LMFDB - Embedded newform 325.6.b.i.274.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.13
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.96479i q^{2} -29.7675i q^{3} +28.1396 q^{4} +58.4868 q^{6} -176.972i q^{7} +118.162i q^{8} -643.103 q^{9} +O(q^{10})$	\(q+1.96479i q^{2} -29.7675i q^{3} +28.1396 q^{4} +58.4868 q^{6} -176.972i q^{7} +118.162i q^{8} -643.103 q^{9} -62.7908 q^{11} -837.645i q^{12} +169.000i q^{13} +347.712 q^{14} +668.304 q^{16} +1825.19i q^{17} -1263.56i q^{18} -2509.78 q^{19} -5268.00 q^{21} -123.371i q^{22} +137.706i q^{23} +3517.37 q^{24} -332.049 q^{26} +11910.0i q^{27} -4979.91i q^{28} -5978.98 q^{29} -5203.47 q^{31} +5094.25i q^{32} +1869.12i q^{33} -3586.12 q^{34} -18096.7 q^{36} +4275.33i q^{37} -4931.19i q^{38} +5030.70 q^{39} +16780.1 q^{41} -10350.5i q^{42} -21508.6i q^{43} -1766.91 q^{44} -270.563 q^{46} -16741.5i q^{47} -19893.7i q^{48} -14511.9 q^{49} +54331.4 q^{51} +4755.59i q^{52} +2749.61i q^{53} -23400.7 q^{54} +20911.2 q^{56} +74709.8i q^{57} -11747.4i q^{58} -16631.5 q^{59} -25005.7 q^{61} -10223.7i q^{62} +113811. i q^{63} +11376.6 q^{64} -3672.44 q^{66} -29425.1i q^{67} +51360.2i q^{68} +4099.16 q^{69} -26909.1 q^{71} -75990.1i q^{72} +33664.0i q^{73} -8400.13 q^{74} -70624.2 q^{76} +11112.2i q^{77} +9884.27i q^{78} +13528.9 q^{79} +198258. q^{81} +32969.4i q^{82} +790.985i q^{83} -148239. q^{84} +42259.9 q^{86} +177979. i q^{87} -7419.47i q^{88} +72223.8 q^{89} +29908.2 q^{91} +3874.99i q^{92} +154894. i q^{93} +32893.6 q^{94} +151643. q^{96} +5458.83i q^{97} -28512.9i q^{98} +40380.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.96479i	0.347329i	0.984805	+	0.173664i	$0.0555608\pi$
$2$	1.96479i	0.347329i	−0.984805	+	0.173664i	$0.944439\pi$
$3$	− 29.7675i	− 1.90958i	−0.297274	−	0.954792i	$-0.596077\pi$
$3$	− 29.7675i	− 1.90958i	0.297274	−	0.954792i	$-0.403923\pi$
$4$	28.1396	0.879363
$5$	0	0
$5$	0	0
$6$	58.4868	0.663254
$7$	− 176.972i	− 1.36508i	−0.730848	−	0.682540i	$-0.760875\pi$
$7$	− 176.972i	− 1.36508i	0.730848	−	0.682540i	$-0.239125\pi$
$8$	118.162i	0.652757i
$9$	−643.103	−2.64651
$10$	0	0
$11$	−62.7908	−0.156464	−0.0782320	−	0.996935i	$-0.524927\pi$
$11$	−62.7908	−0.156464	−0.0782320	+	0.996935i	$0.524927\pi$
$12$	− 837.645i	− 1.67922i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	347.712	0.474132
$15$	0	0
$16$	668.304	0.652641
$17$	1825.19i	1.53175i	0.642992	+	0.765873i	$0.277693\pi$
$17$	1825.19i	1.53175i	−0.642992	+	0.765873i	$0.722307\pi$
$18$	− 1263.56i	− 0.919211i
$19$	−2509.78	−1.59497	−0.797483	−	0.603342i	$-0.793835\pi$
$19$	−2509.78	−1.59497	−0.797483	+	0.603342i	$0.793835\pi$
$20$	0	0
$21$	−5268.00	−2.60674
$22$	− 123.371i	− 0.0543445i
$23$	137.706i	0.0542791i	0.999632	+	0.0271396i	$0.00863985\pi$
$23$	137.706i	0.0542791i	−0.999632	+	0.0271396i	$0.991360\pi$
$24$	3517.37	1.24649
$25$	0	0
$26$	−332.049	−0.0963317
$27$	11910.0i	3.14416i
$28$	− 4979.91i	− 1.20040i
$29$	−5978.98	−1.32018	−0.660088	−	0.751188i	$-0.729481\pi$
$29$	−5978.98	−1.32018	−0.660088	+	0.751188i	$0.729481\pi$
$30$	0	0
$31$	−5203.47	−0.972499	−0.486249	−	0.873820i	$-0.661635\pi$
$31$	−5203.47	−0.972499	−0.486249	+	0.873820i	$0.661635\pi$
$32$	5094.25i	0.879438i
$33$	1869.12i	0.298781i
$34$	−3586.12	−0.532020
$35$	0	0
$36$	−18096.7	−2.32724
$37$	4275.33i	0.513412i	0.966490	+	0.256706i	$0.0826371\pi$
$37$	4275.33i	0.513412i	−0.966490	+	0.256706i	$0.917363\pi$
$38$	− 4931.19i	− 0.553978i
$39$	5030.70	0.529623
$40$	0	0
$41$	16780.1	1.55896	0.779481	−	0.626426i	$-0.215483\pi$
$41$	16780.1	1.55896	0.779481	+	0.626426i	$0.215483\pi$
$42$	− 10350.5i	− 0.905395i
$43$	− 21508.6i	− 1.77395i	−0.461819	−	0.886974i	$-0.652803\pi$
$43$	− 21508.6i	− 1.77395i	0.461819	−	0.886974i	$-0.347197\pi$
$44$	−1766.91	−0.137589
$45$	0	0
$46$	−270.563	−0.0188527
$47$	− 16741.5i	− 1.10548i	−0.833354	−	0.552739i	$-0.813583\pi$
$47$	− 16741.5i	− 1.10548i	0.833354	−	0.552739i	$-0.186417\pi$
$48$	− 19893.7i	− 1.24627i
$49$	−14511.9	−0.863445
$50$	0	0
$51$	54331.4	2.92500
$52$	4755.59i	0.243891i
$53$	2749.61i	0.134457i	0.997738	+	0.0672283i	$0.0214156\pi$
$53$	2749.61i	0.134457i	−0.997738	+	0.0672283i	$0.978584\pi$
$54$	−23400.7	−1.09206
$55$	0	0
$56$	20911.2	0.891066
$57$	74709.8i	3.04572i
$58$	− 11747.4i	− 0.458536i
$59$	−16631.5	−0.622015	−0.311007	−	0.950407i	$-0.600666\pi$
$59$	−16631.5	−0.622015	−0.311007	+	0.950407i	$0.600666\pi$
$60$	0	0
$61$	−25005.7	−0.860429	−0.430214	−	0.902727i	$-0.641562\pi$
$61$	−25005.7	−0.860429	−0.430214	+	0.902727i	$0.641562\pi$
$62$	− 10223.7i	− 0.337777i
$63$	113811.i	3.61270i
$64$	11376.6	0.347187
$65$	0	0
$66$	−3672.44	−0.103775
$67$	− 29425.1i	− 0.800812i	−0.916338	−	0.400406i	$-0.868869\pi$
$67$	− 29425.1i	− 0.800812i	0.916338	−	0.400406i	$-0.131131\pi$
$68$	51360.2i	1.34696i
$69$	4099.16	0.103651
$70$	0	0
$71$	−26909.1	−0.633509	−0.316754	−	0.948508i	$-0.602593\pi$
$71$	−26909.1	−0.633509	−0.316754	+	0.948508i	$0.602593\pi$
$72$	− 75990.1i	− 1.72753i
$73$	33664.0i	0.739365i	0.929158	+	0.369683i	$0.120534\pi$
$73$	33664.0i	0.739365i	−0.929158	+	0.369683i	$0.879466\pi$
$74$	−8400.13	−0.178323
$75$	0	0
$76$	−70624.2	−1.40255
$77$	11112.2i	0.213586i
$78$	9884.27i	0.183954i
$79$	13528.9	0.243891	0.121945	−	0.992537i	$-0.461087\pi$
$79$	13528.9	0.243891	0.121945	+	0.992537i	$0.461087\pi$
$80$	0	0
$81$	198258.	3.35752
$82$	32969.4i	0.541473i
$83$	790.985i	0.0126030i	0.999980	+	0.00630148i	$0.00200584\pi$
$83$	790.985i	0.0126030i	−0.999980	+	0.00630148i	$0.997994\pi$
