LMFDB - Embedded newform 2900.2.c.f.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2900,2,Mod(349,2900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2900.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 580)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$1.66044 + 1.66044i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.f.349.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.32088i q^{3} +1.32088i q^{7} -8.02827 q^{9} +O(q^{10})$	$q-3.32088i q^{3} +1.32088i q^{7} -8.02827 q^{9} +5.32088 q^{11} +5.02827i q^{13} -6.34916i q^{17} +4.34916 q^{19} +4.38650 q^{21} +1.70739i q^{23} +16.6983i q^{27} +1.00000 q^{29} +8.34916 q^{31} -17.6700i q^{33} -6.93438i q^{37} +16.6983 q^{39} -1.02827 q^{41} -10.7357i q^{43} -0.679116i q^{47} +5.25526 q^{49} -21.0848 q^{51} -2.38650i q^{53} -14.4431i q^{57} -10.4431 q^{59} -6.38650 q^{61} -10.6044i q^{63} -5.70739i q^{67} +5.67004 q^{69} -3.61350 q^{71} +6.73566i q^{73} +7.02827i q^{77} +11.3774 q^{79} +31.3684 q^{81} +3.96265i q^{83} -3.32088i q^{87} -2.58522 q^{89} -6.64177 q^{91} -27.7266i q^{93} -15.3209i q^{97} -42.7175 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 22 q^{9}+O(q^{10}) $ 6 * q - 22 * q^9	$ 6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91} - 72 q^{99}+O(q^{100}) $ 6 * q - 22 * q^9 + 16 * q^11 - 16 * q^19 + 32 * q^21 + 6 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 6 * q^49 - 48 * q^51 - 16 * q^59 - 44 * q^61 - 24 * q^69 - 16 * q^71 + 46 * q^81 - 36 * q^89 - 8 * q^91 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 3.32088i	− 1.91731i	−0.284565	−	0.958657i	$-0.591849\pi$
$3$	− 3.32088i	− 1.91731i	0.284565	−	0.958657i	$-0.408151\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.32088i	0.499247i	0.968343	+	0.249624i	$0.0803069\pi$
$7$	1.32088i	0.499247i	−0.968343	+	0.249624i	$0.919693\pi$
$8$	0	0
$9$	−8.02827	−2.67609
$10$	0	0
$11$	5.32088	1.60431	0.802154	−	0.597118i	$-0.203688\pi$
$11$	5.32088	1.60431	0.802154	+	0.597118i	$0.203688\pi$
$12$	0	0
$13$	5.02827i	1.39459i	0.716783	+	0.697296i	$0.245614\pi$
$13$	5.02827i	1.39459i	−0.716783	+	0.697296i	$0.754386\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.34916i	− 1.53990i	−0.638106	−	0.769949i	$-0.720282\pi$
$17$	− 6.34916i	− 1.53990i	0.638106	−	0.769949i	$-0.279718\pi$
$18$	0	0
$19$	4.34916	0.997765	0.498883	−	0.866670i	$-0.333744\pi$
$19$	4.34916	0.997765	0.498883	+	0.866670i	$0.333744\pi$
$20$	0	0
$21$	4.38650	0.957214
$22$	0	0
$23$	1.70739i	0.356015i	0.984029	+	0.178008i	$0.0569652\pi$
$23$	1.70739i	0.356015i	−0.984029	+	0.178008i	$0.943035\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	16.6983i	3.21359i
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	8.34916	1.49955	0.749777	−	0.661691i	$-0.230161\pi$
$31$	8.34916	1.49955	0.749777	+	0.661691i	$0.230161\pi$
$32$	0	0
$33$	− 17.6700i	− 3.07596i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.93438i	− 1.14000i	−0.821643	−	0.570002i	$-0.806942\pi$
$37$	− 6.93438i	− 1.14000i	0.821643	−	0.570002i	$-0.193058\pi$
$38$	0	0
$39$	16.6983	2.67387
$40$	0	0
$41$	−1.02827	−0.160589	−0.0802947	−	0.996771i	$-0.525586\pi$
$41$	−1.02827	−0.160589	−0.0802947	+	0.996771i	$0.525586\pi$
$42$	0	0
$43$	− 10.7357i	− 1.63717i	−0.574383	−	0.818587i	$-0.694758\pi$
$43$	− 10.7357i	− 1.63717i	0.574383	−	0.818587i	$-0.305242\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.679116i	− 0.0990592i	−0.998773	−	0.0495296i	$-0.984228\pi$
$47$	− 0.679116i	− 0.0990592i	0.998773	−	0.0495296i	$-0.0157722\pi$
$48$	0	0
$49$	5.25526	0.750752
$50$	0	0
$51$	−21.0848	−2.95247
$52$	0	0
$53$	− 2.38650i	− 0.327812i	−0.986476	−	0.163906i	$-0.947591\pi$
$53$	− 2.38650i	− 0.327812i	0.986476	−	0.163906i	$-0.0524093\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 14.4431i	− 1.91303i
$58$	0	0
$59$	−10.4431	−1.35957	−0.679785	−	0.733412i	$-0.737927\pi$
$59$	−10.4431	−1.35957	−0.679785	+	0.733412i	$0.737927\pi$
$60$	0	0
$61$	−6.38650	−0.817708	−0.408854	−	0.912600i	$-0.634072\pi$
$61$	−6.38650	−0.817708	−0.408854	+	0.912600i	$0.634072\pi$
$62$	0	0
$63$	− 10.6044i	− 1.33603i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.70739i	− 0.697269i	−0.937259	−	0.348634i	$-0.886646\pi$
$67$	− 5.70739i	− 0.697269i	0.937259	−	0.348634i	$-0.113354\pi$
$68$	0	0
$69$	5.67004	0.682593
$70$	0	0
$71$	−3.61350	−0.428843	−0.214421	−	0.976741i	$-0.568787\pi$
$71$	−3.61350	−0.428843	−0.214421	+	0.976741i	$0.568787\pi$
$72$	0	0
$73$	6.73566i	0.788350i	0.919035	+	0.394175i	$0.128970\pi$
$73$	6.73566i	0.788350i	−0.919035	+	0.394175i	$0.871030\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.02827i	0.800946i
$78$	0	0
