LMFDB - Embedded newform 1935.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1935,2,Mod(1,1935)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1935, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1935.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1935 = 3^{2} \cdot 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1935.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.4510527911$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1933097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 $ x^5 - 2x^4 - 7x^3 + 13x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.50989$ of defining polynomial
Character	$\chi$	$=$	1935.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50989 q^{2} +4.29955 q^{4} -1.00000 q^{5} +3.13519 q^{7} -5.77162 q^{8} +O(q^{10})$	\(q-2.50989 q^{2} +4.29955 q^{4} -1.00000 q^{5} +3.13519 q^{7} -5.77162 q^{8} +2.50989 q^{10} +0.248163 q^{11} -2.67669 q^{13} -7.86899 q^{14} +5.88703 q^{16} +0.902631 q^{17} -6.19474 q^{19} -4.29955 q^{20} -0.622862 q^{22} +7.69647 q^{23} +1.00000 q^{25} +6.71819 q^{26} +13.4799 q^{28} -2.32872 q^{29} +7.73623 q^{31} -3.23256 q^{32} -2.26550 q^{34} -3.13519 q^{35} +3.67425 q^{37} +15.5481 q^{38} +5.77162 q^{40} -9.45452 q^{41} -1.00000 q^{43} +1.06699 q^{44} -19.3173 q^{46} +9.61888 q^{47} +2.82943 q^{49} -2.50989 q^{50} -11.5086 q^{52} +7.09737 q^{53} -0.248163 q^{55} -18.0951 q^{56} +5.84482 q^{58} +3.26173 q^{59} -3.35338 q^{61} -19.4171 q^{62} -3.66068 q^{64} +2.67669 q^{65} -0.613476 q^{67} +3.88091 q^{68} +7.86899 q^{70} +8.46512 q^{71} -1.33737 q^{73} -9.22196 q^{74} -26.6346 q^{76} +0.778039 q^{77} +6.65447 q^{79} -5.88703 q^{80} +23.7298 q^{82} -10.7162 q^{83} -0.902631 q^{85} +2.50989 q^{86} -1.43230 q^{88} +1.34850 q^{89} -8.39193 q^{91} +33.0913 q^{92} -24.1423 q^{94} +6.19474 q^{95} +4.49827 q^{97} -7.10155 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8	$ 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} + 5 q^{13} - q^{14} + 14 q^{16} + 17 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{22} - q^{23} + 5 q^{25} - 22 q^{26} + 26 q^{28} - 6 q^{29} + 6 q^{31} + 7 q^{32} - 5 q^{35} + 5 q^{37} + 16 q^{38} + 3 q^{40} - 2 q^{41} - 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} - 2 q^{50} - 38 q^{52} + 23 q^{53} - 6 q^{55} + 19 q^{56} + 12 q^{58} + q^{59} + 20 q^{61} + 3 q^{62} - 25 q^{64} - 5 q^{65} + 21 q^{67} + 48 q^{68} + q^{70} - 4 q^{71} + 5 q^{73} - 24 q^{74} + 32 q^{76} + 26 q^{77} + 41 q^{79} - 14 q^{80} + 38 q^{82} + 7 q^{83} - 17 q^{85} + 2 q^{86} + 12 q^{88} - 20 q^{89} - 42 q^{91} + 52 q^{92} - 42 q^{94} + 6 q^{95} + 37 q^{97} + 26 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 + 5 * q^13 - q^14 + 14 * q^16 + 17 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^22 - q^23 + 5 * q^25 - 22 * q^26 + 26 * q^28 - 6 * q^29 + 6 * q^31 + 7 * q^32 - 5 * q^35 + 5 * q^37 + 16 * q^38 + 3 * q^40 - 2 * q^41 - 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 - 2 * q^50 - 38 * q^52 + 23 * q^53 - 6 * q^55 + 19 * q^56 + 12 * q^58 + q^59 + 20 * q^61 + 3 * q^62 - 25 * q^64 - 5 * q^65 + 21 * q^67 + 48 * q^68 + q^70 - 4 * q^71 + 5 * q^73 - 24 * q^74 + 32 * q^76 + 26 * q^77 + 41 * q^79 - 14 * q^80 + 38 * q^82 + 7 * q^83 - 17 * q^85 + 2 * q^86 + 12 * q^88 - 20 * q^89 - 42 * q^91 + 52 * q^92 - 42 * q^94 + 6 * q^95 + 37 * q^97 + 26 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.50989	−1.77476	−0.887380	−	0.461038i	$-0.847477\pi$
$2$	−2.50989	−1.77476	−0.887380	+	0.461038i	$0.847477\pi$
$3$	0	0
$3$	0	0
$4$	4.29955	2.14977
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.13519	1.18499	0.592495	−	0.805574i	$-0.298143\pi$
$7$	3.13519	1.18499	0.592495	+	0.805574i	$0.298143\pi$
$8$	−5.77162	−2.04058
$9$	0	0
$10$	2.50989	0.793697
$11$	0.248163	0.0748240	0.0374120	−	0.999300i	$-0.488089\pi$
$11$	0.248163	0.0748240	0.0374120	+	0.999300i	$0.488089\pi$
$12$	0	0
$13$	−2.67669	−0.742380	−0.371190	−	0.928557i	$-0.621050\pi$
$13$	−2.67669	−0.742380	−0.371190	+	0.928557i	$0.621050\pi$
$14$	−7.86899	−2.10308
$15$	0	0
$16$	5.88703	1.47176
$17$	0.902631	0.218920	0.109460	−	0.993991i	$-0.465088\pi$
$17$	0.902631	0.218920	0.109460	+	0.993991i	$0.465088\pi$
$18$	0	0
$19$	−6.19474	−1.42117	−0.710585	−	0.703611i	$-0.751570\pi$
$19$	−6.19474	−1.42117	−0.710585	+	0.703611i	$0.751570\pi$
$20$	−4.29955	−0.961409
$21$	0	0
$22$	−0.622862	−0.132795
$23$	7.69647	1.60482	0.802412	−	0.596770i	$-0.203550\pi$
$23$	7.69647	1.60482	0.802412	+	0.596770i	$0.203550\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.71819	1.31755
$27$	0	0
$28$	13.4799	2.54746
$29$	−2.32872	−0.432432	−0.216216	−	0.976346i	$-0.569372\pi$
$29$	−2.32872	−0.432432	−0.216216	+	0.976346i	$0.569372\pi$
$30$	0	0
$31$	7.73623	1.38947	0.694734	−	0.719266i	$-0.255522\pi$
$31$	7.73623	1.38947	0.694734	+	0.719266i	$0.255522\pi$
$32$	−3.23256	−0.571441
$33$	0	0
$34$	−2.26550	−0.388531
$35$	−3.13519	−0.529944
$36$	0	0
$37$	3.67425	0.604043	0.302021	−	0.953301i	$-0.402339\pi$
$37$	3.67425	0.604043	0.302021	+	0.953301i	$0.402339\pi$
$38$	15.5481	2.52224
$39$	0	0
$40$	5.77162	0.912573
$41$	−9.45452	−1.47655	−0.738274	−	0.674501i	$-0.764359\pi$
$41$	−9.45452	−1.47655	−0.738274	+	0.674501i	$0.764359\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	1.06699	0.160855
