891i	$-0.179271\pi$
$594$	0.182046	0.00746943
$595$	0	0
$596$	1.24736	0.0510939
$597$	16.9968i	0.695632i
$598$	− 1.66272i	− 0.0679935i
$599$	−20.6259	−0.842753	−0.421376	−	0.906886i	$-0.638453\pi$
$599$	−20.6259	−0.842753	−0.421376	+	0.906886i	$0.638453\pi$
$600$	0	0
$601$	−17.7602	−0.724452	−0.362226	−	0.932090i	$-0.617983\pi$
$601$	−17.7602	−0.724452	−0.362226	+	0.932090i	$0.617983\pi$
$602$	− 0.347797i	− 0.0141752i
$603$	− 7.60292i	− 0.309615i
$604$	22.2917	0.907035
$605$	0	0
$606$	−3.08643	−0.125378
$607$	13.0143i	0.528232i	0.964491	+	0.264116i	$0.0850803\pi$
$607$	13.0143i	0.528232i	−0.964491	+	0.264116i	$0.914920\pi$
$608$	6.94118i	0.281502i
$609$	−4.42607	−0.179353
$610$	0	0
$611$	7.69548	0.311326
$612$	− 9.65227i	− 0.390170i
$613$	− 21.9959i	− 0.888407i	−0.895926	−	0.444204i	$-0.853487\pi$
$613$	− 21.9959i	− 0.888407i	0.895926	−	0.444204i	$-0.146513\pi$
$614$	5.85475	0.236278
$615$	0	0
$616$	−0.425671	−0.0171508
$617$	− 33.3091i	− 1.34097i	−0.741922	−	0.670487i	$-0.766085\pi$
$617$	− 33.3091i	− 1.34097i	0.741922	−	0.670487i	$-0.233915\pi$
$618$	− 0.296797i	− 0.0119389i
$619$	27.9714	1.12426	0.562132	−	0.827048i	$-0.309981\pi$
$619$	27.9714	1.12426	0.562132	+	0.827048i	$0.309981\pi$
$620$	0	0
$621$	−6.91259	−0.277393
$622$	6.09268i	0.244294i
$623$	0.381236i	0.0152739i
$624$	4.30276	0.172248
$625$	0	0
$626$	−4.00934	−0.160245
$627$	2.48127i	0.0990925i
$628$	− 5.59634i	− 0.223318i
$629$	50.1690	2.00037
$630$	0	0
$631$	−23.0172	−0.916301	−0.458150	−	0.888875i	$-0.651488\pi$
$631$	−23.0172	−0.916301	−0.458150	+	0.888875i	$0.651488\pi$
$632$	− 2.76024i	− 0.109796i
$633$	− 5.00874i	− 0.199079i
$634$	4.15991	0.165211
$635$	0	0
$636$	7.57551	0.300388
$637$	7.65208i	0.303186i
$638$	− 1.36331i	− 0.0539740i
$639$	15.0566	0.595629
$640$	0	0
$641$	29.1915	1.15300	0.576498	−	0.817099i	$-0.304419\pi$
$641$	29.1915	1.15300	0.576498	+	0.817099i	$0.304419\pi$
$642$	3.12705i	0.123415i
$643$	− 29.6236i	− 1.16824i	−0.811668	−	0.584119i	$-0.801440\pi$
$643$	− 29.6236i	− 1.16824i	0.811668	−	0.584119i	$-0.198560\pi$
$644$	7.99244	0.314946
$645$	0	0
$646$	−2.93912	−0.115638
$647$	− 49.0862i	− 1.92978i	−0.262657	−	0.964889i	$-0.584599\pi$
$647$	− 49.0862i	− 1.92978i	0.262657	−	0.964889i	$-0.415401\pi$
$648$	0.827091i	0.0324912i
$649$	−10.0383	−0.394038
$650$	0	0
$651$	−2.04489	−0.0801454
$652$	27.6045i	1.08108i
$653$	− 11.6946i	− 0.457645i	−0.973468	−	0.228823i	$-0.926512\pi$
$653$	− 11.6946i	− 0.457645i	0.973468	−	0.228823i	$-0.0734876\pi$