$79$	11.3774	1.28006	0.640031	−	0.768349i	$-0.278922\pi$
$79$	11.3774	1.28006	0.640031	+	0.768349i	$0.278922\pi$
$80$	0	0
$81$	31.3684	3.48537
$82$	0	0
$83$	3.96265i	0.434958i	0.976065	+	0.217479i	$0.0697833\pi$
$83$	3.96265i	0.434958i	−0.976065	+	0.217479i	$0.930217\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.32088i	− 0.356036i
$88$	0	0
$89$	−2.58522	−0.274033	−0.137016	−	0.990569i	$-0.543751\pi$
$89$	−2.58522	−0.274033	−0.137016	+	0.990569i	$0.543751\pi$
$90$	0	0
$91$	−6.64177	−0.696247
$92$	0	0
$93$	− 27.7266i	− 2.87511i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 15.3209i	− 1.55560i	−0.628512	−	0.777800i	$-0.716336\pi$
$97$	− 15.3209i	− 1.55560i	0.628512	−	0.777800i	$-0.283664\pi$
$98$	0	0
$99$	−42.7175	−4.29327
$100$	0	0
$101$	11.8688	1.18099	0.590493	−	0.807043i	$-0.298933\pi$
$101$	11.8688	1.18099	0.590493	+	0.807043i	$0.298933\pi$
$102$	0	0
$103$	3.65084i	0.359728i	0.983691	+	0.179864i	$0.0575658\pi$
$103$	3.65084i	0.359728i	−0.983691	+	0.179864i	$0.942434\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.32088i	− 0.514389i	−0.966360	−	0.257195i	$-0.917202\pi$
$107$	− 5.32088i	− 0.514389i	0.966360	−	0.257195i	$-0.0827982\pi$
$108$	0	0
$109$	14.3118	1.37082	0.685411	−	0.728156i	$-0.259622\pi$
$109$	14.3118	1.37082	0.685411	+	0.728156i	$0.259622\pi$
$110$	0	0
$111$	−23.0283	−2.18575
$112$	0	0
$113$	− 2.03735i	− 0.191657i	−0.995398	−	0.0958287i	$-0.969450\pi$
$113$	− 2.03735i	− 0.191657i	0.995398	−	0.0958287i	$-0.0305501\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 40.3684i	− 3.73206i
$118$	0	0
$119$	8.38650	0.768790
$120$	0	0
$121$	17.3118	1.57380
$122$	0	0
$123$	3.41478i	0.307900i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.26434i	− 0.112192i	−0.998425	−	0.0560959i	$-0.982135\pi$
$127$	− 1.26434i	− 0.112192i	0.998425	−	0.0560959i	$-0.0178652\pi$
$128$	0	0
$129$	−35.6519	−3.13897
$130$	0	0
$131$	−1.70739	−0.149175	−0.0745877	−	0.997214i	$-0.523764\pi$
$131$	−1.70739	−0.149175	−0.0745877	+	0.997214i	$0.523764\pi$
$132$	0	0
$133$	5.74474i	0.498132i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.7639i	1.85942i	0.368294	+	0.929709i	$0.379942\pi$
$137$	21.7639i	1.85942i	−0.368294	+	0.929709i	$0.620058\pi$
$138$	0	0
$139$	16.6983	1.41633	0.708166	−	0.706046i	$-0.249523\pi$
$139$	16.6983	1.41633	0.708166	+	0.706046i	$0.249523\pi$
$140$	0	0
$141$	−2.25526	−0.189928
$142$	0	0
$143$	26.7549i	2.23735i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 17.4521i	− 1.43943i
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	6.44305	0.524328	0.262164	−	0.965023i	$-0.415564\pi$
$151$	6.44305	0.524328	0.262164	+	0.965023i	$0.415564\pi$
$152$	0	0
$153$	50.9728i	4.12091i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.7357i	0.856799i	0.903589	+	0.428400i	$0.140922\pi$
$157$	10.7357i	0.856799i	−0.903589	+	0.428400i	$0.859078\pi$
$158$	0	0
$159$	−7.92531	−0.628518
$160$	0	0
$161$	−2.25526	−0.177740
$162$	0	0
$163$	− 3.90611i	− 0.305950i	−0.988230	−	0.152975i	$-0.951115\pi$
$163$	− 3.90611i	− 0.305950i	0.988230	−	0.152975i	$-0.0488854\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.51960i	− 0.427120i	−0.976930	−	0.213560i	$-0.931494\pi$
$167$	− 5.51960i	− 0.427120i	0.976930	−	0.213560i	$-0.0685058\pi$
$168$	0	0
$169$	−12.2835	−0.944888
$170$	0	0
$171$	−34.9162	−2.67011
$172$	0	0
$173$	− 5.80128i	− 0.441063i	−0.975380	−	0.220532i	$-0.929221\pi$
$173$	− 5.80128i	− 0.441063i	0.975380	−	0.220532i	$-0.0707792\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	34.6802i	2.60672i
$178$	0	0
$179$	−14.0565	−1.05064	−0.525318	−	0.850906i	$-0.676054\pi$
$179$	−14.0565	−1.05064	−0.525318	+	0.850906i	$0.676054\pi$
$180$	0	0
$181$	−7.08482	−0.526611	−0.263305	−	0.964713i	$-0.584813\pi$
$181$	−7.08482	−0.526611	−0.263305	+	0.964713i	$0.584813\pi$
$182$	0	0
$183$	21.2088i	1.56780i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 33.7831i	− 2.47047i
$188$	0	0
$189$	−22.0565	−1.60438
$190$	0	0
$191$	−0.547875	−0.0396428	−0.0198214	−	0.999804i	$-0.506310\pi$
$191$	−0.547875	−0.0396428	−0.0198214	+	0.999804i	$0.506310\pi$
$192$	0	0
$193$	− 4.40571i	− 0.317130i	−0.987349	−	0.158565i	$-0.949313\pi$
$193$	− 4.40571i	− 0.317130i	0.987349	−	0.158565i	$-0.0506867\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.64177i	− 0.330712i	−0.986234	−	0.165356i	$-0.947123\pi$
$197$	− 4.64177i	− 0.330712i	0.986234	−	0.165356i	$-0.0528774\pi$
$198$	0	0
$199$	−18.4431	−1.30739	−0.653697	−	0.756757i	$-0.726783\pi$
$199$	−18.4431	−1.30739	−0.653697	+	0.756757i	$0.726783\pi$
$200$	0	0
$201$	−18.9536	−1.33688
$202$	0	0
$203$	1.32088i	0.0927079i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 13.7074i	− 0.952729i