$45$	0	0
$46$	−19.3173	−2.84818
$47$	9.61888	1.40306	0.701529	−	0.712641i	$-0.252501\pi$
$47$	9.61888	1.40306	0.701529	+	0.712641i	$0.252501\pi$
$48$	0	0
$49$	2.82943	0.404204
$50$	−2.50989	−0.354952
$51$	0	0
$52$	−11.5086	−1.59595
$53$	7.09737	0.974899	0.487449	−	0.873151i	$-0.337927\pi$
$53$	7.09737	0.974899	0.487449	+	0.873151i	$0.337927\pi$
$54$	0	0
$55$	−0.248163	−0.0334623
$56$	−18.0951	−2.41806
$57$	0	0
$58$	5.84482	0.767463
$59$	3.26173	0.424641	0.212320	−	0.977200i	$-0.431898\pi$
$59$	3.26173	0.424641	0.212320	+	0.977200i	$0.431898\pi$
$60$	0	0
$61$	−3.35338	−0.429356	−0.214678	−	0.976685i	$-0.568870\pi$
$61$	−3.35338	−0.429356	−0.214678	+	0.976685i	$0.568870\pi$
$62$	−19.4171	−2.46597
$63$	0	0
$64$	−3.66068	−0.457586
$65$	2.67669	0.332002
$66$	0	0
$67$	−0.613476	−0.0749480	−0.0374740	−	0.999298i	$-0.511931\pi$
$67$	−0.613476	−0.0749480	−0.0374740	+	0.999298i	$0.511931\pi$
$68$	3.88091	0.470629
$69$	0	0
$70$	7.86899	0.940524
$71$	8.46512	1.00463	0.502313	−	0.864686i	$-0.332483\pi$
$71$	8.46512	1.00463	0.502313	+	0.864686i	$0.332483\pi$
$72$	0	0
$73$	−1.33737	−0.156528	−0.0782638	−	0.996933i	$-0.524938\pi$
$73$	−1.33737	−0.156528	−0.0782638	+	0.996933i	$0.524938\pi$
$74$	−9.22196	−1.07203
$75$	0	0
$76$	−26.6346	−3.05520
$77$	0.778039	0.0886657
$78$	0	0
$79$	6.65447	0.748686	0.374343	−	0.927290i	$-0.377868\pi$
$79$	6.65447	0.748686	0.374343	+	0.927290i	$0.377868\pi$
$80$	−5.88703	−0.658190
$81$	0	0
$82$	23.7298	2.62052
$83$	−10.7162	−1.17626	−0.588131	−	0.808766i	$-0.700136\pi$
$83$	−10.7162	−1.17626	−0.588131	+	0.808766i	$0.700136\pi$
$84$	0	0
$85$	−0.902631	−0.0979041
$86$	2.50989	0.270648
$87$	0	0
$88$	−1.43230	−0.152684
$89$	1.34850	0.142940	0.0714702	−	0.997443i	$-0.477231\pi$
$89$	1.34850	0.142940	0.0714702	+	0.997443i	$0.477231\pi$
$90$	0	0
$91$	−8.39193	−0.879713
$92$	33.0913	3.45001
$93$	0	0
$94$	−24.1423	−2.49009
$95$	6.19474	0.635567
$96$	0	0
$97$	4.49827	0.456730	0.228365	−	0.973576i	$-0.426662\pi$
$97$	4.49827	0.456730	0.228365	+	0.973576i	$0.426662\pi$
$98$	−7.10155	−0.717365
$99$	0	0
$100$	4.29955	0.429955
$101$	−6.23256	−0.620163	−0.310081	−	0.950710i	$-0.600356\pi$
$101$	−6.23256	−0.620163	−0.310081	+	0.950710i	$0.600356\pi$
$102$	0	0
$103$	7.29211	0.718513	0.359256	−	0.933239i	$-0.383030\pi$
$103$	7.29211	0.718513	0.359256	+	0.933239i	$0.383030\pi$
$104$	15.4488	1.51488
$105$	0	0
$106$	−17.8136	−1.73021
$107$	−5.69841	−0.550886	−0.275443	−	0.961317i	$-0.588825\pi$
$107$	−5.69841	−0.550886	−0.275443	+	0.961317i	$0.588825\pi$
$108$	0	0
$109$	13.2648	1.27053	0.635267	−	0.772292i	$-0.280890\pi$
$109$	13.2648	1.27053	0.635267	+	0.772292i	$0.280890\pi$
$110$	0.622862	0.0593876
$111$	0	0
$112$	18.4570	1.74402
$113$	−8.71451	−0.819792	−0.409896	−	0.912132i	$-0.634435\pi$
$113$	−8.71451	−0.819792	−0.409896	+	0.912132i	$0.634435\pi$
$114$	0	0
$115$	−7.69647	−0.717699
$116$	−10.0124	−0.929631
$117$	0	0
$118$	−8.18658	−0.753636
$119$	2.82992	0.259418
$120$	0	0
$121$	−10.9384	−0.994401
$122$	8.41660	0.762004
$123$	0	0
$124$	33.2623	2.98704
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.3875	1.63163	0.815815	−	0.578313i	$-0.196289\pi$
$127$	18.3875	1.63163	0.815815	+	0.578313i	$0.196289\pi$
$128$	15.6530	1.38355
$129$	0	0
$130$	−6.71819	−0.589224
$131$	−6.10277	−0.533202	−0.266601	−	0.963807i	$-0.585901\pi$
$131$	−6.10277	−0.533202	−0.266601	+	0.963807i	$0.585901\pi$
$132$	0	0
$133$	−19.4217	−1.68407
$134$	1.53976	0.133015
$135$	0	0
$136$	−5.20964	−0.446723
$137$	9.57167	0.817763	0.408882	−	0.912587i	$-0.365919\pi$
$137$	9.57167	0.817763	0.408882	+	0.912587i	$0.365919\pi$
$138$	0	0
$139$	−7.86277	−0.666911	−0.333456	−	0.942766i	$-0.608215\pi$
$139$	−7.86277	−0.666911	−0.333456	+	0.942766i	$0.608215\pi$
$140$	−13.4799	−1.13926
$141$	0	0
$142$	−21.2465	−1.78297
$143$	−0.664255	−0.0555478
$144$	0	0
$145$	2.32872	0.193389
$146$	3.35666	0.277799
$147$	0	0
$148$	15.7976	1.29856
$149$	1.09931	0.0900592	0.0450296	−	0.998986i	$-0.485662\pi$
$149$	1.09931	0.0900592	0.0450296	+	0.998986i	$0.485662\pi$
$150$	0	0
$151$	7.87683	0.641007	0.320504	−	0.947247i	$-0.396148\pi$
$151$	7.87683	0.641007	0.320504	+	0.947247i	$0.396148\pi$
$152$	35.7537	2.90000
$153$	0	0
$154$	−1.95279	−0.157360
$155$	−7.73623	−0.621389
$156$	0	0
$157$	3.04822	0.243274	0.121637	−	0.992575i	$-0.461186\pi$
$157$	3.04822	0.243274	0.121637	+	0.992575i	$0.461186\pi$
$158$	−16.7020	−1.32874
$159$	0	0
$160$	3.23256	0.255556
$161$	24.1299	1.90170
$162$	0	0
$163$	7.04394	0.551724	0.275862	−	0.961197i	$-0.411037\pi$
$163$	7.04394	0.551724	0.275862	+	0.961197i	$0.411037\pi$
$164$	−40.6502	−3.17425
$165$	0	0
$166$	26.8966	2.08758
$167$	−1.89121	−0.146346	−0.0731730	−	0.997319i	$-0.523313\pi$
$167$	−1.89121	−0.146346	−0.0731730	+	0.997319i	$0.523313\pi$
$168$	0	0
$169$	−5.83534	−0.448873
$170$	2.26550	0.173756
$171$	0	0
$172$	−4.29955	−0.327838
$173$	−7.44534	−0.566059	−0.283029	−	0.959111i	$-0.591339\pi$
$173$	−7.44534	−0.566059	−0.283029	+	0.959111i	$0.591339\pi$
$174$	0	0
$175$	3.13519	0.236998
$176$	1.46094	0.110123
$177$	0	0
$178$	−3.38458	−0.253685
$179$	10.1784	0.760771	0.380385	−	0.924828i	$-0.375791\pi$
$179$	10.1784	0.760771	0.380385	+	0.924828i	$0.375791\pi$
$180$	0	0
$181$	13.0309	0.968580	0.484290	−	0.874908i	$-0.339078\pi$