$654$	2.32006	0.0907214
$655$	0	0
$656$	34.0838	1.33075
$657$	2.98798i	0.116572i
$658$	− 0.826404i	− 0.0322166i
$659$	−1.18709	−0.0462424	−0.0231212	−	0.999733i	$-0.507360\pi$
$659$	−1.18709	−0.0462424	−0.0231212	+	0.999733i	$0.507360\pi$
$660$	0	0
$661$	3.72234	0.144782	0.0723912	−	0.997376i	$-0.476937\pi$
$661$	3.72234	0.144782	0.0723912	+	0.997376i	$0.476937\pi$
$662$	− 5.81446i	− 0.225986i
$663$	5.67685i	0.220471i
$664$	8.05016	0.312407
$665$	0	0
$666$	−2.12571	−0.0823698
$667$	51.7672i	2.00443i
$668$	14.6535i	0.566959i
$669$	15.9452	0.616479
$670$	0	0
$671$	11.1082	0.428827
$672$	− 1.43972i	− 0.0555386i
$673$	46.9935i	1.81147i	0.423847	+	0.905734i	$0.360680\pi$
$673$	46.9935i	1.81147i	−0.423847	+	0.905734i	$0.639320\pi$
$674$	−6.64333	−0.255892
$675$	0	0
$676$	22.8421	0.878541
$677$	− 36.4081i	− 1.39928i	−0.714497	−	0.699639i	$-0.753344\pi$
$677$	− 36.4081i	− 1.39928i	0.714497	−	0.699639i	$-0.246656\pi$
$678$	− 2.53102i	− 0.0972034i
$679$	−6.91317	−0.265303
$680$	0	0
$681$	26.8967	1.03069
$682$	− 0.629862i	− 0.0241187i
$683$	4.66507i	0.178504i	0.996009	+	0.0892519i	$0.0284476\pi$
$683$	4.66507i	0.178504i	−0.996009	+	0.0892519i	$0.971552\pi$
$684$	−5.57433	−0.213140
$685$	0	0
$686$	1.68664	0.0643964
$687$	− 2.94598i	− 0.112396i
$688$	− 10.5267i	− 0.401326i
$689$	−4.45543	−0.169738
$690$	0	0
$691$	31.9254	1.21450	0.607249	−	0.794511i	$-0.292273\pi$
$691$	31.9254	1.21450	0.607249	+	0.794511i	$0.292273\pi$
$692$	39.7581i	1.51138i
$693$	− 0.514660i	− 0.0195503i
$694$	−5.79008	−0.219788
$695$	0	0
$696$	6.19394	0.234781
$697$	44.9685i	1.70330i
$698$	2.37144i	0.0897605i
$699$	−23.4054	−0.885275
$700$	0	0
$701$	−3.09598	−0.116933	−0.0584667	−	0.998289i	$-0.518621\pi$
$701$	−3.09598	−0.116933	−0.0584667	+	0.998289i	$0.518621\pi$
$702$	− 0.240534i	− 0.00907838i
$703$	− 28.9733i	− 1.09275i
$704$	−6.06954	−0.228754
$705$	0	0
$706$	2.77546	0.104456
$707$	8.72563i	0.328161i
$708$	− 22.5517i	− 0.847543i
$709$	−47.0490	−1.76696	−0.883481	−	0.468468i	$-0.844806\pi$
$709$	−47.0490	−1.76696	−0.883481	+	0.468468i	$0.844806\pi$
$710$	0	0
$711$	3.33728	0.125158
$712$	− 0.533511i	− 0.0199942i
$713$	23.9169i	0.895696i
$714$	0.609627	0.0228147
$715$	0	0
$716$	−2.21470	−0.0827674
$717$	14.7790i	0.551931i
$718$	− 0.114547i	− 0.00427485i
$719$	−25.7083	−0.958757	−0.479378	−	0.877608i	$-0.659138\pi$
$719$	−25.7083	−0.958757	−0.479378	+	0.877608i	$0.659138\pi$
$720$	0	0
$721$	−0.839072	−0.0312487
$722$	− 2.27469i	− 0.0846553i