$84$	−148239.	−2.29227
$85$	0	0
$86$	42259.9	0.616144
$87$	177979.i	2.52099i
$88$	− 7419.47i	− 0.102133i
$89$	72223.8	0.966508	0.483254	−	0.875480i	$-0.339455\pi$
$89$	72223.8	0.966508	0.483254	+	0.875480i	$0.339455\pi$
$90$	0	0
$91$	29908.2	0.378605
$92$	3874.99i	0.0477310i
$93$	154894.i	1.85707i
$94$	32893.6	0.383965
$95$	0	0
$96$	151643.	1.67936
$97$	5458.83i	0.0589075i	0.999566	+	0.0294538i	$0.00937678\pi$
$97$	5458.83i	0.0589075i	−0.999566	+	0.0294538i	$0.990623\pi$
$98$	− 28512.9i	− 0.299899i
$99$	40380.9	0.414084
$100$	0	0
$101$	−87461.4	−0.853125	−0.426563	−	0.904458i	$-0.640276\pi$
$101$	−87461.4	−0.853125	−0.426563	+	0.904458i	$0.640276\pi$
$102$	106750.i	1.01594i
$103$	− 75287.7i	− 0.699248i	−0.936890	−	0.349624i	$-0.886309\pi$
$103$	− 75287.7i	− 0.699248i	0.936890	−	0.349624i	$-0.113691\pi$
$104$	−19969.3	−0.181042
$105$	0	0
$106$	−5402.41	−0.0467007
$107$	− 77781.0i	− 0.656771i	−0.944544	−	0.328386i	$-0.893495\pi$
$107$	− 77781.0i	− 0.656771i	0.944544	−	0.328386i	$-0.106505\pi$
$108$	335144.i	2.76485i
$109$	−175666.	−1.41619	−0.708096	−	0.706116i	$-0.750446\pi$
$109$	−175666.	−1.41619	−0.708096	+	0.706116i	$0.750446\pi$
$110$	0	0
$111$	127266.	0.980403
$112$	− 118271.i	− 0.890908i
$113$	− 2350.11i	− 0.0173138i	−0.999963	−	0.00865691i	$-0.997244\pi$
$113$	− 2350.11i	− 0.0173138i	0.999963	−	0.00865691i	$-0.00275561\pi$
$114$	−146789.	−1.05787
$115$	0	0
$116$	−168246.	−1.16091
$117$	− 108684.i	− 0.734011i
$118$	− 32677.4i	− 0.216044i
$119$	323007.	2.09096
$120$	0	0
$121$	−157108.	−0.975519
$122$	− 49131.0i	− 0.298852i
$123$	− 499502.i	− 2.97697i
$124$	−146424.	−0.855179
$125$	0	0
$126$	−223614.	−1.25480
$127$	266534.i	1.46637i	0.680031	+	0.733183i	$0.261966\pi$
$127$	266534.i	1.46637i	−0.680031	+	0.733183i	$0.738034\pi$
$128$	185369.i	1.00003i
$129$	−640257.	−3.38750
$130$	0	0
$131$	118703.	0.604341	0.302170	−	0.953254i	$-0.402289\pi$
$131$	118703.	0.604341	0.302170	+	0.953254i	$0.402289\pi$
$132$	52596.4i	0.262737i
$133$	444159.i	2.17726i
$134$	57814.1	0.278145
$135$	0	0
$136$	−215668.	−0.999858
$137$	163850.i	0.745838i	0.927864	+	0.372919i	$0.121643\pi$
$137$	163850.i	0.745838i	−0.927864	+	0.372919i	$0.878357\pi$
$138$	8053.98i	0.0360008i
$139$	−218277.	−0.958231	−0.479115	−	0.877752i	$-0.659042\pi$
$139$	−218277.	−0.958231	−0.479115	+	0.877752i	$0.659042\pi$
$140$	0	0
$141$	−498353.	−2.11100
$142$	− 52870.6i	− 0.220036i
$143$	− 10611.6i	− 0.0433953i
$144$	−429788.	−1.72722
$145$	0	0
$146$	−66142.7	−0.256803
$147$	431983.i	1.64882i
$148$	120306.i	0.451475i
$149$	79310.9	0.292663	0.146331	−	0.989236i	$-0.453253\pi$
$149$	79310.9	0.292663	0.146331	+	0.989236i	$0.453253\pi$
$150$	0	0
$151$	368505.	1.31523	0.657615	−	0.753355i	$-0.271566\pi$
$151$	368505.	1.31523	0.657615	+	0.753355i	$0.271566\pi$
$152$	− 296560.i	− 1.04113i
$153$	− 1.17379e6i	− 4.05378i
$154$	−21833.1	−0.0741846
$155$	0	0
$156$	141562.	0.465731
$157$	− 30957.2i	− 0.100233i	−0.998743	−	0.0501166i	$-0.984041\pi$
$157$	− 30957.2i	− 0.100233i	0.998743	−	0.0501166i	$-0.0159593\pi$
$158$	26581.5i	0.0847103i
$159$	81849.1	0.256756
$160$	0	0
$161$	24370.0	0.0740954
$162$	389535.i	1.16616i
$163$	− 204546.i	− 0.603006i	−0.953465	−	0.301503i	$-0.902512\pi$
$163$	− 204546.i	− 0.603006i	0.953465	−	0.301503i	$-0.0974883\pi$
$164$	472186.	1.37089
$165$	0	0
$166$	−1554.12	−0.00437738
$167$	− 158449.i	− 0.439641i	−0.975540	−	0.219821i	$-0.929453\pi$
$167$	− 158449.i	− 0.439641i	0.975540	−	0.219821i	$-0.0705472\pi$
$168$	− 622475.i	− 1.70157i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	1.61404e6	4.22110
$172$	− 605243.i	− 1.55994i
$173$	230663.i	0.585952i	0.956120	+	0.292976i	$0.0946456\pi$
$173$	230663.i	0.585952i	−0.956120	+	0.292976i	$0.905354\pi$
$174$	−349692.	−0.875613
$175$	0	0
$176$	−41963.4	−0.102115
$177$	495077.i	1.18779i
$178$	141905.i	0.335696i
$179$	−791891.	−1.84728	−0.923640	−	0.383260i	$-0.874801\pi$
$179$	−791891.	−1.84728	−0.923640	+	0.383260i	$0.874801\pi$
$180$	0	0
$181$	−272694.	−0.618699	−0.309350	−	0.950948i	$-0.600111\pi$
$181$	−272694.	−0.618699	−0.309350	+	0.950948i	$0.600111\pi$
$182$	58763.3i	0.131501i
$183$	744358.i	1.64306i
$184$	−16271.5	−0.0354311
$185$	0	0
$186$	−304335.	−0.645014
$187$	− 114605.i	− 0.239663i
$188$	− 471100.i	− 0.972116i
$189$	2.10774e6	4.29203
$190$	0	0
$191$	603069.	1.19614	0.598072	−	0.801442i	$-0.295933\pi$
$191$	603069.	1.19614	0.598072	+	0.801442i	$0.295933\pi$
$192$	− 338653.i	− 0.662982i
$193$	− 59043.0i	− 0.114097i	−0.998371	−	0.0570486i	$-0.981831\pi$
$193$	− 59043.0i	− 0.114097i	0.998371	−	0.0570486i	$-0.0181690\pi$
$194$	−10725.5	−0.0204603
$195$	0	0
$196$	−408360.	−0.759281
$197$	− 815964.i	− 1.49798i	−0.662582	−	0.748989i	$-0.730539\pi$
$197$	− 815964.i	− 1.49798i	0.662582	−	0.748989i	$-0.269461\pi$
$198$	79340.1i	0.143823i
$199$	−490593.	−0.878190	−0.439095	−	0.898441i	$-0.644701\pi$
$199$	−490593.	−0.878190	−0.439095	+	0.898441i	$0.644701\pi$
$200$	0	0
$201$	−875911.	−1.52922
$202$	− 171843.i	− 0.296315i
$203$	1.05811e6i	1.80215i
$204$	1.52886e6	2.57213
$205$	0	0
$206$	147925.	0.242869
$207$	− 88559.0i	− 0.143650i
$208$	112943.i	0.181010i
$209$	157591.	0.249555
$210$	0	0
$211$	593914.	0.918370	0.459185	−	0.888341i	$-0.348142\pi$
$211$	593914.	0.918370	0.459185	+	0.888341i	$0.348142\pi$
$212$	77373.0i	0.118236i
$213$	801015.i	1.20974i
$214$	152823.	0.228116
$215$	0	0
$216$	−1.40731e6	−2.05237
$217$	920866.i	1.32754i