$208$	0	0
$209$	23.1414	1.60072
$210$	0	0
$211$	−4.54787	−0.313089	−0.156544	−	0.987671i	$-0.550035\pi$
$211$	−4.54787	−0.313089	−0.156544	+	0.987671i	$0.550035\pi$
$212$	0	0
$213$	12.0000i	0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.0283i	0.748648i
$218$	0	0
$219$	22.3684	1.51151
$220$	0	0
$221$	31.9253	2.14753
$222$	0	0
$223$	− 1.12217i	− 0.0751459i	−0.999294	−	0.0375730i	$-0.988037\pi$
$223$	− 1.12217i	− 0.0751459i	0.999294	−	0.0375730i	$-0.0119627\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.63270i	0.639345i	0.947528	+	0.319672i	$0.103573\pi$
$227$	9.63270i	0.639345i	−0.947528	+	0.319672i	$0.896427\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	23.3401	1.53566
$232$	0	0
$233$	8.05655i	0.527802i	0.964550	+	0.263901i	$0.0850092\pi$
$233$	8.05655i	0.527802i	−0.964550	+	0.263901i	$0.914991\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 37.7831i	− 2.45428i
$238$	0	0
$239$	−15.6135	−1.00995	−0.504977	−	0.863133i	$-0.668499\pi$
$239$	−15.6135	−1.00995	−0.504977	+	0.863133i	$0.668499\pi$
$240$	0	0
$241$	18.5105	1.19237	0.596184	−	0.802848i	$-0.296683\pi$
$241$	18.5105	1.19237	0.596184	+	0.802848i	$0.296683\pi$
$242$	0	0
$243$	− 54.0757i	− 3.46896i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21.8688i	1.39148i
$248$	0	0
$249$	13.1595	0.833950
$250$	0	0
$251$	−6.67912	−0.421582	−0.210791	−	0.977531i	$-0.567604\pi$
$251$	−6.67912	−0.421582	−0.210791	+	0.977531i	$0.567604\pi$
$252$	0	0
$253$	9.08482i	0.571158i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 26.4249i	− 1.64834i	−0.566342	−	0.824170i	$-0.691642\pi$
$257$	− 26.4249i	− 1.64834i	0.566342	−	0.824170i	$-0.308358\pi$
$258$	0	0
$259$	9.15951	0.569145
$260$	0	0
$261$	−8.02827	−0.496938
$262$	0	0
$263$	− 27.4340i	− 1.69165i	−0.533459	−	0.845826i	$-0.679108\pi$
$263$	− 27.4340i	− 1.69165i	0.533459	−	0.845826i	$-0.320892\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.58522i	0.525407i
$268$	0	0
$269$	10.9717	0.668958	0.334479	−	0.942403i	$-0.391440\pi$
$269$	10.9717	0.668958	0.334479	+	0.942403i	$0.391440\pi$
$270$	0	0
$271$	20.1504	1.22405	0.612026	−	0.790838i	$-0.290355\pi$
$271$	20.1504	1.22405	0.612026	+	0.790838i	$0.290355\pi$
$272$	0	0
$273$	22.0565i	1.33492i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.15951i	0.189837i	0.995485	+	0.0949184i	$0.0302590\pi$
$277$	3.15951i	0.189837i	−0.995485	+	0.0949184i	$0.969741\pi$
$278$	0	0
$279$	−67.0293	−4.01294
$280$	0	0
$281$	−10.3118	−0.615151	−0.307576	−	0.951524i	$-0.599518\pi$
$281$	−10.3118	−0.615151	−0.307576	+	0.951524i	$0.599518\pi$
$282$	0	0
$283$	− 0.423851i	− 0.0251953i	−0.999921	−	0.0125977i	$-0.995990\pi$
$283$	− 0.423851i	− 0.0251953i	0.999921	−	0.0125977i	$-0.00401007\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.35823i	− 0.0801738i
$288$	0	0
$289$	−23.3118	−1.37128
$290$	0	0
$291$	−50.8789	−2.98257
$292$	0	0
$293$	− 19.2462i	− 1.12437i	−0.827010	−	0.562187i	$-0.809960\pi$
$293$	− 19.2462i	− 1.12437i	0.827010	−	0.562187i	$-0.190040\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	88.8498i	5.15559i
$298$	0	0
$299$	−8.58522	−0.496496
$300$	0	0
$301$	14.1806	0.817355
$302$	0	0
$303$	− 39.4148i	− 2.26432i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.7357i	1.29759i	0.760962	+	0.648796i	$0.224727\pi$
$307$	22.7357i	1.29759i	−0.760962	+	0.648796i	$0.775273\pi$
$308$	0	0
$309$	12.1240	0.689712
$310$	0	0
$311$	14.8031	0.839409	0.419704	−	0.907661i	$-0.362134\pi$
$311$	14.8031	0.839409	0.419704	+	0.907661i	$0.362134\pi$
$312$	0	0
$313$	14.9717i	0.846252i	0.906071	+	0.423126i	$0.139067\pi$
$313$	14.9717i	0.846252i	−0.906071	+	0.423126i	$0.860933\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.1222i	1.29867i	0.760502	+	0.649335i	$0.224953\pi$
$317$	23.1222i	1.29867i	−0.760502	+	0.649335i	$0.775047\pi$
$318$	0	0
$319$	5.32088	0.297912
$320$	0	0
$321$	−17.6700	−0.986246
$322$	0	0
$323$	− 27.6135i	− 1.53646i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 47.5279i	− 2.62830i
$328$	0	0
$329$	0.897033	0.0494550
$330$	0	0
$331$	−9.82048	−0.539783	−0.269891	−	0.962891i	$-0.586988\pi$
$331$	−9.82048	−0.539783	−0.269891	+	0.962891i	$0.586988\pi$
$332$	0	0
$333$	55.6711i	3.05076i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8.09389i	− 0.440903i	−0.975398	−	0.220451i	$-0.929247\pi$
$337$	− 8.09389i	− 0.440903i	0.975398	−	0.220451i	$-0.0707530\pi$
$338$	0	0
$339$	−6.76579	−0.367467
$340$	0	0
$341$	44.4249	2.40574
$342$	0	0
$343$	16.1878i	0.874058i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.37743i	− 0.396041i	−0.980198	−	0.198021i	$-0.936549\pi$
$347$	− 7.37743i	− 0.396041i	0.980198	−	0.198021i	$-0.0634513\pi$