$723$	14.3894i	0.535147i
$724$	9.71202	0.360945
$725$	0	0
$726$	−2.14110	−0.0794637
$727$	35.4025i	1.31301i	0.754323	+	0.656504i	$0.227965\pi$
$727$	35.4025i	1.31301i	−0.754323	+	0.656504i	$0.772035\pi$
$728$	0.562432i	0.0208451i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	13.8884	0.513681
$732$	24.9552i	0.922372i
$733$	25.7565i	0.951336i	0.879625	+	0.475668i	$0.157794\pi$
$733$	25.7565i	0.951336i	−0.879625	+	0.475668i	$0.842206\pi$
$734$	6.75560	0.249354
$735$	0	0
$736$	−16.8390	−0.620693
$737$	− 6.62059i	− 0.243873i
$738$	− 1.90536i	− 0.0701374i
$739$	22.1071	0.813222	0.406611	−	0.913601i	$-0.366711\pi$
$739$	22.1071	0.813222	0.406611	+	0.913601i	$0.366711\pi$
$740$	0	0
$741$	3.27847	0.120437
$742$	0.478461i	0.0175648i
$743$	20.2268i	0.742049i	0.928623	+	0.371025i	$0.120993\pi$
$743$	20.2268i	0.742049i	−0.928623	+	0.371025i	$0.879007\pi$
$744$	2.86166	0.104914
$745$	0	0
$746$	3.19314	0.116909
$747$	9.73310i	0.356116i
$748$	− 8.40516i	− 0.307323i
$749$	8.84045	0.323023
$750$	0	0
$751$	−1.23206	−0.0449584	−0.0224792	−	0.999747i	$-0.507156\pi$
$751$	−1.23206	−0.0449584	−0.0224792	+	0.999747i	$0.507156\pi$
$752$	− 25.0126i	− 0.912114i
$753$	18.7342i	0.682714i
$754$	−1.80132	−0.0656002
$755$	0	0
$756$	1.15622	0.0420511
$757$	− 15.2062i	− 0.552678i	−0.961060	−	0.276339i	$-0.910879\pi$
$757$	− 15.2062i	− 0.552678i	0.961060	−	0.276339i	$-0.0891212\pi$
$758$	− 2.42356i	− 0.0880275i
$759$	−6.01945	−0.218492
$760$	0	0
$761$	−32.3209	−1.17163	−0.585817	−	0.810444i	$-0.699226\pi$
$761$	−32.3209	−1.17163	−0.585817	+	0.810444i	$0.699226\pi$
$762$	− 0.113240i	− 0.00410226i
$763$	− 6.55901i	− 0.237452i
$764$	12.6105	0.456231
$765$	0	0
$766$	5.68350	0.205353
$767$	13.2635i	0.478915i
$768$	− 12.6171i	− 0.455279i
$769$	28.8068	1.03880	0.519400	−	0.854531i	$-0.326155\pi$
$769$	28.8068	1.03880	0.519400	+	0.854531i	$0.326155\pi$
$770$	0	0
$771$	6.42218	0.231289
$772$	− 36.0569i	− 1.29771i
$773$	− 38.2458i	− 1.37561i	−0.725897	−	0.687803i	$-0.758575\pi$
$773$	− 38.2458i	− 1.37561i	0.725897	−	0.687803i	$-0.241425\pi$
$774$	−0.588467	−0.0211520
$775$	0	0
$776$	9.67444	0.347292
$777$	6.00959i	0.215593i
$778$	− 7.58510i	− 0.271939i
$779$	25.9700	0.930471
$780$	0	0
$781$	13.1112	0.469156
$782$	− 7.13018i	− 0.254975i
$783$	7.48883i	0.267629i
$784$	24.8715	0.888267
$785$	0	0
$786$	−0.515841	−0.0183994
$787$	− 36.7934i	− 1.31154i	−0.754960	−	0.655771i	$-0.772344\pi$
$787$	− 36.7934i	− 1.31154i	0.754960	−	0.655771i	$-0.227656\pi$