$218$	− 345147.i	− 0.491884i
$219$	1.00209e6	1.41188
$220$	0	0
$221$	−308458.	−0.424830
$222$	250051.i	0.340522i
$223$	− 893543.i	− 1.20324i	−0.798781	−	0.601621i	$-0.794522\pi$
$223$	− 893543.i	− 1.20324i	0.798781	−	0.601621i	$-0.205478\pi$
$224$	901537.	1.20050
$225$	0	0
$226$	4617.48	0.00601359
$227$	− 1.10400e6i	− 1.42201i	−0.703186	−	0.711006i	$-0.748240\pi$
$227$	− 1.10400e6i	− 1.42201i	0.703186	−	0.711006i	$-0.251760\pi$
$228$	2.10230e6i	2.67829i
$229$	662093.	0.834315	0.417158	−	0.908834i	$-0.363026\pi$
$229$	662093.	0.834315	0.417158	+	0.908834i	$0.363026\pi$
$230$	0	0
$231$	330782.	0.407860
$232$	− 706486.i	− 0.861755i
$233$	828333.i	0.999574i	0.866148	+	0.499787i	$0.166588\pi$
$233$	828333.i	0.999574i	−0.866148	+	0.499787i	$0.833412\pi$
$234$	213542.	0.254943
$235$	0	0
$236$	−468003.	−0.546977
$237$	− 402722.i	− 0.465730i
$238$	634641.i	0.726250i
$239$	−359216.	−0.406781	−0.203391	−	0.979098i	$-0.565196\pi$
$239$	−359216.	−0.406781	−0.203391	+	0.979098i	$0.565196\pi$
$240$	0	0
$241$	−928812.	−1.03011	−0.515057	−	0.857156i	$-0.672229\pi$
$241$	−928812.	−1.03011	−0.515057	+	0.857156i	$0.672229\pi$
$242$	− 308685.i	− 0.338826i
$243$	− 3.00750e6i	− 3.26731i
$244$	−703651.	−0.756629
$245$	0	0
$246$	981416.	1.03399
$247$	− 424152.i	− 0.442364i
$248$	− 614851.i	− 0.634805i
$249$	23545.6	0.0240664
$250$	0	0
$251$	−102334.	−0.102526	−0.0512631	−	0.998685i	$-0.516325\pi$
$251$	−102334.	−0.102526	−0.0512631	+	0.998685i	$0.516325\pi$
$252$	3.20259e6i	3.17688i
$253$	− 8646.66i	− 0.00849273i
$254$	−523682.	−0.509312
$255$	0	0
$256$	−158.767	−0.000151412 0
$257$	− 1.17461e6i	− 1.10933i	−0.832075	−	0.554664i	$-0.812847\pi$
$257$	− 1.17461e6i	− 1.10933i	0.832075	−	0.554664i	$-0.187153\pi$
$258$	− 1.25797e6i	− 1.17658i
$259$	756612.	0.700848
$260$	0	0
$261$	3.84510e6	3.49387
$262$	233226.i	0.209905i
$263$	99506.0i	0.0887074i	0.999016	+	0.0443537i	$0.0141229\pi$
$263$	99506.0i	0.0887074i	−0.999016	+	0.0443537i	$0.985877\pi$
$264$	−220859.	−0.195032
$265$	0	0
$266$	−872679.	−0.756224
$267$	− 2.14992e6i	− 1.84563i
$268$	− 828010.i	− 0.704204i
$269$	558985.	0.470999	0.235499	−	0.971874i	$-0.424327\pi$
$269$	558985.	0.470999	0.235499	+	0.971874i	$0.424327\pi$
$270$	0	0
$271$	−503763.	−0.416680	−0.208340	−	0.978056i	$-0.566806\pi$
$271$	−503763.	−0.416680	−0.208340	+	0.978056i	$0.566806\pi$
$272$	1.21978e6i	0.999680i
$273$	− 890291.i	− 0.722979i
$274$	−321930.	−0.259051
$275$	0	0
$276$	115349.	0.0911464
$277$	1.08351e6i	0.848466i	0.905553	+	0.424233i	$0.139456\pi$
$277$	1.08351e6i	0.848466i	−0.905553	+	0.424233i	$0.860544\pi$
$278$	− 428867.i	− 0.332821i
$279$	3.34637e6	2.57373
$280$	0	0
$281$	−1.46593e6	−1.10751	−0.553756	−	0.832679i	$-0.686806\pi$
$281$	−1.46593e6	−1.10751	−0.553756	+	0.832679i	$0.686806\pi$
$282$	− 979158.i	− 0.733213i
$283$	2.11337e6i	1.56859i	0.620390	+	0.784294i	$0.286974\pi$
$283$	2.11337e6i	1.56859i	−0.620390	+	0.784294i	$0.713026\pi$
$284$	−757210.	−0.557084
$285$	0	0
$286$	20849.7	0.0150724
$287$	− 2.96960e6i	− 2.12811i
$288$	− 3.27613e6i	− 2.32744i
$289$	−1.91147e6	−1.34624
$290$	0	0
$291$	162496.	0.112489
$292$	947293.i	0.650170i
$293$	− 2.71115e6i	− 1.84495i	−0.386057	−	0.922475i	$-0.626163\pi$
$293$	− 2.71115e6i	− 1.84495i	0.386057	−	0.922475i	$-0.373837\pi$
$294$	−848756.	−0.572683
$295$	0	0
$296$	−505180.	−0.335133
$297$	− 747842.i	− 0.491947i
$298$	155829.i	0.101650i
$299$	−23272.3	−0.0150543
$300$	0	0
$301$	−3.80641e6	−2.42158
$302$	724035.i	0.456817i
$303$	2.60350e6i	1.62911i
$304$	−1.67730e6	−1.04094
$305$	0	0
$306$	2.30624e6	1.40800
$307$	− 1.55774e6i	− 0.943297i	−0.881787	−	0.471648i	$-0.843659\pi$
$307$	− 1.55774e6i	− 0.943297i	0.881787	−	0.471648i	$-0.156341\pi$
$308$	312693.i	0.187820i
$309$	−2.24113e6	−1.33527
$310$	0	0
$311$	−2.57327e6	−1.50863	−0.754316	−	0.656511i	$-0.772032\pi$
$311$	−2.57327e6	−1.50863	−0.754316	+	0.656511i	$0.772032\pi$
$312$	594436.i	0.345715i
$313$	− 563667.i	− 0.325208i	−0.986691	−	0.162604i	$-0.948011\pi$
$313$	− 563667.i	− 0.325208i	0.986691	−	0.162604i	$-0.0519894\pi$
$314$	60824.3	0.0348139
$315$	0	0
$316$	380698.	0.214468
$317$	1.49727e6i	0.836858i	0.908250	+	0.418429i	$0.137419\pi$
$317$	1.49727e6i	0.836858i	−0.908250	+	0.418429i	$0.862581\pi$
$318$	160816.i	0.0891789i
$319$	375425.	0.206560
$320$	0	0
$321$	−2.31534e6	−1.25416
$322$	47881.9i	0.0257355i
$323$	− 4.58083e6i	− 2.44308i
$324$	5.57890e6	2.95248
$325$	0	0
$326$	401889.	0.209441
$327$	5.22914e6i	2.70434i
$328$	1.98277e6i	1.01762i
$329$	−2.96277e6	−1.50907
$330$	0	0
$331$	2.48241e6	1.24538	0.622692	−	0.782467i	$-0.286039\pi$
$331$	2.48241e6	1.24538	0.622692	+	0.782467i	$0.286039\pi$
$332$	22258.0i	0.0110826i
$333$	− 2.74948e6i	− 1.35875i
$334$	311319.	0.152700
$335$	0	0
$336$	−3.52063e6	−1.70126
$337$	− 2.17057e6i	− 1.04112i	−0.853827	−	0.520558i	$-0.825724\pi$
$337$	− 2.17057e6i	− 1.04112i	0.853827	−	0.520558i	$-0.174276\pi$
$338$	− 56116.4i	− 0.0267176i
$339$	−69957.0	−0.0330622
$340$	0	0
$341$	326730.	0.152161
$342$	3.17126e6i	1.46611i
$343$	− 406164.i	− 0.186409i
$344$	2.54149e6	1.15796
$345$	0	0
$346$	−453204.	−0.203518
$347$	− 3.93089e6i	− 1.75254i	−0.481821	−	0.876269i	$-0.660025\pi$
$347$	− 3.93089e6i	− 1.75254i	0.481821	−	0.876269i	$-0.339975\pi$
$348$	5.00826e6i	2.21686i
$349$	−378915.	−0.166524	−0.0832622	−	0.996528i	$-0.526534\pi$
$349$	−378915.	−0.166524	−0.0832622	+	0.996528i	$0.526534\pi$
$350$	0	0
$351$	−2.01280e6	−0.872032
$352$	− 319872.i	− 0.137600i