$348$	0	0
$349$	−8.82956	−0.472635	−0.236318	−	0.971676i	$-0.575941\pi$
$349$	−8.82956	−0.472635	−0.236318	+	0.971676i	$0.575941\pi$
$350$	0	0
$351$	−83.9637	−4.48165
$352$	0	0
$353$	− 13.7266i	− 0.730593i	−0.930891	−	0.365296i	$-0.880968\pi$
$353$	− 13.7266i	− 0.730593i	0.930891	−	0.365296i	$-0.119032\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 27.8506i	− 1.47401i
$358$	0	0
$359$	1.70739	0.0901126	0.0450563	−	0.998984i	$-0.485653\pi$
$359$	1.70739	0.0901126	0.0450563	+	0.998984i	$0.485653\pi$
$360$	0	0
$361$	−0.0848216	−0.00446429
$362$	0	0
$363$	− 57.4905i	− 3.01747i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 25.9627i	− 1.35524i	−0.735412	−	0.677620i	$-0.763012\pi$
$367$	− 25.9627i	− 1.35524i	0.735412	−	0.677620i	$-0.236988\pi$
$368$	0	0
$369$	8.25526	0.429752
$370$	0	0
$371$	3.15230	0.163659
$372$	0	0
$373$	− 9.92531i	− 0.513913i	−0.966423	−	0.256956i	$-0.917280\pi$
$373$	− 9.92531i	− 0.513913i	0.966423	−	0.256956i	$-0.0827197\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.02827i	0.258969i
$378$	0	0
$379$	−34.2070	−1.75710	−0.878548	−	0.477655i	$-0.841487\pi$
$379$	−34.2070	−1.75710	−0.878548	+	0.477655i	$0.841487\pi$
$380$	0	0
$381$	−4.19872	−0.215107
$382$	0	0
$383$	21.4340i	1.09523i	0.836732	+	0.547613i	$0.184463\pi$
$383$	21.4340i	1.09523i	−0.836732	+	0.547613i	$0.815537\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	86.1888i	4.38123i
$388$	0	0
$389$	36.3684	1.84395	0.921975	−	0.387251i	$-0.126575\pi$
$389$	36.3684	1.84395	0.921975	+	0.387251i	$0.126575\pi$
$390$	0	0
$391$	10.8405	0.548227
$392$	0	0
$393$	5.67004i	0.286016i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 27.4713i	− 1.37875i	−0.724406	−	0.689373i	$-0.757886\pi$
$397$	− 27.4713i	− 1.37875i	0.724406	−	0.689373i	$-0.242114\pi$
$398$	0	0
$399$	19.0776	0.955075
$400$	0	0
$401$	22.3118	1.11420	0.557099	−	0.830446i	$-0.311914\pi$
$401$	22.3118	1.11420	0.557099	+	0.830446i	$0.311914\pi$
$402$	0	0
$403$	41.9819i	2.09127i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 36.8970i	− 1.82892i
$408$	0	0
$409$	−4.95358	−0.244939	−0.122469	−	0.992472i	$-0.539081\pi$
$409$	−4.95358	−0.244939	−0.122469	+	0.992472i	$0.539081\pi$
$410$	0	0
$411$	72.2755	3.56509
$412$	0	0
$413$	− 13.7941i	− 0.678762i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 55.4532i	− 2.71555i
$418$	0	0
$419$	0.198716	0.00970793	0.00485397	−	0.999988i	$-0.498455\pi$
$419$	0.198716	0.00970793	0.00485397	+	0.999988i	$0.498455\pi$
$420$	0	0
$421$	26.4996	1.29151	0.645756	−	0.763544i	$-0.276542\pi$
$421$	26.4996	1.29151	0.645756	+	0.763544i	$0.276542\pi$
$422$	0	0
$423$	5.45213i	0.265091i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.43584i	− 0.408239i
$428$	0	0
$429$	88.8498	4.28971
$430$	0	0
$431$	−21.2835	−1.02519	−0.512596	−	0.858630i	$-0.671316\pi$
$431$	−21.2835	−1.02519	−0.512596	+	0.858630i	$0.671316\pi$
$432$	0	0
$433$	24.8031i	1.19196i	0.802998	+	0.595981i	$0.203237\pi$
$433$	24.8031i	1.19196i	−0.802998	+	0.595981i	$0.796763\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.42571i	0.355220i
$438$	0	0
$439$	32.3118	1.54216	0.771079	−	0.636739i	$-0.219717\pi$
$439$	32.3118	1.54216	0.771079	+	0.636739i	$0.219717\pi$
$440$	0	0
$441$	−42.1907	−2.00908
$442$	0	0
$443$	− 25.3774i	− 1.20572i	−0.797848	−	0.602859i	$-0.794028\pi$
$443$	− 25.3774i	− 1.20572i	0.797848	−	0.602859i	$-0.205972\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	59.7759i	2.82730i
$448$	0	0
$449$	−13.7266	−0.647798	−0.323899	−	0.946092i	$-0.604994\pi$
$449$	−13.7266	−0.647798	−0.323899	+	0.946092i	$0.604994\pi$
$450$	0	0
$451$	−5.47133	−0.257635
$452$	0	0
$453$	− 21.3966i	− 1.00530i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 15.0101i	− 0.702144i	−0.936348	−	0.351072i	$-0.885817\pi$
$457$	− 15.0101i	− 0.702144i	0.936348	−	0.351072i	$-0.114183\pi$
$458$	0	0
$459$	106.020	4.94860
$460$	0	0
$461$	32.7549	1.52555	0.762773	−	0.646666i	$-0.223837\pi$
$461$	32.7549	1.52555	0.762773	+	0.646666i	$0.223837\pi$
$462$	0	0
$463$	24.5369i	1.14033i	0.821531	+	0.570164i	$0.193120\pi$
$463$	24.5369i	1.14033i	−0.821531	+	0.570164i	$0.806880\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.60442i	− 0.398165i	−0.979983	−	0.199083i	$-0.936204\pi$
$467$	− 8.60442i	− 0.398165i	0.979983	−	0.199083i	$-0.0637962\pi$
$468$	0	0
$469$	7.53880	0.348110
$470$	0	0
$471$	35.6519	1.64275
$472$	0	0
$473$	− 57.1232i	− 2.62653i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	19.1595i	0.877254i
$478$	0	0
$479$	−3.26434	−0.149151	−0.0745757	−	0.997215i	$-0.523760\pi$
$479$	−3.26434	−0.149151	−0.0745757	+	0.997215i	$0.523760\pi$
$480$	0	0
$481$	34.8680	1.58984