$181$	13.0309	0.968580	0.484290	+	0.874908i	$0.339078\pi$
$182$	21.0628	1.56128
$183$	0	0
$184$	−44.4211	−3.27477
$185$	−3.67425	−0.270136
$186$	0	0
$187$	0.224000	0.0163805
$188$	41.3569	3.01626
$189$	0	0
$190$	−15.5481	−1.12798
$191$	13.0247	0.942431	0.471216	−	0.882018i	$-0.343815\pi$
$191$	13.0247	0.942431	0.471216	+	0.882018i	$0.343815\pi$
$192$	0	0
$193$	11.6569	0.839083	0.419541	−	0.907736i	$-0.362191\pi$
$193$	11.6569	0.839083	0.419541	+	0.907736i	$0.362191\pi$
$194$	−11.2902	−0.810587
$195$	0	0
$196$	12.1653	0.868947
$197$	15.3026	1.09026	0.545132	−	0.838350i	$-0.316479\pi$
$197$	15.3026	1.09026	0.545132	+	0.838350i	$0.316479\pi$
$198$	0	0
$199$	0.718193	0.0509113	0.0254557	−	0.999676i	$-0.491896\pi$
$199$	0.718193	0.0509113	0.0254557	+	0.999676i	$0.491896\pi$
$200$	−5.77162	−0.408115
$201$	0	0
$202$	15.6430	1.10064
$203$	−7.30097	−0.512428
$204$	0	0
$205$	9.45452	0.660332
$206$	−18.3024	−1.27519
$207$	0	0
$208$	−15.7577	−1.09260
$209$	−1.53730	−0.106338
$210$	0	0
$211$	20.2812	1.39621	0.698107	−	0.715993i	$-0.254026\pi$
$211$	20.2812	1.39621	0.698107	+	0.715993i	$0.254026\pi$
$212$	30.5155	2.09581
$213$	0	0
$214$	14.3024	0.977691
$215$	1.00000	0.0681994
$216$	0	0
$217$	24.2546	1.64651
$218$	−33.2931	−2.25490
$219$	0	0
$220$	−1.06699	−0.0719364
$221$	−2.41606	−0.162522
$222$	0	0
$223$	22.4003	1.50003	0.750017	−	0.661419i	$-0.230045\pi$
$223$	22.4003	1.50003	0.750017	+	0.661419i	$0.230045\pi$
$224$	−10.1347	−0.677153
$225$	0	0
$226$	21.8725	1.45493
$227$	23.0343	1.52884	0.764419	−	0.644720i	$-0.223026\pi$
$227$	23.0343	1.52884	0.764419	+	0.644720i	$0.223026\pi$
$228$	0	0
$229$	12.6648	0.836911	0.418456	−	0.908237i	$-0.362572\pi$
$229$	12.6648	0.836911	0.418456	+	0.908237i	$0.362572\pi$
$230$	19.3173	1.27374
$231$	0	0
$232$	13.4405	0.882410
$233$	30.2611	1.98247	0.991236	−	0.132105i	$-0.0421737\pi$
$233$	30.2611	1.98247	0.991236	+	0.132105i	$0.0421737\pi$
$234$	0	0
$235$	−9.61888	−0.627467
$236$	14.0240	0.912882
$237$	0	0
$238$	−7.10279	−0.460406
$239$	−18.5033	−1.19688	−0.598438	−	0.801169i	$-0.704212\pi$
$239$	−18.5033	−1.19688	−0.598438	+	0.801169i	$0.704212\pi$
$240$	0	0
$241$	−16.0840	−1.03606	−0.518031	−	0.855362i	$-0.673335\pi$
$241$	−16.0840	−1.03606	−0.518031	+	0.855362i	$0.673335\pi$
$242$	27.4542	1.76482
$243$	0	0
$244$	−14.4180	−0.923018
$245$	−2.82943	−0.180765
$246$	0	0
$247$	16.5814	1.05505
$248$	−44.6506	−2.83532
$249$	0	0
$250$	2.50989	0.158739
$251$	12.4459	0.785576	0.392788	−	0.919629i	$-0.371511\pi$
$251$	12.4459	0.785576	0.392788	+	0.919629i	$0.371511\pi$
$252$	0	0
$253$	1.90998	0.120079
$254$	−46.1507	−2.89575
$255$	0	0
$256$	−31.9660	−1.99788
$257$	−4.68647	−0.292334	−0.146167	−	0.989260i	$-0.546694\pi$
$257$	−4.68647	−0.292334	−0.146167	+	0.989260i	$0.546694\pi$
$258$	0	0
$259$	11.5195	0.715785
$260$	11.5086	0.713730
$261$	0	0
$262$	15.3173	0.946305
$263$	−17.3959	−1.07268	−0.536339	−	0.844003i	$-0.680193\pi$
$263$	−17.3959	−1.07268	−0.536339	+	0.844003i	$0.680193\pi$
$264$	0	0
$265$	−7.09737	−0.435988
$266$	48.7463	2.98883
$267$	0	0
$268$	−2.63767	−0.161121
$269$	20.0133	1.22023	0.610115	−	0.792313i	$-0.291123\pi$
$269$	20.0133	1.22023	0.610115	+	0.792313i	$0.291123\pi$
$270$	0	0
$271$	14.0575	0.853933	0.426966	−	0.904267i	$-0.359582\pi$
$271$	14.0575	0.853933	0.426966	+	0.904267i	$0.359582\pi$
$272$	5.31381	0.322197
$273$	0	0
$274$	−24.0238	−1.45133
$275$	0.248163	0.0149648
$276$	0	0
$277$	−10.8297	−0.650695	−0.325347	−	0.945595i	$-0.605481\pi$
$277$	−10.8297	−0.650695	−0.325347	+	0.945595i	$0.605481\pi$
$278$	19.7347	1.18361
$279$	0	0
$280$	18.0951	1.08139
$281$	7.76915	0.463469	0.231734	−	0.972779i	$-0.425560\pi$
$281$	7.76915	0.463469	0.231734	+	0.972779i	$0.425560\pi$
$282$	0	0
$283$	7.32924	0.435678	0.217839	−	0.975985i	$-0.430099\pi$
$283$	7.32924	0.435678	0.217839	+	0.975985i	$0.430099\pi$
$284$	36.3962	2.15972
$285$	0	0
$286$	1.66721	0.0985840
$287$	−29.6417	−1.74970
$288$	0	0
$289$	−16.1853	−0.952074
$290$	−5.84482	−0.343220
$291$	0	0
$292$	−5.75010	−0.336499
$293$	2.25307	0.131626	0.0658129	−	0.997832i	$-0.479036\pi$
$293$	2.25307	0.131626	0.0658129	+	0.997832i	$0.479036\pi$
$294$	0	0
$295$	−3.26173	−0.189905
$296$	−21.2064	−1.23259
$297$	0	0
$298$	−2.75915	−0.159834
$299$	−20.6010	−1.19139
$300$	0	0
$301$	−3.13519	−0.180709
$302$	−19.7700	−1.13763
$303$	0	0
$304$	−36.4686	−2.09162
$305$	3.35338	0.192014
$306$	0	0
$307$	17.5142	0.999586	0.499793	−	0.866145i	$-0.333409\pi$
$307$	17.5142	0.999586	0.499793	+	0.866145i	$0.333409\pi$
$308$	3.34522	0.190611
$309$	0	0
$310$	19.4171	1.10282
$311$	11.8628	0.672676	0.336338	−	0.941741i	$-0.390812\pi$
$311$	11.8628	0.672676	0.336338	+	0.941741i	$0.390812\pi$
$312$	0	0
$313$	−28.1905	−1.59342	−0.796709	−	0.604363i	$-0.793428\pi$
$313$	−28.1905	−1.59342	−0.796709	+	0.604363i	$0.793428\pi$
$314$	−7.65069	−0.431753
$315$	0	0
$316$	28.6112	1.60951
$317$	6.60104	0.370752	0.185376	−	0.982668i	$-0.440650\pi$
$317$	6.60104	0.370752	0.185376	+	0.982668i	$0.440650\pi$
$318$	0	0
$319$	−0.577901	−0.0323563
$320$	3.66068	0.204638
$321$	0	0
$322$	−60.5634	−3.37507
$323$	−5.59156	−0.311123
$324$	0	0
$325$	−2.67669	−0.148476
$326$	−17.6795	−0.979178
$327$	0	0
$328$	54.5679	3.01301
$329$	30.1570	1.66261
$330$	0	0
$331$	−9.68209	−0.532176	−0.266088	−	0.963949i	$-0.585731\pi$