$353$	− 1.13031e6i	− 0.482792i	−0.970427	−	0.241396i	$-0.922395\pi$
$353$	− 1.13031e6i	− 0.482792i	0.970427	−	0.241396i	$-0.0776053\pi$
$354$	−972722.	−0.412554
$355$	0	0
$356$	2.03235e6	0.849911
$357$	− 9.61511e6i	− 3.99286i
$358$	− 1.55590e6i	− 0.641614i
$359$	−3.35579e6	−1.37423	−0.687114	−	0.726550i	$-0.741123\pi$
$359$	−3.35579e6	−1.37423	−0.687114	+	0.726550i	$0.741123\pi$
$360$	0	0
$361$	3.82289e6	1.54391
$362$	− 535787.i	− 0.214892i
$363$	4.67672e6i	1.86284i
$364$	841605.	0.332931
$365$	0	0
$366$	−1.46251e6	−0.570683
$367$	− 1.99093e6i	− 0.771597i	−0.922583	−	0.385799i	$-0.873926\pi$
$367$	− 1.99093e6i	− 0.771597i	0.922583	−	0.385799i	$-0.126074\pi$
$368$	92029.4i	0.0354248i
$369$	−1.07913e7	−4.12581
$370$	0	0
$371$	486603.	0.183544
$372$	4.35866e6i	1.63304i
$373$	1.39554e6i	0.519362i	0.965694	+	0.259681i	$0.0836175\pi$
$373$	1.39554e6i	0.519362i	−0.965694	+	0.259681i	$0.916383\pi$
$374$	225175.	0.0832419
$375$	0	0
$376$	1.97821e6	0.721609
$377$	− 1.01045e6i	− 0.366151i
$378$	4.14126e6i	1.49075i
$379$	−3.81965e6	−1.36592	−0.682960	−	0.730455i	$-0.739308\pi$
$379$	−3.81965e6	−1.36592	−0.682960	+	0.730455i	$0.739308\pi$
$380$	0	0
$381$	7.93403e6	2.80015
$382$	1.18490e6i	0.415456i
$383$	4.30052e6i	1.49804i	0.662547	+	0.749020i	$0.269475\pi$
$383$	4.30052e6i	1.49804i	−0.662547	+	0.749020i	$0.730525\pi$
$384$	5.51796e6	1.90963
$385$	0	0
$386$	116007.	0.0396293
$387$	1.38322e7i	4.69478i
$388$	153609.i	0.0518011i
$389$	−5.63218e6	−1.88713	−0.943567	−	0.331182i	$-0.892553\pi$
$389$	−5.63218e6	−1.88713	−0.943567	+	0.331182i	$0.892553\pi$
$390$	0	0
$391$	−251340.	−0.0831418
$392$	− 1.71475e6i	− 0.563620i
$393$	− 3.53348e6i	− 1.15404i
$394$	1.60320e6	0.520291
$395$	0	0
$396$	1.13630e6	0.364130
$397$	− 1.02692e6i	− 0.327011i	−0.986542	−	0.163505i	$-0.947720\pi$
$397$	− 1.02692e6i	− 0.327011i	0.986542	−	0.163505i	$-0.0522801\pi$
$398$	− 963911.i	− 0.305021i
$399$	1.32215e7	4.15766
$400$	0	0
$401$	−1.07636e6	−0.334269	−0.167134	−	0.985934i	$-0.553451\pi$
$401$	−1.07636e6	−0.334269	−0.167134	+	0.985934i	$0.553451\pi$
$402$	− 1.72098e6i	− 0.531142i
$403$	− 879387.i	− 0.269723i
$404$	−2.46113e6	−0.750206
$405$	0	0
$406$	−2.07896e6	−0.625938
$407$	− 268452.i	− 0.0803304i
$408$	6.41989e6i	1.90931i
$409$	4.48956e6	1.32708	0.663538	−	0.748143i	$-0.269054\pi$
$409$	4.48956e6	1.32708	0.663538	+	0.748143i	$0.269054\pi$
$410$	0	0
$411$	4.87739e6	1.42424
$412$	− 2.11857e6i	− 0.614892i
$413$	2.94330e6i	0.849100i
$414$	174000.	0.0498939
$415$	0	0
$416$	−860928.	−0.243912
$417$	6.49754e6i	1.82982i
$418$	309633.i	0.0866776i
$419$	2.90095e6	0.807245	0.403622	−	0.914926i	$-0.367751\pi$
$419$	2.90095e6	0.807245	0.403622	+	0.914926i	$0.367751\pi$
$420$	0	0
$421$	−5.69840e6	−1.56692	−0.783461	−	0.621441i	$-0.786548\pi$
$421$	−5.69840e6	−1.56692	−0.783461	+	0.621441i	$0.786548\pi$
$422$	1.16692e6i	0.318977i
$423$	1.07665e7i	2.92566i
$424$	−324899.	−0.0877675
$425$	0	0
$426$	−1.57383e6	−0.420177
$427$	4.42530e6i	1.17455i
$428$	− 2.18873e6i	− 0.577540i
$429$	−315882.	−0.0828670
$430$	0	0
$431$	7.25477e6	1.88118	0.940590	−	0.339545i	$-0.110273\pi$
$431$	7.25477e6	1.88118	0.940590	+	0.339545i	$0.110273\pi$
$432$	7.95954e6i	2.05201i
$433$	2.92650e6i	0.750115i	0.927001	+	0.375058i	$0.122377\pi$
$433$	2.92650e6i	0.750115i	−0.927001	+	0.375058i	$0.877623\pi$
$434$	−1.80931e6	−0.461093
$435$	0	0
$436$	−4.94318e6	−1.24535
$437$	− 345611.i	− 0.0865733i
$438$	1.96890e6i	0.490387i
$439$	−2.15422e6	−0.533492	−0.266746	−	0.963767i	$-0.585949\pi$
$439$	−2.15422e6	−0.533492	−0.266746	+	0.963767i	$0.585949\pi$
$440$	0	0
$441$	9.33266e6	2.28512
$442$	− 606054.i	− 0.147556i
$443$	7.40975e6i	1.79388i	0.442148	+	0.896942i	$0.354217\pi$
$443$	7.40975e6i	1.79388i	−0.442148	+	0.896942i	$0.645783\pi$
$444$	3.58121e6	0.862130
$445$	0	0
$446$	1.75562e6	0.417921
$447$	− 2.36088e6i	− 0.558864i
$448$	− 2.01334e6i	− 0.473938i
$449$	1.90284e6	0.445438	0.222719	−	0.974883i	$-0.428507\pi$
$449$	1.90284e6	0.445438	0.222719	+	0.974883i	$0.428507\pi$
$450$	0	0
$451$	−1.05364e6	−0.243921
$452$	− 66131.3i	− 0.0152251i
$453$	− 1.09695e7i	− 2.51154i
$454$	2.16912e6	0.493906
$455$	0	0
$456$	−8.82783e6	−1.98812
$457$	− 2.79693e6i	− 0.626457i	−0.949678	−	0.313229i	$-0.898589\pi$
$457$	− 2.79693e6i	− 0.626457i	0.949678	−	0.313229i	$-0.101411\pi$
$458$	1.30087e6i	0.289782i
$459$	−2.17381e7	−4.81605
$460$	0	0
$461$	4.18307e6	0.916733	0.458366	−	0.888763i	$-0.348435\pi$
$461$	4.18307e6	0.916733	0.458366	+	0.888763i	$0.348435\pi$
$462$	649917.i	0.141662i
$463$	− 2.51126e6i	− 0.544427i	−0.962237	−	0.272213i	$-0.912244\pi$
$463$	− 2.51126e6i	− 0.544427i	0.962237	−	0.272213i	$-0.0877557\pi$
$464$	−3.99578e6	−0.861602
$465$	0	0
$466$	−1.62750e6	−0.347181
$467$	− 2.17175e6i	− 0.460806i	−0.973095	−	0.230403i	$-0.925996\pi$
$467$	− 2.17175e6i	− 0.460806i	0.973095	−	0.230403i	$-0.0740044\pi$
$468$	− 3.05833e6i	− 0.645462i
$469$	−5.20740e6	−1.09317
$470$	0	0
$471$	−921517.	−0.191404
$472$	− 1.96520e6i	− 0.406025i
$473$	1.35054e6i	0.277559i
$474$	791263.	0.161761
$475$	0	0
$476$	9.08930e6	1.83871
$477$	− 1.76828e6i	− 0.355841i
$478$	− 705784.i	− 0.141287i
$479$	9.46791e6	1.88545	0.942726	−	0.333568i	$-0.108253\pi$
$479$	9.46791e6	1.88545	0.942726	+	0.333568i	$0.108253\pi$
$480$	0	0
$481$	−722531.	−0.142395
$482$	− 1.82492e6i	− 0.357788i
$483$	− 725434.i	− 0.141491i
$484$	−4.42097e6	−0.857835
$485$	0	0
$486$	5.90911e6	1.13483
$487$	− 4.87785e6i	− 0.931979i	−0.884790	−	0.465989i	$-0.845698\pi$
