LMFDB - Embedded newform 1875.2.b.f.1249.9

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 22x^{10} + 179x^{8} + 641x^{6} + 869x^{4} + 67x^{2} + 1 $ x^12 + 22x^10 + 179x^8 + 641x^6 + 869x^4 + 67*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.9
Root			$2.01887i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.f.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.01887i q^{2} -1.00000i q^{3} -2.07584 q^{4} +2.01887 q^{6} -1.01887i q^{7} -0.153106i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.01887i q^{2} -1.00000i q^{3} -2.07584 q^{4} +2.01887 q^{6} -1.01887i q^{7} -0.153106i q^{8} -1.00000 q^{9} +4.75961 q^{11} +2.07584i q^{12} -0.103837i q^{13} +2.05697 q^{14} -3.84257 q^{16} +5.83776i q^{17} -2.01887i q^{18} +0.724404 q^{19} -1.01887 q^{21} +9.60904i q^{22} -9.07152i q^{23} -0.153106 q^{24} +0.209634 q^{26} +1.00000i q^{27} +2.11501i q^{28} +3.98847 q^{29} +1.06662 q^{31} -8.06387i q^{32} -4.75961i q^{33} -11.7857 q^{34} +2.07584 q^{36} -4.02621i q^{37} +1.46248i q^{38} -0.103837 q^{39} +7.20977 q^{41} -2.05697i q^{42} +8.62791i q^{43} -9.88018 q^{44} +18.3142 q^{46} +8.19797i q^{47} +3.84257i q^{48} +5.96190 q^{49} +5.83776 q^{51} +0.215549i q^{52} -4.36719i q^{53} -2.01887 q^{54} -0.155995 q^{56} -0.724404i q^{57} +8.05221i q^{58} +4.91285 q^{59} +6.96435 q^{61} +2.15338i q^{62} +1.01887i q^{63} +8.59476 q^{64} +9.60904 q^{66} +9.91998i q^{67} -12.1183i q^{68} -9.07152 q^{69} -10.7866 q^{71} +0.153106i q^{72} +8.63115i q^{73} +8.12840 q^{74} -1.50375 q^{76} -4.84943i q^{77} -0.209634i q^{78} -2.48291 q^{79} +1.00000 q^{81} +14.5556i q^{82} +4.24385i q^{83} +2.11501 q^{84} -17.4186 q^{86} -3.98847i q^{87} -0.728724i q^{88} +18.3752 q^{89} -0.105797 q^{91} +18.8310i q^{92} -1.06662i q^{93} -16.5506 q^{94} -8.06387 q^{96} -6.69876i q^{97} +12.0363i q^{98} -4.75961 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) $ 12 * q - 20 * q^4 - 12 * q^9	$ 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) $ 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * 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 $

$2$	2.01887i	1.42756i	0.700372	+	0.713778i	$0.253018\pi$
$2$	2.01887i	1.42756i	−0.700372	+	0.713778i	$0.746982\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−2.07584	−1.03792
$5$	0	0
$5$	0	0
$6$	2.01887	0.824200
$7$	− 1.01887i	− 0.385097i	−0.981287	−	0.192548i	$-0.938325\pi$
$7$	− 1.01887i	− 0.385097i	0.981287	−	0.192548i	$-0.0616752\pi$
$8$	− 0.153106i	− 0.0541311i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.75961	1.43508	0.717538	−	0.696519i	$-0.245269\pi$
$11$	4.75961	1.43508	0.717538	+	0.696519i	$0.245269\pi$
$12$	2.07584i	0.599243i
$13$	− 0.103837i	− 0.0287992i	−0.999896	−	0.0143996i	$-0.995416\pi$
$13$	− 0.103837i	− 0.0287992i	0.999896	−	0.0143996i	$-0.00458370\pi$
$14$	2.05697	0.549748
$15$	0	0
$16$	−3.84257	−0.960643
$17$	5.83776i	1.41587i	0.706280	+	0.707933i	$0.250372\pi$
$17$	5.83776i	1.41587i	−0.706280	+	0.707933i	$0.749628\pi$
$18$	− 2.01887i	− 0.475852i
$19$	0.724404	0.166190	0.0830949	−	0.996542i	$-0.473520\pi$
$19$	0.724404	0.166190	0.0830949	+	0.996542i	$0.473520\pi$
$20$	0	0
$21$	−1.01887	−0.222336
$22$	9.60904i	2.04865i
$23$	− 9.07152i	− 1.89154i	−0.324834	−	0.945771i	$-0.605308\pi$
$23$	− 9.07152i	− 1.89154i	0.324834	−	0.945771i	$-0.394692\pi$
$24$	−0.153106	−0.0312526
$25$	0	0
$26$	0.209634	0.0411126
$27$	1.00000i	0.192450i
$28$	2.11501i	0.399699i
$29$	3.98847	0.740641	0.370320	−	0.928904i	$-0.379248\pi$
$29$	3.98847	0.740641	0.370320	+	0.928904i	$0.379248\pi$
$30$	0	0
$31$	1.06662	0.191571	0.0957857	−	0.995402i	$-0.469464\pi$
$31$	1.06662	0.191571	0.0957857	+	0.995402i	$0.469464\pi$
$32$	− 8.06387i	− 1.42550i
$33$	− 4.75961i	− 0.828542i
$34$	−11.7857	−2.02123
$35$	0	0
$36$	2.07584	0.345973
$37$	− 4.02621i	− 0.661905i	−0.943647	−	0.330953i	$-0.892630\pi$
$37$	− 4.02621i	− 0.661905i	0.943647	−	0.330953i	$-0.107370\pi$
$38$	1.46248i	0.237245i
$39$	−0.103837	−0.0166273
$40$	0	0
$41$	7.20977	1.12598	0.562988	−	0.826465i	$-0.309652\pi$
$41$	7.20977	1.12598	0.562988	+	0.826465i	$0.309652\pi$
$42$	− 2.05697i	− 0.317397i
$43$	8.62791i	1.31574i	0.753130	+	0.657872i	$0.228543\pi$
$43$	8.62791i	1.31574i	−0.753130	+	0.657872i	$0.771457\pi$
$44$	−9.88018	−1.48949
$45$	0	0
$46$	18.3142	2.70028
$47$	8.19797i	1.19580i	0.801572	+	0.597899i	$0.203997\pi$
$47$	8.19797i	1.19580i	−0.801572	+	0.597899i	$0.796003\pi$
$48$	3.84257i	0.554628i
$49$	5.96190	0.851700
$50$	0	0
$51$	5.83776	0.817451
$52$	0.215549i	0.0298913i
$53$	− 4.36719i	− 0.599880i	−0.953958	−	0.299940i	$-0.903033\pi$
$53$	− 4.36719i	− 0.599880i	0.953958	−	0.299940i	$-0.0969667\pi$
$54$	−2.01887	−0.274733
$55$	0	0
$56$	−0.155995	−0.0208457
$57$	− 0.724404i	− 0.0959497i
$58$	8.05221i	1.05731i
$59$	4.91285	0.639599	0.319799	−	0.947485i	$-0.396384\pi$
$59$	4.91285	0.639599	0.319799	+	0.947485i	$0.396384\pi$
$60$	0	0
$61$	6.96435	0.891694	0.445847	−	0.895109i	$-0.352903\pi$
$61$	6.96435	0.891694	0.445847	+	0.895109i	$0.352903\pi$
$62$	2.15338i	0.273479i
$63$	1.01887i	0.128366i
$64$	8.59476	1.07435
$65$	0	0
$66$	9.60904	1.18279
$67$	9.91998i	1.21192i	0.795496	+	0.605959i	$0.207210\pi$
$67$	9.91998i	1.21192i	−0.795496	+	0.605959i	$0.792790\pi$
$68$	− 12.1183i	− 1.46955i
$69$	−9.07152	−1.09208
$70$	0	0
$71$	−10.7866	−1.28013	−0.640065	−	0.768320i	$-0.721093\pi$
$71$	−10.7866	−1.28013	−0.640065	+	0.768320i	$0.721093\pi$
$72$	0.153106i	0.0180437i
$73$	8.63115i	1.01020i	0.863061	+	0.505100i	$0.168544\pi$
$73$	8.63115i	1.01020i	−0.863061	+	0.505100i	$0.831456\pi$