$331$	−9.68209	−0.532176	−0.266088	+	0.963949i	$0.585731\pi$
$332$	−46.0750	−2.52870
$333$	0	0
$334$	4.74672	0.259729
$335$	0.613476	0.0335178
$336$	0	0
$337$	29.2544	1.59359	0.796794	−	0.604251i	$-0.206527\pi$
$337$	29.2544	1.59359	0.796794	+	0.604251i	$0.206527\pi$
$338$	14.6461	0.796641
$339$	0	0
$340$	−3.88091	−0.210472
$341$	1.91985	0.103966
$342$	0	0
$343$	−13.0756	−0.706013
$344$	5.77162	0.311185
$345$	0	0
$346$	18.6870	1.00462
$347$	−8.34899	−0.448197	−0.224099	−	0.974566i	$-0.571944\pi$
$347$	−8.34899	−0.448197	−0.224099	+	0.974566i	$0.571944\pi$
$348$	0	0
$349$	−22.2663	−1.19189	−0.595944	−	0.803026i	$-0.703222\pi$
$349$	−22.2663	−1.19189	−0.595944	+	0.803026i	$0.703222\pi$
$350$	−7.86899	−0.420615
$351$	0	0
$352$	−0.802202	−0.0427575
$353$	−18.3351	−0.975880	−0.487940	−	0.872877i	$-0.662252\pi$
$353$	−18.3351	−0.975880	−0.487940	+	0.872877i	$0.662252\pi$
$354$	0	0
$355$	−8.46512	−0.449282
$356$	5.79793	0.307290
$357$	0	0
$358$	−25.5467	−1.35019
$359$	28.8857	1.52453	0.762265	−	0.647265i	$-0.224087\pi$
$359$	28.8857	1.52453	0.762265	+	0.647265i	$0.224087\pi$
$360$	0	0
$361$	19.3748	1.01973
$362$	−32.7061	−1.71900
$363$	0	0
$364$	−36.0815	−1.89119
$365$	1.33737	0.0700013
$366$	0	0
$367$	22.7231	1.18613	0.593067	−	0.805153i	$-0.297917\pi$
$367$	22.7231	1.18613	0.593067	+	0.805153i	$0.297917\pi$
$368$	45.3093	2.36191
$369$	0	0
$370$	9.22196	0.479427
$371$	22.2516	1.15525
$372$	0	0
$373$	−30.1880	−1.56308	−0.781539	−	0.623857i	$-0.785565\pi$
$373$	−30.1880	−1.56308	−0.781539	+	0.623857i	$0.785565\pi$
$374$	−0.562214	−0.0290714
$375$	0	0
$376$	−55.5165	−2.86304
$377$	6.23325	0.321029
$378$	0	0
$379$	14.7592	0.758128	0.379064	−	0.925370i	$-0.376246\pi$
$379$	14.7592	0.758128	0.379064	+	0.925370i	$0.376246\pi$
$380$	26.6346	1.36633
$381$	0	0
$382$	−32.6905	−1.67259
$383$	14.0955	0.720248	0.360124	−	0.932904i	$-0.382734\pi$
$383$	14.0955	0.720248	0.360124	+	0.932904i	$0.382734\pi$
$384$	0	0
$385$	−0.778039	−0.0396525
$386$	−29.2576	−1.48917
$387$	0	0
$388$	19.3405	0.981867
$389$	−30.9837	−1.57094	−0.785468	−	0.618902i	$-0.787578\pi$
$389$	−30.9837	−1.57094	−0.785468	+	0.618902i	$0.787578\pi$
$390$	0	0
$391$	6.94707	0.351328
$392$	−16.3304	−0.824808
$393$	0	0
$394$	−38.4078	−1.93496
$395$	−6.65447	−0.334823
$396$	0	0
$397$	31.0726	1.55949	0.779744	−	0.626099i	$-0.215349\pi$
$397$	31.0726	1.55949	0.779744	+	0.626099i	$0.215349\pi$
$398$	−1.80259	−0.0903554
$399$	0	0
$400$	5.88703	0.294351
$401$	4.92956	0.246170	0.123085	−	0.992396i	$-0.460721\pi$
$401$	4.92956	0.246170	0.123085	+	0.992396i	$0.460721\pi$
$402$	0	0
$403$	−20.7075	−1.03151
$404$	−26.7972	−1.33321
$405$	0	0
$406$	18.3246	0.909437
$407$	0.911813	0.0451969
$408$	0	0
$409$	−7.61888	−0.376729	−0.188365	−	0.982099i	$-0.560319\pi$
$409$	−7.61888	−0.376729	−0.188365	+	0.982099i	$0.560319\pi$
$410$	−23.7298	−1.17193
$411$	0	0
$412$	31.3528	1.54464
$413$	10.2261	0.503195
$414$	0	0
$415$	10.7162	0.526040
$416$	8.65256	0.424226
$417$	0	0
$418$	3.85847	0.188724
$419$	−21.8373	−1.06682	−0.533410	−	0.845857i	$-0.679090\pi$
$419$	−21.8373	−1.06682	−0.533410	+	0.845857i	$0.679090\pi$
$420$	0	0
$421$	−13.5012	−0.658009	−0.329005	−	0.944328i	$-0.606713\pi$
$421$	−13.5012	−0.658009	−0.329005	+	0.944328i	$0.606713\pi$
$422$	−50.9036	−2.47795
$423$	0	0
$424$	−40.9633	−1.98935
$425$	0.902631	0.0437840
$426$	0	0
$427$	−10.5135	−0.508783
$428$	−24.5006	−1.18428
$429$	0	0
$430$	−2.50989	−0.121038
$431$	7.29975	0.351617	0.175808	−	0.984424i	$-0.443746\pi$
$431$	7.29975	0.351617	0.175808	+	0.984424i	$0.443746\pi$
$432$	0	0
$433$	10.1310	0.486865	0.243433	−	0.969918i	$-0.421726\pi$
$433$	10.1310	0.486865	0.243433	+	0.969918i	$0.421726\pi$
$434$	−60.8763	−2.92216
$435$	0	0
$436$	57.0326	2.73136
$437$	−47.6776	−2.28073
$438$	0	0
$439$	−17.6650	−0.843103	−0.421552	−	0.906804i	$-0.638514\pi$
$439$	−17.6650	−0.843103	−0.421552	+	0.906804i	$0.638514\pi$
$440$	1.43230	0.0682823
$441$	0	0
$442$	6.06405	0.288437
$443$	−36.7934	−1.74811	−0.874054	−	0.485828i	$-0.838518\pi$
$443$	−36.7934	−1.74811	−0.874054	+	0.485828i	$0.838518\pi$
$444$	0	0
$445$	−1.34850	−0.0639249
$446$	−56.2223	−2.66220
$447$	0	0
$448$	−11.4769	−0.542235
$449$	−0.788959	−0.0372333	−0.0186166	−	0.999827i	$-0.505926\pi$
$449$	−0.788959	−0.0372333	−0.0186166	+	0.999827i	$0.505926\pi$
$450$	0	0
$451$	−2.34626	−0.110481
$452$	−37.4685	−1.76237
$453$	0	0
$454$	−57.8135	−2.71332
$455$	8.39193	0.393420
$456$	0	0
$457$	23.7442	1.11070	0.555352	−	0.831615i	$-0.312583\pi$
$457$	23.7442	1.11070	0.555352	+	0.831615i	$0.312583\pi$
$458$	−31.7872	−1.48532
$459$	0	0
$460$	−33.0913	−1.54289
$461$	−10.5205	−0.489988	−0.244994	−	0.969525i	$-0.578786\pi$
$461$	−10.5205	−0.489988	−0.244994	+	0.969525i	$0.578786\pi$
$462$	0	0
$463$	−33.7411	−1.56808	−0.784042	−	0.620708i	$-0.786845\pi$
$463$	−33.7411	−1.56808	−0.784042	+	0.620708i	$0.786845\pi$
$464$	−13.7092	−0.636435
$465$	0	0
$466$	−75.9521	−3.51841
$467$	−19.2665	−0.891548	−0.445774	−	0.895146i	$-0.647071\pi$
$467$	−19.2665	−0.891548	−0.445774	+	0.895146i	$0.647071\pi$
$468$	0	0
$469$	−1.92336	−0.0888127
$470$	24.1423	1.11360
$471$	0	0
$472$	−18.8254	−0.866511
$473$	−0.248163	−0.0114105
$474$	0	0
$475$	−6.19474	−0.284234
$476$	12.1674	0.557691
$477$	0	0
$478$	46.4412	2.12417