$482$	0	0
$483$	7.48947i	0.340783i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	34.2070i	1.55007i	0.631920	+	0.775033i	$0.282267\pi$
$487$	34.2070i	1.55007i	−0.631920	+	0.775033i	$0.717733\pi$
$488$	0	0
$489$	−12.9717	−0.586602
$490$	0	0
$491$	−8.93438	−0.403203	−0.201601	−	0.979468i	$-0.564615\pi$
$491$	−8.93438	−0.403203	−0.201601	+	0.979468i	$0.564615\pi$
$492$	0	0
$493$	− 6.34916i	− 0.285952i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.77301i	− 0.214099i
$498$	0	0
$499$	−28.6236	−1.28137	−0.640685	−	0.767804i	$-0.721349\pi$
$499$	−28.6236	−1.28137	−0.640685	+	0.767804i	$0.721349\pi$
$500$	0	0
$501$	−18.3300	−0.818922
$502$	0	0
$503$	0.0192012i	0 0.000856140i	1.00000		0.000428070i	$0.000136259\pi$
$503$	0.0192012i	0 0.000856140i	−1.00000		0.000428070i	$0.999864\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	40.7922i	1.81165i
$508$	0	0
$509$	8.25526	0.365908	0.182954	−	0.983121i	$-0.441434\pi$
$509$	8.25526	0.365908	0.182954	+	0.983121i	$0.441434\pi$
$510$	0	0
$511$	−8.89703	−0.393582
$512$	0	0
$513$	72.6236i	3.20641i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.61350i	− 0.158921i
$518$	0	0
$519$	−19.2654	−0.845657
$520$	0	0
$521$	−20.2553	−0.887399	−0.443700	−	0.896176i	$-0.646334\pi$
$521$	−20.2553	−0.887399	−0.443700	+	0.896176i	$0.646334\pi$
$522$	0	0
$523$	− 1.82048i	− 0.0796042i	−0.999208	−	0.0398021i	$-0.987327\pi$
$523$	− 1.82048i	− 0.0796042i	0.999208	−	0.0398021i	$-0.0126728\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 53.0101i	− 2.30916i
$528$	0	0
$529$	20.0848	0.873253
$530$	0	0
$531$	83.8397	3.63833
$532$	0	0
$533$	− 5.17044i	− 0.223957i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	46.6802i	2.01440i
$538$	0	0
$539$	27.9627	1.20444
$540$	0	0
$541$	1.03920	0.0446788	0.0223394	−	0.999750i	$-0.492889\pi$
$541$	1.03920	0.0446788	0.0223394	+	0.999750i	$0.492889\pi$
$542$	0	0
$543$	23.5279i	1.00968i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.7567i	1.61436i	0.590305	+	0.807180i	$0.299008\pi$
$547$	37.7567i	1.61436i	−0.590305	+	0.807180i	$0.700992\pi$
$548$	0	0
$549$	51.2726	2.18826
$550$	0	0
$551$	4.34916	0.185280
$552$	0	0
$553$	15.0283i	0.639067i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.8506i	1.94275i	0.237545	+	0.971376i	$0.423657\pi$
$557$	45.8506i	1.94275i	−0.237545	+	0.971376i	$0.576343\pi$
$558$	0	0
$559$	53.9819	2.28319
$560$	0	0
$561$	−112.190	−4.73666
$562$	0	0
$563$	− 33.3027i	− 1.40354i	−0.712402	−	0.701772i	$-0.752393\pi$
$563$	− 33.3027i	− 1.40354i	0.712402	−	0.701772i	$-0.247607\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	41.4340i	1.74006i
$568$	0	0
$569$	23.6700	0.992300	0.496150	−	0.868237i	$-0.334747\pi$
$569$	23.6700	0.992300	0.496150	+	0.868237i	$0.334747\pi$
$570$	0	0
$571$	14.5671	0.609613	0.304807	−	0.952414i	$-0.401408\pi$
$571$	14.5671	0.609613	0.304807	+	0.952414i	$0.401408\pi$
$572$	0	0
$573$	1.81943i	0.0760077i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 13.7639i	− 0.573000i	−0.958080	−	0.286500i	$-0.907508\pi$
$577$	− 13.7639i	− 0.573000i	0.958080	−	0.286500i	$-0.0924919\pi$
$578$	0	0
$579$	−14.6308	−0.608037
$580$	0	0
$581$	−5.23421	−0.217152
$582$	0	0
$583$	− 12.6983i	− 0.525911i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.04748i	− 0.0432339i	−0.999766	−	0.0216170i	$-0.993119\pi$
$587$	− 1.04748i	− 0.0432339i	0.999766	−	0.0216170i	$-0.00688143\pi$
$588$	0	0
$589$	36.3118	1.49620
$590$	0	0
$591$	−15.4148	−0.634079
$592$	0	0
$593$	− 14.3865i	− 0.590783i	−0.955376	−	0.295391i	$-0.904550\pi$
$593$	− 14.3865i	− 0.590783i	0.955376	−	0.295391i	$-0.0954501\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	61.2472i	2.50668i
$598$	0	0
$599$	5.98080	0.244369	0.122184	−	0.992507i	$-0.461010\pi$
$599$	5.98080	0.244369	0.122184	+	0.992507i	$0.461010\pi$
$600$	0	0
$601$	−7.08482	−0.288996	−0.144498	−	0.989505i	$-0.546157\pi$
$601$	−7.08482	−0.288996	−0.144498	+	0.989505i	$0.546157\pi$
$602$	0	0
$603$	45.8205i	1.86595i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.7175i	1.32796i	0.747749	+	0.663982i	$0.231135\pi$
$607$	32.7175i	1.32796i	−0.747749	+	0.663982i	$0.768865\pi$
$608$	0	0
$609$	4.38650	0.177750
$610$	0	0
$611$	3.41478	0.138147
$612$	0	0
$613$	− 9.22699i	− 0.372675i	−0.982486	−	0.186337i	$-0.940338\pi$
$613$	− 9.22699i	− 0.372675i	0.982486	−	0.186337i	$-0.0596617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.281683i	0.0113401i	0.999984	+	0.00567006i	$0.00180485\pi$
$617$	0.281683i	0.0113401i	−0.999984	+	0.00567006i	$0.998195\pi$
$618$	0	0
$619$	−21.2462	−0.853957	−0.426978	−	0.904262i	$-0.640422\pi$
$619$	−21.2462	−0.853957	−0.426978	+	0.904262i	$0.640422\pi$