$788$	32.9329i	1.17319i
$789$	−14.2071	−0.505788
$790$	0	0
$791$	−7.15543	−0.254418
$792$	0.720227i	0.0255922i
$793$	− 14.6771i	− 0.521199i
$794$	−5.16117	−0.183163
$795$	0	0
$796$	−33.2507	−1.17854
$797$	− 36.6698i	− 1.29891i	−0.760399	−	0.649456i	$-0.774997\pi$
$797$	− 36.6698i	− 1.29891i	0.760399	−	0.649456i	$-0.225003\pi$
$798$	− 0.352068i	− 0.0124631i
$799$	33.0004	1.16747
$800$	0	0
$801$	0.645045	0.0227915
$802$	− 7.20066i	− 0.254264i
$803$	2.60192i	0.0918197i
$804$	14.8736	0.524550
$805$	0	0
$806$	−0.832227	−0.0293139
$807$	− 8.67115i	− 0.305239i
$808$	− 12.2108i	− 0.429576i
$809$	39.2361	1.37947	0.689735	−	0.724062i	$-0.257727\pi$
$809$	39.2361	1.37947	0.689735	+	0.724062i	$0.257727\pi$
$810$	0	0
$811$	6.70444	0.235425	0.117712	−	0.993048i	$-0.462444\pi$
$811$	6.70444	0.235425	0.117712	+	0.993048i	$0.462444\pi$
$812$	− 8.65870i	− 0.303861i
$813$	− 11.5697i	− 0.405766i
$814$	−1.85106	−0.0648797
$815$	0	0
$816$	18.4514	0.645929
$817$	− 8.02076i	− 0.280611i
$818$	2.71557i	0.0949475i
$819$	−0.680012	−0.0237616
$820$	0	0
$821$	−30.1557	−1.05244	−0.526220	−	0.850348i	$-0.676391\pi$
$821$	−30.1557	−1.05244	−0.526220	+	0.850348i	$0.676391\pi$
$822$	1.18007i	0.0411596i
$823$	− 1.90341i	− 0.0663487i	−0.999450	−	0.0331744i	$-0.989438\pi$
$823$	− 1.90341i	− 0.0663487i	0.999450	−	0.0331744i	$-0.0105617\pi$
$824$	1.17422	0.0409057
$825$	0	0
$826$	1.42434	0.0495590
$827$	17.5652i	0.610803i	0.952224	+	0.305402i	$0.0987907\pi$
$827$	17.5652i	0.610803i	−0.952224	+	0.305402i	$0.901209\pi$
$828$	− 13.5231i	− 0.469959i
$829$	24.0291	0.834566	0.417283	−	0.908777i	$-0.362982\pi$
$829$	24.0291	0.834566	0.417283	+	0.908777i	$0.362982\pi$
$830$	0	0
$831$	−19.2540	−0.667914
$832$	8.01958i	0.278029i
$833$	32.8142i	1.13695i
$834$	0.0681596	0.00236017
$835$	0	0
$836$	−4.85410	−0.167883
$837$	3.45991i	0.119592i
$838$	− 3.86999i	− 0.133687i
$839$	−1.07320	−0.0370511	−0.0185255	−	0.999828i	$-0.505897\pi$
$839$	−1.07320	−0.0370511	−0.0185255	+	0.999828i	$0.505897\pi$
$840$	0	0
$841$	27.0826	0.933882
$842$	4.98302i	0.171726i
$843$	3.21917i	0.110874i
$844$	9.79857	0.337281
$845$	0	0
$846$	−1.39826	−0.0480732
$847$	6.05309i	0.207987i
$848$	14.4815i	0.497295i
$849$	30.6086	1.05048
$850$	0	0
$851$	70.2880	2.40944
$852$	29.4551i	1.00912i
$853$	17.2818i	0.591716i	0.955232	+	0.295858i	$0.0956056\pi$
$853$	17.2818i	0.591716i	−0.955232	+	0.295858i	$0.904394\pi$
$854$	−1.57615	−0.0539346
$855$	0	0
$856$	−12.3715	−0.422850