$479$	9.69667	0.443052	0.221526	−	0.975154i	$-0.428896\pi$
$479$	9.69667	0.443052	0.221526	+	0.975154i	$0.428896\pi$
$480$	0	0
$481$	−9.83482	−0.448429
$482$	40.3691	1.83876
$483$	0	0
$484$	−47.0303	−2.13774
$485$	−4.49827	−0.204256
$486$	0	0
$487$	−0.683509	−0.0309728	−0.0154864	−	0.999880i	$-0.504930\pi$
$487$	−0.683509	−0.0309728	−0.0154864	+	0.999880i	$0.504930\pi$
$488$	19.3544	0.876133
$489$	0	0
$490$	7.10155	0.320815
$491$	29.1038	1.31344	0.656718	−	0.754137i	$-0.271944\pi$
$491$	29.1038	1.31344	0.656718	+	0.754137i	$0.271944\pi$
$492$	0	0
$493$	−2.10197	−0.0946680
$494$	−41.6174	−1.87246
$495$	0	0
$496$	45.5434	2.04496
$497$	26.5398	1.19047
$498$	0	0
$499$	30.7637	1.37717	0.688586	−	0.725155i	$-0.258232\pi$
$499$	30.7637	1.37717	0.688586	+	0.725155i	$0.258232\pi$
$500$	−4.29955	−0.192282
$501$	0	0
$502$	−31.2378	−1.39421
$503$	−4.02345	−0.179397	−0.0896983	−	0.995969i	$-0.528590\pi$
$503$	−4.02345	−0.179397	−0.0896983	+	0.995969i	$0.528590\pi$
$504$	0	0
$505$	6.23256	0.277345
$506$	−4.79384	−0.213112
$507$	0	0
$508$	79.0581	3.50764
$509$	−12.8168	−0.568094	−0.284047	−	0.958810i	$-0.591677\pi$
$509$	−12.8168	−0.568094	−0.284047	+	0.958810i	$0.591677\pi$
$510$	0	0
$511$	−4.19292	−0.185484
$512$	48.9252	2.16221
$513$	0	0
$514$	11.7625	0.518823
$515$	−7.29211	−0.321329
$516$	0	0
$517$	2.38705	0.104982
$518$	−28.9126	−1.27035
$519$	0	0
$520$	−15.4488	−0.677476
$521$	10.8754	0.476460	0.238230	−	0.971209i	$-0.423433\pi$
$521$	10.8754	0.476460	0.238230	+	0.971209i	$0.423433\pi$
$522$	0	0
$523$	−35.9454	−1.57178	−0.785892	−	0.618364i	$-0.787796\pi$
$523$	−35.9454	−1.57178	−0.785892	+	0.618364i	$0.787796\pi$
$524$	−26.2392	−1.14626
$525$	0	0
$526$	43.6618	1.90375
$527$	6.98296	0.304183
$528$	0	0
$529$	36.2356	1.57546
$530$	17.8136	0.773774
$531$	0	0
$532$	−83.5045	−3.62038
$533$	25.3068	1.09616
$534$	0	0
$535$	5.69841	0.246364
$536$	3.54075	0.152937
$537$	0	0
$538$	−50.2311	−2.16562
$539$	0.702159	0.0302441
$540$	0	0
$541$	26.1499	1.12427	0.562135	−	0.827045i	$-0.309980\pi$
$541$	26.1499	1.12427	0.562135	+	0.827045i	$0.309980\pi$
$542$	−35.2828	−1.51553
$543$	0	0
$544$	−2.91781	−0.125100
$545$	−13.2648	−0.568200
$546$	0	0
$547$	0.231141	0.00988287	0.00494143	−	0.999988i	$-0.498427\pi$
$547$	0.231141	0.00988287	0.00494143	+	0.999988i	$0.498427\pi$
$548$	41.1539	1.75801
$549$	0	0
$550$	−0.622862	−0.0265589
$551$	14.4258	0.614559
$552$	0	0
$553$	20.8630	0.887186
$554$	27.1814	1.15483
$555$	0	0
$556$	−33.8064	−1.43371
$557$	3.96726	0.168098	0.0840491	−	0.996462i	$-0.473215\pi$
$557$	3.96726	0.168098	0.0840491	+	0.996462i	$0.473215\pi$
$558$	0	0
$559$	2.67669	0.113212
$560$	−18.4570	−0.779949
$561$	0	0
$562$	−19.4997	−0.822546
$563$	−12.1492	−0.512026	−0.256013	−	0.966673i	$-0.582409\pi$
$563$	−12.1492	−0.512026	−0.256013	+	0.966673i	$0.582409\pi$
$564$	0	0
$565$	8.71451	0.366622
$566$	−18.3956	−0.773225
$567$	0	0
$568$	−48.8574	−2.05001
$569$	42.4665	1.78029	0.890143	−	0.455681i	$-0.150604\pi$
$569$	42.4665	1.78029	0.890143	+	0.455681i	$0.150604\pi$
$570$	0	0
$571$	−23.4374	−0.980824	−0.490412	−	0.871491i	$-0.663154\pi$
$571$	−23.4374	−0.980824	−0.490412	+	0.871491i	$0.663154\pi$
$572$	−2.85600	−0.119415
$573$	0	0
$574$	74.3975	3.10529
$575$	7.69647	0.320965
$576$	0	0
$577$	−41.0241	−1.70785	−0.853927	−	0.520393i	$-0.825786\pi$
$577$	−41.0241	−1.70785	−0.853927	+	0.520393i	$0.825786\pi$
$578$	40.6232	1.68970
$579$	0	0
$580$	10.0124	0.415744
$581$	−33.5975	−1.39386
$582$	0	0
$583$	1.76130	0.0729458
$584$	7.71880	0.319406
$585$	0	0
$586$	−5.65496	−0.233604
$587$	29.4970	1.21747	0.608735	−	0.793374i	$-0.291677\pi$
$587$	29.4970	1.21747	0.608735	+	0.793374i	$0.291677\pi$
$588$	0	0
$589$	−47.9239	−1.97467
$590$	8.18658	0.337036
$591$	0	0
$592$	21.6304	0.889004
$593$	−17.7334	−0.728223	−0.364111	−	0.931355i	$-0.618627\pi$
$593$	−17.7334	−0.728223	−0.364111	+	0.931355i	$0.618627\pi$
$594$	0	0
$595$	−2.82992	−0.116015
$596$	4.72655	0.193607
$597$	0	0
$598$	51.7064	2.11443
$599$	16.9852	0.693997	0.346998	−	0.937866i	$-0.387201\pi$
$599$	16.9852	0.693997	0.346998	+	0.937866i	$0.387201\pi$
$600$	0	0
$601$	5.70922	0.232884	0.116442	−	0.993197i	$-0.462851\pi$
$601$	5.70922	0.232884	0.116442	+	0.993197i	$0.462851\pi$
$602$	7.86899	0.320716
$603$	0	0
$604$	33.8668	1.37802
$605$	10.9384	0.444710
$606$	0	0
$607$	4.37105	0.177415	0.0887077	−	0.996058i	$-0.471726\pi$
$607$	4.37105	0.177415	0.0887077	+	0.996058i	$0.471726\pi$
$608$	20.0249	0.812116
$609$	0	0
$610$	−8.41660	−0.340778
$611$	−25.7467	−1.04160
$612$	0	0
$613$	35.0848	1.41706	0.708532	−	0.705679i	$-0.249358\pi$
$613$	35.0848	1.41706	0.708532	+	0.705679i	$0.249358\pi$
$614$	−43.9586	−1.77403
$615$	0	0
$616$	−4.49054	−0.180929
$617$	−23.0537	−0.928108	−0.464054	−	0.885807i	$-0.653606\pi$
$617$	−23.0537	−0.928108	−0.464054	+	0.885807i	$0.653606\pi$
$618$	0	0
$619$	−29.3047	−1.17785	−0.588927	−	0.808186i	$-0.700449\pi$
$619$	−29.3047	−1.17785	−0.588927	+	0.808186i	$0.700449\pi$
$620$	−33.2623	−1.33585
$621$	0	0
$622$	−29.7743	−1.19384
$623$	4.22780	0.169383
$624$	0	0
$625$	1.00000	0.0400000
$626$	70.7550	2.82794
$627$	0	0
$628$	13.1060	0.522985
$629$	3.31649	0.132237
$630$	0	0
$631$	−31.7883	−1.26547	−0.632737	−	0.774367i	$-0.718069\pi$
$631$	−31.7883	−1.26547	−0.632737	+	0.774367i	$0.718069\pi$