$74$	8.12840	0.944907
$75$	0	0
$76$	−1.50375	−0.172491
$77$	− 4.84943i	− 0.552644i
$78$	− 0.209634i	− 0.0237363i
$79$	−2.48291	−0.279350	−0.139675	−	0.990197i	$-0.544606\pi$
$79$	−2.48291	−0.279350	−0.139675	+	0.990197i	$0.544606\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	14.5556i	1.60740i
$83$	4.24385i	0.465823i	0.972498	+	0.232911i	$0.0748252\pi$
$83$	4.24385i	0.465823i	−0.972498	+	0.232911i	$0.925175\pi$
$84$	2.11501	0.230766
$85$	0	0
$86$	−17.4186	−1.87830
$87$	− 3.98847i	− 0.427609i
$88$	− 0.728724i	− 0.0776823i
$89$	18.3752	1.94777	0.973884	−	0.227047i	$-0.0729072\pi$
$89$	18.3752	1.94777	0.973884	+	0.227047i	$0.0729072\pi$
$90$	0	0
$91$	−0.105797	−0.0110905
$92$	18.8310i	1.96327i
$93$	− 1.06662i	− 0.110604i
$94$	−16.5506	−1.70707
$95$	0	0
$96$	−8.06387	−0.823015
$97$	− 6.69876i	− 0.680156i	−0.940397	−	0.340078i	$-0.889546\pi$
$97$	− 6.69876i	− 0.680156i	0.940397	−	0.340078i	$-0.110454\pi$
$98$	12.0363i	1.21585i
$99$	−4.75961	−0.478359
$100$	0	0
$101$	5.66147	0.563337	0.281669	−	0.959512i	$-0.409112\pi$
$101$	5.66147	0.563337	0.281669	+	0.959512i	$0.409112\pi$
$102$	11.7857i	1.16696i
$103$	0.594489i	0.0585767i	0.999571	+	0.0292884i	$0.00932411\pi$
$103$	0.594489i	0.0585767i	−0.999571	+	0.0292884i	$0.990676\pi$
$104$	−0.0158981	−0.00155893
$105$	0	0
$106$	8.81680	0.856363
$107$	− 1.38651i	− 0.134039i	−0.997752	−	0.0670193i	$-0.978651\pi$
$107$	− 1.38651i	− 0.134039i	0.997752	−	0.0670193i	$-0.0213489\pi$
$108$	− 2.07584i	− 0.199748i
$109$	8.85677	0.848325	0.424162	−	0.905586i	$-0.360569\pi$
$109$	8.85677	0.848325	0.424162	+	0.905586i	$0.360569\pi$
$110$	0	0
$111$	−4.02621	−0.382151
$112$	3.91508i	0.369941i
$113$	6.59039i	0.619971i	0.950741	+	0.309986i	$0.100324\pi$
$113$	6.59039i	0.619971i	−0.950741	+	0.309986i	$0.899676\pi$
$114$	1.46248	0.136974
$115$	0	0
$116$	−8.27942	−0.768725
$117$	0.103837i	0.00959975i
$118$	9.91841i	0.913064i
$119$	5.94793	0.545245
$120$	0	0
$121$	11.6539	1.05945
$122$	14.0601i	1.27294i
$123$	− 7.20977i	− 0.650083i
$124$	−2.21414	−0.198836
$125$	0	0
$126$	−2.05697	−0.183249
$127$	7.29144i	0.647011i	0.946226	+	0.323505i	$0.104861\pi$
$127$	7.29144i	0.647011i	−0.946226	+	0.323505i	$0.895139\pi$
$128$	1.22397i	0.108184i
$129$	8.62791	0.759645
$130$	0	0
$131$	−9.55386	−0.834725	−0.417362	−	0.908740i	$-0.637045\pi$
$131$	−9.55386	−0.834725	−0.417362	+	0.908740i	$0.637045\pi$
$132$	9.88018i	0.859959i
$133$	− 0.738074i	− 0.0639991i
$134$	−20.0271	−1.73008
$135$	0	0
$136$	0.893796	0.0766424
$137$	6.19984i	0.529688i	0.964291	+	0.264844i	$0.0853205\pi$
$137$	6.19984i	0.529688i	−0.964291	+	0.264844i	$0.914680\pi$
$138$	− 18.3142i	− 1.55901i
$139$	0.906531	0.0768910	0.0384455	−	0.999261i	$-0.487759\pi$
$139$	0.906531	0.0768910	0.0384455	+	0.999261i	$0.487759\pi$
$140$	0	0
$141$	8.19797	0.690394
$142$	− 21.7767i	− 1.82746i
$143$	− 0.494225i	− 0.0413291i
$144$	3.84257	0.320214
$145$	0	0
$146$	−17.4252	−1.44212
$147$	− 5.96190i	− 0.491729i
$148$	8.35776i	0.687004i
$149$	4.89808	0.401267	0.200633	−	0.979666i	$-0.435700\pi$
$149$	4.89808	0.401267	0.200633	+	0.979666i	$0.435700\pi$
$150$	0	0
$151$	−10.2626	−0.835161	−0.417581	−	0.908640i	$-0.637122\pi$
$151$	−10.2626	−0.835161	−0.417581	+	0.908640i	$0.637122\pi$
$152$	− 0.110911i	− 0.00899603i
$153$	− 5.83776i	− 0.471955i
$154$	9.79036	0.788930
$155$	0	0
$156$	0.215549	0.0172577
$157$	8.89537i	0.709928i	0.934880	+	0.354964i	$0.115507\pi$
$157$	8.89537i	0.709928i	−0.934880	+	0.354964i	$0.884493\pi$
$158$	− 5.01268i	− 0.398788i
$159$	−4.36719	−0.346341
$160$	0	0
$161$	−9.24270	−0.728427
$162$	2.01887i	0.158617i
$163$	11.7791i	0.922607i	0.887242	+	0.461304i	$0.152618\pi$
$163$	11.7791i	0.922607i	−0.887242	+	0.461304i	$0.847382\pi$
$164$	−14.9663	−1.16867
$165$	0	0
$166$	−8.56778	−0.664989
$167$	− 17.3762i	− 1.34461i	−0.740274	−	0.672306i	$-0.765304\pi$
$167$	− 17.3762i	− 1.34461i	0.740274	−	0.672306i	$-0.234696\pi$
$168$	0.155995i	0.0120353i
$169$	12.9892	0.999171
$170$	0	0
$171$	−0.724404	−0.0553966
$172$	− 17.9101i	− 1.36564i
$173$	− 12.9596i	− 0.985298i	−0.870228	−	0.492649i	$-0.836029\pi$
$173$	− 12.9596i	− 0.985298i	0.870228	−	0.492649i	$-0.163971\pi$
$174$	8.05221	0.610436
$175$	0	0
$176$	−18.2892	−1.37860
$177$	− 4.91285i	− 0.369273i
$178$	37.0971i	2.78055i
$179$	−1.09897	−0.0821409	−0.0410704	−	0.999156i	$-0.513077\pi$
$179$	−1.09897	−0.0821409	−0.0410704	+	0.999156i	$0.513077\pi$
$180$	0	0
$181$	−14.9797	−1.11343	−0.556716	−	0.830703i	$-0.687939\pi$
$181$	−14.9797	−1.11343	−0.556716	+	0.830703i	$0.687939\pi$
$182$	− 0.213590i	− 0.0158323i
$183$	− 6.96435i	− 0.514820i
$184$	−1.38890	−0.102391
$185$	0	0
$186$	2.15338	0.157893
$187$	27.7855i	2.03188i
$188$	− 17.0177i	− 1.24114i
$189$	1.01887	0.0741119
$190$	0	0
$191$	−12.7404	−0.921862	−0.460931	−	0.887436i	$-0.652484\pi$
$191$	−12.7404	−0.921862	−0.460931	+	0.887436i	$0.652484\pi$
$192$	− 8.59476i	− 0.620273i
$193$	− 4.38386i	− 0.315557i	−0.987475	−	0.157779i	$-0.949567\pi$
$193$	− 4.38386i	− 0.315557i	0.987475	−	0.157779i	$-0.0504332\pi$
$194$	13.5239	0.970962
$195$	0	0
$196$	−12.3759	−0.883996
$197$	− 3.22165i	− 0.229533i	−0.993392	−	0.114767i	$-0.963388\pi$
$197$	− 3.22165i	− 0.229533i	0.993392	−	0.114767i	$-0.0366120\pi$
$198$	− 9.60904i	− 0.682885i
$199$	2.70518	0.191765	0.0958825	−	0.995393i	$-0.469433\pi$
$199$	2.70518	0.191765	0.0958825	+	0.995393i	$0.469433\pi$
$200$	0	0
$201$	9.91998	0.699701
$202$	11.4298i	0.804196i
$203$	− 4.06374i	− 0.285218i
$204$	−12.1183	−0.848447
$205$	0	0
$206$	−1.20020	−0.0836216