$620$	0	0
$621$	−28.5105	−1.14409
$622$	0	0
$623$	− 3.41478i	− 0.136810i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 76.8498i	− 3.06909i
$628$	0	0
$629$	−44.0275	−1.75549
$630$	0	0
$631$	37.8688	1.50753	0.753766	−	0.657143i	$-0.228235\pi$
$631$	37.8688	1.50753	0.753766	+	0.657143i	$0.228235\pi$
$632$	0	0
$633$	15.1030i	0.600289i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.4249i	1.04699i
$638$	0	0
$639$	29.0101	1.14762
$640$	0	0
$641$	−36.7658	−1.45216	−0.726081	−	0.687609i	$-0.758660\pi$
$641$	−36.7658	−1.45216	−0.726081	+	0.687609i	$0.758660\pi$
$642$	0	0
$643$	24.9728i	0.984830i	0.870360	+	0.492415i	$0.163886\pi$
$643$	24.9728i	0.984830i	−0.870360	+	0.492415i	$0.836114\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.56522i	− 0.140163i	−0.997541	−	0.0700816i	$-0.977674\pi$
$647$	− 3.56522i	− 0.140163i	0.997541	−	0.0700816i	$-0.0223260\pi$
$648$	0	0
$649$	−55.5663	−2.18117
$650$	0	0
$651$	36.6236	1.43539
$652$	0	0
$653$	21.5652i	0.843912i	0.906616	+	0.421956i	$0.138656\pi$
$653$	21.5652i	0.843912i	−0.906616	+	0.421956i	$0.861344\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 54.0757i	− 2.10970i
$658$	0	0
$659$	40.4623	1.57619	0.788093	−	0.615556i	$-0.211069\pi$
$659$	40.4623	1.57619	0.788093	+	0.615556i	$0.211069\pi$
$660$	0	0
$661$	26.1987	1.01901	0.509506	−	0.860467i	$-0.329828\pi$
$661$	26.1987	1.01901	0.509506	+	0.860467i	$0.329828\pi$
$662$	0	0
$663$	− 106.020i	− 4.11749i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.70739i	0.0661104i
$668$	0	0
$669$	−3.72659	−0.144078
$670$	0	0
$671$	−33.9819	−1.31185
$672$	0	0
$673$	− 3.35823i	− 0.129450i	−0.997903	−	0.0647251i	$-0.979383\pi$
$673$	− 3.35823i	− 0.129450i	0.997903	−	0.0647251i	$-0.0206171\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.6965i	− 0.603264i	−0.953424	−	0.301632i	$-0.902469\pi$
$677$	− 15.6965i	− 0.603264i	0.953424	−	0.301632i	$-0.0975314\pi$
$678$	0	0
$679$	20.2371	0.776629
$680$	0	0
$681$	31.9891	1.22582
$682$	0	0
$683$	40.1998i	1.53820i	0.639128	+	0.769101i	$0.279296\pi$
$683$	40.1998i	1.53820i	−0.639128	+	0.769101i	$0.720704\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.64177i	0.253399i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	25.7084	0.977995	0.488998	−	0.872285i	$-0.337363\pi$
$691$	25.7084	0.977995	0.488998	+	0.872285i	$0.337363\pi$
$692$	0	0
$693$	− 56.4249i	− 2.14340i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.52867i	0.247291i
$698$	0	0
$699$	26.7549	1.01196
$700$	0	0
$701$	−8.51775	−0.321711	−0.160855	−	0.986978i	$-0.551425\pi$
$701$	−8.51775	−0.321711	−0.160855	+	0.986978i	$0.551425\pi$
$702$	0	0
$703$	− 30.1587i	− 1.13746i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.6773i	0.589604i
$708$	0	0
$709$	−35.0848	−1.31764	−0.658819	−	0.752301i	$-0.728944\pi$
$709$	−35.0848	−1.31764	−0.658819	+	0.752301i	$0.728944\pi$
$710$	0	0
$711$	−91.3411	−3.42556
$712$	0	0
$713$	14.2553i	0.533864i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	51.8506i	1.93640i
$718$	0	0
$719$	−31.3292	−1.16838	−0.584190	−	0.811617i	$-0.698588\pi$
$719$	−31.3292	−1.16838	−0.584190	+	0.811617i	$0.698588\pi$
$720$	0	0
$721$	−4.82234	−0.179593
$722$	0	0
$723$	− 61.4713i	− 2.28614i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13.9627i	− 0.517846i	−0.965898	−	0.258923i	$-0.916632\pi$
$727$	− 13.9627i	− 0.517846i	0.965898	−	0.258923i	$-0.0833676\pi$
$728$	0	0
$729$	−85.4742	−3.16571
$730$	0	0
$731$	−68.1624	−2.52108
$732$	0	0
$733$	29.5761i	1.09242i	0.837648	+	0.546210i	$0.183930\pi$
$733$	29.5761i	1.09242i	−0.837648	+	0.546210i	$0.816070\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 30.3684i	− 1.11863i
$738$	0	0
$739$	6.66819	0.245293	0.122647	−	0.992450i	$-0.460862\pi$
$739$	6.66819	0.245293	0.122647	+	0.992450i	$0.460862\pi$
$740$	0	0
$741$	72.6236	2.66790
$742$	0	0
$743$	33.7002i	1.23634i	0.786045	+	0.618170i	$0.212126\pi$
$743$	33.7002i	1.23634i	−0.786045	+	0.618170i	$0.787874\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 31.8133i	− 1.16399i
$748$	0	0
$749$	7.02827	0.256808
$750$	0	0
$751$	−48.3129	−1.76296	−0.881481	−	0.472220i	$-0.843453\pi$
$751$	−48.3129	−1.76296	−0.881481	+	0.472220i	$0.843453\pi$
$752$	0	0
$753$	22.1806i	0.808305i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.3985i	1.54100i	0.637440	+	0.770500i	$0.279993\pi$
$757$	42.3985i	1.54100i	−0.637440	+	0.770500i	$0.720007\pi$
$758$	0	0
$759$	30.1696	1.09509
$760$	0	0
$761$	−44.7933	−1.62375	−0.811877	−	0.583828i	$-0.801554\pi$
$761$	−44.7933	−1.62375	−0.811877	+	0.583828i	$0.801554\pi$
$762$	0	0
$763$	18.9043i	0.684380i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 52.5105i	− 1.89605i