$632$	−38.4070	−1.52775
$633$	0	0
$634$	−16.5679	−0.657995
$635$	−18.3875	−0.729687
$636$	0	0
$637$	−7.57349	−0.300073
$638$	1.45047	0.0574246
$639$	0	0
$640$	−15.6530	−0.618741
$641$	−42.7217	−1.68740	−0.843702	−	0.536811i	$-0.819629\pi$
$641$	−42.7217	−1.68740	−0.843702	+	0.536811i	$0.819629\pi$
$642$	0	0
$643$	−5.88234	−0.231977	−0.115988	−	0.993251i	$-0.537004\pi$
$643$	−5.88234	−0.231977	−0.115988	+	0.993251i	$0.537004\pi$
$644$	103.748	4.08823
$645$	0	0
$646$	14.0342	0.552169
$647$	−39.6468	−1.55868	−0.779338	−	0.626604i	$-0.784445\pi$
$647$	−39.6468	−1.55868	−0.779338	+	0.626604i	$0.784445\pi$
$648$	0	0
$649$	0.809440	0.0317733
$650$	6.71819	0.263509
$651$	0	0
$652$	30.2858	1.18608
$653$	39.0899	1.52971	0.764854	−	0.644204i	$-0.222811\pi$
$653$	39.0899	1.52971	0.764854	+	0.644204i	$0.222811\pi$
$654$	0	0
$655$	6.10277	0.238455
$656$	−55.6590	−2.17312
$657$	0	0
$658$	−75.6908	−2.95074
$659$	49.9625	1.94626	0.973131	−	0.230254i	$-0.0739558\pi$
$659$	49.9625	1.94626	0.973131	+	0.230254i	$0.0739558\pi$
$660$	0	0
$661$	−12.3002	−0.478421	−0.239210	−	0.970968i	$-0.576889\pi$
$661$	−12.3002	−0.478421	−0.239210	+	0.970968i	$0.576889\pi$
$662$	24.3010	0.944485
$663$	0	0
$664$	61.8501	2.40025
$665$	19.4217	0.753141
$666$	0	0
$667$	−17.9229	−0.693977
$668$	−8.13134	−0.314611
$669$	0	0
$670$	−1.53976	−0.0594860
$671$	−0.832184	−0.0321261
$672$	0	0
$673$	14.3751	0.554118	0.277059	−	0.960853i	$-0.410640\pi$
$673$	14.3751	0.554118	0.277059	+	0.960853i	$0.410640\pi$
$674$	−73.4253	−2.82824
$675$	0	0
$676$	−25.0893	−0.964975
$677$	19.8543	0.763062	0.381531	−	0.924356i	$-0.375397\pi$
$677$	19.8543	0.763062	0.381531	+	0.924356i	$0.375397\pi$
$678$	0	0
$679$	14.1029	0.541221
$680$	5.20964	0.199781
$681$	0	0
$682$	−4.81861	−0.184514
$683$	−33.8908	−1.29680	−0.648398	−	0.761302i	$-0.724561\pi$
$683$	−33.8908	−1.29680	−0.648398	+	0.761302i	$0.724561\pi$
$684$	0	0
$685$	−9.57167	−0.365715
$686$	32.8182	1.25300
$687$	0	0
$688$	−5.88703	−0.224441
$689$	−18.9974	−0.723745
$690$	0	0
$691$	14.3619	0.546354	0.273177	−	0.961964i	$-0.411926\pi$
$691$	14.3619	0.546354	0.273177	+	0.961964i	$0.411926\pi$
$692$	−32.0116	−1.21690
$693$	0	0
$694$	20.9551	0.795443
$695$	7.86277	0.298252
$696$	0	0
$697$	−8.53394	−0.323246
$698$	55.8860	2.11532
$699$	0	0
$700$	13.4799	0.509493
$701$	−11.8354	−0.447015	−0.223508	−	0.974702i	$-0.571751\pi$
$701$	−11.8354	−0.447015	−0.223508	+	0.974702i	$0.571751\pi$
$702$	0	0
$703$	−22.7610	−0.858448
$704$	−0.908446	−0.0342384
$705$	0	0
$706$	46.0192	1.73195
$707$	−19.5403	−0.734887
$708$	0	0
$709$	−22.9382	−0.861462	−0.430731	−	0.902480i	$-0.641744\pi$
$709$	−22.9382	−0.861462	−0.430731	+	0.902480i	$0.641744\pi$
$710$	21.2465	0.797368
$711$	0	0
$712$	−7.78301	−0.291681
$713$	59.5417	2.22985
$714$	0	0
$715$	0.664255	0.0248417
$716$	43.7626	1.63549
$717$	0	0
$718$	−72.5000	−2.70567
$719$	−40.1157	−1.49606	−0.748032	−	0.663663i	$-0.769001\pi$
$719$	−40.1157	−1.49606	−0.748032	+	0.663663i	$0.769001\pi$
$720$	0	0
$721$	22.8622	0.851431
$722$	−48.6286	−1.80977
$723$	0	0
$724$	56.0270	2.08223
$725$	−2.32872	−0.0864864
$726$	0	0
$727$	10.3290	0.383082	0.191541	−	0.981485i	$-0.438652\pi$
$727$	10.3290	0.383082	0.191541	+	0.981485i	$0.438652\pi$
$728$	48.4350	1.79512
$729$	0	0
$730$	−3.35666	−0.124235
$731$	−0.902631	−0.0333850
$732$	0	0
$733$	−27.0182	−0.997938	−0.498969	−	0.866620i	$-0.666288\pi$
$733$	−27.0182	−0.997938	−0.498969	+	0.866620i	$0.666288\pi$
$734$	−57.0324	−2.10510
$735$	0	0
$736$	−24.8793	−0.917063
$737$	−0.152242	−0.00560791
$738$	0	0
$739$	20.7027	0.761560	0.380780	−	0.924666i	$-0.375655\pi$
$739$	20.7027	0.761560	0.380780	+	0.924666i	$0.375655\pi$
$740$	−15.7976	−0.580732
$741$	0	0
$742$	−55.8491	−2.05029
$743$	−40.3655	−1.48087	−0.740433	−	0.672130i	$-0.765380\pi$
$743$	−40.3655	−1.48087	−0.740433	+	0.672130i	$0.765380\pi$
$744$	0	0
$745$	−1.09931	−0.0402757
$746$	75.7686	2.77409
$747$	0	0
$748$	0.963098	0.0352143
$749$	−17.8656	−0.652795
$750$	0	0
$751$	9.23655	0.337046	0.168523	−	0.985698i	$-0.446100\pi$
$751$	9.23655	0.337046	0.168523	+	0.985698i	$0.446100\pi$
$752$	56.6266	2.06496
$753$	0	0
$754$	−15.6448	−0.569749
$755$	−7.87683	−0.286667
$756$	0	0
$757$	10.6745	0.387973	0.193986	−	0.981004i	$-0.437858\pi$
$757$	10.6745	0.387973	0.193986	+	0.981004i	$0.437858\pi$
$758$	−37.0439	−1.34550
$759$	0	0
$760$	−35.7537	−1.29692
$761$	21.3301	0.773217	0.386608	−	0.922244i	$-0.373647\pi$
$761$	21.3301	0.773217	0.386608	+	0.922244i	$0.373647\pi$
$762$	0	0
$763$	41.5876	1.50557
$764$	56.0002	2.02601
$765$	0	0
$766$	−35.3782	−1.27827
$767$	−8.73063	−0.315245
$768$	0	0
$769$	25.6062	0.923385	0.461692	−	0.887040i	$-0.347242\pi$
$769$	25.6062	0.923385	0.461692	+	0.887040i	$0.347242\pi$
$770$	1.95279	0.0703737
$771$	0	0
$772$	50.1195	1.80384
$773$	21.4476	0.771416	0.385708	−	0.922621i	$-0.373957\pi$
$773$	21.4476	0.771416	0.385708	+	0.922621i	$0.373957\pi$
$774$	0	0
$775$	7.73623	0.277894
$776$	−25.9623	−0.931992
$777$	0	0
$778$	77.7657	2.78803
$779$	58.5683	2.09843
$780$	0	0
$781$	2.10073	0.0751700
$782$	−17.4364	−0.623524
$783$	0	0
$784$	16.6569	0.594890
$785$	−3.04822	−0.108796
$786$	0	0
$787$	−43.6900	−1.55738	−0.778691	−	0.627408i	$-0.784116\pi$
$787$	−43.6900	−1.55738	−0.778691	+	0.627408i	$0.784116\pi$