$487$	− 4.87785e6i	− 0.931979i	0.884790	−	0.465989i	$-0.154302\pi$
$488$	− 2.95472e6i	− 0.561651i
$489$	−6.08881e6	−1.15149
$490$	0	0
$491$	−3.86456e6	−0.723429	−0.361715	−	0.932289i	$-0.617808\pi$
$491$	−3.86456e6	−0.723429	−0.361715	+	0.932289i	$0.617808\pi$
$492$	− 1.40558e7i	− 2.61784i
$493$	− 1.09128e7i	− 2.02217i
$494$	833370.	0.153646
$495$	0	0
$496$	−3.47750e6	−0.634693
$497$	4.76214e6i	0.864791i
$498$	46262.2i	0.00835897i
$499$	−5.75229e6	−1.03416	−0.517082	−	0.855936i	$-0.672982\pi$
$499$	−5.75229e6	−1.03416	−0.517082	+	0.855936i	$0.672982\pi$
$500$	0	0
$501$	−4.71663e6	−0.839532
$502$	− 201064.i	− 0.0356103i
$503$	− 5.93696e6i	− 1.04627i	−0.852249	−	0.523136i	$-0.824762\pi$
$503$	− 5.93696e6i	− 1.04627i	0.852249	−	0.523136i	$-0.175238\pi$
$504$	−1.34481e7	−2.35822
$505$	0	0
$506$	16988.9	0.00294977
$507$	850189.i	0.146891i
$508$	7.50015e6i	1.28947i
$509$	563930.	0.0964785	0.0482393	−	0.998836i	$-0.484639\pi$
$509$	563930.	0.0964785	0.0482393	+	0.998836i	$0.484639\pi$
$510$	0	0
$511$	5.95758e6	1.00929
$512$	5.93149e6i	0.999974i
$513$	− 2.98916e7i	− 5.01482i
$514$	2.30786e6	0.385302
$515$	0	0
$516$	−1.80166e7	−2.97884
$517$	1.05121e6i	0.172968i
$518$	1.48658e6i	0.243425i
$519$	6.86625e6	1.11892
$520$	0	0
$521$	1.09272e6	0.176366	0.0881828	−	0.996104i	$-0.471894\pi$
$521$	1.09272e6	0.176366	0.0881828	+	0.996104i	$0.471894\pi$
$522$	7.55481e6i	1.21352i
$523$	2.47626e6i	0.395861i	0.980216	+	0.197930i	$0.0634220\pi$
$523$	2.47626e6i	0.395861i	−0.980216	+	0.197930i	$0.936578\pi$
$524$	3.34024e6	0.531435
$525$	0	0
$526$	−195508.	−0.0308107
$527$	− 9.49734e6i	− 1.48962i
$528$	1.24914e6i	0.194997i
$529$	6.41738e6	0.997054
$530$	0	0
$531$	1.06957e7	1.64617
$532$	1.24985e7i	1.91460i
$533$	2.83584e6i	0.432378i
$534$	4.22414e6	0.641041
$535$	0	0
$536$	3.47692e6	0.522736
$537$	2.35726e7i	3.52754i
$538$	1.09829e6i	0.163592i
$539$	911215.	0.135098
$540$	0	0
$541$	−5.78515e6	−0.849810	−0.424905	−	0.905238i	$-0.639692\pi$
$541$	−5.78515e6	−0.849810	−0.424905	+	0.905238i	$0.639692\pi$
$542$	− 989788.i	− 0.144725i
$543$	8.11742e6i	1.18146i
$544$	−9.29799e6	−1.34708
$545$	0	0
$546$	1.74923e6	0.251111
$547$	1.26151e7i	1.80269i	0.433102	+	0.901345i	$0.357419\pi$
$547$	1.26151e7i	1.80269i	−0.433102	+	0.901345i	$0.642581\pi$
$548$	4.61067e6i	0.655862i
$549$	1.60813e7	2.27714
$550$	0	0
$551$	1.50059e7	2.10564
$552$	484363.i	0.0676586i
$553$	− 2.39423e6i	− 0.332930i
$554$	−2.12887e6	−0.294697
$555$	0	0
$556$	−6.14222e6	−0.842632
$557$	9.93482e6i	1.35682i	0.734684	+	0.678409i	$0.237330\pi$
$557$	9.93482e6i	1.35682i	−0.734684	+	0.678409i	$0.762670\pi$
$558$	6.57491e6i	0.893931i
$559$	3.63495e6	0.492005
$560$	0	0
$561$	−3.41151e6	−0.457657
$562$	− 2.88025e6i	− 0.384671i
$563$	8.72418e6i	1.15999i	0.814621	+	0.579994i	$0.196945\pi$
$563$	8.72418e6i	1.15999i	−0.814621	+	0.579994i	$0.803055\pi$
$564$	−1.40234e7	−1.85634
$565$	0	0
$566$	−4.15232e6	−0.544816
$567$	− 3.50860e7i	− 4.58328i
$568$	− 3.17962e6i	− 0.413527i
$569$	−5.33272e6	−0.690508	−0.345254	−	0.938509i	$-0.612207\pi$
$569$	−5.33272e6	−0.690508	−0.345254	+	0.938509i	$0.612207\pi$
$570$	0	0
$571$	−1.46290e7	−1.87769	−0.938847	−	0.344334i	$-0.888105\pi$
$571$	−1.46290e7	−1.87769	−0.938847	+	0.344334i	$0.888105\pi$
$572$	− 298608.i	− 0.0381602i
$573$	− 1.79519e7i	− 2.28414i
$574$	5.83465e6	0.739154
$575$	0	0
$576$	−7.31633e6	−0.918834
$577$	− 7.62278e6i	− 0.953177i	−0.879126	−	0.476589i	$-0.841873\pi$
$577$	− 7.62278e6i	− 0.953177i	0.879126	−	0.476589i	$-0.158127\pi$
$578$	− 3.75564e6i	− 0.467590i
$579$	−1.75756e6	−0.217878
$580$	0	0
$581$	139982.	0.0172041
$582$	319270.i	0.0390706i
$583$	− 172650.i	− 0.0210376i
$584$	−3.97780e6	−0.482626
$585$	0	0
$586$	5.32684e6	0.640805
$587$	− 7.92637e6i	− 0.949465i	−0.880130	−	0.474732i	$-0.842545\pi$
$587$	− 7.92637e6i	− 0.949465i	0.880130	−	0.474732i	$-0.157455\pi$
$588$	1.21558e7i	1.44991i
$589$	1.30596e7	1.55110
$590$	0	0
$591$	−2.42892e7	−2.86052
$592$	2.85722e6i	0.335074i
$593$	1.13806e7i	1.32901i	0.747282	+	0.664507i	$0.231358\pi$
$593$	1.13806e7i	1.32901i	−0.747282	+	0.664507i	$0.768642\pi$
$594$	1.46935e6	0.170868
$595$	0	0
$596$	2.23178e6	0.257356
$597$	1.46037e7i	1.67698i
$598$	− 45725.1i	− 0.00522880i
$599$	−7.12203e6	−0.811030	−0.405515	−	0.914088i	$-0.632908\pi$
$599$	−7.12203e6	−0.811030	−0.405515	+	0.914088i	$0.632908\pi$
$600$	0	0
$601$	1.50084e6	0.169492	0.0847458	−	0.996403i	$-0.472992\pi$
$601$	1.50084e6	0.169492	0.0847458	+	0.996403i	$0.472992\pi$
$602$	− 7.47879e6i	− 0.841086i
$603$	1.89234e7i	2.11936i
$604$	1.03696e7	1.15656
$605$	0	0
$606$	−5.11534e6	−0.565839
$607$	1.39958e7i	1.54179i	0.636960	+	0.770897i	$0.280192\pi$
$607$	1.39958e7i	1.54179i	−0.636960	+	0.770897i	$0.719808\pi$
$608$	− 1.27854e7i	− 1.40267i
$609$	3.14972e7	3.44135
$610$	0	0
$611$	2.82932e6	0.306604
$612$	− 3.30299e7i	− 3.56475i
$613$	4.60575e6i	0.495050i	0.968882	+	0.247525i	$0.0796172\pi$
$613$	4.60575e6i	0.495050i	−0.968882	+	0.247525i	$0.920383\pi$
$614$	3.06063e6	0.327634
$615$	0	0
$616$	−1.31303e6	−0.139420
$617$	− 5.55212e6i	− 0.587146i	−0.955937	−	0.293573i	$-0.905156\pi$
$617$	− 5.55212e6i	− 0.587146i	0.955937	−	0.293573i	$-0.0948444\pi$
$618$	− 4.40334e6i	− 0.463779i
$619$	2.71815e6	0.285133	0.142566	−	0.989785i	$-0.454465\pi$
$619$	2.71815e6	0.285133	0.142566	+	0.989785i	$0.454465\pi$
$620$	0	0
$621$	−1.64008e6	−0.170662
$622$	− 5.05592e6i	− 0.523992i
$623$	− 1.27816e7i	− 1.31936i
$624$	3.36204e6	0.345654
$625$	0	0