$207$	9.07152i	0.630514i
$208$	0.399002i	0.0276658i
$209$	3.44788	0.238495
$210$	0	0
$211$	26.4435	1.82044	0.910222	−	0.414122i	$-0.135911\pi$
$211$	26.4435	1.82044	0.910222	+	0.414122i	$0.135911\pi$
$212$	9.06559i	0.622627i
$213$	10.7866i	0.739084i
$214$	2.79918	0.191348
$215$	0	0
$216$	0.153106	0.0104175
$217$	− 1.08675i	− 0.0737736i
$218$	17.8807i	1.21103i
$219$	8.63115	0.583239
$220$	0	0
$221$	0.606177	0.0407759
$222$	− 8.12840i	− 0.545543i
$223$	− 8.03245i	− 0.537893i	−0.963155	−	0.268946i	$-0.913325\pi$
$223$	− 8.03245i	− 0.537893i	0.963155	−	0.268946i	$-0.0866755\pi$
$224$	−8.21604	−0.548957
$225$	0	0
$226$	−13.3051	−0.885044
$227$	− 16.4576i	− 1.09233i	−0.837677	−	0.546166i	$-0.816087\pi$
$227$	− 16.4576i	− 1.09233i	0.837677	−	0.546166i	$-0.183913\pi$
$228$	1.50375i	0.0995880i
$229$	−0.409285	−0.0270463	−0.0135231	−	0.999909i	$-0.504305\pi$
$229$	−0.409285	−0.0270463	−0.0135231	+	0.999909i	$0.504305\pi$
$230$	0	0
$231$	−4.84943	−0.319069
$232$	− 0.610658i	− 0.0400917i
$233$	− 18.4641i	− 1.20962i	−0.796369	−	0.604811i	$-0.793249\pi$
$233$	− 18.4641i	− 1.20962i	0.796369	−	0.604811i	$-0.206751\pi$
$234$	−0.209634	−0.0137042
$235$	0	0
$236$	−10.1983	−0.663852
$237$	2.48291i	0.161283i
$238$	12.0081i	0.778369i
$239$	4.61682	0.298637	0.149319	−	0.988789i	$-0.452292\pi$
$239$	4.61682	0.298637	0.149319	+	0.988789i	$0.452292\pi$
$240$	0	0
$241$	−29.2022	−1.88108	−0.940540	−	0.339682i	$-0.889681\pi$
$241$	−29.2022	−1.88108	−0.940540	+	0.339682i	$0.889681\pi$
$242$	23.5277i	1.51242i
$243$	− 1.00000i	− 0.0641500i
$244$	−14.4569	−0.925506
$245$	0	0
$246$	14.5556	0.928030
$247$	− 0.0752201i	− 0.00478614i
$248$	− 0.163307i	− 0.0103700i
$249$	4.24385	0.268943
$250$	0	0
$251$	0.389664	0.0245954	0.0122977	−	0.999924i	$-0.496085\pi$
$251$	0.389664	0.0245954	0.0122977	+	0.999924i	$0.496085\pi$
$252$	− 2.11501i	− 0.133233i
$253$	− 43.1769i	− 2.71451i
$254$	−14.7205	−0.923645
$255$	0	0
$256$	14.7185	0.919906
$257$	− 8.03324i	− 0.501100i	−0.968104	−	0.250550i	$-0.919389\pi$
$257$	− 8.03324i	− 0.501100i	0.968104	−	0.250550i	$-0.0806114\pi$
$258$	17.4186i	1.08444i
$259$	−4.10219	−0.254898
$260$	0	0
$261$	−3.98847	−0.246880
$262$	− 19.2880i	− 1.19162i
$263$	− 24.2131i	− 1.49304i	−0.665360	−	0.746522i	$-0.731722\pi$
$263$	− 24.2131i	− 1.49304i	0.665360	−	0.746522i	$-0.268278\pi$
$264$	−0.728724	−0.0448499
$265$	0	0
$266$	1.49008	0.0913624
$267$	− 18.3752i	− 1.12454i
$268$	− 20.5923i	− 1.25787i
$269$	−26.4063	−1.61002	−0.805009	−	0.593263i	$-0.797839\pi$
$269$	−26.4063	−1.61002	−0.805009	+	0.593263i	$0.797839\pi$
$270$	0	0
$271$	−10.4088	−0.632287	−0.316144	−	0.948711i	$-0.602388\pi$
$271$	−10.4088	−0.632287	−0.316144	+	0.948711i	$0.602388\pi$
$272$	− 22.4320i	− 1.36014i
$273$	0.105797i	0.00640310i
$274$	−12.5167	−0.756160
$275$	0	0
$276$	18.8310	1.13349
$277$	24.5207i	1.47330i	0.676272	+	0.736652i	$0.263595\pi$
$277$	24.5207i	1.47330i	−0.676272	+	0.736652i	$0.736405\pi$
$278$	1.83017i	0.109766i
$279$	−1.06662	−0.0638572
$280$	0	0
$281$	−14.4228	−0.860393	−0.430197	−	0.902735i	$-0.641556\pi$
$281$	−14.4228	−0.860393	−0.430197	+	0.902735i	$0.641556\pi$
$282$	16.5506i	0.985576i
$283$	2.14925i	0.127760i	0.997958	+	0.0638798i	$0.0203474\pi$
$283$	2.14925i	0.127760i	−0.997958	+	0.0638798i	$0.979653\pi$
$284$	22.3912	1.32867
$285$	0	0
$286$	0.997775	0.0589997
$287$	− 7.34582i	− 0.433610i
$288$	8.06387i	0.475168i
$289$	−17.0795	−1.00468
$290$	0	0
$291$	−6.69876	−0.392688
$292$	− 17.9169i	− 1.04851i
$293$	− 9.02970i	− 0.527521i	−0.964588	−	0.263760i	$-0.915037\pi$
$293$	− 9.02970i	− 0.527521i	0.964588	−	0.263760i	$-0.0849628\pi$
$294$	12.0363	0.701972
$295$	0	0
$296$	−0.616437	−0.0358297
$297$	4.75961i	0.276181i
$298$	9.88860i	0.572831i
$299$	−0.941960	−0.0544750
$300$	0	0
$301$	8.79072	0.506689
$302$	− 20.7189i	− 1.19224i
$303$	− 5.66147i	− 0.325243i
$304$	−2.78358	−0.159649
$305$	0	0
$306$	11.7857	0.673743
$307$	− 5.03454i	− 0.287336i	−0.989626	−	0.143668i	$-0.954110\pi$
$307$	− 5.03454i	− 0.287336i	0.989626	−	0.143668i	$-0.0458898\pi$
$308$	10.0666i	0.573599i
$309$	0.594489	0.0338193
$310$	0	0
$311$	−4.89158	−0.277376	−0.138688	−	0.990336i	$-0.544289\pi$
$311$	−4.89158	−0.277376	−0.138688	+	0.990336i	$0.544289\pi$
$312$	0.0158981i	0 0.000900051i
$313$	− 3.17282i	− 0.179338i	−0.995972	−	0.0896691i	$-0.971419\pi$
$313$	− 3.17282i	− 0.179338i	0.995972	−	0.0896691i	$-0.0285810\pi$
$314$	−17.9586	−1.01346
$315$	0	0
$316$	5.15413	0.289942
$317$	19.9953i	1.12305i	0.827461	+	0.561524i	$0.189785\pi$
$317$	19.9953i	1.12305i	−0.827461	+	0.561524i	$0.810215\pi$
$318$	− 8.81680i	− 0.494422i
$319$	18.9836	1.06288
$320$	0	0
$321$	−1.38651	−0.0773872
$322$	− 18.6598i	− 1.03987i
$323$	4.22890i	0.235302i
$324$	−2.07584	−0.115324
$325$	0	0
$326$	−23.7804	−1.31707
$327$	− 8.85677i	− 0.489780i
$328$	− 1.10386i	− 0.0609503i
$329$	8.35267	0.460498
$330$	0	0
$331$	6.01724	0.330738	0.165369	−	0.986232i	$-0.447119\pi$
$331$	6.01724	0.330738	0.165369	+	0.986232i	$0.447119\pi$
$332$	− 8.80954i	− 0.483486i
$333$	4.02621i	0.220635i
$334$	35.0803	1.91951
$335$	0	0
$336$	3.91508	0.213585
$337$	− 22.8136i	− 1.24274i	−0.783518	−	0.621369i	$-0.786577\pi$
$337$	− 22.8136i	− 1.24274i	0.783518	−	0.621369i	$-0.213423\pi$
$338$	26.2235i	1.42637i
$339$	6.59039	0.357941
$340$	0	0
$341$	5.07672	0.274920
$342$	− 1.46248i	− 0.0790818i
$343$	− 13.2065i	− 0.713084i
$344$	1.32098	0.0712227
$345$	0	0
$346$	26.1637	1.40657
$347$	− 9.79068i	− 0.525591i	−0.964852	−	0.262796i	$-0.915356\pi$
$347$	− 9.79068i	− 0.525591i	0.964852	−	0.262796i	$-0.0846445\pi$