$788$	65.7943	2.34382
$789$	0	0
$790$	16.7020	0.594230
$791$	−27.3217	−0.971446
$792$	0	0
$793$	8.97594	0.318745
$794$	−77.9888	−2.76772
$795$	0	0
$796$	3.08791	0.109448
$797$	46.8097	1.65809	0.829043	−	0.559185i	$-0.188886\pi$
$797$	46.8097	1.65809	0.829043	+	0.559185i	$0.188886\pi$
$798$	0	0
$799$	8.68230	0.307158
$800$	−3.23256	−0.114288
$801$	0	0
$802$	−12.3726	−0.436893
$803$	−0.331886	−0.0117120
$804$	0	0
$805$	−24.1299	−0.850467
$806$	51.9735	1.83069
$807$	0	0
$808$	35.9720	1.26549
$809$	−2.62610	−0.0923288	−0.0461644	−	0.998934i	$-0.514700\pi$
$809$	−2.62610	−0.0923288	−0.0461644	+	0.998934i	$0.514700\pi$
$810$	0	0
$811$	36.1962	1.27102	0.635510	−	0.772093i	$-0.280790\pi$
$811$	36.1962	1.27102	0.635510	+	0.772093i	$0.280790\pi$
$812$	−31.3909	−1.10160
$813$	0	0
$814$	−2.28855	−0.0802136
$815$	−7.04394	−0.246739
$816$	0	0
$817$	6.19474	0.216726
$818$	19.1226	0.668604
$819$	0	0
$820$	40.6502	1.41957
$821$	31.4660	1.09817	0.549085	−	0.835767i	$-0.314976\pi$
$821$	31.4660	1.09817	0.549085	+	0.835767i	$0.314976\pi$
$822$	0	0
$823$	−43.9257	−1.53115	−0.765577	−	0.643344i	$-0.777546\pi$
$823$	−43.9257	−1.53115	−0.765577	+	0.643344i	$0.777546\pi$
$824$	−42.0873	−1.46618
$825$	0	0
$826$	−25.6665	−0.893051
$827$	−28.7775	−1.00069	−0.500346	−	0.865825i	$-0.666794\pi$
$827$	−28.7775	−1.00069	−0.500346	+	0.865825i	$0.666794\pi$
$828$	0	0
$829$	31.2180	1.08425	0.542123	−	0.840299i	$-0.317621\pi$
$829$	31.2180	1.08425	0.542123	+	0.840299i	$0.317621\pi$
$830$	−26.8966	−0.933595
$831$	0	0
$832$	9.79851	0.339702
$833$	2.55393	0.0884883
$834$	0	0
$835$	1.89121	0.0654479
$836$	−6.60972	−0.228602
$837$	0	0
$838$	54.8092	1.89335
$839$	16.8160	0.580554	0.290277	−	0.956943i	$-0.406253\pi$
$839$	16.8160	0.580554	0.290277	+	0.956943i	$0.406253\pi$
$840$	0	0
$841$	−23.5771	−0.813003
$842$	33.8866	1.16781
$843$	0	0
$844$	87.2000	3.00155
$845$	5.83534	0.200742
$846$	0	0
$847$	−34.2940	−1.17836
$848$	41.7824	1.43481
$849$	0	0
$850$	−2.26550	−0.0777062
$851$	28.2787	0.969383
$852$	0	0
$853$	−20.6282	−0.706295	−0.353148	−	0.935568i	$-0.614889\pi$
$853$	−20.6282	−0.706295	−0.353148	+	0.935568i	$0.614889\pi$
$854$	26.3877	0.902967
$855$	0	0
$856$	32.8891	1.12412
$857$	32.8803	1.12317	0.561585	−	0.827419i	$-0.310192\pi$
$857$	32.8803	1.12317	0.561585	+	0.827419i	$0.310192\pi$
$858$	0	0
$859$	−43.1433	−1.47203	−0.736016	−	0.676964i	$-0.763295\pi$
$859$	−43.1433	−1.47203	−0.736016	+	0.676964i	$0.763295\pi$
$860$	4.29955	0.146613
$861$	0	0
$862$	−18.3216	−0.624036
$863$	45.6650	1.55445	0.777227	−	0.629221i	$-0.216626\pi$
$863$	45.6650	1.55445	0.777227	+	0.629221i	$0.216626\pi$
$864$	0	0
$865$	7.44534	0.253149
$866$	−25.4277	−0.864070
$867$	0	0
$868$	104.284	3.53962
$869$	1.65139	0.0560197
$870$	0	0
$871$	1.64208	0.0556399
$872$	−76.5592	−2.59262
$873$	0	0
$874$	119.666	4.04775
$875$	−3.13519	−0.105989
$876$	0	0
$877$	15.9707	0.539294	0.269647	−	0.962959i	$-0.413093\pi$
$877$	15.9707	0.539294	0.269647	+	0.962959i	$0.413093\pi$
$878$	44.3372	1.49631
$879$	0	0
$880$	−1.46094	−0.0492484
$881$	−13.7670	−0.463823	−0.231912	−	0.972737i	$-0.574498\pi$
$881$	−13.7670	−0.463823	−0.231912	+	0.972737i	$0.574498\pi$
$882$	0	0
$883$	12.0482	0.405455	0.202728	−	0.979235i	$-0.435019\pi$
$883$	12.0482	0.405455	0.202728	+	0.979235i	$0.435019\pi$
$884$	−10.3880	−0.349385
$885$	0	0
$886$	92.3475	3.10247
$887$	14.1405	0.474791	0.237395	−	0.971413i	$-0.423706\pi$
$887$	14.1405	0.474791	0.237395	+	0.971413i	$0.423706\pi$
$888$	0	0
$889$	57.6484	1.93347
$890$	3.38458	0.113451
$891$	0	0
$892$	96.3111	3.22474
$893$	−59.5864	−1.99398
$894$	0	0
$895$	−10.1784	−0.340227
$896$	49.0753	1.63949
$897$	0	0
$898$	1.98020	0.0660802
$899$	−18.0155	−0.600850
$900$	0	0
$901$	6.40630	0.213425
$902$	5.88886	0.196078
$903$	0	0
$904$	50.2968	1.67285
$905$	−13.0309	−0.433162
$906$	0	0
$907$	2.93772	0.0975455	0.0487727	−	0.998810i	$-0.484469\pi$
$907$	2.93772	0.0975455	0.0487727	+	0.998810i	$0.484469\pi$
$908$	99.0370	3.28666
$909$	0	0
$910$	−21.0628	−0.698226
$911$	−1.04732	−0.0346992	−0.0173496	−	0.999849i	$-0.505523\pi$
$911$	−1.04732	−0.0346992	−0.0173496	+	0.999849i	$0.505523\pi$
$912$	0	0
$913$	−2.65938	−0.0880125
$914$	−59.5952	−1.97124
$915$	0	0
$916$	54.4528	1.79917
$917$	−19.1334	−0.631839
$918$	0	0
$919$	15.7325	0.518966	0.259483	−	0.965748i	$-0.416448\pi$
$919$	15.7325	0.518966	0.259483	+	0.965748i	$0.416448\pi$
$920$	44.4211	1.46452
$921$	0	0
$922$	26.4053	0.869611
$923$	−22.6585	−0.745813
$924$	0	0
$925$	3.67425	0.120809
$926$	84.6866	2.78297
$927$	0	0
$928$	7.52772	0.247109
$929$	26.5084	0.869711	0.434855	−	0.900500i	$-0.356799\pi$
$929$	26.5084	0.869711	0.434855	+	0.900500i	$0.356799\pi$
$930$	0	0
$931$	−17.5275	−0.574442
$932$	130.109	4.26187
$933$	0	0
$934$	48.3568	1.58228
$935$	−0.224000	−0.00732557
$936$	0	0
$937$	25.7747	0.842021	0.421011	−	0.907056i	$-0.361676\pi$
$937$	25.7747	0.842021	0.421011	+	0.907056i	$0.361676\pi$
$938$	4.82743	0.157621
$939$	0	0
$940$	−41.3569	−1.34891
$941$	33.5036	1.09219	0.546093	−	0.837725i	$-0.316115\pi$
$941$	33.5036	1.09219	0.546093	+	0.837725i	$0.316115\pi$
$942$	0	0
$943$	−72.7664	−2.36960
$944$	19.2019	0.624968
$945$	0	0
$946$	0.622862	0.0202510
$947$	12.5066	0.406411	0.203205	−	0.979136i	$-0.434864\pi$
$947$	12.5066	0.406411	0.203205	+	0.979136i	$0.434864\pi$