$626$	1.10749e6	0.112954
$627$	− 4.69109e6i	− 0.476546i
$628$	− 871122.i	− 0.0881414i
$629$	−7.80331e6	−0.786416
$630$	0	0
$631$	−1.93426e6	−0.193393	−0.0966964	−	0.995314i	$-0.530828\pi$
$631$	−1.93426e6	−0.193393	−0.0966964	+	0.995314i	$0.530828\pi$
$632$	1.59860e6i	0.159201i
$633$	− 1.76793e7i	− 1.75371i
$634$	−2.94182e6	−0.290665
$635$	0	0
$636$	2.30320e6	0.225782
$637$	− 2.45251e6i	− 0.239477i
$638$	737631.i	0.0717443i
$639$	1.73053e7	1.67659
$640$	0	0
$641$	9.58248e6	0.921155	0.460578	−	0.887619i	$-0.347642\pi$
$641$	9.58248e6	0.921155	0.460578	+	0.887619i	$0.347642\pi$
$642$	− 4.54916e6i	− 0.435606i
$643$	2.46659e6i	0.235272i	0.993057	+	0.117636i	$0.0375316\pi$
$643$	2.46659e6i	0.235272i	−0.993057	+	0.117636i	$0.962468\pi$
$644$	685762.	0.0651567
$645$	0	0
$646$	9.00037e6	0.848553
$647$	2.72804e6i	0.256207i	0.991761	+	0.128103i	$0.0408889\pi$
$647$	2.72804e6i	0.256207i	−0.991761	+	0.128103i	$0.959111\pi$
$648$	2.34265e7i	2.19164i
$649$	1.04430e6	0.0973229
$650$	0	0
$651$	2.74119e7	2.53505
$652$	− 5.75584e6i	− 0.530261i
$653$	9.94071e6i	0.912293i	0.889905	+	0.456147i	$0.150771\pi$
$653$	9.94071e6i	0.912293i	−0.889905	+	0.456147i	$0.849229\pi$
$654$	−1.02742e7	−0.939295
$655$	0	0
$656$	1.12142e7	1.01744
$657$	− 2.16494e7i	− 1.95674i
$658$	− 5.82122e6i	− 0.524143i
$659$	−440385.	−0.0395019	−0.0197510	−	0.999805i	$-0.506287\pi$
$659$	−440385.	−0.0395019	−0.0197510	+	0.999805i	$0.506287\pi$
$660$	0	0
$661$	4.85089e6	0.431835	0.215918	−	0.976412i	$-0.430726\pi$
$661$	4.85089e6	0.431835	0.215918	+	0.976412i	$0.430726\pi$
$662$	4.87741e6i	0.432558i
$663$	9.18201e6i	0.811248i
$664$	−93464.0	−0.00822668
$665$	0	0
$666$	5.40215e6	0.471933
$667$	− 823340.i	− 0.0716580i
$668$	− 4.45869e6i	− 0.386604i
$669$	−2.65985e7	−2.29769
$670$	0	0
$671$	1.57013e6	0.134626
$672$	− 2.68365e7i	− 2.29246i
$673$	2.15372e6i	0.183296i	0.995791	+	0.0916479i	$0.0292134\pi$
$673$	2.15372e6i	0.183296i	−0.995791	+	0.0916479i	$0.970787\pi$
$674$	4.26471e6	0.361609
$675$	0	0
$676$	−803695.	−0.0676433
$677$	− 4.93215e6i	− 0.413585i	−0.978385	−	0.206792i	$-0.933698\pi$
$677$	− 4.93215e6i	− 0.413585i	0.978385	−	0.206792i	$-0.0663024\pi$
$678$	− 137451.i	− 0.0114835i
$679$	966058.	0.0804135
$680$	0	0
$681$	−3.28632e7	−2.71545
$682$	641956.i	0.0528499i
$683$	− 1.39352e7i	− 1.14304i	−0.820588	−	0.571521i	$-0.806354\pi$
$683$	− 1.39352e7i	− 1.14304i	0.820588	−	0.571521i	$-0.193646\pi$
$684$	4.54186e7	3.71187
$685$	0	0
$686$	798026.	0.0647451
$687$	− 1.97088e7i	− 1.59320i
$688$	− 1.43743e7i	− 1.15775i
$689$	−464685.	−0.0372915
$690$	0	0
$691$	1.71835e7	1.36904	0.684520	−	0.728994i	$-0.260012\pi$
$691$	1.71835e7	1.36904	0.684520	+	0.728994i	$0.260012\pi$
$692$	6.49076e6i	0.515264i
$693$	− 7.14628e6i	− 0.565258i
$694$	7.72338e6	0.608708
$695$	0	0
$696$	−2.10303e7	−1.64559
$697$	3.06270e7i	2.38793i
$698$	− 744488.i	− 0.0578387i
$699$	2.46574e7	1.90877
$700$	0	0
$701$	4.18640e6	0.321770	0.160885	−	0.986973i	$-0.448565\pi$
$701$	4.18640e6	0.321770	0.160885	+	0.986973i	$0.448565\pi$
$702$	− 3.95472e6i	− 0.302882i
$703$	− 1.07301e7i	− 0.818874i
$704$	−714347.	−0.0543222
$705$	0	0
$706$	2.22082e6	0.167688
$707$	1.54782e7i	1.16458i
$708$	1.39313e7i	1.04450i
$709$	7.30623e6	0.545855	0.272928	−	0.962035i	$-0.412008\pi$
$709$	7.30623e6	0.545855	0.272928	+	0.962035i	$0.412008\pi$
$710$	0	0
$711$	−8.70048e6	−0.645460
$712$	8.53409e6i	0.630895i
$713$	− 716548.i	− 0.0527864i
$714$	1.88917e7	1.38684
$715$	0	0
$716$	−2.22835e7	−1.62443
$717$	1.06930e7i	0.776783i
$718$	− 6.59342e6i	− 0.477309i
$719$	1.22435e7	0.883249	0.441625	−	0.897200i	$-0.354402\pi$
$719$	1.22435e7	0.883249	0.441625	+	0.897200i	$0.354402\pi$
$720$	0	0
$721$	−1.33238e7	−0.954530
$722$	7.51117e6i	0.536246i
$723$	2.76484e7i	1.96709i
$724$	−7.67351e6	−0.544061
$725$	0	0
$726$	−9.18877e6	−0.647017
$727$	− 612630.i	− 0.0429895i	−0.999769	−	0.0214947i	$-0.993157\pi$
$727$	− 612630.i	− 0.0429895i	0.999769	−	0.0214947i	$-0.00684251\pi$
$728$	3.53400e6i	0.247137i
$729$	−4.13490e7	−2.88169
$730$	0	0
$731$	3.92574e7	2.71724
$732$	2.09459e7i	1.44485i
$733$	1.54190e7i	1.05998i	0.848004	+	0.529989i	$0.177804\pi$
$733$	1.54190e7i	1.05998i	−0.848004	+	0.529989i	$0.822196\pi$
$734$	3.91176e6	0.267998
$735$	0	0
$736$	−701508.	−0.0477351
$737$	1.84763e6i	0.125298i
$738$	− 2.12027e7i	− 1.43301i
$739$	−1.10414e7	−0.743724	−0.371862	−	0.928288i	$-0.621280\pi$
$739$	−1.10414e7	−0.743724	−0.371862	+	0.928288i	$0.621280\pi$
$740$	0	0
$741$	−1.26259e7	−0.844731
$742$	956073.i	0.0637502i
$743$	− 1.66010e7i	− 1.10322i	−0.834102	−	0.551611i	$-0.814013\pi$
$743$	− 1.66010e7i	− 1.10322i	0.834102	−	0.551611i	$-0.185987\pi$
$744$	−1.83026e7	−1.21221
$745$	0	0
$746$	−2.74194e6	−0.180390
$747$	− 508684.i	− 0.0333539i
$748$	− 3.22495e6i	− 0.210751i
$749$	−1.37650e7	−0.896546
$750$	0	0
$751$	1.04497e6	0.0676089	0.0338045	−	0.999428i	$-0.489238\pi$
$751$	1.04497e6	0.0676089	0.0338045	+	0.999428i	$0.489238\pi$
$752$	− 1.11884e7i	− 0.721481i
$753$	3.04622e6i	0.195782i
$754$	1.98532e6	0.127175
$755$	0	0
$756$	5.93109e7	3.77425
$757$	− 5.93460e6i	− 0.376402i	−0.982131	−	0.188201i	$-0.939734\pi$
$757$	− 5.93460e6i	− 0.376402i	0.982131	−	0.188201i	$-0.0602656\pi$
$758$	− 7.50481e6i	− 0.474424i
$759$	−257389.	−0.0162176
$760$	0	0
$761$	5.89549e6	0.369027	0.184514	−	0.982830i	$-0.440929\pi$
$761$	5.89549e6	0.369027	0.184514	+	0.982830i	$0.440929\pi$
$762$	1.55887e7i	0.972574i
$763$	3.10879e7i	1.93322i
$764$	1.69701e7	1.05185