$348$	8.27942i	0.443823i
$349$	1.28648	0.0688639	0.0344320	−	0.999407i	$-0.489038\pi$
$349$	1.28648	0.0688639	0.0344320	+	0.999407i	$0.489038\pi$
$350$	0	0
$351$	0.103837	0.00554242
$352$	− 38.3809i	− 2.04571i
$353$	1.16422i	0.0619654i	0.999520	+	0.0309827i	$0.00986368\pi$
$353$	1.16422i	0.0619654i	−0.999520	+	0.0309827i	$0.990136\pi$
$354$	9.91841	0.527158
$355$	0	0
$356$	−38.1439	−2.02162
$357$	− 5.94793i	− 0.314798i
$358$	− 2.21868i	− 0.117261i
$359$	−19.0504	−1.00544	−0.502720	−	0.864449i	$-0.667667\pi$
$359$	−19.0504	−1.00544	−0.502720	+	0.864449i	$0.667667\pi$
$360$	0	0
$361$	−18.4752	−0.972381
$362$	− 30.2421i	− 1.58949i
$363$	− 11.6539i	− 0.611671i
$364$	0.219617	0.0115110
$365$	0	0
$366$	14.0601	0.734935
$367$	− 16.7694i	− 0.875357i	−0.899131	−	0.437679i	$-0.855801\pi$
$367$	− 16.7694i	− 0.875357i	0.899131	−	0.437679i	$-0.144199\pi$
$368$	34.8580i	1.81710i
$369$	−7.20977	−0.375326
$370$	0	0
$371$	−4.44960	−0.231012
$372$	2.21414i	0.114798i
$373$	− 17.0520i	− 0.882920i	−0.897281	−	0.441460i	$-0.854461\pi$
$373$	− 17.0520i	− 0.882920i	0.897281	−	0.441460i	$-0.145539\pi$
$374$	−56.0953	−2.90062
$375$	0	0
$376$	1.25516	0.0647298
$377$	− 0.414152i	− 0.0213299i
$378$	2.05697i	0.105799i
$379$	−1.66611	−0.0855825	−0.0427913	−	0.999084i	$-0.513625\pi$
$379$	−1.66611	−0.0855825	−0.0427913	+	0.999084i	$0.513625\pi$
$380$	0	0
$381$	7.29144	0.373552
$382$	− 25.7212i	− 1.31601i
$383$	3.05361i	0.156032i	0.996952	+	0.0780162i	$0.0248586\pi$
$383$	3.05361i	0.156032i	−0.996952	+	0.0780162i	$0.975141\pi$
$384$	1.22397	0.0624603
$385$	0	0
$386$	8.85045	0.450476
$387$	− 8.62791i	− 0.438581i
$388$	13.9055i	0.705947i
$389$	−28.2725	−1.43347	−0.716737	−	0.697343i	$-0.754365\pi$
$389$	−28.2725	−1.43347	−0.716737	+	0.697343i	$0.754365\pi$
$390$	0	0
$391$	52.9574	2.67817
$392$	− 0.912802i	− 0.0461035i
$393$	9.55386i	0.481928i
$394$	6.50410	0.327672
$395$	0	0
$396$	9.88018	0.496498
$397$	20.4783i	1.02778i	0.857858	+	0.513888i	$0.171795\pi$
$397$	20.4783i	1.02778i	−0.857858	+	0.513888i	$0.828205\pi$
$398$	5.46140i	0.273755i
$399$	−0.738074	−0.0369499
$400$	0	0
$401$	25.5952	1.27816	0.639081	−	0.769139i	$-0.279315\pi$
$401$	25.5952	1.27816	0.639081	+	0.769139i	$0.279315\pi$
$402$	20.0271i	0.998863i
$403$	− 0.110755i	− 0.00551711i
$404$	−11.7523	−0.584698
$405$	0	0
$406$	8.20416	0.407165
$407$	− 19.1632i	− 0.949885i
$408$	− 0.893796i	− 0.0442495i
$409$	12.0402	0.595349	0.297675	−	0.954667i	$-0.403789\pi$
$409$	12.0402	0.595349	0.297675	+	0.954667i	$0.403789\pi$
$410$	0	0
$411$	6.19984	0.305816
$412$	− 1.23406i	− 0.0607979i
$413$	− 5.00556i	− 0.246307i
$414$	−18.3142	−0.900095
$415$	0	0
$416$	−0.837329	−0.0410534
$417$	− 0.906531i	− 0.0443930i
$418$	6.96083i	0.340465i
$419$	34.6045	1.69054	0.845270	−	0.534340i	$-0.179440\pi$
$419$	34.6045	1.69054	0.845270	+	0.534340i	$0.179440\pi$
$420$	0	0
$421$	−14.9525	−0.728738	−0.364369	−	0.931255i	$-0.618715\pi$
$421$	−14.9525	−0.728738	−0.364369	+	0.931255i	$0.618715\pi$
$422$	53.3859i	2.59879i
$423$	− 8.19797i	− 0.398599i
$424$	−0.668643	−0.0324722
$425$	0	0
$426$	−21.7767	−1.05508
$427$	− 7.09577i	− 0.343389i
$428$	2.87816i	0.139121i
$429$	−0.494225	−0.0238614
$430$	0	0
$431$	4.81107	0.231741	0.115871	−	0.993264i	$-0.463034\pi$
$431$	4.81107	0.231741	0.115871	+	0.993264i	$0.463034\pi$
$432$	− 3.84257i	− 0.184876i
$433$	10.3435i	0.497077i	0.968622	+	0.248538i	$0.0799502\pi$
$433$	10.3435i	0.497077i	−0.968622	+	0.248538i	$0.920050\pi$
$434$	2.19401	0.105316
$435$	0	0
$436$	−18.3852	−0.880492
$437$	− 6.57145i	− 0.314355i
$438$	17.4252i	0.832607i
$439$	−30.7640	−1.46829	−0.734143	−	0.678995i	$-0.762416\pi$
$439$	−30.7640	−1.46829	−0.734143	+	0.678995i	$0.762416\pi$
$440$	0	0
$441$	−5.96190	−0.283900
$442$	1.22379i	0.0582099i
$443$	− 17.7545i	− 0.843543i	−0.906702	−	0.421772i	$-0.861408\pi$
$443$	− 17.7545i	− 0.843543i	0.906702	−	0.421772i	$-0.138592\pi$
$444$	8.35776	0.396642
$445$	0	0
$446$	16.2165	0.767873
$447$	− 4.89808i	− 0.231671i
$448$	− 8.75695i	− 0.413727i
$449$	37.2184	1.75645	0.878223	−	0.478251i	$-0.158729\pi$
$449$	37.2184	1.75645	0.878223	+	0.478251i	$0.158729\pi$
$450$	0	0
$451$	34.3157	1.61586
$452$	− 13.6806i	− 0.643480i
$453$	10.2626i	0.482181i
$454$	33.2258	1.55937
$455$	0	0
$456$	−0.110911	−0.00519386
$457$	27.6987i	1.29569i	0.761772	+	0.647846i	$0.224330\pi$
$457$	27.6987i	1.29569i	−0.761772	+	0.647846i	$0.775670\pi$
$458$	− 0.826293i	− 0.0386101i
$459$	−5.83776	−0.272484
$460$	0	0
$461$	9.81742	0.457243	0.228621	−	0.973515i	$-0.426578\pi$
$461$	9.81742	0.457243	0.228621	+	0.973515i	$0.426578\pi$
$462$	− 9.79036i	− 0.455489i
$463$	− 3.72732i	− 0.173223i	−0.996242	−	0.0866115i	$-0.972396\pi$
$463$	− 3.72732i	− 0.173223i	0.996242	−	0.0866115i	$-0.0276039\pi$
$464$	−15.3260	−0.711492
$465$	0	0
$466$	37.2766	1.72680
$467$	7.09925i	0.328514i	0.986418	+	0.164257i	$0.0525227\pi$
$467$	7.09925i	0.328514i	−0.986418	+	0.164257i	$0.947477\pi$
$468$	− 0.215549i	− 0.00996376i
$469$	10.1072	0.466706
$470$	0	0
$471$	8.89537	0.409877
$472$	− 0.752186i	− 0.0346222i
$473$	41.0655i	1.88819i
$474$	−5.01268	−0.230240
$475$	0	0
$476$	−12.3469	−0.565920
$477$	4.36719i	0.199960i
$478$	9.32076i	0.426322i
$479$	28.4451	1.29969	0.649845	−	0.760066i	$-0.274834\pi$
$479$	28.4451	1.29969	0.649845	+	0.760066i	$0.274834\pi$
$480$	0	0
$481$	−0.418070	−0.0190624
$482$	− 58.9555i	− 2.68535i
$483$	9.24270i	0.420557i
$484$	−24.1916	−1.09962
$485$	0	0
$486$	2.01887	0.0915778
$487$	14.7384i	0.667860i	0.942598	+	0.333930i	$0.108375\pi$