$948$	0	0
$949$	3.57973	0.116203
$950$	15.5481	0.504447
$951$	0	0
$952$	−16.3332	−0.529363
$953$	−13.3246	−0.431628	−0.215814	−	0.976435i	$-0.569240\pi$
$953$	−13.3246	−0.431628	−0.215814	+	0.976435i	$0.569240\pi$
$954$	0	0
$955$	−13.0247	−0.421468
$956$	−79.5557	−2.57302
$957$	0	0
$958$	−24.3376	−0.786312
$959$	30.0090	0.969042
$960$	0	0
$961$	28.8493	0.930623
$962$	24.6843	0.795854
$963$	0	0
$964$	−69.1540	−2.22730
$965$	−11.6569	−0.375249
$966$	0	0
$967$	−35.1437	−1.13014	−0.565072	−	0.825042i	$-0.691152\pi$
$967$	−35.1437	−1.13014	−0.565072	+	0.825042i	$0.691152\pi$
$968$	63.1324	2.02915
$969$	0	0
$970$	11.2902	0.362505
$971$	−54.0062	−1.73314	−0.866571	−	0.499054i	$-0.833681\pi$
$971$	−54.0062	−1.73314	−0.866571	+	0.499054i	$0.833681\pi$
$972$	0	0
$973$	−24.6513	−0.790284
$974$	1.71553	0.0549692
$975$	0	0
$976$	−19.7414	−0.631907
$977$	−0.447973	−0.0143319	−0.00716596	−	0.999974i	$-0.502281\pi$
$977$	−0.447973	−0.0143319	−0.00716596	+	0.999974i	$0.502281\pi$
$978$	0	0
$979$	0.334647	0.0106954
$980$	−12.1653	−0.388605
$981$	0	0
$982$	−73.0473	−2.33103
$983$	−43.5886	−1.39026	−0.695130	−	0.718884i	$-0.744653\pi$
$983$	−43.5886	−1.39026	−0.695130	+	0.718884i	$0.744653\pi$
$984$	0	0
$985$	−15.3026	−0.487581
$986$	5.27572	0.168013
$987$	0	0
$988$	71.2925	2.26812
$989$	−7.69647	−0.244733
$990$	0	0
$991$	−47.3742	−1.50489	−0.752445	−	0.658655i	$-0.771126\pi$
$991$	−47.3742	−1.50489	−0.752445	+	0.658655i	$0.771126\pi$
$992$	−25.0078	−0.794000
$993$	0	0
$994$	−66.6119	−2.11280
$995$	−0.718193	−0.0227682
$996$	0	0
$997$	−9.73338	−0.308259	−0.154130	−	0.988051i	$-0.549257\pi$
$997$	−9.73338	−0.308259	−0.154130	+	0.988051i	$0.549257\pi$
$998$	−77.2135	−2.44415
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1935.2.a.u.1.1		5
3.2	odd	2		215.2.a.c.1.5	✓	5
5.4	even	2		9675.2.a.ch.1.5		5
12.11	even	2		3440.2.a.w.1.5		5
15.2	even	4		1075.2.b.h.474.10		10
15.8	even	4		1075.2.b.h.474.1		10
15.14	odd	2		1075.2.a.m.1.1		5
129.128	even	2		9245.2.a.l.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.c.1.5	✓	5	3.2	odd	2
1075.2.a.m.1.1		5	15.14	odd	2
1075.2.b.h.474.1		10	15.8	even	4
1075.2.b.h.474.10		10	15.2	even	4
1935.2.a.u.1.1		5	1.1	even	1	trivial
3440.2.a.w.1.5		5	12.11	even	2
9245.2.a.l.1.1		5	129.128	even	2
9675.2.a.ch.1.5		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1935 = 3^{2} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1935.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50989 q^{2} +4.29955 q^{4} -1.00000 q^{5} +3.13519 q^{7} -5.77162 q^{8} +O(q^{10})\)	\(q-2.50989 q^{2} +4.29955 q^{4} -1.00000 q^{5} +3.13519 q^{7} -5.77162 q^{8} +2.50989 q^{10} +0.248163 q^{11} -2.67669 q^{13} -7.86899 q^{14} +5.88703 q^{16} +0.902631 q^{17} -6.19474 q^{19} -4.29955 q^{20} -0.622862 q^{22} +7.69647 q^{23} +1.00000 q^{25} +6.71819 q^{26} +13.4799 q^{28} -2.32872 q^{29} +7.73623 q^{31} -3.23256 q^{32} -2.26550 q^{34} -3.13519 q^{35} +3.67425 q^{37} +15.5481 q^{38} +5.77162 q^{40} -9.45452 q^{41} -1.00000 q^{43} +1.06699 q^{44} -19.3173 q^{46} +9.61888 q^{47} +2.82943 q^{49} -2.50989 q^{50} -11.5086 q^{52} +7.09737 q^{53} -0.248163 q^{55} -18.0951 q^{56} +5.84482 q^{58} +3.26173 q^{59} -3.35338 q^{61} -19.4171 q^{62} -3.66068 q^{64} +2.67669 q^{65} -0.613476 q^{67} +3.88091 q^{68} +7.86899 q^{70} +8.46512 q^{71} -1.33737 q^{73} -9.22196 q^{74} -26.6346 q^{76} +0.778039 q^{77} +6.65447 q^{79} -5.88703 q^{80} +23.7298 q^{82} -10.7162 q^{83} -0.902631 q^{85} +2.50989 q^{86} -1.43230 q^{88} +1.34850 q^{89} -8.39193 q^{91} +33.0913 q^{92} -24.1423 q^{94} +6.19474 q^{95} +4.49827 q^{97} -7.10155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8	\( 5 q - 2 q^{2} + 8 q^{4} - 5 q^{5} + 5 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} + 5 q^{13} - q^{14} + 14 q^{16} + 17 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{22} - q^{23} + 5 q^{25} - 22 q^{26} + 26 q^{28} - 6 q^{29} + 6 q^{31} + 7 q^{32} - 5 q^{35} + 5 q^{37} + 16 q^{38} + 3 q^{40} - 2 q^{41} - 5 q^{43} + 15 q^{44} - 14 q^{46} + 18 q^{49} - 2 q^{50} - 38 q^{52} + 23 q^{53} - 6 q^{55} + 19 q^{56} + 12 q^{58} + q^{59} + 20 q^{61} + 3 q^{62} - 25 q^{64} - 5 q^{65} + 21 q^{67} + 48 q^{68} + q^{70} - 4 q^{71} + 5 q^{73} - 24 q^{74} + 32 q^{76} + 26 q^{77} + 41 q^{79} - 14 q^{80} + 38 q^{82} + 7 q^{83} - 17 q^{85} + 2 q^{86} + 12 q^{88} - 20 q^{89} - 42 q^{91} + 52 q^{92} - 42 q^{94} + 6 q^{95} + 37 q^{97} + 26 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 8 * q^4 - 5 * q^5 + 5 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 + 5 * q^13 - q^14 + 14 * q^16 + 17 * q^17 - 6 * q^19 - 8 * q^20 - 8 * q^22 - q^23 + 5 * q^25 - 22 * q^26 + 26 * q^28 - 6 * q^29 + 6 * q^31 + 7 * q^32 - 5 * q^35 + 5 * q^37 + 16 * q^38 + 3 * q^40 - 2 * q^41 - 5 * q^43 + 15 * q^44 - 14 * q^46 + 18 * q^49 - 2 * q^50 - 38 * q^52 + 23 * q^53 - 6 * q^55 + 19 * q^56 + 12 * q^58 + q^59 + 20 * q^61 + 3 * q^62 - 25 * q^64 - 5 * q^65 + 21 * q^67 + 48 * q^68 + q^70 - 4 * q^71 + 5 * q^73 - 24 * q^74 + 32 * q^76 + 26 * q^77 + 41 * q^79 - 14 * q^80 + 38 * q^82 + 7 * q^83 - 17 * q^85 + 2 * q^86 + 12 * q^88 - 20 * q^89 - 42 * q^91 + 52 * q^92 - 42 * q^94 + 6 * q^95 + 37 * q^97 + 26 * q^98

Introduction

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