$768$	0	0
$769$	52.2080	1.88267	0.941335	−	0.337473i	$-0.109572\pi$
$769$	52.2080	1.88267	0.941335	+	0.337473i	$0.109572\pi$
$770$	0	0
$771$	−87.7541	−3.16039
$772$	0	0
$773$	24.2179i	0.871058i	0.900175	+	0.435529i	$0.143439\pi$
$773$	24.2179i	0.871058i	−0.900175	+	0.435529i	$0.856561\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 30.4177i	− 1.09123i
$778$	0	0
$779$	−4.47213	−0.160231
$780$	0	0
$781$	−19.2270	−0.687996
$782$	0	0
$783$	16.6983i	0.596749i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.43398i	− 0.0511159i	−0.999673	−	0.0255579i	$-0.991864\pi$
$787$	− 1.43398i	− 0.0511159i	0.999673	−	0.0255579i	$-0.00813623\pi$
$788$	0	0
$789$	−91.1051	−3.24343
$790$	0	0
$791$	2.69110	0.0956845
$792$	0	0
$793$	− 32.1131i	− 1.14037i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14.4732i	− 0.512666i	−0.966588	−	0.256333i	$-0.917486\pi$
$797$	− 14.4732i	− 0.512666i	0.966588	−	0.256333i	$-0.0825144\pi$
$798$	0	0
$799$	−4.31181	−0.152541
$800$	0	0
$801$	20.7549	0.733337
$802$	0	0
$803$	35.8397i	1.26476i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 36.4358i	− 1.28260i
$808$	0	0
$809$	−49.2654	−1.73208	−0.866039	−	0.499976i	$-0.833342\pi$
$809$	−49.2654	−1.73208	−0.866039	+	0.499976i	$0.833342\pi$
$810$	0	0
$811$	−4.69832	−0.164980	−0.0824901	−	0.996592i	$-0.526287\pi$
$811$	−4.69832	−0.164980	−0.0824901	+	0.996592i	$0.526287\pi$
$812$	0	0
$813$	− 66.9173i	− 2.34689i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 46.6911i	− 1.63351i
$818$	0	0
$819$	53.3219	1.86322
$820$	0	0
$821$	34.7730	1.21359	0.606793	−	0.794860i	$-0.292456\pi$
$821$	34.7730	1.21359	0.606793	+	0.794860i	$0.292456\pi$
$822$	0	0
$823$	44.1323i	1.53836i	0.639035	+	0.769178i	$0.279334\pi$
$823$	44.1323i	1.53836i	−0.639035	+	0.769178i	$0.720666\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.6610i	0.787999i	0.919111	+	0.394000i	$0.128909\pi$
$827$	22.6610i	0.787999i	−0.919111	+	0.394000i	$0.871091\pi$
$828$	0	0
$829$	37.9144	1.31682	0.658410	−	0.752659i	$-0.271229\pi$
$829$	37.9144	1.31682	0.658410	+	0.752659i	$0.271229\pi$
$830$	0	0
$831$	10.4924	0.363977
$832$	0	0
$833$	− 33.3665i	− 1.15608i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	139.417i	4.81895i
$838$	0	0
$839$	−30.7175	−1.06049	−0.530243	−	0.847846i	$-0.677899\pi$
$839$	−30.7175	−1.06049	−0.530243	+	0.847846i	$0.677899\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	34.2443i	1.17944i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	22.8669i	0.785716i
$848$	0	0
$849$	−1.40756	−0.0483074
$850$	0	0
$851$	11.8397	0.405859
$852$	0	0
$853$	− 1.25341i	− 0.0429159i	−0.999770	−	0.0214579i	$-0.993169\pi$
$853$	− 1.25341i	− 0.0429159i	0.999770	−	0.0214579i	$-0.00683080\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.85783i	− 0.268418i	−0.990953	−	0.134209i	$-0.957151\pi$
$857$	− 7.85783i	− 0.268418i	0.990953	−	0.134209i	$-0.0428494\pi$
$858$	0	0
$859$	14.4913	0.494438	0.247219	−	0.968960i	$-0.420483\pi$
$859$	14.4913	0.494438	0.247219	+	0.968960i	$0.420483\pi$
$860$	0	0
$861$	−4.51053	−0.153718
$862$	0	0
$863$	9.69646i	0.330071i	0.986288	+	0.165036i	$0.0527739\pi$
$863$	9.69646i	0.330071i	−0.986288	+	0.165036i	$0.947226\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	77.4158i	2.62918i
$868$	0	0
$869$	60.5380	2.05361
$870$	0	0
$871$	28.6983	0.972405
$872$	0	0
$873$	123.000i	4.16293i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.27341i	0.211838i	0.994375	+	0.105919i	$0.0337784\pi$
$877$	6.27341i	0.211838i	−0.994375	+	0.105919i	$0.966222\pi$
$878$	0	0
$879$	−63.9144	−2.15578
$880$	0	0
$881$	19.5953	0.660184	0.330092	−	0.943949i	$-0.392920\pi$
$881$	19.5953	0.660184	0.330092	+	0.943949i	$0.392920\pi$
$882$	0	0
$883$	5.24619i	0.176548i	0.996096	+	0.0882742i	$0.0281352\pi$
$883$	5.24619i	0.176548i	−0.996096	+	0.0882742i	$0.971865\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 32.6044i	− 1.09475i	−0.836888	−	0.547375i	$-0.815627\pi$
$887$	− 32.6044i	− 1.09475i	0.836888	−	0.547375i	$-0.184373\pi$
$888$	0	0
$889$	1.67004	0.0560114
$890$	0	0
$891$	166.907	5.59161
$892$	0	0
$893$	− 2.95358i	− 0.0988378i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	28.5105i	0.951939i
$898$	0	0
$899$	8.34916	0.278460
$900$	0	0
$901$	−15.1523	−0.504796
$902$	0	0
$903$	− 47.0920i	− 1.56712i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.96265i	− 0.0651688i	−0.999469	−	0.0325844i	$-0.989626\pi$
$907$	− 1.96265i	− 0.0651688i	0.999469	−	0.0325844i	$-0.0103738\pi$
$908$	0	0
$909$	−95.2856	−3.16043
$910$	0	0
$911$	50.7066	1.67998	0.839992	−	0.542599i	$-0.182560\pi$
$911$	50.7066	1.67998	0.839992	+	0.542599i	$0.182560\pi$
$912$	0	0