$487$	14.7384i	0.667860i	−0.942598	+	0.333930i	$0.891625\pi$
$488$	− 1.06628i	− 0.0482684i
$489$	11.7791	0.532668
$490$	0	0
$491$	−28.4014	−1.28174	−0.640869	−	0.767650i	$-0.721426\pi$
$491$	−28.4014	−1.28174	−0.640869	+	0.767650i	$0.721426\pi$
$492$	14.9663i	0.674733i
$493$	23.2838i	1.04865i
$494$	0.151860	0.00683248
$495$	0	0
$496$	−4.09858	−0.184032
$497$	10.9901i	0.492974i
$498$	8.56778i	0.383931i
$499$	−26.3842	−1.18112	−0.590559	−	0.806995i	$-0.701093\pi$
$499$	−26.3842	−1.18112	−0.590559	+	0.806995i	$0.701093\pi$
$500$	0	0
$501$	−17.3762	−0.776312
$502$	0.786682i	0.0351113i
$503$	16.6592i	0.742795i	0.928474	+	0.371398i	$0.121121\pi$
$503$	16.6592i	0.742795i	−0.928474	+	0.371398i	$0.878879\pi$
$504$	0.155995	0.00694857
$505$	0	0
$506$	87.1686	3.87512
$507$	− 12.9892i	− 0.576871i
$508$	− 15.1358i	− 0.671544i
$509$	31.7760	1.40845	0.704223	−	0.709979i	$-0.251296\pi$
$509$	31.7760	1.40845	0.704223	+	0.709979i	$0.251296\pi$
$510$	0	0
$511$	8.79403	0.389025
$512$	32.1627i	1.42140i
$513$	0.724404i	0.0319832i
$514$	16.2181	0.715349
$515$	0	0
$516$	−17.9101	−0.788450
$517$	39.0192i	1.71606i
$518$	− 8.28179i	− 0.363881i
$519$	−12.9596	−0.568862
$520$	0	0
$521$	−0.592363	−0.0259519	−0.0129759	−	0.999916i	$-0.504130\pi$
$521$	−0.592363	−0.0259519	−0.0129759	+	0.999916i	$0.504130\pi$
$522$	− 8.05221i	− 0.352436i
$523$	− 17.2298i	− 0.753405i	−0.926334	−	0.376703i	$-0.877058\pi$
$523$	− 17.2298i	− 0.753405i	0.926334	−	0.376703i	$-0.122942\pi$
$524$	19.8323	0.866376
$525$	0	0
$526$	48.8832	2.13141
$527$	6.22670i	0.271240i
$528$	18.2892i	0.795934i
$529$	−59.2924	−2.57793
$530$	0	0
$531$	−4.91285	−0.213200
$532$	1.53212i	0.0664259i
$533$	− 0.748642i	− 0.0324273i
$534$	37.0971	1.60535
$535$	0	0
$536$	1.51881	0.0656025
$537$	1.09897i	0.0474241i
$538$	− 53.3108i	− 2.29839i
$539$	28.3763	1.22226
$540$	0	0
$541$	14.7990	0.636258	0.318129	−	0.948047i	$-0.396945\pi$
$541$	14.7990	0.636258	0.318129	+	0.948047i	$0.396945\pi$
$542$	− 21.0139i	− 0.902626i
$543$	14.9797i	0.642840i
$544$	47.0750	2.01832
$545$	0	0
$546$	−0.213590	−0.00914079
$547$	− 6.50806i	− 0.278264i	−0.990274	−	0.139132i	$-0.955569\pi$
$547$	− 6.50806i	− 0.278264i	0.990274	−	0.139132i	$-0.0444313\pi$
$548$	− 12.8699i	− 0.549773i
$549$	−6.96435	−0.297231
$550$	0	0
$551$	2.88927	0.123087
$552$	1.38890i	0.0591156i
$553$	2.52977i	0.107577i
$554$	−49.5041	−2.10323
$555$	0	0
$556$	−1.88181	−0.0798066
$557$	6.67224i	0.282712i	0.989959	+	0.141356i	$0.0451462\pi$
$557$	6.67224i	0.282712i	−0.989959	+	0.141356i	$0.954854\pi$
$558$	− 2.15338i	− 0.0911597i
$559$	0.895898	0.0378924
$560$	0	0
$561$	27.7855	1.17310
$562$	− 29.1178i	− 1.22826i
$563$	20.0663i	0.845694i	0.906201	+	0.422847i	$0.138969\pi$
$563$	20.0663i	0.845694i	−0.906201	+	0.422847i	$0.861031\pi$
$564$	−17.0177	−0.716573
$565$	0	0
$566$	−4.33906	−0.182384
$567$	− 1.01887i	− 0.0427885i
$568$	1.65149i	0.0692949i
$569$	21.5938	0.905261	0.452631	−	0.891698i	$-0.350486\pi$
$569$	21.5938	0.905261	0.452631	+	0.891698i	$0.350486\pi$
$570$	0	0
$571$	−26.3338	−1.10203	−0.551017	−	0.834494i	$-0.685760\pi$
$571$	−26.3338	−1.10203	−0.551017	+	0.834494i	$0.685760\pi$
$572$	1.02593i	0.0428963i
$573$	12.7404i	0.532237i
$574$	14.8303	0.619003
$575$	0	0
$576$	−8.59476	−0.358115
$577$	9.18240i	0.382268i	0.981564	+	0.191134i	$0.0612165\pi$
$577$	9.18240i	0.382268i	−0.981564	+	0.191134i	$0.938783\pi$
$578$	− 34.4813i	− 1.43423i
$579$	−4.38386	−0.182187
$580$	0	0
$581$	4.32393	0.179387
$582$	− 13.5239i	− 0.560585i
$583$	− 20.7861i	− 0.860874i
$584$	1.32148	0.0546832
$585$	0	0
$586$	18.2298	0.753066
$587$	− 39.9771i	− 1.65003i	−0.565108	−	0.825017i	$-0.691166\pi$
$587$	− 39.9771i	− 1.65003i	0.565108	−	0.825017i	$-0.308834\pi$
$588$	12.3759i	0.510375i
$589$	0.772668	0.0318372
$590$	0	0
$591$	−3.22165	−0.132521
$592$	15.4710i	0.635855i
$593$	− 41.6331i	− 1.70967i	−0.518902	−	0.854834i	$-0.673659\pi$
$593$	− 41.6331i	− 1.70967i	0.518902	−	0.854834i	$-0.326341\pi$
$594$	−9.60904	−0.394264
$595$	0	0
$596$	−10.1676	−0.416482
$597$	− 2.70518i	− 0.110716i
$598$	− 1.90170i	− 0.0777661i
$599$	−39.0726	−1.59646	−0.798232	−	0.602350i	$-0.794231\pi$
$599$	−39.0726	−1.59646	−0.798232	+	0.602350i	$0.794231\pi$
$600$	0	0
$601$	5.46965	0.223112	0.111556	−	0.993758i	$-0.464417\pi$
$601$	5.46965	0.223112	0.111556	+	0.993758i	$0.464417\pi$
$602$	17.7473i	0.723327i
$603$	− 9.91998i	− 0.403973i
$604$	21.3036	0.866829
$605$	0	0
$606$	11.4298	0.464303
$607$	− 42.3108i	− 1.71734i	−0.512527	−	0.858671i	$-0.671291\pi$
$607$	− 42.3108i	− 1.71734i	0.512527	−	0.858671i	$-0.328709\pi$
$608$	− 5.84150i	− 0.236904i
$609$	−4.06374	−0.164671
$610$	0	0
$611$	0.851254	0.0344380
$612$	12.1183i	0.489851i
$613$	− 34.0064i	− 1.37351i	−0.726891	−	0.686753i	$-0.759035\pi$
$613$	− 34.0064i	− 1.37351i	0.726891	−	0.686753i	$-0.240965\pi$
$614$	10.1641	0.410189
$615$	0	0
$616$	−0.742476	−0.0299152
$617$	− 23.3173i	− 0.938717i	−0.883008	−	0.469359i	$-0.844485\pi$
$617$	− 23.3173i	− 0.938717i	0.883008	−	0.469359i	$-0.155515\pi$
$618$	1.20020i	0.0482790i
$619$	−35.2998	−1.41882	−0.709409	−	0.704797i	$-0.751038\pi$
$619$	−35.2998	−1.41882	−0.709409	+	0.704797i	$0.751038\pi$
$620$	0	0
$621$	9.07152	0.364027
$622$	− 9.87547i	− 0.395970i
$623$	− 18.7219i	− 0.750079i
$624$	0.399002	0.0159729
$625$	0	0
$626$	6.40551	0.256016
$627$	− 3.44788i	− 0.137695i
$628$	− 18.4653i	− 0.736847i
$629$	23.5041	0.937169
$630$	0	0
$631$	−3.50433	−0.139505	−0.0697526	−	0.997564i	$-0.522221\pi$
$631$	−3.50433	−0.139505	−0.0697526	+	0.997564i	$0.522221\pi$