$857$	16.2742i	0.555917i	0.960593	+	0.277959i	$0.0896578\pi$
$857$	16.2742i	0.555917i	−0.960593	+	0.277959i	$0.910342\pi$
$858$	− 0.209456i	− 0.00715072i
$859$	11.2429	0.383604	0.191802	−	0.981434i	$-0.438567\pi$
$859$	11.2429	0.383604	0.191802	+	0.981434i	$0.438567\pi$
$860$	0	0
$861$	−5.38664	−0.183576
$862$	− 6.20480i	− 0.211336i
$863$	33.3846i	1.13642i	0.822882	+	0.568212i	$0.192365\pi$
$863$	33.3846i	1.13642i	−0.822882	+	0.568212i	$0.807635\pi$
$864$	−2.43599	−0.0828740
$865$	0	0
$866$	3.76645	0.127989
$867$	7.34391i	0.249412i
$868$	− 4.00040i	− 0.135782i
$869$	2.90609	0.0985825
$870$	0	0
$871$	−8.74768	−0.296404
$872$	9.17883i	0.310834i
$873$	11.6970i	0.395882i
$874$	−4.11778	−0.139286
$875$	0	0
$876$	−5.84536	−0.197497
$877$	16.1140i	0.544130i	0.962279	+	0.272065i	$0.0877066\pi$
$877$	16.1140i	0.544130i	−0.962279	+	0.272065i	$0.912293\pi$
$878$	− 4.39943i	− 0.148473i
$879$	−12.6725	−0.427431
$880$	0	0
$881$	25.2799	0.851701	0.425851	−	0.904793i	$-0.359975\pi$
$881$	25.2799	0.851701	0.425851	+	0.904793i	$0.359975\pi$
$882$	− 1.39037i	− 0.0468163i
$883$	− 31.0260i	− 1.04411i	−0.852913	−	0.522053i	$-0.825166\pi$
$883$	− 31.0260i	− 1.04411i	0.852913	−	0.522053i	$-0.174834\pi$
$884$	−11.1056	−0.373522
$885$	0	0
$886$	0.869420	0.0292087
$887$	− 38.1973i	− 1.28254i	−0.767316	−	0.641270i	$-0.778408\pi$
$887$	− 38.1973i	− 1.28254i	0.767316	−	0.641270i	$-0.221592\pi$
$888$	− 8.40995i	− 0.282220i
$889$	−0.320140	−0.0107372
$890$	0	0
$891$	−0.870796	−0.0291728
$892$	31.1936i	1.04444i
$893$	− 19.0582i	− 0.637758i
$894$	0.133298	0.00445814
$895$	0	0
$896$	3.74066	0.124967
$897$	7.95341i	0.265557i
$898$	1.54547i	0.0515728i
$899$	25.9107	0.864169
$900$	0	0
$901$	−19.1061	−0.636517
$902$	− 1.65918i	− 0.0552447i
$903$	1.66365i	0.0553628i
$904$	10.0135	0.333043
$905$	0	0
$906$	2.38217	0.0791423
$907$	40.9212i	1.35877i	0.733784	+	0.679383i	$0.237753\pi$
$907$	40.9212i	1.35877i	−0.733784	+	0.679383i	$0.762247\pi$
$908$	52.6180i	1.74619i
$909$	14.7636	0.489678
$910$	0	0
$911$	−45.6189	−1.51142	−0.755710	−	0.654906i	$-0.772708\pi$
$911$	−45.6189	−1.51142	−0.755710	+	0.654906i	$0.772708\pi$
$912$	− 10.6560i	− 0.352854i
$913$	8.47554i	0.280500i
$914$	−0.119863	−0.00396473
$915$	0	0
$916$	5.76320	0.190422
$917$	1.45833i	0.0481583i
$918$	− 1.03148i	− 0.0340438i
$919$	−20.6582	−0.681451	−0.340725	−	0.940163i	$-0.610673\pi$
$919$	−20.6582	−0.681451	−0.340725	+	0.940163i	$0.610673\pi$
$920$	0	0
$921$	−28.0055	−0.922813
$922$	0.783265i	0.0257955i