$913$	21.0848i	0.697806i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.25526i	− 0.0744754i
$918$	0	0
$919$	−56.2371	−1.85509	−0.927546	−	0.373710i	$-0.878086\pi$
$919$	−56.2371	−1.85509	−0.927546	+	0.373710i	$0.878086\pi$
$920$	0	0
$921$	75.5025	2.48789
$922$	0	0
$923$	− 18.1696i	− 0.598061i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 29.3100i	− 0.962665i
$928$	0	0
$929$	5.34009	0.175203	0.0876013	−	0.996156i	$-0.472080\pi$
$929$	5.34009	0.175203	0.0876013	+	0.996156i	$0.472080\pi$
$930$	0	0
$931$	22.8560	0.749074
$932$	0	0
$933$	− 49.1595i	− 1.60941i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.01013i	− 0.0983366i	−0.998791	−	0.0491683i	$-0.984343\pi$
$937$	− 3.01013i	− 0.0983366i	0.998791	−	0.0491683i	$-0.0156571\pi$
$938$	0	0
$939$	49.7194	1.62253
$940$	0	0
$941$	−0.829557	−0.0270428	−0.0135214	−	0.999909i	$-0.504304\pi$
$941$	−0.829557	−0.0270428	−0.0135214	+	0.999909i	$0.504304\pi$
$942$	0	0
$943$	− 1.75566i	− 0.0571723i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.3027i	0.692246i	0.938189	+	0.346123i	$0.112502\pi$
$947$	21.3027i	0.692246i	−0.938189	+	0.346123i	$0.887498\pi$
$948$	0	0
$949$	−33.8688	−1.09943
$950$	0	0
$951$	76.7860	2.48996
$952$	0	0
$953$	40.0203i	1.29638i	0.761477	+	0.648192i	$0.224474\pi$
$953$	40.0203i	1.29638i	−0.761477	+	0.648192i	$0.775526\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 17.6700i	− 0.571191i
$958$	0	0
$959$	−28.7476	−0.928310
$960$	0	0
$961$	38.7084	1.24866
$962$	0	0
$963$	42.7175i	1.37655i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 31.0365i	− 0.998068i	−0.866582	−	0.499034i	$-0.833688\pi$
$967$	− 31.0365i	− 0.998068i	0.866582	−	0.499034i	$-0.166312\pi$
$968$	0	0
$969$	−91.7012	−2.94587
$970$	0	0
$971$	−2.66819	−0.0856262	−0.0428131	−	0.999083i	$-0.513632\pi$
$971$	−2.66819	−0.0856262	−0.0428131	+	0.999083i	$0.513632\pi$
$972$	0	0
$973$	22.0565i	0.707100i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.6519i	1.46053i	0.683162	+	0.730267i	$0.260604\pi$
$977$	45.6519i	1.46053i	−0.683162	+	0.730267i	$0.739396\pi$
$978$	0	0
$979$	−13.7557	−0.439633
$980$	0	0
$981$	−114.899	−3.66845
$982$	0	0
$983$	17.4521i	0.556636i	0.960489	+	0.278318i	$0.0897770\pi$
$983$	17.4521i	0.556636i	−0.960489	+	0.278318i	$0.910223\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.97894i	− 0.0948208i
$988$	0	0
$989$	18.3300	0.582859
$990$	0	0
$991$	−4.49960	−0.142935	−0.0714673	−	0.997443i	$-0.522768\pi$
$991$	−4.49960	−0.142935	−0.0714673	+	0.997443i	$0.522768\pi$
$992$	0	0
$993$	32.6127i	1.03493i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 21.1896i	− 0.671083i	−0.942025	−	0.335541i	$-0.891081\pi$
$997$	− 21.1896i	− 0.671083i	0.942025	−	0.335541i	$-0.108919\pi$
$998$	0	0
$999$	115.792	3.66351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.f.349.1		6
5.2	odd	4		2900.2.a.g.1.1		3
5.3	odd	4		580.2.a.c.1.3	✓	3
5.4	even	2	inner	2900.2.c.f.349.6		6
15.8	even	4		5220.2.a.x.1.3		3
20.3	even	4		2320.2.a.m.1.1		3
40.3	even	4		9280.2.a.bw.1.3		3
40.13	odd	4		9280.2.a.bk.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.c.1.3	✓	3	5.3	odd	4
2320.2.a.m.1.1		3	20.3	even	4
2900.2.a.g.1.1		3	5.2	odd	4
2900.2.c.f.349.1		6	1.1	even	1	trivial
2900.2.c.f.349.6		6	5.4	even	2	inner
5220.2.a.x.1.3		3	15.8	even	4
9280.2.a.bk.1.1		3	40.13	odd	4
9280.2.a.bw.1.3		3	40.3	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:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.32088i q^{3} +1.32088i q^{7} -8.02827 q^{9} +O(q^{10})\)	\(q-3.32088i q^{3} +1.32088i q^{7} -8.02827 q^{9} +5.32088 q^{11} +5.02827i q^{13} -6.34916i q^{17} +4.34916 q^{19} +4.38650 q^{21} +1.70739i q^{23} +16.6983i q^{27} +1.00000 q^{29} +8.34916 q^{31} -17.6700i q^{33} -6.93438i q^{37} +16.6983 q^{39} -1.02827 q^{41} -10.7357i q^{43} -0.679116i q^{47} +5.25526 q^{49} -21.0848 q^{51} -2.38650i q^{53} -14.4431i q^{57} -10.4431 q^{59} -6.38650 q^{61} -10.6044i q^{63} -5.70739i q^{67} +5.67004 q^{69} -3.61350 q^{71} +6.73566i q^{73} +7.02827i q^{77} +11.3774 q^{79} +31.3684 q^{81} +3.96265i q^{83} -3.32088i q^{87} -2.58522 q^{89} -6.64177 q^{91} -27.7266i q^{93} -15.3209i q^{97} -42.7175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 22 q^{9}+O(q^{10}) \) 6 * q - 22 * q^9	\( 6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91} - 72 q^{99}+O(q^{100}) \) 6 * q - 22 * q^9 + 16 * q^11 - 16 * q^19 + 32 * q^21 + 6 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 6 * q^49 - 48 * q^51 - 16 * q^59 - 44 * q^61 - 24 * q^69 - 16 * q^71 + 46 * q^81 - 36 * q^89 - 8 * q^91 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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