$632$	0.380149i	0.0151215i
$633$	− 26.4435i	− 1.05103i
$634$	−40.3679	−1.60321
$635$	0	0
$636$	9.06559	0.359474
$637$	− 0.619067i	− 0.0245283i
$638$	38.3254i	1.51732i
$639$	10.7866	0.426710
$640$	0	0
$641$	−9.28841	−0.366870	−0.183435	−	0.983032i	$-0.558722\pi$
$641$	−9.28841	−0.366870	−0.183435	+	0.983032i	$0.558722\pi$
$642$	− 2.79918i	− 0.110475i
$643$	− 0.0291680i	− 0.00115028i	−1.00000		0.000575138i	$-0.999817\pi$
$643$	− 0.0291680i	− 0.00115028i	1.00000		0.000575138i	$-0.000183072\pi$
$644$	19.1863	0.756048
$645$	0	0
$646$	−8.53760	−0.335908
$647$	0.844100i	0.0331850i	0.999862	+	0.0165925i	$0.00528180\pi$
$647$	0.844100i	0.0331850i	−0.999862	+	0.0165925i	$0.994718\pi$
$648$	− 0.153106i	− 0.00601457i
$649$	23.3833	0.917874
$650$	0	0
$651$	−1.08675	−0.0425932
$652$	− 24.4514i	− 0.957591i
$653$	38.0711i	1.48984i	0.667156	+	0.744918i	$0.267511\pi$
$653$	38.0711i	1.48984i	−0.667156	+	0.744918i	$0.732489\pi$
$654$	17.8807	0.699189
$655$	0	0
$656$	−27.7041	−1.08166
$657$	− 8.63115i	− 0.336733i
$658$	16.8630i	0.657387i
$659$	−37.8343	−1.47382	−0.736908	−	0.675993i	$-0.763715\pi$
$659$	−37.8343	−1.47382	−0.736908	+	0.675993i	$0.763715\pi$
$660$	0	0
$661$	−18.5614	−0.721956	−0.360978	−	0.932574i	$-0.617557\pi$
$661$	−18.5614	−0.721956	−0.360978	+	0.932574i	$0.617557\pi$
$662$	12.1480i	0.472147i
$663$	− 0.606177i	− 0.0235420i
$664$	0.649758	0.0252155
$665$	0	0
$666$	−8.12840	−0.314969
$667$	− 36.1815i	− 1.40095i
$668$	36.0702i	1.39560i
$669$	−8.03245	−0.310553
$670$	0	0
$671$	33.1476	1.27965
$672$	8.21604i	0.316941i
$673$	18.1607i	0.700044i	0.936742	+	0.350022i	$0.113826\pi$
$673$	18.1607i	0.700044i	−0.936742	+	0.350022i	$0.886174\pi$
$674$	46.0578	1.77408
$675$	0	0
$676$	−26.9635	−1.03706
$677$	5.64312i	0.216883i	0.994103	+	0.108441i	$0.0345860\pi$
$677$	5.64312i	0.216883i	−0.994103	+	0.108441i	$0.965414\pi$
$678$	13.3051i	0.510981i
$679$	−6.82517	−0.261926
$680$	0	0
$681$	−16.4576	−0.630658
$682$	10.2492i	0.392464i
$683$	30.4468i	1.16502i	0.812825	+	0.582508i	$0.197929\pi$
$683$	30.4468i	1.16502i	−0.812825	+	0.582508i	$0.802071\pi$
$684$	1.50375	0.0574971
$685$	0	0
$686$	26.6622	1.01797
$687$	0.409285i	0.0156152i
$688$	− 33.1534i	− 1.26396i
$689$	−0.453477	−0.0172761
$690$	0	0
$691$	−9.22470	−0.350924	−0.175462	−	0.984486i	$-0.556142\pi$
$691$	−9.22470	−0.350924	−0.175462	+	0.984486i	$0.556142\pi$
$692$	26.9020i	1.02266i
$693$	4.84943i	0.184215i
$694$	19.7661	0.750311
$695$	0	0
$696$	−0.610658	−0.0231469
$697$	42.0889i	1.59423i
$698$	2.59724i	0.0983071i
$699$	−18.4641	−0.698375
$700$	0	0
$701$	−19.9822	−0.754717	−0.377358	−	0.926067i	$-0.623168\pi$
$701$	−19.9822	−0.754717	−0.377358	+	0.926067i	$0.623168\pi$
$702$	0.209634i	0.00791212i
$703$	− 2.91661i	− 0.110002i
$704$	40.9077	1.54177
$705$	0	0
$706$	−2.35042	−0.0884591
$707$	− 5.76830i	− 0.216939i
$708$	10.1983i	0.383275i
$709$	26.8259	1.00747	0.503734	−	0.863859i	$-0.331959\pi$
$709$	26.8259	1.00747	0.503734	+	0.863859i	$0.331959\pi$
$710$	0	0
$711$	2.48291	0.0931166
$712$	− 2.81335i	− 0.105435i
$713$	− 9.67591i	− 0.362366i
$714$	12.0081	0.449391
$715$	0	0
$716$	2.28128	0.0852556
$717$	− 4.61682i	− 0.172418i
$718$	− 38.4602i	− 1.43532i
$719$	−38.8224	−1.44783	−0.723915	−	0.689889i	$-0.757659\pi$
$719$	−38.8224	−1.44783	−0.723915	+	0.689889i	$0.757659\pi$
$720$	0	0
$721$	0.605707	0.0225577
$722$	− 37.2991i	− 1.38813i
$723$	29.2022i	1.08604i
$724$	31.0954	1.15565
$725$	0	0
$726$	23.5277	0.873196
$727$	− 29.5764i	− 1.09693i	−0.836174	−	0.548465i	$-0.815213\pi$
$727$	− 29.5764i	− 1.09693i	0.836174	−	0.548465i	$-0.184787\pi$
$728$	0.0161981i	0 0.000600341i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−50.3677	−1.86292
$732$	14.4569i	0.534341i
$733$	− 48.8782i	− 1.80536i	−0.430315	−	0.902679i	$-0.641598\pi$
$733$	− 48.8782i	− 1.80536i	0.430315	−	0.902679i	$-0.358402\pi$
$734$	33.8553	1.24962
$735$	0	0
$736$	−73.1515	−2.69640
$737$	47.2152i	1.73920i
$738$	− 14.5556i	− 0.535799i
$739$	−21.7603	−0.800466	−0.400233	−	0.916413i	$-0.631071\pi$
$739$	−21.7603	−0.800466	−0.400233	+	0.916413i	$0.631071\pi$
$740$	0	0
$741$	−0.0752201	−0.00276328
$742$	− 8.98317i	− 0.329783i
$743$	− 9.75724i	− 0.357959i	−0.983853	−	0.178979i	$-0.942720\pi$
$743$	− 9.75724i	− 0.357959i	0.983853	−	0.178979i	$-0.0572795\pi$
$744$	−0.163307	−0.00598711
$745$	0	0
$746$	34.4258	1.26042
$747$	− 4.24385i	− 0.155274i
$748$	− 57.6782i	− 2.10892i
$749$	−1.41267	−0.0516178
$750$	0	0
$751$	−17.6413	−0.643741	−0.321871	−	0.946784i	$-0.604312\pi$
$751$	−17.6413	−0.643741	−0.321871	+	0.946784i	$0.604312\pi$
$752$	− 31.5013i	− 1.14873i
$753$	− 0.389664i	− 0.0142002i
$754$	0.836118	0.0304496
$755$	0	0
$756$	−2.11501	−0.0769221
$757$	44.2551i	1.60848i	0.594305	+	0.804240i	$0.297427\pi$
$757$	44.2551i	1.60848i	−0.594305	+	0.804240i	$0.702573\pi$
$758$	− 3.36367i	− 0.122174i
$759$	−43.1769	−1.56722
$760$	0	0
$761$	39.8755	1.44549	0.722743	−	0.691117i	$-0.242881\pi$
$761$	39.8755	1.44549	0.722743	+	0.691117i	$0.242881\pi$
$762$	14.7205i	0.533266i
$763$	− 9.02390i	− 0.326687i
$764$	26.4470	0.956818
$765$	0	0
$766$	−6.16485	−0.222745
$767$	− 0.510137i	− 0.0184200i
$768$	− 14.7185i	− 0.531108i
$769$	−35.2667	−1.27175	−0.635876	−	0.771792i	$-0.719361\pi$
$769$	−35.2667	−1.27175	−0.635876	+	0.771792i	$0.719361\pi$
$770$	0	0
$771$	−8.03324	−0.289310
$772$	9.10018i	0.327523i
$773$	− 23.2417i	− 0.835946i	−0.908459	−	0.417973i	$-0.862741\pi$
$773$	− 23.2417i	− 0.835946i	0.908459	−	0.417973i	$-0.137259\pi$
$774$	17.4186	0.626100
$775$	0	0
$776$	−1.02562	−0.0368176