$923$	− 17.3236i	− 0.570214i
$924$	1.00683	0.0331222
$925$	0	0
$926$	−3.63348	−0.119404
$927$	1.41969i	0.0466289i
$928$	18.2427i	0.598846i
$929$	23.2327	0.762239	0.381119	−	0.924526i	$-0.375539\pi$
$929$	23.2327	0.762239	0.381119	+	0.924526i	$0.375539\pi$
$930$	0	0
$931$	18.9507	0.621084
$932$	− 45.7879i	− 1.49983i
$933$	− 29.1437i	− 0.954121i
$934$	2.09498	0.0685497
$935$	0	0
$936$	0.951625	0.0311048
$937$	− 18.3122i	− 0.598232i	−0.954217	−	0.299116i	$-0.903308\pi$
$937$	− 18.3122i	− 0.598232i	0.954217	−	0.299116i	$-0.0966918\pi$
$938$	0.939397i	0.0306724i
$939$	19.1782	0.625858
$940$	0	0
$941$	17.3075	0.564209	0.282104	−	0.959384i	$-0.408968\pi$
$941$	17.3075	0.564209	0.282104	+	0.959384i	$0.408968\pi$
$942$	− 0.598045i	− 0.0194854i
$943$	63.0020i	2.05163i
$944$	43.1101	1.40311
$945$	0	0
$946$	−0.512434	−0.0166607
$947$	− 18.4829i	− 0.600613i	−0.953843	−	0.300307i	$-0.902911\pi$
$947$	− 18.4829i	− 0.600613i	0.953843	−	0.300307i	$-0.0970890\pi$
$948$	6.52871i	0.212043i
$949$	3.43787	0.111598
$950$	0	0
$951$	−19.8985	−0.645252
$952$	2.41186i	0.0781689i
$953$	20.8872i	0.676604i	0.941038	+	0.338302i	$0.109852\pi$
$953$	20.8872i	0.676604i	−0.941038	+	0.338302i	$0.890148\pi$
$954$	0.809547	0.0262100
$955$	0	0
$956$	−28.9121	−0.935083
$957$	6.52124i	0.210802i
$958$	− 3.89130i	− 0.125722i
$959$	3.33616	0.107730
$960$	0	0
$961$	−19.0290	−0.613840
$962$	2.44578i	0.0788551i
$963$	− 14.9579i	− 0.482011i
$964$	−28.1499	−0.906647
$965$	0	0
$966$	0.854102	0.0274803
$967$	51.2510i	1.64812i	0.566501	+	0.824061i	$0.308297\pi$
$967$	51.2510i	1.64812i	−0.566501	+	0.824061i	$0.691703\pi$
$968$	− 8.47083i	− 0.272263i
$969$	14.0590	0.451639
$970$	0	0
$971$	−38.6012	−1.23877	−0.619385	−	0.785087i	$-0.712618\pi$
$971$	−38.6012	−1.23877	−0.619385	+	0.785087i	$0.712618\pi$
$972$	− 1.95630i	− 0.0627482i
$973$	− 0.192693i	− 0.00617747i
$974$	0.433185	0.0138801
$975$	0	0
$976$	−47.7048	−1.52699
$977$	− 2.91913i	− 0.0933912i	−0.998909	−	0.0466956i	$-0.985131\pi$
$977$	− 2.91913i	− 0.0933912i	0.998909	−	0.0466956i	$-0.0148691\pi$
$978$	2.94992i	0.0943280i
$979$	0.561702	0.0179521
$980$	0	0
$981$	−11.0977	−0.354323
$982$	− 4.64849i	− 0.148339i
$983$	0.980138i	0.0312615i	0.999878	+	0.0156308i	$0.00497563\pi$
$983$	0.980138i	0.0312615i	−0.999878	+	0.0156308i	$0.995024\pi$
$984$	7.53818	0.240308
$985$	0	0
$986$	−7.72456	−0.246000
$987$	3.95301i	0.125826i
$988$	6.41365i	0.204045i
$989$	19.4580	0.618728
$990$	0	0