$765$	0	0
$766$	−8.44961e6	−0.520313
$767$	− 2.81072e6i	− 0.172516i
$768$	4726.10i	0 0.000289134i
$769$	1.30498e7	0.795769	0.397884	−	0.917436i	$-0.369744\pi$
$769$	1.30498e7	0.795769	0.397884	+	0.917436i	$0.369744\pi$
$770$	0	0
$771$	−3.49651e7	−2.11835
$772$	− 1.66145e6i	− 0.100333i
$773$	− 2.29371e7i	− 1.38067i	−0.723491	−	0.690334i	$-0.757464\pi$
$773$	− 2.29371e7i	− 1.38067i	0.723491	−	0.690334i	$-0.242536\pi$
$774$	−2.71774e7	−1.63063
$775$	0	0
$776$	−645025.	−0.0384523
$777$	− 2.25224e7i	− 1.33833i
$778$	− 1.10661e7i	− 0.655456i
$779$	−4.21144e7	−2.48649
$780$	0	0
$781$	1.68964e6	0.0991213
$782$	− 493830.i	− 0.0288775i
$783$	− 7.12099e7i	− 4.15084i
$784$	−9.69838e6	−0.563520
$785$	0	0
$786$	6.94254e6	0.400831
$787$	1.68766e7i	0.971291i	0.874156	+	0.485645i	$0.161415\pi$
$787$	1.68766e7i	0.971291i	−0.874156	+	0.485645i	$0.838585\pi$
$788$	− 2.29609e7i	− 1.31727i
$789$	2.96204e6	0.169394
$790$	0	0
$791$	−415903.	−0.0236348
$792$	4.77148e6i	0.270296i
$793$	− 4.22597e6i	− 0.238640i
$794$	2.01769e6	0.113580
$795$	0	0
$796$	−1.38051e7	−0.772247
$797$	4.13009e6i	0.230311i	0.993348	+	0.115155i	$0.0367366\pi$
$797$	4.13009e6i	0.230311i	−0.993348	+	0.115155i	$0.963263\pi$
$798$	2.59775e7i	1.44407i
$799$	3.05565e7	1.69331
$800$	0	0
$801$	−4.64474e7	−2.55788
$802$	− 2.11482e6i	− 0.116101i
$803$	− 2.11379e6i	− 0.115684i
$804$	−2.46478e7	−1.34474
$805$	0	0
$806$	1.72781e6	0.0936825
$807$	− 1.66396e7i	− 0.899412i
$808$	− 1.03346e7i	− 0.556884i
$809$	−1.01709e7	−0.546369	−0.273185	−	0.961962i	$-0.588077\pi$
$809$	−1.01709e7	−0.546369	−0.273185	+	0.961962i	$0.588077\pi$
$810$	0	0
$811$	−2.22723e7	−1.18909	−0.594543	−	0.804064i	$-0.702667\pi$
$811$	−2.22723e7	−1.18909	−0.594543	+	0.804064i	$0.702667\pi$
$812$	2.97748e7i	1.58474i
$813$	1.49957e7i	0.795686i
$814$	527451.	0.0279011
$815$	0	0
$816$	3.63099e7	1.90897
$817$	5.39818e7i	2.82939i
$818$	8.82104e6i	0.460932i
$819$	−1.92340e7	−1.00198
$820$	0	0
$821$	3.65506e6	0.189250	0.0946251	−	0.995513i	$-0.469835\pi$
$821$	3.65506e6	0.189250	0.0946251	+	0.995513i	$0.469835\pi$
$822$	9.58305e6i	0.494680i
$823$	− 7.29867e6i	− 0.375616i	−0.982206	−	0.187808i	$-0.939862\pi$
$823$	− 7.29867e6i	− 0.375616i	0.982206	−	0.187808i	$-0.0601383\pi$
$824$	8.89612e6	0.456439
$825$	0	0
$826$	−5.78296e6	−0.294917
$827$	− 3.19608e7i	− 1.62500i	−0.582958	−	0.812502i	$-0.698105\pi$
$827$	− 3.19608e7i	− 1.62500i	0.582958	−	0.812502i	$-0.301895\pi$
$828$	− 2.49201e6i	− 0.126321i
$829$	−9.74730e6	−0.492604	−0.246302	−	0.969193i	$-0.579215\pi$
$829$	−9.74730e6	−0.492604	−0.246302	+	0.969193i	$0.579215\pi$
$830$	0	0
$831$	3.22534e7	1.62022
$832$	1.92265e6i	0.0962923i
$833$	− 2.64871e7i	− 1.32258i
$834$	−1.27663e7	−0.635550
$835$	0	0
$836$	4.43455e6	0.219449
$837$	− 6.19736e7i	− 3.05769i
$838$	5.69976e6i	0.280380i
$839$	6.75711e6	0.331403	0.165701	−	0.986176i	$-0.447011\pi$
$839$	6.75711e6	0.331403	0.165701	+	0.986176i	$0.447011\pi$
$840$	0	0
$841$	1.52371e7	0.742867
$842$	− 1.11962e7i	− 0.544238i
$843$	4.36371e7i	2.11489i
$844$	1.67125e7	0.807580
$845$	0	0
$846$	−2.11539e7	−1.01617
$847$	2.78037e7i	1.33166i
$848$	1.83758e6i	0.0877519i
$849$	6.29096e7	2.99535
$850$	0	0
$851$	−588738.	−0.0278675
$852$	2.25402e7i	1.06380i
$853$	2.85698e7i	1.34442i	0.740362	+	0.672209i	$0.234654\pi$
$853$	2.85698e7i	1.34442i	−0.740362	+	0.672209i	$0.765346\pi$
$854$	−8.69479e6	−0.407957
$855$	0	0
$856$	9.19073e6	0.428712
$857$	− 2.35895e6i	− 0.109715i	−0.998494	−	0.0548577i	$-0.982529\pi$
$857$	− 2.35895e6i	− 0.109715i	0.998494	−	0.0548577i	$-0.0174705\pi$
$858$	− 620642.i	− 0.0287821i
$859$	−1.07995e7	−0.499368	−0.249684	−	0.968327i	$-0.580327\pi$
$859$	−1.07995e7	−0.499368	−0.249684	+	0.968327i	$0.580327\pi$
$860$	0	0
$861$	−8.83976e7	−4.06380
$862$	1.42541e7i	0.653388i
$863$	− 2.27989e7i	− 1.04204i	−0.853543	−	0.521022i	$-0.825551\pi$
$863$	− 2.27989e7i	− 1.04204i	0.853543	−	0.521022i	$-0.174449\pi$
$864$	−6.06728e7	−2.76509
$865$	0	0
$866$	−5.74995e6	−0.260537
$867$	5.68998e7i	2.57077i
$868$	2.59128e7i	1.16739i
$869$	−849491.	−0.0381601
$870$	0	0
$871$	4.97284e6	0.222105
$872$	− 2.07570e7i	− 0.924429i
$873$	− 3.51059e6i	− 0.155899i
$874$	679053.	0.0300694
$875$	0	0
$876$	2.81985e7	1.24155
$877$	− 3.23124e7i	− 1.41863i	−0.704890	−	0.709317i	$-0.749003\pi$
$877$	− 3.23124e7i	− 1.41863i	0.704890	−	0.709317i	$-0.250997\pi$
$878$	− 4.23258e6i	− 0.185297i
$879$	−8.07041e7	−3.52309
$880$	0	0
$881$	1.40863e7	0.611447	0.305723	−	0.952120i	$-0.401102\pi$
$881$	1.40863e7	0.611447	0.305723	+	0.952120i	$0.401102\pi$
$882$	1.83367e7i	0.793688i
$883$	4.14803e7i	1.79036i	0.445705	+	0.895180i	$0.352953\pi$
$883$	4.14803e7i	1.79036i	−0.445705	+	0.895180i	$0.647047\pi$
$884$	−8.67988e6	−0.373579
$885$	0	0
$886$	−1.45586e7	−0.623068
$887$	6.64591e6i	0.283626i	0.989893	+	0.141813i	$0.0452931\pi$
$887$	6.64591e6i	0.283626i	−0.989893	+	0.141813i	$0.954707\pi$
$888$	1.50379e7i	0.639965i
$889$	4.71689e7	2.00171
$890$	0	0
$891$	−1.24488e7	−0.525331
$892$	− 2.51439e7i	− 1.05809i
$893$	4.20175e7i	1.76320i
$894$	4.63864e6	0.194110
$895$	0	0
$896$	3.28050e7	1.36512
$897$	692757.i	0.0287475i
$898$	3.73869e6i	0.154713i
$899$	3.11115e7	1.28387
$900$	0	0
$901$	−5.01858e6	−0.205953
$902$	− 2.07018e6i	− 0.0847210i
$903$	1.13307e8i	4.62422i
$904$	277693.	0.0113017
$905$	0	0
$906$	2.15527e7	0.872331
$907$	− 1.12437e7i	− 0.453827i	−0.973915	−	0.226914i	$-0.927137\pi$
$907$	− 1.12437e7i	− 0.453827i	0.973915	−	0.226914i	$-0.0728635\pi$