$777$	4.10219i	0.147165i
$778$	− 57.0786i	− 2.04637i
$779$	5.22279	0.187126
$780$	0	0
$781$	−51.3399	−1.83709
$782$	106.914i	3.82324i
$783$	3.98847i	0.142536i
$784$	−22.9091	−0.818180
$785$	0	0
$786$	−19.2880	−0.687980
$787$	− 6.83095i	− 0.243497i	−0.992561	−	0.121749i	$-0.961150\pi$
$787$	− 6.83095i	− 0.243497i	0.992561	−	0.121749i	$-0.0388502\pi$
$788$	6.68762i	0.238237i
$789$	−24.2131	−0.862010
$790$	0	0
$791$	6.71475	0.238749
$792$	0.728724i	0.0258941i
$793$	− 0.723159i	− 0.0256801i
$794$	−41.3430	−1.46721
$795$	0	0
$796$	−5.61551	−0.199036
$797$	− 53.9287i	− 1.91025i	−0.296198	−	0.955126i	$-0.595719\pi$
$797$	− 53.9287i	− 1.91025i	0.296198	−	0.955126i	$-0.404281\pi$
$798$	− 1.49008i	− 0.0527481i
$799$	−47.8578	−1.69309
$800$	0	0
$801$	−18.3752	−0.649256
$802$	51.6734i	1.82465i
$803$	41.0809i	1.44971i
$804$	−20.5923	−0.726233
$805$	0	0
$806$	0.223601	0.00787599
$807$	26.4063i	0.929544i
$808$	− 0.866804i	− 0.0304941i
$809$	18.4104	0.647275	0.323637	−	0.946181i	$-0.395094\pi$
$809$	18.4104	0.647275	0.323637	+	0.946181i	$0.395094\pi$
$810$	0	0
$811$	−11.7910	−0.414040	−0.207020	−	0.978337i	$-0.566376\pi$
$811$	−11.7910	−0.414040	−0.207020	+	0.978337i	$0.566376\pi$
$812$	8.43565i	0.296033i
$813$	10.4088i	0.365051i
$814$	38.6880	1.35601
$815$	0	0
$816$	−22.4320	−0.785279
$817$	6.25009i	0.218663i
$818$	24.3076i	0.849895i
$819$	0.105797	0.00369683
$820$	0	0
$821$	−12.3261	−0.430185	−0.215093	−	0.976594i	$-0.569005\pi$
$821$	−12.3261	−0.430185	−0.215093	+	0.976594i	$0.569005\pi$
$822$	12.5167i	0.436569i
$823$	2.81852i	0.0982475i	0.998793	+	0.0491237i	$0.0156429\pi$
$823$	2.81852i	0.0982475i	−0.998793	+	0.0491237i	$0.984357\pi$
$824$	0.0910197	0.00317082
$825$	0	0
$826$	10.1056	0.351618
$827$	37.4251i	1.30140i	0.759335	+	0.650700i	$0.225524\pi$
$827$	37.4251i	1.30140i	−0.759335	+	0.650700i	$0.774476\pi$
$828$	− 18.8310i	− 0.654422i
$829$	−30.9434	−1.07471	−0.537355	−	0.843356i	$-0.680576\pi$
$829$	−30.9434	−1.07471	−0.537355	+	0.843356i	$0.680576\pi$
$830$	0	0
$831$	24.5207	0.850613
$832$	− 0.892455i	− 0.0309403i
$833$	34.8042i	1.20589i
$834$	1.83017	0.0633736
$835$	0	0
$836$	−7.15724	−0.247538
$837$	1.06662i	0.0368679i
$838$	69.8620i	2.41334i
$839$	26.6975	0.921701	0.460851	−	0.887478i	$-0.347544\pi$
$839$	26.6975	0.921701	0.460851	+	0.887478i	$0.347544\pi$
$840$	0	0
$841$	−13.0921	−0.451451
$842$	− 30.1871i	− 1.04032i
$843$	14.4228i	0.496748i
$844$	−54.8923	−1.88947
$845$	0	0
$846$	16.5506	0.569023
$847$	− 11.8738i	− 0.407989i
$848$	16.7813i	0.576271i
$849$	2.14925	0.0737620
$850$	0	0
$851$	−36.5239	−1.25202
$852$	− 22.3912i	− 0.767109i
$853$	29.2600i	1.00184i	0.865493	+	0.500921i	$0.167005\pi$
$853$	29.2600i	1.00184i	−0.865493	+	0.500921i	$0.832995\pi$
$854$	14.3254	0.490207
$855$	0	0
$856$	−0.212282	−0.00725566
$857$	33.5284i	1.14531i	0.819797	+	0.572654i	$0.194086\pi$
$857$	33.5284i	1.14531i	−0.819797	+	0.572654i	$0.805914\pi$
$858$	− 0.997775i	− 0.0340635i
$859$	25.8243	0.881115	0.440557	−	0.897725i	$-0.354781\pi$
$859$	25.8243	0.881115	0.440557	+	0.897725i	$0.354781\pi$
$860$	0	0
$861$	−7.34582	−0.250345
$862$	9.71293i	0.330824i
$863$	− 28.8886i	− 0.983378i	−0.870771	−	0.491689i	$-0.836380\pi$
$863$	− 28.8886i	− 0.983378i	0.870771	−	0.491689i	$-0.163620\pi$
$864$	8.06387	0.274338
$865$	0	0
$866$	−20.8822	−0.709605
$867$	17.0795i	0.580050i
$868$	2.25592i	0.0765710i
$869$	−11.8177	−0.400888
$870$	0	0
$871$	1.03006	0.0349023
$872$	− 1.35602i	− 0.0459207i
$873$	6.69876i	0.226719i
$874$	13.2669	0.448759
$875$	0	0
$876$	−17.9169	−0.605355
$877$	15.1004i	0.509905i	0.966954	+	0.254953i	$0.0820598\pi$
$877$	15.1004i	0.509905i	−0.966954	+	0.254953i	$0.917940\pi$
$878$	− 62.1086i	− 2.09606i
$879$	−9.02970	−0.304564
$880$	0	0
$881$	36.6216	1.23381	0.616906	−	0.787037i	$-0.288386\pi$
$881$	36.6216	1.23381	0.616906	+	0.787037i	$0.288386\pi$
$882$	− 12.0363i	− 0.405284i
$883$	− 34.5459i	− 1.16256i	−0.813703	−	0.581281i	$-0.802552\pi$
$883$	− 34.5459i	− 1.16256i	0.813703	−	0.581281i	$-0.197448\pi$
$884$	−1.25832	−0.0423220
$885$	0	0
$886$	35.8441	1.20421
$887$	− 33.0360i	− 1.10924i	−0.832104	−	0.554620i	$-0.812864\pi$
$887$	− 33.0360i	− 1.10924i	0.832104	−	0.554620i	$-0.187136\pi$
$888$	0.616437i	0.0206863i
$889$	7.42903	0.249162
$890$	0	0
$891$	4.75961	0.159453
$892$	16.6741i	0.558289i
$893$	5.93864i	0.198729i
$894$	9.88860	0.330724
$895$	0	0
$896$	1.24706	0.0416614
$897$	0.941960i	0.0314511i
$898$	75.1392i	2.50743i
$899$	4.25420	0.141886
$900$	0	0
$901$	25.4947	0.849350
$902$	69.2789i	2.30674i
$903$	− 8.79072i	− 0.292537i
$904$	1.00903	0.0335597
$905$	0	0
$906$	−20.7189	−0.688340
$907$	44.2708i	1.46999i	0.678074	+	0.734994i	$0.262815\pi$
$907$	44.2708i	1.46999i	−0.678074	+	0.734994i	$0.737185\pi$
$908$	34.1634i	1.13375i
$909$	−5.66147	−0.187779
$910$	0	0
$911$	−38.8844	−1.28830	−0.644148	−	0.764901i	$-0.722788\pi$
$911$	−38.8844	−1.28830	−0.644148	+	0.764901i	$0.722788\pi$
$912$	2.78358i	0.0921734i
$913$	20.1991i	0.668492i
$914$	−55.9201	−1.84967
$915$	0	0
$916$	0.849608	0.0280719
$917$	9.73414i	0.321450i
$918$	− 11.7857i	− 0.388986i
$919$	28.5303	0.941127	0.470563	−	0.882366i	$-0.344051\pi$
$919$	28.5303	0.941127	0.470563	+	0.882366i	$0.344051\pi$
$920$	0	0
$921$	−5.03454	−0.165894
$922$	19.8201i	0.652740i
$923$	1.12005i	0.0368668i
$924$	10.0666	0.331168
$925$	0	0
$926$	7.52497	0.247286
$927$	− 0.594489i	− 0.0195256i
$928$	− 32.1625i	− 1.05579i
$929$	−36.3886	−1.19387	−0.596936	−	0.802289i	$-0.703615\pi$
$929$	−36.3886	−1.19387	−0.596936	+	0.802289i	$0.703615\pi$