$991$	37.0913	1.17824	0.589122	−	0.808044i	$-0.299474\pi$
$991$	37.0913	1.17824	0.589122	+	0.808044i	$0.299474\pi$
$992$	8.42830i	0.267599i
$993$	27.8128i	0.882613i
$994$	−1.86035	−0.0590068
$995$	0	0
$996$	−19.0408	−0.603331
$997$	− 16.7675i	− 0.531031i	−0.964107	−	0.265515i	$-0.914458\pi$
$997$	− 16.7675i	− 0.531031i	0.964107	−	0.265515i	$-0.0855421\pi$
$998$	− 5.91031i	− 0.187088i
$999$	10.1681	0.321705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.d.1249.5		8
5.2	odd	4		1875.2.a.g.1.2	yes	4
5.3	odd	4		1875.2.a.f.1.3	✓	4
5.4	even	2	inner	1875.2.b.d.1249.4		8
15.2	even	4		5625.2.a.j.1.3		4
15.8	even	4		5625.2.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.3	✓	4	5.3	odd	4
1875.2.a.g.1.2	yes	4	5.2	odd	4
1875.2.b.d.1249.4		8	5.4	even	2	inner
1875.2.b.d.1249.5		8	1.1	even	1	trivial
5625.2.a.j.1.3		4	15.2	even	4
5625.2.a.m.1.2		4	15.8	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.209057i q^{2} -1.00000i q^{3} +1.95630 q^{4} +0.209057 q^{6} -0.591023i q^{7} +0.827091i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.209057i q^{2} -1.00000i q^{3} +1.95630 q^{4} +0.209057 q^{6} -0.591023i q^{7} +0.827091i q^{8} -1.00000 q^{9} -0.870796 q^{11} -1.95630i q^{12} +1.15057i q^{13} +0.123557 q^{14} +3.73968 q^{16} +4.93395i q^{17} -0.209057i q^{18} +2.84943 q^{19} -0.591023 q^{21} -0.182046i q^{22} +6.91259i q^{23} +0.827091 q^{24} -0.240534 q^{26} +1.00000i q^{27} -1.15622i q^{28} +7.48883 q^{29} +3.45991 q^{31} +2.43599i q^{32} +0.870796i q^{33} -1.03148 q^{34} -1.95630 q^{36} -10.1681i q^{37} +0.595693i q^{38} +1.15057 q^{39} +9.11409 q^{41} -0.123557i q^{42} -2.81486i q^{43} -1.70353 q^{44} -1.44512 q^{46} -6.68842i q^{47} -3.73968i q^{48} +6.65069 q^{49} +4.93395 q^{51} +2.25085i q^{52} +3.87238i q^{53} -0.209057 q^{54} +0.488830 q^{56} -2.84943i q^{57} +1.56559i q^{58} +11.5277 q^{59} -12.7564 q^{61} +0.723318i q^{62} +0.591023i q^{63} +6.97010 q^{64} -0.182046 q^{66} +7.60292i q^{67} +9.65227i q^{68} +6.91259 q^{69} -15.0566 q^{71} -0.827091i q^{72} -2.98798i q^{73} +2.12571 q^{74} +5.57433 q^{76} +0.514660i q^{77} +0.240534i q^{78} -3.33728 q^{79} +1.00000 q^{81} +1.90536i q^{82} -9.73310i q^{83} -1.15622 q^{84} +0.588467 q^{86} -7.48883i q^{87} -0.720227i q^{88} -0.645045 q^{89} +0.680012 q^{91} +13.5231i q^{92} -3.45991i q^{93} +1.39826 q^{94} +2.43599 q^{96} -11.6970i q^{97} +1.39037i q^{98} +0.870796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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