$908$	− 3.10660e7i	− 1.25046i
$909$	5.62466e7	2.25781
$910$	0	0
$911$	2.28294e7	0.911377	0.455689	−	0.890139i	$-0.349393\pi$
$911$	2.28294e7	0.911377	0.455689	+	0.890139i	$0.349393\pi$
$912$	4.99289e7i	1.98776i
$913$	− 49666.6i	− 0.00197191i
$914$	5.49538e6	0.217587
$915$	0	0
$916$	1.86310e7	0.733666
$917$	− 2.10070e7i	− 0.824974i
$918$	− 4.27109e7i	− 1.67275i
$919$	−3.58183e7	−1.39899	−0.699497	−	0.714635i	$-0.746593\pi$
$919$	−3.58183e7	−1.39899	−0.699497	+	0.714635i	$0.746593\pi$
$920$	0	0
$921$	−4.63699e7	−1.80130
$922$	8.21885e6i	0.318408i
$923$	− 4.54763e6i	− 0.175704i
$924$	9.30807e6	0.358657
$925$	0	0
$926$	4.93410e6	0.189095
$927$	4.84177e7i	1.85057i
$928$	− 3.04584e7i	− 1.16101i
$929$	1.37009e7	0.520847	0.260424	−	0.965494i	$-0.416138\pi$
$929$	1.37009e7	0.520847	0.260424	+	0.965494i	$0.416138\pi$
$930$	0	0
$931$	3.64217e7	1.37716
$932$	2.33090e7i	0.878988i
$933$	7.65996e7i	2.88086i
$934$	4.26704e6	0.160051
$935$	0	0
$936$	1.28423e7	0.479131
$937$	− 2.45732e7i	− 0.914352i	−0.889376	−	0.457176i	$-0.848861\pi$
$937$	− 2.45732e7i	− 0.914352i	0.889376	−	0.457176i	$-0.151139\pi$
$938$	− 1.02315e7i	− 0.379691i
$939$	−1.67789e7	−0.621013
$940$	0	0
$941$	3.53611e7	1.30182	0.650912	−	0.759153i	$-0.274387\pi$
$941$	3.53611e7	1.30182	0.650912	+	0.759153i	$0.274387\pi$
$942$	− 1.81059e6i	− 0.0664801i
$943$	2.31072e6i	0.0846191i
$944$	−1.11149e7	−0.405952
$945$	0	0
$946$	−2.65353e6	−0.0964043
$947$	− 4.90338e7i	− 1.77673i	−0.459140	−	0.888364i	$-0.651842\pi$
$947$	− 4.90338e7i	− 1.77673i	0.459140	−	0.888364i	$-0.348158\pi$
$948$	− 1.13324e7i	− 0.409545i
$949$	−5.68922e6	−0.205063
$950$	0	0
$951$	4.45699e7	1.59805
$952$	3.81671e7i	1.36489i
$953$	− 1.81532e7i	− 0.647472i	−0.946147	−	0.323736i	$-0.895061\pi$
$953$	− 1.81532e7i	− 0.647472i	0.946147	−	0.323736i	$-0.104939\pi$
$954$	3.47431e6	0.123594
$955$	0	0
$956$	−1.01082e7	−0.357708
$957$	− 1.11755e7i	− 0.394444i
$958$	1.86025e7i	0.654872i
$959$	2.89967e7	1.01813
$960$	0	0
$961$	−1.55304e6	−0.0542466
$962$	− 1.41962e6i	− 0.0494578i
$963$	5.00212e7i	1.73815i
$964$	−2.61364e7	−0.905844
$965$	0	0
$966$	1.42532e6	0.0491441
$967$	− 5.23618e7i	− 1.80073i	−0.435135	−	0.900365i	$-0.643299\pi$
$967$	− 5.23618e7i	− 1.80073i	0.435135	−	0.900365i	$-0.356701\pi$
$968$	− 1.85642e7i	− 0.636777i
$969$	−1.36360e8	−4.66527
$970$	0	0
$971$	2.95200e7	1.00477	0.502387	−	0.864643i	$-0.332455\pi$
$971$	2.95200e7	1.00477	0.502387	+	0.864643i	$0.332455\pi$
$972$	− 8.46299e7i	− 2.87315i
$973$	3.86287e7i	1.30806i
$974$	9.58395e6	0.323703
$975$	0	0
$976$	−1.67114e7	−0.561551
$977$	1.21759e7i	0.408098i	0.978961	+	0.204049i	$0.0654102\pi$
$977$	1.21759e7i	0.408098i	−0.978961	+	0.204049i	$0.934590\pi$
$978$	− 1.19632e7i	− 0.399946i
$979$	−4.53499e6	−0.151224
$980$	0	0
$981$	1.12971e8	3.74797
$982$	− 7.59304e6i	− 0.251268i
$983$	− 1.46087e6i	− 0.0482200i	−0.999709	−	0.0241100i	$-0.992325\pi$
$983$	− 1.46087e6i	− 0.0482200i	0.999709	−	0.0241100i	$-0.00767520\pi$
$984$	5.90220e7	1.94324
$985$	0	0
$986$	2.14413e7	0.702360
$987$	8.81942e7i	2.88169i
$988$	− 1.19355e7i	− 0.388998i
$989$	2.96186e6	0.0962883
$990$	0	0
$991$	3.16160e7	1.02264	0.511320	−	0.859390i	$-0.329157\pi$
$991$	3.16160e7	1.02264	0.511320	+	0.859390i	$0.329157\pi$
$992$	− 2.65078e7i	− 0.855252i
$993$	− 7.38951e7i	− 2.37817i
$994$	−9.35660e6	−0.300367
$995$	0	0
$996$	662564.	0.0211631
$997$	3.93744e7i	1.25452i	0.778812	+	0.627258i	$0.215823\pi$
$997$	3.93744e7i	1.25452i	−0.778812	+	0.627258i	$0.784177\pi$
$998$	− 1.13020e7i	− 0.359195i
$999$	−5.09194e7	−1.61425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.13		22
5.2	odd	4		325.6.a.k.1.5	yes	11
5.3	odd	4		325.6.a.j.1.7	✓	11
5.4	even	2	inner	325.6.b.i.274.10		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.7	✓	11	5.3	odd	4
325.6.a.k.1.5	yes	11	5.2	odd	4
325.6.b.i.274.10		22	5.4	even	2	inner
325.6.b.i.274.13		22	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.96479i q^{2} -29.7675i q^{3} +28.1396 q^{4} +58.4868 q^{6} -176.972i q^{7} +118.162i q^{8} -643.103 q^{9} +O(q^{10})\)	\(q+1.96479i q^{2} -29.7675i q^{3} +28.1396 q^{4} +58.4868 q^{6} -176.972i q^{7} +118.162i q^{8} -643.103 q^{9} -62.7908 q^{11} -837.645i q^{12} +169.000i q^{13} +347.712 q^{14} +668.304 q^{16} +1825.19i q^{17} -1263.56i q^{18} -2509.78 q^{19} -5268.00 q^{21} -123.371i q^{22} +137.706i q^{23} +3517.37 q^{24} -332.049 q^{26} +11910.0i q^{27} -4979.91i q^{28} -5978.98 q^{29} -5203.47 q^{31} +5094.25i q^{32} +1869.12i q^{33} -3586.12 q^{34} -18096.7 q^{36} +4275.33i q^{37} -4931.19i q^{38} +5030.70 q^{39} +16780.1 q^{41} -10350.5i q^{42} -21508.6i q^{43} -1766.91 q^{44} -270.563 q^{46} -16741.5i q^{47} -19893.7i q^{48} -14511.9 q^{49} +54331.4 q^{51} +4755.59i q^{52} +2749.61i q^{53} -23400.7 q^{54} +20911.2 q^{56} +74709.8i q^{57} -11747.4i q^{58} -16631.5 q^{59} -25005.7 q^{61} -10223.7i q^{62} +113811. i q^{63} +11376.6 q^{64} -3672.44 q^{66} -29425.1i q^{67} +51360.2i q^{68} +4099.16 q^{69} -26909.1 q^{71} -75990.1i q^{72} +33664.0i q^{73} -8400.13 q^{74} -70624.2 q^{76} +11112.2i q^{77} +9884.27i q^{78} +13528.9 q^{79} +198258. q^{81} +32969.4i q^{82} +790.985i q^{83} -148239. q^{84} +42259.9 q^{86} +177979. i q^{87} -7419.47i q^{88} +72223.8 q^{89} +29908.2 q^{91} +3874.99i q^{92} +154894. i q^{93} +32893.6 q^{94} +151643. q^{96} +5458.83i q^{97} -28512.9i q^{98} +40380.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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