$930$	0	0
$931$	4.31883	0.141544
$932$	38.3284i	1.25549i
$933$	4.89158i	0.160143i
$934$	−14.3325	−0.468973
$935$	0	0
$936$	0.0158981	0.000519645 0
$937$	47.3354i	1.54638i	0.634175	+	0.773189i	$0.281340\pi$
$937$	47.3354i	1.54638i	−0.634175	+	0.773189i	$0.718660\pi$
$938$	20.4051i	0.666249i
$939$	−3.17282	−0.103541
$940$	0	0
$941$	52.9254	1.72532	0.862659	−	0.505787i	$-0.168798\pi$
$941$	52.9254	1.72532	0.862659	+	0.505787i	$0.168798\pi$
$942$	17.9586i	0.585123i
$943$	− 65.4035i	− 2.12983i
$944$	−18.8780	−0.614426
$945$	0	0
$946$	−82.9059	−2.69550
$947$	− 18.9330i	− 0.615239i	−0.951509	−	0.307619i	$-0.900468\pi$
$947$	− 18.9330i	− 0.615239i	0.951509	−	0.307619i	$-0.0995324\pi$
$948$	− 5.15413i	− 0.167398i
$949$	0.896234	0.0290930
$950$	0	0
$951$	19.9953	0.648392
$952$	− 0.910662i	− 0.0295147i
$953$	− 11.4499i	− 0.370899i	−0.982654	−	0.185450i	$-0.940626\pi$
$953$	− 11.4499i	− 0.370899i	0.982654	−	0.185450i	$-0.0593742\pi$
$954$	−8.81680	−0.285454
$955$	0	0
$956$	−9.58377	−0.309961
$957$	− 18.9836i	− 0.613652i
$958$	57.4270i	1.85538i
$959$	6.31683	0.203981
$960$	0	0
$961$	−29.8623	−0.963300
$962$	− 0.844030i	− 0.0272126i
$963$	1.38651i	0.0446795i
$964$	60.6191	1.95241
$965$	0	0
$966$	−18.6598	−0.600370
$967$	36.9926i	1.18960i	0.803873	+	0.594801i	$0.202769\pi$
$967$	36.9926i	1.18960i	−0.803873	+	0.594801i	$0.797231\pi$
$968$	− 1.78428i	− 0.0573490i
$969$	4.22890	0.135852
$970$	0	0
$971$	−30.2897	−0.972043	−0.486022	−	0.873947i	$-0.661552\pi$
$971$	−30.2897	−0.972043	−0.486022	+	0.873947i	$0.661552\pi$
$972$	2.07584i	0.0665825i
$973$	− 0.923638i	− 0.0296105i
$974$	−29.7549	−0.953408
$975$	0	0
$976$	−26.7610	−0.856600
$977$	− 16.6410i	− 0.532394i	−0.963919	−	0.266197i	$-0.914233\pi$
$977$	− 16.6410i	− 0.532394i	0.963919	−	0.266197i	$-0.0857672\pi$
$978$	23.7804i	0.760413i
$979$	87.4588	2.79520
$980$	0	0
$981$	−8.85677	−0.282775
$982$	− 57.3388i	− 1.82975i
$983$	26.1897i	0.835320i	0.908603	+	0.417660i	$0.137150\pi$
$983$	26.1897i	0.835320i	−0.908603	+	0.417660i	$0.862850\pi$
$984$	−1.10386	−0.0351897
$985$	0	0
$986$	−47.0069	−1.49700
$987$	− 8.35267i	− 0.265868i
$988$	0.156145i	0.00496762i
$989$	78.2682	2.48878
$990$	0	0
$991$	−24.8373	−0.788982	−0.394491	−	0.918900i	$-0.629079\pi$
$991$	−24.8373	−0.788982	−0.394491	+	0.918900i	$0.629079\pi$
$992$	− 8.60112i	− 0.273086i
$993$	− 6.01724i	− 0.190951i
$994$	−22.1876	−0.703749
$995$	0	0
$996$	−8.80954	−0.279141
$997$	− 25.0863i	− 0.794490i	−0.917713	−	0.397245i	$-0.869966\pi$
$997$	− 25.0863i	− 0.794490i	0.917713	−	0.397245i	$-0.130034\pi$
$998$	− 53.2662i	− 1.68611i
$999$	4.02621	0.127384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.f.1249.9		12
5.2	odd	4		1875.2.a.j.1.2		6
5.3	odd	4		1875.2.a.k.1.5		6
5.4	even	2	inner	1875.2.b.f.1249.4		12
15.2	even	4		5625.2.a.p.1.5		6
15.8	even	4		5625.2.a.q.1.2		6
25.3	odd	20		375.2.g.c.76.1		12
25.4	even	10		375.2.i.d.49.2		24
25.6	even	5		375.2.i.d.199.2		24
25.8	odd	20		375.2.g.c.301.1		12
25.17	odd	20		75.2.g.c.61.3	yes	12
25.19	even	10		375.2.i.d.199.5		24
25.21	even	5		375.2.i.d.49.5		24
25.22	odd	20		75.2.g.c.16.3	✓	12
75.17	even	20		225.2.h.d.136.1		12
75.47	even	20		225.2.h.d.91.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.16.3	✓	12	25.22	odd	20
75.2.g.c.61.3	yes	12	25.17	odd	20
225.2.h.d.91.1		12	75.47	even	20
225.2.h.d.136.1		12	75.17	even	20
375.2.g.c.76.1		12	25.3	odd	20
375.2.g.c.301.1		12	25.8	odd	20
375.2.i.d.49.2		24	25.4	even	10
375.2.i.d.49.5		24	25.21	even	5
375.2.i.d.199.2		24	25.6	even	5
375.2.i.d.199.5		24	25.19	even	10
1875.2.a.j.1.2		6	5.2	odd	4
1875.2.a.k.1.5		6	5.3	odd	4
1875.2.b.f.1249.4		12	5.4	even	2	inner
1875.2.b.f.1249.9		12	1.1	even	1	trivial
5625.2.a.p.1.5		6	15.2	even	4
5625.2.a.q.1.2		6	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)

\(f(q)\)	\(=\)	\(q+2.01887i q^{2} -1.00000i q^{3} -2.07584 q^{4} +2.01887 q^{6} -1.01887i q^{7} -0.153106i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.01887i q^{2} -1.00000i q^{3} -2.07584 q^{4} +2.01887 q^{6} -1.01887i q^{7} -0.153106i q^{8} -1.00000 q^{9} +4.75961 q^{11} +2.07584i q^{12} -0.103837i q^{13} +2.05697 q^{14} -3.84257 q^{16} +5.83776i q^{17} -2.01887i q^{18} +0.724404 q^{19} -1.01887 q^{21} +9.60904i q^{22} -9.07152i q^{23} -0.153106 q^{24} +0.209634 q^{26} +1.00000i q^{27} +2.11501i q^{28} +3.98847 q^{29} +1.06662 q^{31} -8.06387i q^{32} -4.75961i q^{33} -11.7857 q^{34} +2.07584 q^{36} -4.02621i q^{37} +1.46248i q^{38} -0.103837 q^{39} +7.20977 q^{41} -2.05697i q^{42} +8.62791i q^{43} -9.88018 q^{44} +18.3142 q^{46} +8.19797i q^{47} +3.84257i q^{48} +5.96190 q^{49} +5.83776 q^{51} +0.215549i q^{52} -4.36719i q^{53} -2.01887 q^{54} -0.155995 q^{56} -0.724404i q^{57} +8.05221i q^{58} +4.91285 q^{59} +6.96435 q^{61} +2.15338i q^{62} +1.01887i q^{63} +8.59476 q^{64} +9.60904 q^{66} +9.91998i q^{67} -12.1183i q^{68} -9.07152 q^{69} -10.7866 q^{71} +0.153106i q^{72} +8.63115i q^{73} +8.12840 q^{74} -1.50375 q^{76} -4.84943i q^{77} -0.209634i q^{78} -2.48291 q^{79} +1.00000 q^{81} +14.5556i q^{82} +4.24385i q^{83} +2.11501 q^{84} -17.4186 q^{86} -3.98847i q^{87} -0.728724i q^{88} +18.3752 q^{89} -0.105797 q^{91} +18.8310i q^{92} -1.06662i q^{93} -16.5506 q^{94} -8.06387 q^{96} -6.69876i q^{97} +12.0363i q^{98} -4.75961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) \) 12 * q - 20 * q^4 - 12 * q^9	\( 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) \) 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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